MHD numerical simulations in a cosmological context Ludwig-Maximilians-Universit¨ at

MHD numerical simulations in a cosmological context Ludwig-Maximilians-Universit¨ at
Ludwig-Maximilians-Universität
Sigillum Universitatis Ludovici Maximiliani
MHD numerical simulations
in a cosmological context
Dissertation der Fakultät für Physik
Dissertation of the Faculty of Physics
der Ludwig-Maximilians-Universität München
at the Ludwig Maximilian University of Munich
für den Grad des
for the degree of
Doctor rerum naturalium
vorgelegt von Federico Andrés Stasyszyn
presented by
aus Córdoba, Argentina
from
München, 15.03.2011
Sigillum Universitatis Ludovici Maximiliani
1. Gutachter: Prof. Dr. Simon D. M. White
referee:
2. Gutachter: Prof. Dr. Harald Lesch
referee:
Tag der mündlichen Prüfung: 27.05.2011
Date of the oral exam:
Contents
Contents
I
Zusammenfassung
VII
Abstract
I
IX
Introduction
1
1 Astrophysics and Magnetic Fields
1.1 Extragalactic physics basics . . . . . . . .
1.2 Observation of Magnetic Fields . . . . . .
1.2.1 Synchrotron Emission . . . . . . .
1.2.2 Faraday rotation . . . . . . . . . .
1.2.3 Zeeman Splitting . . . . . . . . . .
1.2.4 Polarization of Optical Starlight . .
1.3 Magnetic field in galaxy clusters . . . . . .
1.4 Large scale magnetic fields in the Universe
1.5 Numerical simulations of cluster formation
1.6 Open questions . . . . . . . . . . . . . . .
II
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Numerical Methods
3
4
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2 MHD - Smoothed Particle Hydrodynamics
2.1 Introduction to SPH . . . . . . . . . . . . . . . . . . .
2.1.1 Interpolants . . . . . . . . . . . . . . . . . . . .
2.2 Building our set of Equations . . . . . . . . . . . . . .
2.2.1 Co-moving variables and units system . . . . . .
2.2.2 Magnetic signal velocity . . . . . . . . . . . . .
2.2.3 Induction equation . . . . . . . . . . . . . . . .
2.2.4 Magnetic force . . . . . . . . . . . . . . . . . .
2.3 Instability corrections . . . . . . . . . . . . . . . . . . .
2.3.1 Improvements in the underlying SPH formalism
2.3.2 Divergence force subtraction . . . . . . . . . . .
2.3.3 Divergence Cleaning: Dedner Method . . . . . .
2.3.4 Smoothing the magnetic field . . . . . . . . . .
I
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27
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2.4
2.5
III
2.3.5 Artificial magnetic dissipation . . . . .
2.3.6 Euler potential . . . . . . . . . . . .
Test problems . . . . . . . . . . . . . . . . . .
2.4.1 Brio-Wu shock-tube . . . . . . . . . .
2.4.2 The effect of field regularization . . . .
2.4.3 Shock tube problems . . . . . . . . . .
2.4.4 Multi dimensional Tests - Planar Tests
Discussion . . . . . . . . . . . . . . . . . . . .
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Applications
3 Galaxy cluster and magnetic fields
3.1 Galaxy cluster slices . . . . . . . .
3.2 Magnetic field profiles . . . . . . .
3.3 Synthetic RM Measurements . . . .
3.4 Structure Functions . . . . . . . . .
3.5 Discussions . . . . . . . . . . . . .
Conclusions
5 Summary and outlook
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4 Large Scale cosmological magnetic fields
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Simulations . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 The cosmological MHD simulations . . . . . . . . .
4.2.2 Artificial MHD models . . . . . . . . . . . . . . . .
4.2.3 Synthetic RM catalogs . . . . . . . . . . . . . . . .
4.2.4 Magnetic depth of the universe . . . . . . . . . . .
4.3 Evaluating the cosmological Cross-Correlation Signal . . .
4.3.1 Estimators . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Evaluating uncertainties . . . . . . . . . . . . . . .
4.3.3 Magnetic depths . . . . . . . . . . . . . . . . . . .
4.3.4 Magnetic field models . . . . . . . . . . . . . . . .
4.4 Simulating the observational process . . . . . . . . . . . .
4.4.1 Foreground removal procedure . . . . . . . . . . . .
4.4.2 Adding observational noise . . . . . . . . . . . . . .
4.4.3 Adding Galactic foreground . . . . . . . . . . . . .
4.5 The simulated observational cross-correlations: an example
4.5.1 Signal pile-up . . . . . . . . . . . . . . . . . . . . .
4.5.2 Differentiating cosmic magnetization . . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
IV
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V
Appendices
109
A Magnetohydrodynamics and Plasma Physics
B Finding optimal numerical parameters
B.1 Divergence threshold . . . . . . . . . . . . . . .
B.2 Regularization by smoothing the magnetic field
B.3 Artificial dissipation . . . . . . . . . . . . . . .
B.3.1 Time dependent artificial dissipation . .
B.4 Dedner Method . . . . . . . . . . . . . . . . . .
111
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115
. 116
. 116
. 116
. 117
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C Resolution convergence tests
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D ∇ · B distribution analysis
123
VI
127
VII
Acknowledgments
Bibliography
131
List of Figures
List of Figures
V
1.1
1.2
1.3
1.4
Cluster images: Abell 1689 . . . . . . . . . . . . . .
Radio Image of Coma cluster . . . . . . . . . . . .
Rotation measurement as function of cooling rate. .
Intergalactic contribution to the RMig as a function
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zs .
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. 7
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
Test 5A without any correction. . . . . . . . . . . . . . .
Test 5A with the standard implementation. . . . . . . . .
Test 5A with Dedner cleaning scheme. . . . . . . . . . .
Test 5A smoothing the magentic field. . . . . . . . . . .
Test 5A including artificial dissipation. . . . . . . . . . .
Test 5A including time dependent artificial dissipation. .
Representative plot of eight shock tube problems. . . . .
Magnetic modulus in Fast Rotor test. . . . . . . . . . . .
Density cut of the Fast Rotor test. . . . . . . . . . . . . .
Density in Strong Blast test. . . . . . . . . . . . . . . . .
Density cut of the Strong Blast test. . . . . . . . . . . . .
Magnetic energy results for the Orszang-Tang Vortex. . .
Pressure cut of the Orszang-Tang Vortex. . . . . . . . . .
∇ · B error comparison in the Orszang-Tang Vortex . . .
Comparison plot between all the implementation in tests.
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3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
Zoom into the simulated cluster . . . . . . . . . . . . . . . . . .
Density slice cut trough the galaxy cluster . . . . . . . . . . . .
Cluster slice of the magnetic energy . . . . . . . . . . . . . . . .
cluster slice in the ∇ · B errors . . . . . . . . . . . . . . . . . . .
Resolution comparison . . . . . . . . . . . . . . . . . . . . . . .
Magnetic field profile for different methods. . . . . . . . . . . . .
Profiles comparing different parameters. . . . . . . . . . . . . .
Magnetic field profiles for different resolutions. . . . . . . . . . .
Synthetic RM maps for different schemes . . . . . . . . . . . . .
Synthetic RM maps for different resolutions . . . . . . . . . . .
Comparison maps between simulations and observations. . . . .
Structure functions comparing between theory and observations
Structure and correlation function in different schemes . . . . .
Structure and correlation function in different resolution. . . . .
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V
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of redshift
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4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
Mean cosmic magnetic field as a function of density. . . . . . . . . . .
Full sky maps of the local universe. . . . . . . . . . . . . . . . . . . .
Cross-correlation functions of different estimators . . . . . . . . . . .
Cross-correlation for different magnetic depths. . . . . . . . . . . . . .
Cross-correlation for different models. . . . . . . . . . . . . . . . . . .
Cross-correlation varying the removal size. . . . . . . . . . . . . . . .
Cross-correlations for different noise strength. . . . . . . . . . . . . .
Full sky maps of synthetic and observed RMs . . . . . . . . . . . . .
Cross-correlation piling-up the GF, CS and foreground removal. . . .
Cross-correlation piling-up the GF, CS, noise and foreground removal.
Cross-correlation for different models and all the effects studied. . . .
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B.1
B.2
B.3
B.4
Parameter
Parameter
Parameter
Parameter
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studies
studies
studies
studies
for
for
for
for
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the
the
the
magnetic field smoothing. . . . . . .
Artificial dissipation. . . . . . . . . .
time dependent artificial dissipation.
Dedner cleaning scheme. . . . . . . .
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C.1 Density cut of the Orzang-Tang for different resolutions in Eulerian codes. 121
C.2 Density cut of the Orzang-Tang for different resolutions in GADGET-3. . . . 122
D.1
D.2
D.3
D.4
Magnetic vs density profiles / Evolution in time of ∇ · B errors.
Histogram distribution of properties at z = 0. . . . . . . . . . .
Magnetic field as function of density/ ∇ · B errors as function of
∇ · B errors as function of the density . . . . . . . . . . . . . .
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βkin .
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124
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Zusammenfassung
Magnetfelder sind in fast überall im beobachtbaren Universums zu finden. Ihre Präsenz
im intergalaktischen Medium und innerhalb von Galaxienhaufen (intra-cluster medium,
ICM) wurde durch Beobachtungen von diffuser Radioemission und Faraday-Rotation
polarisierten Lichts von Radio-Quellen, die innerhalb oder hinter dem magnetisierten
Medium liegen, bestätigt. Trotz dieser Beobachtungen ist die dynamische Bedeutung von
Magnetfeldern in astrophysikalischen Systemen nach wie vor nicht verstanden, und es
gibt viele Prozesse bei denen der Einfluss von Magnetfeldern weitere Untersuchungen
benötigt. Astrophysikalische Systeme sind komplex und in hohem Maße nichtlinear,
weswegen numerische Simulationen eine gute Möglichkeit darstellen um diese Systeme
zu studieren. Die Berücksichtigung von Magnetfeldern in solchen Simulationen ist jedoch
nicht einfach, da es oft nicht trivial ist die Bedingung ∇ · B = 0 und die Stabilität des
Verfahrens in verschiedenen Anwendungen zu gewährleisten.
Während meiner Doktorarbeit habe ich die MHD Implementierung innerhalb des Nbody/SPH Programms GADGET betrachtet und weiterentwickelt. Ich habe verschiedene
Methoden zur Regulierung des Magnetfeldes und zur Vermeidung von Instabilitäten
untersucht. Diese Methoden beinhalten die Verwendung von Euler Potentialen zur
Beschreibung des Magnetfeldes als auch Verfahren zur Verminderung des numerischen ∇ ·
B Fehlers. Mit Hilfe des GADGET Programms habe ich die Entwicklung des Magnetfeldes
während der kosmologischen Entstehung von Galaxienhaufen untersucht. Der Vergleich
zwischen den verschiedenen numerischen Methoden zeigt, dass die numerischen ∇ · B
Fehler nicht zu einer unphysikalischen Verstärkung des Magnetfeldes führen.
Der Vergleich synthetischer Karten der Faraday-Rotation-Verteilung unserer
simulierten Galaxienhaufen mit Beobachtungen zeigt, dass die relevante Physik nur
durch ein Turbulenzmodell beschrieben werden kann, das viele Längenskalen abdeckt.
Das bedeutet, dass der vom kosmologischen Entstehungsprozess getriebene turbulente
Dynamo effizient arbeitet. Dabei werden die grundlegenden Beobachtungsmerkmale
als auch Details wie Struktur-Funktionen (die bei steigender Auflösung gegen die
Beobachtungen konvergieren) gut wiedergegeben. Damit wird deutlich, dass moderne
numerische Verfahren zusammen mit einer hohen Auflösung sehr gut dafür geeignet sind,
die Eigenschaften des ICM zu studieren.
Darüber hinaus habe ich den Zusammenhang zwischen der großräumigen
kosmologischen Struktur und dem Magnetfeld untersucht. Im Allgemeinen ist das
beobachtete Rotations-Signal von dichten Regionen (d.h.
Galaxienhaufen und gruppen) dominiert, weshalb es unklar bleibt, wie sich das Magnetfeld in weniger
dichten Bereichen des Universums verhält. Deshalb müssen statistische Methoden
wie Korrelationsfunktionen verwendet werden. Auf diese Weise habe ich mit Hilfe
kosmologischer Simulationen die möglichen Beobachtungsfehler untersucht. Es zeigt
sich, dass die Form der Korrelationsfunktion bei Verwendung eines normalisieren
Schätzwertes der Rotation die zugrundeliegende Verteilung des Magnetfeldes innerhalb
der kosmischen Struktur gut wiederspiegelt. Derzeitige Beobachtungsfehler unterdrücken
das Rotationssignal jedoch in einer Art, die es leider unmöglich macht, die Amplitude
der Korrelationsfunktion mit der Magnetisierung der großräumigen Struktur in
Zusammenhang zu bringen (Stasyszyn et al., 2010).
VII
Abstract
Magnetic fields in the Universe are found in almost all studied environments. In
particular, their presence in the inter-galactic medium and in the intra-cluster medium
is confirmed by diffuse radio emission as well as by observations of Faraday Rotation
Measures towards polarized radio sources within or behind the magnetized medium.
Besides the observations, their dynamical importance in astrophysical systems is poorly
constrained, therefore there are still plenty of processes in which the role of magnetic fields
are not fully understood.
Astrophysical systems are complex and highly nonlinear. Therefore, numerical
simulations have demonstrated to be a useful tool to study those problems. However,
the inclusion of magnetic fields in numerical implementations is not easy to achieve.
Mainly because of the difficulties to keep the ∇ · B constraint low, and to have a stable
implementation in different circumstances.
We study and developed a cosmological MHD code in SPH. We study different possible
schemes to regularize the magnetic field, and avoid instabilities. Those schemes included
the use of Euler potentials to build the magnetic field, as well as cleaning schemes for the
numerical ∇ · B errors.
We studied the magnetic field evolution in the context of cosmological structure
formation of galaxy clusters. We compare different numerical schemes leading us to
the conclusion that the ∇ · B terms do not drive the evolution and growth of the
magnetic field in galaxy clusters. We made synthetic rotation measure maps and study
the reversals of the magnetic field in comparison with observations. The comparison
between observations and high resolution simulations, suggests that the physics may be
described by a multi scale turbulence model. This means that the turbulent dynamo
driven by the cosmological cluster formation process works effectively, reproducing basic
properties from observations, even to details shown in structure functions and converging
to the observation when we increase the resolution. We clearly demonstrates that using
advanced schemes together with very high resolution allow to probe the properties of the
ICM.
Additionally, we investigate the magnetic fields and their relation with the cosmic
structure in which they are embedded. In general, the observed rotation measure signal
is strongly dominated by denser regions (e.g. those populated by galaxy clusters and
groups), and in unclear how is their transition to low density regions, because there is
difficult to acquire direct magnetic field information of those regions.
Therefore statistical tools, such as correlation functions have to be used. To do so,
we use cosmological simulations and try to mimic all the possible observation biases to
constrain actual measurements. We find that the shape of the cross-correlation function
using a normalized estimator (in absence of any noise or foreground signal) nicely reflects
the underlying distribution of magnetic field within the large scale structure.
However, current measurement errors suppress the signal in such a way that it is
impossible to relate the amplitude of the cross-correlation function to the underlying
magnetization of the large scale structure (Stasyszyn et al., 2010).
IX
Part I
Introduction
1
Chapter 1
Astrophysics and Magnetic Fields
Truth is always strange, stranger than fiction.
Lord Byron (1788-1824)
Since early in history, humans were attracted to magnetic fields due to their ‘magical’
nature. Aristotle was the first who scientifically approached magnetism in a discussion
with Thales around 625 BC in Greece.
“The lodestone makes iron come or it attracts it” is one of the earliest references of
magnetism in literature is in a book from the 4th century BC, titled Book of The Devil
Valley Master. 14 centuries later, the Chinese scientist Shen Kuo (1031-1095) described
the concept of magnetic needle and how it can improve the navigation accuracy. By the
XII century the Chinese were known to have huge exploration and merchant fleets using
the lodestone compass to orientate.
In 1187, Alexander Neckham was the first in Europe to describe the compass and its
use for navigation. Later on, in 1269, Peter Peregrinus De Maricourt wrote the Epistola de
magnete, the first extant treatise describing the properties of magnets. The properties of
magnets and the dry compass were discussed by Al-Ashraf, a Yemeni physicist, astronomer
and geographer, in 1282.
In 1600, William Gilbert published De Magnete, Magneticisque Corporibus, et de
Magno Magnete Tellure (On the Magnet and Magnetic Bodies, and on the Great Magnet
the Earth), in which he describes many of his experiments with an earth model called the
terrella. From his experiments, he concluded that the Earth is magnetic itself and that
this is the reason compasses point north (previously, some believed that it was the pole
star, Polaris, or a large magnetic island on the north pole that attracted the compass).
The discovery of the relationship between electricity and magnetism began in 1819
with work by Hans Christian Oersted, professor at the University of Copenhagen, when by
accident he observed that an electric current could influence a compass needle (nowadays
this is called Oersted’s experiment). After that, scientists from that epoch put effort
in discovering new properties from these interactions. In 1865, James Clerk Maxwell
was able to compile all the phenomenological insights found to the epoch, into the socalled Maxwell’s equations, unifying electricity, magnetism and optics in the theory of
electromagnetism. He compiled a set of 14 integral equations, which are not easy to use.
Oliver Heaviside, 20 years later, re-formulated these equations in the vectorial form and
using partial derivatives, and which we nowadays use. In 1905, Albert Einstein used these
3
4
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
laws of electromagnetism in motivating his theory of special relativity, requiring that the
laws held true in all inertial reference frames.
While humankind was discovering this ‘new’ phenomena, in nature several living beings
already made use of it. Animals like sharks and rays are able to feel the field lines
propagating in salty water by indirectly feeling the field’s force using particular receptors.
Birds are also known to sense magnetic fields, it helps them to orient themselves in their
seasonal trips. Another example is the Chiton, a primitive marine mollusc, that has teeth
coated with magnetite. It is believed that this aids them in their homing behaviour,
perceiving magnetic fields for orientation. In other biological kingdoms, some bacteria
are capable of synthesizing magnetite and using it to find out at which level from the
surface of the water they are, as to stay at the correct depth where there is less oxygen,
allowing them to survive easily. Also there are plenty of insects and other species which
have tracers for magnetite and it is believed that they use them for orientation. However,
it is still unclear how these senses work in the majority of the cases.
The understanding of the role of magnetic fields in nature contrasts with the open
enigma that remains regarding magnetic fields in astrophysics. The way magnetic fields
sustain or generate themselves in astrophysics is an open issue. As an example, most
theories explaining Earth magnetic field agree that it should be imprinted from the protoplanetary stages of the solar system, but how it survived afterwards given the proper
timescales of this system is still unknown. Additionally, the primordial magnetic field
in the proto-planetary disc is linked to the star formation process that created the Sun,
which supports the existence of magnetic field in such processes and scales. We have good
measurements of the magnetic field on earth, in laboratories and even from the Sun. In
general we rely on indirect methods to infer them (see section 1.2), this makes it difficult
to observe them in large scales. Besides the observations, their dynamical importance
for many astrophysical systems is poorly constrained, therefore there are still plenty of
processes in which the role of magnetic fields are not fully understood.
In the next sections we will first describe briefly the astrophysical background of the
objects that we plan to study. Then, how galactic and extragalactic magnetic fields can
be estimated and briefly summarize how they are embedded in those objects and on the
largest scales of the Universe. We will also spent some time on numerical simulations that
have shown to be useful to study this astrophysical problems. Finally, we summarize the
discussion on magnetic fields which we address in more detail within the next chapters.
For a detailed discussion on extra-galactic magnetic field, dynamo effects and observations
we refer the reader to Kronberg (1994), Beck et al. (1996), Moss and Shukurov (1996)
and Widrow (2002). A detailed review on numerical simulations and clusters within a
cosmological context can be found in Borgani and Kravtsov (2009).
1.1
Extragalactic physics basics
The most common state of ordinary matter in the Universe is plasma, i.e. atoms are
separated and free protons, electrons and other particles fill the space. Outer space is
the best vacuum environment that we are aware of, it is filled with approximately one
Hydrogen atom per cubic meter on cosmological scales, and about 5 particles per cubic
meter in Earth neighborhood. Therefore, in general astrophysical plasmas in the Universe
can be characterized as dilute plasmas, except to the small fraction present inside stars
1.1. EXTRAGALACTIC PHYSICS BASICS
5
and other compact objects. There is no net charge of these plasma and charges can move
almost freely, justifying an “Ideal” treatment of the Magnetohydrodynamical regime (see
appendix A where we derive the corresponding equations). Hence, electric fields are
negligible and matter and magnetic fields are strongly coupled.
A consequence from the laws of Magnetohydrodynamics (see appendix A) is that
magnetic fields need a source of any kind to sustain themselves. Space is permeated
with magnetic fields, and is electrically neutral given the high conductivity of the plasma,
not allowing to have large dynamos from electric currents.
The way magnetic fields evolve in time, coupled with the dynamics of baryons, growing
or vanishing (never completely) in some regions is an open issue. It is unclear whether
complex mechanisms are needed, such as galactic dynamos, super-nova explosions, or just
a gentle baryon flow to recover the magnetic flux to the measured values (see section 1.2).
Magnetic fields are tightly coupled with baryons, and thus to the structures that build
the Universe. But, how those structures assembled? The current knowledge in cosmology
is based on the so-called standard cosmological model. In this scenario, soon after the
“big bang“, the main constituents of the Universe were radiation and matter that, during
the early stages, were in thermodynamical equilibrium yielding to emission of energy with
black-body spectrum that we still observe today 1 . The presence of density and energy
disomogeneities capable of growing under gravitational instabilities is fundamental for
the structure formation process. Nowadays the presence of primordial fluctuations of the
order of 10−5 the mean density is well established from the CMB experiment. In more
quantitative terms, one can define a density contrast δ(x, t), as a function of the cosmic
time and spatial position
ρ(x, t) − ρb (t)
δ(x, t) ≡
,
(1.1)
ρb (t)
being ρ(x, t) the density at point x and instant t and ρb (t) the background mean density
as function of time. Probably these fluctuation originated in the very early phases of the
Universe undergoing an exponential expansion (“inflation”), becoming the actual seeds
of structure formation. When these small over-densities are present, they can grow only
if the gravitational forces are stronger than the pressure or dispersion forces.
A simple linear evolution would predict present-day structures with density contrasts
δ ∼ 10−2 (Peebles, 1980). Nevertheless, we observe over-densities (i.e. galaxies) with
δ ≫ 1, meaning that their evolution must have been strongly nonlinear. Additionally,
once structure growth proceeds and enters an advanced stage, the different accretion
process and histories particular of each halo make the whole picture more complex, even
possibly implying that baryons physics can modify the pure gravitational interaction
(Pedrosa et al., 2010).
Most of those structures reached an equilibrium between gravitational energy and other
energies (i.e. kinetic, thermal) stopping their collapse, when objects reach that point in
the evolution are called “viralized“. However, even if they are in a dynamical equilibrium
inside , they still are prompt to interact with surrounding objects. Given their sizes and
the age of the Universe, galaxy clusters are young objects, therefore we can find some of
them not virialized yet, as well as structures at scales beyond them.
Still, galaxy clusters are the largest bound structures of which we are aware. Theses
1 Actually this is the best measurement from black body spectrum at the present-day, it is known as cosmic microwave
background (CMB), with an effective temperature of about 2.725 ◦ K.
6
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
structures are one of the busiest places in the Universe. Their formation process leads
to an accretion of a huge amount of matter, which heats up the plasma to levels
where X-ray emission occurs. The intra-cluster medium has typically temperatures of
T ≃ 107 − 108 ◦ K and a typical particle number density of 105 − 102 m−3 , compared
with the T ≃ 105 − 107 ◦ K and 10 − 100 m−3 in the solar neighborhood. The ICM is
characterized by an emission of about 1043 –1045 erg s−1 The main contributions to this
emission are the free-free interactions of electrons and ions and emission lines by ions of
heavy elements (i.e. iron). From X-ray observations we can trace the hot plasma while
from radio emission and Faraday rotation measurements reveal the presence of magnetic
fields (see section 1.2). The cross information from the different observation methods in
galaxy clusters make them ideal laboratories to test theories of the origin and evolution
of extragalactic magnetic fields.
Dynamical constrains from measurements of the velocities of galaxies in clusters have
provided the first determination of the mass involved in such structures, which typically
falls in the range2 1014 –1015 M⊙ . Baryons constitute only a small fraction of this mass,
however galaxy clusters are charaterize to have almost the same baryon fraction of the
Universe independent of their total mass (Gonzalez et al., 2007). Rich clusters appear
to be in hydrostatic equilibrium with velocities ∼ 1000 km s−1 (see, for example, Sarazin,
1986). In those cases, the temperature of the ICM is consistent with velocities of galaxies
and indicates that both galaxies and gas are in equilibrium within a common gravitational
potential well. The mass inferred from the galaxies and ICM are not sufficient to explain
the depth of the gravitational potential well, which implies that most of the mass in
clusters is in a form of dark matter.
Fig. (1.1) shows an overlay of the optical and X-ray images of the massive cluster Abell
1689. We can recognize the main components of the clusters of galaxies: stars in galaxies,
the hot ICM via its X-ray emission, and the presence of the unseen dark matter through
the distorted shapes of background galaxies because of gravitational lensing effects. Bright
elliptical galaxies are typically located at the cluster center. Such central galaxies show
little evidence of ongoing star formation, despite their extremely large masses and being
in the center of a crowded environment.
Form the mass range of galaxy clusters, we can infer that they are the result of the
collapse of primordial density fluctuations of comoving scales of the order of ∼ 10Mpc.
On scales roughly above 10Mpc, the main acting force is gravity and therefore is the main
driver of the evolution of the structure. These scales can be studied analytically, and
tested by cosmological N-body simulations. We can follow the evolution and growth, due
to gravitational instabilities, of the fluctuations that we seed in the denisty field as initial
conditions defined at an early epoch of the Universe.
Since gravity is a force scale-free, we can try to build up a galaxy cluster model being
self-similar, meaning that clusters of different mass are scaled versions of each other, and
their mass is the only parameter that determines their properties(see Kaiser, 1986, for a
review on this model).
To define such model we take into account the cosmic evolution, defining Mδc as the
mass contained within the radius rδc at redshift z, enclosing a mean over-density δc , similar
as defined in Eq. (1.1). Then Mδc is proportional to ρc (z)δc rδ3c 3 . The critical density of
2M
33
⊙ ≈ 1.99 × 10 g is
3 Note that ρ (z) is the
c
the mass of the Sun.
critical density of the Universe defined by the Friedmann equations which governs the expansion
of the Universe. ρc (z) = 3 H(z)/8 G π
1.1. EXTRAGALACTIC PHYSICS BASICS
7
Figure 1.1: Composite X–ray/optical image of the galaxy cluster Abell 1689, located at redshift z = 0.18.
The map shows an area of 556 Kpc on a side. The purple diffuse halo shows the distribution of gas at
a temperature of about 108 ◦ K, as revealed by the Chandra X-ray Observatory. Images of galaxies in
the optical band, colored in yellow, are from observations performed with the Hubble Space Telescope.
(Credits:X-ray: NASA/CXC/MIT; Optical: NASA/STScI; Composite:Borgani and Kravtsov (2009)).
the universe scales with redshift as ρc (z) = ρc0 E 2 (z), where
1/2
E(z) = H(z)/H0 = (1 + z)3 Ωm + (1 + z)2 Ωk + ΩΛ
(1.2)
gives the evolution of the Hubble parameter H(z), taking into account the expansion of
the Universe. Ωk = 1 − Ωm − ΩΛ is the contribution from curvature of the Universe, Ωm
is the baryon density ratio needed to stop the expansion of the Universe only by matter
and ΩΛ is the so called cosmological constant that interacts as a repulsive force in the
expantion. Therefore, the cluster size rδc is only function of time by z and mass by Mδc .
If we assume that the gas heated during the infall is in equilibrium within the
gravitational potential Φ of the cluster, the temperature should follow kB T ∝ Φ ∝
Mδc /rδc . Therefore the above relation between radius and mass can be linked to the
temperature
(1.3)
Mδc ∝ T 3/2 E −1 (z)
This relation between mass and temperature can be used as a scaling relation when
analyzing observations.
8
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
The X-ray luminosity, as an example, scales with the effective emissivity times the
volume that is emitting inside the cluster. Therefore, assuming a thermal bremsstrahlung
process (see section 1.2) dominating the emission of a given electron number density
ne (e.g., Peterson and Fabian, 2006), we have LX ∝ n2e T 1/2 rδ3c . Then, ne scales as
∝ M/rδ3c = const and T ∝ Mδc /rδc , which (using the M − T relation Eq. (1.3) above)
gives LX ∝ T 2 E(z).
At small scales (i.e. < 1 Mpc), baryon physics begin to play an important role besides
gravity. The structure formation via sequencial mergers and accretion of smaller systems,
implies that the intergalactic gas is heated and disturbed by compression and shocks.
Once this gas cools and is dense enough, it can be transformed into stars or accreted
onto super massive black holes inside the cluster galaxies. The process as a whole implies
different baryon physics as cooling, star formation and super massive black holes activity,
which produces an energetic feedback, injecting and mixing the baryons throughout the
cluster. The complex star formation process, includes the generation and spread of heavy
elements by the end of the life cycle of stars, in super nova events.
As an example of the intriguing physics happening in cluster cores, we should mention
that some of them show an X-ray emission extremely high. This emission should finish
when the system reaches a thermal equilibrium. However, this cooling time is shorter than
the dynamical time of the system (by size, mass, etc.). Additionally, when the cool gas
is compressed, flow towards the center, following the gravitational force and the pressure
from the surrounding hot environment (Fabian et al., 1984). These “cooling flows” are
also found elliptical galaxies and groups of galaxies. In particular in some galaxy clusters
a sharp peak in the X-ray surface brightness distribution is observed, implying the density
rise towards the cluster center, therefore confirming the presence of cooling events (see,
Fabian, 1994).
Accretion flows and the motions of groups and galaxies, which perturb their medium
by gravitational and hydrodynamical interactions, can generate substantial stochastic
motions in the ICM. Those phenomena are substantially more important than gravity
in the central regions of the clusters, breaking the self–similarity in these regions and
interacting with the “cool flow” event. Energy of the large-scale turbulent eddies can
cascade down to smaller scales resulting in power law turbulent energy and velocity
spectra. In fact, numerical simulations of cluster formation generally show that subsonic
random flows are ubiquitous even in apparently relaxed clusters This turbulence can
have several important effects on the ICM. It can facilitate mixing of gas at different
radii and correspondingly exchange of thermal energy, diffusion of heavy elements which
would tend to broaden the centrally peaked abundance profile (Rebusco et al., 2005).
Turbulence is also believed to maintain and amplify cluster magnetic fields via dynamo
processes (Subramanian et al., 2006), and to contribute to the acceleration of cosmic-rays
4
in the ICM (e.g., Brunetti and Lazarian, 2007).
Galaxy clusters are objects where astrophysics and cosmology exists side by side. The
overall dynamics is dominated by gravity with cosmological properties, given by their
formation history and size. Astrophysical processes at the galactic scale are highly related
with the surrounding environment and interactions with other galaxies. Given that we
can probe each of their components separately with observations, clusters are powerful
laboratories where to study the processes operating during galaxy formation and their
4 Cosmic-rays
are charged sub atomic particles originated in astrophysical processes.
9
1.2. OBSERVATION OF MAGNETIC FIELDS
effects on the surrounding intergalactic medium.
1.2
Observation of Magnetic Fields
The presence of magnetic fields in the Universe is confirmed by diffuse radio emission in
the inter-galactic medium (see Beck, 2009, for a recent review) and in the intra-cluster
medium, and by observations of Faraday rotation measures towards polarized radio sources
within or behind the magnetized medium (e.g. Govoni, 2006). When studying magnetic
fields in astrophysics, we face the problem of the physical complexity of those systems and
the uncertainties inherent in the observations. In order to understand these limitations,
we first introduce some observational techniques that are used to infer magnetics fields
on astrophysical scales.
A more thorough discussion of observational techniques can be found in various
references including Ruzmaikin, A. A. and Sokolov, D. D. and Shukurov A.M. (1988), we
will briefly summarize in the same way as done by Widrow (2002)
1.2.1
Synchrotron Emission
When relativistic particles are in the presence of a magnetic field they change their energy
states while spiraling in the field lines, therefore release photons in the so called form of
“synchrotron” emission. Given this emission, it can be used to study magnetic fields in
astrophysical sources ranging from pulsars to clusters of galaxies. The total synchrotron
luminosity from a source give us a primary estimates for the strength of magnetic fields
in galaxies and clusters. To be able to study synchrotron sources, we rely on a basic
assumptions on the energy distributions between magnetic and electrons, which commonly
we assumed to be in equipartition. The first studies commenting on the possibility to use
this emission and their polarization degree of this emission as an indicator of the structure
the field, were carried out by (Ginzburg and Syrovatskii, 1969).
For a single electron in a magnetic field B, the emissivity as a function of frequency ν
and electron energy E is
J(ν, E) ∝ B⊥
mc2 ν
νL
1/3 2 mc ν
f
EνL
(1.4)
where B⊥ is the component of the magnetic field perpendicular to the line of sight,
νL = (eB⊥ /2πmc) is the Larmor frequency 5 , and f (x) is a cut-off function which
approaches unity for x → 0 and vanishes rapidly for x ≫ 1.
The total synchrotron emission from a given source depends on the energy distribution
of electrons, ne (E). A commonly used class of models is based on a power-law distribution
ne (E)dE = ne0
E
E0
−γ
dE
(1.5)
assumed to be valid over several ranges in energy. The exponent γ is called the spectral
index while the constant ne0 ≡ ne (E0 ) sets the normalization of the distribution.
5 This frequency given the precession of the particle by the torque between the magnetic field and the magnetic momentum
of particle
10
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
R
The synchrotron emissivity is jν ≡
J(ν, E)ne (E)dE. Eq. (1.4) shows that
the synchrotron emission at a frequency ν is dominated by electrons with energy
E ≃ me c2 (ν/νL )1/2 , i.e., ν ≃ νc , so that to a good approximation, we can write
J(ν, E) ∝ B⊥ νc δ (ν − νc ). Giving us a power-law distribution of Eq. (1.5)
(1+γ)/2
jν ∝ ne0 ν (1−γ)/2 B⊥
.
(1.6)
Assuming equipartition of the magnetic field with relativistic electron density ǫre one
can use Eq. (1.6) to have an estimate from the synchrotron emission of the magnetic field.
As a consequence, B/Beq sets the scale for the thickness of radio synchrotron halos. A field
as small as 0.1Beq requires higher particle energies to explain the synchrotron emission
data. However, high energies imply large propagation lengths and hence an extended
radio halo (scale height ∼ 30 Kpc) in conflict with observations of typical spiral galaxies.
Conversely, a field as large as 10Beq would confine particles to a thin disk (∼ 300 pc)
again in conflict with galaxy observations.
In our Galaxy, the validity of the equipartition assumption can be tested because we
have direct measurements of the local cosmic-ray electron energy density and estimates
of the local cosmic-ray proton density. A combination of the radio synchrotron emission
measurements with these results yields a field strength in excellent agreement with results
derived from equipartition arguments (Beck et al., 2002).
While synchrotron radiation from a single electron is elliptically polarized, the emission
from an ensemble of electrons is partially polarized because of the stochasticity of
the emission. For a regular magnetic field and power-law electron distribution, the
polarization will be constrained by the spectral index γ, giving more steep distributions
high polarizations. In the same line of thought, one can imagine that as much polarized
the source much stronger the magnetic field. However, if the field is highly tangled, the
polarization also will be low.
On the largest scales, like those of filaments, magnetic fields are notoriously difficult
to measure by this method, because we require either a high density of electrons and the
presence of relativistic particles, which is not necessary the case in low density regions.
1.2.2
Faraday rotation
Electromagnetic waves, propagating through a region with magnetic field, experience
Faraday rotation because circular polarized states travel with different phase velocities.
Polarized light propagating through an astrophysical medium will be affected, particularly
inside galaxy clusters or galaxies. This effect is similar to the birefringence of polarized
light transmitted through some media, but in this case is caused by the magnetic field
component in the propagation direction. For linearly polarized radiation, this results in
a rotation of the electric field vector along the path length by an angle
e3 λ2
ϕ=
2πm2e c4
Z
ls
ne (l)Bk (l)dl + ϕ0
(1.7)
0
where me is the mass of the electron, λ the wavelength of the radiation, ϕ0 the initial
polarization angle, and Bk the line-of-sight component of the magnetic field. Here, ne (l)
is the density of thermal electrons along the line of sight from the source (l = ls ) to the
observer (l = 0). ϕ is usually written in terms of the rotation measure, RM:
11
1.2. OBSERVATION OF MAGNETIC FIELDS
ϕ = (RM) λ2 + ϕ0
(1.8)
R ls
ne (l)Bk (l)dl
R ls
≃ 810 0 ne Bk dl
(1.9)
RM = RMg + RMs + RMig
(1.10)
where
RM ≡
e3
2πm2e c4
0
being in the last equation the RM in rad m−2 , ne in cm−3 , the magnetic field in µG and
the distances in Kpc. In general, the polarization change has to be measured at three
or more wavelengths in order to determine RM accurately and remove the ϕ ≡ ϕ ± nπ
degeneracy.
By convention, RM is positive for a magnetic field directed towards the observer.
The Faraday rotation angle includes contributions from all magnetized regions along the
line-of-sight. Following Kronberg and Perry (1982) we decompose RM into three basic
components:
where RMg , RMs , and RMig are the contributions to the rotation measure due to the
Galaxy, the source itself, and the intergalactic medium respectively. When we mention
RMs we refer as the effect in the local environment of the source, because the source
itself emits polarized independent on the wavelength, without any Faraday effect. From
magnetic field observation in other galaxies and from radio sources inside our Galaxies
we are able to build simplified magnetic field models of our Galaxy (Spergel et al.,
2007; Sun et al., 2008). Using those models, Waelkens et al. (2009) developed mock
observations of Faraday rotation measurements and other observables, helping us to
distinguish between the different components of the total signal.
Note that in the case of studies of objects with a small angular size compared with
the typical 3 degree reversal scales found in our Galaxy, it is usually assumed to have a
constant or smooth contribution. Therefore, in the cases of galaxy clusters as RMs , the
variations have a small angular size and studies focus en relative values, the contribution
of the effects from our Galaxy and the RMig are neglected.
Measurements of the magnetic field strength have been successful mainly in high
density regions (e.g. galaxies and galaxy clusters), and fields significantly below the µG
level can hardly be detected, because of the low electron densities of such environments.
In order to study such low magnetic field strengths, even more indirect methods have to
be used. Thereby, statistical approaches are a valuable tool, as they allow to measure very
weak signals, otherwise not be detectable. Such statistical approaches (like correlation
functions) measure the integrated signal over the whole sample of available RMs, see
section 1.4.
1.2.3
Zeeman Splitting
In vacuum, the energy levels of the electrons of an atom are independent of the direction
of its angular momentum vector. A magnetic field lifts this degeneracy by defining a
particular spatial direction. If the total angular momentum of an atom is J = spin S plus
12
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
orbital angular momentum L there will be 2j + 1 levels where j is the quantum number
associated with J. The splitting between neighboring levels is ∆E = gµB where g is the
Lande factor which relates the angular momentum of an atom to its magnetic moment
and µB is the Bohr magneton 6 . This effect, known as Zeeman splitting, is of historical
importance as it was used by Hale (1908) to discover magnetic fields in sunspots, providing
the first evidence for extraterrestrial magnetic fields.
Zeeman splitting is the most direct method to observe astrophysical magnetic fields.
Once ∆E is measured, B can be determined without additional assumptions. Moreover,
Zeeman splitting is sensitive to the magnetic field at location of the emission, without
any integrated or projection effects, besides the contribution from all the emitters in that
particular place.
Unfortunately, the Zeeman effect is extremely difficult to observe. The line shift
associated with the energy splitting is
∆ν
B
= 1.4g
(1.11)
ν
ν
This effect compete with the Doppler broadening due thermal movements, being
frequently observed as an abnormal7 broadening of the line. Therefore, positive detections
have to be constraint to regions of low temperature and high magnetic field.
Within our Galaxy, Zeeman effect measurements have provided information on the
magnetic field in star forming regions and near the Galactic center. Of particular interest
are studies of Zeeman splitting in water and OH masers performed by Reid and Silverstein
(1990). Their results are consistent with those found in radio observations and, as they
stress, provide in situ measurements of the magnetic field contrary to the integrated field
along the line-of-sight of radio observations.
Robishaw et al. (2008) studied a set of ultra-luminous infrared galaxies and found
significant Zeeman splitting and line-of-sight magnetic field strengths ranging from
∼ 0.5 − 18µG. This detections are consistent with synchrotron observations and the
field strengths are similar to those measured in Galactic OH masers. This suggest that
the local process of massive star formation is similar, despite the variety of conditions.
1.2.4
Polarization of Optical Starlight
Polarized light from stars can reveal the presence of large-scale magnetic fields in our
Galaxy and those nearby. Hiltner (1949) attempted to observe polarized radiation
produced in the atmosphere of stars by studying eclipsing binary systems. He expected
to find time-variable polarization levels of 1 − 2%. Instead, he found polarization levels as
high as 10% for some of the stars but not all. While the polarization degree for individual
stars did not show the expected time-variability, polarization levels appeared to correlate
with position in the sky. This observation led to the conjecture that a new property of
the interstellar medium (ISM) had been discovered. Coincidentally, it was just at this
time that Alfvén (1949) proposed the existence of a galactic magnetic field to confine
the cosmic rays propagation. The connection between polarized starlight and a galactic
magnetic field was made by Davis and Greenstein (1951). They suggested that elongated
dust grains would have a preferred orientation in a magnetic field: for prolate grains, one
6
Which is defined by µB = e~/2me c = 9.3 × 10−21 erg G−1 , and represent the magnetic moment of an electron.
the thermally broadened line
7 From
1.3. MAGNETIC FIELD IN GALAXY CLUSTERS
13
of the short axes would coincide with the direction of the magnetic field. The grains, in
turn, preferentially absorb light polarized along their long axis, i.e., perpendicular to the
field. The net result is that the transmitted radiation has a polarization direction parallel
to the magnetic field.
Polarization of optical starlight has limited value as a probe of extragalactic magnetic
fields, for several reasons. There is at least one other effect that can lead to polarization of
starlight, namely anisotropic scattering in the ISM. Also, the starlight polarization effect
is self-obscuring since it depends on extinction. There is approximately one magnitude
of visual extinction for each 3% of polarization. In other words, a 10% polarization effect
must go hand in hand with a factor of 20 reduction in luminosity, making this method
extremely difficult to use. Still, the precise mechanism by which dust grains are oriented
in a magnetic field is not well understood.
1.3
Magnetic field in galaxy clusters
Cluster magnetic fields have been treated as secondary topics in reviews of cluster
atmospheres (Sarazin, 1988; Fabian, 1994), and in general reviews of cosmic magnetic
fields (Kronberg, 1996; Ruzmaikin, A. A. and Sokolov, D. D. and Shukurov A.M., 1988).
The only dedicated review on cluster magnetic fields is Carilli and Taylor (2002).
We will attempt to synthesize various of these works, showing the develop of a general
picture for cluster magnetic fields, having in mind that there is a significant differences
between clusters, and even within a given cluster atmosphere.
Radio halos
A fraction (∼, 30%) of rich clusters have observable radio halos. These radio-halo clusters
share a number of properties, such as a large homogeneous hot intra-cluster medium and
the absence of a central dominant galaxy (the so called cD galaxy).
Large et al. (1959) discovered a radio source in the Coma cluster that was extended
even when observed with large beam size (a 45 arcmin resolution). Later, this source
(Coma C) was studied by Willson (1970) who found that it had a steep spectral index
and meaning that this emissions could not be made up of individual sources, but instead
was a smooth “radio halo” with no structure on scales less than 30 arcmin. Furthermore, if
there is equipartition between the energy stored in particles and magnetic fields, Burbidge
(1959), inferred that the emission mechanism was likely to be synchrotron with a magnetic
field strength of 2 µG, The equations for deriving minimum energy fields from radio
observations are given in Miley (1980). Estimates for minimum energy magnetic field
strengths in cluster halos range from 0.1 to 1 µG (Feretti et al., 1999a). One of the best
studied halos is in Coma, for which Giovannini et al. (1993); Bonafede et al. (2010) report
a minimum energy magnetic field of 0.4 µG. In Fig. (1.2) we show an image obtained of
the radio halo in the Coma cluster, taken from Feretti et al. (1995). Easily we observe
the extended emission (gray area), compared with the radio sources (back areas).
Using the Northern VLA Sky Survey and X-ray selected samples as starting
points Giovannini and Feretti (2000) have performed moderately deep VLA observations
(integrations of a few hours) which have more than doubled the number of known radio
halo sources. Several new radio halos have also been identified since then. These radio
halos typically have sizes ∼ 1Mpc, steep spectral indices, low polarizations, low surface
14
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
brightnesses, and centroids close to the cluster center defined by the X-ray emission. A
strong correlation between cluster X-ray and radio halo luminosity has been found, as well
as a correlation between radio and X-ray surface brightnesses in clusters (Govoni et al.,
2001).
28 40
30
DECLINATION (B1950)
20
Coma C
10
00
27 50
40
30
20
10
12 59
1253+275
58
57
56
55
54
RIGHT ASCENSION (B1950)
53
52
Figure 1.2: WSRT radio image of the Coma cluster region at 90 cm, with angular resolution of 55′′ ×
125′′ (HPBW, RA × DEC) from Feretti et al. (1995) Labels refer to the halo source Coma C and the
relic source 1253+275. The grey scale range displays total intensity emission from 2 to 30 mJy/beam
while contour levels are drawn at 3, 5, 10, 30, and 50 mJy/beam. The bridge of radio emission connecting
Coma C to 1253+275 is resolved and visible only as a region with an apparent higher positive noise. The
Coma cluster is at a redshift of 0.023, such that 1′ = 27 Kpc.
Brunetti et al. (2001) present a method for estimating magnetic fields in the Coma
cluster radio halo independent of minimum energy assumptions. They base their
analysis on considerations of the observed radio and X-ray spectra, the electron inverse
Compton and synchrotron radiative lifetimes, and reasonable mechanisms for particle reacceleration. They conclude that the fields vary smoothly from 2±1 µG in the cluster
center, to 0.3±0.1 µG at radius of 1 Mpc. Recently, Donnert et al. (2010) constraint these
conclusions with numerical simulations, finding agreement with current observations.
1.3. MAGNETIC FIELD IN GALAXY CLUSTERS
15
Radio relics
A possibly related phenomena to radio halos is a class of sources found in the outskirts
of clusters known as radio relics. Like the radio halos, these are very extended sources
without an identifiable host galaxy. Radio relics are often elongated or irregular in shape
and are strongly polarized. One of the first explanations to explain these objects was that
these are the remnants of a radio jet associated with an active galactic nucleus (therefore
the name), that since stop its activity the jet continue traveling in the ICM. A problem
with this idea is that, once the energy source is removed, the radio source is expected
to fade on a timescale ≪ 108 years due to adiabatic expansion, inverse Compton, and
synchrotron losses. This short timescale precludes significant motion of the host galaxy
from the vicinity of the radio source.
Another explanation is that the relics are the result of relativistic particles accelerated
in shocks produced during cluster mergers (Ensslin et al., 1998), or are radio sources
revived by compression associated with those mergers (Enßlin and Gopal-Krishna, 2001).
In these cases equipartition field strengths for relics range from 0.4−2.7µG (Ensslin et al.,
1998).
Cluster center sources
Rotation measures in cluster centers can be derived, as we already mentioned, from multifrequency polarimetric observations of sources by measuring the position angle of the
polarized radiation as a function of frequency. However thought these effect, only the
magnetic field component along the line-of-sight is measured, so the results depend on
the assumed magnetic field topology.
The Cygnus A observations were the first to show that the large RM must arise in
an external screen of magnetized, ionized plasma, but cannot be Galactic in origin.
Dreher et al. (1987) considered a number of options for the Faraday screen toward Cygnus
A, and concluded that the most likely site was the X-ray emitting cluster atmosphere
enveloping the radio source. They infer values for magnetic fields around 2 ∼ 10 µG to
explain the observed RMs.
Since those observations, RM studies of cluster center radio sources have become a
standard tool to measure cluster core magnetic fields. RM studies of radio galaxies in
clusters can be divided into studies of cooling-flow and non-cooling-flow clusters. Typical
mass cooling flow rates are 100 M⊙ yr−1 . Cooling-flow clusters are more dynamically
relaxed than non-cooling flow clusters which often show evidence of cluster mergers
(Markevitch et al., 2002).
Radio galaxies in cooling flow clusters attracted some of the first detailed RM studies
by virtue of their anomalously high RMs. Out of a sample of 14 cooling-flow clusters with
strong embedded radio sources Taylor et al. (2002), found that 10 of 14 sources display
RMs in excess of 800 rad m−2 . Current data are consistent with all radio galaxies at
the center of cooling flow clusters having extreme RMs, with the magnitude of the RMs
roughly proportional to the cooling flow rate (see Fig. 1.3).
From the shapes of the RM distributions is found that the magnetic fields are not
regularly ordered on cluster scales, however they have characteristic coherence lengths
of scales up to several Kpc (one example is shown in chapter 3). The RM distributions
at the centers of cooling flow clusters tend to have reversals with coherence lengths of
16
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
3C295
Hyd-A
A2029
A2597
Cyg-A
A1795
Virgo
A2199
Centaurus A4059
A2052
3C129.1
A119
Figure 1.3: The maximum absolute RM plotted as a function of the estimated cooling flow rate, Ẋ, for
a sample of X-ray luminous clusters with measured RMs from Taylor et al. (2002). Both RM and Ẋ are
expected to depend on density to a positive power, in that sense the correlation is expected.
5 ∼ 10Kpc. Larger “patches” up to 30 Kpc are seen for example in Cygnus A. In both
Cygnus A and Hydra A one can find stripes of alternating high and low RM. Such bands
are also found in the non-cooling flow cluster sources (Eilek and Owen, 2002), along with
slightly larger coherence lengths of 15 ∼ 30Kpc. In Hydra A there is a strong trend for all
the RMs to the north of the nucleus to be positive and to the south negative. To explain
this requires a field reversal and implies a large-scale (100Kpc) ordered component to the
cluster magnetic fields in Hydra A.
Minimum cluster magnetic field strengths can be estimated by assuming a constant
magnetic field along the line-of-sight through the cluster and, form to the X-ray
observations, is fitted a radial electron density distribution, commonly using a modified
King sphere model (Cavaliere and Fusco-Femiano, 1976). Such estimates usually lead to
magnetic field strengths of 5−−10 µG in cooling flow clusters, and the half in non-cooling
flow clusters.
In a reanalysis of the Abell 119 measurements, Dolag et al. (2001b) found that the
field scales as n0.9
e . This dependence is marginally steeper than that expected assuming
2/3
flux conservation, for which the tangled field scales as ne , and significantly steeper than
that expected assuming a constant ratio between magnetic and thermal energy density,
1.4. LARGE SCALE MAGNETIC FIELDS IN THE UNIVERSE
17
1/2
for which the tangled field scales as ne for an isothermal atmosphere. This can lead us
to weight different the inner from the outer regions in the cluster, if the density scales up
steeply, and should be taken in consideration.
Extreme rotation measurements have been observed in a variety of different
morphologies. This means that high rotation measurements are not a particular
phenomena from strong interactions between galaxies and its environment, but are more
likely to be linked to cluster scales characteristic of the inter cluster medium.
1.4
Large scale magnetic fields in the Universe
Spatially coherent magnetic fields are ubiquitous in galaxies and galaxies clusters. From
the theoretical side, it is not well understood whether these fields are flux-frozen primordial
fields or they originated by the dynamo amplification from other phenomena. If those
large scale (≪ 1 Mpc) fields exist unambiguously, they can flavor one of those scenarios,
depending on their strength. Nevertheless, their existence might be of great importance
to understand the way galaxies and clusters formed and accreted their baryons.
There is evidence of magnetic fields in galaxies at early times, setting up a serious
challenge for the galactic dynamo hypothesis, since it would imply that there was a short
amount of time for an effective field amplification. At present, the most convincing
observations of galactic magnetic fields at intermediate redshifts come from rotation
measurement studies of radio galaxies and quasars. However, as we already mentioned,
Wolfe et al. (2008) estimated a magnetic field strength of 86µG at redshift z ∼ 0.7, using
the Zeeman splitting technique and obtaining one of the highest values for high redshift
galaxies.
Athreya et al. (1998) studied 15 high redshift (z > 2) radio galaxies at multiple
frequencies in polarized radio emission, founding significant rotation measurements in
almost all of the sample, and even with several of the sources having RM > 1000 rad m−2 .
The highest RM in the sample is 6000 rad m−2 for the z ≃ 2.17. Rotation measurements
of this magnitude require micro-gauss fields that are coherent over several Kpc.
Faraday rotation of radio emission from high redshift sources can be used to study
cosmological magnetic fields. For a source at a cosmological distance ls , the rotation
measure is given by the generalization of Eq. (1.9) appropriate to an expanding Universe:
RM ≃ 8.1 × 10
5
Z
ls
ne (l)Bk (l)(1 + z)−2 dl
(1.12)
0
The factor of (1 + z)−2 accounts for the redshift dilution of the electromagnetic fluxes
as they propagate to the observer. We consider the contribution to this integral from
cosmological magnetic fields. If the magnetic field and electron density are homogeneous
across our observable volume, an all-sky rotation measurement map will result in a dipole
distribution, with the amplitude depending on the evolution of B and ne . The simplest
assumption is that the co-moving magnetic flux and electron density are constant, i.e.,
B(z) = B0 (1 + z)2 and ne (z) = n0 (1 + z)3 . The cosmological component of the RM is
then
RMig ∼ ×104 h−1 cos(θ)n0 B0 , FΩm ,ΩΛ (z)
(1.13)
18
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
where θ is the amplitude angle between the source and the magnetic field orientation,
FΩm ,ΩΛ (z) concentrates all the cosmological factors and the integration in the line of sight
Z
dl
H0 zs
dz (1 + z)3
(1.14)
F (z) =
c 0
dz
Chosen a cosmological model and taking into the account of the expansion of the
dl
universe Eq. (1.2) the dz
takes the form
c
dl
=
dz
H0 (1 + z) E(z)
(1.15)
In Fig. (1.4), we plot F and RMig as a function of zs for selected cosmological models.
The path length to a source and hence the cosmological contribution to the RM are
increasing functions of zs , as is evident in Fig. (1.4). We can see that the nowadays
flavored cosmological model (the solid line in the plot) show an increase of the order of
20 just for cosmological effects.
Combining Eq. (1.13) and Eq. (1.15), with rotation measurement data from highredshift galaxies and quasars, we can constrain the strength of Hubble-scale magnetic
fields. The difficulty is that the source and the Galaxy contributions to the RM are
not completely known. However, it can be modeled as done by Waelkens et al. (2009)
or one can try to suppress the Galaxy contribution. Kronberg and Simard-Normandin
(1976) found that even at high galactic latitude, some objects have RM > 200 rad m−2.
In particular, the high Galactic latitude subsample that they considered was evidently
composed of two populations, one with hRM 2 i1/2 ≃ 50 rad m−2 and another with
hRM 2 i1/2 ≃ 200 rad m−2. The High-RM subsample is more likely to be saturated by
the source phenomena and not from the inter galactic contribution, and given that we
ensure low galactic contamination at high latitudes, they conclude that the best chances
to constrain Hubble-scale magnetic fields are studying the galactic polar sectors and lowRM subsample. The galaxies in this work extended to z ≃ 3.6 though most of the objects
were at z < 2.
In an not so ideal case, the pattern of the cosmological contribution across the sky will
be more complicated than a simple dipole, because the electron density or/and magnetic
field can vary. However, still the average cosmological rotation measurement over the sky
2
will be zero and the variance σRM
will increase with redshift. Kronberg and Perry (1982)
considered a simple model in which clouds of uniform electron density and magnetic field
are scattered at random throughout the Universe.
The Kronberg-Perry model was motivated by spectroscopic observations of QSOs which
reveal countless hydrogen absorption lines spread out in frequency by the expansion of
the Universe. This dense series of lines, known as the lyman-α forest, are thought to
be a large number of neutral hydrogen clouds at cosmological distances. Clearly, the
limits derived from rotation measurement data will depend on the model one assumes
for the lyman-α clouds. (Blasi et al., 1999) suggested another model for the intergalactic
medium in which the Universe is divided into cells of uniform electron density following
some particular distribution and the magnetic field is parametrized by its coherence length
and mean field strength. Random lines of sight are simulated for various model universes.
The results suggest that a detectable variance in RM is possible for magnetic fields as
low as B0 ≃ 6 µG. The enhanced sensitivity relative to the Kronberg and Perry (1982)
1.4. LARGE SCALE MAGNETIC FIELDS IN THE UNIVERSE
19
Figure 1.4: Intergalactic contribution to the rotation measure (RMig ) as a function of source redshift zs .
The left-hand vertical axis gives the function F (zs ) as defined in Eq. (1.14). The right-hand axis gives
RMig assuming a 1 µG field, h75 = 1, ne0 = 10−5 cm−3 , and θ = 0. Curves shown are for the standard
cold dark matter (CDM) model (Ωm = 1), two LCDM models (CDM with a cosmological constant)
(Ωm = 0.15; ΩΛ = 0.85 and and Ωm = 0.3; ΩΛ = 0.7) and an open CDM model (Ωm = 0.3; ΩΛ = 0).
Taken from Widrow (2002)
observations is primarily due to the large filling factor assumed for the clouds. Recently,
Dolag et al. (2010) found that maybe this is the case. However, Stasyszyn et al. (2010)
showed that given the current instrumentation, we are still unable to identify this signal
(see chapter 4).
Further improvements in the use of Faraday rotation to probe cosmological magnetic
fields may be achieved by looking for correlations between them (Kolatt, 1998). The
correlation function of the position of the sources and its RM strength increases as Nc2
2
where Nc is the average number of clouds along the line of sight. By contrast, σRM
increases linearly with Nc , meaning that the signal in a correlation map can be enhanced
by an order of magnitude or more, depending on the number of clouds that we are
considering. However this methods can be affected by other issues that we study in
chapter 4.
20
1.5
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
Numerical simulations of cluster formation
The first simulations of cluster formation have been performed in the 1970s (White, 1976)
and were focused in testing the idea of gravitational instability. The first simulations of
galaxy clusters which followed the dynamics of both baryons and dark matter where
performed in the 90s (Katz and White, 1993a). The predictions of the self–similar
model have been tested by a number of authors against hydrodynamical simulations,
which include only the effect of gravitational heating of baryons(e.g., Navarro et al., 1995;
Nagai et al., 2007). These simulations generally confirmed the scaling relations, as defined
by Eq. (1.3), for the average quantities in radial profiles, as well as their evolution
with redshift. However, the self–similar model is far too simple to capture all of the
complexities of cluster formation and the detailed observations. Numerical cosmological
simulations carried out on parallel supercomputers represent the modern tool to describe
these complexities. Such simulations begins at a sufficiently early epoch of the Universe
when density fluctuations were small and can be specified using analytical theories of
primordial perturbations. Those initial conditions are usually generated as realizations
of a density field with statistical properties of the adopted cosmological model. The
simulations co-evolve the collisionless dark matter and normal baryonic components from
the initial conditions, by integrating numerically the equations governing the dynamics of
those components represented by density and velocity fields.
Observational probes, such as the Cosmic Microwave Background, the statistics of
the population of galaxies and galaxy clusters, and the properties of absorption systems
in the spectra of distant quasars, have provided tight constraints on the cosmological
model (e.g., Komatsu et al., 2009, and references therein). The main challenge for
actual simulations is to follow the dynamics driven by gravitational instability and the
gas-dynamical processes that contributes in different forms to the correct evolution of the
cosmic baryons, when this evolutions becomes highly non linear.
In simulations we are able to only treat a subset of equations that will characterize
the system, by solving them discretizing and sampling the initial phase space and then
integrating their equations of motion. The dynamics of diffuse baryonic matter is expected
to be highly collisional and therefore we are able to use hydrodynamics techniques to model
them.
The addition of baryons in the simulations allow to predict properties of clusters in Xrays. Those simulations confirmed that gas is heated up to emitting temperatures during
cluster formation, only due to the adiabatic compression and shocks. Early simulations are
successful in reproduce general morphological characteristics of the X-ray observations.
The simulations also demonstrated that gas in the inner regions of clusters is close to
be in hydrostatic equilibrium, confirming the predictions of simple models from X-ray
observations and self–similar models. Further progress in simulations show that while the
outer regions of clusters are quite regular, nearly self–similar, the inner regions do not
converge to a simple solution and show evidence of galaxy formation effects and energy
injection.
As for the numerical schemes behind the hydrodynamics simulations the two most
commonly used approaches are grid-less smoothed particle hydrodynamics (SPH) and
grid-based Eulerian schemes. In the SPH, fluid elements describing the system are
sampled and represented by particles discretizing the mass, and the dynamic equations
are obtained from the Lagrangian form of the hydrodynamic conservation laws (see
1.5. NUMERICAL SIMULATIONS OF CLUSTER FORMATION
21
Monaghan, 2005, for a recent review, and chapter 2). The main advantage of SPH lies
in its Lagrangian nature as there is no grid to constrain the dynamic range in spatial
resolution or the global geometry of the modeled systems.
Eulerian methods, sample the space of our physical system into grid cells, and thus
follow the exchange of different properties between them. These scheme perform well for
flows with shocks, contact discontinuities and for moderately subsonic and turbulent flows.
Additionally the adoption of adaptive mesh refinement (AMR) techniques in cosmological
simulations has changed the situation in the last ten years (e.g., Kravtsov et al., 2002;
Teyssier, 2002; Loken et al., 2002), opening new frontiers for Eulerian schemes. The AMR
is used to increase the spatial and temporal resolution of numerical simulations beyond the
limits imposed by the available hardware and dynamically to adapt to the astrophysical
system. This is done by using fine mesh only in regions of interest (as high-density
regions or regions of steep gradients in gas properties) and coarse meshes in other regions
(low-density regions or smooth flows).
It is important to keep in mind that although different numerical schemes aim to solve
the same sets of equations, the actual discretized versions solved are not the same. This
does lead to differences in the numerical solution (Agertz et al., 2007). However, the
use of both techniques is useful and complementary in highlighting possible systematic
errors associated to a given technique. Such systematics errors can only be identified in
comparisons of simulations performed with different techniques.
In the case of MHD simulations, Eulerian approaches leaded the development from
many years. Their flux constrain and shock capturing capabilities, make more easy to
deal with MHD instabilities. Early tries in SPH (Phillips and Monaghan, 1985) had
only good results for low βplasma 8 and low Mach number shocks. However, the adaptive
characteristics made the studies to continue (Balsara and Spicer, 1999; Omang et al.,
2006; Dolag et al., 1999; Price and Monaghan, 2005), and nowadays there are plenty of
schemes to reduce magnetic tensile instabilities in SPH. Particularly, in our case we take
the advantage of the improvements in cosmological simulations given by SPH (Springel,
2005).
As we already discussed, magnetic fields are indeed observed in galaxy cluster cores
(e.g., Eilek and Owen, 2002; Carilli and Taylor, 2002) and there is evidence towards their
existence in whole cluster volume (e.g., Govoni and Feretti, 2004; Eilek et al., 2006),
therefore can contribute as a non-thermal component. By the use of simulations,
magnetic fields are also predicted to be effectively amplified from the primordial values
in mergers during cluster formation (Dolag et al., 1999, 2002; Brüggen and Hoeft, 2006;
Subramanian et al., 2006; Dolag and Stasyszyn, 2009). The questions that then arise
are: how dynamically important the magnetic fields actually are?, how they affect the
turbulent motions? and which magneto-hydrodynamic effects are important in the ICM?
MHD cosmological simulations suggests that magnetic fields generally provide
a minor contribution to the total pressure (Dolag et al., 2001a), while they
can become dynamically non-negligible in peculiar cases involving strong mergers
(Dolag and Schindler, 2000) or in strong cooling flows (Dubois and Teyssier, 2008).
Xu et al. (2009) feed the magnetic fields from AGNs, being further amplified by the
ICM turbulence through small-scale dynamo processes, showing that can be the origin of
cluster-wide magnetic fields. The strong link between turbulence processes and magnetic
8β
plasma
is defined as the thermal over magnetic energy ratio
22
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
field modifies the spectral slope of those quantities, therefore studying them we can
determine how this phenomena take place (Li et al., 2008a; Brüggen and Hoeft, 2006;
Dolag et al., 2002). However, the inclusion of magnetic effects has not yet became
standard in cosmological simulations.
1.6
Open questions
Magnetic fields are present everywhere, independent of the environment or related
physical phenomena. As we discussed, baryons and magnetic fields in space are usually
tightly coupled. However, from simulations and observations, we know that, at first
approximation we do not need to follow the MHD equations or explicit baryonic evolution
to understand the cosmic structure of the Universe. The overall structure, voids and
groups, are reasonably well determined by gravity alone. However, baryon physics become
important on small scales, which later on manifest themselves in galactic scales (i.e. by
wind, AGN feedback, etc.), modifying the environment, as we are able to observe.
In the preceding sections we briefly commented the techniques which are used to carry
out cosmological simulations of cluster formation. Many of the observed properties of
clusters, such as scaling relations between observables and total mass, radial profiles
of entropy and density of the intra-cluster gas and radial distribution of galaxies are
reproduced quite well. In particular, the outer regions of clusters exhibit scaling with
mass close to the scaling expected for a self-similar model. However, simulations generally
do not reproduce the observed “cool core” structure of clusters. The amount of cooling
that simulated clusters generally show in their central regions, make the temperature and
energy profiles incorrect. These discrepancies point towards an important role played by
additional physical processes, beyond those already included in the simulations.
The richness of observational datasets which are quickly becoming available need to
be properly modeled by cosmological simulations. Although most of the processes that
operate in clusters are not well understood, the comparison between observations and
simulations, will be extremely useful to advance in their interpretation and to build
theoretical models.
Nowadays, we reach the accuracy in studies that start to involve additional physics,
such is the case of MHD. We can follow the complete structure formation (from an
adiabatic collapse of the initial clouds) that in the case of magnetic fields cinch the field
lines, and increases the magnetic fluxes, but how much this will affect the star formation
process or baryon accretion is unknown. As we mention, there is a strong link between
turbulence and magnetic fields. Current theoretical models favor the growth of magnetic
fields by turbulent motions. Turbulence contributes to the chemical mixing and energy
transport in the ICM, wherefore the study of magnetic field shed light on other physics
where the magnetic fields are not directly linked.
Magnetic field observations of the Coma cluster (Bonafede et al., 2010) and of radio
galaxies at redshift z ≃ 2, points towards the existence of a widespread cosmological
field. The origin of this magnetic field is under debate, being the simplest hypothesis a
primordial field which is amplified by the collapse of the proto-galaxy and subsequently
maintained by some dynamo action, becoming the micro-gauss field observed in presentday galaxies and clusters. This simple model involves different process on different
scales which should be proven, and in most of the cases can be modified by other
1.6. OPEN QUESTIONS
23
physical phenomena. Nowadays, observations and theory are reaching the point to make
prediction.
To continue these studies we need to build an sufficiently accurate numerical scheme
to compute these high non-linear systems. Even with simple models and assumptions, we
can try to answer the following open questions
• How are magnetic fields distributed on scales larger than galaxy clusters? It is
possible to measure them?
• Are models beyond self-similar capable to explain observations? Are they consistent
with magnetic field profiles and Faraday rotation measurement?
• If the numerical ∇ · B = 0 constraint is not satisfied by simulations, which effects
does this implies?
• Can we explain current observations with “ideal” MHD and non-radiative simulations
only?
• Are magnetic field coupled to other processes happening in the inter galactic medium?
How much?
Cosmological MHD simulations are already performed using finite-volume and finitedifference methods. Astrophysical fluid dynamics usually involve changes over several
orders of magnitude in spacial, temporal and energetic scales. Thus, adaptivity is essential
to study these problems. In that sense Smooth Particle Hydrodynamics, offers an elegant
solution. However, it was not until recently that it was possible to successfully include
MHD equations into a cosmological SPH formalism (Dolag and Stasyszyn, 2009). Such
simulations can be used to either follow a primordial magnetic field (Li et al., 2008b) or
the creation of magnetic fields in shocks through the so-called Biermann battery effect
(Kulsrud et al., 1997; Ryu et al., 1998), on which a subsequent turbulent dynamo may
operate. The latter process predicts magnetic field strengths in filaments with somewhat
higher values (see Sigl et al., 2004) than predicted by simulations which start from a
primordial magnetic seed field. However, they are in concordance with predictions of
magnetic field values from turbulent amplification (Ryu et al., 2008).
Therefore, further investigations are needed to clarify the structure, evolution and
origin of magnetic fields in the largest structures of the Universe, their observational
signatures, as well as their interplay with other processes acting in galaxy clusters and
the large scale structure of the Universe.
In the present work, we present the achievements done in the SPH-MHD formalism in a
cosmological context, to clarify the possible effects of numerical errors done in simulations.
We start in chapter 2, by briefly explaining how our numerical simulations are done and the
different characteristics of different implementations. We applied these numerical schemes
to galaxy clusters, showing the astrophysical differences and implications in chapter 3. In
chapter 4 we show how simulations can help to estimate the values of the large scale
magnetic field. Finally, we summarize our work in chapter 5.
24
CHAPTER 1. ASTROPHYSICS AND MAGNETIC FIELDS
Part II
Numerical Methods
25
Chapter 2
MHD - Smoothed Particle
Hydrodynamics
A computer shall not waste your time or require you to do more work than is strictly necessary.
Jef Raskin (1945-2005)
One standard approach to solving the equations of fluid dynamics numerically is to
discretize the space which the fluid occupies, often on a regular grid, defining the fluid
properties in each cell and computing derivatives using finite volume differentiation. In
astrophysical problems, the fluid dynamics usually is under changes in spatial, temporal
and density scales over several orders of magnitude. Adaptivity is essential to face these
problems and fix grid approaches might be too limited. Theses needs where faced with the
development of procedures like adaptive mesh refinement (AMR), which gives additional
space refinements in the regions where the physical processes needed to be calculated more
accurately 1 . The implementation of such procedures is far from being trivial, although
there are plenty codes these that offer that kind of solution. However, these methods still
suffer from artifacts because astrophysical problems are frequently highly asymmetric,
deriving in substantial artificial diffusion when solving derivatives or due errors in the
non proper solution of Galilean invariance, independent of adaptive or fixed grids (see
Springel, 2010, for instance).
An alternative to these methods is to follow the fluid not by discretizing space but
matter. This means following portions of fluid defined by their masses, resulting in
methods which are naturally adaptive. In this approach fluid quantities are carried by
a set of interpolation tracers, so called particles, which follow the fluid motion. Each
point particle is defined by a characteristic mass, the volume that represents is defined
by the local spatial distribution of tracer particles, which also are used to calculate the
interpolants for derivatives. 2
The particle method is called Smoothed Particle Hydrodynamics (SPH) and was
introduced by Lucy (1977) and Gingold and Monaghan (1977). It has found widespread
use in astrophysics due to its ability to handle a wide range of problems involving complex,
asymmetric phenomena with relative ease. SPH concepts are simple, the equations are
1 Other approaches to this problem are to use unstructured grids (where typically the grid is reconstructed at each new
time-step) or Lagrangian grid methods, where the grid shape deforms according to the flow pattern.
2 To be precise derivatives can be evaluated either by interpolation over neighbouring particles, or via an hybrid approach
by interpolation to an overlaid grid (referred to as particle-mesh methods).
27
28
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
self-consistently derived from physical principles, and thus the incorporation of additional
physics modules is relatively straight forward. Adaptivity and scalability are natural
features of the methods.
The SPH method, whose formal derivation will be given later in this chapter, ensures
that changes in density and flow morphology are automatically taken into account. These
properties, to some extent, allow the code to be naturally prepared to resolve high density
regions as presented in many astrophysical problems, without much loss in computational
effort in regions physically unimportant for the problem. A particular highlight of SPH
is the natural coupling with N-Body techniques, that makes a big advantage in the
calculation of gravity interactions and enables the use of well known procedures used
in astrophysics.
In particular in N-Body techniques (namely via tree-codes) already construct the
neighbour list to calculate the gravitational force. So this techniques offers a natural
skeleton for SPH without any extra load. SPH has not reached a mainstream status as
finite volume technique. In SPH the vector nature of magnetic fields, in comparisons
with other scalar fields (i.e. temperature), can involve anisotropic stresses and lead
to instabilities. The reason for this is that the magnetic stress can become negative
depending on the morphology of the magnetic field lines and the numerical errors in
the evolution, leading to the clumping of particles, and thus spurious forces arise. That
was a mayor issue in the development of magnetohydrodynamics in SPH that resulted in
instabilities that were not possible to solve properly until recent years. However, progress
has been done in the treatment of those instabilities (Morris and Monaghan, 1997;
Børve et al., 2001; Price and Monaghan, 2005). In particular non-radiative simulations
of galaxy clusters within a cosmological environment which follow the evolution of a
primordial magnetic seed field have been performed using SPH codes (Dolag et al.,
1999, 2002, 2005a). We also gained insight in the field of galaxies studying their
isolated evolution (Kotarba et al., 2009) and in interacting environments (Kotarba et al.,
2010a,b). The difficulty here was again the dynamical ranges involved. Other works (i.e.
Elstner et al., 2000) were done using finite boxes to focus on a small section, and then
extrapolate that into the whole galaxy. All simulations have in common that instabilities
that come from MHD dynamics. We are able now to handle them properly and we learnt
were they came from.
In this chapter we provide an overview of the MHD-SPH method, including several
improvements to regularize the field from instabilities. The chapter is organised as follows:
In section 2.1 and 2.1.1 we will try to introduce some basic numerical concepts of SPHMHD, in section 2.3 we focus on the changes needed to stabilize the code, later we will
show some test cases in section 2.4 that help us to understand the code behavior and
grade of success of our implementation, that finally we discuss in section 2.5.
2.1
Introduction to SPH
Cosmological investigations demand an enormous dynamic range in space and time. For
example in studies of galaxy clustering, one would like to resolve until scales of 10 Kpc
for individual galaxies in regions of hundreds of Mpc. System with crossing times of less
than 106 years must be followed for 1010 years. The strength of TreeSPH in these kind
of numerical simulations lies in its dynamical range and balance between accuracy and
29
2.1. INTRODUCTION TO SPH
resources like memory and time.
We have implemented the MHD equations in the cosmological SPH code GADGET
(Springel et al., 2001; Springel, 2005), allowing us to explore the full size and dynamical
range of state of the art cosmological simulations. Note that the implementation
therefore is fully parallelized and benefits from many optimizations within the general
parts of the code, especially the calculation of self gravity and optimization in data
structures as well as work-load balancing. Therefore, this implementation is an ideal
tool to follow the evolution of magnetic fields in astrophysics. GADGET also allows us
to turn on the treatment of many additional physical processes which are of interest
for structure formation and make interesting links with the treatment of magnetic
fields for future studies. This includes thermal conduction (Jubelgas et al., 2004;
Dolag et al., 2004b), physical viscosity (Sijacki and Springel, 2006), cooling and starformation (Springel and Hernquist, 2003), detailed modelling of the stellar population
and chemical enrichment (Tornatore et al., 2004, 2007) and a self consistent treatment of
cosmic rays (Enßlin et al., 2007; Pfrommer et al., 2007). The MHD implementation at
first approximation should be fully compatible with all these extensions but we are aware
that the need of additional astrophysics and proper modifications of the algorithms to take
into account the effects of the magnetic fields. All such processes are expected to increase
the complexity and lead to a complicated interplay with the evolution of the magnetic
field and would make it impossible to critically check the numerical effects caused by the
different SPH-MHD implementations or the astrophysics itself. As a first step we want
to investigate the numerical behavior of the different schemes implemented. Therefore,
our work is mainly focused on non-radiative simulations, to emphasize the effects of the
methods and exhaustively analyze their behavior.
2.1.1
Interpolants
The basis of the SPH approach is given by Monaghan (1992); Lucy (1977);
Gingold and Monaghan (1977). For a recent review one can see Monaghan (2005) but we
will try to briefly summarize the basics in this section. If we start with the trivial identity
Z
A(ra ) = A(ra )δ(|ra − rb |)drb ,
(2.1)
where A is any variable defined on the spatial co-ordinates vector r 3 and δ refers to the
Dirac delta function. This integral is then approximated by replacing the delta function
with a smoothing kernel W with characteristic width h 4 , such that
lim W (ra − rb , ha ) = δ(ra − rb ),
ha →0
thus
A(ra ) =
Z
A(rb )W (|ra − rb |, ha )drb + O(ha 2 ).
The kernel function is normalised to unity as
Z
W (ra − rb , ha )drj = 1.
3 The
4 This
indexes a,b represent different spatial positions, later on will this will be identified as the particles itself
variable is called smoothing length - Hsml.
(2.2)
(2.3)
(2.4)
30
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
Finally the integral (2.3) is discretized onto a finite set of interpolation points (the
particles) by replacing the integral by a summation and the mass element ρdV with
the particle mass m, i.e.
Z
A(rb )
W (|ra − rb |, h)ρ(rb )drb + O(ha 2 ),
A(ra ) =
ρ(rb )
N
X
Ab
≈
mb W (|ra − rb |, ha ),
(2.5)
ρ
b
b=1
where the subscript a refers to the quantity evaluated at the position of particle a. This
summation interpolant is the basis of all SPH schemes. Gradient terms may be easily
obtained by taking the analytic derivative of (2.5),
Z
∂
A(rb )
∇A(ra ) =
W (|ra − rb |, ha )ρ(rb )drj + O(h2a ),
(2.6)
∂ra
ρ(rb )
X Ab
(2.7)
≈
mb ∇a Wab ,
ρb
b
where we have assumed that the gradient is evaluated at another particle b, so we define
∇a ≡ ∂r∂a and Wab ≡ W (|ra − rb |, ha ) ∼ W (|rab |, ha ). Note that given a Kernel, these
r
∂Wab that depends only in the derivatives of the
derivative takes the form of ∇a Wab ∼ |r|
Kernel and the smoothing length, and thus this is independent of any other calculation.
By doing a Taylor expansion and identifying the error terms, one can subtract the first
error term, giving
X (Ab − Aa )
∇a Wab ,
(2.8)
∇Aa =
mb
ρ
b
b
Since the first error term in the Taylor expansion is removed, the interpolation is exact for
constant functions and indeed this is obvious from the form of (2.8). For further reading
and interesting discussions about error constrains, I refer the reader to Monaghan (2005).
2.2
Building our set of Equations
To build our set of continuous fluid quantities equations, one first needs to define the
smoothing kernel introduced in previous section 2.1.1. The most frequently used kernel
W (|r|, h) is the B2 -Spline (Monaghan and Lattanzio, 1985), which can be written as

2
3
0 ≤ xh ≤ 0.5
1 − 6 hx + 6 hx

8
3
W (x, h) =
(2.9)
2 1 − hx
0.5 ≤ xh ≤ 1
πh3 
x
0
1≤ h
It is worth stressing that, contrary to other SPH implementation, GADGET uses the
notation in which the kernel W (x, h) reaches zero at x = h and not at x = 2h. The
density ρa at each particle position ra can be estimated via
X
hρa i =
mb W (|rab |, ha ),
(2.10)
b
2.2. BUILDING OUR SET OF EQUATIONS
31
where the smoothing length ha found solving iteratively the implicit equation
4π 3
h ρa = Nma .
(2.11)
3 a
A typical value for N is in the range of 32-64, which correspond to the number of neighbors
which are traditionally chosen in SPH implementations. We need to discretize our set of
physical equations in the form 2.3 As an example, we can derive the proper motion
equations starting from the Lagrangian (as Price and Monaghan (2004a); Monaghan
(2005) already shown)
Z 1 2
L=
ρv − ρu dV,
(2.12)
2
where u is the internal energy per unit mass. Again, in SPH form this is re-written in
summation form as
X
1 2
L=
ma va − ua (ρa , sa ) ,
(2.13)
2
a
where as before we have replaced the volume element ρdV with the mass per SPH particle
m. We regard the particle co-ordinates as the canonical variables. Being able to specify
all of the terms in the Lagrangian directly in terms of these variables, and taking into
account the invariance of the Lagrangian to transformations, the conservation laws will
be automatically satisfied. Since the equations of motion result then the Euler-Lagrange
equations
d ∂L
∂L
−
= 0.
(2.14)
dt ∂va
∂ra
The internal energy is regarded as a function of the particle’s density, which in turn is
specified as a function of the co-ordinates by (2.10). The terms in (2.14) are therefore
given by
∂L
= ma va ,
∂va
X
∂L
∂ub ∂ρb
=
mb
.
∂ra
∂ρ
b s ∂ra
b
From the first law of thermodynamics in the absence of dissipation we have
Pb
∂ub = 2,
∂ρb s ρb
(2.15)
(2.16)
(2.17)
and using (2.10) we have
such that
∂ρa X
=
mc ∇a Wbc (δba − δca ) ,
∂rb
c
X Pb X
∂L
=
mb 2
mc ∇a Wbc (δba − δca ) ,
∂ra
ρ
b
a
b
X
Pa Pb
+ 2 ∇a Wab ,
= ma
mb
2
ρ
ρb
a
b
(2.18)
(2.19)
(2.20)
32
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
where we have used the fact that the gradient of the kernel is anti-symmetric (i.e.
∇a Wab = −∇a Wba ). The SPH equation of motion in the absence of dissipation is therefore
given by
X
Pa Pb
dva
mb
+ 2 ∇a Wab ,
=−
(2.21)
dt
ρ2a
ρb
b
which can be seen to explicitly conserve momentum since the contribution of the
summation to the momentum of particle a is equal and opposite to that given to particle
b (given the antisymmetry of the kernel gradient).
In GADGET, the equation 2.21 for the SPH particles are implemented based on this
derivation (Springel and Hernquist, 2002) and take the form
X
Pa
Pb
dva
mb fa 2 ∇a Wa + fb 2 ∇a Wb ) .
=−
(2.22)
dt hyd
ρ
ρ
a
b
b
The coefficients fa are defined by
ha ∂ρa
fa = 1 +
3ρa ∂ha
−1
,
(2.23)
and reflect the full, self-consistent correction terms arising from varying the particle
smoothing length. The abbreviation stands Wa/b = W (|ra − rb |, ha/b ) for the two kernels
of the interacting particles. The pressure of each particle is given by Pa = Aa ργa , where
the entropic function Aa stays constant for each particle in the absence of shocks or other
sources of heat.
One of the most interesting aspects of SPH is the lack of any kind of numerical diffusion
of the method. This as first thought is an advantage, but in the other hand does not
help to regularize instabilities, or properly handle physical sharp discontinuities in the
medium. For example to capture shocks properly, artificial viscosity is needed. Therefore,
in GADGET the viscous force is implemented as
N
X
dva
mb Πab ∇a Wab ,
(2.24)
=−
dt visc
b=1
where Πab ≥ 0 is non-zero only when particles approach each other in physical space.
In the case of an ideal gas equation of state where
P = A(s)ργ ,
the entropic function A(s) evolves according to
γ − 1 du P dρ
dA
,
=
− 2
dt
ργ−1 dt
ρ dt
γ − 1 du
=
.
ργ−1 dt diss
(2.25)
(2.26)
This has the advantage of placing strict controls on sources of entropy, since A is constant
in the absence of dissipative terms. The thermal energy is evaluated using
u=
A γ−1
ρ .
γ−1
(2.27)
33
2.2. BUILDING OUR SET OF EQUATIONS
This formulation of the energy equation has been advocated in an SPH context by
Springel and Hernquist (2002).
The viscosity generates entropy at a rate
1γ −1X
dAa
mb Πab vab · ∇a W̄ab ,
=
dt
2 ργ−1
a
b
(2.28)
Here, the symbol W̄ab denotes the arithmetic mean of the two kernels Wa and Wb and vab
denotes the relative velocity between the particles.
For the parametrization of the artificial viscosity we use a formulation proposed by
Monaghan (1997) based on an analogy with Riemann solutions of compressible gas
dynamics. The resulting viscosity term can be written as
Πab = −
sig
α vab
µab
2 ρab
(2.29)
for rab · vab ≤ 0 and Πab = 0 otherwise, i.e. the pair-wise viscosity is only non-zero if
the particles are approaching each other. Here µab = vab · rab /|rab | is the relative velocity
projected onto the separation vector and the signal velocity is estimated as
sig
vab
= ca + cb − βµab ,
(2.30)
p
with ca = γPa /ρa denoting the sound velocity. In GADGET the values α = 1 and β = 3
are commonly used for the dimensionless parameters within the artificial viscosity. Note
that in Eq. (2.29) usually a viscosity-limiter is used. In our case this create tensile
instabilities between the particles so is discarded.
This also leads naturally to a Courant-like hydrodynamical time-step
∆thyd
=
a
CCourant ha
,
sig
maxa (vab
)
(2.31)
where CCourant is a numerical constant, typically chosen to be in the range 0.15 − 0.2.
2.2.1
Co-moving variables and units system
The equations of motion are integrated using a leap-frog integration making use of a
kick-drift-kick scheme. Within this scheme, all the pre-factors due to the cosmological
background expansion are taken into account within the calculation of the kick- and driftfactors (see Quinn et al., 1997; Springel, 2005). For the integration of the entropy within
dt
a cosmological simulation, a factor H −1 = da
is present in all the equations with time
derivatives to take into account that the internal time variable in GADGET is the expansion
parameter a. The formulation of the MHD equations within GADGET has to be adapted
accordingly to this choice of variables.
In GADGET the unit system is defined by setting LENGTH, MASS and VELOCITY factors
with respect to the CGS system. Thereby the internal units of ENERGY and TIME are
defined. The magnetic field units are defined as GAUSS. Therefore µ0 within the MHD
equations becomes to
[TIME]2 [LENGTH]
µ0 =
.
(2.32)
4π[MASS]
34
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
For cosmological simulations, the parametrization of the units for LENGTH, MASS and
contain the Hubble constant h and therefore µ0 has to be modified to be
TIME
µ0 =
[TIME]2 [LENGTH]
.
4π[MASS]h2
(2.33)
Note that the magnetic field in the code is always physical for historical reasons. This
means that all the evolution in cosmological simulations has to take additional the noncomoving terms and factors accordingly.
2.2.2
Magnetic signal velocity
signal
A natural generalization of the signal velocity vab
in the framework of MHD is to replace
the sound velocity ca by the fastest magnetic wave as suggested by Price and Monaghan
(2004a). Therefore the sound velocity ca gets replaced by
v
u
s
2
u
2
1 t 2
c2 (B · r̂ab )2
Ba
Ba2
mag
2
·
ca = √
−4 a
(2.34)
ca +
ca +
µ0 ρa
µ0 ρa
µ0 ρa
2
As this new definition of the signal velocity also enters the time step criteria (2.31),
no extra time-step criteria due to the magnetic field has to be defined. If we choose
more conservative settings within the Courant condition, we still see improvements in the
solution of the test problems. Therefore we generally use CCourant = 0.075, which is half
the value usually used in pure hydrodynamical problems. Different authors also propose
to use different values for α and β within the artificial viscosity definition (2.29). Whereas
typically α = 1 is chosen, Monaghan (1997) proposed to use β = 3. Price and Monaghan
(2004b) propose to use β = 2 or β = 1 respectively. We find slight improvements in our
test problems when using α = 2 and β = 1.5, which are the values that we use if not
noted otherwise.
2.2.3
Induction equation
In appendix A we set the proper equations that ideal MHD should follow, in particular
the evolution of the magnetic field is given by the induction equation,
dB
= (B · ∇)v − B(∇ · v)
dt
(2.35)
if Ohmic dissipation is neglected and the constraint ∇ · B = 0 is used. In a cosmological
framework there are additional terms and factors because of the dilution of the space and
the fact that magnetic fields behave like fluxes. This equation takes then the form
dB(z)
dB 1
=
− 2H(z)B(z)
dt
dt a2
If we construct the SPH equivalent as described in section 2.1.1, the full induction equation
takes the form
"
#
1 fa X
dBa
=
·
mb (Ba (vab · ∇a Wab ) − vab (Ba · ∇a Wab )) − 2Ba
(2.36)
dt
Ha2 ρa
b
2.2. BUILDING OUR SET OF EQUATIONS
35
where H −1 takes into account that the internal time variable in GADGET is the expansion
parameter a. Note that here, by construction, only the kernel Wa and its derivative are
used. All the cosmological pre factor and terms are only present in the cosmological
simulations and absent in the code evaluation presented in section 2.4. In component5
form the induction equation reads
#
"
i
X
dBa
1 fa
j
i
(2.37)
mb (vab
Baj − Bai vab
)∂Wa r̂jab − 2Bia
=
2
dt
Ha ρa b
Note that, as also suggested by Price and Monaghan (2004b), we wrote down the
equations including the correction factor fa which reflects the correction terms arising from
the variable particle smoothing length. Unfortunately it is not possible to directly infer
the exact form of the correction factors from first principles for the induction equation.
However, Price and Monaghan (2004b) showed that, if not chosen to be the same as
the one used for the Lorenz force, inconsistency between the induction equation and
magnetic force results. The effect of these factors in the induction equation is quite
small, but nevertheless one notices improvements in test problems when they are included.
Therefore, we included them for all applications presented in this work.
2.2.4
Magnetic force
The magnetic field acts on the plasma via the Lorenz force shown in equation. Eq. (A.3),
which can be written in a symmetric, conservative form involving the magnetic stress
tensor (Phillips and Monaghan, 1985)
1
ij
i j
2 ij
Ma = Ba Ba − |Ba | δ
.
(2.38)
2
The magnetic contribution to the acceleration of the a-th particle can therefore be written
as
a3γ X
Ma
Mb
dva
=
mb fa 2 · ∇a Wa + fb 2 · ∇b Wb
(2.39)
dt mag
µ0 b
ρa
ρb
Where the factor a3γ is needed to transform to the internal variables for cosmological
simulations and is set to one in all other cases. Also µ0 has to be chosen properly
as described in section 2.2.1. The factors fa reflect the correction terms as discussed
previously. In component form the equation reads
"
#
i
ij
dva
a3γ X
Maij
M
=
mb fa 2 ∂Wa r̂jab + fb 2b ∂Wb r̂jab
(2.40)
dt mag
µ0 b
ρa
ρb
The main problem with this formulation is that it becomes unstable for situations in
which magnetic forces are dominating (Phillips and Monaghan, 1985). The reason for
this is that the magnetic stress can become negative depending on the morphology of the
magnetic field lines and the numerical errors in the evolution, leading to the clumping of
particles, and thus spurious forces arise. Therefore, some additional measures have to be
taken to suppress the onset of this instability.
5 We
denote with i, j, k the different components
36
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
2.3
Instability corrections and regularization schemes
There are several methods proposed in the literature to suppress the onset of the clumping
instability which is caused by the implementation of the magnetic force. However their
performance was found to depend on the details of the simulation setup. Also the
requirement to have ∇ · B close to zero is not only physically motivated but numerical
in the sense that large ∇ · B will perturb the Lorentz equation with spurious force. The
numerical origin of this errors is simply the coarseness, either in the integration or due to
SPH operator, and those evolution is highly non linear and unstable.
Beside theses instabilities, noise (e.g. fluctuations of the magnetic field imprinted by
numerical effects when integrating the induction equation) is a source of errors in SPHMHD implementations. Therefore there are regularization schemes to obtain a magnetic
field which does not show strong fluctuations below the smoothing length. This can
be achieved indirectly through improvements in the underlying SPH formalism (as the
instability corrections work) as well as in reformulation of the interactions to reduce the
creation of small irregularities arising from numerical effects. Besides direct smoothing of
the magnetic field, there is the possibility to dissipate small irregularities by introducing
artificial terms in same way as the artificial viscosity is implemented.
It is important to note that the only nature of dissipation in SPH is that all the values
come from the neighboring summation. This effect underlay inside the smoothing length,
therefore it is useful to consider this as the “resolution” of our simulation. If one scheme
uses smoothed values to infer the primitive ones, then we fall in an additional smooth or
dissipative process that is larger than the smoothing length, therefore noticeable in the
“resolution” given by it.
We briefly comment on only a couple of possibilities that turn out to be successful in
the context of building up an implementation for cosmological simulations.
2.3.1
Improvements in the underlying SPH formalism
The entropy conserving formalism (Springel and Hernquist, 2002) of the underlying SPH
implementation contributes to a significant improvement of the MHD formalism compared
to previous MHD implementations in SPH by generally improving the density estimate
and the calculation of derivatives. It has to be noted, that this is not only due to the
dW
terms, but in large part also from the new formalism for calculating of the smoothing
dh
length itself. As described before, the smoothing length h for each particle is no longer
calculated by counting neighbors within the sphere, but by solving equation Eq. (2.11)
for a predefined number N, there exists only one unambiguous value of h. This equation
is solved iteratively and note that there is no restriction on the value of N beside the
convergence 6 . It is usual to give some allowed range of N for the equation to converge,
however in our case we choose the range smaller than 1 and typically we use N = 64 ±0.1.
2.3.2
Divergence force subtraction
Børve et al. (2001) suggested explicitly subtracting the effect of any numerically nonvanishing divergence of the magnetic field. Therefore, one can explicitly subtract the
6 In
other implementations N was usually an integer value, without iterative solution.
2.3. INSTABILITY CORRECTIONS
term as an error, that by definition shouldn’t exist
i corr
X
dva
Ba
Bb
3γ 1
= −a
β̂
mb fa 2 · ∇a Wa + fb 2 · ∇b Wb
dt mag
µ0 b
ρa
ρb
37
(2.41)
from the momentum equation. To be consistent with the other formulations, we included
the dW
terms. The factor β̂ is introduced in the original work (Børve et al., 2001), to
dh
have some control over this correction and a value of β̂ = 1 was chosen.
In principle, this term breaks the momentum conserving form of the MHD formulation.
However, in practice, this seems to be a minor effect. In general this correction should be
always small, Børve et al. (2004) argued that stability for linear waves in 2D can be safely
reached even when not subtracting the full term but choosing β̂ < 1; e.g. they suggested
β̂ = 0.5 to further minimizing the non-conservative contribution. However, it is not clear
if this stays true for 3D setups and in the non-linear regime. Additionally, Børve et al.
(2004) used a higher order kernel and therefore it is also not clear if that is still true in
our case.
We noticed, that under certain circumstances, the correction itself can drive
instabilities, specially when the amplitude of the correction is not longer small compared
with the acting Lorentz force.
These problems can be overcome by introducing a numerical limiter. Basically we
redefine β̂ to be particle dependent (i.e. β̂a ), and constrained by the ratio between the
magnitudes of the Lorentz force and the correction by the following threshold
i
dva dt mag (2.42)
γ̂a = i corr ≥ 2
dva
dt mag therefore, if the Lorentz force is not at most twice the correction we renormalise the
correction to ensure so. We tested the robustness of this parameter showing that is not
so sensitive to variations, see appendix B. In general we find that this correction term,
even without the threshold, significantly improves all results in our test simulations and
effectively suppresses the onset of the clumping instability. It was also already successfully
used in previous, cosmological applications (e.g. Dolag et al., 2004a; Rordorf et al., 2004;
Dolag et al., 2005a). However, the introduction of the threshold shown to be useful in
simulations of colliding galaxies (Kotarba et al., 2010a) and star formation (Bürzle et al.,
2010).
2.3.3
Divergence Cleaning: Dedner Method
Dedner et al. (2002) introduced a novel form to control ∇·B errors. Their idea is to create
a scalar field to follow the numerical ∇ · B and by these means correct the magnetic field
to guarantee a divergence free field. To do so they studied different possible fields and
how they should evolve. The resulting evolution implies a mixed scheme, that allows to
propagate the errors outside the boundary or used region of interest, and damp these
effects at the same time. That method is widely used in Eulerian codes and also has been
suggested by Price and Monaghan (2005) to be used in SPH.
38
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
The first step is to modify the induction equation and adding a term in the form of a
scalar field
∇φa
dBa
=−
(2.43)
dt Dedner
Ha
The factor H(z)−1 takes into account the cosmological derivatives in time and the
co-moving coordinates. One should note that for historical reasons in cosmological
simulations the units of φ are mixed between co-moving and physical 7 .
Changes in the energy introduced by this artificial magnetic field in the induction
equation should be taken into account in the energy equation.
dAa
dt
Dedner
=−
γ − 1 Ba · (∇φa )
Ha2
µ0 ργ−1
a
(2.44)
Dedner et al. (2002) also show how to construct and evolve this additional field. The
evolution equation is given by
∂φa
ch 2
2
= − ch ∇ · B a + 2 φ a
(2.45)
∂t
cp
In 2.45 there are two constant values that will determine the evolution of that equation.
ch is related to ∇ · B as a source term and its wave propagation velocity. cp is introduced
as a damp factor in the evolution. Price and Monaghan (2005) study for the first time
the implementation of such correction applied to Lagrangian codes. They suggest that
it can be useful to, instead of a constant value for ch , use the maximum signal velocity
between particles. In our case it is given by the maximum between the sound velocity and
the Alvfén velocity in the media, as described in section 2.2.2. For the damping time or
time decay term also they propose to scale it with the signal velocity and the smoothing
length as:
ch 2 = σ(cmag
)2
a
ch 2
1
πcmag
a
=
=
2
cp
ha
τa
(2.46)
σ and π are dimensionless parameters to be chosen depending on the problem, to balance
the source and propagation parts of the evolution. With π = 1.0 and σ = 1.0 we recover
the best solution in Price and Monaghan (2005). But we study this in detail in section
2.4.
So the total derivative and taking the cosmological dependence one has
dφa
1 ∂φa
=
− Hφa
dt
a ∂t
σcmag
1
dφa
a
mag 2
π(ca ) ∇ · Ba +
=−
φa − φa
dt
Ha
h
Similar to the ∇ · B subtraction in the Lorentz force, this method has as drawback that
the correction can lead to instabilities in the case when the change introduced is larger
with the one applied by the induction equation alone.
7 The actual units of φ are [V elocity][Gauss] where the velocity is co-moving and the magnetic field is physical, due our
special implementation inside the code.
2.3. INSTABILITY CORRECTIONS
39
To avoid this, we also here introduced a numerical limiter β̃a modifying Eq. (2.43) by
∇φa
dBa
= −β̃a
(2.47)
dt Dedner
Ha
there β̃a will renormalise the correction to be a fraction Q of the original induction equation
in magnitude but allowing the change of the field.
In this case we need to add a new parameter Q that will define at which threshold we
renormalise
∇φa dBa (2.48)
Ha < Q dt We study extensively possible constrains of this threshold in addition with the correction
itself, but a Q = 0.5 seems to be adequate (see apendix B for details). This method
successfully regularizes the field locally, which implies that, compared with other methods,
it strictly follows the ideal MHD equations. This is well studied in section 2.4.
2.3.4
Smoothing the magnetic field
Another method to remove small scale fluctuations and to regularize the magnetic field is
to smooth the magnetic field periodically. As suggested by Børve et al. (2001), one can
calculate a smoothed magnetic field hBi for each particle,
P
Ba
b mb ρb Wa
hBa i = P
.
(2.49)
Wa
b mb ρb
Then, in periodic intervals, one can calculate a new, regularized magnetic field by
Bnew = q hBi + (1 − q)B.
(2.50)
Note that this, in principal, acts similar to the mixing process on resolution scale present
in Eulerian schemes. However, introduced in this way, the amount of mixing (e.g.
dissipation) of magnetic field depends on the frequency with which this procedure is
applied and the value of q chosen. Typically, we set q to one and perform the smoothing
at every 15th -20th main time-step. It is worth to mention that implemented in this
form, total energy is not conserved (as magnetic field fluctuations on scales smaller
than the smoothing length are just removed) and, as the time-steps depend on the
chosen resolution, this method is even resolution dependent. Never the less it leads
to improvements in the results of our test problems, without strongly smoothing sharp
features. It also works without problem in 3D and has already been used in cosmological
simulations (Dolag et al., 2004a, 2005a).
2.3.5
Artificial magnetic dissipation
Another possibility to regularize the magnetic field was presented by Price and Monaghan
(2004b), who suggested including an artificial dissipation for the magnetic field, analogous
to the artificial viscosity used in SPH. They suggested that the dissipation terms be
constructed based on the magnetic field component perpendicular to the line joining the
interacting particles. However, to better suppress the small scale fluctuations within the
40
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
magnetic field which appear due to numerical effects especially in multi-dimensional tests,
Price and Monaghan (2005) suggested to base the artificial dissipation on the change of
the total magnetic field rather than on the perpendicular field components only. We
also found this to work significantly better in our test cases and therefore use the latter
implementation only. Such an artificial dissipation term can be included in the induction
equation as
sig
1 ρa αB X
vab
dBa
=
mb 2 (Ba − Bb )r̂ab · ∇a Wa .
(2.51)
dt diss
Ha2 2
ρ¯ab
b
The parameter αB is used to control the strength of the effect, typical values are suggested
to be around αB ∼ 0.5. Similar to the artificial viscosity, this will create entropy at the
rate
dAa
v sig
γ − 1 αB X
mb ab2 (Ba − Bb )2 r̂ab · ∇a Wa .
= − γ−1
(2.52)
dt diss
4µ0 b
ρ¯ab
ρi
The pre-factor (γ−1)/(ργ−1
) properly converts the dissipation term to a change in entropy.
i
This method reduces noise significantly. However, depending on the choice of αB , it
can also lead to smearing of sharp features. Additionally, to avoid this outside of strong
shocks (e.g. where this is needed), Price and Monaghan (2005) proposed evolving αB for
each particle, similar to the handling of the time dependent viscosity as suggested by
Morris and Monaghan (1997). Then, evolution of αB for each particle will be followed by
integrating
min
dαB
(αB − αB
)
=−
+ S,
(2.53)
dt
τ
where the source term S can be chosen as
|∇ × B| |∇ · B|
(2.54)
S = S0 max
, √
√
µ0 ρ
µ0 ρ
(see Price and Monaghan, 2005). The time-scale τ defines how fast the dissipation
constant decays. Taking the signal velocity, one can translate this directly into a distance
to the shock over which the dissipation constant decays. A useful choice of τ can be
written as
h
τ=
,
(2.55)
C v sig
where the constant C typically is chosen to be around 0.2, allowing the dissipation constant
to decay within a time that corresponds to the shock travelling 5 kernel lengths (see
Price and Monaghan, 2004a),
2.3.6
Euler potential
Besides the implementations to follow the magnetic field evolution using the induction
equation 2.35, there are other methods to represent the magnetic fields that implies the
use of potential fields (either vector or scalars). Theses methods have the advantage
of ∇ · B = 0 by construction. However, this constraint is still under the effects of the
underlying scheme (i.e. SPH) and other numerical artifacts (i.e. particle noise), that will
define the level in which we can sample this condition.
2.4. TEST PROBLEMS
41
One very elegant way to implement the MHD equations in Lagrangian codes is the
usage of so called Euler potentials (see Rosswog and Price, 2007, and references therein).
They are build as two independent variables α and β and correspond to an implicit choice
of a gauge for the vector potential 8 . They can be thought of as labels of magnetic field
lines and will be advected with the flow. In this formulation, the magnetic field at any
time can be represented as
B = ∇α × ∇β.
(2.56)
In principle, having obtained the magnetic field, one could use this magnetic field also
in the equation of motion as before. However, this would mean that the magnetic force
is based on the second derivative of a variable. This is usually quite noisy and not
recommendable unless regularization schemes are implemented for this as well as for
example done in Rosswog and Price (2007).
The main drawback of this method is that the final magnetic field in a region of
interest usually comes from several field windings. In such situations, the advection of
Euler potentials with the fluid elements folds improperly the underlaying magnetic field
below the resolution, breaking down the scheme, if no re-mapping of the quantities is
applied (as implicitly done when follow the induction equation). Therefore, after several
“winding-up” times9 , the resulting magnetic field corresponds to the one produced by the
stochastic distribution of the potentials due the advection of the fluid.
Rosswog and Price (2007) suggested to evolve the potentials using a similar
implementation as the artificial dissipation. We study this possibility and didn’t find
any improvement in our results. Furthermore, Brandenburg (2010) showed that this kind
of implementation is incorrect in the limits of turbulent boxes, showing that the use of an
dissipative version of Euler potentials will break the gauge with vector potentials, making
them unphysical.
Therefore we use this simple description only as a check in cosmological simulations,
to investigate the influence of ∇ · B driven errors, because this implementation should be
divergenceless by construction. In such cases the evolution of the magnetic field predicted
when using Euler potentials can be seen as an upper bound on the amplification processes
in lack of back reaction which could lead to saturation effects. Therefore they are a useful
tool to check influence of the numerical scheme applied on the results for the cosmological
simulations where we have no other means to verify the results. Note that even with this
method the ∇ · B constraint is numerically not fulfilled. As we already mentioned the
sampling and the numerical errors due interpolants are always present, this translates to
numerical ∇ · B even if by construction it should be zero. As an example, we found that
using the ∇ · B subtraction method improves our results even with the use of this scheme
10
.
2.4
Test problems with stable MHD
The solutions of our astrophysical problems are very complex and non linear. From
the original configuration to the final stage, our system will have gone through
8 Actually what we are representing is just a family of the possible Vector potential, therefore not all the possible magnetic
field can be represented by the Euler potencials.
9 Time when the magnetic flux had been re-connected in normal cases.
10 Even it become to be necessary to stabilise the solutions in our studies in star formation (Bürzle et al., 2010), which
made us that is the reason why previous studies didn’t succeed.
42
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
Figure 2.1: Test 5A at time t = 7 with the MHD implementation similar to that used in Dolag et al.
(1999, 2002), but already including the instability correction due to subtraction of the ∇ · B term in the
force equation and in a fully three-dimensional setup. Shown in the first row are the density (left panel),
total energy and pressure (middle panel) and the x,z component of the velocity field (right panel). The
second row shows the y-component in the velocity field (left panel), the three components of the magnetic
field (middle panel) and the measure of the ∇ · B error, see equation Eq. (2.57), in the right panel. The
red lines with error bars are show the SPH results and blue just for the pressure case, the black lines are
the reference results obtained with ATHENA in a 1D setup.
advection, rarefactions, hydrodynamical, MHD shocks, etc. will happen in different
regimes and epochs. To be confident about our results one have to test properly
the method used and compare with known solutions. To know the limitations of our
implementations we need to know how they perform, and thus even find the optimal
numerical parameters for different schemes. Therefore, we performed a series of shocktube problems as presented by Ryu and Jones (1995). In particular test 5A, which is
also used in Brio and Wu (1988) was used to show the effects of different numerical
treatments. Additionally we performed several 2D test cases including the Fast Rotor
test (Toth, 2000; Londrillo and Del Zanna, 2000; Balsara and Spicer, 1999), a Strong
Blast (Londrillo and Del Zanna, 2000; Balsara and Spicer, 1999) and the Orszang-Tang
Vortex (Orszag and Tang, 1979; Dai and Woodward, 1994; Picone and Dahlburg, 1991;
Londrillo and Del Zanna, 2000). To obtain results under realistic circumstances, we
performed all the tests by setting up a fully three-dimensional particle distribution. Note
that three degrees of freedom in each particle are known to arise instabilities, because
of the added complexity. There are some regularization methods that show good results
in 1/2 dimensions and when tested in 3 dimensions didn’t succeed. The astrophysical
problems that we want to face have already high complexity and need to be studied in 3
43
2.4. TEST PROBLEMS
dimensions, therefore we focus directly on this kind of test setups. We also avoid starting
from regular grids but used glass-like (White, 1996) initial particle distributions instead.
To obtain such a configuration the particles are originally distributed in an random fashion
within the volume and then relaxed until they saddle in a equilibrium distribution which
is quasi force free and homogeneous in density. This is similar to the distribution of atoms
in an amorphous structure like a glass. Compared to a distribution of the particles based
on a grid this guarantees that all the kernel averages in the SPH formalism sample the
kernel in a uniform way and not several times at the same distances which furthermore
are fractions of the underlying grid spacing. For all tests we try to use the same particle
masses11 , independent of the initial density. Therefore, typical initial particle distributions
for the shock-tube tests where based on 53 particles in low density and 103 particles in
high density regions within unit volume. Usually, these unit volumes are then replicated
35 times along the x-direction each to avoid boundary problems. For some test cases with
strong (and therefore fast) shocks, we evolved the simulations longer. In such cases we
doubled the simulation setup size in the x-direction.
We assume ideal gas (e.g. γ = 5/3) and, as described before, use an equivalent of
N = 64 neighbors for calculating the SPH smoothing length. This ensures that, in the
low density regions, SPH particles get smoothed over a region corresponding to a unit
length. The number of resolution elements corresponding to a unit length therefore ranges
from 1 to 4, depending whether one associates the smoothed region or the mean interparticle distance with the effective resolution in SPH. In general, SPH converges somewhat
slower compared to grid codes when comparing simulations with the same number of grid
cells as SPH particles (see section C for an example).
For the SPH results we usually plot the mean within a 3D slab corresponding to the
smoothing length and (as error bars) the RMS over the individual particles within this
volume. The reference solution was obtained using ATHENA (Stone et al., 2008) with
typically 10-20 resolution elements per unit length, depending on the individual test. As
one criteria of the goodness of the SPH simulation result we use the usual measure for
the non-vanishing divergence of the magnetic field,
E∇·B = |∇ · B|
2.4.1
h
.
|B|
(2.57)
Brio-Wu shock-tube
The most commonly used MHD shock-tube test is the one studied first time by
Brio and Wu (1988) (Test 5A in Ryu and Jones, 1995, work). This is a good test, because
it involves a shock and a rarefaction of the same family moving together. Therefore it
allows simultaneous testing of the code in different regimes.
Fig. (2.1) shows the result for a code implementation similar to the first implementation
used to study galaxy clusters (e.g. see Dolag et al., 1999, 2002). In addition, the
instability correction due to subtraction of the ∇ · B term was used in the force equation.
Various hydrodynamical variables at the final time (e.g. t = 7 in this case) are shown. The
red lines with error bars show the SPH-MHD result, the black lines are the reference result
obtained with ATHENA in a 1D setup (our ideal solution). Shown are (from upper left to
11 In principle we use exactly the same mass per particle, but in some cases to have the correct density we modify a little
bit the masses but less than 1%.
44
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
Figure 2.2: As figure Fig. (2.1), but including the magnetic waves in the signal velocity The main
advantage are a significant reduction in the noise, specifically in the velocity, but also in the magnetic
field. Also the ∇ · B errors are reduced by a factor of ≈ 2.
the bottom right panel) the density, total energy and pressure, the x- and z-component
of the velocity field, the y-component in the velocity field, the three components of the
magnetic field and the measure of the ∇ · B error, obtained from equation (2.57). Note
for the SPH pressure is shown in blue to differentiate with the total energy. Here we also
switched back to the conventional formulation of the artificial viscosity as for example
described in Monaghan (1992), not the one based on signal velocity as used in GADGET.
Although the SPH-MHD results in general follow the solution obtained with ATHENA,
there is a large scatter in the individual particle values within the 3D volume elements, as
well as some instability, especially in the low-density part. Note that although the mean
values for the internal energy, as well as the velocity or magnetic field, can locally show
some systematic deviations from the ideal solution, the total energy shows much better,
nearly unbiased, behaviour. This demonstrates the conservative nature of the symmetric
formulations in SPH-MHD.
Noticeable reduction of noise is obtained when using the signal-velocity based artificial
viscosity and including the magnetic waves in the calculation of the signal velocity.
Therefore, the magnetic waves are directly captured for the time step calculation
and in the artificial viscosity, needed to capture shocks. In general, the SPH-MHD
implementation gains from the new formulation of SPH, including the dW
terms and
dh
the new way to determine the SPH smoothing length, both contributing to a reduction
of noise (and ∇ · B) in the general treatment of hydrodynamics. In figure Fig. (2.2)
we include the magnetic waves in the signal velocity. Theses version already has the
2.4. TEST PROBLEMS
45
Figure 2.3: As figure Fig. (2.2), but including now the Dedner cleaning scheme. The main advantage
that we found is the suppression of the divergence but not features smearing of any kind.
thresholds in the ∇ · B subtraction 12 . We will refer to this implementation of SPH-MHD
as Standard SPH-MHD from now on.
In the Fig. (2.3) we can see how the cleaning scheme is working properly with a proper
reduction of the ∇ · B error and lowering the noise. To find the numerical parameters that
give best results and increase the stability of the code, we complete a rigorous tests suite
performing parameters studies. Theses studies takes into account the quality in terms
of noise and accuracy as the proper divergence (see Appendix B). The final parameters
range are σ = 2 ∼ 5 and π ∼ 1. The interpretation of these numbers (see Eq. (2.3.3))
are that the ∇ · B errors propagate at the signal velocity and they will be damped at
approximately 2 ∼ 5h.
2.4.2
The effect of field regularization
As described in section (2.3), there are several suggestions for regularization of the
magnetic field. We show results obtained by two regularization methods, namely
smoothing the magnetic field in regular intervals and including an artificial dissipation.
To test theses methods we only use the ∇ · B subtraction and the magnetic signal velocity
for properly stabilize in the Lorentz force equation (the Standard SPH-MHD runs).
In the smoothing field case, we use the same kernel as used for the normal SPH
calculations. In this case, there are two numerical parameters one can choose. One
is q in equation (2.50), which quantifies the weight with which the smoothed component
12 To find the proper numerical parameters we used the same test suit as shown in the next section, the proper studies
are shown in Appendix B.
46
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
Figure 2.4: As Fig. (2.2), but including the regular smoothing of the magnetic field as a regularization
scheme. This SPH-MHD implementation basically reflects the one used in Dolag et al. (2004a, 2005a).
The main advantages are a further, significant reduction in the noise as well as a strong reduction of the
∇ · B errors by a factor of ≈ 10.
enters into the updated magnetic field. We always use q = 1 here, which means that we
completely replace the magnetic field by the smoothed value. The second is TBS , which is
the time interval at which the smoothing is done. Here we use a value corresponding to
smoothing every 30th global time step. This corresponds to the SPH-MHD implementation
used to study the magnetic field in clusters and large scale structure within the local
universe(see Dolag et al., 2004a, 2005a). Fig. (2.4) shows the result for the same shocktube test as before. Clearly, the noise in the individual quantities is strongly reduced.
Also the error in ∇ · B is reduced by more than one order of magnitude. Note that the
error bars for the SPH-MHD implementation are of the size of the line width or smaller
in most of the cases and therefore no longer clearly visible. However, one can notice some
small effect of smearing sharp features. Additionally, some states 13 converge to values
which have small but systematic deviations from the exact solution.
The magnetic field can be dissipated in the same way as artificial dissipation works
in the hydrodynamics. Here the numerical parameter one has to chose is the strength of
this artificial, magnetic dissipation αB in equation (2.51) and (2.52). Fig. (2.5) shows the
result for the same shock-tube test as before using αB = 0.1. Similar to the first method
presented, the noise in the individual quantities is strongly reduced and also the error in
∇·B is reduced by one order of magnitude. Again, the error bars are smaller than the line
width nearly everywhere. Also, some small effects of smearing sharp features are visible
13 like the region with the negative x-component of the velocity behind the the fast rarefaction wave propagating to the
right.
47
2.4. TEST PROBLEMS
TEST Nr.
— 1A —
— 1B —
— 2A —
— 2B —
— 3A —
— 3B —
— 4A —
— 4B —
— 4C —
— 4D —
— Brio Wu —
Side
R
L
R
L
R
L
R
L
R
L
R
L
R
L
R
L
R
L
R
L
R
L
ρ
1.00
1.00
1.00
0.10
1.08
1.000
1.00
0.100
1.00
0.100
0.10
1.000
1.00
0.200
0.40
1.000
0.65
1.000
1.00
0.300
1.00
0.125
V
[10.0, 0.0, 0.0]
[−10.0, 0.0, 0.0]
[0.0, 0.0, 0.0]
[0.0, 0.0, 0.0]
[1.2, 0.01, 0.5]
[0.0, 0.0, 0.0]
[0.0, 0.0, 0.0]
[0.0, 2.0, 1.0]
[50.0, 0.0, 0.0]
[0.0, 0.0, 0.0]
[−1.0, 0.0, 0.0]
[1.0, 0.0, 0.0]
[0.00, 0.0, 0.0]
[0.0, 0.0, 0.0]
[−0.669, 0.986, 0.0]
[0.0, 0.0, 0.0]
[0.667, −0.257, 0.0]
[0.4, −0.94, 0.0]
[0.0, 0.0, 0.0]
[0.0, 0.0, 0.0]
[0.0, 0.0, 0.0]
[0.0, 0.0, 0.0]
B
[5.0, 5.0, 0.0]/(4π)2
[5.0, 5.0, 0.0]/(4π)2
[3.0, 5.0, 0.0]/(4π)2
[3.0, 2.0, 0.0]/(4π)2
[2.0, 3.6, 2.0]/(4π)2
[2.0, 4.0, 2.0]/(4π)2
[3.0, 6.0, 0.0]/(4π)2
[3.0, 1.0, 0.0]/(4π)2
−[0.0, 1.0, 2.0]/(4π)2
[0.0, 1.0, 2.0]/(4π)2
[0.0, 1.0, 0.0]
[0.0, 1.0, 0.0]
[1.0, 1.0, 0.0]
[1.0, 0.0, 0.0]
[1.3, 0.0025293, 0.0]
[1.3, 1.0, 0.0]
[0.75, 0.55, 0.0]
[0.75, 0.00001, 0.0]
[7.0, 0.001, 0.0]
[7.0, 1.0, 0.0]
[0.75, 1.0, 0.0]
[0.75, −1.0, 0.0]
P
20.0
1.00
1.0
10.0
0.95
1.00
1.0
10.0
0.4
0.20
1.0
1.00
1.0
0.10
0.5247
1.00
0.50
0.75
1.0
0.20
1.0
0.10
Table 2.1: Summary table with the initial conditions of the left and right side of the shock tubes.
as well as some small but systematic deviations from the exact solution. In general, this
method works slightly better than the smoothing of the magnetic field, but the differences
are generally small.
One idea to reduce the unwanted side effects of such regularization schemes
(Price and Monaghan, 2005) is based on a modification of the artificial, magnetic
dissipation constant αB . Therefore every particle evolves its own numerical constant, so
that this value can decay where it is not needed and therefore the effects of the artificial
dissipation are suppressed. Fig. (2.6) shows the same test as before, but this time
where αB evolves for each particle, as shown in the lower right panel. Clearly, the values
are strongly reduced outside the regions associated with sharp features (e.g. shocks),
but the effect of smearing sharp features and the small offset of some states are not
significantly reduced. This is because in the region in which these side effects originate,
the dissipation is still working with its maximum numerical value. On the other hand,
due to the suppression of the artificial magnetic dissipation constant outside the shock
region, the regularization after the shock passes is nearly switched off. Therefore it does
not longer work as efficient as before in the post shock region, visible as increase in the
∇ · B error compare to the run with a constant, artificial magnetic dissipation.
2.4.3
Shock tube problems
As it can be seen in figures (2.4),(2.5) and (2.3), the side effects of smoothing features by
the different regularization methods depend on the details of the underlying structure of
the shock-tube test. Even more interesting, the states where one can see small deviations
48
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
Figure 2.5: As Fig. (2.2), but including artificial magnetic dissipation as a regularization scheme. Similar
to the smoothing of the magnetic field, significant reduction in the noise as well as a strong reduction of
the ∇ · B errors by a factor of ≈ 10 is obtained compared to the basic SPH-MHD implementation.
from the ideal solution are different for the different regularization methods. Therefore
we performed the full set of different shock-tube tests as presented in Ryu and Jones
(1995) to test the overall performance of the different implementations under different
circumstances. The four test families deal with different complexities of velocity and
magnetic field structures, leading to different kinds of waves propagating. A summary of
the results of these tests can be found in Fig. (2.7). Plotted are the total energy (left
panels), the velocity along the x-direction (middle panels) and the magnetic field along the
y-direction (right panels). The red lines reflect the ideal solution obtained with ATHENA,
the black lines with error bars mark the results from the SPH-MHD implementation using
the magnetic field smoothing every 30th main time step. Note that the error bars in most
cases are smaller than the line width. The initial setups for the shock-tube tests can be
found in table 2.1, which lists the state vector of the left and right states for the different
shock tube tests.
The first family of tests (1A/1B) has no structure in the tangential direction of the
propagating shocks in magnetic field and velocity, e.g. Bz = vz = 0. As we expect,
in the 1A test, the strong shock (large jump in vx ) leads to some visible noise in the
magnetic field component By , also translating into significant noise in the total energy.
The regularization method here suppresses the formation of the intermediate state in By
in the SPH-MHD implementation, as can be seen in figure 2.7(a). The second case, the
1B test, the weak shock is captured well. Again in some regions some smearing of sharp
features due to the regularization method is clearly visible.
2.4. TEST PROBLEMS
49
Figure 2.6: As Fig. (2.2), but including time dependent artificial magnetic dissipation as a regularization
scheme. No significance improvement is obtained. Note that here in the lower right panel the artificial
dissipation constant (αB ) is shown. The effect of suppressing the dissipation is clearly visible, and the
maximum value is only reached in peaks associated with the region of strong shocks. However the
improvement in the smearing of sharp features is not very significant.
The second class of shocks (2A/2B) involve three dimensional velocity structures, where
the plane of the magnetic field rotates. All features (e.g. fast/slow shocks, rotational
discontinuity and fast/slow rarefaction wave, for details see Ryu and Jones (1995)), are
well captured, see figure 2.7(c) and 2.7(d). Some of the features are clearly smoothed by
the regularization method.
The third class of tests (3A/3B) shows handling of magnetosonic structures. The first
has a pair of magnetosonic shocks with zero parallel field and the second are magneto
sonic rarefactions. Although there is slightly more noise present, all states are captured
extremely well, except the numerical feature left at the position dividing the two states
initially, see figure 2.7(e) and 2.7(f).
The fourth test family (4A/4B/4C/4D) deals with the so-called switch-on and switchoff structures. The tangential magnetic field turns on in the region behind switch-on
fast shocks and switch-on slow rarefaction. Conversely, in the switch-off slow shocks and
switch-off fast rarefaction the tangential magnetic field turns off. Again, all structures are
captured well with the exception of one feature in figure 2.7(h) as well and maybe 2.7(j)
too, where clearly the regularization leads to the washing out of a state. Otherwise the
regularization leads to smoothing of some structures similar to the tests presented before.
In general, figure 2.7 demonstrates that all these different situations have to be included
when trying to measure the performance and quality of different implementations of
regularization methods. The utility of the amount of tests comes when we make a
50
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
(a) Test 1A
(b) Test 1B
(c) Test 2A
(d) Test 2B
(e) Test 3A
(f) Test 3B
(g) Test 4A
(h) Test 4B
(i) Test 4C
(j) Test 4D
Figure 2.7: Representative plots of the additional 10 shock-tube tests from Ryu and Jones (1995). Shown
for each test are the total energy (left panels), the velocity along the x-direction (middle panels) and the
magnetic field along the y-direction (right panels).
parameter study of our implementations described in section 2.3. When doing that one
can see how the implementations stress some features with some tests and are almost
indifferent to others, allowing to find the optimal numerical parameters as shown in
appendix B.
2.4.4
Multi dimensional Tests - Planar Tests
Besides the one dimensional shock tube test described in the previous section, two
dimensional (e.g. planar) test problems are a good test-bed to check code performance.
Such higher dimensional tests include additional interaction between the evolving
2.4. TEST PROBLEMS
51
components with non-trivial solution. These can be quite complex (with several classes of
waves propagating in several directions) such as the Orszang-Tang Vortex or simple (but
with strong MHD discontinuities) such as Strong Blast or Fast Rotor.
Fast Rotor
This test problem was introduced by Balsara and Spicer (1999), to study star formation
scenarios, in particular the strong torsional Alfvén waves and is also commonly used to
validate MHD implementations (for example see Toth, 2000; Londrillo and Del Zanna,
2000; Price and Monaghan, 2005; Børve et al., 2006). The test consists of a fast rotating
dense disk embedded in a low density, static and uniform media, with a initial constant
magnetic field along the x-direction (e.g. Bx = 2.5π −1/2 ). In the initial conditions, the
disk with radius r = 0.1, density ρ = 10 and pressure P = 1 is spinning with an angular
velocity ω = 20. It is embedded in a uniform background with ρ = P = 1. Again we
setup the initial conditions by distributing the particles on a glass like distribution in 3D,
using 700 × 700 × 5 particles and periodic boundaries in all directions for the background
particles. The disk is created by removing all particles which fall inside the radius of
the disk and replacing this space with a denser representation of particles of the same
mass (mimicking realistic numerical conditions used in cosmological simulations). As an
ideal solution to compare with, we again used the result of a simple, two dimensional
ATHENA run with 400 × 400 cells. The solutions for the different implementations and the
comparison with ATHENA are shown in Fig. (2.8). The shape, positions and amplitudes
correspond quite well, although the GADGET runs appears slightly more smoothed, which
is more evident in the cases of dissipation (panel 2.8(c)) and B-field smoothing (panel
2.8(e)).
Fig. (2.9) presents a more quantitative comparison. Shown is a diagonal cut through
the Fast Rotor at t = 0.1 showing the density. The different lines show the results obtained
with ATHENA (black line) and for the implementations in GADGET, in blue the standard
and in red the artificial dissipation. Very small errors are reflected by the absence of
wiggles of those values. Note that our case the simulation is done fully in 3D, whereas
the ATHENA solution is two dimensional. The results show remarkable agreement between
the two simulations and also compare well with results quoted in the literature (e.g.
Londrillo and Del Zanna, 2000).
Note that although we perform our calculations in three dimensions and without a
regularization scheme, the implementation produce a result, which has the same quality
as other schemes in two dimensions with regularization (e.g. Price and Monaghan, 2005;
Børve et al., 2006).
Strong Blast
The Strong Blast test consists of the explosion of a circular hot gas in a static
magnetized medium and is also regularly used for MHD code validation (see for example
Londrillo and Del Zanna, 2000; Balsara and Spicer, 1999). The initial conditions consist
of a constant density ρ = 1 where a hot disk of radius r0 = 0.125 is embedded, which is
hundred times over-pressured, e.g. the pressure in the disk is set to Pd = 100 whereas
the pressure outside the disk is set to P0 = 1. Additionally there is initially an overall
homogeneous magnetic field in the x-direction, with a strength of Bx = 10. The system
52
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
(a) Fast Rotor - ATHENA
(b) Fast Rotor - Standard
(c) Fast Rotor - Dissipation
(d) Fast Rotor - Dedner
(e) Fast Rotor - B-field Smoothing
Figure 2.8: The magnetic field strength in the Fast Rotor test at t = 0.1. The ATHENA solution of
the test problem is shown in the panel 2.8(a) whereas the other panel show the result obtained with
the different schemes in GADGET. All the main features are reproduced in the GADGET runs. The
shape, positions and amplitudes correspond quite well, although the GADGET run appears slightly more
smoothed, which is more evident in the cases of dissipation (panel 2.8(c)) and B-field smoothing (panel
2.8(e)), see also Fig. (2.9).
2.4. TEST PROBLEMS
53
Figure 2.9: Cut through the diagonal (x = y) of Fast Rotor at t = 0.1 showing the density. In black
result obtained with ATHENA. The blue line show the GADGET solution using the standard SPH-MHD
implementation. The red line show the solution using the artificial dissipation scheme. In general, the
standard SPH-MHD result shows an excellent agreement in all the features, however the dissipation run
shown a visible over-smoothing at the outer edges and lower values at the peaks.
is evolved until time t = 0.02 and an outgoing shock wave is visible which, due to the
presence of the magnetic field, is no longer circular but propagates preferentially along
the field lines. Fig. (2.10) shows the density at the final time, comparing the ATHENA
results with the ones from the different schemes in GADGET. Although the setup is a
strong blast wave, there is no visible difference between the SPH-MHD implementations
and the ATHENA results. This is quantitatively confirmed in Fig. (2.11) which shows
a horizontal cut (at y = 0.5) of the density through the Strong Blast test, comparing
the ATHENA (black line) with the GADGET results. In blue we show the standard SPHMHD, in red the artificial dissipation and in green the Dedner scheme. Besides very small
variations there is no significant difference between the two results and all features are
well reproduced by the SPH-MHD implementation. Note that the errors of the GADGET
results again are in all cases smaller than the shown line width.
Orszang-Tang Vortex
This planar test problem, introduced by Orszag and Tang (1979), is well known to study
the interaction between several classes of shock waves (at different velocities) and the
transition to MHD turbulence. Also, this test is commonly used to validate MHD
implementations (for example see Dai and Woodward, 1994; Picone and Dahlburg, 1991;
Londrillo and Del Zanna, 2000; Price and Monaghan, 2005; Børve et al., 2006). The
initial conditions for an ideal gas with γ = 5/3 are constructed within a unit-length
domain (e.g. x = [0, 1], y = [0, 1]) with periodic boundary conditions. The velocity field
is defined by vx = − sin(2πy) and vy = sin(2πx). The initial magnetic field is set to
be Bx = B0 vx and By = B0 sin(4πx). The initial density is ρ = γP and the pressure
is set to P = γB02 . This system is evolved until t = 0.5. Fig. (2.12) shows the final
54
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
(a) Strong Blast - ATHENA
(b) Strong Blast - Standard
(c) Strong Blast - Dissipation
(d) Strong Blast - Dedner
(e) Strong Blast - B-field smoothing
Figure 2.10: Shown is the resulting density distribution for the Strong Blast test at t = 0.02. The panel
2.10(a) shows the results obtained with ATHENA, the results obtained with the different implementations
in GADGET are shown in the rest of the plots. There is no visible difference between the different schemes,
however we note slight smooth in sharp features compared with the ATHENA solution (see also Fig. 2.11).
2.4. TEST PROBLEMS
55
Figure 2.11: Horizontal cut through the Strong Blast test (x = [0.0; 1.0], y = 0.5) showing the density.
The black line is from the ATHENA simulation, the blue line reflects the standard, the red one the artificial
dissipation and in green we show the Dedner implementation. The overall behavior is excellent, with only
very small differences between the solutions in the sharp edges.
result for the magnetic pressure for the ATHENA run, in panel 2.12(a), and the different
implementations in GADGET, including the Euler case, in panel 2.12(f). Visually the results
are quite comparable, however the GADGET results look slightly more smeared, which is
the imprint of the underlying SPH method. This impression is confirmed in figure Fig.
(2.13), which shows a cut through the simulations, made to be comparable with other cuts
done in the literature (Børve et al., 2006). In this case, the black line shows the ATHENA
result, in blue in the standard, in red the artificial dissipation, in green the Dedner and
in pink the Euler potential representation. In general there is a reasonable agreement,
however all the SPH-MHD results clearly show a smoothing of some features. However,
the Dedner and standard implementations, tends to match better some regions that the
dissipative schemes over-smooth (e.g. region near x ∼ 1 in Fig. 2.13), even better that
the Euler scheme. A proper comparison is difficult, mainly because this test includes
the propagation of several types of magneto sonic waves, this implies that if one miss
the correct velocity (i.e. by some dissipative effect) of a particular wave, the result will
diverge between implementations. This is shown even by the Eulerian schemes, when we
compare the solutions from the AMR code RAMSES and ATHENA in different resolutions
and there is no complete convergence with the solution.
The adaptive nature of the SPH-MHD implementation allows the central density peak
to be resolved whereas in ATHENA is can only be resolved by increasing the number of
grid cells. Never-the-less the SPH-MHD implementation seems to converge slower when
increasing the resolution (see Appendix C).
Note that this test, because of their characteristic periodicity, is the one that we use the
check the Euler potentials formalism. We found a perfect agreement with other authors
(i.e. Rosswog and Price, 2007). Our major interest in this scheme is the possibility to
measure the errors that arise from the interpolation itself or the coarseness of our initial
56
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
(a) Orszang-Tang Vortex- ATHENA
(b) Orszang-Tang Vortex- Standard
(c) Orszang-Tang Vortex- Dissipation
(d) Orszang-Tang Vortex- Dedner
(e) Orszang-Tang Vortex- B-field smoothing
(f) Orszang-Tang Vortex- Euler scheme
Figure 2.12: The magnetic field strength of the Orszang-Tang Vortex at t = 0.5. The panel 2.12(a) shows
the ATHENA solution and other panels show the different implementation in GADGET. In the panel
2.12(f) we shown the solution from the Euler scheme. Some of the sharp features are smoothed in all the
SPH-MHD implementations but overall the results compares very well (see also Fig. 2.13). Interestingly
some reconnection features are better captured in the SPH cases (see appendix C).
2.5. DISCUSSION
57
Figure 2.13: A t = 0.5 cut through the pressure in the Orszang-Tang Vortex at y = 0.4277.As before, the
black line reflect the results obtained with ATHENA, in blue the standard, in red the artificial dissipation,
in green the Dedner and in pink the Euler potentials implementation. The cut is chosen to compare with
results from the literature, e.g. Børve et al. (2006). Note the variations in the solution between schemes,
mainly due to the difference in the dissipative characteristics. However, the overall solutions are in
reasonable agreement.
conditions. Having this information we can distinguish better which issues comes from
the SPH formulation itself and which ones come from the MHD part.
In Fig. (2.14) we plot the ∇ · B errors as defined by Eq. (2.57). First, the errors from
the Dedner implementation, in the middle the results from the Euler formulation, and
finally an example from the dissipative method. Note, that as by construction, the Euler
scheme should be divergenceless, this shows that with the Dedner implementation we are
already at the numerical limit. Most of the errors will come from the stochasticity and
sampling noise. The only way to get rid of this is to use an extension to the “ideal” MHD
as are the magnetic field smoothing or artificial dissipation.
2.5
Discussion
We presented the implementation of MHD in the cosmological, SPH code GADGET.
We performed various test problems and discussed several instability correction and
regularization schemes. We studied demonstrated the role of the ∇ · B = 0 constrain
and schemes with resolution and comparing them to Eulerian solutions.
Our main findings are:
• The combination of many improvements in the SPH implementation, like the
correction terms for the variable smoothing length (Springel and Hernquist, 2002)
as well as the usage of the signal velocity in the artificial viscosity (Monaghan,
1997) together with its generalization to the MHD case (Price and Monaghan, 2004a)
improve the handling of magnetic fields in SPH significantly.
58
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
(a) Dedner cleaning scheme
(b) Euler potencials
(c) Artifitial dissipation
Figure 2.14: ∇ · B errors as defined by Eq. (2.57). First, the errors from the Dedner implementation, in
the middle the results from the Euler formulation, and finally an example from the dissipative method.
Note, that as by construction, the Euler scheme should be divergenceless, this shows that with the Dedner
implementation we are already at the numerical limit. Most of the errors will come from the stochasticity
and sampling noise. The only way to get rid of this is to use an extension to the “ideal” MHD as are the
magnetic field smoothing or artificial dissipation.
• Correcting the instability by explicitly subtracting the contribution of a numerical
non-zero divergence of the magnetic field to the Lorenz force from the Maxwell
tensor as suggested by Børve et al. (2001) seems to perform well. To avoid spurious
instabilities due to sampling problems, particularly in the front shocks, a maximum
threshold in the correction improves the performance and quality. Specifically in
three dimensional setups where it seems to work much better than other suggestions
in the literature. This feature turn out to be fundamental to progress in works
like Bürzle et al. (2010) and Kotarba et al. (2010b), where previously the numerical
instabilities dominate and such problems could not be investigated.
• The SPH-MHD implementation performs very well on our multidimensional shock
tube tests as well as on commonly used planar test problems. We performed all
tests in a fully three-dimensional setup and find excellent agreement of the results
obtained with the SPH-MHD implementation compared to the results obtained with
ATHENA in one or two dimensions only.
2.5. DISCUSSION
59
Figure 2.15: Comparison plot between all the implementation in tests. Clearly the dissipative schemes
have lower ∇ · B errors, however Dedner cleaning schemes outstands, as the lowers the ∇ · B keeping the
diffusion at Hsml scale. Also is shown when the tests are improved compared with correct solutions. We
can see the trend when the dissipative effects make the final result differ by smoothing features.
• The stability of our implementation corresponds to the expected values for the tests,
without bias and low inter particle scatter.
• We successfully implemented for first time a multidimensional divergence cleaning
method in SPH. We showed that Dedner scheme does not affect strongly the shape
of shocks, but reduces the ∇ · B errors. This is important in physical meaningful
situations. Also the evolution of this correction is completely local and only affected
by the SPH interpolants.
• Regularization schemes help to further suppresses noise and ∇ · B errors in the
test simulations, however one has to carefully select the numerical parameters to
avoid too strong smoothing of sharp features. Performing a full set of shock tube
tests allows one to tune the numerical schemes and to determine optimal values in
different conditions (i.e. rarefaction, magnetic-sonic regimes, etc.).
• The magnetic field smoothing succeeds in terms of stability, however in general it
over-smooths features. Also is parametrised by global time-steps of all the particles,
therefore it will not adapt to dynamical changes.
• The artificial dissipation implementation also increases the stability and behaves
similar to the magnetic field. In this case, the over-smoothing of features comes from
the changes in the magnetic field morphology itself, as expected from the scheme.
60
CHAPTER 2. MHD - SMOOTHED PARTICLE HYDRODYNAMICS
There is no dependence in global time-steps, and it will only depend on the local
magnetic field gradients.
• We implemented an Euler Potential approach to evolve the magnetic field without
the need of the induction equation. This has the benefits of fulfill the constraint
∇ · B = 0 by construction, up to the numerical level. We demonstrated that our
schemes reached these errors levels when measure them in the tests (see Fig. 2.14).
The most important finding of our studies is the comparison to a divergenceless-byconstruction scheme (i.e. Euler scheme). It shows tat we already reached the ∇ · B error
levels inherent from the SPH scheme itself, and any further cleaning or regularization will
require to numerically dissipate the magnetic field. However this does not meant that we
are error-free, thus still some ∇ · B related instabilities can arise that we need to avoid.
Additionally this shows the great improvements done in the underlying schemes since the
first MHD-SPH implementations, where this instabilities dominate the SPH behavior.
Further more our code has different features with each regularization or cleaning scheme
used. The success of ∇ · B cleaning bring our simulations closer to the “ideal” MHD state.
These flavor concentrates all the efforts of cleaning locally, by measuring the “local” error
and subtracting it in the Induction equation.
In Fig. (2.15) we show the comparison between all the implementations in different
tests. Clearly the dissipative schemes have lower ∇ · B errors, however Dedner cleaning
schemes out stands, lowering the ∇ · B keeping the diffusion at Hsml scale, and in most
of the cases improving the results. In contrast there some tests show that to lower ∇ · B
errors the scheme over smooth features, therefore enhancing differences with the correct
solution.
Other regularization schemes (e.g. as smoothing and artificial dissipation) depend
strongly on the gradients of the magnetic field. Both cases can be seen as mimics of
Ohmic dissipation. Therefore, leading us away from “ideal” MHD, fact which can be
an issue for further development of additional physics. Furthermore, a non dissipative
scheme will allow more complex morphological structures as expected in complex regions
as galaxy cluster cores. From the other side, additional physics like Ohmic dissipation are
expected to be important (Bonafede et al., 2010), being an interesting case for the physics
to be added, and making important to know how our numerical dissipation works.
Part III
Applications
61
Chapter 3
Galaxy cluster and magnetic fields
A subtle thought that is in error may yet give rise to fruitful inquiry that can establish truths of great value.
Isaac Asimov (1920-1991)
We already discussed some properties and open issues of galaxy clusters in the first
chapter especially in sections 1.1 and 1.3. As we mentioned galaxy clusters are complex
systems, with a long history from their hierarchical build up within the large-scale
structure of the Universe. Therefore, to study their formation it is necessary to follow
a large volume of the Universe. We also mentioned that simulations described fairly
well the self-similar scenario, but we must go down to to relatively small scales, thus
spanning 5 to 6 orders of magnitude in size, to describe them properly. The complexity
of the cluster atmosphere reflects the in-fall of thousands of smaller objects and their
subsequent destruction or survival within the cluster potential. Being the source of
shocks and turbulence, these processes directly act on the magnetic field causing redistribution and amplification. Therefore realistic modelling of these processes critically
depends on the ability of the simulation to resolve and follow correctly this dynamics in
galaxy clusters.
To perform our studies we use a re-simulation of a Lagrangian region selected from
a lower resolution dark matter only cosmological box. This parent simulation has a
box–size of 684 Mpc, and assumed a flat ΛCDM cosmology with Ωm = 0.3 for the
matter density parameter, H0 = 70 for the Hubble constant, fbar = 0.13 for the baryon
fraction and σ8 = 0.9 for the normalization of the power spectrum. The cluster has
a final mass of 1.5 × 1014 M⊙ and was re-simulated at 3 different particle masses for
the high resolution region. Using the “Zoomed Initial Conditions” (ZIC) technique
(Katz and White, 1993b; Tormen et al., 1997), these regions were re-simulated with
higher mass and force resolution by populating their Lagrangian volumes with a larger
number of particles, while appropriately adding additional high–frequency modes drawn
from the same power spectrum. To optimize the setup of the initial conditions, the high
resolution region was sampled with a 163 grid, where only sub-cells are re-sampled at high
resolution to allow for quasi arbitrary shapes of the high resolution region. The exact
shape of each high–resolution regions was iterated by repeatedly running dark-matter
only simulations, until the targeted objects are clean of any lower–resolution boundary
particle out to 3 ∼ 5 virial radii. The initial particle distributions, before adding any
Zeldovich displacement, were taken from a relaxed glass configuration (White, 1996). The
63
64
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
three resolutions used correspond to a mass of the dark matter particles of 1.6 × 109 M⊙ ,
2.5 × 108 M⊙ and 1.6 × 108 M⊙ for the 1x, 6x and 10x simulation. The gravitational
softening corresponds to 7, 3.9 and 3.2 Kpc respectively. For simplicity we assumed an
initially homogeneous magnetic field of 10−11 G co-moving as also used in previous work
(Dolag et al., 1999, 2002). Furthermore we tested our different implementations. We
applied the regularization by smoothing the magnetic field in the same way than we did
in previous work (Dolag et al., 2004a, 2005a). We also checked the effects of regularization
by artificial dissipation varying values of αB and in the case of the cleaning scheme, we
test a small set of parameters. Always in correspondence with the results shown in the
appendix B.
Fig. (3.1) shows a zoom-in from the full cosmological box down to the cluster. The
structures in the outer parts get less pronounced due to the decrease in resolution, which
is designed to capture only the very largest scales of the simulation volume. Each panel
shows (in clockwise order) a zoom-in by a factor of ten. Finally the elongated box in the
lower left panel marks the size of the observational frame shown on the left. The dynamical
range of the simulation spawns more than five orders of magnitude in spatial dimension.
The size of the underlying box is 6 and 5 times larger than the AMR simulations presented
in Dubois and Teyssier (2008) and Brüggen et al. (2005), respectively. Still the resolution
of the underlying dark matter distribution is, respectively, 2 and 5 times better than these
AMR simulations and the cluster is resolved with more than one million dark matter
particles within the virial radius at the 10x resolution.
We mainly focus on the aspects of the magnetic fields itself. Given the complexity
inherent from MHD simulations, our the studies shown here are non-radiative to
understand better how the different implementations them selfs capture the magnetic
field features without any confusion possible from additional physics.
3.1
Galaxy cluster slices
We study the inner part of the object making slices trough the center, giving us broad
information in the magnetic field distribution. Fig. (3.2) shows a density slice for two
different resolutions 1x and 10x, easily is seen that smaller structures are resolved in
the higher resolution slightly changing the dynamics of the system. However, we don’t
observe important difference between the different schemes. In figure Fig. (3.3) we
see characteristic shapes of the magnetic field distribution inside of the galaxy cluster,
compared with Fig. (3.2) one can observe the expected correlation between the density
and the magnetic field. In high β plasmas as the ones we are simulating, the magnetic field
lines are “frozen” to it, being compressed and advected all together. Smaller structures
are resolved in the higher resolution slightly changing the dynamics of the system.
The diffusive schemes, panels c and d in Fig. (3.3), show less growth of the magnetic
field towards the center due the transport of the field to the cluster surroundings. The
Standard, Euler and Dedner implementations are able to keep the magnetic field in
the central part. We study two cases for the Euler potential runs, one following the
Lorentz force, while other run is completely hydrodynamical but we build up the final
magnetic field from the Euler potentials. Remember that the Euler scheme is valid only
at approximately the firsts winding of the field, basically it breaks when the reconnection
should occur, and the scheme is unable to handle it. Therefore the panel e from Fig.
65
3.1. GALAXY CLUSTER SLICES
DEC
Observation
684 Mpc
68.4 Mpc
684 kpc
6.84 Mpc
Simulation
3C449
Feretti et al. 1999
RA
RM
(counts)
(RAD/M/M)
Figure 3.1: Zoom into the cluster simulated within the cosmological box. Clockwise, each panel displays
a factor 10 increase in imaging magnification, starting from the full box (684 Mpc) down to the cluster
center (680 Kpc). On the very large scale, the density of the dark matter particles are shown, whereas
in the high resolution region the temperature of the gas is rendered to emphasize the presence and
dynamics of the substructure. The last zoom extracts a region of the same size of an observed radio jet
(3C449) with measured rotation measure (Feretti et al., 1999b). Both, the simulated and the observed
map are displayed using a linear color-scale based on the minimum and maximum values in the maps.
The synthetic RM map is clipped to the shape of the observations.
(3.3) represents the distribution from a random magnetic field but with the dynamic
of the system modified by an additional magnetic pressure. The final panel f shows the
magnetic field is build from the advection of the potentials taking only the hydrodynamical
forces into account. The result should be considered as magnetic field from a complete
hydrodynamical turbulence case. In Fig. (3.4) we see the same cut but showing the
∇ · B errors for different regularization schemes. The first thing to notice is that we
are still in a low percentage error, compared with the magnetic field strength. In the
standard case and Dedner case the values are higher than in the other runs, however,
the errors are smaller with the Dedner scheme. In the high density regions are the
lower values, with the exception of the Euler potential cases where the errors are almost
constant trough the slice. This is explained, in the Standard case, because the numerical
∇ · B errors in the center are temporal due to shock fronts passing thought the ICM
66
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
(a) Standard implementation 1x
(b) Standard implementation 10x
Figure 3.2: Density slice trough the center of the galaxy cluster at redshift z = 0 for two resolutions.
Smaller structures are resolved at high resolution. This can slightly modify the dynamics of the system.
However, the overall structure is similar and the radial profiles generally converge. When comparing the
schemes, we don’t found differences, even in density profiles.
(in the same way as shown in the test shock tubes). Meanwhile in the outer regions
the dynamical times are longer and additionally the baryons properties are mostly driven
by the kinetic energy (see appendix D). However, this facts does not apply to the Euler
potential simulations because of the nature on how we build the magnetic field, from the
final configuration of the adverted potentials. Additionally, is interesting to note that even
if the representations of those fields have different origins ,one completely hydrodynamical
and other magnetohydrodynamical (into some extent), there are no effects in in the ∇ · B
distribution, but there is change in the core magnetic field strength.
Besides the density comparison, we study the magnetic field in different resolutions
runs. Slices comparing two resolutions and the magnetic properties are shown in Fig.
(3.5). The amplification of the magnetic field is stronger as we increase the resolution,
as a consequence of resolving better the turbulence at smaller scales. However, we don’t
see an enhancement of the values of the ∇ · B errors, getting even a lower average.
This fact demonstrates that the growth of the magnetic field in SPH schemes is not
dominated by ∇ · B terms that generate instabilities and unphysical dynamo effects
(Brackbill and Barnes, 1980). In the inner region of the cluster, the turbulent motions
and shocks from the incoming halos that steer the ICM can lead to complex configurations
of magnetic fields, being a source of ∇ · B errors to appear temporally. These kind of
errors are harmless for the overall evolution of the code because, being just a temporary
stage located in the front shocks (as seen also in the tests), and do not propagate, and any
further spurious effect is controlled by the improvements in our implementations (section
2). As is seen in the shock tube and test problems, in the front shocks itself there is a
numerical ∇ · B because a numerical incapacity in properly resolve that particular region
between two divergenceless stages. However as soon as the front shock went trough that
region the field keeps evolving to the correct solution of the shock problem. Different is
the case of the outermost regions. There the magnetic field energy is quite low compared
with thermal or kinetic 1 ,making the magnetic field be dominated by the kinematics of
1 We
are always in a high βplasma = 8π · u/B 2 regime, in this cases we refer higher than 109 , see apendix D to see an
67
3.1. GALAXY CLUSTER SLICES
(a) Standard implementation
(b) Dedner cleaning scheme
(c) Art. Dissipation
(d) B-Smoothing
(e) Euler Potentials
(f) Euler Pot. without Lorentz Force
Figure 3.3: Magnetic field energy density slice trough the center of the galaxy cluster at redshift 0 for
the 1x resolution runs. Different panels show the magnetic regularization/cleaning schemes. One can
see several differences in magnetic field distribution. Also there is a decrease of the center values for the
schemes more diffusive. In the case of Euler Potentials is due the lack of reconnection. The Standard and
Dedner, slices (a) and (b), show the large scatter in low density regions that will be the main source of
the ∇ · B errors. Finally we show two runs of the Euler potentials implementations with and without the
back reaction given by the Lorentz force. The run without force represents a magnetic field distribution
completely build from the Hydrodynamical turbulence, while the run with force build a magnetic field
from a almost randomly distributed magnetic field.
68
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
(a) Standard implementation
(b) Dedner cleaning scheme
(c) Art. Dissipation
(d) B-Smoothing
(e) Euler Potentials
(f) Euler Pot. without Lorentz Force
Figure 3.4: ∇ · B errors cut trough the center of the galaxy cluster at redshift 0 and 1x resolution. We
present the same resolution with different regularization schemes. One can see several substructures in
the ∇ · B errors, that anti-correlates with density, with exception of the Euler potential cases. The lower
∇ · B errors are obtained with the dissipative schemes, however we are still in small faction of the total
magnetic field strength.
69
3.2. MAGNETIC FIELD PROFILES
(a) Magnetic Energy (1x)
(b) Magnetic Energy (10x)
(c) ∇ · B errors (1x)
(d) ∇ · B errors (10x)
Figure 3.5: In the first row we show the magnetic energy slices trough the cluster center at redshift
z = 0 for two different resolutions. We are able resolve smaller scales and follow the turbulent motions
accordingly, showing an enhancement of the magnetic field. In the bottom row are shown ∇ · B errors
slices. Note that increasing the resolution we decrease the errors in the inner region, demonstrating the
fact that the inner errors are consequence of the temporary unresolved shock fronts, as found in shock
tube tests.
the baryons. Additionally the non-diffusive characteristic of SPH, makes the ∇ · B errors
difficult to disappear (see Appendix D). Finally, density and magnetic field profiles have
different slopes in the outskirts of the cluster or in the halos in the simulation, being the
magnetic field profile steeper. This facts in the outskirts of the cluster, can be a source
of instabilities that can be solved implementing high order Kernels or taking special care
in the time step integration for those particles.
3.2
Magnetic field profiles
In Fig. (3.6), the radial magnetic field profiles (left panel) as well as the numerical
∇ · B errors to (right panel) are shown. For galaxy clusters, only the shape of the
predicted magnetic profiles converge (with the exception of the central part of clusters)
distribution of βplasma .
70
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
with resolution and are in good agreement with previous studies (Dubois and Teyssier,
2008; Brüggen et al., 2005), and in correspondence with the self-similar models. The
shape of the magnetic field is basically unchanged when including the Dedner cleaning
scheme, however, the numerical ∇ · B error drops almost one order of magnitude in the
center (see right panel in Fig. 3.6). These values are at the same level of the Euler case in
the center, therefore we reach the numerical threshold given the resolution. In addition,
we show the Euler implementation without any back reaction from the Lorentz force
dynamics. Therefore, the final stage of the magnetic field comes from the construction of
the advected quantities that only follow hydrodynamical forces. In this configuration, as
we lack of the magnetic pressure, the magnetic field continues rising towards the center.
Note that the Dedner method is characterized to transport and dump the numerical errors
outside our region of interest (the halo). In the case of a periodic box (as it happened
in the Vortex test) the most important parameter in that scheme is π, which control the
dumping term, however in the case of open boundaries and collapsing regions we note that
σ, which control the propagation, should be high enough otherwise we cumulate errors
in the center of the cluster. Previous works in star formation works, using the Dedner
method, this velocity propagation was fix and not studied, reporting no success for this
implementation (Price and Monaghan, 2005). The dissipative regularization schemes,
namely Art. Dissipation and B-smoothing, does get lower errors but at the cost of
suppressing the turbulent growth of the field.
Comparing the panels in Fig. (3.6), we demonstrate that the amplification of the
magnetic field in is not significantly influenced by the non-zero numerical ∇ · B. The
central ∇ · B errors for the dissipative schemes and Dedner are similar (even for the case
of hydrodynamical case of Euler) but the central magnetic field differs in several orders
of magnitude.
As already noted in earlier work (Dolag et al., 2002), in Fig. (3.8) shows the
dependence of the amplification of magnetic fields with resolution. In addition, although
the absolute value of the amplification is not converged yet with resolution, the shape of
the predicted magnetic field profile appears to be converged. Note that this convergence,
as usual for all hydro-dynamical quantities, is only reached at radii significant larger than
the size of the gravitational softening, that in our cases is ∼ 10 Kpc, ∼ 7 Kpc and ∼ 6 Kpc
for the 1x, 6x and 10x respectively.
The situation changes when using artificial magnetic dissipation, as shown in Fig. (3.7).
In the left panel we show the magnetic field profiles for three values of αB compared and
two frequency values for the magnetic field smoothing. Clearly a normal value (αB ∼ 0.5)
for artificial magnetic dissipation leads to a large dissipation of magnetic field over the
simulation time (e.g. close to the Hubble time). The right panels show the profiles
artificially normalized at large radius (∼ 1Mpc). Clearly the self similarity of the profiles
is lost. Therefore it appears that the use of artificially dissipation as a regularization
scheme is not a good choice for cosmological simulations. Additionally it points out that
true physical dissipation might play an important role in determining the shape of the
magnetic field profile in galaxy clusters. Here transport processes, the role of cosmic rays,
turbulence (specially at scales below the resolution) as well as reconnection of magnetic
field lines are not well understood.
Within the ICM there the micro-physics mentioned are far outside the resolution scales
which can be ever reached by cosmological simulations, future work will have to include
them as sub-grid models. This means, additional physic models to be followed inside
3.3. SYNTHETIC RM MEASUREMENTS
71
Figure 3.6: In the left magnetic field profiles obtained for the galaxy cluster using different regularization
methods. Additionally, results from Eulerian codes are shown. In the right panel the profile of the
∇ · B errors are shown. Note how the results improves with the methods towards the center, reaching
values below one. The Euler potentials case without the Lorentz force mimics the case of a random field
dominated by the turbulence hydrodynamic. Therefore the ∇ · B errors are almost constant inside the
cluster and the profile keeps growing towards the center.
the Hsml kernel, and motivated and corroborated by small scale numerical experiments.
The existing implementations of radiative cooling, star formation and AGN feedback do
not take into account yet the effects of magnetic fields. Magnetic fields can act as an
additional reservoir of energy, therefore a self-consistent implementation is needed. In
the case of radiative cooling, this addition might not only give a more complete physical
scenario but also contribute to prevent the excessive clumping of the baryons that needs
to be re-heated by some other mechanism (e.g. Supernova feedback).
The results obtained with dissipative regularization schemes in cosmological
simulations indicate that physical dissipation could play a crucial role in determining
the exact shape of the predicted, magnetic field profiles in galaxy clusters. Future work,
especially when including more physical processes at work in galaxy cluster – as can be
done easily with our SPH-MHD implementation – might reveal an interesting interplay
between the dynamics of the cluster atmosphere and amplification of magnetic fields.
Thus having the potential to shed light on many, currently unknown aspects of cluster
magnetic fields, their structure and their evolution.
3.3
Synthetic RM Measurements
For comparison with observations we produced a synthetic Faraday rotation map from
the simulation, additionally this studies give an indication of the structures resolved by
our simulations. The simulations were not specially designed to compare with an object
in particular, therefore in these comparisons we focus on the main features of the schemes
contrasted with a real astrophysical system. As we shown the schemes have different
magnetic field growth rates, but the baryon distribution is quite similar. Therefore we
focus on the spatial distribution and structures obtained. As in observations, a Faraday
rotation map give us spatial information in the line of sight distribution of magnetic fields
and constrains on the morphological properties. As we discussed early, in Section 1.3,
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CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
Figure 3.7: The magnetic field profiles obtained for the galaxy cluster using different Art. Dissipation
constants and Smoothing the magnetic field at different frequencies. The right panel shows the normalized
profiles at ∼ 0.8Mpc. Therefore the changes in the shape of the profile as stronger is the diffusivity is
shown.
Figure 3.8: The magnetic field profiles for a resolution study of our cluster. The profiles are normalized
to the value of the (1x) simulation at ∼ 0.8Mpc. In dashed lines we see the limit resolution given by the
Hsml. We can observe how increases the central values of the magnetic field.
the inner regions of the galaxy cluster environment presents a complex mixture of physics
that imprints all the formation history of the observed object. In our case we compare
with simulations simple physics and magnetic fields, that in any case were meant to
match an specific object. However, the comparison are needed as a starting point to
validate our schemes, at least to broadly reproduce basic observation features. In section
1.2 we presented Faraday rotation measures, basically this observation that depends on
the electron density and the strength of the magnetic field in the line of sight. It is
an integrated quantity, therefore there is the cumulative effect during the path to the
observer. Given the information (e.g. density, temperature, magnetic field) that we
have from our numerical simulations we are able to build these line of sight integration.
However only in the volume that our simulation occupies, therefore we neglect the effects
from the foreground. In our case this hypothesis is not so bad, given the distance of the
sources, the angular diameter is small and therefore the expected contribution from the
close foreground sources will be constant or at least smooth. The fact that we don’t know
3.3. SYNTHETIC RM MEASUREMENTS
73
Figure 3.9: Synthetic RM maps for the single resolution and several regularization schemes. Note how
the dissipative schemes diminish the amount of rehearsals and the extension. All the plots have an scale
of 0.8 Kpc/pix, so we cover the central part of the cluster.
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CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
Figure 3.10: Synthetic RM maps for different resolutions with Dedner regularization scheme. Note that
as we increase the resolution we resolve more rehearsals in the field. Therefore we increase the amount
of structures, but the morphology seems to be quite similar. All the plots have an scale of 0.8 Kpc/pix,
so we cover the central and the outer part of the cluster.
exactly the foreground is not an issue, as we don’t study the absolute values but their
distribution inside the Faraday screen. Different is the case if we consider larger volumes
as we do in chapter 4.
To make our synthetic maps from our simulations we use “Smac” (Dolag et al., 2005b),
that basically reads our simulation output, project it to a given distance to match the
observation size of the object, and integrates the magnetic field on the line sight, weighed
by the electron density. This integration takes into account all the cosmological factors
and effects due to the distances from the object, giving us as a result a mock observation.
Similar to plots in previous sections, in Fig. (3.9) we compare the effects of the
regularization schemes at the same resolution. Note that the scale in the figures is
0.8 Kpc/pix, giving a total size for the image ∼ 0.8Mpc. This is half of the size of the slices
shown in Fig. (3.3), properly centered on the cluster, and clearly showing the effects of the
different numerical schemes. We observe the reversals of the magnetic field as bubbles or
blobs in the figures. Note that we expect to have similar density distributions, therefore
the patterns shown here are produced by the magnetic fields. The rotation measurements
varies from hundreds rad m−2 in the Standard run to tens rad m−2 in the dissipative runs,
therefore the plots are adjusted to remark the reversal sizes. The dissipative schemes
diminish the amount of rehearsals and extend them for various Kpc, Consequently the
other schemes have smaller reversals, with the exception of the Euler potentials. This
case is particular, due the fact that tries to conserve the morphology of the field, therefore
suppresses small reversals Formally speaking the Euler potentials have null helicity 2 , this
is a topological property of the field and is null from the way we re-construct the field.
However, is an open issue if this property can evolve or is constant in time, recent studies
concludes that is not a conserved (Brandenburg et al., 2009). However, our schemes using
the induction equation do not suffer from this restriction.
In Fig. (3.10) we show the synthetic maps that come from the Dedner scheme applied at
three different resolutions. We note an increase in the number of reversals with resolution.
This is expected as we resolve more smaller structures and therefore the inverse turbulent
2 Defined
as
R
A · BdV
75
3.3. SYNTHETIC RM MEASUREMENTS
3c449 (Obs)
1x
6x
10x
Figure 3.11: Shown is a comparison of the predicted RM signal from the cluster simulation at various
resolutions compared to the observational signal. One can see that we still not reach the amount of small
structure seen in the observation but, when we increase the resolution it begin to appear. In all this cases
the projection scale is 0.17 Kpc/pix to be comparable with observations.
cascade, that we show enhances the magnetic field. Note that we reach coherence lengths
of few Kpc, showing the complexity in morphology that the magnetic field lines can
acquire.
To compare with observations of real objects, we need to build up RM maps, projecting
the galaxy cluster to the observed resolution available, and using a mask with the shape
of the Faraday screen. This help us to mimic any kind of bias that we can introduce in the
latter comparison procedure. Note that the shape of the images is given by distribution of
polarized emission of the observation. This can be quite irregular for several reasons: the
shape of the source (in general a jet-like structure), the noise of the observation and the
sensitivity of our instrument. The observation that we use is from Feretti et al. (1995)
of 3c449, a radio galaxy. The soft X-ray emission from this object is dominated by an
extended halo with a scale comparable to that of the radio source. Observations suggests
that the jets are strongly influenced by the external medium. The minimum pressure in
the radio lobe is considerably lower than estimates of the pressure in the external medium,
meaning that is not in equilibrium (Hardcastle et al., 1998). The radio galaxy is located
at z = 0.1711, giving us a resolution of 0.17 Kpc/pix in the image. The comparison
results from our procedure are shown in Fig. (3.11).
76
3.4
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
Structure Functions
To obtain a more quantitative comparison we calculated the projected structure function
E
D
2
(3.1)
S(d) = (RM(r ′ ) − RM(r ′ + d)) ,
p
from both the observed and synthetic Faraday rotation maps, with d = ∆x2 + ∆y 2
being the radial offsets from a pixel at position r = (x, y). The resulting matrix is then
averaged in radial bins to obtain the structure functions.
Kraichnan (1965) formulate one of the first phenomenological theory of MHD
turbulence. They argued that in the presence of a strong mean magnetic field, the
Alfven waves will interact weakly, deriving an energy spectra that is proportional to
∼ k −3/2 . However, if the magnetic field is weak, as used in most of the mean field
theories, the expected magnetic energy spectra should be Kolmogorov like and expected
to be proportional to ∼ k −5/3 . Those phenomenologies assume isotropic turbulence, that
may be not the case in some astrophysical systems, yielding to a suppression of the
energy cascade along the direction of the mean magnetic field. However, observations in
galactic scales (Han et al., 2004) and numerical simulations (Mason et al., 2008), flavors
the Kolmogorov slope in the spectrum, meaning that there is a energy transfer from larger
to small scales.
Following Ruzmaikin, A. A. and Sokolov, D. D. and Shukurov A.M. (1988), we can
derive rough estimates for our structure functions for the rotation measurements. There
will be a coherence length l0 , which will determine when the RM are not longer specially
correlated. Thus, for scales d ≪ l0
2/3
d
2
S(d) ≃ RM0
(3.2)
l0
for the Kolmogorov spectrum and
S(d) ≃
RM02
1/3
d
l0
(3.3)
for the Kraichnan spectrum. At scales larger than l0 the structure functions should be
have a constant value RM02 and the rotation measurements should be uncorrelated.
In Fig. (3.12) we compare the theoretical predictions for the structure functions with
the observed RM (shown in Fig. (3.11)). The black line is the result from the observation.
In red is shown the expected shape for a Kolmogorov like spectrum and in green one
expected from a Kraichnan spectrum. As we mentioned, at larger distances the RM should
be uncorrelated and therefore have a constant structure function. We show the results
for two different parameter sets for the predictions. In dotted lines for RM0 = 35radm−2
and l0 = 18Kpc and in dashed line for the smaller values RM0 = 27 and l0 = 11Kpc. It is
really difficult to flavor any of the phenomenological models. In the case of larger values
for the parameters (dotted line), it seems that the Kolmogorov like dependence fits better,
but if we change the parameters to lower values flavors the Kraichnan slope. Note that
given the dispersion in the image, and accuracy of the measurements all this parameters
are arbitrary chosen. And even if fixed the errors from the measurement makes impossible
to distinguish between the models. At scales larger than 60 Kpc we observe the effects
due to the shape of our Faraday screen, therefore the function gets very irregular.
3.4. STRUCTURE FUNCTIONS
77
Figure 3.12: Structure functions of the rotation measurement of the observations (shown in Fig. (3.11))
and theoretical predictions. The black line is the result from the observation. In red the expected shape
for the Kolmogorov spectrum and in green one expected from a Kraichnan spectrum. At larger distances
the RM should be uncorrelated and therefore have a constant value. We show two different fittings. In
dotted lines, for the parameters RM0 = 35 rad m−2 and l0 = 18 Kpc and in dashed line for the smaller
values RM0 = 27 rad m−2 and l0 = 11 Kpc.
Fig. (3.13) shows a comparison of the obtained structure function from the observations
(black line) and the simulations. For each simulated cluster we calculated the synthetic
rotation measure maps, clipped accordingly to the shape of the observed map. To obtain
different realizations of the same simulation we produced nine different maps where we
shifted the clipped region by ±20 Kpc in both spacial directions within the original,
cluster centered maps. The thick lines mark the mean structure function over these
maps, whereas the thin lines show the RMS scatter between the different maps (shown
only for one scheme). To be properly comparable with the observations, we scaled up the
synthetic maps with the amplitude at large scales, given that at those scales the RM should
be uncorrelated. Doing so, the amplitude of the structure function should be the same.
Due to the large scatter in the values at large scales we make this normalization averaging
in values between 30 Kpc and 100 Kpc The effects of the different implementation will
also we reflected in those functions. The most dissipative the scheme will tend to suppress
small scale structures, therefore is expected that the Euler simulation without the Lorentz
force should have large values at small d because is only driven by the hydrodynamical
turbulence. However, this is not the case it has the same power than the Dedner, which
is a scheme that takes into account the magnetic pressure of the system.
It is interesting to note that in the Euler potential case have the lower power of all
the 1x simulations. This is due the low final magnetic field, and the small difference
between the reversals. For all the 1x simulations we have a minimum resolution constrain
78
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
Figure 3.13: Shown are the structure functions (S) calculated from the observed (black line) and from
the synthetic Faraday Rotation maps (colored lines) for the different implementations. The thick lines
correspond to the mean calculated over 9 realizations of the maps, whereas the thin lines mark the RMS
scatter between the different maps (only shown for the Euler simulation).
of Hsml ∼ 15 Kpc, explaining the different slope compared with the observations for lower
radii. Note that the synthetic maps are the result of a projection, where each pixel is the
integrate value from several particles, that contributes during the path. This fact allow
us to observe the features even a smaller radii than the Hsml, as is shown by the structure
function.
In Fig. (3.14) show the effects with resolution. As we already mentioned, increasing
the resolution we are able to resolve smaller scales in the RM maps, and this is shown
in the functions. We don’t observe considerable differences between 6x and 10x, but we
see the tendency to better agreement with observations in the internal regions. Although
the total magnetic field in both simulations are different, reversals have similar structure,
independent of the underlying magnetic strength. Interestingly, the 10x simulation, seems
to match the features at observations at 20 Kpc and the drop at 6 Kpc given by the
resolution. We can explain this changes in the slope by considering as transition between
different regimes, i.e. from a uncorrelated to a Kolmogorov, and then to a Kraichnan for
small scales. However, given the scatter from the observations and uncertainties from
the simulations (remember that this set of simulations were not meant to be compared
with any observation in particular), we need more information to arrive to any conlution.
However clearly demonstrates that using such advanced scheme together with very high
resolution allows to probe the properties of the ICM and comparing synthetic and observed
maps one can estimate which physics are more important in the RM phenomena.
3.5. DISCUSSIONS
79
Figure 3.14: Shown are the structure functions (S) calculated from the observed (black line) and from
the synthetic Faraday Rotation maps (colored lines) for different resolutions. The thick lines correspond
to the mean calculated over 9 realizations of the maps, whereas the thin lines mark the RMS scatter
between the different maps (only shown for the 1x simulation).
3.5
Discussions
We run cosmological simulations of galaxy clusters with different underlying SPH schemes
of magnetic field evolution. The magnetic field profiles in the different implementations
show similar properties, even in comparison with Eulerian codes.
The main difference comes from the inner region (i.e. radius < 0.5Mpc), where the
slopes are quite different in each of the methods. In the most dissipative cases, there is a
flatter slope due all the losses in the small scale growth. SPH schemes are known to have
a less numerical mixing than other schemes. However, inside the smoothing length, there
is a intrinsic softening of the properties. In schemes as, Dedner Cleaning, Standard and
Euler potentials, reconnection occurs at this level, thus, keeping the numerical dissipative
processes as low as the scheme permits. The comparison between different schemes and
common properties lead us to the conclusion that the non vanishing numerical ∇ · B
terms are not important for the evolution or growth of the magnetic field. But, they help
to determine where numerical instabilities can happen, therefore a correct treatment of
these quantities is needed. We know, from the successful tests studies (see chapter 2),
that we can reproduce the correct properties from the shocks. Exactly the front shock and
contact discontinuity are a transient stages between two different ones. Thus, depending
on those final stages, temporally can show some numerical inaccuracy that will disappear
when the system evolves. The astrophysical case is even more complicated, having a
80
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
variety of shocks configurations and possible interactions that are impossible to test 3 . In
non-radiative simulations, we found that the numerical MHD implementation is stable
enough to properly simulate galaxy clusters in cosmological context, showing that the
scheme successfully do not generate or propagates spurious ∇ · B in our region of interest,
and lowering the numerical ∇ · B errors with increasing resolution as expected.
It is known that the growth of the magnetic field having only the adiabatic collapse
as dynamo mechanism is unlikely. One of the reasons is the strong tangling of
the field lines, that leads to instabilities and thus a complex morphology of the
field (Ruzmaikin, A. A. and Sokolov, D. D. and Shukurov A.M., 1988). An exclusive
characteristic of SPH is the adaptivity, thus the particles can follow the properties in
different scales. Therefore, the turbulent cascade is well described, integrating correctly
the growth of the magnetic field. Note that even without knowing exactly the proper
value of the initial magnetic field strength for our simulations, we showed in other works
that the astrophysical systems reach an equipartition between the magnetic energy and
the turbulent, in accordance with turbulent dynamo theories (Kotarba et al., 2010b).
In the case of Euler potentials we stress the fact that this representation lacks from
important features in their evolution. This, implies that the structure represented
afterwards is mostly driven by turbulent motions and a magnetic pressure term. The final
field obtained is related to the stochastic events that derives the field to that particular
configuration, but the causality from the previous time steps is not followed properly. In
the case without the back reaction from the magnetic field (i.e. Lorentz force), the final
field structure is build by the advection of the Euler potentials to the final configuration
completely driven by the hydrodynamical turbulence. In this case the final magnetic field
is higher than the ones that take into account the magnetic force. The ∇·B errors in both
cases are similar because they are constraint by numerical precision of the building up
each time step. The lack of reconnection, characteristic from Euler potentials, is shown
in the fact that the magnetic field is accumulated in the center, but does not grow as in
the induction cases.
In the case of smoothing the magnetic field at the point we smooth the field we break
the idealisation of the MHD and let the field diffuse with the surroundings. The same
principle applies to Dissipation. A numerical dissipation is difficult to translate to a
determined physical diffusion value, because they depend on different local properties4
making comparisons and predictions more complicated.
When we study synthetic RM maps we can see how the reversals scales in the
dissipation and smoothing case are extreme. They are represented by just a few big
reversals zones of the orders of ∼ 100 Kpc. In the other cases, there are more zones, with
sizes relatives to of the order ∼ 10 Kpc, which is closer to the values from observations.
Additionally the dissipative cases, the magnetic field is propagated far away from the
cluster center, which is difficult to constrain by observations.
From simple isotropic turbulent theories, one can try to estimate the shape of the
structure functions. However, when comparing theory with observations, we are not
able to distinguish between models, because the uncertainties in observations. When
we study the structure functions of the synthetic maps, we observe that in general we
can’t reproduce the features in observations at all scales, but we confirm a trend to the
observed results when we increase the resolution. This comparison between observations
3 At
most we can test different shocks families as done in section 2.4.
the signal velocity for the numerical case and resistivity on the physical one.
4 Being
3.5. DISCUSSIONS
81
and high resolution simulations, suggests that the physics may be described by a multi
scale turbulence model. The improvements in observations and numerical power in the
next years will allow us to constrain different features in the structure function, and build
a compressive model for the physics involved at different scales.
We clearly demonstrates that using advanced SPH MHD schemes together with very
high resolution allow to probe the properties of the ICM. More effects that can help to
steer-up the inter-cluster medium, therefore helping to generate turbulence and/or power
in the magnetic field structure function in different scales. Additionally we show that
comparing synthetic RM maps with observations can be used to estimate which physics
are relevant to understand the underlaying phenomena, contributing to the global picture
of the galaxy cluster physics.
82
CHAPTER 3. GALAXY CLUSTER AND MAGNETIC FIELDS
Chapter 4
Large Scale cosmological magnetic
fields
I was thinking the day most splendid till I saw what the not-day exhibited, I was thinking this globe enough till
there sprang out so noiseless around me myriads of other globes.
Night on the Prairies - Walt Whitman (1819-1892)
4.1
Introduction
Recently an interesting attempt to constrain the value of large scale cosmic magnetic
fields was done by Lee et al. (2009).
These authors detected a positive crosscorrelation signal between the distribution of galaxies from the SDSS Sixth Data Release
(Adelman-McCarthy et al., 2008) and the RM values extracted from the Taylor et al.
(2009) catalog. Statistical tools, as the correlation functions, demonstrated to be sufficient
show a positive signal due to the RM quality constrains, increasing the signal to noise
ratio (see section 1.4). Using the amplitude of the correlation signal, together with a
simplified model for the magnetic fields configuration in the Universe (estimated from
its mean electron density), let us to calculate the typical RM values expected from this
coherent field in a given length scale. In this way they were able to derive limits for the
corresponding cosmic magnetic fields, at scales where it is difficult to infer the magnetic
fields strengths from other methods.
In this scene we investigated: (i) to what extent a self-consistent treatment of
the cosmological RM signal based on magneto-hydrodynamical (MHD) simulations of
structure formation changes the expected shape and amplitude of such a correlation signal,
and (ii) how such an approach is affected by the presence of the Galactic foreground (GF)
and noise in the final RM signal. Both points are of extreme importance, if robust field
properties are to be derived from any observed signal. Furthermore, the appearance of
magnetic field reversals (as observed in galaxy clusters at various length scales) will alter
the cosmological signal magnitude and shape, whereas the residuals of any foreground
and measurement errors will bias the relation between the amplitude of the correlation
function and the underlying cosmological field. In order to self-consistently treat cosmic
magnetic fields, we make use of several cosmological MHD simulations which compute
the resulting magnetization of the cosmological structures (e.g. amplitude and structure)
following different models for the origin and seeding process of such magnetic fields. We
83
84
CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
also construct magnetic field models with much higher magnetization amplitude in the
low density regions to test how the resulting signatures of more extreme models affect our
results. Here we scale up the predicted amplitude of the magnetic field in filaments by
several orders of magnitude to test if such strong magnetic fields in low density regions
significantly effect the expected correlation signal. By introducing GF and adding noise to
the signal on top of the underlying cosmological signal, we can study how the shape and
amplitude of the cross-correlation function would be modified when considering actual
observations. To avoid further complications we ignore the cosmological evolution of
magnetic fields, which, in principle, would be consistently treated within our cosmological
MHD simulations. Hence, we neglect the evolution of the cosmic magnetic field seen in
the simulation as a result of the structure formation process, and assume the present day
magnetization of the simulated universe to be present up to the redshift of the sources.
This chapter is organized as follows. In Section 4.2 we describe the cosmological MHD
simulations used and how we compute the synthetic RM catalogs. In section 4.3 we
discuss the cross-correlation estimators used, the estimation of the intrinsic uncertainties
due to the limited number of lines of sight probing the magnetization of the cosmological
structures, the different signals expected for the various magnetization of the universe, as
well as the uncertainties induced by the redshift distribution of the sources. In section
4.4 we show how the shape and amplitude of the signal is affected by the recipe normally
used to remove the foreground signal, due to observational noise and to the Galaxy itself.
In section 4.5 we summarize the combination of all the effects, and present the resulting
observable signal of the different magnetic field models. Finally, our conclusions are given
in Section 4.6.
4.2
4.2.1
The Simulations
The cosmological MHD simulations
We used results from one of the constrained, cosmological MHD simulations presented in
Dolag et al. (2005a) and Donnert et al. (2009). In both simulations, the initial conditions
for a constrained realization of the local Universe were the same as used in Mathis et al.
(2002). The initial conditions were obtained based on the the IRAS 1.2-Jy galaxy survey
(see Dolag et al., 2005a, for more details). Its density field was smoothed on a scale of
7 Mpc, evolved back in time to z = 50 using the Zeldovich approximation, and used as
an Gaussian constraint (Hoffman and Ribak, 1991) for an otherwise random realization
of a ΛCDM cosmology (ΩM = 0.3, Λ = 0.7, h = 0.7). The IRAS observations constrain
a volume of ≈ 115 Mpc centered on the Milky Way. In the evolved density field, many
locally observed galaxy clusters can be identified by position and mass. The original initial
conditions were extended to include gas by splitting dark matter particles into gas and
dark matter, obtaining particles of masses 6.9 × 108 M⊙ and 4.4 × 109 M⊙ respectively.
The gravitational softening length was set to 10 Kpc.
The magnetic field was followed by our MHD simulations through the turbulent
amplification driven by the structure formation process. For the magnetic seed fields,
the first simulation (labeled MHD) followed a cosmological seed field (see Fig. 4.1),
while in the second (labeled MHD Gal) used a semi-analytic model for galactic winds.
In particular, we considered the result of the 0.1 Dipole simulation from Donnert et al.
(2009). In both simulations, the resulting magnetic field at z = 0 reproduce the observed
85
4.2. THE SIMULATIONS
Figure 4.1: Mean cosmic magnetic field as a function of density (in units of the mean cosmic baryon
density) obtained from two, fully self-consistent, cosmological MHD simulations for different magnetic
field origins (MHD and MHD Gal), as well as three models, where we artificially scaled-up the magnetic
field intensity at low densities to obtain scenarios with extreme values in filaments (Model 1, Model 2 and
Model 3). For more details on these models see the text. Additionally, the primordial seed fields of the
MHD simulation and that obtained by Lee et al. (2009) are shown.
Model
MHD Gal
MHD
Model 1
Model 2
Model 3
|RM|z=0.03
[rad m−2 ]
0.025
0.018
0.018
0.025
0.040
∆|RM|z=0.03
[rad m−2 ]
0.010
0.010
0.008
0.010
0.013
|RM|z=0.52
[rad m−2 ]
0.12
0.09
0.09
0.12
0.20
∆|RM|z=0.52
[rad m−2 ]
0.05
0.05
0.04
0.05
0.06
|RM|z=1.03
[rad m−2 ]
0.21
0.15
0.15
0.21
0.34
∆|RM|z=1.03
[rad m−2 ]
0.09
0.09
0.07
0.09
0.11
Table 4.1: Mean and standard deviations of RM absolute values for the different catalogs at z = 0.03
and estimates for z = 0.52 and z = 1.03. The catalogs were constructed in each case using four different
realizations (see text). MHD is our fiducial structure formation model. Models 1, 2 and 3 are its scaled-up
versions. MHD Gal includes a semi-analytic model for Galactic winds to seed magnetic fields at z = 4.1.
rotation Measure in galaxy clusters very well. A visual impression for the magnetic field
within the two different simulations and their corresponding galaxy distribution is shown
in Fig. 4.2.
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CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
A3627
Coma
Centaurus
Perseus
Virgo
Hydra
−7.0
−5.0
−3.0
−1.0
1.0
log(|B|) [ µG]
Figure 4.2: Full sky maps of the local universe in super-galactic coordinates for the projected magnetic
field in the MHD run (upper panel) and in the MHD Gal run (middle panel). The galaxy distribution
expected from the corresponding hydrodynamical run is shown in the bottom panel. Galaxies are colourcoded from blue to red using their B − V colors (0.3 < B − V < 1; see Nuza et al. (2010).
87
4.2. THE SIMULATIONS
4.2.2
Artificial MHD models
As is clearly visible in Fig. 4.2, such cosmological simulations usually predict relatively
low magnetic fields in low density regions. To explore more extreme models, we scaled-up
the magnetic field of the MHD simulation by a factor
B1,2,3 = BMHD ×
ρ
ρscale
α
,
(4.1)
with α being 1/3 (Model 1), 1/2 (Model 2) and 2/3 (Model 3). Here ρscale denotes density
scale for fixing the magnetic field, which we choose to be 104 times the mean cosmic
baryon density. The resulting behavior of the mean magnetic field as a function of baryon
density for the original runs, as well as for the scaled-up models, are shown in Fig. 4.1.
Note that the lines shown reflect the mean value of the magnetic field at the corresponding
over density, while the dispersion of its amplitude can span several orders of magnitude
in each density bin (see Dolag et al. 2005). The ratio between the magnetic and kinetic
energy in the simulations is on average at the percent level, which results in a magnetic
field counterpart at least 2 orders of magnitude below the equipartition value, meaning
that the magnetic field is not dynamically important, even in cluster cores. It stays
below equipartition also for the models where the magnetic field is scaled up, with the
exception of Model 3, where the magnetic field nearly reaches equipartition in the cluster
outskirts. We want to stress that such scaled-up magnetic fields are artificial models, as
the primordial field needed to generate them would be well above current cosmological
constraints (e.g. from CMB). Such strong seed fields would lead to an over-prediction
of the magnetic field amplitude in galaxy clusters by the simulations and it is quite
unclear which physical process could be responsible to avoid this. We also remark that
the scaled-up models lead to slightly lower central values for the magnetic field inside
of galaxy clusters. Donnert et al. (2010) show a direct comparison of observed magnetic
field radial profiles and simulations, in particular they consider a similar scaled-up version
of the magnetic field than Model 2 used in this work. This is qualitatively in agreement
with what is needed to fit the observed RM signal within the Coma galaxy cluster (see
Bonafede et al., 2010).
4.2.3
Synthetic RM catalogs
For each of the 5 models, we construct full sky RM catalogs, sampling the whole sky using
3072 different lines of sight (i.e. ∼ 3◦ resolution) making use of the HEALPix (Górski et al.,
2005) tessellation of the sphere. Therefore, our RM catalog contains roughly half as many
number of lines of sight to probe the RM signal of the large scale structure than the catalog
used as Lee et al. (2009).
Although we are only reproducing much shorter lines of sight than expected in the
real Universe (due to the limited volume of the underlying simulation) we believe that
the region probed reflects a fair representation of the present large scale distribution
of galaxies (Nuza et al., 2010), and therefore, we do not expect the amplitude of any
normalized correlation signal to be strongly affected by a lack of fluctuation power.
88
4.2.4
CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
Magnetic depth of the universe
On its way to the observer, the polarized radio emission of the observed sources will pass
several times through the cosmological filamentary structures. The final RM value does
accumulates in a random walk. Examples of the magnetic field structure along some lines
of sight through the simulated local universe can be found in previous work as Fig. 12
in Dolag et al. (2005a) or Fig. 10 in Dolag and Stasyszyn (2009). The magnitude of the
observed RMs, and thus, the mean of the RM absolute values, will strongly depend on
the magnetic depth (given by the redshift range probed) accessible to the observed sample
of radio sources. In addition, if the magnetic field changes during the formation of the
universe, such changes have to be convolved with the redshift distribution of the observed
sources. For simplicity, we assume that the magnetization of the universe at the time of
interest was the same as today and that all sources towards the RMs are measured at the
same redshift, e.g. all lines of sight used probe the same magnetic depth of the universe.
Unfortunately, our cosmological MHD simulation is much smaller than is required to
compare with observations directly and therefore we have to extrapolate our calculated
RMs to the redshift of the real observed sources. With the size of our simulation box,
we can probe only out to z = 0.03. Because of this, we account for the increase of
the RM values due to a random walk process towards higher redshift by assuming the
same contribution of cosmic structures to estimate the cosmological RMs. This is done
by replicating the original volume 15 and 22 times (corresponding to redshifts out to
z = 0.52 and z = 1.03). As shown in Table 4.2.1, the associated RM amplification factors,
including the shift of the rest-frame frequencies due to the cosmological expansion for each
replication of the box, are 4.97 and 8.54 respectively. We will use such expected amplified
RM signals in the following analysis, indicating this by adding the redshift used for the
magnetic depth together with the model name. Note that to increase the RM signal by a
factor of 100 one must probe cosmic structures up to z ≈ 8.
Even for the extreme scaled-up models and extrapolation out to z = 1.03, the expected
RM signal is still one order of magnitude smaller than the reported value by Lee et al.
(2009) in their simplified model (i.e. |RM| ≈ 2 rad m−2 ). This emphasizes the fact that
simulations which properly take the cosmological structures into account are needed to
relate any possible correlation signal to global magnetic field values. Note also that such
small signals are expected to be very sensitive to measurement errors which will scale
with the even much larger foreground signal imposed by our galaxy. We explore these
problems in the following sections.
4.3
4.3.1
Evaluating the cosmological Cross-Correlation Signal
Estimators
We compute the cross-correlation signal between the RM computed along 3072 lines
of sight using a HEALPix tessellation of the sky and the angular positions of simulated
galaxies. For every direction, we count the number of galaxies lying at an angle between
θ and θ + dθ, weighting the counts with the corresponding absolute RM value. Formally,
the cross-correlation function between |RM| and the galaxy density n is defined as follows
4.3. EVALUATING THE COSMOLOGICAL CROSS-CORRELATION SIGNAL
89
Figure 4.3: Angular cross-correlation functions, for the full sky map based on the original MHD Gal
simulation, using the two estimators presented in the text (see Eqs. 4.2 and 4.3). In both panels, the
results of different magnetic field realizations can be seen as blue dotted lines. The average result is
shown as black filled circles. Error bars denote the 1-σ dispersion due to the different realizations. The
Grey area indicates the null signal obtained by reshuffling the RMs.
ωRM (θ) ≡
h∆n(θ)|RM|i
n̄|RM|
,
(4.2)
where ∆n measures the fluctuations around the mean value of n, n̄ is the mean density
of the galaxy sample, |RM| is the mean of the |RM| catalog, and h. . .i denotes ensemble
average. If the distributions of n and RM are Gaussian, this estimator is insensitive to the
addition of an uncorrelated signal (like noise and/or foreground). While for the galaxy
density n a Gaussian distribution is still a reasonable assumption, the RM absolute values
are strongly non-Gaussian. Therefore, it cannot be easily predicted how this estimator
will behave once the observational process is included. In fact, Lee et al. (2009) used a
different estimator, where the normalization by |RM| is not evaluated, i.e.
h∆n(θ)|RM|i
.
(4.3)
n̄
Note that this estimator is likely quite sensitive to processes which change the value of
|RM|, such as the magnetic depth probed by the redshift distribution of the sources.
ξRM (θ) ≡
4.3.2
Evaluating uncertainties
We investigate two types of errors for our simulated cross-correlation functions. To
estimate the significance of our obtained correlation signal, we shuffled the RM data
twenty times. By doing this, we randomly change the direction of the lines of sight, while
keeping the value distribution. This procedure guarantees that the synthetic RM values
and the simulated galaxy distribution will be uncorrelated, while keeping the mean value
of the RM non-Gaussian distribution unchanged. As a consequence, the average value of
the different realizations will correspond to the null signal.
This allow us to take into account the variance given by the used sampling and reveals
the significance of the correlation itself. We will indicate this as the expected level for
a null signal in the figures. A second source of errors is given by the magnetic field
realizations available inside the simulated volume. Since we are still using a small number
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CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
Figure 4.4: Angular cross-correlation function, based on the original MHD Gal cosmological signal (CS),
evaluated for different magnetic depths probed by the RMs, using the two estimators presented in the
text (left and right panels).
of RM points (i.e. 3072), this also introduces significant noise to the correlation functions
obtained. It is beyond the scope of the present work to produce several independent
simulations based on different realizations of the initial magnetic seed field. Therefore,
we assessed this by calculating the RM signal of each particle using either the x, y and z
component or the radially projected magnetic field component. Note that the first three
components are only statistically equivalent to the radially projected component once a
isotropic distribution of the magnetic field is assumed. Next we use these four realizations
of the RM signal to estimate the uncertainty according to the underlying cosmic magnetic
field realization in the simulations. This uncertainty is then added as error bars to the
individual models.
Fig. 4.3 shows the correlation function signal obtained from the MHD Gal model for
the two estimators (ωRM and ξRM ). It shows the individual signal for the four different
realizations of the magnetic fields and the resulting mean signal with their corresponding
error bars. The null signal obtained from reshuffling the RMs twenty times is indicated by
the grey area. Whereas the uncertainties coming from the magnetic field variance in the
different realizations changes the amplitude of ωRM (θ) only by 10%, it is clearly visible
that using the unnormalized ξRM (θ) estimator introduces much larger uncertainties in its
amplitude (by roughly a factor ∼ 2). It also highlights the fact that using this estimator
together with the mean of the |RM| signal to infer the underlying magnetic field will
generate large uncertainties in the estimation. It is important to keep in mind that the
ratio between both estimators (in every scale) for each of the individual realizations is
given by the mean value of the corresponding absolute RMs. However, the normalization
of the different realizations of each magnetic field model can vary significantly (up to a
factor of ∼ 3). Whereas in the normalized cross-correlation this is naturally absorbed, in
the unnormalized case it enters in the error bars when building the assembly average over
different realizations.
As a final remark, we expect that both errors will decrease similarly as Poissonian
errors do when the number of lines of sight to probe the RM signal is increased.
4.3. EVALUATING THE COSMOLOGICAL CROSS-CORRELATION SIGNAL
91
Figure 4.5: Comparison between the angular cross-correlation functions, for the full sky maps in different
models, using the two estimators presented in the text (left and right panels). Black solid line indicates
the MHD model, while blue, light green and red indicate Model 1, 2 and 3 respectively. Pink solid line
indicates the MHD Gal model. In both panels, the light Grey shaded area indicates the randomly shuffled
region, while the dark Grey area indicates the magnitude of typical errors present at a given scale due to
the different RM realizations.
4.3.3
Magnetic depths
As mentioned before, to correctly interpret the cross-correlation signal we must consider
the magnetic depth probed by the RMs. Fig. 4.4 shows the expected signal, assuming
a non-evolving magnetic field probed by the sources up to a certain given redshift. As
expected, the normalized correlation function signal ωRM (θ) is insensitive to the particular
probed volume and its amplitude reflects the underlying magnetic field distribution,
independent on the redshift distribution of the sources. On the other hand, the signal
given by the unnormalized estimator ξRM (θ) increases as a function of the magnetic depth
as expected. In this case, the amplitude changes by more than a factor of two if the redshift
distribution of the radio sources is changed from z = 0.52 to z = 1.03. This means that
one has to consider the redshift distribution of the radio sources towards the observed
RMs in order to relate the amplitude of the signal to the underlying magnetization of the
large scale structures. Therefore, it is difficult to interpret such an observed signal (as
done e.g. by Lee et al., 2009).
4.3.4
Magnetic field models
Fig. 4.5 shows the cross-correlation function signal obtained from the five different models
investigated. The two shaded regions indicate the contribution of the two errors discussed
before. The shape and the ordering of the correlation signal of ωRM reflects the scaling
of the underlying magnetic field models with density. In particular, the crossover of the
correlation function reflects the one seen in the underlying magnetic field models very
well (see Fig. 4.1). The correlation signal thus indeed carries information about the
strength and distribution of the cosmic magnetization. It is expected that using more
lines of sight will reduce the statistical errors enough to make our extreme models clearly
distinguishable. This would be in principle possible with the available number of line of
sights in current data (e.g. Taylor et al., 2009).
The unnormalized correlation function ξRM leads to a larger relative change of the
amplitude of the correlation signal for the different magnetic field models, especially for
92
CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
the ones with very high magnetic fields in filaments. However, the resulting signal comes
with much larger errors (coming mainly from the different magnetic field realizations in
the same models) and is therefore less significant. Also, the ordering of the magnetic field
models with less extreme magnetic field values in filaments is not longer reflected in the
correlation amplitude, particularly towards smaller impact parameters.
In summary, we conclude that the correlation signal for ωRM inherits a clear signal
from the cosmological magnetization, whereas the correlation function ξRM (as used in
Lee et al., 2009) is very difficult to interpret. Because of the missing normalization,
changes in the underlying RM distribution (caused by different realizations of the
same magnetic field model or the magnetic depth probed by the radio sources) are not
compensated for.
4.4
Simulating the observational process
To test for the effects caused by the observational process on the cross-correlation functions
it is important to use an underlying scenario which reflects the expected amplitude of the
RM signal. Therefore, we use (unless specifically stated) an underlying magnetic depth of
the universe of z = 1.
4.4.1
Foreground removal procedure
For RM observations, the removal of the foreground imposed by our galaxy is a major
problem. Usually one assumes that the foreground varies on (much) larger scales than the
ones of interest and removes the GF by subtracting a smoothed signal from the original
data. Here we test how such a removal procedure affects the underlying cosmological
signal traced using correlation functions following exactly the same procedure applied in
Lee et al. (2009). At every point, we subtract the mean of the RM absolute values within
a given radius (excluding the central value). Specifically, we tested three different angular
sizes for the removal (i.e. 3◦ , 6◦ and 9◦ ). Fig. 4.6 shows the result of such foreground
subtraction technique on the normalized correlation function for the cosmological signal
using the normalized estimator ωRM . At small distances, this procedure leads to a
significant suppression of the correlation signal, even up to a factor of ∼ 2 for angular
distances below ∼ 2◦ , almost independently of the size of the removing radius. At larger
distances, the amplitude of the correlation function is slightly increased (10 – 20%) starting
from scales larger than the smoothing radius.
4.4.2
Adding observational noise
Another problem for the observed RMs are the measurement errors by themselves. For
example, since the data recently published by Taylor et al. (2009) is based on only two
different frequency bands the resulting RMs will be affected by a significant uncertainty.
We also note that these errors are not reduced by the smoothing involved when removing
the foreground. The typical error of the observational RMs (as inferred from comparison
with a data subset which was observed at more frequency bands) turns out to be around
10 to 20 rad m−2 (see Taylor et al., 2009, Fig. 2). In order to estimate the effect of the
observational errors, we added random values to our simulated RM signal, which were
drawn from a Gaussian distribution with a dispersion given by σRM . We explored values
4.4. SIMULATING THE OBSERVATIONAL PROCESS
93
Figure 4.6: Angular cross-correlation function, based on the MHD Gal cosmological signal (CS), assuming
a magnetic depth of z = 0.03. Shown are the results obtained by subtracting to each |RM| the average
of its neighbors (excluding itself) within a radius of 3◦ , 6◦ and 9◦ for the ωRM estimator.
of 0.001, 0.01, 0.1, 1.0 and 10 rad m−2 for σRM . Note that most of these values are
much more optimistic than what is expected from current instruments. However, future
instruments, like e.g. SKA and ASKAP will achieve an RM accuracy of a few rad m−2
(Beck and Gaensler, 2004).
Fig. 4.7 shows the impact of such measurement errors onto the resulting correlation
function. Even σRM = 0.01 (e.g. a hundredth of the actual measurement error) leads to
a sizable (ca. 50%) reduction of the correlation signal. Furthermore, σRM = 0.1 (e.g. ten
percent of the actual measurement error) reduces the signal by a factor of ∼ 5 and σRM = 1
(e.g. nearly the present measurement errors) makes the correlation very close to the one
of the corresponding null signal. From this it is clear that using the normalized estimator
ωRM (θ) will be quite problematic. The presence of even small measurement errors (far
smaller than what can be reached currently) will affect the shape and amplitude of the
correlation function in a way that the information on the cosmic magnetization is basically
lost.
4.4.3
Adding Galactic foreground
In recent years, different models for the Galactic magnetic field were proposed (e.g.
Han et al., 2006; Page et al., 2007; Jansson et al., 2008; Sun et al., 2008). To estimate the
influence of the GF on the cosmological cross-correlations we produced a synthetic map
of the RM signal expected for our galaxy using the publicly available code HAMMURABI
94
CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
Figure 4.7: Angular cross-correlation function for the MHD Gal cosmological signal (CS) using the ωRM
estimator. Same as Fig. 4.3 (left panel), but including the effects of different random noise (N) scenarios
at magnetic depth of z = 0.03.
(Waelkens et al., 2009), where we made use of the Galactic magnetic model given by
Sun et al. (2008). The original model was constructed to give a good representation of
the synchrotron emission of the Milky Way but, by missing possible reversals within
the model magnetic field, it overproduces the RM signal by a significant factor. We
therefore scaled the original model down to obtain a better representation of the observed
RMs. We also note that such reversals could lead to significant small scale structures
in the RM signal due to the GF, as shown by Sun and Reich (2009). Such fluctuations
could significantly compromise the cosmological signal, as they would be present on scales
smaller than the one used to filter the GF. However, we do not currently include this effect
in our foreground model.
In Fig. 4.8 we show the obtained RM map models (left column) compared to the
observed RM signal (right column) taken from Taylor et al. (2009). From top to bottom
we show the original GF model with noise and the RM data-set, a smoothed version
of the maps (within 8◦ ), and the residuals when applying the foreground subtraction as
described above for 3◦ . All the synthetic maps are imprinted with an observational error
of σ = 10 rad m−2 . The last row shows the synthetic residual map when reducing the
noise level to σ = 1 rad m−2 as expected for future instruments. The close-ups show
the remaining signal of prominent galaxy clusters in the residual maps. Note that the
signal of other prominent clusters, lying behind the Galactic plane, are not longer visible
after the foreground subtraction was applied. As expected, when adding such a large,
plain foreground signal to the cosmological one, the cross-correlation function vanishes.
Therefore, we also applied the GF removal technique described before. The results can be
95
4.4. SIMULATING THE OBSERVATIONAL PROCESS
(a) CS+GF+N(σ = 10.0 rad m−2 )
(b) RM data
(c) CS+GF+N(σ = 10.0 rad m−2 ) smoothed in 8◦
(d) RM data smoothed in 8◦
(e) CS+GF+N(σ = 10.0 rad m−2 ) with GF removal in 3◦
(f) RM data with GF removal in 3◦
(g) CS+GF+N(σ = 1.0 rad m−2 ) with GF removal in 3◦
Figure 4.8: Full sky maps in galactic coordinates for the total synthetic signal (left column) and for the
observed RMs (right column). The first row shows the Galactic foreground (GF) map generated with the
HAMMURABI code of Waelkens et al. (2009) including the cosmological signal (CS) from the MHD Gal
simulation with a magnetic depth of z = 1 and an imprinted observational error (N) of σ = 10 rad m−2
compared with the plain RM data given by Taylor et al. (2009). The second row shows the same maps
but smoothed by 8◦ (as in Fig. 4 of Taylor et al., 2009) and the third row shows the resulting residual
maps when foreground removal is applied (within 3◦ ). The lower left plot shows the former synthetic
map where the noise was reduced to σ = 1 rad m−2 , as it is expected for future observations. In the
lower right we show 40◦ × 40◦ wide close-ups of three prominent clusters in the simulation (from left to
right: Perseus-Pisces, Virgo and Coma).
96
CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
Figure 4.9: Angular cross-correlation function for the combined cosmological signal (CS) from the MHD
Gal simulation and the Galactic foreground (GF), including the foreground removal in 3◦ as described in
the text. Shown is the signal using both estimators as presented in the text (see Eqs. 4.2 and 4.3).
seen in Fig. 4.9, where the angular cross-correlation function of the combined maps for
both estimators is shown. For comparison, we show the expected signal from the plain
MHD Gal simulation (e.g. assuming a very small magnetic depth of z = 0.03), the GF
signal alone, and the MHD Gal combined with the GF signal for a large magnetic depth
(i.e. z = 1.03), as well as for a extreme magnetic depth corresponding to z ≈ 8. In all
cases we applied the foreground removal using a radius of 3◦ . Even for the extreme case
of magnetic depth, despite the foreground removal applied, the normalized estimator ωRM
drops further by a factor of ∼ 3 when adding the GF, and drops by a factor of ∼ 30 for the
most optimistic cosmological signal. On the contrary, the unnormalized estimator ξRM
turns out to be quite insensitive to the foreground provided that the removal technique is
applied. The combined signal still corresponds roughly to the original, cosmological one,
as can be seen when comparing with Fig. 4.4.
We conclude, that although the normalized estimator ωRM in principle contains a
much more unbiased and reliable imprint of the cosmological magnetization, once GF
and observational noise are added, the underlying cosmological signal is completely lost.
In contrast, the unnormalized estimator ξRM is relatively insensitive to the GF and to the
noise. However, as seen before, the interpretation of its amplitude and shape is extremely
challenging, as it is quite biased by the underlying magnetic depth probed by the redshift
distribution of the radio sources used in the RM measurements.
4.5
The simulated observational cross-correlations: an example
To study if it is possible to infer the underlying cosmological signal through an
observational process which includes GF (and its removal technique) as well as
measurement errors by cross-correlating the |RM| signal with the galaxy density we
assume:
• An optimistic magnetic depth of the universe of z = 1.03;
• A GF according to the model presented in Section 4.4.3 together with the foreground
subtraction technique presented in Section 4.4.1 using the mean value of the |RM|
map within 3◦ ;
4.5. THE SIMULATED OBSERVATIONAL CROSS-CORRELATIONS: AN EXAMPLE
97
Figure 4.10: Changes in the cross-correlation functions for the two estimators ωRM (left panel) and ξRM
(right panel) when gradually including all described steps to the cosmological signal (CS) using the MHD
Gal simulation with amagnetic depth of the universe of z = 1.03. GF and N stand for Galactic foreground
and observational noise respectively.
• A measurement error distribution consistent with a Gaussian having σRM = 1 rad
m−2 , i.e. a dispersion similar to the magnitude of typical errors achievable by future
instruments.
4.5.1
Signal pile-up
Fig. 4.10 shows how the resulting signal changes when gradually adding all effects
described before, using the MHD gal model. As already seen in the individual steps above,
the normalized estimator ωRM gives a much more significant signal, but its amplitude and
shape suffers dramatically from the inclusion of noise and addition of the GF (despite of
the subtraction technique applied). On the other hand, the unnormalized estimator ξRM
gives a much less significant signal only mildly changed by all the contributions to the
total signal, mainly at larger distances.
4.5.2
Differentiating cosmic magnetization
Fig. 4.11 summarizes the results for such a combination of contributions to the total signal
for our five different magnetic field models. The signal for the normalized estimator ωRM
(left panel) is reduced by a factor of ∼ 50 and the shape does not represent the underlying
magnetic field models as well as when applied to the cosmological signal itself (e.g.
compare with Fig. 4.5). The amplitude of the unnormalized estimator ξRM corresponds to
the underlying cosmological signal, but here the original ordering due to the magnetic field
models is no longer present. In general, for both estimators, the significance of the total
signal is only marginal and the differences between the different magnetic field models lie
far inside the error bars. Note that if we would only consider the null signal and ignore
the errors from the magnetic field realization both estimators would give highly significant
detections. The errors coming from the magnetic field realizations are not accessible from
the observations and, therefore, the observationally obtained significance of the signal
can be misleading unless compared to detailed simulations. As well, both errors can be
significantly reduced by using higher number of RMs. The results can also be improved
by avoiding the Galactic region (e.g. cutting the region of the maps lying inside ±10◦ ).
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CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
Figure 4.11: Cross-correlation functions for the different magnetic field models, using an optimistic
magnetic depth for the cosmological signal (CS) (corresponding to a universe magnetized out to z = 1.03),
taking into account the Galactic foreground (GF), assuming a Gaussian noise (N) with σRM = 1 rad m−2
(as expected for the next generation of instruments) and applying the foreground subtraction in 3◦ . Left
panel shows the result for normalized estimator ωRM whereas the right panel shows the result for the
unnormalized estimator ξRM .
In that case, the unnormalized estimator will strongly reduce the power excess seen at
distances larger than 3◦ . Such a cut would also remove the artificial but significant signal
seen for separations larger than 10◦ in both estimators.
4.6
Conclusions
Using cosmological MHD simulations of the magnetic field in galaxy clusters and filaments
we evaluated the possibility to infer the magnetic field strength in filaments by measuring
cross-correlation functions between RMs and the galaxy density field.
We find that the shape of the cross-correlation function using the normalized estimator
ωRM (in absence of any noise or foreground signal) nicely reflects the underlying
distribution of magnetic field within the large scale structure. However, a very large
number of lines of sight probed by RM measurements (much more than the 3072 used in
this investigation) are needed to overcome the statistical noise induced by the particular
magnetic field realization within the cosmic structures, in order to distinguish between
the wide range of models we used here. In general, the RM signal is strongly dominated
by the denser regions (e.g. those populated by galaxy clusters and groups) and not by
the low density ones, like filaments. On this point, the magnetic field associated with
filaments already changes by several orders of magnitudes within the different models
used here.
Additionally, the normalized estimator ωRM is extremely sensitive to measurement
errors and to the presence of the GF (despite attempts to remove it by subtracting
a smoothed map). It is fair to say that given the current measurement errors in
the available RMs and our knowledge of the GF, present studies cannot determine
the magnetization magnitude of the Universe based only on the cross-correlation ωRM ,
whatever the significance of the measured signal is. On the contrary, the shape of the
unnormalized estimator ξRM (the same as used by Lee et al. 2009) is relatively insensitive
against the presence of measurement errors for the RMs and for the presence of the GF (as
long as the described removal technique is used). Its amplitude, however, is quite strongly
4.6. CONCLUSIONS
99
affected by measurement uncertainties. Current measurement errors (as for example those
inherited by the Taylor’s published sample) suppress the signal by a significant amount in
such a way that it is impossible to relate the amplitude of the cross-correlation function
to the underlying magnetization of the the large scale structure. However, we expect that
future radio telescopes will be able of reaching error magnitudes of order of 1 rad m−2
that could make the correction of the signal possible.
Unfortunately, this estimator does not nicely encode in its shape the details of the
magnetization of the large scale structure and, especially, its amplitude is extremely
sensitive to the magnetic depth of the Universe. Therefore, any interpretation of an
observed signal is limited by our knowledge of the redshift distribution of the sources
(towards the RM signals measured), as well as by our knowledge of the distribution and
evolution of the cosmic universal magnetization. Future observational data will help to
put better constraints on theoretical models for the origin of cosmological magnetic fields
which, in return, can be implemented in next generation of MHD cosmological simulations
in order to draw a self-consistent picture that can be compared against observations.
In summary, we conclude that current RM observations cannot constrain the amplitude
and distribution of magnetic fields within the large scale structure. On the other hand,
future data-sets, based on a larger number of observations with more accurate RMs,
might be able to shed light on the magnetic field distribution and evolution within
these structures. However, very detailed model predictions are needed in order to
compare with any observed cross-correlation signal. It will be a quite demanding task for
future cosmological simulations to provide detailed enough information of the large scale
structure magnetization process within a large enough volume to produce useful templates
of such correlation functions which can then be compared directly to the observations.
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CHAPTER 4. LARGE SCALE COSMOLOGICAL MAGNETIC FIELDS
Part IV
Conclusions
101
Chapter 5
Summary and outlook
Some say that the universe is made so that when we are about to understand it, it changes into something even
more incomprehensible. And then there are those who say that this has already happened.
The Hitchhikers guide to the galaxy - Douglas Addams (1952-2001)
Magnetic fields are found in almost every environment in the studied Universe.
However, they are notoriously difficult to measure because of the lack of possible direct
observations at large astrophysical distances. Nevertheless, progress has been made, and
their presence in the inter-galactic medium and intra-cluster medium has been confirmed
by observations of diffuse radio emission as well as Faraday rotation measurements towards
polarized radio sources within or behind the magnetized medium (e.g. Govoni, 2006).
Recent studies (Dolag et al., 2010) even show considerable magnetic filling factors in
cosmological volumes, like those in filaments or voids.
Given the high non-linearity of many of the physical processes involved in astrophysical
problems, the use of numerical codes has become a powerful technique and widespread
approach for studies of structure formation. Astrophysical fluid dynamics usually involve
changes on spacial, temporal and density scales over several orders of magnitude. Thus,
adaptivity is essential to study these problems. An elegant solution to face this if offered
by Smooth Particle Hydrodynamics (SPH). However, it was not until recently that it
became possible to solve MHD equations using the SPH formalism. This unique tool can
now be extensively used to constrain the interaction of magnetic fields with the baryon
component of the Universe in several environments, from star forming regions to galaxy
clusters, including galaxy evolution and cosmic ray propagation.
Cosmological MHD Simulations
We focused in novel implementations of MHD inside the well-known cosmological
SPH code Gadget-3. Thereby, we were able to implement and study different MHD
approximations and schemes, either building the field from potentials or evolving the
magnetic field itself. The vector nature of magnetic fields translates into an extra level
of complexity compared with known numerical problems already solved for scalar fields,
like density or temperature fields. There are known issues related with SPH interpolants
that can be problematic in the case of MHD. For instance, the constraint ∇ · B = 0 in
the Maxwell equations is not ensured for the numerical implementation itself. This is not
103
104
CHAPTER 5. SUMMARY AND OUTLOOK
only a problem from the physical point of view, but being not zero this divergence can
grow instabilities by adding an artificial contribution to the Lorentz force and possibly
drive artificial dynamo effects.
Such issues are now well understood and can be controlled (Dolag and Stasyszyn,
2009). The combination of many improvements in the underlying SPH algorithm with
its generalization to the MHD case enhances the handling of magnetic fields in SPH
significantly. We found that correcting the instability by explicitly subtracting the
contribution of a numerical non-zero ∇ · B to the Lorenz force from the Maxwell tensor, is
useful in terms of stability 1 . Placing a limiter in this correction improves the performance
and quality of the numerical scheme. This limiter is needed because in certain situations
the ∇ · B source term can be temporally high, thus driving instabilities if taken into
account directly. This improvement is particularly efficient in three dimensional setups,
where it seems to work much better than other suggestions in the literature.
We studied two regularization schemes: smoothing the magnetic field and including
an artificial dissipation. The first scheme basically regularizes the magnetic field by
smoothing it every desired time interval. The second scheme regularizes strong variations
in the field by artificially introducing diffusivity terms in the induction equation. The
schemes demonstrated to further minimize inter-particle noise and ∇ · B errors. However,
one has to carefully select the numerical parameters to avoid excessive smoothing of
sharp features. In particular, the magnetic field smoothing succeeds in terms of stability,
however, in general it tends to over-smooth features independent of the dynamics of the
system. The artificial dissipation also increases the stability (behaving similarly to the
magnetic field smoothing). Yet, in this case, the over-smoothed features come from the
changes in the magnetic field morphology itself. Because there is no dependence in global
time-steps and it depends on local gradients, this scheme turns out to be desirable for
physical runs. However, there are cases in which those gradients are correct and the
magnetic field configuration is over suppressed. From a physical point of view, both
regularisation schemes mimic the effects of diffusion by numerical means.
We also implemented a ∇ · B cleaning scheme (namely Dedner), which constructs
and follows a scalar field build up from the ∇ · B errors. From this field we correct
the original magnetic field to the numerical threshold given by our resolution. The
successful implementation of a multidimensional Dender cleaning method in SPH, does
not significantly affect the shape of the front shocks while reducing the numerical ∇ · B
generated. The evolution of this correction is completely local within the Hsml and
therefore only affected by the quality of SPH derivatives. The success of ∇ · B cleaning
brings our simulations closer to the “ideal” MHD state (Stasyszyn et al., 2011), with only
fast reconnection happening inside the representative volume of each particle.
Finally we study the magnetic field evolved from the advection of two scalars, the so
called Euler potentials, from which we construct the final magnetic field. The implemented
potential approach has the benefits of fulfilling the constraint ∇ · B = 0 by construction,
up to a numerical threshold. Hence, we are able to compare results between schemes,
demonstrating that schemes using the induction equation reach similar error levels when
measured in the tests. In the astrophysical cases, Euler potentials schemes have the
drawback of being unable to change the topology of the field. Therefore no reconnection
occurs, losing representativity of the astrophysical system. In those cases the magnetic
1 Even
when using Euler potentials to construct the magnetic field
105
field obtained by this method is the representation of the stochastic processes driven
by the dynamics of the system. Therefore helping us to test different phenomenological
approaches to drive the magnetic field evolution.
We studied the magnetic field evolution in the context of cosmological structure
formation of galaxy clusters. Given the complexity inherent to MHD simulations, our
studies were non-radiative, to focus on how magnetic fields can reproduce observations
without any additional physics. We compare different numerical schemes leading us to
the conclusion that the ∇ · B terms are not important for the evolution or growth of the
magnetic field in galaxy clusters. As found in the tests, the special care taken for the
∇·B terms demonstrated to be useful to lower inter particle scatter, improving the overall
performance of the code. We conclude that the turbulent cascades are well modeled in the
simulations, which is manifested in the growth of the magnetic field. This was concretely
studied by the Euler potentials runs, where the reconstructed magnetic field is given by
the final turbulent driven configuration. We observe that despite this facts, the ∇ · B
errors between schemes in our region of interest are similar, meaning that this turbulence
is phenomenological and not numerically driven. Due to the lack of diffusion in these
runs, the magnetic field accumulates in the center of the cluster and is not distributed to
the outermost parts of the halo, especially in the case when we do not take into account
the Lorentz force. From the magnetic field profiles we conclude that the magnetic field
distribution in inner region of the cluster, depends strongly on the numerical dissipation of
the underlaying scheme. However, the outer part agree with the self-similar model when
compared between different schemes and resolutions. Furthermore, a non dissipative
scheme allows for complex morphological structures as expected in galaxy cluster or even
in galaxies. On the other hand, additional physics like Ohmic dissipation are expected to
be important (Bonafede et al., 2010). Therefore, a numerical scheme like Dedner, with low
numerical diffusion, is interesting for future studies of the interplay of physical diffusion
and magnetic field morphology in the ICM.
We made synthetic rotation measure maps and study the reversals of the magnetic
field in comparison with observations. The artificial dissipation and smoothing cases are
extreme in this context. They present just a few big reversals zones on scales of ∼ 100 Kpc.
In the other runs, we resolve a larger number of reversals, with spacial scales of the order
of ∼ 10 Kpc, which is closer to the observed values. Additionally, in the case of Euler
potentials with Lorentz force, we observe a synthetic map comparable with dissipative
runs. This implies that a complete stochastic representation of MHD turbulence does not
properly reproduce the observations in terms of reversals. We observe that, as we enhance
the resolution the number of reversals increases, demonstrating a better resolution of the
turbulent cascade.
With the synthetic maps we are able to build structure functions, which can be
a quantitative comparison with observations. Additionally, simple isotropic turbulent
theories make predictions on the shape of these structure functions. However, when
comparing theory with observations, we are not able to distinguish between models
because of the uncertainties in observations and the small difference in the scale
dependence of the slopes. When studying the structure functions of the synthetic maps,
we reproduce the overall behavior. We have not yet reached the resolution to reproduce
all the features from observations. However, we see how the results tend to the observed
ones at the highest resolution, which is a promising result. The comparison between
observations and high resolution simulations suggests that the physics may be described
106
CHAPTER 5. SUMMARY AND OUTLOOK
by a multi scale turbulence model. This means that the turbulent dynamo driven by the
cosmological cluster formation process works effectively, reproducing basic observational
properties, even to details shown in structure functions.
We clearly demonstrate that using advanced schemes together with high resolution
allow to probe the properties of the ICM. Additional astrophysical effects (such as star
formation, AGN feedback, etc.) will contribute to steer-up the inter-cluster medium,
therefore helping to generate turbulence and power in the magnetic field structure function
on different scales. Additionally, we show that the comparison of synthetic RM maps
with observations can be used to assess which physics are relevant in the underlaying
phenomena, contributing to the global picture of the galaxy cluster physics, as can be a
multi-scale turbulence model.
In addition to the cosmological results, a stable numerical scheme is useful in several
astrophysical problems. One major example is the first simulation of the collapse
of protostellar cores including ideal-MHD following the induction equation with SPH
(Bürzle et al., 2010), showing the effects of the magnetic fields in multiple star formation.
Another example is given by the studies of galaxies (Kotarba et al., 2009, 2010a,b), which
clearly demonstrates how the magnetic field reaches equipartition with the turbulent
energy. These studies are a particularly exciting subject given the impressive quality
of observational data which we can compare with (e.g Beck, 2009), and which provide
important dynamo theory challenges.
Statistical studies
Having demonstrated that structure formation simulations are able to reproduce
properties of magnetic fields in galaxy clusters, especially in the outer regions of haloes,
we investigate the relation between the magnetic field and the structure in which it is
embedded.
Recently new large observational data sets (e.g. Taylor et al., 2009) became available,
requiring simulations of cosmological volumes with magnetic field to test their properties
of cosmic volumes. These large amounts of data open the possibility for statistical studies
of the distributions of baryons and magnetic field. We use a set of cosmological MHD
simulations to evaluate the possibility of inferring the magnetic field strength by statistical
means, particularly measuring cross-correlation functions between rotation measures and
the galaxy density field.
To take into account the uncertainties in the possible dynamo implicated in our
simulations, we study the effects of several models of scale-up the magnetic field with
density. We use the same field in the center of the clusters (that is well constrained as
shown before) and different field strengths in low density regions (like filaments and voids).
The use of different magnetic field models associated to the cosmological structures, allows
us to compare variations of several orders of magnitude in the magnetic values in large
scales. Additionally, we compare with observational results, and try to mimic all the
possible observational bias. Therefore, we build a synthetic foreground map of the galaxy
and apply different estimators used in observations to suppress spurious effects. Finally,
we also added noise expected from current instruments and made predictions for future
instrumental facilities. Taking into account all the steps in the process separately allowed
us to understand each of them and to find out which are most important.
107
In general, the observational rotation measure signal is strongly dominated by denser
regions (e.g. those populated by galaxy clusters and groups), and the transition of
the magnetic field to low density regions is unclear. We find that the shape of the
cross-correlation function using a normalized estimator ωRM (in absence of any noise
or foreground signal) nicely reflects the underlying distribution of magnetic field within
the large scale structure.
The normalized estimator ωRM is extremely sensitive to measurement errors and to
the presence of the galactic foreground (despite attempts to remove it by subtracting
a smoothed map). It is fair to say that given the current measurement errors in the
available observed rotation measurements and our knowledge of the galactic foreground,
present studies cannot determine the magnetization magnitude of the Universe based
only on the cross-correlation ωRM , whatever the significance of the measured signal is.
On the contrary, the shape of the unnormalized estimator ξRM is relatively insensitive
against the presence of measurement errors dur to noise and for the presence of the
rotation measurement foreground (as long as a removal technique is used). Its amplitude,
however, is quite strongly affected by measurement uncertainties. Current measurement
errors (as for example those inherited by the sample published by Taylor) suppress the
signal by a significant amount in such a way that it is impossible to relate the amplitude of
the cross-correlation function to the underlying magnetization of the large scale structure
(Stasyszyn et al., 2010).
Future Prospects
It is known that clusters with high cooling rates seem to have stronger magnetic
fields (Taylor et al., 1994). This leads us to a full set of predicted observables that can
be constrained from numerical studies. helping to understand which effects are more
important to prevent the catastrophic collapse of cooling clusters. However, this also
means that the magnetic field in such regions can be dynamically important, and thus
we need a stable numerical implementation, such as the one presented, to start to study
those regimes. Additionally, even in regions where magnetic field are not dynamically
important, we can use them as tracers to study turbulence and dissipative processes
known to be part of the ICM, and in combination with X-ray measurements (Zhuravleva,
2011) trace their importance and effects.
Transport process inside clusters are sensitive to magnetic fields, and simulations help
to construct recipes to a proper treatment of these processes (Donnert et al., 2011).
Therefore, to continue exploring the effects of magnetic fields, we need to include its
effects self-consistently in the sub-grid physical methods. Magnetic fields are expected to
have important effects and can act as an additional energy depending on the environment
properties. This has to be taken into account into the energy balance of the system and
linked to other process. The use of a self-consistent implementation implies that energy
will be redistributed, which is important in studies of cosmic ray propagation or cluster
relics and cavities.
As we mentioned, the current measurements of magnetic field in different environments
are not completely explained by primordial magnetic fields and their growth due to only
gravitational collapse, signaling that additional dynamo actions should occur naturally
in the Universe. Besides the magnetic field seeding there are possible dynamo actions
108
CHAPTER 5. SUMMARY AND OUTLOOK
that include non-ideal MHD.Within the present development status of MHD in Gadget3, possible numerical artifacts are well constrained. Therefore this numerical code is an
ideal ground for implementations of theoretical dynamo models or additional sub-grid
physics. How dynamo mechanism work is an important question that can be applied on
different scales, from galaxy discs to collapsing clouds in star forming regions or AGN
accretion disks.
Magnetic fields are known to play a supporting role in the formation and evolution of
several astrophysical objects (e.g. stars). Even if true that magnetic fields are not essential
for our understanding of the large-scale dynamics of the Universe, a total knowledge of
structure formation process seems to be not possible without solving also the physics on
smaller scales. Therefore, in our seek for understanding magnetic fields in the Universe,
we can find answers to other astrophysical questions.
Part V
Appendices
109
Appendix A
Magnetohydrodynamics and Plasma
Physics
Magnetohydrodynamics (MHD) and plasma physics describe the interaction between
electromagnetic fields and non-viscous conducting fluids (see, for example, Jackson, 1975;
Parker, 1979). Basic fluid properties are velocity V(x, t), pressure p(x, t), and current
density J(x, t). In Gaussian units, the relevant Maxwell equations take the form
4π
J
(A.1)
c
which is called Ampère Law. It was taken into account that moving charges generate
a measurable Magnetic Field (this effect was also know in the case of constant current as
Biot-Savart law).
Then there is a relation between the variation in the magnetic field fluxes and the
electromotive energy.
1 ∂B
∇×E+
=0
(A.2)
c ∂t
which is known as Faraday induction equation. Note that this equation is similar to Eq.
(A.1), and even more interesting is the fact that the change in flux can be given by moving
magnets either by electric fields. If we choose a special reference frame, these equations
will lead to a relation between velocities that the magnetic field is moving and a force
that will appear trying to satisfy the relation in Eq. (A.2).
∇×B=
Fmag = J × B
(A.3)
This is know as the Lorentz Force 1 .
Another interesting fact in the magnetic field studies are the absence of monopole
sources.
All manifestations of magnetic fields comes in the form of dipoles or
representations of higher order expansions. This is represented by
∇·B=0
(A.4)
This is supported by experiments, that have constrained their possible existence to almost
null probability. But, let’s suppose hypothetically that Eq. (A.4) is not valid, if so this
1 The total energy will have another term referring to the electric field F = qE that we neglect, because in our problem
E is completely neglected.
111
112
APPENDIX A. MAGNETOHYDRODYNAMICS AND PLASMA PHYSICS
will imply that these magnetic charges have similar properties as the electric charges.
Then one can derive that electromagnetic waves should not exist. And the fact that we
can use them in everyday life is one of the most important supporting facts to Eq. (A.4).
For further insights about this subject I recommend to follow the discussions in Parker
(2007).
Free charges moving are subject to be trapped by the medium. This is taken by Ohm’s
law
J = σE
(A.5)
where σ is the conductivity in the rest frame of the fluid. Most astrophysical fluids are
electrically neutral and non-relativistic so Eq. (A.5) becomes
V×B
J=σ E+
(A.6)
c
being the term with the cross product of the B times V the result from the Lorentz force.
This can be combined with Eq. (A.2) and Eq. (A.1), to yield the ideal MHD equation:
∂B
= ∇ × (V × B) + η∇2 B.
(A.7)
∂t
To derive this equation, the molecular diffusion coefficient, η ≡ c2 /4πσ, is assumed to
be constant in space.
To have a complete set of dynamical equations we must define a continuity equation
for the fluid
dρ
= −ρ∇ · V,
(A.8)
dt
which is important because relate the density of our fluid ρ with the pressure ∇ · V.
In the limit of infinite conductivity, magnetic diffusion is ignored and the induction
equation becomes
∂B
= ∇ × (V × B)
∂t
(A.9)
or equivalently
dB
= (B · ∇) V − B (∇ · V)
(A.10)
dt
where d/dt = ∂/∂t + V · ∇ is the convective derivative, so-called Lagrangian and we use
Eq. (A.4).
The first term in Eq. (A.10) describes the stretching of magnetic field lines that
occurs in flows with shear and vorticity. As an illustrative example, consider an initial
magnetic field B = B0 x̂ subject to a velocity field with ∂Vy /∂x = constant. Over a
time t, B develops a component in the y-direction and its strength increases by a factor
1/2
.
1 + (t ∂Vy /∂x)2
The second term describes the adiabatic compression or expansion of magnetic field
that occurs when ∇ · V 6= 0. Consider, for example, a region of uniform density ρ and
volume V that is undergoing homogeneous collapse or expansion so that the first term
is null and ∇ · V = C where C = C(t) is a function of time but not position. Using
113
the continuity equation and Eq. (A.10) one find that B ∝ ρ2/3 ∝ V −2/3 . Thus, magnetic
fields in a system that is undergoing gravitational collapse are amplified while cosmological
fields in an expanding universe are diluted.
Combining Eq. (A.10) and Eq. (A.8) one can find the following alternative form
d B
B
=
· ∇ V.
(A.11)
dt ρ
ρ
The formal solution of this equation is
Bi (x, t)
Bj (ξ, 0) ∂xi
=
ρ (x, t)
ρ (ξ, 0) ∂ξj
where ξ is the Lagrangian coordinate for the fluid:
Z t
xi (t) = ξi +
Vi (s)ds .
(A.12)
(A.13)
0
It follows that if a “material curve” coincides with a magnetic field line at some initial
time then, in the limit η = 0, it will coincide with the same field line for all subsequent
times. Thus, the evolution of a magnetic field line can be determined by following the
motion of a material curve (in practice, traced out by test particles) as it is carried along
by the fluid. This is called that the field is frozen-in with the plasma.
The equation of motion for the fluid is given by
1
1
∂V
+ (V · ∇) V = − ∇p − ∇Φ +
(J × B) + ν∇2 V
(A.14)
∂t
ρ
cρ
where ν is the hydrodynamic viscosity coefficient of the fluid and Φ the gravitational
potential. In many astrophysical situations, the fields are weak and the Lorentz term in
Eq. (A.14) can be ignored. This is the kinematic regime. In the limit that the pressure
term is also negligible, the vorticity ζ ≡ ∇ × V obeys an equation that is similar, in form,
to Eq. (A.7):
∂ζ
= ∇ × (V × ζ) + ν∇2 ζ.
(A.15)
∂t
Moreover, if viscosity is negligible, then ζ satisfies the Cauchy equation (Moffatt, 1978):
ζj (ξ, 0) ∂xi
ζi (x, t)
=
.
ρ (x, t)
ρ (ξ, 0) ∂ξj
(A.16)
However, Eq. (A.16) is not a solution of the vorticity equation so much as a restatement
of Eq. (A.15) since ∂x/∂ξ is determined from the velocity field which, in turn, depends
on x. By contrast, in the kinematic regime and in the absence of magnetic diffusion, Eq.
(A.12) provides an explicit solution of Eq. (A.10).
The magnetic energy density associated with a field of strength B is ǫB = B2 /8π.
For reference, we note that the energy density of a 1 G field is ≃ 0.040 erg cm−3 . A
magnetic field that is in equipartition with turbulent energy of a fluid of density ρ
1/2
and rms velocity v has a field strength B ≃ (4πρv 2 ) . In a fluid in which magnetic
and kinetic energies are comparable, hydromagnetic waves either propagate parallel or
perpendicular to the direction of the magnetic field. The first ones are the so-called
114
APPENDIX A. MAGNETOHYDRODYNAMICS AND PLASMA PHYSICS
1/2
Alfvén waves and propagates at a characteristic speed vA ≡ (B 2 /4πρ) . The second
ones are called magnetosonic waves and have similar characteristics than sonic waves but
with the contribution of the magnetic pressure.
It is often useful to isolate the contribution to the magnetic field associated with a
particular length scale L. Following Rees and Reinhardt (1972) we write
2 Z
B
B(L)2 dL
=
(A.17)
8π
8π L
V
where hB 2 /8πi is the magnetic field energy density averaged over some large volume
V. B(L) is roughly the component of the field with characteristic scale between L and
R
1/2
2L. Formally, B(L) = (k 3 /2π 2 V) Bk where Bk ≡ d3 x exp (ik · x)B(x) is the Fourier
component of B associated with the wave number k = 2π/L.
In the ideal MHD limit, magnetic fields are distorted and amplified (or diluted) but no
net flux is created. A corollary of this statement is that if at any time B is zero everywhere,
it must be zero at all times. This conclusion follows directly from the assumption that
charge separation effects are negligible. When this assumption breaks down, currents
driven by non-electromagnetic forces can create magnetic fields even if B is initially zero.
Therefore, to have a dynamo effect working, we must apart from an ideally MHD.
Appendix B
Finding optimal numerical
parameters
All the methods described in section 2.3, had their particular numerical parameters that
modify how the schemes will behave. Those parameters are difficult to choose, because in
some situation their effects are minimal, while in other cases can be counterproductive. To
find the optimum numerical parameters, we performed 11 shock-tube tests with different
settings for the parameters in the regularization methods and evaluated the quality of
the result obtained with the SPH-MHD implementation, and confirmed them with a
smaller span in the planar tests. To have a quantitative measure of the tests, we used
two estimators. First, we have chosen the mean of all ∇ · B errors within the simulation
region, as defined by
h
.
(B.1)
∆∇·B = ∇ · B
|B| x
Second, we measured the discrepancy of the SPH-MHD result for the magnetic field
relative to the results obtain by ATHENA. Therefore we calculate first
2
i
i
δBi (x) = { BSPH
(x) − BAthena
(x) }RMS2Bi (x)
(B.2)
for each component i of the magnetic field B within each 3D slab corresponding to the
smoothing length. The RMS of B i reflects the noise of Bi within the chosen slab. We
then calculate
!
!
X
X
ˆ Bi =
(B.3)
∆
δBi (x)
RMS2Bi (x) ,
x
x
for each component of the magnetic field. This includes both contributions, the deviation
of the SPH-MHD from the ideal solution as well as the noise within each 3D slab of the
SPH-MHD implementation. To judge the improvement of the regularisation methods we
sum up all three components and further relate this measurement to the value obtained
with the basic SPH-MHD implementation, e.g.
P
ˆ Bi
∆
∆B = P i std − 1.
(B.4)
ˆ i
∆
i
B
We will use these two error estimators, ∆∇·B and ∆B , to measure the quality of the
individual SPH-MHD implementations.
115
116
B.1
APPENDIX B. FINDING OPTIMAL NUMERICAL PARAMETERS
Divergence threshold
There are no significant effects in the threshold used in normal tests. We only can test
this effects in extreme situations and by using the threshold to avoid the ∇ · B generate
unstable interactions. Therefore for this parameter, that in principle is free, we observe
that in simulations is enough to ask if the change that will be done is of the same order
of the force (i.e. parameter ∼ 1). However, just to ensure that the correction is not
dominating the dynamics we use a value of 2 as shown in Eq. (2.42). The ∇ · B errors
are usually a couple of orders of magnitude below the change, and when one approach to
the level of given by the Lorentz force equation, the correction is dominant and breaking
the MHD equations. Therefore when applied to test cases, it does not contribute in the
performance. However, as it helps to stabilize the situations, it improved the numerical
values ranges where some tests can be performed. As an example for the test 1A was
unstable with Dedner for π < 2, and now the are able to test it until 10. That’s because
this implementation uses the ∇ · B as source for the correction, and this particular test
has the stronger shock fronts.
B.2
Regularization by smoothing the magnetic field
Choosing the time interval between smoothing the magnetic field is a compromise between
reducing the noise in the magnetic field components, as well as ∇ · B) (by smoothing
more often) and preventing sharp features from being smeared out. Figure B.1 shows
a summary of the results of the individual shock-tube test computed with different
smoothing intervals. As expected, when using shorter smoothing intervals the error in
∇ · B reduces. For the quality measure of the SPH-MHD implementation the situation
changes. Short smoothing intervals generally increase the discrepancy, many of them
even to larger values than the basic SPH-MHD run. Specifically 4B and 4C show strong
deviations due to smearing of sharp features. Note that the non monotonic behavior
shown in some tests usually relates to some residual resonances between the magnetic
waves and the smoothing intervals in the noise. Some tests show a minimum in the
differences at smoothing intervals around 20. The test 3A seems to prefer even shorter
smoothing intervals. In general, values around 20-30 seem to be a good choice.
B.3
Artificial dissipation
As before, choosing the value for the artificial magnetic dissipation constant αB is a
compromise between reducing the noise in the magnetic field components (as well as
reducing ∇ · B) and preventing sharp features from smearing out due to the effect of
the dissipation. Figure B.2 shows a summary of the results of the individual shock-tube
tests computed with different values for the artificial magnetic dissipation. As expected,
using larger values reduces the error in ∇ · B significantly. Similar to before, using larger
values also generally results in an increase of the discrepancy between the SPH MHD
implementation and the true solution, again usually to even larger values than in the
basic SPH-MHD run. As before, especially the shock-tube test 4B and 4C show strong
deviations due to smearing of sharp features. Note that here less non-monotonic behavior
is visible (except for test 4B). The main reason is that dissipation is a continuous process,
B.3. ARTIFICIAL DISSIPATION
117
Figure B.1: Shown are the mean error in divergence (upper panel) and the measure of the quality (lower
panel) as defined in equation (B.4) obtained by the SPH-MHD implementation for different values of the
smoothing interval. The different lines are for the 11 different shock-tube tests as indicated by the labels.
so resonances between dissipation and the magnetic waves cannot be very pronounced.
Taking all tests into account, a good choice for αB seems to be around 0.1.
B.3.1
Time dependent artificial dissipation
One idea to reduce the effect of the artificial dissipation is to make the artificial magnetic
dissipation constant αB time dependent. The idea here is that, if the evolution of αB is
properly controlled, dissipation will happen only at the places where it is needed and it
will be suppressed in all other parts of the simulation volume. The evolution of αB is
controlled by the two parameters S0 (source term) and C (decay term) where we have
min
max
chosen αB
and αB
as 0.01 and 0.5 respectively. Figure B.3 shows the result for varying
these two parameters. As before, generally, the larger the dissipation is (e.g. large source
term or small decay time) the smaller the noise and the error in ∇ · B becomes. However,
as soon as these parameters have values which drive αB in the shocks to the maximum
allowed value, there is marginally no gain in quality, although the values for αB outside
the shocks can still be quite small. Therefore, the time dependent method does not
improve the results significantly, as the regions in which the artificial dissipation constant
is suppressed do not significantly contribute to the smearing of sharp features.
118
APPENDIX B. FINDING OPTIMAL NUMERICAL PARAMETERS
Figure B.2: Similar to figure B.1 but for different values of the artificial magnetic dissipation constant
αB .
B.4
Dedner Method
Basically in this scheme one has two major constant to be chosen. Namely ch and cp . ch
basically is the velocity in which the ∇ · B errors will propagate away from the source. cp
is a dumping length in which we fade away the errors. Using Eq. (2.46) we can rewrite
the equations in terms of the magnetic signal velocity and the smoothing length. In this
scheme the first constant is σ that is related with the source (i.e. ∇ · B). If one let the
errors propagate at the magnetic signal velocity seems to be a proper choice. Faster than
that reduces the sensitivity of the scheme, and slower can make the errors to accumulate
in some specific places. For the π case we needed to know how the different shocks
were performing. In Fig. (B.4) we show the comparison for various π values, finding as
optimum values π ∼ 2 − 3. Note that it is not a clear trend, so the chosen value is not
the best in all cases, but is a good compromise between all tests. Additionally to these
constants, we added a threshold similar to the one used in the ∇ · B subtraction. These
study is also shown in Fig. (B.4). The best value corresponds to Q = 0.5, thus when
the correction is the half of the change given by the induction equation, this correction is
rescaled. Note that this studies were actually done in all the parameter space, but we just
show the proper cuts to that parameter space in Fig. (B.4) at Q = 0.5 and π = 3. Again
this values were ensured to be consistent in planar tests, were significantly enhanced the
performance.
B.4. DEDNER METHOD
119
Figure B.3: Similar to figure B.1 but for different values of the source term S0 (left panels) and the decay
term C (right panel) of the time dependent, artificial magnetic dissipation.
120
APPENDIX B. FINDING OPTIMAL NUMERICAL PARAMETERS
Figure B.4: Similar to figure B.1 but for different values of the dumping term π (left panels) and the
threshold Q (right panel) of the Dedner scheme. When we vary π we have fixed Q = 0.5 and when
varying Q we use
Appendix C
Resolution convergence tests
Numerical experiments are normally restricted by the resolution one can technically
(in terms of computing/memory requirements) achieve. Therefore tests as presented in
section 3 are usually at nominally better resolution than can be obtained in relevant (in
this case cosmological) simulations. Never-the-less an interesting question is, how good
do the numerical methods used converge if one further increase the resolution? Figure C.1
and C.2 show this for ATHENA and the basic SPH-MHD implementation respectively. We
repeated the Orszang-Tang Vortex test problem with ATHENA on a 1922 , 4002 and 8002
grid. Figure C.1 shows a cut through the density of the Orszang-Tang Vortex, comparing
with the result obtained with the AMR code Ramses (Teyssier, 2002). Clearly, the results
obtained with ATHENA when increasing the resolution approaches the results obtained with
Ramses. Figure C.2 shows the same for setups with 3502 × 5, 7002 × 5 and 14002 × 5
particles. The SPH-MHD implementation also converges towards the Ramses results with
increasing resolution. However, although the central feature is better resolved in the SPHMHD implementation than in the ATHENA run with comparable resolution, some other
Figure C.1: A cut through the density for the Orszang-Tang Vortex test (see Figure 13/14). Shown
in black is the result obtained with Ramses, compared to the results obtained with ATHENA using 3
different resolutions.
121
122
APPENDIX C. RESOLUTION CONVERGENCE TESTS
Figure C.2: Same than figure (C.1), but showing the results obtained with the basic SPH-MHD
implementation at two different resolutions compared to the results obtained with Ramses.
features can be seen to converge slower in the SPH-MHD implementation when increasing
the resolution. Specifically, in some very smoothed features there are small but systematic
differences between the SPH and the true solution. Here the SPH results seem to converge
only extremely slowly (if at all).
Appendix D
∇ · B distribution analysis
One of the major objections to MHD-SPH simulations comes from the ∇ · B constrain,
given the freedom inherent of particle schemes, this constrain can be subtle to be broken
in regimes where the field is kinetically disturbed.
We analyze this in the framework of MHD tests in chapter 2. But cosmological
simulations are not so easy to describe.
The shocks and conditions that the
implementation has to solve are countless, therefore we study statistically how this errors
evolve.
In the left panel of Fig. (D.1) we can show the evolution of the ∇·B errors and magnetic
energy in redshift. We plot the averaged quantities in the whole simulation in physical
units. At early times, we observe how the magnetic field is diluted by the cosmological
expansion of the universe until z ∼ 10, when the structure formation process start to take
place and the dilution is canceled and reverted. Additionally, we observe some merger
events as bursts in the mean magnetic energy. At early times, the ∇·B errors are small, as
the simulation evolves they grow following the disorder proper from the hydrodynamical
motions. At redshift z ∼ 10 we the structure formation process starts and now the
magnetic field begin to try to drive the errors. At redshift z < 4 the errors arise to
values > 1E − 2 that epoch corresponds to the most agitated in the Universe history
where the due to rapid structure formation and halo collapses. In those processes, that
will continue during the rest of the simulation, the there are shocks from the accretion of
matter in halos and there is a difference on the shape the magnetic profile and density.
However, both magnetic field and ∇ · B errors, reach a saturation level at z < 0.5 due to
the equipartition between magnetic and turbulent energy.
This evolution dependence of the ∇ · B errors can be originated from the difference
in how hydrodynamical and magnetic forces scale with distance. As an example in in
the right panel of Fig. (D.1), we show the final (z = 0) density profile of our cluster,
superposing the magnetic field. One can observe a steeper slope for the magnetic field,
mostly around ∼ 0.8Mpc. In general this is not a problem per se, SPH should be able
to deal with it. However, be observe certain situations where the induction equation can
derive a wrong magnetic field, usually high.It will not affect overall evolution of the system
because of the low numerical mixing of SPH, and unless it is corrected or advected to the
high magnetic field regions, it will remain with high ∇ · B with a slow decay until the end
of the simulation.
In the first panel of Fig. (D.2) we show the distribution of the magnetic (black), kinetic
(blue) and thermal (green) energy densities. We can see how in general the thermal and
123
124
APPENDIX D. ∇ · B DISTRIBUTION ANALYSIS
Figure D.1: In the left panel we shown the evolution of the mean numerical ∇ · B errors (black)
and magnetic field energy (red) in the whole cosmological simulation. When first structures collapse
(z < 10)the numerical errors arise. In the right panel we shown a comparison between density and
magnetic profiles. The magnetic field is multiplied by a factor of 10−20 to show the shape change
between them. One can observe that the magnetic field is steeper at ∼ 1Mpc.
Figure D.2: In the first panel we show the distribution of the magnetic (black), kinetic (blue) and thermal
(green) energy densities. We can see how in general the thermal and kinetic energy densities are several
orders of magnitude higher. The upper right panel shows the density over density-critical ratio histogram
seen the huge dynamical range that SPH allow us to cover, and magnetic fields are distributed in. In the
bottom left corner we show the distribution of the ∇·B errors. The average and most of the particles have
low than one error, arising the question where are those particles located. Finally we show a distribution
of the different energy ratios β.
125
Figure D.3: In the left panel we show magnetic field as function of density for the particles slitted in
samples of ∇ · B higher and lower than 0.1. In general we see that the high ∇ · B particles are the ones
with less magnetic energy at a given density. Note that for ρ/ρcrit arround 10−2 both samples have the
same values. In the right panel we show ∇ · B errors as function of the βkin , showing a clear correlation
to higher ratios.
kinetic energy densities are several orders of magnitude higher, however there is a energy
region where all of them overlaps. In the bottom left corner we show the distribution
of the ∇ · B errors. The average and most of the particles have below unity. However,
the slopes for higher or lower errors are different, arising the question where are those
particles located. Finally we show a distribution of the different energy ratios β, in the
bottom right panel of Fig. (D.2). We can see that in general either kinetic or thermal
energy ratios are higher, and in terms of thermal-kinetic ratio, the kinetic energy densities
are higher.
In general the ∇ · B errors don’t seem to correlate with other variables. However there
is a slight tendency to correlate with the strength of the magnetic field (Bürzle et al.,
2010; Kotarba et al., 2010a). To continue the analysis we separate the particles in two
samples, in terms of high and low ∇ · B error, we use ∇ · B = 0.1 as the threshold. In the
left panel of Fig. (D.3) we shown how particles with high ∇ · B errors use to have lower
magnetic fields for a given density. This fact give us the hint that the final force that they
suffer will be not magnetic dominated 1 compared with the low ∇ · B case. Note that
both samples have the same magnetic field average at densities of ρ/ρcrit ∼ 10−2 which
is the characteristic density at r ∼ 0.5 − 0.8Mpc. Interestingly if we analyze the relation
between these energies we saw again a correlation.
To confirm this, we study the ∇ · B errors against their kinetic/thermal to magnetic
energy density ratio (also called β in the literature) for the set including all the particles.
These plot is shown in the right panel of Fig. (D.3), where a strong correlation is found.
Historically, instabilities happened in the opposite case (Phillips and Monaghan, 1985)
Meaning that the high beta are particles dominated by kinetic motions, therefore the
magnetic field configuration is more difficult to be restored. Note that when we talk
about low and high in the β samples is relative to the whole sample. Statistically most
of the particles have high β (see Fig. (D.2)).
Finally we also plot in Fig. (D.4) the ∇ · B errors in density bins for two β samples,
larger and smaller than 107 . Surprisingly we don’t find any correlation of the effect with
1 However,
he magnetic energy are always a couple or orders of magnitude bellow the thermal and kinetic
126
APPENDIX D. ∇ · B DISTRIBUTION ANALYSIS
Figure D.4: ∇ · B errors as function of density, separated in two samples of βkin larger or smaller than
7. Clearly there is a correlation between the higher ratio and lower ∇ · B errors, and surprisingly there
is no trend with density, meaning that possibly this is a drawback from the scheme itself.
density. This means that the effect is not caused by strong hydrodynamic shocks in the
center of halos, leading to the open question which kind of shocks generate this ∇ · B. On
the other hand explains why the use of threshold in our schemes is so useful, the changes
in the stress tensor given by ∇ · B terms can be driven by completely hydrodynamical
effects, not related with real numerical artifacts, therefore the correction is not needed.
Part VI
Acknowledgments
127
Acknowledgments
I want to thank, specially, my wife, Michi, because she join me in this journey to the
unknown, called life. My family, for all the continuous support thought space and time.
Klaus, to take a chance with me. Hanna, Julius and Florian to the support in the writing
and reading, and to MHD considerations. Fer, Leo, Sebas, Ceci, Paula, Jesús, Eva, José,
Tom and Damian (the most acknowledged boy of the world), and all the friends that
I’ve done in Germany, for the supportive true friendship build thought the years. Dante,
Nelson, Yami, for being myself in south America. The Peralta brothers, for the friendship
and reading. My friends in Argentina (Facu, Mario, Negro, Dani, GuilleK, Leo, Jose) to
build what I am. My office mates (Umberto, Mona, Qi, Andre) and people from the MPA
for the warm welcomed. Galileo-mobile project, to teach us, that plenty of projects can
be done, while working together.
129
130
Part VII
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