Olb8888a

Olb8888a
A Gallery of Simple Models
from Climate Physics
Dirk Olbers
Abstract. The climate system of the earth is one of the most complex systems
presently investigated by scientists. The physical compartments – atmosphere,
hydrosphere and cryosphere – can be described by mathematical equations
which result from fundamental physical laws. The other ’nonphysical’ parts
of the climate system, as e.g. the vegetation on land, the living beings in the
sea and the abundance of chemical substances relevant to climate, are represented by mathematical evolution equations as well. Comprehensive climate
models spanning this broad range of coupled compartments are so complex
that they are mostly beyond a deep reaching mathematical treatment, in particular when asking for general analytical solutions. Solutions are obtained
by numerical methods for specific boundary and initial conditions. Simpler
models have helped to construct these comprehensive models, they are also
valuable to train the physical intuition of the behavior of the system and guide
the interpretation of the results of numerical models. Simple models may be
stand-alone models of subsystems, such stand-alone general circulation models of the ocean or the atmosphere or coupled models, with reduced degrees
of freedom and a reduced content of the physical processes. They exist in a
wide range of structural complexity but even the simplest model may still
be mathematically highly complicated due to nonlinearities of the evolution
equations. This article presents a selection of such models from ocean and
atmosphere physics. The emphasis is placed on a brief explanation of the
physical ingredients and a condensed outline of the mathematical form.
Contents
1. Introduction
2. Fluid dynamics and thermodynamics
2.1. Equations of motion for ocean and atmosphere
2.2. Coupling of ocean and atmosphere
2.3. Building a climate model
3. Reduced physics equations
3.1. The wave branches
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To appear: Progress in Probability Vol. 49, Eds.: P. Imkeller and J. von Storch, Birkhäuser Verlag,
p. 3-63.
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Dirk Olbers
3.2. The quasigeostrophic branch
3.3. The geostrophic branch
3.4. Layer and reduced gravity models
4. Integrated models
4.1. Energy balance models and the Daisy World
4.2. A radiative-convective model of the atmosphere
4.3. The ocean mixed layer
4.4. ENSO models
4.5. The wind- and buoyancy-driven horizontal ocean circulation
4.6. The thermohaline-driven meridional ocean circulation
4.7. Symmetric circulation models of the atmosphere
5. Low-order models
5.1. Benard convection
5.2. A truncated model of the wind-driven ocean circulation
5.3. The low frequency atmospheric circulation
5.4. Charney-DeVore models
5.5. Low order models of the thermohaline circulation
5.6. The delayed ENSO oscillator
ReferencesREFERENCES
Coordinates and constants
The forcing functions of the wave equations
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1. Introduction
The evolution of the climate system is governed by physical laws, most of which
arise from mechanical and thermodynamical conservation theorems applying to the
fluidal envelopes of the earth. They constitute a coupled set of partial differential
equations, boundary and initial conditions, of the form
∂
Bϕ + Lϕ + N [ϕ, ϕ] = Σ.
∂t
(1)
The state of the system is described by ϕ(X, t) which generally is a vector function.
There are linear or nonlinear differential operators B, L, N acting on the spatial
dependence of ϕ, and Σ denotes sources of the property ϕ. Externally prescribed
forces and coefficients may also enter the operators B, L and N . Nonlinearity,
indicated above by the N -term, is an inherent and important property of climate
dynamics. It mostly arises from transport of ϕ by the fluid motion (e.g. advection of
heat in the ocean and atmosphere by the fluid circulation which in turn depends
on the distribution of temperature). Nonlinearity not only defeats solutions of
complex models by analytical means. It also introduces a coupling in the broad
range of scales of the climate system, and may lead to multiple equilibria and
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chaotic behavior, thus rendering a separation of the system into a manageable
aggregate of subsystems difficult.
The conservation equations, coming from basic physics, govern motions for
a vast range of space-time scales, and climate models of ocean and atmosphere
circulation must necessarily disregard a high frequency-wavenumber part of the
spectrum of motions to describe the evolution of a slow manifold. Climate physicists do this by averaging and filtering techniques. If (1) is considered the result of
such procedures – i.e. if ϕ represents an averaged and slowly varying state – the
source term Σ contains contributions from the field ϕ0 representing the subrange
of scales, generally referred to as turbulence, and terms which couple the resolved
component ϕ to the filtered variables χ (the fast manifold). The source would then
be of the form
Σ = −N [ϕ0 , ϕ0 ] − N1 [χ, ϕ] − N2 [χ, χ] + F,
(2)
where the overbar indicates averaging over the turbulent components and F is an
external source. For practical reasons climate models must be closed with respect
to the turbulence which is usually done by invoking some parameterization relating
the mean turbulent source to the resolved fields,
N [ϕ0 , ϕ0 ] = P[γ, ϕ].
(3)
The parameterization operator P may be nonlocal in space and time but in most
practical cases one deals with simple local and linear relations with constant parameters γ. As an example, the divergence of turbulent fluxes of heat is frequently
represented by Fickian diffusion.
In the view of a climate physicist, stochastic elements enter the problem (1),
(2) and (3) where variables or coefficients appear which are not well known and
should be considered as members of some random ensemble. Depending on the
problem the random variable could represent the initial conditions ϕ(X, t = 0),
the external forcing F , the turbulent field ϕ0 (in form of the parameters γ) or the
fast manifold χ of the system.
Evidently, with this concept in mind, problem (1) is a nonlinear Langevin
equation which was the starting point of Hasselmann’s stochastic climate model
(Hasselmann 1976). It was practically applied by Hasselmann and coworkers to
explain the observed redness of climate spectra in terms of white noise forcing.
Examples are the sea surface temperature variability on time scales of weeks to
decades, treated by a Langevin model of ocean mixed-layer physics (Frankignoul
and Hasselmann 1977), long-term climate variations of the global temperature,
treated by a Langevin global energy balance model (Lemke 1977), and similar
treatment of sea-ice variations (Lemke et al. 1980). A review of various applications has recently been given by Frankignoul (1995) and an even more recent
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investigation of that framework in the wind-driven ocean circulation is found in
Frankignoul et al. (1997) and Frankignoul (1999).
Almost any compartment of the global climate system can be viewed through
a stochastic frame. For each separate compartment one can identify an external
driving force, and when this is varying in a stochastic way the system’s response
will be a stochastic process as well, with physically and mathematically interesting
properties if the system has a rich interesting ’life’, for instance in form of nonlinearities, resonances, time delay, instabilities and other ingredients of complex
dynamics.
In the sections 2 and 3 I give a brief review of the basic fluid mechanical and
thermodynamical equations used in ocean and atmosphere physics. There are various important aspects where geophysical fluid dynamics differ from conventional
fluid mechanics. All models presented in the paper can be derived from this fundament, at least in principle. In section 4 I introduce the concept of filtering which
breaks the equations into those describing fast and slow manifolds of evolution.
Basically, the equations split into two subsets, representing fast modes of motion
with adjustment mechanisms due to gravity, and slow modes whose time scales are
governed by the differential rotation of the earth, as contained in the latitudinal
dependence of the Coriolis frequency.
Every part of the climate system shows a rich variability in space and time, so
besides filtering in time oceanographers and meteorologists have developed various
techniques to reduce the spatial degrees of freedom. Most of these techniques are
brute force actions: the system dynamics is integrated or averaged in some spatial
directions – examples of integrated models are collected in section 5 – and/or the
state vector of the system is expanded into a set of spatial structure functions
with subsequent heavy truncation down to a manageable number of variables –
examples of these low-order models are collected in section 6. Though being brute
the techniques are applied in an intelligent way in order to include the interesting
physical mechanisms and arrive at a meaningful physical system.
To my knowledge only very few of the dynamical systems collected in this
’gallery’ have been investigated within a stochastic framework, for some of them
the interest was originally addressed exclusively to steady state solutions. It should
be clear that the models are extremely simple crooks used by climate physicists to
move on the complex terrain of the climate system in the search for understanding of bits and pieces. Most of them, however, are still so complex that general
analytical solutions are not known. In fact, they are stripped-down and simplified parts of complex climate models which are cast into numerical coding and
solved on computers. Occasionally, numerical climate modelers have driven their
codes by artificial white noise forcing to study the long-term red response (see
e.g. Mikolajewicz and Maier-Reimer 1990, Eckert and Latif 1997).
Only models of ocean and atmosphere systems will be introduced. I will
briefly explain their physics and point out where random elements might be attached. In most cases, however, this is obviously the prescribed forcing which in
total or in part can be considered as a random variable. I should like to point
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out that the choice of the models is rather subjective and the presentation rather
limited: the emphasis lies on the model equations rather than on physical or mathematical results. In any case, I strongly recommend to consult the original or text
book literature for a deeper understanding of the model context and applicability.
Suggestions for further reading are given in each section.
2. Fluid dynamics and thermodynamics
The evolution of the atmosphere or ocean is governed by the conservation of momentum, total and partial masses, and internal energy. The state of the system
is completely described by a 7-dimensional state vector which is usually taken as
(V , T, p, %, m = [S or q]) where V is the 3-dimensional velocity of the fluid, T the
temperature, p the pressure, % is the density of total mass, and m the concentration
of partial mass (such that %m is the density of the respective substance). For each
of the fluids – seawater or air – there are only two dynamically relevant partial
masses. In the ocean we have a mixture of pure water and various salts which are
combined into one salinity variable m = S (measured in kg salt per kg sea water).
The air of the earth’s atmosphere is considered as a mixture of dry air (basically
oxygen and nitrogen) and water vapor with concentration m = q (measured in kg
vapor per kg moist air)1 . The concentration of the complementary partial mass is
then 1 − m. For a binary fluid, i.e. a fluid composed from two partial masses, the
thermodynamic state is described by three thermodynamic state variables which
are usually taken as T, m and p. This implies that any thermodynamic potential
can be expressed in these three variables. In particular, since the density % is a
thermodynamic variable, there is a relation
%
=
F (T, m, p),
(4)
which is referred to as equation of (thermodynamic) state. For the atmosphere,
(4) is the ideal gas law, expressed in this context for the mixture of the two ideal
gases dry air and water vapor. For the ocean, various approximate formulae are
used (see e.g. Gill 1982).
The above mentioned conservation theorems may be expressed as a set of
partial differential equations for the state vector, written here in a rotating coordinate system fixed to the earth, with angular velocity Ω, and with consideration
of the gravitational and centrifugal acceleration combined in g,
1 Other
constituents of the air such as water droplets, ice and radioactive trace gases can be
neglected in the mass balance.
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DV
Dt
D%
Dt
Dm
%
Dt
T Dθ
%cp
θ Dt
%
=
−2%Ω × V − ∇p + %g + F
(5)
=
−%∇ · V
(6)
=
Gm
=
%cp
(7)
·
¸
Dp
DT
−Γ
= Gθ .
Dt
Dt
(8)
The advection operator is
·
¸
Dϕ
∂
∂
%
=%
+ V · ∇ ϕ = (%ϕ) + ∇ · (V %ϕ),
Dt
∂t
∂t
(9)
where the first of these relations defines the time rate of change following the
motion of a fluid parcel (Lagrangian or material derivative) and the second gives
the equivalent Eulerian form where the effect of the flow appears now as the
divergence of the advective flux V %ϕ of the property ϕ. Equations (4) to (8) form
a complete set of evolution equations.
In (8) the use of the potential temperature θ enables to express the conservation of internal energy in a simpler form by separating the adiabatic heating
(derived from adiabatic expansion work contained in the second term in the brackets) and the diabatic heating rate Gθ in the heating rate of the ordinary (measured
in-situ) temperature T . The potential temperature is defined by dθ = dT − Γdp for
an infinitesimal adiabatic displacement in the thermodynamic phase space. Here
Γ = αT /(%cp ) is the adiabatic temperature gradient, i.e. dT /dp = Γ if Gθ = 0. The
thermal expansion coefficient α and the specific heat cp are known thermodynamic
functions of T, m, p. The potential temperature also depends on a constant reference pressure p0 ; if the fluid parcel is moved adiabatically from its pressure p to
this reference pressure p0 and temperature is measured there, its value equals the
potential temperature θ of the parcel. The differential relation dθ = dT − Γdp may
be integrated to express the potential temperature in terms of the thermodynamic
state variables in the form θ = θ(T, m, p). This relation may be used to replace the
ordinary temperature T by θ to simplify the equations. Then, an equation of state
% = G(θ, m, p) = F (T (θ, m, p), m, p) is appropriate. In fact when meteorologists
or oceanographers refer to temperature in a dynamical model they usually mean
potential temperature2 .
2 It
should noted that the difference between T and θ is small in oceanic conditions (about 0.5 K
at most), it is usually ignored in the term on the lhs of (8). Furthermore, the specific heat cp in
this term is taken at the value of the reference pressure while cp in the next part of the equation
is taken at the in-situ pressure (for an ideal gas cp is constant so that this difference is irrelevant
for the atmosphere).
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The source/sink terms F , Gm and Gθ of momentum, partial mass and internal
energy contain an important part of the physics. In general we may write the
source/sink terms as the sum of a flux divergence – which describes the transport
of property through the boundaries of fluid parcels – and source/sink terms which
are proportional to the volume, e.g. Gq = −∇ · J q + Cq where J q is the diffusive
flux of vapor q and Cq represents the source/sink due to evaporation of water
droplets in clouds or condensation of vapor to liquid water. Boundary conditions
will be discussed later in section 2.1 for a simplified set of equations.
When the fluxes appearing in F , Gm and Gθ are taken according to molecular
theory Navier-Stokes equations are obtained as balance of momentum and Fickian
diffusion of substances and heat is considered. In this case the equations (4) to (8)
describe the full spectrum of atmospheric and oceanic motions, including sound
waves with time scales of milliseconds to the thermohaline circulation of the ocean
with periods of up to thousands of years. Clearly, such a range of variability is
not the aim of a climate model: solving equations over climate time scales with
resolution down to the sound waves is intractable and certainly not meaningful.
Fortunately, the coupling of large-scale oceanic and atmospheric motions with
motions at very small spatial and temporal scales of sound is very weak and can
safely be ignored. The elimination of sound waves from the evolution equations
and other approximations are outlined in the next section. Further wave filtering
of (4) to (8) is demonstrated in the sections 3.1 and 3.2.
2.1. Equations of motion for ocean and atmosphere
Climate is defined as an average of the state of ocean and atmosphere and the
other parts of the climate system over space and time. A climate model of any
complexity level will always have to abandon to resolve a certain range of small
scales. It must cut off the resolved part of variability somewhere at the high frequencies and wavenumbers and we must look for a slow manifold of solutions.
In geophysical fluid dynamics two concepts are employed to handle the cut-off
procedure: filtering and averaging. Filtering eliminates some part of variability by
analytical treatment of the equations of motion with the aim to derive equations
which describe a slow manifold of solutions. In contrast, averaging is a brute force
action: defining cut-off scales for space and time any field ϕ(X, t) is split into a
mean ϕ̄(X, t) over the subscale range, and the deviation ϕ0 (X, t) (the turbulent
component). Then, equations are derived for the mean fields by averaging the original equations (named Reynolds averaging after O. Reynolds). Equations for higher
order moments of ϕ0 are considered as well to close the system. To simplify the
work arising from the non-commutativity of averaging and differential operators,
the averaging procedure is frequently
formulated in terms of an ensemble of states
R
ϕ(X, t; λ) such that ϕ̄(X, t) = dP (λ)ϕ(X, t; λ) is the expectation with respect
to a probability measure dP (λ) and ϕ0 = ϕ − ϕ̄ is the deviation of a particular
realization.
As a consequence of nonlinearity the averaged equations are not closed: the
advection terms introduce divergences of fluxes V 0 ϕ0 supported by the motion in
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the subrange of scales. These Reynolds fluxes override the molecular fluxes by far
(except in thin layers on the fluid boundary) and the latter are usually neglected.
For ocean and atmosphere circulation models various elaborate closure schemes
have been worked out to relate the Reynolds fluxes to resolved fields. Here, we
shall only consider the simplest one: all Reynolds fluxes will be expressed by a
diffusive parameterization,
V 0 ϕ0 = −D · ∇ϕ̄,
(10)
with a diagonal diffusion tensor D = diag(Dh , Dh , Dv ). We omit the overbar of
the mean fields in the following.
An obvious way to eliminate sound waves from the system is to consider
the fluid as incompressible, i.e. to ignore the pressure dependence in the equation
of state (4). Geophysical fluid dynamicists have less stringent approximations, the
anelastic and the Boussinesq approximation. The density and pressure fields are expressed as a perturbation ρ, π about a hydrostatically balanced state3 %r (z), pr (z)
such that % = %r (z) + ρ, p = pr (z) + π and dpr /dz = −g%r and pr (z = 0) = 0.
For wave and QG problems (see sections 3.1 and 3.2 below) %r (z) is the horizontal
mean of density in the area of interest, or some standard profile. It is associated
with some θr (z) and mr (z) such that %r (z) = G(θr (z), mr (z), pr (z)). In models
which should predicted the complete stratification θr and mr (but not %r ) are
taken constant. The perturbation fields are generally small compared to the reference state variables.
Apparently, the pressure pr (z) is – together with the corresponding gravity
force g%r (z) – inactive in the momentum equations, they may there be eliminated.
Sound waves are filtered by realizing that the time rate of change of density ρ due
to diabatic effects and compressibility is much smaller than that due to change of
volume (given by the flow divergence). In the anelastic approximation the mass
conservation (6) is then replaced by
∇ · (%r V ) = %r ∇ · V + w
d%r
= 0,
dz
(11)
where w is the vertical component of the velocity vector V . Notice that the equation of state – equation (4) is now expressed by the perturbation density ρ – still
describes the complete compressibility of the medium. Furthermore, the density %
3 The
coordinate system is chosen with z-direction parallel and opposite to the gravity acceleration
vector g and z = 0 at the mean sea level. We will use ’horizontal’ coordinates λ and φ in a
spherical coordinate system attached to the earth; λ is longitude, φ is latitude. In the β-plane
approximation used below these spherical coordinates are then approximated as local Cartesian
coordinates by dx = adφ, dy = a cos φ0 dλ where a is the earth radius and φ0 the reference
latitude.
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as factor in the inertial terms (all terms on the lhs of (5) to (8)) is replaced by the
reference density %r .
There is a suite of further approximations which finally casts the equations
(4) to (8) into the form representing the large-scale oceanic and atmospheric flow
in climate models. These include:
• the hydrostatic approximation which realizes that pressure and gravity
forces approximately balance in the vertical (not only for the reference
fields but for the perturbation fields as well),
• the traditional approximation which drops all forces arising from the meridional component Ω cos φ of the angular velocity Ω = (0, Ω cos φ, Ω sin φ)
of the earth.
Hiding the Reynolds fluxes and other sources in Φ, Γm and Γθ , the equation of
motion for ocean and atmosphere then become in the anelastic approximation
Du
Dt
∂π
∂z
∂
(%r w)
∂z
Dm
%r
Dt
T Dθ
%r cp
θ Dt
ρ
∇ · (%r u) +
1
∇π + Φ
%r
=
−f k × u −
(12)
=
−gρ
(13)
=
0
(14)
=
Γm
(15)
=
Γθ
(16)
=
G(θ, m, pr + π) − G(θr , mr , pr ).
(17)
Here u is the horizontal and w the vertical velocity, k is a vertical unit vector,
f = 2Ωsinφ is the Coriolis frequency, and ∇ is the horizontal gradient or divergence operator. These equations are referred to as shallow water equations; when
expressed in spherical coordinates of the earth and the radius is taken constant in
the metric coefficients they are called primitive equations. Apart for the complete
elimination of sound waves the shallow water system also deforms the kinematics
of high frequency gravity waves.
The set (12) to (17) is one of many ways to represent the evolution equations of the atmosphere. Frequently, the complete mass conservation (6) is used
instead of (14) – in particular in numerical models where the analytical convenience of simple equations is not needed – which takes into account an incomplete
filtering of sound waves (the hydrostatic approximation alone filters only vertically
propagating sound waves).
The last term in (11) is of order gH/c2r where cr is the speed of sound of the
reference profile and H the depth of the fluid. The Boussinesq approximation is
applied to the ocean where we find gH/c2r ¿ 1. It omits therefore the %r -term in
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(11), arriving at ∇ · V = 0, and replaces % in the inertial terms by a constant %0
since density varies only little in the ocean. In addition the perturbation pressure
is omitted from the equation of state (17). The Boussinesq equations then become
Du
Dt
∂
(π/%0 )
∂z
∂w
∇·u+
∂z
Dm
%0
Dt
T Dθ
%0 cp
θ Dt
ρ
π
+Φ
%0
=
−f k × u − ∇
(18)
=
−gρ/%0
(19)
=
0
(20)
=
Γm
(21)
=
Γθ
(22)
=
G(θ, m, pr ) − G(θr , mr , pr ).
(23)
The most important contributions to Φ, Γm and Γθ for oceanic or atmospheric
flows arise from Reynolds fluxes of momentum, partial mass and heat in both media
– generally expressed by diffusive laws – and additionally from the radiative flux
and phase transitions of water in the atmosphere, thus
Φ
=
Γm
=
Γθ
=
∂
∂u
Av
∂z
∂z
∂
∂m
∇ · (Kh ∇m) +
Kv
+ Cm /(%r cp )
∂z
∂z
·
¸
∂
∂θ
%r cp ∇ · (Kh ∇θ) +
Kv
− ∇ · J rad + Λm Cm .
∂z
∂z
∇ · (Ah ∇u) +
(24)
In the ocean the divergence of the radiative flux vector J rad may be neglected
below the top few meters. The terms involving Cm appear only in the atmospheric
balances, they describe the effect of evaporation e and condensation c of water
with Cq = e − c. Furthermore, Λq is the latent heat of evaporation. There is no
source of salt in the ocean so that CS ≡ 0. In the simplest form the (eddy) diffusion
coefficients Ah , Av , Kh and Kv are taken constant.
Finally, we should realize that – due to the hydrostatic approximation – the
equations (12) to (17) do not describe vertical convection which occurs in the ocean
by increasing the buoyancy by cooling or evaporation and in the atmosphere by
decreasing the buoyancy by heating the air. For the latter case meteorologists
have developed complex convection parameterizations whereas the oceanic case is
treated quite simple by taking very large vertical diffusion coefficients to mimic
the increased vertical mixing resulting from unstable stratification.
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2.2. Coupling of ocean and atmosphere
The boundary conditions of each medium must express the physical requirement
of continuity of the fluxes of momentum, partial masses and internal energy across
(and normal to) the boundaries. For instance, the net vertical heat flux leaving
the atmosphere at the air–sea interface must be taken up by the advective and
diffusive fluxes of heat in the ocean. We will describe the simple physical ideas of
parameterization of boundary fluxes (for details cf. Gill 1982, Peixoto and Oort
1992).
If the topography of the sea surface is ignored – for simplicity we make this
approximation in this section – the continuity of the heat flux at the interface
at z = 0 of ocean and atmosphere would be expressed by Jθ (z = 0+ ) = Jθ (z =
0− ) where Jθ is the total vertical flux of internal energy. The simple diffusive
parameterization of Jθ and the other fluxes in (24) are, however, not valid in
the proximity of the air-sea interface as gradients of properties may become very
small due to the action of enhanced turbulence. Meteorologists have developed
alternative and more accurate parameterizations of surface fluxes in terms of ’bulk
formulae’. Observations have shown that vertical fluxes of momentum, matter
and energy are constant within a shallow layer of a few meters above the surface
and empirical laws have been elaborated to relate these fluxes to the values of
velocity, partial masses and temperature at the upper boundary of this ’constant
flux layer’ (the standard level is 10 meters height) and the corresponding sea
surface properties.
The conductive heat flux QH is parameterized by the difference of surface air
and water temperature, and a similar relation is taken for the rate of evaporation
E,
QH = %air cp CH Uair (θs − θair )
E = %air CE Uair (qs − qair ),
(25)
with dimensionless coefficients CH and CE of order 10−3 . The variables θair , qair
and Uair are the air temperature, specific humidity and wind speed, taken at the
standard level, E is the rate of evaporation/condensation (in kg water vapor per m2
and s), and qs the saturation value of humidity at temperature θs . The momentum
flux is parameterized by a drag law relating the tangential surface stress – the wind
stress – to the 10 m wind speed in the form
τ 0 = %air CW Uair uair ,
(26)
again with a drag coefficient CW ∼ 10−3 .
The only driving of the climate system occurs by the radiative heat flux
coming from the sun and entering the atmosphere at its outer edge with a value
of S0 = 1372 Wm−2 , the solar ’constant’ which is, however, not constant on long
time scales because of changing orbital parameters of the earth, and on small time
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scales because of changing solar activity. The heat flux from the earth interior is
negligible. At the interface of the atmosphere and the ocean, as well as atmosphere–
land, heat is exchanged by radiation, i.e. short-wave solar radiation and long-wave
radiation. The latter is determined by the surface temperature and thus mainly
in the infrared range. Furthermore, there is heat loss associated with evaporation
(the ’latent’ heat flux QL = Λq E) and by heat conduction (the ’sensible’ heat flux
QH ). Above the constant flux layer and below it in the ocean the fluxes are carried
further as parameterized by the diffusive approximations4 . The sum of the above
described heat fluxes has to match the oceanic diffusive heat flux at sea surface,
¸
·
∂θ
= −QSW (1 − αs ) + QLW + QL + QH .
− %cp Kv
∂z ocean
(27)
A similar relation holds for the atmospheric flux above the constant flux layer.
Here, QSW is the incident energy flux of short-wave radiation (computed from a
radiation model which is an essential part of a full climate model, see section 4.2),
αs is the sea surface albedo, QLW = ²σTs4 follows from the Stefan-Boltzmann law
(² is the emissivity of the surface, σ the Stefan-Boltzmann constant and Ts the sea
surface temperature).
The vapor entering the atmosphere by evaporation is carried further by turbulence so that
·
∂q
− %Kv
∂z
¸
= E.
(28)
atm
Exchange of other partial masses is irrelevant, in particular the total (diffusive
plus advective) flux of salt through the sea surface is assumed to vanish,
·
¸
∂S
− %Kv
= S(P − E).
∂z ocean
(29)
Here, P is the rate of precipitation (in kg water per m2 and second and (P − E)
equals the total mass flux entering the ocean from the atmosphere. Notice that (29)
expresses the vanishing of the sum of the advective and diffusive salt fluxes across
the air-sea interface. In the ocean the stress exerted by the wind is transferred
from the surface by turbulent diffusion, thus
·
%Av
4 For
∂u
∂z
¸
= %air CW Uair uair .
(30)
ocean
simplicity we take the diffusive parameterization for the oceanic fluxes in this section. A
flavor of the complex physics of the oceanic side of the air-sea interface is discussed in section
4.3.
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At the upper boundary of the atmosphere and the bottom of the ocean the
usual conditions assume the vanishing of the normal fluxes of partial masses and
heat. The requirement of zero flux of total mass implies that the normal velocity
vanishes at the ocean bottom. There is a suite of differing stress conditions, with
the limiting cases of no-slip (i.e. vanishing of the tangential velocity) and free-slip
(i.e. vanishing of the tangential stress).
2.3. Building a climate model
A complete climate model needs more than ocean and atmosphere modules. As
mentioned above, a radiation model is needed which calculates the short- and longwave radiation field in the atmosphere from the incoming solar flux at the top of
the atmosphere. A sea-ice module is needed to simulated the freezing/melting and
storage of frozen water as well as the transport of sea ice. Boundary conditions
over land surfaces and sea ice are required, possibly including a hydrological model
which organizes water storage on land and transport by rivers into the sea. For
long-term climate simulations the building and decay of ice-sheets must be included. Various empirical parameters have to be specified, as e.g. the albedo of the
land and sea ice surface, turbulent diffusivities and exchange coefficients. One even
has demand for model components of the terrestrial and marine ecosystems, atmospheric chemistry and ocean biogeochemical tracers. A complete climate model
should ultimately even include interactions between socioeconomic variables and
climate.
Climate physicists have developed a wide suite of techniques to circumvent
the enormous problems of treating the complete system. They
• use stand-alone models of the compartments of the climate system,
• simplify the dynamics,
• decrease the degrees of freedom
to arrive at manageable and understandable systems. In the following we find
various examples of such undertaking.
Further reading for section 2: Gill (1982), Müller and Willebrand (1986), Washington and Parkinson (1986), Peixoto and Oort (1992), Trenberth (1992), Olbers
et al. (1999)
3. Reduced physics equations
A dynamical system governed by an equation like (1) can be transformed to a
state space in which the linear part of the evolution operator, i.e. B −1 L for (1),
is diagonal. For not too nonlinear fluid systems this implies the isolation of the
internal time scales appearing in the system’s evolution, which mean a classification
of the wave branches. The following sections are largely dedicated to this problem.
14
3.1. The wave branches
In wave theory of fluid motions the balance of partial mass and potential temperature are conveniently combined into a balance of buoyancy, determined from
(15) to (17). Taking for simplicity the Boussinesq approximation the buoyancy is
b = −gρ/%0 . Separating out nonlinear advection terms, the linearized equations of
motion become
∂u
∂p
− fv +
= Fu
∂t
∂x
∂v
∂p
+ fu +
= Fv
∂t
∂y
∂b
+ wN 2 = Gb
∂t
∂p
−b=0
∂z
∂w
∇·u+
= 0,
∂z
(31)
(32)
(33)
(34)
(35)
where we have absorbed the constant density %0 in the pressure5 . The terms on
the rhs contain the nonlinearities and the source/sink terms6 in the corresponding
equations (12) to (16), e.g. (F u , F v ) = Φ/%0 −u·∇u−w∂u/∂z. The Brunt-Väisälä
frequency N (z), defined by
·
¸
g d%r
dθr
g2
dSr
N =−
− 2 = g α∗
− β∗
,
%0 dz
cr
dz
dz
2
(36)
is the only relic of the stratification of the reference density field (α∗ and β∗ are
the coefficients of thermal and haline expansion when the equation of state is
expressed in terms of potential temperature). Atmosphere and ocean are wave
guides because of the vertical boundaries and the mean stratification entering the
theory via N (z). For purpose of demonstration we will consider here the ocean case
with the kinematic boundary conditions Dζ/Dt − w = E − P at the sea surface
z = ζ, and w = −u · ∇h at the bottom z = −h, and the dynamic boundary
condition p = patm at z = ζ. Here, the rate of evaporation minus precipitation,
E − P , and the atmospheric pressure patm enter as external forcing. We expand
these conditions about the mean sea surface z = 0 and the mean bottom z = −H
and get
5 We
use the Boussinesq form of the scaled pressure π/%0 → p and density ρ/%0 → ρ in the rest
of this paper. Also stresses and surface mass fluxes will be scaled by %0 so that stresses and
pressure are measured in units of m2 s−2 and surface mass fluxes in ms−1
6 All source/sink terms - written in calligraphic type - appearing on the rhs in the linearized
equations in this section are given in the appendix.
15
∂ζ
−w
∂t
p − gζ
w
=
Z
at z = 0
(37)
=
=
P
W
at z = 0
at z = −H,
(38)
(39)
where Z, P, W contain the forcing terms and the nonlinear terms arising in this
expansion (see appendix). Notice that H is constant.
The wave state is described by a 3-d state vector. Taking (u, v, p) as state
vector the remaining fields follow from diagnostic equations: (34) determines b,
(35) together with the kinematic bottom boundary condition determines w, and
(38) determines ζ as functionals of (u, v, p). A prognostic equation for the pressure
to supplement (31) and (32) is obtained from (33), (34) and (37) to (39). After
some mathematical work one arrives at
∂p
+ M ∇ · u = Q,
∂t
(40)
where the operator
Z
0
M =g
Z
dz 00 +
−H
0
Z
dz 0 N 2 (z 0 )
z
z0
dz 00
(41)
−H
acts only on the vertical structure. The eigenvalue problem M ϕ = c2n ϕ is of the
standard Liouville form (a differential form with appropriate boundary conditions
is easily derived). There is a discrete spectrum (c2n , n = 0, 1, 2, 3 · · · ) with an
orthogonal and complete set of eigenfunctions ϕn (z). The eigenvalue c2n determines
the n-th Rossby radius λn√= cn /f . The eigenvalues should be ordered according to
c0 > c1 > · · · , then λ0 ≈ gH/f
√ (this is the barotropic mode ϕ0 which is almost
vertically constant) and λ1 ≈ g 0 H/f with the ’reduced gravity’ g 0 = g∆%r /%0
(this is the first baroclinic mode ϕ1 which has one zero in the vertical, ∆%r is the
vertical range of %r ). It is found that the barotropic Rossby radius λ0 exceeds the
baroclinic radii λi , i ≥ 1 by far since g 0 ¿ g.
3.1.1. Midlatitude waves For midlatitude waves it turns out that the projection of the prognostic equations (31), (32) and (40) onto the wave branches requires
to form the divergence δ and vorticity η of the horizontal momentum balances,
and therefore
δ=
∂u ∂v
+
∂x ∂y
and
η=
∂v
∂u
−
∂x ∂y
(42)
16
are used to replace7 u = ∇−2 (∂δ/∂x − ∂η/∂y), v = ∇−2 (∂η/∂x + ∂δ/∂y). The
evolution equations become
∂η
+ f δ + βv
∂t
∂δ
− f η + βu + ∇2 p
∂t
∂p
+ Mδ
∂t
=
=
=
∂F v
∂F u
−
∂x
∂y
∂F u
∂F v
+
∂x
∂y
Q,
(43)
(44)
(45)
where β = df /dy arises from the differential rotation due to the change of the
Coriolis parameter f with latitude8 . Projection onto a scalar wave equation is
easily done for the so called f -plane case with uniform rotation where f = f0 is
considered constant and β=0. Then, for the pressure,
·
·
¸
¸
∂ ∂2
∂2Q
∂D
2
2
2
+ f0 (1 − M∇ ) p =
+ f0 M
− f0 C
∂t ∂t2
∂t2
∂t
(46)
with M = M/f02 is obtained. Plane wave solutions p ∼ ϕ(z) exp i(K · x − ωt) with
K = (k, `) are found for the unforced equation with three branches (i.e. different
£
¤1/2
(1,2)
= ±f0 1 + (Kλn )2
(the gravity
wave types). The dispersion relations are ωn
(3)
branches) and ωn = 0 (the geostrophic branch). The latter is degenerated on the
f -plane: it describes a steady geostrophically balanced current where Coriolis and
pressure forces balance, −f0 v = −∂p/∂x and f0 u = −∂p/∂y. The degeneracy is
relieved on the β-plane where both f and β are retained and considered constant
in the above equations. This approximation can be justified by a proper expansion
into various small parameters (see QG approximation in the next section); here we
simply assume βL/f ¿ 1 where L is a typical horizontal scale, and ω ¿ f . The
first order correction of the geostrophic branch can then be derived by extracting
the geostrophic balances from (31) and (32), evaluating the geostrophic vorticity
as η = (1/f0 )∇2 p and using (43) and (45) to get
¢
∂ ¡
∂p
1 − M∇2 p − βM
= f0 C.
∂t
∂x
(47)
£
¤
(3)
We deduce ωn = −βk/ k 2 + `2 + λ−2
which is the dispersion relation of linear
n
planetary Rossby waves. Another way to filter out gravity waves is to neglect the
tendency in the divergence equation (44).
7 The use of the inverse of the Laplace operator ∇2 in this section and other operators involving
∇2 is the sloppy notation of a physicist.
8 We use here the β-plane approximation which projects the equations onto a plane tangent to the
earth at a central latitude φ0 but has a nonuniform rotation: the Coriolis parameter is expanded
as f = f0 + βy with f0 = 2Ω sin φ0 , β = (2Ω/a) cos φ0 and x, y are Cartesian coordinates.
17
The separation of the wave spectrum of motions is not completed with the
above analysis: (46) describes all wave branches though one is degenerate while
(47) yields the correct form of the geostrophic wave response. A complete separation into the wave branches requires a proper diagonalization of the linear matrix
operator appearing in the system (43) to (45). With the same approximations
used above to get (47) this goal can strictly be achieved. Let us consider the wave
evolution equations, written in terms of the state vector
ψ1 = δ
ψ2 = η
ψ3 =
1 2
∇ p.
f0
(48)
a = 1, 2, 3
(49)
It is found in the form9
∂ψa
+ i(Hab + Bab )ψb = qa
∂t
where qa derives from the above source terms and operators

Hab
0
= −if0  1
M∇2

−1 1
0 0 
0 0

Bab
∂/∂x −∂/∂y
= −iβ∇−2  ∂/∂y ∂/∂x
0
0

0
0  (50)
0
appear. Because Bab /Hab = O(βL/f0 ) (where L is a typical length scale) and
Hab is easily diagonalized we will approach the problem by expansion in terms of
βL/f0 . To lowest order in βL/f0 the eigenvalue problem of the evolution operator
is Hab Rsb = Ω(s) Rsa (no s-summation) which is solved by
¡
¢1/2
Ω(s) = sΩ = sf0 1 − M∇2
s = ±, 0
Rs2 = 1
Rs1 = isΩ/f0
(51)
Rs3 = 1 − (sΩ/f0 )2
The branches s = ± describe gravity waves and s = 0 is the steady geostrophic
flow. The Rsa achieve the representation of the state vector ψa in terms of the wave
branch vector ψ s , i.e. ψa = Rsa ψ s . The inverse transformation is achieved by left
eigenvectors of the evolution operator, P sa Hab = Ω(s) P sb . which are found to be
£ ±¤
Pa
£ 0¤
Pa
9 Summation
=
=
1
T
(Ω/f0 )−2 [−iΩ/f0 , 1, −1]
2
−2
(Ω/f0 )
£
¤T
0, (Ω/f0 )2 − 1, 1 .
over repeated lower or upper indices is used in the following.
(52)
18
Projection onto the wave state is thus performed by ψ s = P sa ψa . The representation and projection operators are mutually orthogonal, P sa Rta = δ st , Rsa P sb = δab .
s
Finally, the first order correction of the eigenvalue problem is Bab Rsb + Hab R̃b =
s
Ω(s) R̃as + Ω̃(s) Ra where the Bab and the tilde terms are first order in βL/f0 .
It yields corrections to the representation operator (which we do not use) and
Ω̃(s) = P sa Bab Rsb (no s-summation). For the geostrophic branch we thus get
Ω̃(0) =
iβM
∂
1 − M∇2 ∂x
(53)
as correction to the geostrophic evolution operator. Denoting from now Ω̃(0) by
Ω(0) the wave evolution is then governed by the three diagonal problems
∂ψ s
+ iΩ(s) ψ s = q s
∂t
s = ±, 0.
(54)
These abstract equations of wave evolution can be cast into a more familiar form.
It can easily be verified that ψ = ∇−2 ψ 0 is the streamfunction of the geostrophic
part of the motion and f0 ψ is the corresponding pressure part. From (54) we find
¢
∂ ¡ 2
∂ψ
∇ − M−1 ψ + β
=C
∂t
∂x
(55)
for the s = 0-equation, describing the evolution of the Rossby wave branch. The
gravity wave branch fields are conjugate to each other, ψ + = (ψ − )∗ . Introducing
real and imaginary parts by ψ + + ψ − = $ and ψ + − ψ − = γ the evolution of the
gravity branch is found as
∂2$
+ f02 (1 − M∇2 )$
∂t2
µ
=
µ
2
∂ γ
+ f02 (1 − M∇2 )γ
∂t2
Ω
f0
=
−
¶−2
Ω
f0
∂C
− f0 D
∂t
¶−1 ½
¾
(56)
∂D
+ f0 C .
∂t
The gravity wave pressure is given by f0 M$ and the total pressure thus by p =
f0 (M$ + ψ).
For practical applications the branch amplitude fields ψ and $, γ should
be expanded in the eigenfunctions of M, which simply replaces M by λ2n and
e.g. the Rossby wave field ψ(x, y, z, t) by the modal amplitudes ψn (x, y, t). Kinematic conditions of zero normal velocity, n · u = 0, must be satisfied at solid
19
boundaries. Separate equations for long or short waves may be generated by assuming M∇2 ∼ (Kλ)2 ¿ 1 for length scales which are large compared to the
respective Rossby radius, or M∇2 ∼ (Kλ)2 À 1 for the opposing case.
3.1.2. Equatorial waves At the equator the Coriolis parameter vanishes and
a special wave theory must be developed. The equatorial β-plane uses f = βy with
constant β = 2Ω/a in (31), (32) and (40). The system supports gravity and Rossby
type waves which are trapped vertically as in midlatitudes but also meridionally.
Equations for the wave branches are obtained by vertical decomposition (replace√
Scaling of time by (2βc)1/2 = 2c/λe
ment of M by c2n ; we will omit the index n).
√
and space coordinates by (c/2β)1/2 = λe / 2 and use of the state vector (q, v, r),
with q = p/c+u and r = p/c−u, is convenient. Here λe = (c/β)1/2 is the equatorial
Rossby radius (of vertical mode n). Then
µ
¶
∂
∂
+
q + E +v = F u + Q = F
∂t ∂x
¢
∂v 1 ¡ −
+
E q + E + r = 2F v = H
2
µ ∂t ¶
∂
∂
−
r + E − v = −F u + Q = G
∂t ∂x
(57)
(58)
(59)
with
E± =
1
∂
∓ y.
∂y 2
(60)
Notice the following properties:
1
1
∂2
1
= E −E + + =
− y2
2
2
∂y 2
4
√
1 2
` = 0, 1, 2, · · ·
D` (y) = 2−`/2 e− 4 y H` (y/ 2)
E = E +E − −
1
ED` = −(` + )D`
2
E + D` = −D`+1
(61)
E − D` = `D`−1 .
The D` (y) and H` (y) are parabolic cylinder functions and Hermite polynomials,
respectively. The operators E ± thus excite or annihilate one quantum of the meridional mode index `. Apparently, to satisfy (57) to (58), we must take v ∼ D` , r ∼
D`−1 , q ∼ D`+1 . Wave solution (q, v, r) ∼ (q, vE − , r(E − )2 )D` (y) exp i(kx − ωt)
are obtained which are oscillatory in a band of width of the Rossby radius about the
equator but decay exponentially away from there. These are ω0 = k (Kelvin wave),
ω1 = k/2+(k 2 /4+1/2)1/2 (Yanai wave) and for ` ≥ 2 we have ω 2 −k 2 −(1/2)k/ω =
` − 1/2 with approximate solutions ω = ±(k 2 + ` + 1/2)1/2 (two gravity waves) and
20
ω` = −k/(2k 2 +2`−1) (Rossby wave). The corresponding eigenvectors (q, v, r) are
easily evaluated. Notice that Kelvin and Yanai waves travel eastward and Rossby
waves westward while gravity waves exist for both directions.
There is a simple procedure to filter out the gravity and Yanai waves from
the system: this is performed by omitting the time derivative in (58). Eliminating
v from (57) and (59) yields a prognostic equation
E
∂ϕ 1 ∂ϕ
−
=
∂t
2 ∂x
µ
¶
µ
¶
1
1
∂
1 ∂
= E+
E − F − (E − )E + G −
+E
H
2
2
2 ∂t
∂x
(62)
for the equatorial potential vorticity variable ϕ = E − q − E + r = yp + 2∂u/∂y.
Because ϕ ≡ 0 for the Kelvin wave, only long Rossby waves (with ω = −k/(2`−1))
are retained in (62). The Kelvin wave has v = r ≡ 0 and is therefore described by
a simple equation for the amplitude q0 (x, t),
µ
∂
∂
+
∂t ∂x
¶
q0 = F0 .
(63)
By projecting onto the meridional modes various problems of trapped equatorial wave motion can be formulated. If the system is zonally unbounded or periodic the waves propagate independently. Interesting problems arise in a zonally
bounded wave guide (e.g. the Pacific Ocean) since wave reflection at zonal boundaries couple the wave branches. The reflection process is complicated. Depending
on the frequency it may involve a large number of modes of long and short waves
(at the eastern boundary the solution even requires further coastal trapped waves).
We abandon an exact treatment to give a simple example: assume that the only
waves present are Kelvin waves (traveling eastward with speed ω/k = 1) and the
long Rossby waves (traveling westward with speed ω/k = −1/3) of mode number
` = 2 and consider an ocean contained zonally in the interval x = 0 and x = xe .
The Rossby wave amplitude ϕ2 (x, t) and the Kelvin wave amplitude q0 (x, t) are
coupled by requiring zero flux of mass through the boundaries. Because the zonal
velocity is given by u = (q − r)/2 we get the constraint (assuming H = 0 in (58))
1
( d3 + d1 )ϕ2 + d0 q0 = 0
3
at x = 0, xe ,
(64)
√
where d` = 2` π`! is the normalization constant of D` . The system is then governed by (62), projected on ` = 2, and (63), they are coupled by (64). These waves
dominate the ocean component of ENSO (see the sections 4.4 and 5.6).
21
Further reading: Hasselmann (1976), Gill (1982), Olbers (1986), Frankignoul et
al. (1997)
3.2. The quasigeostrophic branch
The evolution equation (55) of Rossby waves is the linearized version of the quasigeostrophic (QG) potential vorticity equation10
dQ
∂Q
∂ Gb
=
+ u · ∇Q = curl Φ + f0
,
dt
∂t
∂z N 2
(65)
which states the conservation of the QG potential vorticity11
Q = ∇2 Ψ +
∂ f02 ∂Ψ
+ f0 + βy
∂z N 2 ∂z
(66)
along the path of the fluid elements (the path is projected onto the horizontal
plane because vertical advection can be neglected for QG motions). The advection
velocity is given by the geostrophic part of the current, u = k × ∇Ψ where Ψ =
p/f0 is the QG streamfunction. The constituents of the QG potential vorticity (66)
are identified with the relative vorticity of the horizontal velocity, the stretching
vorticity and the planetary vorticity. Eq. (65) may be derived from (55) by noting
that Q = −M−1 (1 − M∇2 )ψ 0 + f0 + βy. Indeed, extracting the geostrophic
advection term and the frictional and buoyancy sources from the source term
on the rhs of (55) (and neglecting the remaining terms) we get (65). A more
precise derivation starts with (31) to (35) in spherical coordinates and performs
an expansion in various small parameters. The most important are the Rossby
number Ro = U/(2ΩL), the planetary scale ratio L/a, and the aspect ratio H/L
and Ekman numbers Ek = Ah /(2ΩL2 ) or Av /(2ΩH 2 ), where U, H, L are scales
of the horizontal velocity and vertical and horizontal lengths, Ω is the angular
velocity of the earth, and a is the radius of the earth. For QG theory all these
parameters are assumed to be of the same small magnitude. The theory is thus
restricted to a geostrophic horizontal flow (to zero order) with a small vertical
circulation (the vertical velocity w = −(f0 /N 2 )d(∂Ψ/∂z)/dt is first order) and a
slowly evolving geostrophic pressure field governed by (65) which is a first order
balance. Boundary conditions at top and bottom follow from (37) to (39), they
are expressed by
10 curl,
acting on a 2-d vector, is a sloppy notation for curl Φ = ∂Φ(y) /∂x − ∂Φ(x) /∂y.
theory is presented here for a Boussinesq fluid. For the anelastic approximation the stretching term in the potential vorticity has an extra factor 1/%r (z) in front of the first vertical derivative and a factor %r (z) behind.
11 QG
22
d ∂Ψ N 2 dΨ
Gb
N 2 dpatm
+
= 2−
dt ∂z
g dt
N
gf0 dt
at z = 0
(67)
2
d ∂Ψ
N
=
u · ∇h
dt ∂z
f0
at z = −h.
The condition at the upper boundary is expanded about the mean sea surface
z = 0, the condition at the bottom is exact (though ∇h should be small of order
of the Rossby number).
QG motions are a projection of the complete flow onto a slow manifold.
The equations could be supplemented by the so far neglected sources arising from
the fast manifold, basically the gravity wave field. It should also be emphasized
that more elaborate theories exist for slow manifolds. They arise from a different
ordering of the small parameters mentioned above.
Further reading: Pedlosky (1986)
3.3. The geostrophic branch
The most important nonlinearity in the dynamics of large scale flow stems from
the advection of heat and partial mass. The balance of momentum is well described
by the geostrophic and hydrostatic equations as utilized in the quasigeostrophic
theory. While this theory aims to describe perturbations on a given background,
a theory of the establishment of the oceanic stratification must consider the complete balance of advection and diffusion for temperature and partial mass. In the
atmosphere convection and heating by phase transitions and by radiation is of
overwhelming importance. Theories of the thermohaline stratification of the ocean
in general simplify the problem by using the perturbation density ρ as thermohaline variable and ignore its compressibility. Expressed in spherical coordinates we
are dealing then with the planetary geostrophic equations (in Boussinesq form)
−f v
=
fu
=
pz
1
[uλ + (v cos φ)φ ] + wz
cos φ
u
v
ρt +
ρλ + ρφ + wρz
a cos φ
a
=
1
pλ
a cos φ
1
− pφ
a
−gρ
=
0
(71)
=
(Kv ρz )z + ∇ · (Kh ∇ρ),
(72)
−
(68)
(69)
(70)
23
where we use indices to denote partial derivatives. As noted by Needler (1967)
these equations may be reduced to a single nonlinear differential equation for the
pressure. A simpler form was found by Welander (1971), defining the M -function
Zz
p dz 0 + M0 (λ, φ),
M (λ, φ, z) =
z0
which allows to express all fields as partial derivatives of one variable,
p
=
Mz
u
=
−
1
Mφz
fa
1
ρ = − Mzz
g
1
v=
Mλz .
f a cos φ
(73)
The vertical velocity may as well be expressed by M : from (71) we find the planetary vorticity relation f wz = βv and thus, by integration,
w=
β
f
Z
z
v dz 0 + w0 =
0
β
f 2 a cos φ
(M − M0 )λ + w0 ,
(74)
Rλ
where w0 (λ, φ) = w(λ, φ, z0 ). Thus, with M0 = 2Ωa2 sin2 φ 0 w0 dλ0 , the thermohaline density equation (72) results in a nonlinear partial differential equation of
second degree and fourth order for M ,
a2 Ω sin 2φ Mzzt +
∂(Mz , Mzz )
+ cot φ Mλ Mzzz =
∂(λ, φ)
= a2 Ω sin 2φ [(Kv Mzzz )z + ∇ · (Kh ∇Mzz )] .
(75)
The M -equation is thought to describe the evolution of the oceanic thermocline
as response to pumping of water with surface characteristics ρs (λ, φ) = ρ(λ, φ, z0 )
to depth, at a rate given by the pumping velocity w0 . The level z0 is placed at the
bottom of the turbulent layer which is immediately influenced by wind and surface
wave breaking (roughly the upper 50 to 100 meters, see section 4.3). The pumping
velocity is then determined by the divergence of the wind-induced transport in
that layer, i.e. w0 is the Ekman pumping velocity
w0 = wE = curl
τ0
,
f
where τ 0 is the wind stress. Boundary conditions for (75) are thus
(76)
24
Mλz = 2Ωa2 sin2 ϕ wE (λ, ϕ)
and
Mzz = −gρs (λ, ϕ)
at z = z0
(77)
Mλ = 0
and
Mzzz = 0
at z = −H
assuming for simplicity a flat bottom. The last two conditions express the vanishing
of w and the buoyancy flux at the bottom.
Further reading: Pedlosky (1987), Salmon (1998)
3.4. Layer and reduced gravity models
Special treatment of the vertical dependence of field variables was demonstrated
in section 3.1 where we have used decomposition into vertical normal modes. Another popular projection of the overwhelmingly horizontally layered structure of
ocean and atmosphere is that of layer models. In the simplest concept the fluid
is considered as a stack of immiscible layers, each with a constant density12 %i .
The index i = 1(=top layer ), · · · , n(=bottom layer) identifies the layer of vertical
height di (x, t) with vertically constant horizontal velocity ui = ui (x, t). This can
be justified by the Taylor-Proudman theorem according to which vertical shears
∂ui /∂z are weak if the fluid is homogeneous, rapidly rotating and hydrostatic. The
pressure pi is evaluated from the hydrostatic balance (13)
P as sum of the masses in
the layers on top of the respective one so that ∇pi = g j<i %j ∇dj . The evolution
of the system is then governed by the conservation of momentum and mass, in the
flux form this is expressed by
∂
di ui + ∇ · (ui di ui ) + f k × di ui = −di ∇pi + τ i−1 − τ i + Ri
∂t
(78)
∂
di + ∇ · di ui = 0,
∂t
where τ i is the horizontal stress at the bottom of the i-th layer, τ 0 is the stress
at the surface of the fluid (the wind stress in case of the ocean), and Ri denotes
the divergence of lateral stresses, Ri = Ah ∇2 di ui for simple lateral eddy diffusion
of momentum, Ri = −²i−1 di−1 (ui − ui−1 ) − ²i di (ui − ui+1 ) for linear interfacial
friction where ²0 = 0 and un+1 = 0 for the bottom layer i = n. The coefficient
12 The
concept of immiscible layers should actually be based on an adiabatically conserved property such as potential temperature or potential density. The latter is defined as %θ = G(θ, m, p0 ),
i.e. the density taken for the parcel’s potential temperature and partial mass but at the reference
pressure p0 , using the equation of state outlined in section 2. Both properties, θ and %θ , are
conserved along streamlines if diabatic processes – such as diffusion, radiative heating etc – are
absent. In-situ density is not conserved but enters the calculation of the pressure force. This
conflict between in-situ and potential density is inherent for all layer models.
25
²n describes bottom friction. The simplest stratified model has two layers with
d1 = ζ + ξ, d2 = h − ξ where ζ is the elevation of the surface and z = −ξ and
z = −h give the positions of the layer interface and the bottom, respectively.
The ocean abyss has a very sluggish flow. Applied to the ocean circulation, the
assumption of a motionless deep layer is therefore a coarse but fairly acceptable
approximation in certain areas. It yields the reduced gravity model where u2 ≡
0, τ 2 ≡ 0 requires ∇p2 ≡ 0 or gζ = g 0 ξ with g 0 = g(%2 − %1 )/%1 (reduced gravity).
The resulting model
∂
u1 + u1 · ∇u1 + f k × u1 = −g 0 ∇ξ + τ 0 /d1 + R1
∂t
(79)
∂
ξ + ∇ · ξu1 = 0
∂t
is equivalent to the first vertical mode of the wave model (31), (32) and (40) if
these equations are supplemented by the nonlinear terms.
For frictionless conditions the conservation of the potential vorticity (ηi +
f )/di with relative vorticity ηi = ∂vi /∂x − ∂ui /∂y is easily proven from (78).
Straightforward QG perturbation theory yields the layered version of the QG potential vorticity theory. For a two-layer system we find
∂Qi
+ ui · ∇Qi = Fi /Hi
i = 1, 2
∂t
f0
Q 1 = ∇2 Ψ 1 −
[ζ − ξ] + f0 + βy
H1
f0
Q 2 = ∇2 Ψ 2 −
[ξ − b] + f0 + βy,
H2
(80)
where ζ = (f0 /g)Ψ1 , ξ = (f0 /g 0 )(Ψ2 −Ψ1 ) are the elevations of the surface and the
interface, respectively, and Ψi is the streamfunction (ui = k ×∇Ψi ). Furthermore,
H1 is the undisturbed thickness of the upper layer and H2 − b the thickness of
the lower layer with b(x) as topographic elevation. The forcing and friction are
contained in the source terms Fi . For forcing a QG ocean by wind stress one uses
F1 = curl τ 0 , linear bottom friction has the form F2 = −²2 H2 ∇2 Ψ2 . Additional
interfacial friction is needed in case that the model does not adequately resolve
the quasigeostrophic turbulence. Then a term −²1 H1 ∇2 (Ψ1 − Ψ2 ) must appear
in F1 and a corresponding term with opposite sign in F2 . The QG model is also
applied to the zonal atmospheric circulation. The domain is then zonally periodic
and the forcing specified by relaxation of the interface to a prescribed zonal flow
in form of meridional temperature or interface displacement ξR (y). In this case,
Fi = ∓f0 [ξ − ξR (y)] /tR .
Further reading: Pedlosky (1986), Wolff et al. (1991), Pavan and Held (1996)
26
4. Integrated models
Integrating any of the conservation equations, discussed in section 2, over any piece
of volume of the system yields a budget of the corresponding physical quantity ϕ
in that volume in terms of storage (rate of change of the content of ϕ in the volume), fluxes across the boundaries and sources in the interior. It is also meaningful
to consider budgets which apply to integration over certain spatial directions, say
the vertical direction or the horizontal domain. In general, one cannot expect that
such a reduction of the spatial degrees of freedom would yield a problem which
is mathematically well-posed for determination of the integrated state variables,
i.e. parameterizations must be considered to get a problem which is closed with
respect to integrated variables. The method of spatial integration (or averaging)
is nonetheless a frequently used crook to produce simplified models. The methodological similarity and overlapping with the low-order models discussed further
below is obvious.
4.1. Energy balance models and the Daisy World
The most popular and simplest climate model reflects the energy balance of the
earth in integrated form. The zero-dimensional model integrates the balance of
heat (essentially (16) with boundary conditions discussed in section 2.2) vertically
and laterally over the whole earth, such that the incoming and reflected solar
radiation must balance the outgoing infrared radiation,
c∗
∂T
1
= S0 (1 − αpl ) − ²∗ σT 4 .
∂t
4
(81)
Here, T is thought to represent the mean surface temperature and the heat capacity
c∗ = %cp H is adjusted to some global mean value, H is a measure of the vertical
extent of the climate system, S0 is the solar constant (downward solar flux at
the outer rim of the atmosphere), and αpl the albedo. In the radiative cooling the
Stefan-Boltzmann law is corrected by the factor ²∗ < 1 to account for the difference
between T and the radiative temperature of the earth, it can also be used to correct
for the greenhouse effect (then ²∗ ≈ 0.62). The model is easily extended to a onedimensional version by omitting the meridional integration, the energy balance
(81) then includes a term to represent the divergence of the meridional transport
of heat. These models have extensively been reviewed in the literature (e.g. North
1981, Ghil 1981, see also chapters by Fraedrich and Imkeller in this book).
The interesting physics and mathematical complexity enter the models via
feedback and nonlinearity implemented into the albedo dependence αpl (T ) on temperature (in the one-dimensional case the albedo depends on the latitude of the
ice extent and this depends on the local temperature). But the albedo would not
only change if the surface becomes ice-covered, it also changes with changing plant
cover. A very popular and philosophically stimulating model has been formulated
27
by Lovelock to illustrate the Gaia Hypothesis (cf. Watson and Lovelock 1983) according to which the biosystem on earth creates its own – eventually optimal –
climate.
Consider a planet covered partly with white daisies (fraction Fw ), black
daisies (fraction Fb ) and bare land (fraction F` ) such that F` + Fw + Fb = 1
and the albedo becomes αpl = α` F` + αw Fw + αb Fb . The areas change in time due
to growth and death of daisies. To put it more specific, we take Lotka–Volterra
dynamics to describe these processes,
∂Fw
∂t
=
−γFw + β(Tw )F` Fw
(82)
∂Fb
∂t
=
−γFb + β(Tb )F` Fb .
Here, γ is the death rate (it can be taken constant) and β the growth rate which
depend on the local temperature, e.g. a function with optimal growth at Topt is
µ
β(T ) = 1 −
T − Topt
T∗
¶2
(83)
but other more complicated forms could be taken. The time in (82) is scaled by the
value of the optimal growth rate. The energy balance is given by (81) but we include
additionally a lateral heat exchange Ni of each compartment i = w, b, ` with the
environment, thus Nw Fw + Nb Fb + N` F` = 0. The Ni must be related to other
variables in the problem. Lovelock considers the parameterization Ni = q ∗ (αi −αpl )
which yields
c
∂Ti
∂t
=
S(1 − αi ) − Ni − ²∗ σTi4 =
=
S(1 − αpl ) + q(αpl − αi ) − ²∗ σTi4
i = w, b, `,
(84)
where c takes into account the time scaling of (82), and S = S0 /4 and q =
q ∗ + S. The value of q controls the importance of the heat exchange: for q = 0
one gets a steady state where the heat exchange produces a uniform temperature,
for q = S one gets a local equilibrium with zero exchange, i.e. Ni = 0. Thus
0 ≤ q ≤ S is the interesting range and the stationary points of (82) and (84) are
easily determined. The thermostat property of the daisies can be demonstrated by
numerical integration. Typical parameter values of this model are γ = 0.3, Topt =
295.5 K, T∗ = 17.5 K, q = 0.2S.
The Daisy World could be used to bring more life into the higher dimensional
energy balance models. Other interesting dynamics are obtained by coupling to
28
an ice sheet module which, however requires one or two spatial dimensions (see
e.g. Källen et al. 1980).
Further reading: Lemke (1977), Lovelock (1989), Olbers et al. (1997)
4.2. A radiative-convective model of the atmosphere
The balance of the horizontally averaged internal energy of the atmosphere is
obtained from (16) and (24). For simplicity we take an atmosphere consisting
of dry air, thus ignoring phase transitions. Written in terms of the horizontally
averaged potential temperature θ(z, t) we get
%cp
∂θ
∂Jrad
∂
∂θ
=−
+
%cp Kv ,
∂t
∂z
∂z
∂z
(85)
Here, Jrad is the vertical flux of radiant energy. In the radiative-convective models
the diffusive term should lead to a very efficient mixing if the air column is convectively unstable and thus, it is generally assumed that Kv = 0 where ∂θ/∂z > 0
(stable stratification) and Kv 6= 0 with a very large value where ∂θ/∂z < 0 (unstable stratification).
The flux Jrad follows from a radiative transfer equation determining the radiant intensity Iν (ϕ, ϑ) at each frequency ν. The amount of radiant energy traversing
a unit area per unit time from the solid angle dω = sin ϑdϑdϕ in the frequency
interval (ν, ν + dν) is given by Iν dνdω. To obtain the flux Jrad the intensity has
to be integrated over frequencies and solid angle, after projection of the rays onto
the vertical direction,
Z
Z
∞
Jrad =
dν
0
Z
2π
dϕ
0
π
dϑ sin ϑ cos ϑ Iν .
(86)
0
We assume for simplicity that scattering can be neglected. The radiative transfer
equation then reduces to the Schwarzschild equation
cos ϑ
∂
Iν = %κν [−Iν + Bν (T )] ,
∂z
(87)
where κν is the absorption coefficient of the air (it is a sum over the cross sections
of all radiatively active gases weighted by their relative mass fractions in the
atmosphere) and Bν (T ) is Planck’s function of the intensity of black body radiation
at temperature T . Equation (87) describes the attenuation of the intensity along
the vertical direction due to absorption, and the gain of radiant energy due to
radiation from the heated air.
The temperature profile is then obtained by simultaneous solution of (85)
and (87), using Poisson’s formula θ = T (p0 /p)γ to relate the in-situ temperature
with the potential temperature (γ = R/cp = 2/7 where R is the gas constant and
29
p0 a constant reference pressure). The problem is highly complicated because the
absorption coefficient is a very complicated function of frequency, reflecting a vast
number of narrow absorption bands of the atmospheric gas constituents.
We consider a highly simplified radiation model, the grey atmosphere where
κν = κ is assumed constant. Integrating (87) over the frequencies of the long-wave
(infrared) radiant energy, the horizontal plane and then separately over the upper
part 0 < ϑ < π/2 and the lower part π/2 < ϑ < π the two-stream model is
obtained
∂F ↑
∂z
=
¤
£
β%κ −F ↑ + σT 4
=
£
¤
β%κ F ↓ − σT 4 .
(88)
∂F
∂z
↓
The factor β = 5/3 arises due to an approximation of the integrals (the problem
is not closed exactly), and F ↑ and F ↓ are the total upward and downward flux of
long-wave radiant energy, respectively, such that Jrad = F ↑ − F ↓ + FSW , where
the last summand is the net contribution from the solar (short-wave) radiation.
Boundary conditions for the two-stream model are easily given: at the top of the
atmosphere (z = ∞) the downward flux is zero and at the bottom (z = 0) the
upward flux is given by the infrared radiation of the surface, thus
F↓
=
0
=
σTs4
at z = ∞
(89)
F
↑
at z = 0,
where Ts = T (z = 0) is the surface temperature. The boundary conditions for the
heat balance (85) are simply Kv ∂θ/∂z = 0 at the top and (27), expressed here for
the atmospheric side,
−Kv %cp
∂θ
= F ↑ − F ↓ + FSW = σTs4 − F ↓ + FSW
∂z
at z = 0.
(90)
Finally, a model of the solar part must be specified. The simplest version assumes that FSW penetrates undiminished from top to bottom and thus FSW =
−(1 − αpl )S0 /4 where αpl is the planetary albedo. A more complex but still simple
model assumes an exponential decrease of the short-wave radiation, ∂FSW /∂z =
−κSW FSW , with upper boundary condition FSW (z = ∞) = −(1 − αpl )S0 /4. It
is easy to show that a temperature profile in equilibrium with the radiation must
30
yield zero net radiant energy flux at the top, thus F ↑ (z = ∞) = (1 − αpl )S0 /4.
Notice that the solar constant S0 is the only driving force of this model.
Further reading: Ramanathan and Coakley (1978), Salby (1996)
4.3. The ocean mixed layer
The limitation of applicability of the parameterizations (24) in the proximity of
the sea surface has been touched in section 2.2 where, for the atmospheric side,
the concept of the constant flux layer was introduced. Also oceanographers have
developed much more elaborate parameterizations of the fluxes in the oceanic layer
adjacent to the sea surface. Diffusive parameterizations do not work there because
the layer is very well mixed by the action of the wind and breaking surface waves.
Nevertheless, heat and substances pass this layer to enter the ocean interior.
The structure of the oceanic near-surface layer is well represented by a completely mixed fluid of temperature T0 of vertical extent from the surface z = 0
to a depth z = −h, residing over stratified water which, immediately below the
mixed layer, has a thin sharp thermocline (a layer with strong vertical gradient
of temperature), followed by a more gradual decrease of temperature down to the
abyss. The structure is thus given by T = T0 (t) for 0 ≤ z ≤ −h(t), and a linear increase of T from T0 to T? between z = −h(t) and z = −h? to represent
the thermocline. The deep temperature T? must be specified in this model, the
value ∆ = h? − h is assumed constant and small, ∆ ¿ h. It will not enter the
model equations explicitly. The physics of the mixed layer model should determine
T0 (t) and h(t) from the specified surface heat flux and the characteristics of the
turbulence in that layer.
We step back to the Reynolds form of turbulent transports and concentrate
on the heat balance (a more extended version may consider the salt budget in
addition)
∂Q
∂T
=
,
∂t
∂z
(91)
where Q = −w0 T 0 is the turbulent flux of heat (apart from a factor %cp ). Evidently,
since T is constant in the mixed layer, Q(z) must be a linear function of depth,
Q(z) = Q0 +
z
(Q0 − Qh ),
h
(92)
where Q0 is the surface heat flux and Qh = Q(z = −h). The surface heat flux is a
prescribed forcing, the flux Qh at the mixed layer base will be parameterized in a
rather indirect way as shown below. Integrating (91) over the mixed layer we find
h
∂T0
= Q0 − Qh ,
∂t
(93)
31
whereas the integration over the intermediate layer −h < z < −h? yields approximately
(T0 − T? )
∂h
= Qh − Q? .
∂t
(94)
We assume here that Q? = Q(z = −h? ) is small, it could easily be retained as a
diffusive flux below the mixed layer. In practice, T0 − T? will always be positive.
Thus, according to (94), the mixed layer will deepen if the heat flux Qh is positive,
i.e. downward. In this case, fluid of temperature T? from below the mixed layer is
warmed to T0 and mixed (’entrained’) into the mixed layer. As we shall see below,
this mixing consumes turbulent energy, or – to be more specific – it can only take
place if turbulence energy is available for mixing fluid from below into the mixed
layer.
Now, the basic assumption of mixed layer physics is that Qh originates from
turbulent processes above and should therefore not become negative: there is no
cooling of fluid in the mixed layer from below and no ’unmixing’. Instead, if Qh
drops to zero, there is not enough turbulent energy to mix at all at the mixed
layer base. Mixing and deepening of the mixed layer then stops and a new mixed
layer depth is established at a shallower depth. Its depth is determined from Qh =
f ct[h, · · · ] = 0 where f ct[h, · · · ] is the parameterization of the flux Qh in terms
of h and other parameters which determine the mixing properties. This function
2
is found from the budget of turbulent kinetic energy T KE = u0 /2 in the mixed
layer, given by
∂F
∂
T KE =
− αgQ − ²,
∂t
∂z
(95)
where F = −(u0 2 /2 + p0 )w0 is the turbulent flux of T KE, ² is the dissipation of
T KE, and αgQ = −αgw0 T 0 = −w0 b0 is the exchange of T KE with the turbulent
potential energy (α is the coefficient of thermal expansion). Remember from section
3.1 that b0 = −gρ0 = αgT 0 is the buoyancy fluctuation (salinity is here neglected).
Lifting of fluid with density anomaly ρ0 > 0 increases the potential energy at the
expense of T KE, then w0 ρ0 > 0 and thus αgQ > 0, as incorporated in (95). The
T KE usually equilibrates within a few minutes and integration of (95) over the
mixed layer then yields, using (92),
h
0 = F0 − Fh − αg (Q0 + Qh ) −
2
Z
0
² dz.
(96)
−h
A few further assumptions (parameterizations) close the problem: Fh is assumed
small and neglected, F0 is related to the wind stress τ0 (which excites surface waves
which by breaking create T KE). By dimensional arguments one postulates F0 =
32
c|τ0 |3/2 with a dimensionless coefficient c of order unity. Finally, the dissipation
term must be positive and it is simply assumed that it ’eats’ away a certain fraction
of F0 (which is positive) and the term involving the surface heat flux Q0 . This is a
gain of T KE for cooling (Q0 < 0) because the potential energy of the cooled and
heavier fluid is converted to T KE. There is no eating from the Q0 -term in case of
heating. Hence, the dissipation term is expressed as
Z
0
h
² dz = r1 F0 + r2 αg (|Q0 | − Q0 )
4
−h
(97)
with 0 < r1 , r2 < 1. This relation, together with (96), finally leads to the required
parameterization of Qh in the form
i
h
hh
r2
r2
αg Qh = (1 − r1 )c|τ0 |3/2 − αg (1 − )Q0 + |Q0 | .
2
2
2
2
(98)
The recipe to get T0 (t) and h(t) is now as follows: τ0 and Q0 are specified
forcing terms and if Qh from (98) is positive the two functions T0 (t) and h(t)
follow from (93) and (94). If Qh from (98) becomes negative we determine h from
(98) by setting the rhs to zero, and T0 from (93) with Qh = 0. Though this recipe
appears rather technical the physics of the model should be quite clear and there
are only two adjustable parameters, (1 − r1 )c and r2 . Typical parameter values are
c = 1, r1 = 0.1, r2 = 0.9, τ0 = 10−4 m2 s−2 , Q0 = 2.5 × 10−5 Kms−1 , α = 10−4 K−1 .
Further reading: Frankignoul and Hasselmann (1977), Kraus (1977), Lemke (1986)
4.4. ENSO models
The strongest known variability of the climate system is the El Niño/Southern
Oscillation (ENSO) phenomenon. It has time scales of months to a few years
and, though being centered in the tropical Pacific, there are dramatic effects all
over the globe in the climate variables as temperature and rainfall but also – as a
consequence – in the economy of many states in the tropical belt (see e.g. Philander
1990)
El Niño is an aperiodic warming of the surface layer of the equatorial Pacific
Ocean, occurring roughly every four years and lasting four some months, with
largest amplitudes in sea surface temperature around Christmas on the Peruvian
coast. It is closely connected to another large scale variability, the Southern Oscillation, which is a seesaw of atmospheric mass motions across half of the globe,
already detected in 1920th by Sir Gilbert Walker and visible in the sealevel pressure variation between Djakarta and Tahiti. ENSO is known today as a coupled
ocean-atmosphere mode of climate variability.
33
4.4.1. Coupled instabilities The upper ocean wave system propagates in a
surface layer above the equatorial thermocline of mean thickness d (roughly 150
m), residing over a deep motionless layer. The dynamics are represented by a
reduced gravity model (cf. section 3.4), or equivalently, by a baroclinic wave model,
adjusted to the equatorial belt,
∂p
∂u
− βyv +
∂t
∂x
∂v
∂p
+ βyu +
∂t
∂y
∂u
∂v
∂p
+ c2o (
+
)
∂t
∂x ∂y
=
−νo u + τ x /d
(99)
=
−νo v + τ y /d
(100)
=
−µo p.
(101)
with c2o = g 0 d. It is driven by a wind stress (τ x , τ y ) and damped by a simple linear
damping law with coefficients νo and µo . The pressure is here only an anomaly,
related to the interface displacement anomaly ξ − d by p = g 0 (ξ − d). The atmosphere is a corresponding wave system in a two-layer troposphere in a state of the
first baroclinic mode described by amplitudes U, V, P and governed by
∂P
∂U
− βyV +
∂t
∂x
∂V
∂P
+ βyU +
∂t
∂y
∂P
∂U
∂V
+ c2a (
+
)
∂t
∂x
∂y
=
−νa U
(102)
=
−νa V
(103)
=
−µa P − Q.
(104)
Here c2a = N 2 D2 , where D is a vertical scale of the model. It is forced by anomalous
heating Q (evaporation or precipitation). The simplest coupling assumes that the
wind stress is linearly related to the atmospheric wind and the heating to the
oceanic interface anomaly, thus
τ x /d = γU
τ y /d = γV
Q = α(ξ − d).
(105)
with parameters α and γ. The system can be solved analytically (various simplifications can be made, such as filtering of gravity waves, see previous section;
the atmospheric system can even be considered stationary because it adjusts on a
much faster time scale than the ocean system). Coupled wave modes are obtained
which become unstable for certain ranges of the parameters. The instability has
a simple physical explanation: suppose that the oceanic interface deepens somewhere by an amount ∆ξ, this will increase the anomalous heating Q (equation
(105)) which in turn excites a convergent low level wind field (equation (104)),
leading to convergent ocean currents (equations (105) and (99) to (100)). These in
34
turn further increase ξ via (101), leading to a positive feedback. This instability is
a fundamental ingredient of most ENSO models.
4.4.2. The Zebiak-Cane model of ENSO To simulate the more complex
ENSO phenomenon only a few more physical accessories have to be implemented.
At first it is evident that the heating Q is only fairly indirectly related by the
interface anomaly, it should rather depend on the sea surface saturation humidity
or temperature (cf. the bulk formulae in (25)). This requires to consider the heat
balance of the oceanic surface layer which is assumed to be well mixed in the vertical. Furthermore, this layer – which is active in exchanging heat and momentum
with the atmosphere – is generally shallower than the upper ocean layer considered
above for the wave propagation. We thus distinguish between the ’wave layer’ of
depth d and the ’active top layer’ of depth h1 such that d = h1 + h2 . Zebiak and
Cane (1987, henceforth ZC) thus implement an additional surface Ekman layer of
depth h1 ≈ 50m with velocities (uE , vE ), governed by frictional dynamics,
−βyvE
=
−νo uE + τ x /h1
(106)
+βyuE
=
y
−νo vE + τ /h1 ,
such that u1 = u + h2 uE /d, u2 = u − h1 uE /d are the velocities in the upper
(active) and lower parts of the oceanic wave layer, respectively. Hence h1 u1 +
h2 u2 = du. The heat balance of the top active layer is given by
∂T
+ u1 · ∇T + Θ(w1 )w1 (T − Ts ) = −²T (T − T̄ ),
∂t
(107)
where T is the total sea surface temperature, w1 = h1 ∇ · u1 is the upwelling
velocity at the base of the top layer, Θ(w) the Heaviside function and the Newtonian cooling term on the rhs is the heating rate of the layer, it brings T back
to an equilibrium value T̄ which represents the seasonal cycle. The vertical advection term mimics numerical differencing. The upwelled water has a temperature
Ts = (1 − ϑ)T + ϑTd , it is considered to be a mixture of water in the top layer
and subsurface water of a temperature Td at the base depth z = −d. The latter
is parameterized as a nonlinear function Td (ξ) of the interface displacement. ZC
take Td (ξ) = T ∗ tanh b∗ ξ with constants T ∗ and b∗ which take different values for
positive or negative ξ. Finally, the heating rate Q of the atmosphere in (104) – and
thereby the coupling the mixed layer temperature to the wind – must be specified.
It consists of heating due to local evaporation and due to low-level moisture convergence in the atmosphere. ZC define Q∗ (T ) = µT exp [(T − 30◦ C)/16.7◦ C], where
T is measured in degree Celsius and µ is a constant, but feed Q = Q∗ (T ) − Q∗ (T̄ )
into the atmosphere. Notice that only the atmospheric and oceanic anomalous
state is given by (99) to (104).
35
The seasonal cycle T̄ is then prescribed from observations. ZC apply the
filtering concept explained in section 3.1: they omit all time derivatives in the
atmospheric model and filter the gravity waves for the ocean part which is then
given by the wave equations (62) and (63). Also the no-flow boundary conditions
explained there are used in the ZC model. It is obvious that running the ZC
model requires to specify a huge set of empirical parameters and tuning to seasonal
variations of the equatorial upper ocean.
Further reading: Philander (1990), McCreary and Anderson (1991), Neelin and
Latif (1994), Neelin et al. (1998)
4.5. The wind- and buoyancy-driven horizontal ocean circulation
The first analytical models of the wind-driven ocean circulation (Stommel 1948,
Munk 1950) have ignored the stratification of the fluid and nonlinearity. They
assume that a wind-driven flow regime resides in a layer of uniform depth h which
has no communication with underlying water (alternatively a flat-bottom ocean of
depth h may be considered). The vertically integrated two-dimensional flow can
R0
be represented by a streamfunction of the mass transport, −h u dz = k × ∇Ψ,
if the rigid lid approximation (w = 0 at the mean sea surface z = 0) is made.
Integration of the horizontal momentum balances (31) and (32) and elimination of
the pressure gradient term by cross-differentiation yields the Stommel-Munk model
of the circulation in a homogeneous ocean,
∂ 2
∂Ψ
∇ Ψ+β
= curl τ 0 + Ah ∇4 Ψ − Rb ∇2 Ψ,
∂t
∂x
(108)
with the appropriate conditions Ψ = const at lateral boundaries. The second term
on the rhs derives from lateral diffusion of momentum, the third from linear bottom friction. Only one of these processes is necessary to extract the momentum
imparted by the wind stress. The solution consists of forced, damped long Rossby
waves which set up a gyre circulation in a basin with a narrow frictionally controlled western boundary current (where friction and the β-term balance) and a
broad recirculation in the interior (the Sverdrup regime where the β-term and
wind forcing balance).
Equations for the total mass transport from top to bottom of the ocean,
Z
ζ
U=
u dz,
−h
are readily derived from (12) to (17) by vertical integration,
(109)
36
∂U
+ f k × U + g(h + ζ)∇ζ
∂t
=
−h∇pclin
− ∇² + τ 0 − Rb U + Ah ∇2 U = F
b
(110)
∂ζ
+∇·U
∂t
=
E − P = X.
They take the wave response of the surface elevation into account and consider
topography and stratification. The bottom pressure pb = gζ + pclin
has been split
b
into the surface component gζ and the baroclinic part given by
Z
=g
pclin
b
Z
0
ρ dz
−h
and
ζ
²=g
zρ dz.
(111)
−h
The latter is the potential energy (referred to the surface). We have taken the
same friction laws and neglected nonlinear terms from the momentum advection
as above in (108). The correspondence of (110) to the wave problem in section
3.1 is obvious: we are dealing here with the gravest mode describing barotropic
gravity and Rossby waves forced by wind stress τ 0 , surface mass flux E − P , and
the baroclinic pressure fields as given by the gradient of the baroclinic bottom
pressure pclin
and the potential energy ².
b
Obviously, if h = const and ζ ¿ h and if the mass conservation in (110)
is approximated by ∇ · U = 0 the Stommel-Munk model (108) is obtained from
(110). A few other interesting equations with more elaborate physics are derived
from (110):
• The topographic Stommel-Munk problem follows if an ocean with varying
depth h is considered. Equation (108) then takes the form
µ
¶
µ
¶
µ
¶
∂
1
∂(Ψ, f /h)
1
2
+ Rb ∇ ·
∇Ψ +
= k × ∇(τ /h) + Ah ∇ ·
∇∇ Ψ . (112)
∂t
h
∂(x, y)
h
The characteristics are the f /h-contours (called ’geostrophic contours,
they replace the f -contours valid for (108)). The h-dependence of the
friction terms may be ignored since they are anyhow only parameterizations of unknown turbulent transports of vorticity (i.e. h in the last term
of (112) should be replaced by a constant). Solutions for simple patterns
of topography can be found e.g. in Salmon (1998).
• As in case of the wave problem (43) to (45) the equations (110) of the
barotropic mode may be cast into evolution equations for the vorticity
Z = k × ∇ · U , the divergence D = ∇ · U and the surface elevation ζ,
37
∂Z
+ f D + βV
∂t
∂D
− f Z + βU + g∇(h + ζ)∇ζ
∂t
∂ζ
+D
∂t
=
k×∇·F
(113)
=
∇·F
(114)
=
X.
(115)
The Helmholtz representation of U by a potential and a streamfunction,
U = ∇Φ + k × ∇Ψ, implies Z = ∇2 Ψ and D = ∇2 Φ. For a large scale
flow the non-divergent part (described by Ψ) dominates but corrections
by the potential flow may have to be considered. The divergence equation
(114) is then approximated by −f Z + βU + g∇h∇ζ ≈ 0 (this eliminates
the gravity waves), or
∇ · (f ∇Ψ) = g∇ · (h∇ζ),
(116)
which is the linear balance equation. The approximate solution is Ψ ≈
(gh/f )ζ. Taking a constant h = H for simplicity the vorticity equation
(113) and mass conservation (115) then combine to the linear balance
model
∂
∂t
¶
µ
∂Ψ
1
∇2 − 2 Ψ + β
= curl τ 0 + Ah ∇4 Ψ − Rb ∇2 Ψ.
λ0
∂x
(117)
Compared to (108) this vorticity balance considers the effect of the elevation of the surface (i.e. the ’rigid lid approximation’ is not applied). It
finds its manifestation in the stretching term −Ψ/λ20 adding to the vorticity ∇2 Ψ and it yields the correct form of the long barotropic
√ Rossby
waves (long compared with the barotropic Rossby radius λ0 = gH/f ).
Another approximation of (113) to (115) neglects the change of surface elevation in (115), so that the divergence is determined by the diagnostic relation D = X. This is a filtering of barotropic gravity and Rossby
waves which can be applied in the ocean if time scales longer than a few
days are considered and if the time evolution arising from the propagation
of these waves is not of interest. We arrive at
∂ 2
∂Ψ
∇ Ψ+β
= −f (E − P ) + curl τ 0 + Ah ∇4 Ψ − Rb ∇2 Ψ.
∂t
∂x
(118)
Compared to (108) the generation of barotropic vorticity by the surface
mass flux E − P is included. The ratio f (E − P )/curl τ is fairly small
38
(of order 0.01) but it is interesting that (118) was solved already in 1933
by Goldsbrough in his study of ocean currents forced by evaporation and
precipitation (Goldsbrough 1933) and also by wind (Goldsbrough 1934),
well before the dynamical regimes of the wind-forced ocean circulation in
the Stommel-Munk model (108) was rediscovered in the oceanographic
community (see also Stommel 1984).
• The above described theories of the vertically integrated circulation have
neglected the effect of the baroclinic pressure forces in (110) altogether.
The effect can be investigated by a simple barotropic-baroclinic interaction
model (cf. Olbers and Wolff, 2000). Let us assume for simplicity that the
balance of total mass (the second equation in (110)) is approximated by the
rigid lid form ∇ · U = 0. We also abandon lateral diffusion of momentum
for simplicity. Taking the curl of the momentum balance (110) yields
µ
¶
µ
¶
∂
1
∂(Ψ, f /h)
∂(²/h2 , h)
+ Rb ∇ ·
∇Ψ +
= k × ∇(τ /h) −
.
∂t
h
∂(x, y)
∂(x, y)
(119)
Compared to the topographic - completely wind stress forced – StommelMunk problem (112) we realize a second vorticity source stemming from
the baroclinic pressure term of the baroclinic potential energy ε: the last
term on the rhs of (119) is called the JEBAR-term (Joint Effect of Baroclinicity and bottom Relief) or the baroclinic bottom torque. Coupling to
the stratification only occurs where the bottom is not flat. Estimation of
this term shows that it is of overwhelming importance compared to the
wind stress curl unless ²-contours follow closely the contours of h.
The bottom torque can be considered as a prescribed source in (119)
but, in fact, it is determined from the thermohaline balances (15) to (17)
of the full dynamical problem of the ocean circulation. We expand the
density about a reference field, described by the Brunt-Väisälä frequency
N (z), as in section 3.1, and assume N to be constant. Projecting (15) to
(17) on the baroclinic potential energy and retaining in the advection only
the barotropic flow we get a coupled set of equations for the streamfunction
Ψ and the potential energy ² = ²0 − (1/3)N 2 h3 , given by (119) and
∂² 0
∂(Ψ, ²0 /h2 ) 1
∂(Ψ, h)
+h
− hN 2
= Q + Kh ∇2 ²0 .
∂t
∂(x, y)
3
∂(x, y)
(120)
According to this simplified balance, potential energy is provided by a surface buoyancy flux Q and advected by the barotropic flow (the second and
the third term on the lhs, the latter is the vertical advection of the reference state). We have additionally included lateral diffusion of density in
the last term on the rhs. Whereas the restricted barotropic problem (112)
contains only barotropic Rossby waves, the coupled barotropic-baroclinic
39
problem (119) and (120) additionally contains a baroclinic Rossby wave.
It also allows forcing of the circulation by fluxes of heat and freshwater
(combining to Q) at the ocean surface. Notice that the bottom torque
only arises from the perturbation potential energy ²0 , i.e. ε in (119) may
be replaced by ε0 . Notice also that the consideration of the stratification
– in concert with varying topography – makes the determination of the
integrated circulation a nonlinear problem.
• The generalization (108) to nonlinear advection is obvious. Equations
(110) are in fact Laplace tidal equations if the forcing is replaced by the
tidal forcing (the momentum is forced by the tidal potential arising from
attraction of the ocean water by moon, sun and planets). Laplace derived
the equations for a homogeneous ocean where they read with full nonlinearities
∂u
+ u · ∇u + f k × u + g∇ζ = F /h
∂t
(121)
∂ζ
+ ∇ · [(h + ζ)u] = X.
∂t
To derive (121) it must be assumed that u = U /(h + ζ) is vertically
constant.
• The nonlinear Stommel-Munk problem is derived from the above equations
for h = const, ζ ¿ h and neglecting the tendency and surface flux in the
mass balance, i.e. ∇ · u = 0 and thus u = k × ∇ϕ with a streamfunction
ϕ. Then
∂ 2
∂(ϕ, ∇2 ϕ + f )
∇ ϕ+
= curl τ 0 /h + Ah ∇4 ϕ − Rb ∇2 ϕ.
∂t
∂(x, y)
(122)
• The nonlinear balance equations are obtained by forming vorticity and
divergence of (121). Neglecting the rate of change of the divergence, as
above for the linear problem, one finds three coupled non-linear equations,
µ
¶
∂
+ ∇2 φ + u · ∇ (∇2 ψ + f ) = curl F /h
∂t
∂(∂ψ/∂x, ∂ψ/∂y)
∇ · (f ∇ψ) + 2
= g∇2 ζ
∂(x, y)
µ
¶
∂
+ u · ∇ ζ + h∇2 φ = X,
∂t
(123)
40
where u = k × ∇ψ + ∇φ. These equations were first discussed by Bolin
(1955) and Charney (1955). Solutions – even numerical – are rather difficult to obtain. The nonlinear balance equation have recently regained some
interest in oceanographic applications (Gent and McWilliams 1983a,b).
Notice that some of these linear two-dimensional circulation problems are
easily reduced to one spatial dimension, namely those with constant coefficients.
Consider the circulation in a rectangular box, driven by a wind stress that is
sinusoidal in the meridional direction, curl τ 0 = T (x, t)Sin(y), where Sin(y) is an
eigenfunction of ∂ 2 /∂y 2 (with eigenvalue −`2 ), which vanishes on the southern
and northern boundaries of the box ocean. With Ψ(x, y, t) = P (x, t)Sin(y), the
Stommel-Munk problem (108) reduces to
∂
∂t
µ
¶
µ 4
¶
µ 2
¶
∂2
∂P
∂
∂
2
4
2
−` P +β
= T + Ah
+ ` P − Rb
−` P
∂x2
∂x
∂x4
∂x2
(124)
with boundary conditions P = 0 on the western and eastern boundaries of the
box. A similar reduction is possible for (117) and (118).
Further reading: Pedlosky (1986), Salmon (1998), Frankignoul et al. (1997)
4.6. The thermohaline-driven meridional ocean circulation
The vertical integral of the equations of motion emphasizes the wind-driven part
of the ocean circulation. The effects of stratification appear as forcing in the equations of the horizontal mass transport. A complementary view is gained from zonal
integration. Marotzke et al. (1988) and Stocker and Wright (1991) and numerous
authors thereafter have used this framework to study the thermohaline (or overturning) circulation in a simplified model of ocean circulation. Zonally integrated
diagnostics and models are quite common in atmospheric studies (cf. next section),
for the investigation of the oceanic overturning they recently got attention, mainly
because they are considerable less expensive than full 3-d simulations.
We take the planetary geostrophic equations (68) to (72), supplemented by
vertical friction to couple directly to the wind forcing at the ocean surface. For simplicity we stick to the thermohaline density equation (72) though a more complete
model should use the balances of heat and salt separately and apply the complete
equation of state. We consider a closed ocean basin with no islands (an idealized
Atlantic Ocean closed by a southern coast) of zonal width
R ∆λ(φ) at latitude φ and
define zonal averages of all fields, e.g. ρ̃(φ, z) = (1/∆λ) ρ(λ, φ, z)dλ. The zonally
averaged equations become
41
−f ṽ
=
f ũ
=
p̃z
1
(ṽ cos φ)φ + w̃z
a cos φ
ṽ
ρ̃t + ρ̃φ + w̃ρ̃z
a
=
1 ∆p
+ Av ũzz
a cos φ ∆λ
1
− p̃φ + Av ṽzz
a
−g ρ̃
=
0
=
(Kv ρ̃z )z +
−
(125)
(126)
(127)
(128)
1
(Kh ρ̃φ cos φ)φ + q.
a2 cos φ
(129)
The meaning of q is outlined below. The dynamics of this model may be condensed
to two coupled nonlinear differential equations for the density ρ̃ and the meridional
overturning streamfunction Λ(φ, z), which is introduced on the basis of (128),
ṽ = −
1 ∂Λ
cos φ ∂z
w̃ =
1 ∂Λ
.
a cos φ ∂φ
(130)
One finds
f 2 Λz + A2v Λzzzzz
=
1 ∂(Λ, ρ̃)
a cos φ ∂(φ, z)
=
ρ̃t +
g
f ∆p
Av cos φ ρ̃zφ −
a
a∆λ
1
(Kh ρ̃φ cos φ)φ + (Kv ρ̃z )z + q.
2
a cos φ
(131)
(132)
This set of equations is not closed. At first, the term q on the Rrhs of the thermohaline balance contains the divergence of Reynolds-type fluxes (v − ṽ)(ρ − ρ̃)dλ.
At the present stage of the 2-d thermohaline models these fluxes are ignored. Secondly, the pressure difference ∆p(φ, z) is not known, it cannot be ignored and must
be parameterized in terms of the resolved (zonally averaged) fields. Marotzke et
al. (1988) effectively replace (131) by a modified version
A∗v Λzzzz
g
cos φ ρ̃φ ,
a
=
(133)
with rescaled friction coefficient A∗v of order A∗v ∼ Av (1+(f h2 /Av )2 ). The equation
postulated a linear relation a cos φA∗v ṽzz = p̃φ . Stocker and Wright (1991) use the
parameterization
Z
∆p = g²0 sin 2φ
0
ρφ dz,
z
(134)
42
with ²0 = 0.3, derived from experiments with full 3-d dynamics.
The system (131) and (132) needs boundary conditions at the top, the bottom, and the northern and southern restrictions. The kinematic condition of zero
normal velocity is Λ = const on all boundaries. For a ’box’ ocean of constant
depth H the thermohaline balance is supplemented by flux conditions Kh ρ̃φ = 0
at lateral (north and south) boundaries, and Kv ρ̃z = Qρ at z = 0 and Kv ρ̃z = 0 at
z = −H, where Qρ is the density flux established by heat and freshwater transfer
at the ocean surface. Finally, frictional boundary conditions regulate the transfer
of stresses across the models interfaces at top and bottom. Various possible combinations of stress or no-slip conditions can be used, a typical example is Av ṽz = 0
leading to Λzz = 0 at z = 0, −H, Av ũz = τ λ at z = 0, Av ũz = 0 at z = −H, where
the vertical shear translates into ũz = g ρ̃φ /(af ) − Av Λzzz /(f cos φ), and τ λ is the
zonal wind stress. Alternatively, the no-slip condition Λz = 0 may be taken at the
bottom. The condition on the zonal stress cannot be incorporated into Marotzke’s
model.
It should be mentioned that the model might yield an unstable density stratification, in the sense that heavier water resides on top of lighter water. This is
a consequence of the hydrostatic approximation which has canceled the vertical
acceleration as natural reaction to such a situation. For practical applications,
ocean models implant a very strong vertical mixing of density (heat and salt)
at corresponding locations. These are hidden in the term q in the thermohaline
balance.
Further reading: Broecker (1991), Rahmstorf et al. (1996)
4.7. Symmetric circulation models of the atmosphere
With slight simplifications the model also describes an important aspect of the
atmospheric circulation. Here we consider averaging around complete latitude circles so that ∆p = 0 and the associated parameterization problem does not exist.
While atmospheric fields are far from being zonally symmetric (i.e. independent on
longitude; actually the oceanic circulation is even more ’asymmetric’) the concept
of a symmetric atmospheric state has a long history (see e.g. Lorenz (1967) for
a review) and even today many aspects of data interpretation uses zonal averaging (cf. Peixoto and Oort 1992). Various attempts have been made to construct
a corresponding symmetric model (e.g. Schneider and Lindzen 1977, Held and
Hou 1980). For the atmospheric case, equation (129) should be replaced by the
heat balance (16), considered in the above investigations with simplified heating in
form of restoring to a prescribed climatology θe (φ, z) of the radiative equilibrium
temperature distribution. The heat balance then reads
1
θ̃ − θe
ṽ
(Kh θ̃φ cos φ)φ −
,
θ̃t + θ̃φ + w̃θ̃z = (Kv θ̃z )z + 2
a
a cos φ
tR
(135)
where θ̃ refers to the potential temperature. A simplified form of θe is given by
43
·
µ
¶
¸
1 2
θe (φ, z) = θ0 1 − ∆h
+ P2 (sin φ) + ∆v (z − H/2) ,
3 3
(136)
where ∆h is the relative temperature drop from equator to pole, ∆v the drop from
the height H to the ground, and P2 the Legendre polynomial of second degree. The
equation of state must be used to relate ρ̃ and θ̃, also here the system is simplified
using ρ̃ = −θ̃/θ0 (remember that ρ is the dimensionless Boussinesq variable). In
contrast to the oceanic case where forcing by the thermohaline boundary conditions
spreads its effect in the interior by advection and diffusion, the dominating balance
in (135) is between the local heating and advection. The most simple version even
omits the meridional advection and linearizes the vertical term, so that we obtain
f 2 Λz + A2v Λzzzzz
=
1
Λφ Θz
a cos φ
=
θ̃t +
g
Av cos φ θ̃zφ
aθ0
θ̃ − θc
−
,
tR
−
(137)
(138)
with constant and prescribed Θz .
Further reading: Lindzen (1990), James (1994)
5. Low-order models
The models considered in the previous sections are described by partial differential equations, some cases are even nonlinear. Analytical solutions are known only
for the most simple, fairly restrictive conditions. In some cases even numerical
solutions are difficult to obtain. To gain insight into the behavior of the climate
system on a more qualitative level low-order models are developed. They resolve
the spatial structures in a truncated aspect but allow nonlinearities to be considered in detail. The construction is simple: the spatial structure of the fields is
represented by a set of prescribed structure functions with time dependent amplitudes. Projection of the evolution equations then yields a set of coupled ordinary
differential equations for the amplitudes. Proper selection of these spatial functions is of course the most delicate and important problem in the construction
of a low-order model. Most of such models apply to atmospheric systems. The
oceans are embedded in rather irregular basins and even simple box-type oceans
develop dynamically important boundary layers (as the Gulf Stream) which defies
representation by simple structure functions. Nevertheless, we have some oceanic
low-order models as well.
An early example of a nonlinear low-order model is found in Lorenz (1960)
where the philosophy and truncation method is explained for a barotropic QG flow
for atmospheric conditions. The expansion of the streamfunction ψ into a complete
set of orthogonal function is truncated to an interacting triad
44
ψ = −(A/`2 ) cos `y − (F/k 2 ) cos kx − 2G/(k 2 + `2 ) sin `y sin kx
(139)
with zonal and meridional wavenumbers k and `. The flow consists of mean zonal
and meridional components with amplitudes A and F , respectively, and a wave
mode with amplitude G. The system is governed by
µ
Ȧ
=
Ḟ
=
Ġ
=
¶
1
1
−
k` F G − µA + X
k2
k 2 + `2
¶
1
1
−
k` AG − µF + Y
`2
k 2 + `2
µ
¶
1 1
1
−
−
k` AF − µG + Z
2 k2
`2
−
µ
(140)
with forcing X, Y, Z and dissipation by linear friction included. If these are absent
the energy (A2 /`2 +F 2 /k 2 +2G2 /(k 2 +`2 ))/4 and the enstrophy (A2 +F 2 +2G2 )/2
of the system are conserved so an analytical solution of the equations (in terms
of elliptic functions) is possible. Periodic solutions arise entirely due to nonlinear
interaction of the triad. Notice that only the aspect ratio α = k/` of the wave
vector is relevant. A stochastic variant with white noise X, Y, Z is discussed in
Egger (1999). Typical parameter values are α = 0.9, µ = 10−6 s−1 and white noise
with < X 2 >1/2 = 10−10 s−2 .
5.1. Benard convection
A fluid which is heated from below develops convective motions. The linear stage
of instability is treated in the classical monograph of Chandrasekhar (1961), a
low-order model for the nonlinear evolution is Lorenz’ famous chaotic attractor
(Lorenz 1963).
Consider a layer of vertical extent H where the temperature at top and
bottom is held fixed, θ(x, y, z = 0, t) = θ0 + ∆θ and θ(x, y, z = H, t) = θ0 . We
assume for simplicity invariance in the y-direction and introduce a streamfunction
Ψ(x, z, t) with u = −∂Ψ/∂z and w = ∂Ψ/∂x and the temperature perturbation
Θ(x, z, t) about a linear profile with amplitude ∆θ,
³
z´
θ(x, z, t) = θ0 + ∆θ 1 −
+ Θ(x, z, t).
H
(141)
Eliminating the pressure from the x- and z-component of (5) (without rotation),
assuming a linear equation of state, % = %0 (1 − α(θ − θ0 )), and inserting (141) into
(8) we get
45
∂ 2
∇ Ψ =
∂t
−
∂(Ψ, ∇2 Ψ)
∂Θ
+ gα
+ ν∇4 Ψ
∂(x, z)
∂x
(142)
∂
Θ =
∂t
∂(Ψ, Θ) ∆θ ∂Ψ
−
+
+ κ∇2 Θ,
∂(x, z)
H ∂x
where α is the thermal expansion coefficient, ν is the kinematic viscosity and κ
the thermal conductivity. Furthermore, ∇ denotes here the (x, z)-gradient.
A low-order model of these equations was proposed by Lorenz (1963), it
became an icon of chaotic behavior. The Lorenz equations are found by taking
boundary conditions Θ = 0, Ψ = 0, ∇2 Ψ = 0 at z = 0, H and using the truncated
representation of Θ and Ψ by three modes,
υ
Ψ =
κ(1 + υ 2 )
√
πRa
Θ =
Rac ∆T
√
2X(t) sin
³ πυ ´
³π ´
x sin
z
H
H
µ
¶
³ πυ ´
³π ´
2π
2Y (t) cos
x sin
z − Z(t) sin
z ,
H
H
H
(143)
with amplitudes X, Y, Z. Here υ is the aspect ratio of the roles and a Rayleigh
number Ra = gαH 3 ∆θ/(κν) is the introduced with critical value Rac = π 4 υ −2 (1+
υ 2 )3 (this value controls the linear stability problem, see Chandrasekhar 1961).
Introducing (143) into (142) one finds (the original Lorenz model has F = 0)
Ẋ
Ẏ
=
=
− Pr X + Pr Y + F cos ϑ
−XZ + rX − Y + F sin ϑ
Ż
=
XY − bZ.
(144)
The derivative is with respect to the scaled time π 2 H −2 (1 + υ 2 )κt, dimensionless
control parameters are the Prandtl number Pr = ν/κ, a geometric factor b =
4(1 + υ 2 )−1 and r = Ra/Rac ∝ ∆θ as measure of the heating. Lorenz investigated
the system for Pr = 10, b = 8/3 and positive r. Palmer (1998) considers a forced
version of the Lorenz model (with F 6= 0 and various values of ϑ), reviving the
notion of the ’index cycles’ of the large-scale atmospheric circulation as result of
a chaotic evolutionary process. The index cycle is the irregular switching of the
zonal flow between quasisteady regimes with strong and more zonal conditions and
weak and less zonal (more wavy) conditions (see also section 5.4).
46
5.2. A truncated model of the wind-driven ocean circulation
The same year that Edward Lorenz’ chaotic attractor was published George Veronis applied the truncation technique to an oceanic circulation problem, the winddriven barotropic circulation in a rectangular shaped basin (Veronis 1963). The
system is governed by the Stommel-Munk model (108), for simplicity with Ah = 0.
A square ocean box with depth H and lateral size L in the domain 0 ≤ x ≤ π, 0 ≤
y ≤ π is considered. The coordinates are scaled by L and time by 1/(Lβ). The
ocean is forced by a wind stress with curl τ 0 = −(W/L) sin x sin y and the response
is represented by the truncated scaled streamfunction
ψ
=
20H 2 β 3 L3
[A sin x sin y +
9W 2
+B sin 2x sin y + C sin x sin 2y + D sin 2x sin 2y].
(145)
A particular problem is the projection of the β-term: to meet the boundary condition the streamfunction must consist of sine-terms and then all terms in the
vorticity balance are sine-terms with exception of the β-term which is a cosineterm. Veronis arrives at
Ȧ
=
Ḃ
=
Ċ
=
Ḋ
=
4
9
B − ²A + Ro
3π
40
8
A + AC − ²B
15π
8
−
D − AB − ²C
15π
1
C − ²D,
3π
−
(146)
(147)
(148)
(149)
where ² = Rb /(βL) and Ro = W/(β 2 HL3 ) are the nondimensional friction coefficient and wind stress amplitude, respectively. The β-term is found in the first
terms on the rhs (leading to a linear oscillatory behavior), the other terms are
readily identified as derived from the nonlinear and friction terms. There may
be three steady state solutions, one corresponding (for small friction) to the familiar Sverdrup balance where the β-term and the wind curl balance in (146),
B ≈ 27πRo/160, and all other coefficients are small. If friction is small and the
Rossby number is sufficiently large (strong wind stress), Ro > 0.32, a frictionally
controlled solution is possible where A ≈ 9Ro/(40²). If ² > 0.3 only one solution
exists regardless of the value of Ro. Not all solutions are stable, however: if there is
only one steady solution it is stable, if there are three only the one with maximum
A is stable. The time dependent system has damped oscillating solutions (settling
towards the Sverdrup balance) but also very complicated limit cycles (e.g. for
² = 0.01, Ro = 0.3).
47
5.3. The low frequency atmospheric circulation
Any time series of atmospheric data shows variability, no matter what frequencies
are resolved. In fact, the power spectra of atmospheric variables are red which
means that amplitudes of fluctuations increase with increasing period. A wide
range of processes is responsible for this irregular and aperiodic behavior, they
overlap and interact in the frequency domain and therefore it is difficult to extract
signatures of specific processes from data. A major part of the climate signals
derive from the interaction of ocean and atmosphere (as e.g. ENSO, cf. section 4.4
and 5.6), others derive from the internal nonlinearity in the atmospheric dynamics
alone. Prominent processes are wave-mean flow and wave-wave interactions and
the coupling of the flow to the orography of the planet. Examples of low-order
models of these features are presented in the next two sections.
Besides the Lorenz attractor another low-order model with chaotic properties
was introduced by Lorenz (1984) to serve as an extremely simple analogue of the
global atmospheric circulation. The model is defined by three interacting quantities: the zonal flow X represents the intensity of the mid-latitude westerly wind
current (or, by geostrophy, the meridional temperature gradient) in the northern
and southern hemisphere, and a wave component exists with Y and Z representing
the cosine and sine phases of a chain of vortices superimposed on the zonal flow.
The horizontal and vertical structures of the zonal flow and the wave are specified,
the zonal flow may only vary in intensity and the wave in longitude and intensity.
Relative to the zonal flow, the wave variables are scaled so that X 2 + Y 2 + Z 2 is
the total scaled energy (kinetic plus potential plus internal). Lorenz considers the
dynamical system
Ẋ
= −(Y 2 + Z 2 ) − a(X − F )
(150)
Ẏ
= −bXZ + XY − Y + G
(151)
Ż
= bXY + XZ − Z.
(152)
The system bears similarity with the Lorenz attractor (144) (as many other loworder systems derived from fluid mechanics) but additional terms appear. The
vortices are linearly damped by viscous and thermal processes, the damping time
defines the time unit and a < 1 is a Prandtl number. The terms XY and XZ in
(151) and (152) represent the amplification of the wave by interaction with the
zonal flow. This occurs at the expense of the westerly current: the wave transports
heat poleward, thus reducing the temperature gradient, at a rate proportional to
the square of the amplitudes, as indicated by the term −(Y 2 + Z 2 ) in (150). The
total energy is not altered by this process. The terms −bXZ and bXY represent
the westward (if X > 0) displacement of the wave by the zonal current, and b > 1
allows the displacement to overcome the amplification. The zonal flow is driven by
the external force aF which is proportional to the contrast between solar heating
at low and high latitudes. A secondary forcing G affects the wave, it mimics the
48
contrasting thermal properties of the underlying surface of zonally alternating
oceans and continents. The model may be derived from the equations of motion
by extreme truncation along similar routes as demonstrated above for the Lorenz
attractor.
When G = 0 and F < 1, the system has a single steady solution X =
F, Y = Z = 0, representing a steady Hadley circulation. This zonal flow becomes
unstable for F > 1, forming steadily progressing vortices. For G > 0 the system
clearly shows chaotic behavior (Lorenz considers a = 1/4, b = 4, F = 8 and G =
O(1)). Long integrations (see e.g. James 1994) reveals unsteadiness, even on long
timescales of tens of years, with a typical red-noise spectrum.
One fairly complex but still handy low-order model was recently investigated by Kurgansky et al. (1996). It includes wave-mean flow interaction and
orographic forcing. The problem is formulated in spherical coordinates, all quantities are scaled by taking the earth’s radius a as unit length and the inverse of the
earth’s rotation rate Ω as unit of time,
¢
∂χ
∂ ¡ 2
∇ ψ − ψ/L2 + u · ∇(k∇2 ψ + 2 sin φ + H/L2 ) + L−2
=
∂t
∂t
−2
2
= L k × ∇χ · ∇H + ν∇ (χ − ψ)
∂χ
∂ψ
+ u · ∇χ − ε
= −εu · ∇H + κ(χ∗ − χ).
∂t
∂t
(153)
(154)
√
Here H = gh/( 2a2 Ω2 ) is the scaled topography height, L = λ1 /a is the scaled
baroclinic Rossby radius, ε = R/(R + cp ) = 2/9 where R is the gas constant of dry
air and cp the specific heat capacity, ν and κ are scaled Ekman and Newtonian
damping coefficients and κχ∗ the scaled heating rate. The coefficient k = 4/3
is introduced to improve the model’s vertical representation. The state variables
ψ and χ are scaled as well, they represent the streamfunction and the vertically
averaged temperature field. Hence u = k × ∇ψ.
The equations are derived from the basic equations of motion by vertical averaging and assuming only slight deviations from a barotropic (vertically constant)
state. Horizontal inhomogeneities of temperature are accounted for, and in this respect the above equations generalize the barotropic models considered in section
4.5 and the quasigeostrophic models considered in the sections 3.2 and 3.4. For
more details we refer to Kurgansky et al. (1996). Basically, (153) is the balance of
potential vorticity and (154) is the balance of heat. The effect of the topography
on the flow is seen in the terms involving H (’orographic forcing terms’). Notice
also the correspondence to the Charney-DeVore model discussed below in section
5.4.
The model may be taken as a coupled set for ψ and χ in the two-dimensional
domain of the sphere, with specified thermal forcing κχ∗ (φ, λ, t). Kurgansky et
al. (1996) reduce the degrees of freedom by constructing a low-order model, based
on the representation
49
ψ = −α(t)µ + F (t)PN0 (µ) + A(t)Pnm (µ) sin mλ + U (t)Pnm (µ) cos mλ
(155)
χ = −β(t)µ +
G(t)PN0 (µ)
+
B(t)Pnm (µ) cos mλ
+V
(t)Pnm (µ) sin mλ,
with µ = sin φ and Pnm denoting associated Legendre functions. Furthermore, the
topography and the thermal forcing are specified as
H = H0 Pnm (µ) sin mλ
χ∗ = −χ0 (t)µ,
(156)
where the amplitude χ0 (t) describes a seasonal cycle. The system is thus reduced to a zonal flow represented by (α, β) and (F, G) and a wave represented
by (A, B, U, V ). It is governed by eight coupled differential equations for these
amplitudes. We refer to Kurgansky et al. (1996) because they are rather lengthy.
In their experiments they adopt m = 2, n = 5 and N = 3.
The model shows a rich low-frequency time variability, with and without
seasonal forcing. Fluctuations are predominantly caused by interaction of the orographically excited standing wave and the zonal mean flow. Spectra are red up to
periods of decades and chaotic behavior shows up as well.
A simplified version is obtained if the zonal contributions to ψ and χ, described by the amplitudes α and β, are considered as given constants, and orographic and thermal forcing is omitted. The model then represents the response of
the wave system to the coupling of the wave to the mean flow and and wave–wave
interaction. The six remaining amplitudes follow from
Ḟ
=
Ġ
=
Ȧ
=
1
− Π(U V − AB)
2
1
− Ξ(U V − AB)
2
−ΓU + ∆B − Π(BF − U G)
(157)
U̇
=
ΓA − ∆V + Π(V F − AG)
Ḃ
=
ΥA − ΣV + Ξ(V F − AG)
V̇
=
−ΥU + ΣB − Ξ(BF − U G).
£
¤
Time is scaled as n(n + 1) + L−2 (1 − ε) t, furthermore k = 1, N = n, H0 =
0, £and the following abbreviations
Π = mq/L2 , ∆ = αm/L2 , Γ =
¤ are made:
£
¤
2
2
m 2(1 +£ α) + α/L − αn(n
+ 1) , Υ = £m 2ε(1 + α) + α/L
+ αn(n + 1)(1 − ε) ,
¤
¤
Ξ = mq n(n + 1) + L−2 , and Σ = mα n(n + 1) + L−2 , and q is a triple integral
50
R
of the Legendre functions, q = (PNm )2 (dPN0 /dµ)dµ. Typical parameter values are
m = 2, n = 3, q = 3.6, α = 6 × 10−2 , ε = 2/9, L−2 = 5.7. In this version dissipative
terms have omitted as well and the system then yields self-sustained non-linear
oscillations. In fact, Kurgansky et al. (1996) describe a solution of (157) in terms
of elliptic functions. The model produces an interesting torus-type portrait in the
phase space. The zonal thermal forcing (156), however, does not enter the equations
of the six wave amplitudes and, thus, for studies of forced and dissipative problems
either the complete model has to invoked or a direct thermal forcing of the wave
must be considered.
Further reading: James (1994)
5.4. Charney-DeVore models
The state of the atmosphere in midlatitudes of the northern hemisphere shows
long persisting anomalies (’Großwetterlagen’) during which the movement of irregular weather variability across the Atlantic seems to be blocked. It is appealing
to connect these ’Großwetterlagen’ with the steady regimes of a low-order subsystem of the atmospheric dynamics and explain transitions by interaction with
shorter waves simply acting as white noise. Starting with the work of Egger (1978)
and Charney and DeVore (1979) the concept of multiple equilibria in a severely
truncated ’low-order’ image (the CdV model) of the atmospheric circulation was
put forward. The observational evidence for dynamically disjunct multiple states,
particularly with features of the CdV model, in the atmospheric circulation is however sparse (see the collection of papers in Benzi et al. 1986) and the applicability
has correctly been questioned (see e.g. Tung and Rosenthal 1985).
The simplest CdV model describes a barotropic zonally unbounded flow over
a sinusoidal topography in a zonal channel with quasigeostrophic dynamics. The
flow is governed by the barotropic version of (65) or, in layer form, by (80). The
vorticity balance of such a flow
·
¸
∂ 2
f0 b
2
∇ Ψ + u · ∇ ∇ Ψ + βy +
= R∇2 (Ψ∗ − Ψ)
∂t
H
(158)
needs an additional constraint to determine the boundary values of the streamfunction Ψ on the channel walls. The vorticity concept has eliminated the pressure
field and its reconstruction in a multiconnected domain requires in addition to
(158) the validity of the momentum balance, integrated over the whole domain,
f0
∂Ψ
∂U
= R (U ∗ − U ) +
<b
>.
∂t
H
∂x
(159)
51
Here, U is the zonally and meridionally averaged zonal velocity and R∇2 Ψ∗ =
−R∂U ∗ /∂y is the vorticity and RU ∗ the zonal momentum imparted into the system, e.g. by thermal forcing or, in an oceanic application, by wind stress. Furthermore, R is a coefficient of linear bottom friction. The last term in the latter
equation is the force exerted by the pressure on the bottom relief, called bottom
form stress (the cornered brackets denoted the average over the channel domain).
The momentum input RU ∗ is thus balanced by bottom friction and bottom form
stress.
The depth of the fluid is H −b and the topography height b is taken sinusoidal,
b = b0 cos Kx sin Ky with K = 2πn/L where L is the length and L/2 the width of
the channel. A heavily truncated expansion
Ψ = −U y +
1
[A cos Kx + B sin Kx] sin Ky
K
(160)
represents the flow in terms of the zonal mean U and a wave component with sine
and cosine amplitudes A and B. It yields the low-order model
U̇
=
Ȧ
=
Ḃ
=
1
R (U ∗ − U ) + δB
4
−KB (U − cR ) − RA
1
KA (U − cR ) − δU − RB.
2
(161)
where cR = β/2K 2 is the barotropic Rossby wave speed and δ = f0 b0 /H. The
steady states are readily determined: the wave equations yield for the form stress
(the wave component which is out of phase with respect to the topography)
1
1
Rδ 2 U
δB[U ] = −
2
4
2 R + K 2 (U − cR )2
(162)
and equating this with R(U ∗ −U ), three equilibria are found if U ∗ is well above cR .
The three possible steady states can be classified according to the size of the mean
flow U compared to the wave amplitudes: the high zonal index regime is frictionally
controlled, the flow is intense and the wave amplitude is low; the low zonal index
regime is controlled by form stress, the mean flow is weak and the wave is intense.
The intermediate state is transitional, it is actually unstable to perturbations. This
’form stress instability’ works obviously when the slope of the resonance curve is
below the one associated with friction, i.e. ∂(RU − 14 δB[U ])/∂U > 0, so that
a perturbation must run away from the steady state. Typical parameter values
for this model are R = 10−6 s−1 , K = 2π/L, L = 10000 km, b0 = 500 m, H =
5000 m, U ∗ = 60 ms−1 . Stochastic versions of the CdV-model have been studied
by Egger (1982) and De Swart and Grasman (1987).
52
In realistic atmospheric applications of the CdV model the parameter window (topographic height, forcing and friction parameters) for multiple solutions is
quite narrow, due to the dispersiveness of the barotropic Rossby wave it may even
not exist at all for more complex topographies where the resonance gets blurred
because cR is a function of wave length. For realistic values of oceanic parameters
multiple states do not exist because here U ∗ ¿ cR . Extending the model, however,
to baroclinic conditions (a two layer quasigeostrophic model described by (80)),
interesting behavior is found which can be applied to the dynamical regime of the
Antarctic Circumpolar Current (Olbers and Völker 1996). The Circumpolar Current is due to its zonal unboundedness the only oceanic counterpart (with dynamic
similarity) of the zonal atmospheric circulation. The resonance occurs when the
barotropic current U is of order of the baroclinic Rossby wave speed βλ2 . The
model allows for complex topographies since long baroclinic Rossby waves are free
of dispersion, the location of the resonance is thus independent of the wavenumber
K.
In its simplest form the model is derived by expanding the barotropic and
baroclinic streamfunctions Ψ = Ψ1 + Ψ2 and Θ = Ψ1 − Ψ2 (assuming equal layer
depths for simplicity) again into a small number of modes:
Ψ
= −U y + E sin 2y + 2[A cos x + B sin x] sin y
(163)
Θ
= −uy + G + F sin 2y + 2[C cos x + D sin x] sin y.
All variables are scaled using a time scale 1/|f0 | and a length scale Y /π where Y
is the channel width. From constraints on the zonal momentum balance similar to
(151) and the condition of no mass exchange between the layers one easily arrives
at the conditions E = U/2, F = u/2 and G = uπ/2 that can be used to eliminate
these variables. Inserting the expansion into the potential vorticity balances (81)
and projection then yields prognostic equations for U, u, A, B, C, and D. These are,
however, strongly simplified by neglecting the relative vorticity term ∇2 Ψi and the
surface elevation ζ in the potential vorticity. This approximation is equivalent to
reducing the dynamics to the slow baroclinic mode alone, assuming infinitely fast
relaxation of the barotropic mode (the fast mode is ’slaved’ by the slow mode).
Due to this approximation the barotropic low-order equations become diagnostic
relations
0
=
0
=
0
=
−²(U − u) + b(A − C) + τ
3
−²(A − C) − βB − b(U − u)
4
−²(B − D) + βA,
(164)
53
while the baroclinic ones still contain a time derivative,
3
u̇ =
2
Ċ
=
Ḋ
=
−4σµu + 2στ − 2(AD − BC)
3
−4σµC − σβ(B + D) + (U D − uB)
2
3
−4σµD + σβ(A + C) − (U C − uA).
2
(165)
Here, ² = R/|f0 | is the scaled coefficient of a linear bottom friction, µ is the
scaled coefficient of a linear interfacial friction that is meant to mimic the momentum exchange between the layers caused by small-scale eddies (see section 3.4),
β = β ∗ Y /(π|f0 |) is the scaled form of the dimensioned gradient β ∗ of the Coriolis parameter, and σ = Y 2 /(πλ)2 is the scaled squared inverse of the baroclinic
Rossby radius λ. The system is forced by a zonal wind stress with scaled amplitude
τ = τ0 /(HY f02 ) where τ0 is the dimensioned stress amplitude, the meridional dependence is given by τ x = τ sin2 y. The scaled height is defined as b = −(π/2)b0 /H
with the same topography as before in the barotropic CdV-model. Typical parameter values are ² = 10−3 , r = 2², b0 = 600 m, H = 5000 m, Y = 1500 km, λ =
31 km, τ0 = 10−4 m2 s−2 . The system produces aperiodic oscillations, it contains
parameter windows with chaotic behavior (there is Shil’nikov attractor in the range
µ = 2 · · · 3 × 10−3 , b0 = 600 · · · 700m).
Further reading: Ghil and Childress (1987), James (1994), Völker (1999)
5.5. Low order models of the thermohaline circulation
Stommel in his seminal paper of 1961 was the first to point out that the thermohaline circulation in the ocean might have more than one state in equilibrium with
the same forcing by input of heat and freshwater at the surface. The notion has
found ample interest in recent years in context with the ocean’s role in climate
change. Numerous papers have demonstrated Stommel’s mechanisms with numerical 2-d and 3-d circulation models (see e.g. Marotzke et al. 1998, Rahmstorf et
al. 1996).
Stommel’s simple model is a two-box representation of the thermal and haline
state in the midlatitude and polar regions in the North Atlantic. In terms of the
differences ∆θ and ∆S of the temperatures and salinities of the two boxes (well
mixed and of equal volume) the evolution equations are
d∆θ
dt
=
γ(∆θ∗ − ∆θ) − 2|q|∆θ
(166)
d∆S
dt
=
2F − 2|q|∆S.
54
It is assumed that the flow rate between the boxes is proportional to their density
difference,
q = k∆ρ = k(−α∆θ + β∆S),
(167)
which assumes hydraulic dynamics where the flow is proportional to pressure
(and thus density) differences. Surface heating is parameterized by restoring to
an atmospheric temperature ∆θ∗ whereas salt changes are due to a prescribed
flux F of freshwater (it is actually virtual flux of salinity related to freshwater flux E (in ms−1 ) by F = S0 E/H where S0 is constant reference value of
salinity and H is the ocean depth; the factor 2 arises because the amount F
is taken out of the southern box and imparted into the northern box). These
’mixed’ boundary conditions reflect the fact that heat exchange with the atmosphere depends on the sea surface temperature but the freshwater flux E from
the atmosphere to the ocean does not depend on the water’s salinity. Besides
the forcing F there are two more parameters in the model: γ is the inverse of a
thermal relaxation time, and the hydraulic coefficient k measures the strength of
the overturning circulation (α and β are the thermal and haline expansion coefficients of sea water). The ratio γ/q determines the role of temperature in this
model. If γ À q the temperature ∆θ adjusts very quickly to the atmospheric
value ∆θ∗ and the salinity balance alone determines the dynamical system. It has,
however, still interesting properties, in particular multiple equilibria. With γ ∼ q
we get a truly coupled thermohaline circulation. Typical parameter values are
k = 2 × 10−8 s−1 , α = 1.8 × 10−4 K −1 , β = 0.8 × 10−3 , ∆θ∗ = 10 K, F = 10−14 s−1 .
Evidently, γ or k may be eliminated by appropriate time scaling.
Steady states are easily found: for F > 0 (net precipitation in the polar
box) there are three equilibria, two of which are stable: a fast flowing circulation (poleward in the surface layer with sinking in the polar box) driven mainly
by temperature contrast, and a slow circulation flowing reversely (sinking in the
tropics) which is driven by salinity contrast. There is a threshold for F where a
saddle-node bifurcation occurs which leaves only the haline mode alive. A detailed
description of a phase space perspective of Stommel’s model has recently been
given by Lohmann and Schneider (1999), noise induced transitions are investigated in Timmermann and Lohmann (1999).
There is no limit cycle associated with the unstable steady state in Stommel’s model and the system cannot support self-sustained oscillations. Various
routes may be pursued to refine the model, in fact there is great variety of simple
thermohaline oscillators which have, in an idealized fashion, a relation to processes
in the coupled ocean-atmosphere system. A gallery of thermal and thermohaline
oscillators has been collected by Welander (1986).
Presentation of the ocean by more boxes increases the structural complexity
(see e.g. Welander 1986, Marotzke 1996, Rahmstorf et al. 1996, Kagan and Maslova
1991) but does not increase the physical content. An obvious weakness lies in the
55
representation of the ’hydraulic dynamics’ (167) of the circulation relating the
meridional flow q to the meridional density gradient ∆ρ, which is contrary to
the notion of the geostrophic balance (but it parallels the 2-d closure (133)). A
qualitative improvement can be seen in the low-order model of Maas (1994) who
considers more complete dynamics in form of the conservation of the 3-d angular
momentum
L=
1
V
Z
X × V d3 x
(168)
of a rectangular box ocean with volume V and size L, B, H in the x, y, z-direction.
The coordinate system has its origin in the center (x eastward, y northward, z
upward). The rate of change of angular momentum can be derived from (5) by
straightforward integration (we use here the full 3-d balance of momentum but
employ the rest of the Boussinesq approximation, i.e. extract the hydrostatically
balanced reference field and replace the density in the inertial terms by a constant).
A balance by torques due to Coriolis, pressure, buoyancy and frictional forces
arises. Under quite ’mild’ assumptions and with representation of the density field
by plane isopycnal surfaces,
ρ = xρ̄x (t) + y ρ̄y (t) + z ρ̄z (t) + (x2 − L2 /3)ρ̄xx (t) · · ·
(169)
(the last term and all of higher order are ignored), Maas derives
R a set ofR six autonomous coupled equations for the vectors Li and Ri = ∇ρ̄ = xi ρ d3 x/ x2i d3 x.
These are
Pr−1 L̇ + Ek−1 k × L
=
−R2 i + R1 j − (L1 , L2 , rL3 ) + T T
(170)
Ṙ + R × L
=
−(R1 , R2 , µR3 ) + Ra Q.
(171)
Scaling has been applied: Pr = Ah /(12Kh ) is a Prandtl number, Ek = 2Ah /(f L2 )
is an Ekman number, µ = Kv L2 /(Kh H 2 ) and r = Av L2 /(Ah H 2 ) are diffusive and
frictional coefficients, Ra = gδ%e HL2 /(2Ah Kh ) is a Rayleigh number (it measures
the buoyancy input by an externally imposed density difference δ%e ), and T =
τ0 L3 /(2HAh Kh ) measures the torque exerted by the wind stress τ 0 . Time is scaled
by L2 /(12Kh ). The scaled forcing moments are
56
Z
T
=
(−τ2 /2, τ1 /2, −1) dx dy
(172)
Z
Q
=
(x, y, 1)B dx dy,
where (τ1 , τ2 ) is the wind stress vector (scaled by τ0 ) and B the buoyancy flux
(scaled by δ%e HKh /L2 ) entering a the surface. Typical values of the dimensionless
numbers are Pr = 2 × 104 , Ek = 0.02, r = 1, µ = 1, Ra = 106 , T = 500, the time
scale L2 /12Kh is of order of 500 years.
Evidently, the system is easily expanded to nine equations in case that temperature and salinity are used to replace the combined density balance (171) (see
Schrier and Maas, 1998). We also point out that in the balance of the zonal component L1 of angular momentum we indeed find the terms of the simplified Stommel
dynamics (167): neglecting the rate of change and the Coriolis and wind moments
we have L1 ≈ −R2 which appears here as the frictional balance between the
meridional overturning and the north-south density gradient.
Maas finds a rich suite of regimes in his model. For the case f = 0, T =
0, Q1 = 0 the Lorenz’63 attractor is found, for the case T = 0, Q1 = 0, Pr → ∞
the Lorenz’84 attractor is found. Typical parameter values for the ocean avoid the
chaos which these equation may exhibit. However, multiple equilibria are possible
and self-sustained oscillations (with interesting phase portraits) with time scales
of order 500 years are obtained.
Further reading: Colin de Verdiere (1993)
5.6. The delayed ENSO oscillator
Simple conceptual models of the oscillatory behavior of ENSO (cf. section 4.4)
were suggested by Suarez and Schopf (1988) and Battisti and Hirst (1989). They
are represented by a one-dimensional state variable T which could be any of the
variables of the coupled ocean-atmosphere ENSO system, for instance the thermocline depth anomaly or the anomalies of the sea surface temperature or the wind
stress amplitude. The model combines the physics of wave propagation and the
unstable coupled mode, explained in the sections 3.1 and 4.4, into one equation.
An example is the delayed oscillator
Ṫ = cT − bT (t − τ ) − eT 3 .
(173)
The first term on the rhs represents the positive feedback (with c > 0), the second
term the delayed effect of the Kelvin and Rossby wave propagation across the
equatorial basin. The Rossby waves are excited in the interaction region, later
reflected at the western boundary and then – after the time delay τ – return
REFERENCES
57
their signal to the east via a Kelvin wave. The cubic term limits the growing
unstable mode. Twop
of the four parameters (c and e) can be eliminated by scaling
time by c and T by c/e. Depending on the values of the remaining parameters,
b/c and cτ , steady or oscillatory solutions are possible (a detailed discussion of
(173) and other low-order models of ENSO is given by McCreary and Anderson
(1991), take e.g. c = 1, e = 1, b = 0.5, τ = 1 for a steady case, or c = 1, e =
1, b = 1.5, τ = 3 for oscillating case). Note that these models can be extended to
seasonally varying parameters. The solution is very sensitive to the parameters so
that in case of stochastic forcing an interesting switching of regimes occurs which
might be relevant to the ENSO phenomenon.
Acknowledgements I appreciate discussions with Hartmut Borth, Andreas
Hense, Mojib Latif, Christoph Völker and, in particular, with Jürgen Willebrand.
Josef Egger and an anonymous reviewer helped to improve the manuscript. I am
grateful to Klaus Hasselmann, my teacher in physics over three decades. Contribution No. 1697 from the Alfred-Wegener-Institute for Polar and Marine Research.
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Coordinates and constants
t
time
x, y, z x eastward, y northward, z upward
of Cartesian coordinates
λ, φ
longitude and latitude
of spherical coordinates
a
radius of the earth
g
gravitational acceleration
f, f0
Coriolis frequency
f = 2Ω sin φ = f0 + βy
β
differential rotation
midlatitudes 2Ω cos φ0 /a
equator 2Ω/a
Ω
angular velocity of the earth
S0
solar constant
σ
Stefan-Boltzmann constant
R
gas constant of dry air
cp
specific heat
ocean (variable)
atmosphere (dry air) cp = (7/2)R
N
Brunt Väisälä frequency
ocean
atmosphere
6.371 × 106 m
9.806 ms−2
10−4 s−1 for midlatitudes
2 × 10−11 m−1 s−1
2.289 × 10−11 m−1 s−1
7.292 × 10−5 s−1
1.372 × 103 Wm−2
5.67 × 10−8 Wm−2 K−4
287.04 Jkg−1 K−1
4217 Jkg−1 K−1
1005 Jkg−1 K−1
2π/N ∼ 30 min
2π/N ∼ 5 min
The forcing functions of the wave equations
(F u , F v )
=
Gb
=
P
=
Z
=
W
=
Φ/%0 − u · ∇u − w
∂u
∂z
∂b
−(g/%0 ) [−α∗ Γθ + β∗ ΓS ] − u · ∇b − w
∂z
µ
¶
∂p 1 2 2
− ζ
+ N ζ + ···
+ patm
∂z
2
z=0
µ
¶
∂w
ζ
+ ···
∂z
z=0
µ
¶
∂w
u · ∇η − η
− ···
∂z
z=−H
Q
=
C
=
D
=
Z 0
∂P
+ g2 Z −
Gb dz
∂t
z
∂F v
∂F u
1
−
− M−1 Q
∂x
∂y
f0
∂F u
∂F v
+
∂x
∂y
REFERENCES
63
where h = H − η, i.e. η is the elevation of the bottom above the mean depth H.
Dirk Olbers
Alfred-Wegener-Institute for Polar and Marine Research
25757 Bremerhaven, Germany
E-mail address: [email protected]
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