Los2009b

Los2009b
On the formulation of sea-ice models. Part 1:
Effects of different solver implementations and
parameterizations
Martin Losch a,1 , Dimitris Menemenlis b , Jean-Michel Campin c ,
Patick Heimbach c , and Chris Hill c
a Alfred-Wegener-Institut
b Jet
für Polar- und Meeresforschung, Postfach 120161, 27515
Bremerhaven, Germany
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA
c Department
of Earth, Atmospheric, and Planetary Sciences, Massachusetts
Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Abstract
This paper describes the sea ice component of the Massachusetts Institute of Technology general circulation model (MITgcm); it presents example Arctic and Antarctic results from a realistic, eddy-admitting, global ocean and sea ice configuration;
and it compares B-grid and C-grid dynamic solvers and other numerical details of
the parameterized dynamics and thermodynamics in a regional Arctic configuration. Ice mechanics follow a viscous-plastic rheology and the ice momentum equations are solved numerically using either line-successive-over-relaxation (LSOR) or
elastic-viscous-plastic (EVP) dynamic models. Ice thermodynamics are represented
using either a zero-heat-capacity formulation or a two-layer formulation that conserves enthalpy. The model includes prognostic variables for snow thickness and for
sea ice salinity. The above sea ice model components were borrowed from currentgeneration climate models but they were reformulated on an Arakawa C grid in order
to match the MITgcm oceanic grid and they were modified in many ways to permit
efficient and accurate automatic differentiation. Both stress tensor divergence and
advective terms are discretized with the finite-volume method. The choice of the
dynamic solver has a considerable effect on the solution; this effect can be larger
than, for example, the choice of lateral boundary conditions, of ice rheology, and
of ice-ocean stress coupling. The solutions obtained with different dynamic solvers
typically differ by a few cm s−1 in ice drift speeds, 50 cm in ice thickness, and order
200 km3 yr−1 in freshwater (ice and snow) export out of the Arctic.
Key words: NUMERICAL SEA ICE MODELING, VISCOUS-PLASTIC
RHEOLOGY, EVP, COUPLED OCEAN AND SEA ICE MODEL, STATE
Preprint submitted to Elsevier
29 January 2010
ESTIMATION, ADJOINT MODELING, CANADIAN ARCTIC
ARCHIPELAGO, SEA-ICE EXPORT, SENSITIVITIES
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1
Introduction
It is widely recognized that high-latitude processes are an important component of the climate system (Lemke et al., 2007, Serreze et al., 2007). As
a consequence, these processes need to be accurately represented in climate
state estimates and in predictive models. Sea ice, though only a thin layer
between the air and the sea, has strong and numerous influences within the
climate system; it influences radiation balance due to its high albedo, surface heat and mass fluxes due to its insulating properties, freshwater fluxes
due to transport and ablation, ocean mixed layer processes, and human operations. Sea ice variability and long term trends are distinctly different in
the polar regions of the Northern and of the Southern hemispheres (Cavalieri
and Parkinson, 2008, Parkinson and Cavalieri, 2008). These differences and
their interaction with the global climate system are still poorly represented in
state-of-the-art general circulation models (Holloway et al., 2007, Kwok et al.,
2008). In addition, the atmospheric and oceanic states, which are needed to
drive sea ice models, are still highly uncertain. Sea ice in turn constrains the
state of both ocean and atmosphere near the surface so that observations of
sea ice contain valuable information about the state of the coupled system.
One way to reduce the model and boundary-condition uncertainties and to
improve the representation of coupled ocean and sea ice processes is via coupled ocean and sea ice state estimation, that is, by using ocean and sea ice
data to constrain a numerical model of the coupled system in order to obtain
a dynamically consistent ocean and sea ice state with closed property budgets.
This paper describes a new sea ice model designed to be used for coupled ocean
and sea ice state estimation. While many of its features are “conventional” (yet
for the most part state-of-the-art), the model is different from previous models in that it is tailored for the generation of efficient adjoint code for coupled
ocean and sea ice simulations by means of automatic (or algorithmic) differentiation (AD, Griewank, 2000). Sensitivity propagation in coupled systems is
highly desirable as it permits both ocean and sea ice observations to be used
as simultaneous constraints, leading to a truly coupled estimation problem.
For example this approach is being used in planetary scale ocean and sea-ice
monitoring and measuring activities, such as Heimbach (2008), Stammer et al.
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corresponding author, email: [email protected],
ph: ++49 (471) 4831-1872, fax: ++49 (471) 4831-1797
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(2002) and Menemenlis et al. (2005).
Our work is presented in two parts. Part 1 (this paper) outlines the dynamic
and thermodynamic sea ice model that has been coupled to the MITgcm
ocean, with special emphasis on examining the influence of sea-ice rheology
solvers and on model behavior. Part 2 (a companion paper) is devoted to the
development of an efficient and accurate coupled ocean and sea ice adjoint
model by means of automatic differentiation and to using adjoint sensitivity
calculations to understand model sea ice dynamics.
Most standard sea-ice models are discretized on Arakawa B grids (e.g., Hibler,
1979, Harder and Fischer, 1999, Kreyscher et al., 2000, Zhang et al., 1998,
Hunke and Dukowicz, 1997), probably because early numerical ocean models
were formulated on the Arakawa B grid and because of the easier (implicit)
treatment of the Coriolis term. As model resolution increases, more and more
ocean and sea ice models use an Arakawa C grid discretization (e.g., Marshall
et al., 1997a, Ip et al., 1991, Tremblay and Mysak, 1997, Lemieux et al., 2008,
Bouillon et al., 2009). The new MITgcm sea ice model is formulated on an
Arakawa C grid, and two different solvers (LSOR and EVP) are implemented;
a previous version of the LSOR solver on a B grid is also available. It is used
here for comparison with the new C grid implementation.
From the perspective of coupling a sea ice-model to a C-grid ocean model, the
exchange of fluxes of heat and freshwater pose no difficulty for a B-grid sea
ice model (e.g., Timmermann et al., 2002). Surface stress, however, is defined
at velocity points and thus needs to be interpolated between a B-grid sea ice
model and a C-grid ocean model. Smoothing implicitly associated with this
interpolation may mask grid scale noise and may contribute to stabilizing the
solution. Additionally, the stress signals are damped by smoothing, which may
lead to reduced variability of the system. By choosing a C grid for the sea-ice
model, we avoid this difficulty altogether and render the stress coupling as
consistent as the buoyancy flux coupling.
A further characteristic of the C-grid formulation is apparent in narrow straits.
In the limit of only one grid cell between coasts, there is no flux allowed for
a B grid (with no-slip lateral boundary conditions, which are natural for the
B grid) and models have used topographies with artificially widened straits
in order to avoid this problem (Holloway et al., 2007). The C-grid formulation, however, allows a flux through narrow passages even if no-slip boundary
conditions are imposed (Bouillon et al., 2009). We examine the quantitative
impact of this effect in the Canadian Arctic Archipelago (CAA) by exploring differences between the solutions obtained on either the B or the C grid,
with either the LSOR or the EVP solver, and under various options for lateral
boundary conditions (free-slip vs. no-slip). Compared to the study of Bouillon
et al. (2009), which was carried out using a grid with minimum horizontal grid
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spacing of 65 km in the Arctic Ocean, this study includes discussion of the
LSOR solver and the sensitivity experiments are carried out on an Arctic grid
with uniform 18-km horizontal grid spacing.
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The remainder of this paper is organized as follows. Section 2 describes the
dynamics and thermodynamics components, which have been incorporated in
the MITgcm sea ice model. Section 3 presents example Arctic and Antarctic
results from a realistic, eddy-admitting, global ocean and sea ice configuration. Section 4 compares B-grid and C-grid dynamic solvers under different
lateral boundary conditions and investigates other numerical details of the parameterized dynamics and thermodynamics in a regional Arctic configuration.
Discussion and conclusions follow in Section 5.
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Sea ice model formulation
The MITgcm sea ice model is based on a variant of the viscous-plastic (VP)
dynamic-thermodynamic sea-ice model of Zhang and Hibler (1997) first introduced by Hibler (1979, 1980). Many aspects of the original codes have been
adapted; these are the most important ones:
• the model has been rewritten for an Arakawa C grid, both B- and C-grid
variants are available; the finite-volume C-grid code allows for no-slip and
free-slip lateral boundary conditions,
• two different solution methods for solving the nonlinear momentum equations, LSOR (Zhang and Hibler, 1997) and EVP (Hunke, 2001, Hunke and
Dukowicz, 2002), have been adopted,
• ice-ocean stress can be formulated as in Hibler and Bryan (1987) as an
alternative to the standard method of applying ice-ocean stress directly,
• ice concentration and thickness, snow, and ice salinity or enthalpy can be
advected by sophisticated, conservative advection schemes with flux limiters.
The sea ice model is tightly coupled to the ocean component of the MITgcm
(Marshall et al., 1997b,a). Heat, freshwater fluxes and surface stresses are
computed from the atmospheric state and modified by the ice model at every
time step. The remainder of this section describes the model equations and
details of their numerical realization. Further documentation and model code
can be found at http://mitgcm.org.
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2.1
Dynamics
Sea-ice motion is driven by ice-atmosphere, ice-ocean and internal stresses;
and by the horizontal surface elevation gradient of the ocean. The internal
stresses are evaluated following a viscous-plastic (VP) constitutive law with
an elliptic yield curve as in Hibler (1979). The full momentum equations for the
sea-ice model and the solution by line successive over-relaxation (LSOR) are
described in Zhang and Hibler (1997). Implicit solvers such as LSOR usually
require capping very high viscosities for numerical stability reasons. Alternatively, the elastic-viscous-plastic (EVP) technique following Hunke (2001)
regularizes large viscosities by adding an extra term in the constitutive law
that introduces damped elastic waves. The EVP-solver relaxes the ice state
towards the VP rheology by sub-cycling the evolution equations for the internal stress tensor components and the sea ice momentum solver within one
ocean model time step. Neither solver requires limiting the viscosities from
below (see Appendix A for details).
For stress tensor computations the replacement pressure (Hibler and Ip, 1995)
is used so that the stress state always lies within the elliptic yield curve by
definition. In an alternative to the elliptic yield curve, the so-called truncated
ellipse method (TEM), the shear viscosity is capped to suppress any tensile
stress (Hibler and Schulson, 1997, Geiger et al., 1998).
The horizontal gradient of the ocean’s surface is estimated directly from
ocean sea surface height and pressure loading from atmosphere, ice and snow
(Campin et al., 2008). Ice does not float on top of the ocean, instead it depresses the ocean surface according to its thickness and buoyancy.
Lateral boundary conditions are naturally “no-slip” for B grids, as the tangential velocities points lie on the boundary. For C grids, the lateral boundary
condition for tangential velocities allow alternatively no-slip or free-slip conditions. In ocean models free-slip boundary conditions in conjunction with
piecewise-constant (“castellated”) coastlines have been shown to reduce to
no-slip boundary conditions (Adcroft and Marshall, 1998); for coupled ocean
sea-ice models the effects of lateral boundary conditions have not yet been
studied (as far as we know). Free-slip boundary conditions are not implemented for the B grid.
Moving sea ice exerts a surface stress on the ocean. In coupling the sea-ice
model to the ocean model, this stress is applied directly to the surface layer
of the ocean model. An alternative ocean stress formulation is given by Hibler
and Bryan (1987). Rather than applying the interfacial stress directly, the
stress is derived from integrating over the ice thickness to the bottom of the
oceanic surface layer. In the resulting equation for the combined ocean-ice
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momentum, the interfacial stress cancels and the total stress appears as the
sum of wind stress and divergence of internal ice stresses (see also Eq. 2 of
Hibler and Bryan, 1987). While this formulation tightly embeds the sea ice
into the surface layer of the ocean, its disadvantage is that the velocity in the
surface layer of the ocean that is used to advect ocean tracers is an average over
the ocean surface velocity and the ice velocity, leading to an inconsistency as
the ice temperature and salinity are different from the oceanic variables. Both
stress coupling options are available for a direct comparison of their effects on
the sea-ice solution.
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The finite-volume discretization of the momentum equation on the Arakawa
C grid is straightforward. The stress tensor divergence, in particular, is discretized naturally on the C grid with the diagonal components of the stress
tensor on the center points and the off-diagonal term on the corner (or vorticity) points of the grid. With this choice all derivatives are discretized as
central differences and very little averaging is involved (see Appendix B for
details). Apart from the standard C-grid implementation, the original B-grid
implementation of Zhang and Hibler (1997) is also available as an option in
the code.
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2.2
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Thermodynamics
Upward conductive heat flux through the ice is parameterized assuming a
linear temperature profile and a constant ice conductivity implying zero heat
capacity for ice. This type of model is often referred to as a “zero-layer” model
(Semtner, 1976). The surface heat flux is computed in a similar way to that
of Parkinson and Washington (1979) and Manabe et al. (1979).
The conductive heat flux depends strongly on the ice thickness h. However,
the ice thickness in the model represents a mean over a potentially very heterogeneous thickness distribution. In order to parameterize a sub-grid scale
distribution for heat flux computations, the mean ice thickness h is split into
seven thickness categories Hn that are equally distributed between 2h and a
h for n ∈ [1, 7]. The
minimum imposed ice thickness of 5 cm by Hn = 2n−1
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heat fluxes computed for each thickness category are area-averaged to give the
total heat flux (Hibler, 1984).
The atmospheric heat flux is balanced by an oceanic heat flux. The oceanic
flux is proportional to the difference between ocean surface temperature and
the freezing point temperature of seawater, which is a function of salinity. This
flux is not assumed to instantaneously melt or create ice, but a time scale of
three days is used to relax the ocean temperature to the freezing point. While
this parameterization is not new (it follows the ideas of, e.g., Mellor et al.,
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1986, McPhee, 1992, Lohmann and Gerdes, 1998, Notz et al., 2003), it avoids
a discontinuity in the functional relationship between model variables, which
improves the smoothness of the differentiated model (see Fenty, 2010, for
details). The parameterization of lateral and vertical growth of sea ice follows
that of Hibler (1979, 1980).
On top of the ice there is a layer of snow that modifies the heat flux and
the albedo as in Zhang et al. (1998). If enough snow accumulates so that its
weight submerges the ice and the snow is flooded, a simple mass conserving
parameterization of snow ice formation (a flood-freeze algorithm following
Archimedes’ principle) turns snow into ice until the ice surface is back at
sea-level (Leppäranta, 1983).
The concentration c, effective ice thickness (ice volume per unit area, c · h),
effective snow thickness (c·hs ), and effective ice salinity (in g m−2 ) are advected
by ice velocities. From the various advection schemes that are available in
the MITgcm (MITgcm Group, 2002), we choose flux-limited schemes, that
is, multidimensional 2nd and 3rd-order advection schemes with flux limiters
(Roe, 1985, Hundsdorfer and Trompert, 1994), to preserve sharp gradients
and edges that are typical of sea ice distributions and to rule out unphysical
over- and undershoots (negative thickness or concentration). These schemes
conserve volume and horizontal area and are unconditionally stable, so that
no extra diffusion is required.
There is considerable doubt about the reliability of a “zero-layer” thermodynamic model — Semtner (1984) found significant errors in phase (one month
lead) and amplitude (≈50% overestimate) in such models — so that today
many sea ice models employ more complex thermodynamics. The MITgcm
sea ice model provides the option to use the thermodynamics model of Winton (2000), which in turn is based on the 3-layer model of Semtner (1976) and
which treats brine content by means of enthalpy conservation. This scheme
requires additional state variables, namely the enthalpy of the two ice layers
(instead of effective ice salinity), to be advected by ice velocities. The internal
sea ice temperature is inferred from ice enthalpy. To avoid unphysical (negative) values for ice thickness and concentration, a positive 2nd-order advection
scheme with a SuperBee flux limiter (Roe, 1985) is used in this study to advect all sea-ice-related quantities of the Winton (2000) thermodynamic model.
Because of the non-linearity of the advection scheme, care must be taken in
advecting these quantities: when simply using ice velocity to advect enthalpy,
the total energy (i.e., the volume integral of enthalpy) is not conserved. Alternatively, one can advect the energy content (i.e., product of ice-volume and
enthalpy) but then false enthalpy extrema can occur, which then leads to unrealistic ice temperature. In the currently implemented solution, the sea-ice
mass flux is used to advect the enthalpy in order to ensure conservation of
enthalpy and to prevent false enthalpy extrema.
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In Section 3 and 4 we exercise and compare several of the options, which have
been discussed above; we intercompare the impact of the different formulations
(all of which are widely used in sea ice modeling today) on Arctic sea ice
simulation (Proshutinsky and Kowalik, 2007).
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Global Ocean and Sea Ice Simulation
One example application of the MITgcm sea ice model is the eddy-admitting,
global ocean and sea ice state estimates, which are being generated by the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2) project
(Menemenlis et al., 2005). One particular, unconstrained ECCO2 simulation,
labeled cube76, provides the baseline solution and the lateral boundary conditions for all the numerical experiments carried out in Section 4. Figure 1
shows representative sea ice results from this simulation.
The simulation is integrated on a cubed-sphere grid, permitting relatively even
grid spacing throughout the domain and avoiding polar singularities (Adcroft
et al., 2004). Each face of the cube comprises 510 by 510 grid cells for a mean
horizontal grid spacing of 18 km. There are 50 vertical levels ranging in thickness from 10 m near the surface to approximately 450 m at a maximum model
depth of 6150 m. The model employs the rescaled vertical coordinate “z∗ ”
(Adcroft and Campin, 2004) with partial-cell formulation of Adcroft et al.
(1997), which permits accurate representation of the bathymetry. Bathymetry
is from the S2004 (W. Smith, unpublished) blend of the Smith and Sandwell
(1997) and the General Bathymetric Charts of the Oceans (GEBCO) one arcminute bathymetric grid. In the ocean, the non-linear equation of state of
Jackett and McDougall (1995) is used. Vertical mixing follows Large et al.
(1994) but with meridionally and vertically varying background vertical diffusivity; at the surface, vertical diffusivity is 4.4 × 10−6 m2 s−1 at the Equator,
3.6 × 10−6 m2 s−1 north of 70◦ N, and 1.9 × 10−5 m2 s−1 south of 30◦ S and
between 30◦ N and 60◦ N, with sinusoidally varying values in between these
latitudes; vertically, diffusivity increases to 1.1 × 10−4 m2 s−1 at a depth of
6150 m as per Bryan and Lewis (1979). A 7th-order monotonicity-preserving
advection scheme (Daru and Tenaud, 2004) is employed and there is no explicit
horizontal diffusivity. Horizontal viscosity follows Leith (1996) but is modified
to sense the divergent flow (Fox-Kemper and Menemenlis, 2008). The global
ocean model is coupled to a sea ice model in a configuration similar to the case
C-LSR-ns (see Table 1 in Section 4). The values of open water, dry ice, wet
ice, dry snow, and wet snow albedos are, respectively, 0.15, 0.88, 0.79, 0.97,
and 0.83. These values are relatively high compared to observations and they
were chosen to compensate for deficiencies in the surface boundary conditions
and to produce realistic sea ice extent (Figure 1).
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Fig. 1. Effective sea ice thickness distribution (color, in meters) averaged over the
years 1992–2002 from an eddy-admitting, global ocean and sea ice simulation. The
ice edge estimated as the 15% isoline of modeled ice concentration is drawn as a
white dashed line. The white solid line marks the ice edge, defined as the 15%
isoline of ice concentrations, retrieved from passive microwave satellite data for
comparison. The top row shows the results for the Arctic Ocean and the bottom
row for the Southern Ocean; the left column shows distributions for March and the
right column for September.
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The simulation is initialized in January 1979 from rest and from temperature
and salinity fields derived from the Polar Science Center Hydrographic Climatology (PHC) 3.0 (Steele et al., 2001). Surface boundary conditions are derived
from the European Centre for Medium-Range Weather Forecasts (ECMWF)
40 year re-analysis (ERA-40) (Uppala et al., 2005). Six-hourly surface winds,
temperature, humidity, downward short- and long-wave radiation, and precipitation are converted to heat, freshwater, and wind stress fluxes using the Large
and Yeager (2004) bulk formulae. Shortwave radiation decays exponentially
with depth as per Paulson and Simpson (1977). Low frequency precipitation
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has been adjusted using the pentad (5-day) data from the Global Precipitation Climatology Project (GPCP, Huffman et al., 2001). The time-mean river
run-off from Large and Nurser (2001) is applied globally, except in the Arctic Ocean where monthly mean river runoff based on the Arctic Runoff Data
Base (ARDB) and prepared by P. Winsor (personal communication, 2007) is
specified.
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The remainder of this article discusses results from forward sensitivity experiments in a regional Arctic Ocean model, which operates on a sub-domain of,
and which obtains open boundary conditions from, the cube76 simulation just
described.
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Arctic Ocean Sensitivity Experiments
This section presents results from regional coupled ocean and sea ice simulations of the Arctic Ocean that exercise various capabilities of the MITgcm
sea ice model. The objective is to compare the old B-grid LSOR dynamic
solver with the new C-grid LSOR and EVP solvers. Additional experiments
are carried out to illustrate the differences between different lateral boundary
conditions, ice advection schemes, ocean-ice stress formulations, and alternate
sea ice thermodynamics.
The Arctic Ocean domain has 420 by 384 grid boxes and is illustrated in Figure 2. For each sensitivity experiment, the model is integrated from January 1,
1992 to March 31, 2000. This time period is arbitrary and for comparison purposes only: it was chosen to be long enough to observe systematic differences
due to details of the model configuration and short enough to allow many
sensitivity experiments.
Table 1 gives an overview of all the experiments discussed in this section. In
all experiments except for DST3FL ice is advected with the original second
order central differences scheme that requires small extra diffusion for stability
reasons. The differences between integrations B-LSR-ns and C-LSR-ns can be
interpreted as being caused by model finite dimensional numerical truncation.
Both the LSOR and the EVP solvers aim to solve for the same viscous-plastic
rheology; while the LSOR solver is an iterative scheme with a convergence
criterion the EVP solution relaxes towards the VP solution in the limit of
infinite intergration time. The differences between integrations C-LSR-ns, CEVP-10, and C-EVP-03 are caused by fundamentally different approaches to
regularize large bulk and shear viscosities; LSOR and other iterative techniques need to clip large viscosities, while EVP introduces elastic waves that
damp out within one sub-cycling sequence. Both LSOR and EVP solutions
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Fig. 2. Bathymetry and domain boundaries of Arctic Domain, cut-out from the
global solution. The white line encloses what is loosely referred to as the Canadian Arctic Archipelago in the text. The letters label sections in the Canadian
Archipelago, where ice transport is evaluated: A: Nares Strait; B: Peary Channel;
C: Prince Gustaf Adolf Sea; D: Ballantyne Strait; E: M’Clure Strait; F: Amundsen
Gulf; G: Lancaster Sound; H: Barrow Strait W.; I: Barrow Strait E.; J: Barrow
Strait N.; K: Fram Strait. The sections A through F comprise the total Arctic
inflow into the Canadian Archipelago. The white labels denote Ellesmere Island
of the Queen Elizabeth Islands (QEI), Svalbard (SB), Franz Joseph Land (FJL),
Severnaya Zemlya (SZ), and the New Siberian Islands (NSI).
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represent approximations to true viscous-plastic rheology and neither will be
considered “truth” in our comparisons: On the one hand, LSOR (and other
implicit solvers) requires many so-called pseudo time steps to fully converge
in a non-linear sense (Lemieux and Tremblay, 2009), which makes this type of
solver very expensive. We use only 2 (customary) pseudo time steps. On the
other hand, the elastic wave energy in EVP damps out completely only after
an infinite time compared to the damping time scale, so that in practice the
rheology is not completely viscous-plastic.
For the EVP solver we use two different damping time scales and sub-cycling
time steps. In the C-EVP-10 experiment, the damping time scale is one third
of the ocean model times step; the EVP model is sub-cycled 120 times within
each 1200 s ocean model time step resulting in ∆tevp = 10 s. In the C-EVP-03
experiment, we reduce the damping time scale to a tenth of the ocean model
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Table 1
Overview of forward model sensitivity experiments in a regional Arctic Ocean domain.
Experiment
Description
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C-LSR-ns
The LSOR solver discretized on a C grid with no-slip lateral
boundary conditions (implemented via ghost-points), advection
of ice variables with a 2nd-order central difference scheme plus
explicit diffusion for stability.
B-LSR-ns
The original LSOR solver of Zhang and Hibler (1997) on an
Arakawa B grid, implying no-slip lateral boundary conditions
(u = 0 exactly).
C-EVP-10
The EVP solver of Hunke (2001) on a C grid with no-slip lateral
boundary conditions and ∆tevp = 10 s (=
b 120 subcycling steps).
C-EVP-03
The EVP solver of Hunke (2001) on a C grid with no-slip lateral
boundary conditions and ∆tevp = 3 s (=
b 400 subcycling steps).
C-LSR-fs
The LSOR solver on a C grid with free-slip lateral boundary
conditions (no lateral stress on coast lines).
DST3FL
C-LSR-ns with a third-order flux limited direct-space-time advection scheme for thermodynamic variables (Hundsdorfer and
Trompert, 1994).
TEM
C-LSR-ns with a truncated ellipse method (TEM) rheology (Hibler and Schulson, 1997).
HB87
C-LSR-ns with ocean-ice stress coupling according to Hibler and
Bryan (1987).
WTD
C-LSR-ns with 3-layer thermodynamics following Winton
(2000).
time step to achieve faster damping of elastic waves. In this case, the EVP
model is sub-cycled 400 times within an ocean model time step with a time
step of 3 seconds in order to resolve the shorter damping time scale. Table 2
shows timings for these cases. Note that in our configuration on 36 CPUs of a
SGI Altix 3700 the EVP technique is faster than LSOR for the 10 seconds time
step (C-EVP-10); the shorter time step of 3 seconds was chosen to arrive at
approximately the same computational effort as for C-LSR-ns. For comparison
purposes, Hunke (2001) used a sub-cycling time step of 30 s for an ocean model
time step of 3600 s and a damping time scale of 1296 s.
Lateral boundary conditions on a coarse grid (coarse compared to the roughness of the true coast line) are ill-defined so that comparing a no-slip solution
(C-LSR-ns) to a free-slip solution (C-LSR-fs) gives another measure of uncertainty in the sea ice model. The sensitivity experiments also explore the
response of the coupled ocean and sea ice model to different numerics and
12
Table 2
Integration throughput on 36 CPUs of a SGI Altix 3700.
Wall clock per integration month (2232 time steps)
339
340
341
342
343
344
345
346
347
348
349
Experiment
ice dynamics
entire model
C-LSR-ns
600 sec
2887 sec
C-EVP-10
262 sec
2541 sec
C-EVP-03
875 sec
3159 sec
physics, that is, to changes in advection and diffusion properties (DST3FL), in
rheology (TEM), in stress coupling (HB87), and in thermodynamics (WTD).
Comparing the solutions obtained with different realizations of the model dynamics is difficult because of the non-linear feedback of the ice dynamics and
thermodynamics. Already after a few months the model trajectories have diverged far enough so that velocity differences are easier to interpret within the
first 3 months of the integration while the ice distributions are still comparable. The effect on ice-thickness of different numerics tends to accumulate along
the time integration, resulting in larger differences - also easier to interpret at the end of the integration. We choose C-LSR-ns as the reference run for all
comparisons bearing in mind that any other choice is equally valid.
351
Tables 3 and 4 summarize the differences in drift speed and effective ice thickness for all experiments. These differences are discussed in detail below.
352
4.1
350
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
Ice velocities in JFM 1992
Figure 3 shows ice velocities averaged over January, February, and March
(JFM) of 1992 for the C-LSR-ns solution; also shown are the differences between this reference solution and various sensitivity experiments. The velocity
field of the C-LSR-ns solution (Figure 3a) roughly resembles the drift velocities of some of the AOMIP (Arctic Ocean Model Intercomparison Project)
models in a cyclonic circulation regime (Martin and Gerdes, 2007, their Figure 6) with a Beaufort Gyre and a Transpolar Drift shifted eastwards towards
Alaska.
The difference between experiments C-LSR-ns and B-LSR-ns (Figure 3b) is
most pronounced (∼ 2 cm/s) along the coastlines, where the discretization
differs most between B and C grids. On a B grid the tangential velocity lies
on the boundary, and is thus zero through the no-slip boundary conditions,
whereas on the C grid it is half a cell width away from the boundary, thus
allowing more flow. The B-LSR-ns solution has less ice drift through the Fram
Strait and along Greenland’s East Coast; also, the flow through Baffin Bay and
13
Table 3
Overview over drift speed differences (JFM of first year of integration) and effective
ice thickness differences (JFM of last year of integration) relative to C-LSR-ns. For
reference the corresponding values for C-LSR-ns are given in the first line.
speed (cm/s)
mean
rms
median
max
3.295
4.711
2.502
28.599
B-LSR-ns
-0.236
0.714
-0.071
14.355
C-EVP-10
0.266
0.513
0.213
10.506
C-EVP-03
0.198
0.470
0.143
10.407
C-LSR-fs
0.160
0.472
0.084
9.921
DST3FL
0.035
0.301
0.008
10.251
TEM
0.027
0.168
0.014
8.922
HB87
0.184
0.316
0.169
9.175
WTD
0.354
1.418
0.039
26.298
mean
rms
median
max
C-LSR-ns (ref)
1.599
1.941
1.542
10.000
B-LSR-ns
0.065
0.175
0.049
2.423
C-EVP-10
-0.082
0.399
-0.020
5.993
C-EVP-03
-0.069
0.374
-0.014
5.688
C-LSR-fs
-0.037
0.289
-0.005
3.947
DST3FL
0.014
0.338
-0.018
9.246
TEM
-0.020
0.138
-0.001
2.541
HB87
-0.052
0.114
-0.029
2.520
WTD
0.518
0.667
0.528
4.144
C-LSR-ns (ref)
thickness (m)
368
369
370
371
372
373
374
Davis Strait into the Labrador Sea is reduced with respect to the C-LSR-ns
solution.
The C-EVP-10 solution with ∆tevp = 10 s allows for increased drift by order
1 cm/s in the Beaufort Gyre and in the Transpolar Drift. In general, drift
velocities tend towards higher values in the EVP solution with a root-meansquare (rms) difference of 0.51 cm/s. As the number of sub-cycling time steps
increases, the EVP approximation converges towards VP dynamics: the C14
Table 4
Root-mean-square differences for drift speed (JFM of first year of integration)
and effective thickness (JFM of last year of integration) for the “Candian Arctic
Archipelago” defined in Figure 2 and the remaining domain (“rest”). For reference
the corresponding values for C-LSR-ns are given in the first line.
rms(speed) (cm/s)
375
376
377
378
379
380
381
382
383
rms(thickness) (m)
total
CAA
rest
total
CAA
rest
C-LSR-ns (ref)
4.711
1.425
5.037
1.941
3.304
1.625
B-LSR-ns
0.714
0.445
0.747
0.175
0.369
0.117
C-EVP-10
0.513
0.259
0.543
0.399
1.044
0.105
C-EVP-03
0.470
0.234
0.497
0.374
0.982
0.095
C-LSR-fs
0.472
0.266
0.497
0.289
0.741
0.099
DST3FL
0.301
0.063
0.323
0.338
0.763
0.201
TEM
0.168
0.066
0.179
0.138
0.359
0.040
HB87
0.316
0.114
0.337
0.114
0.236
0.079
WTD
1.418
1.496
1.406
0.667
1.110
0.566
EVP-03 solution with ∆tevp = 3 s (Figure 3d) is closer to the C-LSR-ns solution (root-mean-square of 0.47 cm/s and only 0.23 cm/s in the CAA). Both
EVP solutions have a stronger Beaufort Gyre as in Hunke and Zhang (1999).
As expected the differences between C-LSR-fs and C-LSR-ns (Figure 3e) are
also largest (∼ 2 cm/s) along the coastlines. The free-slip boundary condition
of C-LSR-fs allows the flow to be faster, for example, along the East Coast of
Greenland, the North Coast of Alaska, and the East Coast of Baffin Island, so
that the ice drift for C-LSR-fs is on average faster than for C-LSR-ns where
for B-LSR-ns it is on average slower.
394
The more sophisticated advection scheme of DST3FL (Figure 3f) has the
largest effect along the ice edge (see also Merryfield and Holloway, 2003),
where the gradients of thickness and concentration are largest and differences
in velocity can reach 5 cm/s (maximum differences are 10 cm/s at individual
grid points). Everywhere else the effect is very small (rms of 0.3 cm/s) and
can mostly be attributed to smaller numerical diffusion (and to the absence
of explicit diffusion that is required for numerical stability in a simple second
order central differences scheme). Note, that the advection scheme has an
indirect effect on the ice drift, but a direct effect on the ice transport, and
hence the ice thickness distribution and ice strength; a modified ice strength
then leads to a modified drift field.
395
Compared to the other parameters, the ice rheology TEM (Figure 3g) also has
384
385
386
387
388
389
390
391
392
393
15
(a) C-LSR-ns
(b) B-LSR-ns − C-LSR-ns
(c) C-EVP-10 − C-LSR-ns
(d) C-EVP-03 − C-LSR-ns
Fig. 3. (a) Ice drift velocity of the C-LSR-ns solution averaged over the first 3 months
of integration (cm/s); (b)-(h) difference between the C-LSR-ns reference solution
and solutions with, respectively, the B-grid solver, the EVP-solver with ∆tevp = 10 s,
the EVP-solver with ∆tevp = 3 s, free lateral slip, a different advection scheme
(DST3FL) for thermodynamic variables, the truncated ellipse method (TEM), and
a different ice-ocean stress formulation (HB87). Color indicates speed or differences
of speed and vectors indicate direction only. The direction vectors represent block
averages over eight by eight grid points at every eighth velocity point. Note that
color scale varies from panel to panel.
396
397
398
399
400
401
402
403
404
405
406
407
a very small (mostly < 0.5 cm/s and the smallest rms-difference of all solutions) effect on the solution. In general the ice drift tends to increase because
there is no tensile stress and ice can drift apart at no cost. Consequently,
the largest effect on drift velocity can be observed near the ice edge in the
Labrador Sea. Note in experiments DST3FL and TEM the drift pattern is
slightly changed as opposed to all other C-grid experiments, although this
change is small.
By way of contrast, the ice-ocean stress formulation of Hibler and Bryan (1987)
results in stronger drift by up to 2 cm/s almost everywhere in the computational domain (Figure 3h). The increase is mostly aligned with the general
direction of the flow, implying that the Hibler and Bryan (1987) stress formulation reduces the deceleration of drift by the ocean.
16
(e) C-LSR-fs − C-LSR-ns
(f) DST3FL − C-LSR-ns
(g) TEM − C-LSR-ns
(h) HB87 − C-LSR-ns
Fig. 3. Continued.
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
4.2
Integrated effect on ice volume during JFM 2000
Figure 4a shows the effective thickness (volume per unit area) of the C-LSR-ns
solution, averaged over January, February, and March of year 2000, that is,
eight years after the start of the simulation. By this time of the integration,
the differences in ice drift velocities have led to the evolution of very different
ice thickness distributions (as shown in Figs. 4b–h) and concentrations (not
shown) for each sensitivity experiment. The mean ice volume for the January–
March 2000 period is also reported in Table 5.
The generally weaker ice drift velocities in the B-LSR-ns solution, when compared to the C-LSR-ns solution, in particular through the narrow passages in
the Canadian Arctic Archipelago, where the B-LSR-ns solution tends to block
channels more often than the C-LSR-ns solution, lead to a larger build-up of
ice (2 m or more) north of Greenland and north of the Archipelago in the Bgrid solution (Figure 4b). The ice volume, however, is not larger everywhere.
Further west there are patches of smaller ice volume in the B-grid solution,
most likely because the Beaufort Gyre is weaker and hence not as effective in
transporting ice westwards. There is no obvious explanation, why the ice is
17
(a) C-LSR-ns
(b) B-LSR-ns − C-LSR-ns
(c) C-EVP-10 − C-LSR-ns
(d) C-EVP-03 − C-LSR-ns
Fig. 4. (a) Effective thickness (volume per unit area) of the C-LSR-ns solution, averaged over the months January through March 2000 (m); (b)-(h) difference between
the C-LSR-ns reference solution and solutions with, respectively, the B-grid solver,
the EVP-solver with ∆tevp = 10 s, the EVP-solver with ∆tevp = 3 s, free lateral slip,
a different advection scheme (DST3FL) for thermodynamic variables, the truncated
ellipse method (TEM), and a different ice-ocean stress formulation (m).
425
426
427
428
429
430
431
432
433
434
435
436
437
438
thinner in the western part of the Canadian Archipelago. We attribute this
difference to the different effective slipperiness of the coastlines in the two
solutions, because in the free-slip solution the pattern is reversed. There are
also dipoles of ice volume differences with more ice on the upstream side and
less ice on the downstream side of island groups, for example, of Franz Josef
Land, of Severnaya Zemlya, of the New Siberian Islands, and of the Queen
Elizabeth Islands (see Figure 2 for their geographical locations). This is because ice tends to flow less easily along coastlines, around islands, and through
narrow channels in the B-LSR-ns solution than in the C-LSR-ns solution.
The C-EVP-10 solution with ∆tevp = 10 s has thinner ice in the Candian
Archipelago and in the central Arctic Ocean than the C-LSR-ns solution
(Figure 4c); the rms difference between C-EVP-10 and C-LSR-ns ice thickness is 40 cm. Thus it is larger than the rms difference between B- and CLSR-ns, mainly because within the Canadian Arctic Archipelago more drift
18
(e) C-LSR-fs − C-LSR-ns
(f) DST3FL − C-LSR-ns
(g) TEM − C-LSR-ns
(h) HB87 − C-LSR-ns
Fig. 4. Continued.
Table 5
Arctic ice volume averaged over Jan–Mar 2000, in km3 . Mean ice transport (and
standard deviation in parenthesis) for the period Jan 1992 – Dec 1999 through the
Fram Strait (FS), the total northern inflow into the Canadian Arctic Archipelago
(CAA), and the export through Lancaster Sound (LS), in km3 y−1 .
Volume
Sea ice transport (km3 yr−1 )
Experiment
(km3 )
FS
CAA
LS
C-LSR-ns
24,769
2196 (1253)
70 (224)
77 (110)
B-LSR-ns
23,824
2126 (1278)
34 (122)
43 (76)
C-EVP-10
22,633
2174 (1260)
186 (496)
133 (128)
C-EVP-03
22,819
2161 (1252)
175 (461)
123 (121)
C-LSR-fs
23,286
2236 (1289)
80 (276)
91 (85)
DST3FL
24,023
2191 (1261)
88 (251)
84 (129)
TEM
23,529
2222 (1258)
60 (242)
87 (112)
HB87
23,060
2256 (1327)
64 (230)
77 (114)
WTD
31,634
2761 (1563)
23 (140)
94 (63)
19
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
in C-EVP-10 leads to faster ice export and to reduced effective ice thickness.
With a shorter time step (∆tevp = 3 s) the EVP solution converges towards
the LSOR solution in the central Arctic (Figure 4d). In the narrow straits in
the Archipelago, however, the ice thickness is not affected by the shorter time
step and the ice is still thinner by 2 m or more, as it is in the EVP solution
with ∆tevp = 10 s.
Imposing a free-slip boundary condition in C-LSR-fs leads to much smaller
differences to C-LSR-ns (Figure 4e) than the transition from the B grid to the
C grid, except in the Canadian Arctic Archipelago, where the free-slip solution
allows more flow (see Table 4). There, it reduces the effective ice thickness by
2 m or more where the ice is thick and the straits are narrow (leading to an
overall larger rms-difference than the B-LSR-ns solution, see Table 4). Dipoles
of ice thickness differences can also be observed around islands because the
free-slip solution allows more flow around islands than the no-slip solution.
The differences in the Central Arctic are much smaller in absolute value than
the differences in the Canadian Arctic Archipelago although there are also
interesting changes in the ice-distribution in the interior: Less ice in the Central
Arctic is most likely caused by more export (see Table 5).
The remaining sensitivity experiments, DST3FL, TEM, and HB87, have the
largest differences in effective ice thickness along the north coasts of Greenland
and Ellesmere Island in the Canadian Arctic Archipelago. Although using the
TEM rheology and the Hibler and Bryan (1987) ice-ocean stress formulation
has different effects on the initial ice velocities (Figure 3g and h), both experiments have similarly reduced ice thicknesses in this area. The 3rd-order advection scheme (DST3FL) has an opposite effect of similar magnitude, pointing
towards more implicit lateral stress with this numerical scheme. The HB87 experiment shows ice thickness reduction in the entire Arctic basin greater than
in any other experiment, possibly because more drift leads to faster export of
ice.
Figure 5 summarizes Figures 3 and 4 by showing histograms of sea ice thickness
and drift velocity differences to the reference C-LSR-ns. The black line is the
cumulative number grid points in percent of all grid points of all models where
differences up to the value on the abscissa are found. For example, ice thickness
differences up to 50 cm are found in 90% of all grid points, or equally differences
above 50 cm are only found in 10% of all grid points. The colors indicate the
distribution of these grid points between the various experiments. For example,
65% to 90% of grid points with ice thickness differences between 40 cm and
1 m are found in the run WTD. The runs B-LSR-ns, C-EVP-10, and HB87
only have a fairly large number of grid points with differences below 40 cm.
B-LSR-ns and WTD dominate nearly all velocity differences. The remaining
contributions are small except for small differences below 1 cm/s. Only very
few points contribute to very large differences in thickness (above 1 m) and
20
Fig. 5. Histograms of ice thickness and drift velocity differences relative to C-LSR-ns;
the bin-width is 2 cm for thickness and 0.1 cm/s for speed. The black line is the
cumulative number of grid points in percent of all grid points. The colors indicate
the distribution of these grid points between the various experiments in percent of
the black line.
482
velocity (above 4 cm/s) indicated by the small slope of the cumlative number
of grid point (black line).
483
4.3
481
484
485
486
487
488
489
490
491
492
493
494
495
Ice transports
The difference in ice volume and in ice drift velocity between the various
sensitivity experiments has consequences for sea ice export from the Arctic
Ocean. As an illustration (other years are similar), Figure 6 shows the 1996
time series of sea ice transports through the northern edge of the Canadian
Arctic Archipelago, through Lancaster Sound, and through Fram Strait for
each model sensitivity experiment. The mean and standard deviation of these
ice transports, over the period January 1992 to December 1999, are reported
in Table 5. In addition to sea ice dynamics, there are many factors, e.g., atmospheric and oceanic forcing, drag coefficients, and ice strength, that control sea
ice export. Although calibrating these various factors is beyond the scope of
this manuscript, it is nevertheless instructive to compare the values in Table 5
with published estimates, as is done next. This is a necessary step towards con21
Fig. 6. Transports of sea ice during 1996 for model sensitivity experiments listed in
Table 1. Top panel shows flow through the northern edge of the Canadian Arctic
Archipelago (Sections A–F in Figure 2), middle panel shows flow through Lancaster
Sound (Section G), and bottom panel shows flow through Fram Strait (Section K).
Positive values indicate sea ice flux out of the Arctic Ocean. The time series are
smoothed using a monthly running mean. The mean range, i.e., the time-mean
difference between the model solution with maximum flux and that with minimum
flux, is computed over the period January 1992 to December 1999.
496
497
498
499
500
501
straining this model with data, a key motivation for developing the MITgcm
sea ice model and its adjoint.
The export through Fram Strait for all the sensitivity experiments is consistent
with the value of 2300 ± 610 km3 yr−1 reported by Serreze et al. (2006, and
references therein). Although Arctic sea ice is exported to the Atlantic Ocean
principally through the Fram Strait, Serreze et al. (2006) estimate that a
22
502
503
504
505
506
507
508
509
510
511
512
considerable amount of sea ice (∼ 160 km3 yr−1 ) is also exported through the
Canadian Arctic Archipelago. This estimate, however, is associated with large
uncertainties. For example, Dey (1981) estimates an inflow into Baffin Bay of
370 to 537 km3 yr−1 but a flow of only 102 to 137 km3 yr−1 further upstream in
Barrow Strait in the 1970’s from satellite images; Aagaard and Carmack (1989)
give approximately 155 km3 yr−1 for the export through the CAA. The recent
estimates of Agnew et al. (2008) for Lancaster Sound are lower: 102 km3 yr−1 .
The model results suggest annually averaged ice transports through Lancaster
Sound ranging from 43 to 133 km3 yr−1 and total northern inflow of 34 to
186 km3 yr−1 (Table 5). These model estimates and their standard deviations
cannot be rejected based on the observational estimates.
522
Generally, the EVP solutions have the highest maximum (export out of the
Arctic) and lowest minimum (import into the Arctic) fluxes as the drift velocities are largest in these solutions. In the extreme of the Nares Strait, which
is only a few grid points wide in our configuration, both B- and C-grid LSOR
solvers lead to practically no ice transport, while the EVP solutions allow
200–500 km3 yr−1 in summer (not shown). Tang et al. (2004) report 300 to
350 km3 yr−1 and Kwok (2005) 130 ± 65 km3 yr−1 . As as consequence, the import into the Canadian Arctic Archipelago is larger in all EVP solutions than
in the LSOR solutions. The B-LSR-ns solution is even smaller by another
factor of two than the C-LSR solutions.
523
4.4
513
514
515
516
517
518
519
520
521
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
Thermodynamics
The last sensitivity experiment (WTD) listed in Table 1 is carried out using
the 3-layer thermodynamics model of Winton (2000). This experiment has
different albedo and basal heat exchange formulations from all the other experiments. Although, the upper-bound albedo values for dry ice, dry snow, and
wet snow are the same as for the zero-layer model, the ice albedos in WTD are
computed following Hansen et al. (1983) and can become much smaller as a
function of thickness h, with a minimum value of 0.2 exp(−h/0.44 m). Further
the snow age is taken into account when computing the snow albedo. With
the same values for wet snow (0.83), dry snow (0.97), and dry ice (0.88) as
for the zero-heat-capacity model (see Section 3), this results in albedos that
range from 0.22 to 0.95 (not shown). Similarly, large differences can be found
in the basal heat exchange parameterizations. For this reason, the resulting
ice velocities, volume, and transports have not been included in the earlier
comparisons. However, this experiment gives another measure of uncertainty
associated with ice modeling. The key difference with the “zero-layer” thermodynamic model is a delay in the seaice cycle of approximately one month in the
maximum sea-ice thickness and two months in the minimum sea-ice thickness.
This is shown in Figure 7, which compares the mean sea-ice thickness seasonal
23
Fig. 7. Seasonal cycle of mean sea-ice thickness (cm) in a sector in the western Arctic
(75◦ N to 85◦ N and 180◦ W to 140◦ W) averaged over 1992–2000 of experiments
C-LSR-ns and WTD.
547
cycle of experiments with the zero-heat-capacity (C-LSR-ns) and three-layer
(WTD) thermodynamic model. The mean ice thickness is computed for a sector in the western Arctic (75◦ N to 85◦ N and 180◦ W to 140◦ W) in order to
avoid confounding thickness and extent differences. Similar to Semtner (1976),
the seasonal cycle for the “zero-layer” model (gray dashed line) is almost twice
as large as for the three-layer thermodynamic model.
548
5
542
543
544
545
546
549
550
551
552
553
554
555
556
Conclusions
We have shown that changes in discretization details, in boundary conditions,
and in sea-ice-dynamics formulation lead to considerable differences in model
results. Notably the sea-ice-dynamics formulation, e.g., B-grid versus C-grid or
EVP versus LSOR, has as much or even greater influence on the solution than
physical parameterizations, e.g., free-slip versus no-slip boundary conditions.
This is especially true
• in regions of convergence (see ice thickness north of Greenland in Fig. 4),
• along coasts (see eastern coast of Greenland in Fig. 3 where velocity differ24
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
ences are apparent),
• and in the vicinity of straits (see the Canadian Arctic Archipelago in Figs. 3
and 4).
These experiments demonstrate that sea-ice export from the Arctic into both
the Baffin Bay and the GIN (Greenland/Iceland/Norwegian) Sea regions is
highly sensitive to numerical formulation. Changes in export in turn impact
deep-water mass formation in the northern North Atlantic. Therefore uncertainties due to numerical formulation might potentially have wide reaching
impacts outside of the Arctic.
The relatively large differences between solutions with different dynamical
solvers is somewhat surprising. The expectation was that the solution technique should not affect the solution to a higher degree than actually modifying
the equations. The EVP solutions tend to produce effectively “weaker” ice that
yields more easily to stress than the LSOR solutions, similar to the findings
in Hunke and Zhang (1999). The differences between LSOR and EVP can, in
part, stem from incomplete convergence of the solvers due to linearization and
due to different methods of linearization (Hunke, 2001, and B. Tremblay, pers.
comm. 2008). We note that the EVP-to-LSOR differences decrease with decreasing sub-cycling time step but that the difference remains significant even
at a 3-second sub-cycling period. For the LSOR solutions we use 2 pseudo
time steps so that the convergence of the non-linear momentum equations
may not be complete. This effect is most likely reduced and constrained to
small areas as in Lemieux and Tremblay (2009) because of the small time step
that we used. Whether more pseudo time steps make the LSOR solution generate weaker ice requires further investigation. Preliminary tests indicate that
the viscosity increases with increasing number of LSOR pseudo time steps,
especially in areas of thick ice (not shown).
Other numerical formulation choices that were tested include switching from
one horizontal grid staggering (C-grid) to another (B-grid). This change significantly affects narrow straits, for example, in the Canadian Arctic Archipelago,
and subsequent conditions upstream and downstream of the straits. It also
affects flows of ice along the West Greenland coast. Similar, but smaller, differences between B-grid and C-grid sea ice solutions were noted in the coarserresolution study of Bouillon et al. (2009). The differences between the no-slip
and free-slip lateral boundary conditions are also most significant near the
coast. As in the case of oceanic boundary conditions (Adcroft and Marshall,
1998), we expect that the changes are due to the effective “slipperiness” of
the coastline boundary condition.
The flux-limited scheme without explicit diffusion (DST3FL) is recommended.
This is because the flux-limited scheme preserves sharp gradients and edges
that are typical of sea ice distributions and because it avoids unphysical (neg25
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ative) values for ice thickness and concentration (see also Merryfield and Holloway, 2003). The flux limited scheme conserves volume and horizontal area
and is unconditionally stable, so that no extra diffusion is required.
Changing the ice rheology to the truncated ellipse method (TEM) primarily impacts the solution in the Canadian Arctic Archipelago and the West
Greenland coast as does altering the stress formulation on the ice solution.
We interpret this result as indicating that the CAA and West Greenland current are regions of high-sensitivity. Here, more ice leads to a rigid structure
that inhibits ice flow and yields ice accumulation upstream.
Although the Hibler and Bryan (1987) stress formulation appears more natural
for advecting sea ice, the advection of oceanic properties is problematic: Thermodynamic and passive tracers in the top ocean model level are advected with
a velocity that is the average over ice drift and ocean currents rather than an
average of surface oceanic currents alone. For our purposes, the preferred iceocean coupling uses the rescaled vertical coordinates of Campin et al. (2008),
which allows the ice to depress the ocean surface according to its thickness
and buoyancy.
A few comments regarding the robustness of our results against choice of
forcing, integration period, and horizontal resolution follow. Strictly speaking,
our results refer to an 8-year integration with 18 km horizontal grid spacing.
We find that the differences between the solutions have an obvious trend after
the first season but that this trend flattens out after a few seasons. We do
not expect the differences to increase dramatically with additional integration
time, since the simulated multi-year sea ice has reached a quasi equilibrium.
Surface atmospheric conditions are specified every 6 hours. Models with weaker
ice can react more quickly to a change in wind forcing, therefore we speculate
that the differences between EVP and LSOR integrations would change with
different forcing: less variable wind forcing would lead to smaller differences,
while larger fluctuations in the forcing would increase them. In the same way,
we expect that with coarser grids, the ocean component is much less variable
so that in this case one will only find smaller differences between ice models.
The MITgcm sea ice model enables, within the same code, the direct comparison of various widely used dynamics and thermodynamics model components.
What sets apart the MITgcm sea ice model from other current-generation sea
ice models is the ability to derive an accurate, stable, and efficient adjoint
model using automatic differentiation source transformation tools. This capability is the topic of a companion, second paper. The adjoint model greatly
facilitates and enhances exploration of the model’s parameter space. It lays
the foundation for coupled ocean and sea ice state estimation.
26
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A
Dynamics
For completeness we provide more details on the ice dynamics of the sea-ice
model. The momentum equations are
m
Du
= −mf k × u + τ air + τ ocean − m∇φ(0) + F,
Dt
(A.1)
where m = mi + ms is the ice and snow mass per unit area; u = ui + vj is the
ice velocity vector; i, j, and k are unit vectors in the x, y, and z directions,
respectively; f is the Coriolis parameter; τ air and τ ocean are the wind-ice
and ocean-ice stresses, respectively; g is the gravity acceleration; ∇φ(0) is the
gradient (or tilt) of the sea surface height; φ(0) = gη +pa /ρ0 +mg/ρ0 is the sea
surface height potential in response to ocean dynamics (gη), to atmospheric
pressure loading (pa /ρ0 , where ρ0 is a reference density) and a term due to
snow and ice loading (Campin et al., 2008); and F = ∇ · σ is the divergence of
the internal ice stress tensor σij . Advection of sea-ice momentum is neglected.
The wind and ice-ocean stress terms are given by
τ air =ρair Cair |Uair − u|Rair (Uair − u),
τ ocean =ρocean Cocean |Uocean − u|Rocean (Uocean − u),
where Uair/ocean are the surface winds of the atmosphere and surface currents of the ocean, respectively; Cair/ocean are air and ocean drag coefficients;
ρair/ocean are reference densities; and Rair/ocean are rotation matrices that act
on the wind/current vectors. In this paper both rotation angles are set to zero.
For an isotropic system the stress tensor σij (i, j = 1, 2) can be related to the
ice strain rate and strength by a nonlinear viscous-plastic (VP) constitutive
law (Hibler, 1979, Zhang and Hibler, 1997):
σij = 2η(˙ij , P )˙ij + [ζ(˙ij , P ) − η(˙ij , P )] ˙kk δij −
P
δij .
2
(A.2)
The ice strain rate is given by
1
˙ij =
2
∂ui ∂uj
+
∂xj
∂xi
!
.
The maximum ice pressure Pmax , a measure of ice strength, depends on both
thickness h and compactness (concentration) c:
Pmax = P ∗ c h e[C
∗ ·(1−c)]
,
(A.3)
with the constants P ∗ and C ∗ ; we use P ∗ = 27 500 N m−2 and C ∗ = 20. The
nonlinear bulk and shear viscosities η and ζ are functions of ice strain rate
invariants and ice strength such that the principal components of the stress
27
lie on an elliptical yield curve with the ratio of major to minor axis e equal to
2; they are given by:
Pmax
, ζmax
ζ = min
2 max(∆, ∆min )
ζ
η= 2
e
!
with the abbreviation
∆=
667
668
669
670
671
672
673
674
h
i1
˙211 + ˙222 (1 + e−2 ) + 4e−2 ˙212 + 2˙11 ˙22 (1 − e−2 )
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
.
In the simulations of this paper, the bulk viscosities are bounded above by imposing both a minimum ∆min = 10−11 s−1 and a maximum ζmax = Pmax /∆∗ ,
where ∆∗ = (5 × 1012 /2 × 104 ) s−1 . For stress tensor computation the replacement pressure P = 2 ∆ζ (Hibler and Ip, 1995) is used so that the stress state
always lies on the elliptic yield curve by definition.
In the so-called truncated ellipse method (experiment TEM) the shear viscosity η is capped to suppress any tensile stress (Hibler and Schulson, 1997,
Geiger et al., 1998):


675
2
P
− ζ(˙11 + ˙22 ) 
ζ
.
η = min  2 , q 2
e
(˙11 + ˙22 )2 + 4˙212
(A.4)
In the current implementation, the VP-model is integrated with the semiimplicit line successive over relaxation (LSOR)-solver of Zhang and Hibler
(1997), which allows for long time steps that, in our case, are limited by the
explicit treatment of the Coriolis term. The explicit treatment of the Coriolis
term does not represent a severe limitation because it restricts the time step
to approximately the same length as in the ocean model where the Coriolis
term is also treated explicitly.
Hunke and Dukowicz (1997) introduced an elastic contribution to the strain
rate in order to regularize Eq. A.2 in such a way that the resulting elasticviscous-plastic (EVP) and VP models are identical at steady state,
1 ∂σij
1
η−ζ
P
+ σij +
σkk δij + δij = ˙ij .
E ∂t
2η
4ζη
4ζ
(A.5)
The EVP-model uses an explicit time stepping scheme with a short time step.
According to the recommendation of Hunke and Dukowicz (1997), the EVPmodel is stepped forward in time O(120) times within the physical ocean model
time step, to allow for elastic waves to disappear. Because the scheme does
not require a matrix inversion it is fast in spite of the small internal time step
28
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693
694
695
696
697
and simple to implement on parallel computers (Hunke and Dukowicz, 1997).
For completeness, we repeat the equations for the components of the stress
tensor σ1 = σ11 + σ22 , σ2 = σ11 − σ22 , and σ12 . Introducing the divergence
DD = ˙11 + ˙22 , and the horizontal tension and shearing strain rates, DT =
˙11 − ˙22 and DS = 2˙12 , respectively, and using the above abbreviations, the
equations A.5 can be written as:
σ1
P
P
∂σ1
+
+
=
DD
∂t
2T
2T
2T ∆
∂σ2 σ2 e2
P
+
=
DT
∂t
2T
2T ∆
∂σ12 σ12 e2
P
+
=
DS
∂t
2T
4T ∆
698
699
700
701
(A.6)
(A.7)
(A.8)
Here, the elastic parameter E is redefined in terms of a damping time scale T
for elastic waves
E=
ζ
.
T
705
T = E0 ∆t with the tunable parameter E0 < 1 and the external (long) time
step ∆t. In experiment C-EVP-10 use E0 = 31 which is close to value of 0.36
1
used by Hunke (2001). In experiment C-EVP-03 we use E0 = 10
resulting in
T = 120 s for our choice of ∆t.
706
B
702
703
704
707
708
709
710
711
712
713
714
715
716
Finite-volume discretization of the stress tensor divergence
On an Arakawa C grid, ice thickness and concentration and thus ice strength
P and bulk and shear viscosities ζ and η are naturally defined at C-points in
the center of the grid cell. Discretization requires only averaging of ζ and η to
Z
vorticity or Z-points at the bottom left corner of the cell to give ζ and η Z .
In the following, the superscripts indicate location at Z or C points, distance
across the cell (F), along the cell edge (G), between u-points (U), v-points
(V), and C-points (C). The control volumes of the u- and v-equations in the
s
grid cell at indices (i, j) are Aw
i,j and Ai,j , respectively. With these definitions
(which follow the model code documentation at http://mitgcm.org except
that vorticity or ζ-points have been renamed to Z-points in order to avoid
29
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718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
confusion with the bulk viscosity ζ), the strain rates are discretized as:
˙11 = ∂1 u1 + k2 u2
ui+1,j − ui,j
C vi,j+1 + vi,j
+ k2,i,j
=> (11 )C
i,j =
F
∆xi,j
2
˙22 = ∂2 u2 + k1 u1
vi,j+1 − vi,j
C ui+1,j + ui,j
+ k1,i,j
=> (22 )C
i,j =
F
∆yi,j
2
!
1
˙12 = ˙21 =
∂1 u2 + ∂2 u1 − k1 u2 − k2 u1
2
1 vi,j − vi−1,j ui,j − ui,j−1
=> (12 )Zi,j =
+
U
2
∆xVi,j
∆yi,j
!
Z vi,j + vi−1,j
Z ui,j + ui,j−1
− k1,i,j
− k2,i,j
,
2
2
(B.1)
(B.2)
(B.3)
so that the diagonal terms of the strain rate tensor are naturally defined at
C-points and the symmetric off-diagonal term at Z-points. No-slip boundary
conditions (ui,j−1 + ui,j = 0 and vi−1,j + vi,j = 0 across boundaries) are implemented via “ghost-points”; for free slip boundary conditions (12 )Z = 0 on
boundaries.
For a spherical polar grid, the coefficients of the metric terms are k1 = 0 and
k2 = − tan φ/a, with the spherical radius a and the latitude φ; ∆x1 = ∆x =
a cos φ∆λ, and ∆x2 = ∆y = a∆φ. For a general orthogonal curvilinear grid
as used in this paper, k1 and k2 can be approximated by finite differences of
the cell widths:
G
G
− ∆yi,j
1 ∆yi+1,j
F
∆yi,j
∆xFi,j
G
1 ∆xG
i,j+1 − ∆xi,j
=
F
∆xFi,j
∆yi,j
C
C
− ∆yi−1,j
1 ∆yi,j
=
U
∆yi,j
∆xVi,j
C
1 ∆xC
i,j − ∆xi,j−1
=
U
∆xVi,j
∆yi,j
736
C
k1,i,j
=
(B.4)
737
C
k2,i,j
(B.5)
738
Z
k1,i,j
739
Z
k2,i,j
740
741
742
743
744
(B.6)
(B.7)
The stress tensor is given by the constitutive viscous-plastic relation σαβ =
2η ˙αβ + [(ζ − η)˙γγ − P/2]δαβ (Hibler, 1979). The stress tensor divergence
(∇σ)α = ∂β σβα , is discretized in finite volumes. This conveniently avoids dealing with further metric terms, as these are “hidden” in the differential cell
30
745
widths. For the u-equation (α = 1) we have:
(∇σ)1 :
746
1 Z
(∂1 σ11 + ∂2 σ21 ) dx1 dx2
Aw
i,j cell
(B.8)
747
x2 +∆x2 )
x1 +∆x1 Z
(
x1 +∆x1
1 Z x2 +∆x2
σ21 dx1 +
σ11 dx2 = w
Ai,j x2
x
1
x2
x1
748
1
1
≈ w ∆x2 σ11 Ai,j
x1
749
1
C
= w (∆x2 σ11 )C
i,j − (∆x2 σ11 )i−1,j
Ai,j
750
(∆x1 σ21 )Zi,j+1
x +∆x1
(
x2 +∆x2 )
+∆x1 σ21 x2
(
)
+
−
(∆x1 σ21 )Zi,j
751
752
753
with
754
755
756
757
758
759
760
761
762
ui+1,j − ui,j
∆xFi,j
F
C vi,j+1 + vi,j
+ ∆yi,j
(ζ + η)C
i,j k2,i,j
2
v
−
v
i,j+1
i,j
F
+ ∆yi,j
(ζ − η)C
i,j
F
∆yi,j
F
C ui+1,j + ui,j
+ ∆yi,j
(ζ − η)C
i,j k1,i,j
2
F P
− ∆yi,j
2
V Z ui,j − ui,j−1
= ∆xi,j η i,j
U
∆yi,j
vi,j − vi−1,j
+ ∆xVi,j η Zi,j
∆xVi,j
Z ui,j + ui,j−1
− ∆xVi,j η Zi,j k2,i,j
2
v
+
vi−1,j
i,j
Z
− ∆xVi,j η Zi,j k1,i,j
2
F
C
(∆x2 σ11 )C
i,j = ∆yi,j (ζ + η)i,j
(∆x1 σ21 )Zi,j
31
(B.9)
(B.10)
763
Similarly, we have for the v-equation (α = 2):
1 Z
(∂1 σ12 + ∂2 σ22 ) dx1 dx2
Asi,j cell
(∇σ)2 :
764
(B.11)
765
x2 +∆x2 )
x1 +∆x1 Z
(
x1 +∆x1
1 Z x2 +∆x2
σ22 dx1 +
σ12 dx2 = s
Ai,j x2
x
1
x2
x1
766
1
1
≈ s ∆x2 σ12 Ai,j
x1
x +∆x1
(
x2 +∆x2 )
+∆x1 σ22 x2
(
=
767
1
(∆x2 σ12 )Zi+1,j − (∆x2 σ12 )Zi,j
Asi,j
)
+
768
(∆x1 σ22 )C
i,j
−
(∆x1 σ22 )C
i,j−1
769
770
771
with
(B.12)
(∆x2 σ22 )C
i,j
(B.13)
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
ui,j − ui,j−1
U
∆yi,j
U Z vi,j − vi−1,j
+ ∆yi,j
η i,j
∆xVi,j
U Z Z ui,j + ui,j−1
η i,j k2,i,j
− ∆yi,j
2
v
+
vi−1,j
i,j
U Z Z
− ∆yi,j
η i,j k1,i,j
2
F
C ui+1,j − ui,j
= ∆xi,j (ζ − η)i,j
∆xFi,j
C vi,j+1 + vi,j
+ ∆xFi,j (ζ − η)C
i,j k2,i,j
2
F
C vi,j+1 − vi,j
+ ∆xi,j (ζ + η)i,j
F
∆yi,j
C ui+1,j + ui,j
+ ∆xFi,j (ζ + η)C
i,j k1,i,j
2
P
− ∆xFi,j
2
U Z
(∆x1 σ12 )Zi,j = ∆yi,j
η i,j
Acknowledgements
We thank Jinlun Zhang for providing the original B-grid code and for many
helpful discussions. ML thanks Elizabeth Hunke for multiple explanations and
Sergey Danilov and Rüdiger Gerdes for comments on the manuscript. This
work was supported by NSF award ARC-0804150, DOE award DE-FG0208ER64592, and NASA award NNG06GG98G. It is a contribution to the
ECCO2 project sponsored by the NASA Modeling Analysis and Prediction
32
790
(MAP) program and to the ECCO-GODAE project sponsored by the National
Oceanographic Partnership Program (NOPP). Computing resources were provided by NASA/ARC, NCAR/CSL, and JPL/SVF.
791
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