5946
JOURNAL OF CLIMATE
VOLUME 20
The Granular Sea Ice Model in Spherical Coordinates and Its Application to a Global
Climate Model
JAN SEDLACEK, JEAN-FRANÇOIS LEMIEUX,
AND
LAWRENCE A. MYSAK
Earth System Modelling Group, Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada
L. BRUNO TREMBLAY
Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada, and Lamont-Doherty Earth
Observatory, Columbia University, Palisades, New York
DAVID M. HOLLAND
Courant Institute of Mathematical Sciences, New York University, New York, New York
(Manuscript received 1 September 2006, in final form 11 April 2007)
ABSTRACT
The granular sea ice model (GRAN) from Tremblay and Mysak is converted from Cartesian to spherical
coordinates. In this conversion, the metric terms in the divergence of the deviatoric stress and in the strain
rates are included. As an application, the GRAN is coupled to the global Earth System Climate Model from
the University of Victoria. The sea ice model is validated against standard datasets. The sea ice volume and
area exported through Fram Strait agree well with values obtained from in situ and satellite-derived
estimates. The sea ice velocity in the interior Arctic agrees well with buoy drift data. The thermodynamic
behavior of the sea ice model over a seasonal cycle at one location in the Beaufort Sea is validated against
the Surface Heat Budget of the Arctic Ocean (SHEBA) datasets. The thermodynamic growth rate in the
model is almost twice as large as the observed growth rate, and the melt rate is 25% lower than observed.
The larger growth rate is due to thinner ice at the beginning of the SHEBA period and the absence of
internal heat storage in the ice layer in the model. The simulated lower summer melt is due to the
smaller-than-observed surface melt.
1. Introduction
Sea ice dynamics plays an important role in shaping
the sea ice cover in polar regions. In the last few decades, several models have been developed to represent the dynamics of sea ice. Crucial to the model representation of dynamics is the formulation of the rheology, that is, the relationship between applied stresses
and the resulting deformations.
In these models, sea ice has been modeled either as a
continuum (e.g., Coon et al. 1974) or as a collection of
discrete particles (e.g., Hopkins 1996). Observations
during the Arctic Ice Dynamics Joint Experiment
(AIDJEX) campaign suggested that on a scale of about
Corresponding author address: Jan Sedlacek, Earth System
Modelling Group, Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, QC H3A 2K6, Canada.
E-mail: jan.sedlacek@mail.mcgill.ca
DOI: 10.1175/2007JCLI1664.1
© 2007 American Meteorological Society
JCLI4340
100 km, there are enough randomly distributed cracks,
ridges, and leads to allow sea ice to be treated as an
isotropic material and as a continuous medium (Coon
et al. 1974). Until recently, most sea ice models were
based on the continuum approach and assumed that sea
ice is an isotropic material. Recent studies, however,
are questioning these assumptions (e.g., Coon et al.
2006), and anisotropic/discontinuous models are being
developed (e.g., Hibler and Schulson 2000; Schreyer et
al. 2007).
The models assuming isotropy and a continuous medium can be divided into two categories: (i) the elastic–
plastic models (e.g., Coon et al. 1974; Pritchard 1975)
and (ii) the viscous–plastic models (e.g., Hibler 1979;
Hunke and Dukowicz 1997; Tremblay and Mysak 1997;
Zhang and Rothrock 2005).
In the elastic–plastic approach, sea ice undergoes
small elastic strains under low stress conditions, while
plastic deformations occur when the stresses reach criti-
15 DECEMBER 2007
SEDLACEK ET AL.
cal values defined by a yield curve (Coon et al. 1974).
Since the elastic part of the deformation is reversible,
the history of the strain has to be known. This usually
requires a Lagrangian description of the motion. Furthermore, the time step has to be small in order to
resolve elastic waves. In the viscous–plastic approach,
the elastic strains are neglected (Hibler 1977). When
the deformations are small, sea ice behaves as a very
viscous fluid (i.e., exhibits creeping flow). Since the viscous–plastic approach can be used with an Eulerian
description and is numerically easier to implement,
these models are now widely used for large-scale sea ice
simulations. Nevertheless, recent work shows promising developments with an elastic–decohesive model
(Schreyer et al. 2006) using an efficient numerical
scheme in a Lagrangian framework (Sulsky et al. 2007),
which is argued to be as numerically efficient as the
Hunke and Dukowicz (1997) model.
To complete the formulation of the viscous–plastic
rheology, a yield curve, which marks the transition between the viscous phase and the plastic phase, and a
flow rule, which describes the relation between the
strain rates and the stresses on the yield curve, have to
be specified. Owing to the difficulties in measuring in
situ large-scale stresses, the yield curve and the flow
rule that are the most realistic on geophysical scales
remain unclear. For this reason, various yield curves
have been used in sea ice modeling. For instance, current models use an ellipse (Hibler 1979; Hunke and
Dukowicz 1997), a lens (Zhang and Rothrock 2005), a
square (Ip et al. 1991), a teardrop (Zhang and Rothrock
2005), a line (Flato and Hibler 1992), or a triangle (Ip et
al. 1991; Flato and Hibler 1992; Tremblay and Mysak
1997).
For an elastic–plastic rheology, the principle of plastic irreversibility states that the yield curve should be
convex and that the resulting deformation should be
normal to the yield curve (Goodier and Hodge 1958).
Although it is not required for a viscous–plastic rheology to ensure proper energy dissipation (Hibler and
Schulson 2000), most of the viscous–plastic models use
a normal flow rule (e.g., Hibler 1979). But others, like
the ones based on a Mohr–Coulomb failure criterion
(triangle), use a nonnormal flow rule (e.g., Ip et al.
1991; Tremblay and Mysak 1997).
The most widely used viscous–plastic model is the
one developed by Hibler (1979), and modifications
thereof. This model uses an elliptical yield curve with a
normal flow rule. Hunke and Dukowicz (1997) added a
simplified elastic component to the viscous–plastic approach. This elastic part is not intended to represent the
physical elastic strains but is introduced to allow an
explicit numerical scheme with a large sub–time step.
5947
This explicit numerical scheme is therefore suitable for
parallelization. In the cavitating fluid approach (Flato
and Hibler 1992), sea ice is assumed to have no shear
strength. This way of representing ice interactions is
mathematically simple and easy to implement. However, the lack of shear strength induces too-large sea ice
drifts and led some authors to question the validity of
this model (Steele et al. 1997; Arbetter et al. 1999).
Flato and Hibler also described how shear stresses can
be added to the cavitating fluid model using a Mohr–
Coulomb failure criterion. As an extension to the work
of Flato and Hibler (1992), Tremblay and Mysak (1997)
developed a viscous–plastic sea ice model based on the
physics of a slowly deforming granular material. The
failure in shear is represented by a Mohr–Coulomb failure criterion. Reorganization of the granules (ice floes)
during shear deformation can lead to densification or
dilatation of the material. This phenomenon is referred
to as the dilatancy effect. With the continuum approach, the microphysical dilatancy effect is taken into
account by allowing convergence (densification) or divergence (dilatation) during shear deformation. Hence,
the dilatancy effect modifies the flow rule of a yield
curve using a Mohr–Coulomb failure criterion. In this
model, the macroscopic angle of friction is expressed in
terms of the angle of dilatancy and the microphysical
angle of friction. This clearly indicates the link between
the deformations and the resistance to deformations.
There is currently a tendency among the different
global climate modeling groups to adopt the Hibler
(1979) rheology or modified versions thereof. Of the 23
models participating in the Intergovernmental Panel on
Climate Change (IPCC) fourth assessment, 4 have no
dynamics or a very crude representation of the rheology, 4 use the cavitating fluid model (i.e., without shear
strength), and the remaining 15 models use the Hibler
(1979) model or modified versions (e.g., Hunke and
Dukowicz 1997). Many authors have shown that the
choice of yield curve and flow rule has a significant
effect on the simulated thickness and velocity fields (Ip
et al. 1991; Arbetter et al. 1999; Zhang and Rothrock
2005). The use of different rheologies would raise different questions and further our understanding of the
role of sea ice rheology on the sea ice drift and its
impact on the climate system in general.
The main purpose of this work is to present a spherical coordinate version of the granular sea ice model
(GRAN) of Tremblay and Mysak (1997). The yield
curve in this model is based on a Mohr–Coulomb failure criterion as compared to an ellipse in the Hibler
(1979) model. Recent laboratory measurements and
field experiments suggest the use of a Coulombic yield
curve (e.g., Overland et al. 1998; Hibler and Schulson
5948
JOURNAL OF CLIMATE
2000). In GRAN, a nonnormal flow rule with the dilatancy effect is used.
Starting from the Cartesian coordinate version of the
GRAN (Tremblay and Mysak 1997), we derive the governing equations in spherical coordinates. Due to the
curvature of the grid, extra terms1 arise in the formulation of the divergence of the stress tensor and of the
strain rates in spherical coordinates owing to the differentiation of the scale factors. The metric terms are important near the poles, and can have a significant contribution to the simulated thickness and velocity fields
(Hunke and Dukowicz 2002). The metric terms are incorporated into the model presented here.
As an application, the spherical coordinates version
of GRAN is coupled to the Earth System Climate
Model of the University of Victoria (UVic ESCM;
Weaver et al. 2001). Currently, this model includes the
elastic–viscous–plastic (EVP) dynamics of Hunke and
Dukowicz (1997). However, it is important to note that
the GRAN presented in this paper could be coupled to
any global or regional atmosphere–ocean model. The
GRAN will be part of a future release of the UVic
ESCM (M. Eby 2006, personal communication). This
will provide a platform to perform climate simulations
using two different rheologies.
Various aspects of the GRAN simulations of sea ice
in the Arctic are validated with and compared to the
standard datasets of recent sea ice observations such as
ice thickness distribution, sea ice drift, and Fram Strait
ice volume and ice area fluxes. The dynamics of sea ice
is strongly coupled to its thermodynamics through the
sea ice thickness and concentration-dependent compressive and tensile strengths. Accordingly, we also
validate the thermodynamical part of the model against
the year-long datasets from the Surface Heat Budget of
the Arctic Ocean (SHEBA) field experiment (see Uttal
et al. 2002 for a summary). While these datasets are for
single point measurements only, they are useful verification tools due to their high temporal resolution.
This paper is structured as follows. In section 2, we
give a general overview of the GRAN and describe the
UVic ESCM. The transformation of the sea ice model
equations to spherical coordinates and the coupling of
the GRAN to the UVic ESCM are presented in section
3. At the end of this section we compare the convergence and the numerical efficiency of the GRAN with
that of the EVP. Various simulation results of the new
coupled model and their comparison with data and
simulations of the EVP model are shown in the first
VOLUME 20
three parts of section 4. In the fourth part of section 4,
the simulation results for the sea ice seasonal heat budget
are compared with data from the SHEBA campaign.
The main conclusions are summarized in section 5.
2. Models
a. The granular sea ice model
In GRAN both the advection and the acceleration
terms in the momentum equation are neglected. These
assumptions are valid for length scales larger than a
couple of kilometers and for forcing time scales of
about half a day and longer. The remaining terms in the
momentum equation (which is now diagnostic) are the
Coriolis force, wind stress, water drag, divergence of
the stress tensor, and sea surface tilt, namely,
⫺␳i hf k ⫻ ui ⫹ A共␶a ⫺ ␶w兲 ⫹ ⵱ · ␴ ⫺ ␳i hg⵱Hd ⫽ 0,
where ␳i is the density of the ice, h the sea ice thickness,
f the Coriolis parameter, ui the sea ice velocity, A the
sea ice concentration, ␶a the wind stress, ␶w the water
drag, ␴ the internal ice stress tensor, g the gravity, and
Hd is the sea surface height.
In GRAN, sea ice is considered to be a slowly deforming granular material (Balendran and NemmatNasser 1993). The continuum approach is adopted; individual ice floes are not resolved but the microphysical
dilatancy effect is represented in the macrophysical
constitutive equation. Hence, the present formulation
allows for convergence (densification) or divergence
(dilatation) of the pack ice during shear deformation.
The dilatancy effect is determined by ␦, the angle of
dilatancy, which is the angle between the microscopic
plane of motion of the individual floes and the macroscopic sliding plane (see Tremblay and Mysak 1997,
their Fig. 6). A positive angle of dilatancy leads to dilatation, while a negative ␦ characterizes densification.
Observations show that, on average, shear deformations are associated with dilatation (Stern et al. 1995).
The yield curve is represented in stress invariant
space by a triangle (Fig. 1). The two stress invariants
are the pressure p, which is the (negative) average of
the normal stresses, and q, the maximum shear stress at
a point. The triangular yield curve is based on a Mohr–
Coulomb failure criterion for which the maximum allowable q (qmax) is proportional to the ice pressure p.
The Mohr–Coulomb failure criterion is defined by the
following equation:
qmax ⫽ p sin␾f ,
1
Sometimes they are referred to as metric terms (e.g., Hunke
and Dukowicz 2002).
共1兲
共2兲
where ␾f is the angle of friction and qmax defines the
largest shear stress that sea ice can sustain as a function
15 DECEMBER 2007
5949
SEDLACEK ET AL.
From these considerations and upon neglecting the
much smaller elastic deformations (Coon et al. 1974),
the internal ice stresses can be written as
␴ij ⫽ ⫺p␦ij ⫺ ␩⑀˙ kk␦ij ⫹ 2␩⑀˙ ij, i, j ⫽ 1, 2,
共5兲
where ␦ij is the Kronecker delta and ⑀˙ ij are the strain
rates. See Tremblay and Mysak (1997) for a detailed
derivation of Eq. (5). The shear viscosity ␩ is given by
冋公
␩ ⫽ min
FIG. 1. Yield curve with a Mohr–Coulomb failure criterion. The
quantities p and q are the stress invariants, and Pmax is the maximum pressure, Pmin is the cohesion, and ␾f is the angle of friction.
of pressure. When q ⫽ qmax, plastic deformation occurs,
consisting of shearing along sliding lines plus some convergence/divergence determined by ␦ and the rate of
shearing.
Failure in compression occurs when p reaches the
maximum allowable pressure denoted by Pmax. This
pressure is parameterized, according to Hibler (1979),
by the following equation:
Pmax ⫽ P*h exp关⫺C共1 ⫺ A兲兴,
共3兲
where P* is the ice strength per meter, A is the ice
concentration, and C is the ice concentration parameter, an empirical constant characterizing the dependence of the compressive strength of sea ice on the ice
concentration. When the pressure reaches Pmax, sea ice
can no longer sustain the compressive load; in this case,
convergence (ridging) occurs.
GRAN also allows for cohesion or tensile stresses.
Failure in tension occurs when p reaches the tensile
strength Pmin. In this case, the ice drifts freely.
To close the system of equations, a flow rule must be
specified. We assume that the plane of maximum (minimum) normal strain rate is aligned with the plane of
maximum (minimum) normal stress, that is, the principal strain rates are aligned with the principal stresses. In
Flato and Hibler (1992), the sea ice pressure is calculated assuming that the flow is nondivergent when the
pressure is between Pmax and Pmin. In GRAN, the divergence (⑀˙ kk) is related to the angle of dilatancy and
the rate of shearing according to the equation
⑀˙ kk ⫽ ⑀˙ 11 ⫹ ⑀˙ 22 ⫽ 2␥˙ tan␦, when Pmin ⬍ p ⬍ Pmax,
共4兲
where ␥˙ is the rate of shearing.
p sin␾f
共⑀˙ 11 ⫺ ⑀˙ 22兲2 ⫹ 4⑀˙ 212
册
, ␩max .
共6兲
When the strain rates are small, the denominator in
Eq. (6) tends to zero and the shear viscosity is capped
at ␩max. When the shear viscosity is equal to ␩max, the
state of stress lies inside the yield curve and sea ice
behaves as a very viscous fluid. When the shear viscosity is smaller than ␩max, the state of stress is on one of
the two limbs of the triangle and plastic deformations
occur (shear plus divergence/convergence).
b. The UVic ESCM
The UVic ESCM belongs to the family of Earth System Models of Intermediate Complexity (EMIC) (see
Claussen et al. 2002). This means that some components of the earth system are described in a less sophisticated way than in coupled general circulation models
(GCMs); that is, they include fewer processes or use a
coarser resolution. The reduced complexity allows transient model runs over long periods with reasonable
computational time. The first release of the UVic
ESCM is described in Weaver et al. (2001). The UVic
ESCM used in this study is version 2.6, which consists of
oceanic, atmospheric, and sea ice components. The
resolution of the model is 3.6° zonally and 1.8° meridionally, and the time step used is 1.25 days for the ocean
and 0.625 days for the atmosphere; the coupling of the
two components is done every 2.5 days.
The ocean model is the 3D primitive equation GCM
Modular Ocean Model version 2.2 (MOM 2.2; Pacanowski 1995) with 19 unequally spaced layers in the
vertical. The near-surface layer is 50 m thick and the
bottom layer is 518 m thick. The atmospheric model is
an energy moisture balance model (EMBM) (Fanning
and Weaver 1996) with heat and moisture transport
parameterized by a diffusion process. In this model, the
moisture can also be advected. The wind stress forcing
and the atmospheric CO2 concentration are prescribed.
The sea ice component is a dynamic–thermodynamic
model. The thermodynamics is calculated with a zerolayer Semtner model (Bitz et al. 2001) and the original
dynamical part of the UVic ESCM is the EVP model
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JOURNAL OF CLIMATE
VOLUME 20
(Hunke and Dukowicz 1997). The ice advection is calculated with a first-order upstream advection scheme.
In the UVic ESCM, the computational grid can be
rotated to avoid the singularity at the poles due to the
convergence of the meridians of longitude. In the rotated system, the North Pole is in Greenland and the
South Pole is over East Antarctica. Iceland is removed
to allow for a faster North Atlantic drift, and the Bering
Strait is closed owing to the coarse resolution.
latter term can be rewritten using the geostrophic approximation for the ocean currents. The divergence of
the stress tensor can be split into two parts: the gradient
of the pressure p and the divergence of the deviatoric
stress tensor (⵱ · ␴⬘). The gradient of the pressure in
spherical coordinates is given by
3. Transformation and coupling
where ␭ is the longitude, ␾ is the latitude, Re is the
radius of the earth, and ␭ and ␸ are unit normal vectors.
The divergence of a two-dimensional tensor in spherical coordinates can be found in Schade (1997). Using
Eq. (5), the divergence of the deviatoric stress tensor
can be written as follows:
a. Transformation
The steady-state sea ice momentum equation without
advection contains two terms with derivatives: the rheology and the sea surface tilt terms [see Eq. (1)]. The
⵱ · ␴⬘ ⫽
冢
1 ⭸p
⭸p
1
␭⫹
␸,
Re cos␾ ⭸␭
Re ⭸␾
2
1
⭸
⭸
关␩共⑀˙ 11 ⫺ ⑀˙ 22兲兴 ⫹
共cos2␾␩⑀˙ 12兲
2 ⭸␾
Re cos␾ ⭸␭
Re cos ␾
1
2
⭸
⭸
共␩⑀˙ 12兲 ⫹
关cos2␾␩共⑀˙ 22 ⫺ ⑀˙ 11兲兴
Re cos␾ ⭸␭
Re cos2␾ ⭸␾
where the ⑀˙ ij (i, j ⫽ 1, 2) are the strain rates. The top
line in Eq. (8) is the zonal component and the bottom
line is the meridional component. Owing to the curvature of the grid, metric terms arise in the right-hand side
of Eq. (8) upon carrying out the differentiation with
respect to ␾.
The strain rates ⑀˙ 11, ⑀˙ 22, and ⑀˙ 12 in spherical coordinates are given by
⑀˙ 11 ⫽
1
⭸u
␷
⫺
tan␾,
Re cos␾ ⭸␭ Re
⑀˙ 22 ⫽
1 ⭸␷
,
Re ⭸␾
共10兲
⑀˙ 12 ⫽
1 ⭸u
u
1
⭸␷
⫹
tan␾ ⫹
.
2Re ⭸␾ 2Re
2Re cos␾ ⭸␭
共11兲
共9兲
The strain rate term is symmetric, that is, ⑀˙ 12 ⫽ ⑀˙ 21
(Tremblay and Mysak 1997). In spherical coordinates,
the strain rates also include metric terms [i.e., the second term on the right-hand side of Eqs. (9) and (11)].
b. Coupling
The GRAN sea ice dynamics is coupled to the atmospheric component of the UVic ESCM and uses the
same resolution and time step as the latter. The thermodynamics and advection are calculated with the
original UVic ESCM model. The dynamical part can
冣
,
共7兲
共8兲
now be calculated using the original EVP sea ice dynamics of the UVic ESCM or the GRAN model.
The numerical scheme of the latter is the same as in
the Cartesian version of the model (Tremblay and
Mysak 1997). Step 1 calculates the free-drift solution as
a first approximation for the velocity field. In step 2, the
pressure term is added and the initial velocities and
pressure (assumed equal to zero in step 1) are corrected. During this correction, the off-diagonal terms
(such as, e.g., the Coriolis term) are considered constant. In step 3, the shear viscosities are calculated from
the strain rates and ice pressure calculated in step 2.
This linearizes the momentum equation. Finally, in step
4, the velocities are updated taking into account the
complete rheology; however, the shear viscosities are
kept constant.
Because the momentum equation is linearized, steps
2, 3, and 4 are repeated several times. We refer to this
as the “superloop” iteration. The number of superloops
is analogous to the number of pseudo time steps in
Zhang and Rothrock (2000) and the sub–time step in
Hunke and Dukowicz (1997). To ensure sufficient convergence of the velocity field, the spherical version of
GRAN needs at least six superloop iterations. With
sufficient convergence we mean that the total kinetic
energy (KE) of the Arctic sea ice is within 10% of the
final value if an infinite number of superloops were
used.
15 DECEMBER 2007
5951
SEDLACEK ET AL.
The pressure p in step 2 is calculated in a similar way
as described in Flato and Hibler (1992). A Reynolds
decomposition for the velocity and pressure fields is
inserted into the momentum equation and the basic
state is subtracted. The shear stresses, the deviatoric
stresses, and the off-diagonal terms are assumed to be
constant. The pressure perturbation is then calculated
as a function of the divergence of the sea ice velocity
field.
The free-drift velocities are computed on an Arakawa B grid, whereas the velocity updates are computed on an Arakawa C grid. At the end of the dynamics calculation the ice velocities are interpolated again
onto an Arakawa B grid and passed to the UVic
ESCM. The other variables such as p, ⑀˙ , and ␩ are
calculated at the tracer point, in the center of the grid
cell.
The analysis of the numerical efficiency and convergence depends highly on the convergence criteria. Inspired by Zhang and Rothrock (2000), we use the total
Arctic KE as a measure of the convergence. A simulation of 50 days is performed. We consider the GRAN to
have converged when the KE has reach 10% of its final
value. Similarly, the necessary length of the sub–time
steps for the EVP is determined. Using this criterion,
GRAN needs six superloops and the EVP sub–time
step has to be approximately 90 s, on average. The
corresponding mean CPU times are of similar magnitude with 2400 ms per time step for the GRAN and
2200 ms per time step for the EVP.
4. Simulation results
The spinup of the model was done in two phases. We
started from an existing equilibrium run provided by
the UVic ESCM with an atmospheric CO2 concentration of 280 ppm. The first 200 years of spinup was done
with a constant CO2 concentration. The second spinup
phase between 1800 and 1948 was done with a linearly
increasing CO2 concentration going from 280 to 316
ppm in 1959. During this phase the daily wind stress
fields are specified from a random permutation of the
years from 1950 to 2000 of the National Centers for
Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis. This was
done in order to reduce any biases due to different
modes in the atmospheric circulation. The actual run
from 1948 onward was done with a CO2 concentration
following the Mauna Loa record (Keeling and Whorf
2005) and the actual daily reanalysis wind stress fields
from NCEP–NCAR.
In Table 1 the physical parameters and constants
used for this simulation are listed. The diagnostic vari-
TABLE 1. Values of the physical parameters and constants used
in the simulation.
Variable
Symbol
Value
Angle of dilatancy
Internal angle of friction
Maximum shear viscosity
Water density
Water turning angle
Ice concentration parameter
Air drag coefficient
Water drag coefficient
Ice strength per meter
Cohesion
␦
␾f
␩max
␳w
⌰w
C
Cda
Cdw
P*
Pmin
10°
30°
1 ⫻ 1015 (g s⫺1)
1.026 (g cm⫺3)
25°
20
1.2 ⫻ 10⫺3
5.5 ⫻ 10⫺3
3 ⫻ 105 (dyn cm⫺2)
0 (dyn cm⫺2)
ables calculated by the sea ice dynamics model are the
sea ice velocities and pressure. The sea ice thermodynamics and sea ice advection are calculated with the
original UVic model. In the following three sections
(4a–c), we compare the GRAN output with some standard observations and with an EVP run. In section 4b,
the simulation results with the GRAN are compared
with data from the SHEBA campaign.
a. Fram Strait sea ice export
The major export site for sea ice into the North Atlantic is Fram Strait. The freshwater flux associated
with this export can affect the meridional overturning
circulation (Mauritzen and Häkkinen 1997; Holland et
al. 2001; Mysak et al. 2005) and hence plays an important role in determining the local and global climate.
This flux through Fram Strait involves the interplay
between the thermodynamics (ice thickness) and the
dynamics (velocity) of sea ice and, hence, its simulation
provides a measure of the overall performance of the
sea ice model.
Vinje et al. (1998) computed the ice volume flux
through Fram Strait from August 1990 to July 1996 at
around 79°N. A velocity profile across Fram Strait was
determined from buoys and satellite measurements. To
get a continuous velocity record, the drift was related to
the pressure gradient across Fram Strait and a climatological ocean velocity. The ice thickness profile was obtained from upward-looking sonars deployed in Fram
Strait. From observations at four draft measurement
stations between September 1992 and March 1993, a
relationship was derived in order to get a thickness profile from the single-point measurement time series from
the other years. Kwok and Rothrock (1999) provide an
ice area flux record derived from satellite measurements at 81°N. The ice motion record spans the period
from October 1979 up to May 1996. The summer velocities obtained from the satellite observations are less
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FIG. 2. ⌱ce volume flux through Fram Strait (km3 month⫺1) simulated by the GRAN (solid
curve), simulated by the EVP model (dotted curve), estimated by Vinje et al. (1998) (dashed
curve), and estimated by Kwok and Rothrock (1999) (dash–dotted curve).
reliable; thus these velocities are parameterized in
terms of the sea level pressure gradient. In addition,
Kwok and Rothrock (1999) used the same thickness
profiles as in Vinje et al. to compute the ice volume flux
through Fram Strait at 79°N from October 1990 to May
1996.
Figure 2 shows the simulated ice volume flux at
79.2°N (solid), along with the estimates from Vinje et
al. (1998) (dashed), Kwok and Rothrock (1999) (dash–
dotted), and the EVP simulation (dotted). The GRAN
flux is in good agreement with the estimations. The
correlation coefficient between the simulated ice volume flux and the Vinje et al. data is 0.58. For the Kwok
and Rothrock estimations the correlation coefficient is
0.65. The EVP has a correlation coefficient of 0.60 with
Vinje et al.’s estimates and 0.70 with Kwok and Rothrock’s data. The EVP and GRAN curves are remarkably similar, and indeed have a correlation coefficient
of 0.97. All correlation coefficients are well above the
99% significance level. Köberle and Gerdes (2003),
who employed a higher-resolution Hibler-type sea ice
model forced by NCEP–NCAR reanalysis data, obtained a correlation coefficient of 0.68 with Vinje et al.’s
estimates.
The winter (October–May) sea ice area flux is also
compared with that of Kwok and Rothrock (1999).
These fluxes are considered more reliable as they are
based on satellite measurements only. The amplitude of
the simulated total area flux is generally in good agreement with the observations from Kwok and Rothrock
(1999; Fig. 3). Owing to the model’s land–ocean configuration, the simulated area flux is calculated at
79.2°N in comparison with 81°N for the observations.
The simulated mean winter area of ice exported over
the whole period is 574 ⫻ 103 km2, which is about 15%
smaller than the observed mean area of 669 ⫻ 103 km2.
The correlation coefficient between the two time series
is 0.79, significant at the 99% level. The correlation with
the EVP run is 0.76. Köberle and Gerdes (2003) reported a correlation coefficient of 0.84 over the same
period, and Hilmer and Jung (2000) obtained a correlation coefficient of 0.88.
b. Sea ice thickness
In 1991 and 1995 the European Remote Sensing Satellites ERS-1 and ERS-2 were launched. Laxon et al.
(2003) used the measurements from the radar altimeter
to estimate the ice freeboard from 1993 to 2001 in the
latitudinal band from 65° to 81.5°N. The ice freeboard,
which is the thickness above sea level, was converted to
ice thickness using fixed ice and seawater densities in
addition to a climatological snow cover. The resulting
uncertainty is of the order of 40 cm for the mean thickness. Their altimeter estimates excluded thicknesses below 1 m.
Figure 4 compares the simulated thickness distribution (Fig. 4a) with the winter thickness climatology for
1993–2001 derived from satellite observations (Fig. 4b).
The winter here is defined as the months between October and March, inclusive. Qualitatively, the model
agrees well with the observations. The thickest simulated ice of 3–5 m is north of Greenland and north of
the Canadian Archipelago. In these regions the observed ice thickness ranges from 3.5 to 4.5 m. Overall,
the thickness distribution in the Arctic Ocean is well
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SEDLACEK ET AL.
5953
FIG. 3. Total winter ice area exported (October–May) through Fram Strait (km2): area flux
simulated by the GRAN (solid curve), the flux simulated by the EVP model (dotted curve),
and observations from Kwok and Rothrock (1999) (dashed curve).
captured. Further details on the simulated seasonal ice
thickness in the Beaufort Sea will be given in section 4d.
Figure 4a also reveals that the simulated ice edge
extends well into the Norwegian and Barents Seas. This
is a common problem in coarse-resolution ocean models. This discrepancy occurs because the simulated
North Atlantic Drift and the associated ocean heat
transport are too weak. This causes a nonrealistic sea
ice edge position, which is also the case for the UVic
ESCM with the EVP model (Mysak et al. 2005).
c. Sea ice velocity
The Arctic sea ice atlas from the Environmental
Working Group (EWG) provides sea ice motion data.
The data were compiled from measurements taken
from the International Arctic Buoy Program (IABP),
FIG. 4. Mean sea ice thickness for the winters 1993–2001 (October–March): (a) the simulated sea ice thickness and (b)
satellite observations of the thickness (adapted from Laxon et al. 2003). Note the different contour levels and the blind
spot north of 81.5°N in the observations.
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FIG. 5. Winter sea ice velocity for (a) the GRAN model and (b) for EWG for 1984–88 (anticyclonic regime).
the Russian Drifting Automatic Radiometeorological
Stations (DARMS) and manned research stations. In
this dataset, the climatological ice motion was computed for the years 1950 up to 1996. Then, the monthly
mean velocity distribution was calculated as a deviation
from the climatological field. For each particular
month, a certain region around an actual trajectory
measurement was selected. A weighting function was
used to compute the deviatoric velocity around the
measurement site, whereas for the remaining velocity
field the climatological ice motion was used. The motion data are available on a CD-ROM (Arctic Climatology Project 2000).
To compare the simulated velocities with the EWG
ice motion data we chose two periods. The mean velocities from December to March were first calculated
for the years 1984–88. Then, the winter velocities, as
above, were calculated for the period 1989–93. These
two periods correspond to, respectively, the anticyclonic (1984–88) and cyclonic (1989–93) regimes described in Proshutinsky and Johnson (1997). The anticyclonic ice motion in the EWG data (Fig. 5b) shows a
large Beaufort gyre (BG). The observed velocities are
approximately 1 cm s⫺1 in the BG. The Transpolar
Drift Stream (TDS) has a fairly straight trajectory from
the Laptev and East Siberian Seas with a velocity of
about 2 cm s⫺1. The simulated anticyclonic regime (Fig.
5a) has a BG that is shifted too far west. However, the
location of the BG agrees with the results from
Proshutinsky and Johnson (1997). The simulated velocities are about 1 cm s⫺1 in the BG, which compare
favorably with the observations. The simulated velocities north of Alaska also agree with the observations.
The TDS starts mainly in the Laptev Sea and its speed
is slightly higher than the one observed. As mentioned
above, EWG data used climatological data if there
were no observations for a certain period. Therefore,
the average velocities were calculated only in the interior of the Arctic basin, where there are sufficient buoy
trajectories. The EWG mean velocity is 1.9 cm s⫺1 and
the simulated mean velocity is 2.1 cm s⫺1. This difference is due to a faster simulated TDS.
The observed cyclonic regime (Fig. 6b) shows a
smaller BG as compared to the anticyclonic regime, as
described in Proshutinsky and Johnson (1997), and the
center is closer to the northern Canadian coastline. The
velocities are again about 1 cm s⫺1 in the BG. The TDS
starts off moving from the Siberian coast toward the
BG and then turns toward Fram Strait, producing the
cyclonic motion. The observed velocities of the TDS
are higher than in the anticyclonic regime with magnitudes around 3 cm s⫺1. The simulated velocities for the
cyclonic regime are shown in Fig. 6a. The BG is smaller
but again shifted to the west. However, the simulated
location also agrees with the results from Proshutinsky
and Johnson (1997). The simulated magnitude of the
ice motion is comparable to the observed speeds. The
simulated TDS is similar to the observed trajectories
from the Laptev Sea toward Fram Strait. Contrary to
the observations, however, the simulated velocities of
the TDS are lower in the cyclonic regime than in the
anticyclonic regime. The average speed in the interior
15 DECEMBER 2007
SEDLACEK ET AL.
5955
FIG. 6. Winter sea ice velocity for (a) the GRAN model and (b) the EWG for 1989–93 (cyclonic regime).
of the Arctic basin is 2.0 cm s⫺1 for the EWG data and
1.7 cm s⫺1 for the simulated ice motion. Again, the discrepancy is caused by a different TDS speed. Similar
findings for the cyclonic regime were obtained by
Mysak et al. (2005) with the UVic ESCM and using the
EVP rheology.
d. Comparison with SHEBA data
During the SHEBA field experiment, the Canadian
icebreaker Des Groseilliers was frozen in the Beaufort
Sea from October 1997 until October 1998. One of the
observational programs of the field campaign included
measuring the mass balance of sea ice (Perovich et al.
2003).
From autumn 1997 to autumn 1998 the SHEBA ice
camp drifted from 75°N, 142°W to 80°N, 166°W. To
compare the simulated data with the observations, we
determined the time and position of the field station in
relation to the model grid. Although the dataset is a
point measurement, its temporal resolution is high, providing a good opportunity to validate the thermodynamic component of a sea ice model. A detailed study
on the energy balance of the SHEBA dataset for model
validation purposes is given by Huwald et al. (2005a,b).
1) SEA
ICE THICKNESS
The Pittsburgh site was located on a multiyear ice
floe. We chose data from gauges 53, 69, and 71 to compare with our simulation. The data from these gauges
were used by Huwald et al. (2005a) to produce a “best
guess” ice and snow thickness time series. The two time
series were adopted by the Sea Ice Model Intercomparison Project, Phase 2 (SIMIP2; violet line in Fig. 7 is
the “best guess” thickness series). In the following we
compare the simulated ice thickness evolution against
the best-guess estimate of Huwald et al. The individual
gauge time series provide an envelope of temporal evolutions of the ice thicknesses at the Pittsburgh site.
In the UVic ESCM, four processes contribute to seasonal ice thickness changes: 1) the melt/growth associated with the ocean and net atmospheric heat fluxes, 2)
ice growth over open water, 3) sublimation/deposition,
and 4) horizontal ice advection (i.e., ridging and lead
opening). During the SHEBA campaign, the growth
over open water is small in this region since the open
water fraction is nearly zero during this period (Fig. 8).
Furthermore, the sublimation/deposition is small. The
simulated contribution to the thickness change due to
advection is also small (Fig. 9). Since the SHEBA data
are point measurements where ice only grows/melts
thermodynamically and processes 2, 3, and 4 stated
above are small, we examine the growth/melt of the sea
ice due to thermodynamic processes only.
The initial simulated thickness in October 1997
(around Julian day 300) is 35 cm smaller than the observed best-guess initial thickness (Fig. 7). The simulated growth rate, however, is larger than that observed
(as seen in the individual gauges and the best-guess
estimate). As a consequence, the simulated maximum
thickness is about 30 cm greater than the best-guessobserved thickness at around Julian day 500. Note that
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FIG. 7. Sea ice thickness from October 1997 to October 1998 at the location of the Pittsburgh site from the SHEBA field experiment. The black line is the modeled thickness. The
three blueish lines are the thicknesses measured at different gauges at the Pittsburgh site
during the SHEBA field experiment (see legend for the color coding of the individual gauges).
The violet line is the best-guess thickness from Huwald et al. (2005a) for the Pittsburgh site.
the sudden rises and falls in the simulated thickness are
mainly due to the position change of the SHEBA camp
from one model grid cell to another. The onset of the
melting phase starting in June 1998 (around Julian day
540) is close to that observed. The simulated melt rate
is 25% lower than the best-guess melt rate. Huwald et
al. (2005b) estimated that about 60%–70% of the total
melt at the Pittsburgh site was caused by surface melting. In the UVic ESCM the surface and ocean heat
budgets contribute equally to the melt.
2) SURFACE
ENERGY BUDGET
The growth season can be separated into two parts:
1) the polar night, when the incident shortwave radiation is essentially zero (Julian days 300–410), and 2) the
polar day, when shortwave radiation is present (Julian
days 410–540). During the polar night, the net flux at
the surface compares well with the observed mean flux
(Fig. 10f). However, the simulated net flux does not
exhibit the observed synoptic-scale variability associated with frontal activity, changes in cloud cover, and
hence the net longwave radiation flux. The higher sea
ice growth rate in early winter can be explained in part
by the thinner sea ice, which grows faster, and the relatively low ocean heat flux (see Fig. 11). In addition, the
ice thermodynamic model does not consider internal
heat storage. This leads to a higher amplitude of the
seasonal cycle and a faster growth rate in winter (Bitz
and Lipscomb 1999). In the real world, the heat lost at
the surface is used for brine pocket freeze up, which
reduces the growth rate. Holland et al. (1993) observed
a decrease in mean ice thickness with the inclusion of
internal heat storage.
After the onset of the polar day (Julian day 410), the
surface ice temperature is higher than observed (not
shown), owing to the lower albedo value used in the
15 DECEMBER 2007
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SEDLACEK ET AL.
FIG. 8. Simulated sea ice concentration at the SHEBA camp
location.
UVic ESCM (Fig. 10a) and the fact that the thermodynamic model has no internal heat storage. As a consequence, the net heat flux emitted from the surface is
higher than the observed flux (Fig. 10f). This also
causes a much higher sensible heat flux after Julian day
450 when the downwelling shortwave radiation starts to
become important (Fig. 10d). In spring, the air temperature begins to rise, but the northward transport of
moisture in the model is smaller than in reality (not
FIG. 9. Simulated thickness changes (⌬h) due to advection
(dashed line) and thermodynamics only (solid line).
shown). This causes a larger simulated latent heat flux
(Fig. 10e) as well. Note that the specific humidity in the
model represents an average over the lower 1.8 km of
the atmosphere (Weaver et al. 2001), whereas at
SHEBA the specific humidity of the atmosphere is
taken at 10 m above ground (Persson et al. 2002). In
addition, the heat transfer coefficient in the model,
which is a function of wind speed and temperature, is
generally twice as large as the values typically used by
other modeling groups. This issue has been resolved in
version 2.8 of the UVic ESCM (M. Eby 2006, personal
communication).
In the model, the ice layer starts to melt at Julian day
540 (25 June) (Fig. 7), a few days after the insolation
reached its maximum value. From then on the incoming
shortwave radiation is decreasing. This, together with
the constant albedo value in the UVic ESCM, decreases the net shortwave radiation after the onset of
melt (Fig. 10b). The sensible heat flux drops to almost
zero 20 days later (Fig. 10d) as both the surface air and
ice layer are around 0°C. However, the net longwave
radiation is overestimated by about 20 W m⫺2 and the
latent heat by about 35 W m⫺2 (Figs. 10c and 10e).
The overly large simulated net longwave radiation is
mainly caused by a low simulated downward longwave
radiation. The higher latent heat loss is due to a continued drier atmosphere in the model. The maximum
observed magnitude of the net surface heat flux is
about 75 W m⫺2 as compared to about 30 W m⫺2 in the
model (Fig. 10f).
3) BASAL
ENERGY BUDGET
The observed ocean heat flux shows a large peak
around Julian day 440 (Fig. 11). This peak is associated
with the ice station drifting into shallow waters during a
storm event and entraining warmer, deeper water (Perovich and Elder 2002). A second large change in measured ocean heat flux starting around Julian day 500 is
again associated with a large storm event. Perovich and
Elder (2002) calculated a mean ocean heat flux of 7.5 W
m⫺2 over the entire period at the Pittsburgh site. This
number is almost two to four times the values found in
earlier studies (i.e., Maykut and McPhee 1995; Perovich
et al. 1997). For instance, Perovich et al. (1997) found
that in the Beaufort Sea the ocean heat flux in winter is
close to zero and in summer it is around 5 W m⫺2.
In the UVic ESCM the sensible heat flux of an icecovered grid cell is calculated with the temperature difference between the ice base, set at the freezing point of
water for a given salinity, and the surface ocean. In the
following this sensible heat flux is referred to as the
ocean heat flux. On the other hand, when the sea ice
concentration is lower than 100%, heat that enters the
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FIG. 10. (a) Albedo, (b) absorbed shortwave radiation, (c) net longwave radiation, (d) sensible heat, (e)
latent heat, and (f) net heat budget at the ice surface. The observed time series from SHEBA are the dotted
lines and the model output is the solid line. Turbulent fluxes are defined positive upward.
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SEDLACEK ET AL.
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again, heat that enters the open water is directly used
for melting. The total available net ocean heat flux during the melting phase reaches a maximum of 22 W m⫺2
(solid line in Fig. 11).
5. Summary
FIG. 11. The simulated net ocean heat flux (solid line), the
ocean heat flux only (dashed line), and the observed net ocean
heat flux (dotted line). An upward heat flux is defined as negative.
surface waters from the atmosphere (radiation, turbulent) is used directly for basal ice melt. This is equivalent to assuming an infinite sensible heat transfer coefficient. Together with the ocean heat flux, this is referred to as the net ocean heat flux.
In the model, the ocean heat flux is on the order of
0.5 W m⫺2 (upward) during the snow/ice accumulation
period (Fig. 11). After Julian day 490 the simulated
ocean heat flux changes sign. At that point the snow
layer starts to melt. In the model, the melted freshwater
is released into the upper ocean and thus decreases its
salinity. This increases the freezing point temperature
(i.e., the ice base temperature), yet the actual surface
ocean temperature remains nearly the same. This
causes the ocean heat flux to change sign and basal ice
to form (dashed line in Fig. 11). The second positive
bulge of the ocean heat flux is associated with the melting of the ice layer. The positive ocean heat flux (from
the ice to the ocean) in the model is associated with
summer ice formation (“false bottoms”). These false
bottoms were observed during the SHEBA campaign
(Eicken et al. 2002; Perovich et al. 2003). The large
simulated ocean heat flux pulse starting at Julian day
575 and ending at Julian day 605 is related to a sudden
increase in ocean surface temperature. At this point the
simulated SHEBA camp moves from shallow to deep
bathymetry. The increase in temperature is presumably
caused by diffusion or convection from below.
Around Julian day 520 the ice pack opens up (Fig. 8)
and the sea ice concentration falls below 100%. From
this date until Julian day 610 when the sea ice closes up
The choice of rheology can lead to significant differences in the simulated sea ice thickness and velocity
distributions (Ip et al. 1991; Arbetter et al. 1999; Zhang
and Rothrock 2005). Which rheology is the most accurate to describe the sea ice dynamics on a large scale is
still under debate. Therefore, it is useful to test different rheologies in global climate simulations. This can
further our understanding of the role of sea ice rheology and its impact on the climate system and help raise
new questions. Thus, in this paper we presented the
transformation to spherical coordinates of the GRAN
model (Tremblay and Mysak 1997). The GRAN is
coupled to the UVic ESCM as an application and it will
be part of a future release of this model (M. Eby 2006,
personal communication). Its output is compared to
different observational datasets and an EVP model
simulation.
The simulated volume export through Fram Strait is
in good agreement with the estimates from Vinje et al.
(1998) and Kwok and Rothrock (1999). The comparison with the ice area flux estimated by Kwok and Rothrock (1999) shows that the area flux follows the estimated behavior, although the mean is 15% lower. The
winter climatological thickness distribution corresponds well with the data provided by Laxon et al.
(2003). The sea ice thickness distribution in the central
Arctic Ocean is captured. The cyclonic and anticyclonic
ice drift regimes found by Proshutinsky and Johnson
(1997) are well captured and the simulated velocities
are comparable to the buoy velocities (Arctic Climatology Project 2000). However, we note that the center of
the simulated BG is located too far west in both regimes compared to the observations. Nonetheless, the
location of the center in each regime agrees with that
shown in Proshutinsky and Johnson (1997). The simulated mean velocity of the sea ice in the central Arctic
is slightly too large for the anticyclonic regime and
slightly too small in the cyclonic regime.
Finally, the thickness evolution in the Beaufort Sea
was compared against the SHEBA datasets. The net
growth of the ice layer compared to SHEBA data is
more than twice as large. This is mainly caused by the
absence of heat storage in the ice layer, the lower ocean
heat flux, and the thinner simulated ice at the beginning
of the SHEBA observation period. However, the simulated onset of melting coincides with the observed on-
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set. The total melt in late summer is about 25% too low
compared with SHEBA.
During the growing phase in winter, when the sun is
below the horizon, the net surface heat flux is comparable with the SHEBA data. However, owing to the
absence of internal heat storage in the ice layer and the
initially thin ice, the growth rate is too large. During the
polar day, the ice layer absorbs too much shortwave
radiation due to the lower albedo value. Again, owing
to the missing internal heat storage, the surface ice temperature increases too quickly, causing high sensible
and latent heat fluxes. For the latent heat, this is exacerbated by the low atmospheric relative humidity simulated by the model. During the melting phase the net
surface heat balance is lower than observed.
The simulated ocean heat flux is low during the growing season. At the onset of the snowmelt the model
simulates the formation of false bottoms, similar to
what was observed during SHEBA (Eicken et al. 2002;
Perovich and Elder 2002). After breakup of the pack
ice, an additional bottom melt effect appears in order to
account for the heating of the mixed layer. In the
model, the surface and bottom melt contribute equal
amounts to the total simulated melt, but it is too low
compared to observations.
Acknowledgments. We thank M. Eby for his considerable help during the coupling of the GRAN to the
UVic ESCM and H. Huwald for providing the skin
temperatures at the SHEBA site. Furthermore, we
thank the two anonymous reviewers for their comments
and suggestions, which helped to improve the paper.
This work was supported by research grants awarded to
L. A. Mysak from NSERC, the Canadian Institute of
Climate Studies (Arctic Node), and the Canadian
Foundation for Climate and Atmospheric Sciences (Canadian CLIVAR Research Network). L. B. Tremblay
was supported by an NSERC Discovery Grant and the
National Science Foundation Grants OPP-0230264,
OPP-1230325, and ARC-05-20496.
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