Los2007c

Los2007c
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,
Modeling Ice Shelf Cavities in a z -Coordinate Ocean General Circulation
Model
M. Losch
Alfred-Wegener-Institut für Polar- und Meeresforschung, Bremerhaven, Germany
Abstract. Processes at the ice shelf-ocean interface and in particular in ice shelf cavities around
Antarctica have an observable effect on the solutions of basin scale to global coupled ice-ocean
models. Despite this, these processes are not routinely represented in global ocean and climate models. It is shown that a new ice shelf cavity model for z-coordinate models can reproduce results from an intercomparison project of earlier approaches with vertical σ- or isopycnic coordinates. As a proof of concept, ice shelves are incorporated in a 100 year global integration of a z-coordinate model. In this simulation, glacial melt water can be traced as far
as north as 15◦ S. The observed effects of processes in the ice shelf cavities agree with previous results from a σ-coordinate model, notably the increase in sea ice thickness. However,
melt rates are overestimated probably because the parameterization of basal melting does not
suit the low resolution of this configuration.
spurious motion due to the numerical representation of the pressure
gradient terms. Therefore, a priori, one expects the resulting errors
to be large near the ice shelf edges where σ-coordinates are “bent”
from surface values to approximately 200 m depth. For this reason,
Beckmann et al. [1999] modified the ice shelf edge topography to
allow the σ-coordinates to vary smoothly across this discontinuity.
In a different approach, Grosfeld et al. [1997] introduced hybrid coordinates (z-coordinates near the surface and σ-coordinates beneath
the ice shelves) to resolve the steep topography at the ice shelf front.
Isopycnic coordinate models, on the other hand, are a natural
choice for density-driven flows in ice shelf cavities (and the ocean
in general). However, these models require a non-isopycnic layer to
admit (surface or ice-ocean interface) buoyancy forcing. With such
a layer, some of the benefits of isopycnic coordinates are lost [Holland and Jenkins, 2001]. Further, isopycnic models are traditionally
formulated in terms of potential density. Difficulties associated with
this choice (in particular the absence of a unique reference point for
the potential density) have probably prevented the general use of
isopycnic coordinate models in a global context. Newer formulations with hybrid coordinates have overcome these limitations [e.g.,
HYCOM, Bleck, 2002].
To date, many global ocean models are still formulated in zcoordinates, so that previous ice shelf cavity implementations for
σ- or isopycnic coordinates cannot be used for these models. For
this reason, Beckmann and Goosse [2003] proposed a parameterization of ice shelf-ocean interaction based on the experience with
high resolution σ-coordinate models [Beckmann et al., 1999; Timmermann et al., 2002a]. In this paper, the first explicit model of
ice shelf cavities in a z-coordinate general circulation model is presented. Z-coordinate models are known to have difficulties representing bottom boundary layers and downslope flow due spurious
diabatic mixing. Legg et al. [2006] found that downslope flow of
dense plumes depends strongly on horizontal and vertical resolution: only at intermediate to high horizontal and vertical resolution
a z-coordinate model gives results similar to an isopycnic model.
Similar difficulties are anticipated for the representation of light
plumes rising along the shelf ice topography. On the other hand,
z-coordinate models do not not suffer from pressure gradient errors. The partial cell treatment of topography [Adcroft et al., 1997]
ensures an accurate representation of topography in the ice shelf
cavities.
Section 2 describes the Massachusetts Institute of Technology
general circulation model [MITgcm, MITgcm Group, 2002] and
specifically the modifications necessary to include ice shelf cavities. In Section 3, model performance is compared to the results of
ISOMIP (Ice-Shelf Ocean Model Intercomparison Project) test experiments [Holland et al., 2003]. A coarse but realistic global model
with ice shelf cavities is presented in Section 4, where the impact
of ice shelf cavities on the general circulation is also demonstrated.
Conclusions are presented in Section 5.
1. Introduction
About 50 % of the Antarctic continental margin is covered by ice
shelves [Fox and Cooper, 1994]. Ice shelf cavities are found where
the continental ice sheets and outlet glaciers reach the ocean and ice
masses float, owing to their lower density compared to sea water.
Ice shelf thicknesses vary from more than a thousand meters near
the grounding line to about two hundred meters near the ice shelf
edge. Depending on the location of the ice shelf, warm deep water
(e.g., Circumpolar Deep Water at Pine Island Glacier) and/or High
Salinity Shelf Water (e.g., Filchner-Ronne Ice Shelf) can enter the
cavities. There it melts the ice shelf base, rises with the melt water
along the inclined ice shelf base and — in some parts — (re-)freezes
due to the pressure dependence of the freezing point of sea water [0.753◦ K/1000 dBar, Millero, 1978], forming Ice Shelf Water
(ISW). The circulation associated with these processes is known as
the ice pump [Lewis and Perkins, 1986]. Basal melt water around
Antarctica contributes approximately 28 mSv (1 Sv = 106 m3 s−1
and 1 mSv = 103 m3 s−1 ) of freshwater to the Southern Ocean
[Hellmer, 2004]. Weddell Sea Bottom Water forms by mixing between Ice Shelf Water and Weddell Deep Water [Foldvik et al.,
1985] so that ice shelf processes contribute to the deep branch of
the global ocean circulation. Hellmer [2004] claimed that models
without ice shelf cavities and the freshwater deficit associated with
them tend not only to underestimate sea ice thickness but more importantly to increase the bottom water formation and overturning.
Improved estimates of the ocean state from both regional and global
ocean models would therefore require inclusion of the effects of ice
shelf-ocean interaction [Hellmer et al., 2005; Thoma et al., 2006;
Schodlok et al., 2007].
Up to now, models with an explicit treatment of Antarctic ice
shelves employ either bottom-following (σ) or isopycnic coordinates [Beckmann et al., 1999; Timmermann et al., 2002a; Hunter
et al., 2004; Holland and Jenkins, 2001; Jenkins and Holland, 2002;
Grosfeld et al., 1997; Gerdes et al., 1999; Thoma et al., 2006]. The
vertical levels of a σ-coordinate model follow the ice shelves’ base
and thus provide a convenient way of resolving the ice shelf topography. Ice shelf processes can be included in σ-models with
little technical effort: The surface layer is subducted beneath the ice
shelf and all ice-ocean interactions are applied to the surface level.
However, models using bottom-following coordinates suffer from
Copyright 2008 by the American Geophysical Union.
0148-0227/08/$9.00
1
X-2
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
2. Model Description: Ice Shelf-Water Interaction
in a z -Level Model
The M.I.T. General Circulation Model (MITgcm) is a general
purpose grid-point algorithm that solves the Boussinesq form of the
Navier-Stokes equations for an incompressible fluid, hydrostatic or
fully non-hydrostatic, with a spatial finite-volume discretization on
a curvilinear computational grid (in the present context on a threedimensional longitude, latitude, depth grid). The model algorithm
is described in Marshall et al. [1997]; for online documentation and
access to the model code, see MITgcm Group [2002].
2.1. Dynamics
The dynamical core of the MITgcm has been used in z-coordinate
modeling of the ocean circulation and pressure coordinate modeling of the atmosphere [Marshall et al., 2004]. Losch et al. [2004]
used the same concept as Marshall et al. for constructing an ocean
model in pressure coordinates. In this configuration, the free surface, which is at the top of the water column in z-coordinates, is
interpreted as bottom pressure (anomaly) along the castellated bottom topography or surface orography. The shelf ice topography
on top of the water column has a similar role as (and in the language of Marshall et al. [2004] is “isomorphic” to) the orography
and the pressure boundary conditions at the bottom of the fluid for
atmospheric and oceanic models in pressure coordinates. For this
reason, the code infrastructure of the MITgcm requires only very
little change for the inclusion of ice shelf topography. The following details of the MITgcm are relevant to implementing an ice shelf
cavity model: treatment of hydrostatic (and non-hydrostatic) pressure in conjunction with a linear or a non-linear free surface and
partial cells for representing topography accurately.
The total pressure ptot in the ocean can be divided into the pressure at the top of the water column ptop , the hydrostatic pressure
and the non-hydrostatic pressure contribution pNH :
ptot = ptop +
Z
η−h
g ρ dz + pNH ,
(1)
−h
Underneath the ice shelf, the “sea-surface height” η is the deviation
from the “reference” ice-shelf draft h. During a model integration, η
adjusts so that the isostatic equilibrium is maintained for sufficiently
slow and large scale motion.
In the MITgcm, the total pressure anomaly p0tot which is used for
pressure gradient computations is defined by substracting a purely
depth dependent contribution −gρ0 z with a constant reference density ρ0 from ptot. Eq. (1) becomes
ptot = ptop − g ρ0 (z + h)+g ρ0 η +
Z
η−h
g (ρ − ρ0 ) dz + pNH ,
z
(4)
n−1
X
ρ∗k0 ∆zk0 + pa
(5)
k0 =1
where n is the vertical index of the first (at least partially) “wet”
cell and ∆zk0 is the thickness of the k0 -th layer (counting downwards). The pressure anomaly for evaluating the pressure gradient
is computed in the center of the “wet” cell k as
«
k „
X
1 + H(k0 − k)
0
0
=
+ gρn η + g
(ρk − ρ0 )∆zk
2
k0 =n
(6)
where H(k0 − k) = 1 for k0 < k and 0 otherwise.
The partial cell method addresses a severe limitation of zcoordinate models. Without partial cells, the “staircase” representation of topography in such models becomes only reasonable at very
high resolution. With partial cells, topography (of the sea floor)
can be approximated more accurately than with full cells and this
leads to generally smoother solutions that compare favorably with
solutions of a terrain-following model [Adcroft et al., 1997].
The application of partial cells for representing topography and
fluxes along topography of an ice shelf is completely analogous to
that of bottom topography as reported in Adcroft et al. [1997]. Grid
p0k
p0top
ice−ocean interaction
layer k
ice shelf
layer k+1
+
B
+
+
water
A
∆ z k hk
C
+
∆zk (1−hk )
∆z k+1 −∆zk (1−hk )
z
x
Figure 1. Schematic of a vertical section of the grid at the base of
an ice shelf. Grid lines are thin, the thick line is the model’s representation of the ice shelf-water interface. Plus-signs mark the
position of pressure points for pressure gradient computations.
The letters A, B, and C, mark specific grid cells for reference in
the text. hi,j,k is the fractional cell thickness, so that hi,j,k ∆zk
is the actual cell thickness, horizontal indices (i, j) are dropped
in the schematic.
Table 1. Model mixing and friction parameters, GM = Gent
and McWilliams [1990], Leith=Leith [1996], KPP = Large et al.
[1994]
parameter name
viscosities and diffusivities
vertical viscosity
vertical diffusivity
horizontal viscosity
horizontal diffusivity
quadratic stress coefficients
Z η−h
ice-ocean stress
g (ρ − ρ0 ) dz + pNH , bottom stress
+g ρ0 η +
z
(3)
p0tot = p0top
ptop = g
z
with the gravitational acceleration g, the density ρ, the vertical coordinate z (positive upwards), and the dynamic sea-surface height η.
For the open ocean, ptop = pa (atmospheric pressure) and h = 0.
Underneath an ice-shelf that is assumed to be floating in isostatic
equilibrium, ptop at the top of the water column is the atmospheric
pressure pa plus the weight of the ice-shelf. It is this weight of
the ice-shelf that has to be provided as a boundary condition at the
top of the water column. The weight is conveniently computed by
integrating a density profile ρ∗ , that is constant in time and corresponds to the sea-water replaced by ice, from z = 0 to a “reference”
ice-shelf draft at z = −h [Beckmann et al., 1999], so that
Z 0
ptop = pa +
g ρ∗ dz.
(2)
and after rearranging
with p0tot = ptot + g ρ0 z and p0top = ptop − g ρ0 h. The nonhydrostatic pressure contribution pNH is neglected in the following.
In practice, the ice shelf contribution to ptop is computed by integrating Eq. (2) from z = 0 to the bottom of the last fully dry cell
within the ice shelf:
ISOMIP 1
0.001 m2 s−1
0.00005 m2 s−1
600 m2 s−1
100 m2 s−1
0.0025 m−1
0.0025 m−1
global (Section 4)
KPP
KPP
Leith
GM (≤ 600 m2 s−1 )
0.0 m−1
0.0 m−1
X-3
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
a) barotropic stream function [Sv]
a) barotropic stream function [Sv]
−70
−70
0
0
−72
−72
0
0.12
6
15
−80
°
longitude [ E]
0.2
−78
8
0.1
6.12
0.10
0.08
0 0.04
0
0.0
4
4
0.28
8
0.0
−76
0
0.04
0.08
0.0
−74
0.2
0
0.04
0.08
0.
12
6
0.0
1
.
−76 0
4
0.1
6
0.2
0.284
0.12
−78
0.08
0.04
0
0
−80
0
5
10
latitude [°N]
°
0
0
−74
0.2
latitude [ N]
0
0
5
10
15
longitude [°E]
b) overturning stream function [Sv]
b) overturning stream function [Sv]
0
0
−100
0
−200
−900
−80
−78
−76
−74
2
0.
−700
0
0.0 .06
4
0.02
−800
−72
latitude [°N]
0.02
04
0.
−600
2
0.0
0.0
0.0
0.02 4
−800
depth [m]
0
0.0.04
6
06
0.
−500
06
0.
04
−700
−400
0.
−600
0
0.02
−300
2
4
8
0.0
−500
0.0
0.0
.08
0.02
−400
02
0.
02
−300
depth [m]
−100
0
−200
−900
−80
−78
−76
−74
−72
°
latitude [ N]
c) melt rate [m/yr]
−70
c) melt rate [m/yr]
−70
8
−76
0
−74
0.4
0
0.
−76
0
0
8
0
−78
0
−78
0
5
10
15
Figure 2. Steady state solution ISOMIP Experiment 1. Top:
Stream function of the vertically integrated mass transport (in
Sv); middle: (Overturning) Stream function for the zonally integrated mass transport (in Sv); bottom: Freshwater flux (in m/y)
with contour interval 0.4 m/y. The freshwater flux in the southwest corner is negative (melting) and contour labels are omitted
for better visibility. Maximum and minimum freshwater fluxes
are 1.62 m/y (freezing) and −1.56 m/y (melting), with a mean
of −0.054 m/y under the inclined ice shelf base.
cells above the base of the ice shelf topography h are “dry” in the
same sense as cells below the sea floor. The ice shelf base topography can be approximated more accurately by allowing cells that are
partially filled from the bottom; cell A in Figure 1 is an example. In
this case, the pressure for evaluating horizontal pressure gradients
is still computed for the center (scalar) point (plus signs in Figure 1)
of the full cell according to Eq. (6) [in the same way as for bottom
−80
0
°
longitude [ E]
0
5
−0.4
10
−0−. 0.4
8
−80
0
4
1.2
0
0.
latitude [°N]
°
−72
−74
0.
latitude [ N]
−72
15
longitude [°E]
Figure 3. Same as Figure 2, but with the simple boundary layer
scheme. The solution is more energetic in the spin-up phase,
because the enhanced mixing allows more water to be cooled
by the ice shelf. The noise, in particular in the freshwater flux,
is reduced. Maximum/minimum/mean of freshwater flux are
1.77/−2.42/−0.054 m/y.
topography, see Adcroft et al., 1997], even if the cell center lies in
the dry part of the cell. With this formulation non-zero pressure
gradients in the absence of horizontal density gradients are avoided
in a way that is standard for the partial cell method.
2.2. Thermodynamics
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
a) barotropic stream function [Sv]
−70
0
−72
−74
0.04
−78
0.24.2
0
16
0.0.
12
0
5
0
0.08
0.04
0
10
15
longitude [°E]
b) overturning stream function [Sv]
0
−100
0
−200
0.02
04
0.
−400
−500
02
0.0
6
depth [m]
−300
0.
−600
0.08
The water is initially at rest and has uniform potential temperature (−1.9 ◦ C) and salinity (34.4). As described by Grosfeld et al. [1997] the thermodynamic processes along the sloping ice-water interface cause cold but fresh water to rise along
the ice shelf base thereby stretching the water column. This vorticity forcing drives a gyre with a western boundary current of
0.31 Sv underneath the sloping ice shelf base and a meridional
overturning of 0.095 Sv (Figure 2). Both shape and magnitude of
the circulation are comparable to ISOMIP results which are available at http://fish.cims.nyu.edu/project_oisi/
isomip/experiments/phase_I/overview.html. The
z-coordinate solution agrees with the solutions of the σ- and isopycnic models where the western boundary current transport ranges
from 0.3 to 0.36 Sv and the overturning from 0.08 to 0.09 Sv.
The magnitude of melt rates is also comparable to other ISOMIP
results. There is melting of approximately 1 m/y in the south-east
corner of the domain, where warm water from the interior is brought
0.04
0.08
0.12
−76
3. ISOMIP experiments
3.1. Experiment 1
0
0
−80
4
−700
0.0
0.06
−800
2
0.0
0.02
−900
−80
−78
−76
−74
−72
°
latitude [ N]
c) melt rate [m/yr]
−70
−72
−74
0
−76
0.8
latitude [°N]
0
4
0.
0
−78
0
The Ice Shelf–Ocean Model Intercomparison Project (ISOMIP)
is an international effort to identify systematic errors in subice shelf cavity ocean models (see http://fish.cims.
nyu.edu/project_oisi/isomip/overview.html and
Hunter [2006] for details). Currently, only data for the “Experiment 1” of “Phase I” are available for comparison. Specifications
for an “Experiment 2” are also available. Based on the description
of the experiment a rectangular domain with four closed boundaries
on the sphere is set up with a horizontal resolution of 0.3◦ in longitude and 0.1◦ in latitude covering a region from 0◦ E to 15◦ E and
80◦ S to 70◦ S; hence, the horizontal grid spacing ranges from approximately 6 to 11 km. 30 layers of thickness 30 m cover the depth
of 900 m. The ice-shelf base is at 200 m depth north of 76◦ S and
increases linearly to 700 m depth towards the southern boundary so
that the entire domain is covered with an ice shelf. The bottom is flat
at 900 m depth [see also Grosfeld et al., 1997]. Mixing and friction
parameters are given in Table 1.
Density is computed by a nonlinear equation of state [Jackett and
McDougall, 1995]. The pressure dependence of density is approximated by a depth dependence (p(z) ≈ −gρ0 z). While this (very
common) approximation can lead to errors of up to 3 Sv in GCMs
[Dewar et al., 1998], it simplifies specifying the initial pressure at
the base of the ice shelf. For the realistic simulations in Section 4 this
approximation is replaced by an exact treatment of pressure [Losch
et al., 2004]. In the case of unstable conditions, vertical convection is parameterized by a simple convective adjustment scheme.
Other schemes are available [e.g., the so-called implicit diffusion,
and KPP, Large et al., 1994] but do not alter the present results
significantly.
For comparison with other ISOMIP experiments and Grosfeld
et al. [1997] the horizontal stream function, the meridional overturning stream function and basal melt rates are shown at day 10,000
(27.8 years) of the integration when the system is in quasi-steady
state.
0
0.08
Freezing and melting form a boundary layer between ice shelf
and ocean which is parameterized following Grosfeld et al. [1997]
or Hellmer and Olbers [1989]. Both parameterizations yield an
effective heat flux and a virtual salt flux (i.e., a freshwater flux without volume change in the ocean) between ice shelf and water, that
can be conveniently applied at the base of the ice shelf, that is, as
a tendency term in the ocean at the appropriate depth. The simpler parameterization similar to Grosfeld et al. [1997] is required
for the ISOMIP experiments. The more realistic parameterization
of Hellmer and Olbers [1989] takes into account the dependence
of the freezing point of seawater on salinity and generally leads to
smaller melt rates than the former. In the present implementation,
the conservative formulation of Jenkins et al. [2001] is used. Both
schemes are outlined in the appendix.
latitude [°N]
X-4
−80
0
−0.4
5
10
−0.4
−0.8
15
longitude [°E]
Figure 4. Same as Figure 2, but with the C-D grid and boundary layer scheme. The noise in the stream functions has disappeared. Maximum/minimum/mean of freshwater flux are
1.41/−1.74/−0.048 m/y.
into contact with the ice shelf, and freezing of approximately 1 m/y
along the western boundary current, where cold water rises and
refreezes because of the pressure-dependent freezing point of seawater.
There is a striking difference between the previous ISOMIP solutions and the results of the present z-coordinate model. The zcoordinate solution is very noisy in space, which can be seen particularly in the melt rates and to a lesser extent in the overturning
stream function. The noise in the melt rates is clearly aligned with
individual grid cells and can be explained as follows: The partial
X-5
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
θk = θk hk + θk+1 (1 − hk )
(7)
where hk ∈ (0, 1] is the fractional layer thickness of the k-th
layer. The original contributions due to ice shelf-ocean interaction gθ to the total tendency terms Gθ in the time-stepping equation
θn+1 = f (θn , ∆t, Gn
θ ) are
gθ,k =
Q
and gθ,k+1 = 0
ρ0 cp hk ∆zk
(8)
for layers k and k + 1 (cp is the heat capacity). Averaging these
terms over a layer thickness ∆zk (e.g., extending from the ice shelf
base down to the dashed line in cell C) and applying the averaged
tendency to cell A (in layer k) and to the appropriate fraction of
cells C (in layer k + 1) yields
Q
ρ0 cp ∆zk
Q
∆zk (1 − hk )
=
.
ρ0 cp ∆zk
∆zk+1
∗
=
gθ,k
∗
gθ,k+1
(9)
(10)
Eq. (10) describes averaging over the part of the grid cell k + 1 that
∗
and the part with
is part of the boundary layer with tendency gθ,k
no tendency. Salinity is treated in the same way. The momentum
equations are not modified.
Figure 3 shows the effect of the boundary layer scheme. Through
averaging, the scheme increases the vertical diffusion near the icewater interface so that more warm water reaches the interface and
melting increases. As a result, the ocean is cooler and fresher
than without the boundary layer scheme. Further, the circulation
is stronger in the spin-up phase (not shown), but is very similar to
the original circulation in steady state. The noise in the buoyancy
flux fields is reduced.
The noise in the velocity fields (stream functions) is typical for
C-grid models that do not resolve the Rossby radius. The necessary averaging of the Coriolis term on a C-grid leads to a dispersion
relation of waves with false minima and hence incorrect group velocities for some short waves. When grid scale energy is generated,
for example near topography, grid scale waves can feed energy into
short-scale perturbations thus allowing standing grid scale noise to
persist [e.g., Adcroft et al., 1999]. Adcroft et al. [1999] suggested a
coupled C-D grid, where horizontal velocities are evaluated on both
3.5
max melt
max freeze
mean x 10
3
melt rate [m/y]
cell treatment of the ice shelf topography results in adjacent cells
with decreasing thicknesses within one layer (see the schematic in
Figure 1). The heat flux, or equivalently the melt rate (Eq. (A3) in
the appendix), is proportional to the difference between the temperature in ocean and at the ice shelf base TW − Tb , but the resulting
temperature tendency is proportional to the heat flux per grid-cell
volume, so that larger (thicker) cells are cooled more slowly than
smaller (thinner) cells. Hence, for a Tb fixed at the freezing point,
TW − Tb remains larger in thicker cells and this leads to larger heat
fluxes (or melt rates) at the ice-water interface. Thus, following the
ice shelf topography downwards, one encounters decreasing cell
thicknesses (decreasing temperatures and heat fluxes) until a layer
is crossed where the cell thickness jumps from thin to thick (small
heat flux to large heat flux, e.g. cell A and B in Figure 1). This jump
is present at each instance in which the topography intersects a layer
leading to repeated local maxima in heat flux and hence noise.
In a σ-model, the thickness of the layer along the ice shelf base
varies slowly (and more importantly monotonically in the ISOMIP
configuration) so that “noise” due to jumps in layer thickness can
not develop. In principle, vertical water mass exchange between
grid cells should reduce this problem in z-coordinate models, but in
practice this vertical exchange is too slow.
Introducing a simple boundary layer reduces the noise problem
at the cost of increased vertical mixing. For this purpose the water
temperature at the k-th layer abutting ice shelf topography (e.g.,
cell A in Figure 1) for use in the heat flux parameterizations is computed as a mean temperature θk over a boundary layer of the same
thickness as the layer thickness ∆zk :
2.5
2
1.5
1
0.5
0
450 m
225 m
100 m
75 m
∆z
45 m
30 m
20 m
10 m
Figure 5. Mean, minimum (maximal freezing), and maximum
melt rate (in m/y) as a function of vertical resolution. Mean melt
rates are scaled by a factor of ten and refer to the mean under the
inclined ice shelf base. ∆z = 30 m corresponds to the solution
in Figure 4.
a C-grid and a D-grid to avoid averaging for the evaluation of the
Coriolis term. This algorithm is implemented in the MITgcm and
the computational modes are damped by relaxing the D-grid velocities to the C-grid velocities with a relaxation time scale of 400,000
seconds. With these parameters, the noise in the stream functions
disappears completely (Figure 4) at the cost of additional dissipation which reduces the maxima of the horizontal and overturning
stream functions to 0.27 Sv and 0.089 Sv, respectively. Instead of
the C-D scheme, bi-harmonic viscosity may be used to damp the
noise effectively (not shown).
In order to understand the noise in the overturning stream function, consider a region where the ice shelf topography intersects a
grid layer (Figure 1). Cells A and B are subject to melting and
freezing at the ice-water interface, whereas cell C is not. Therefore,
a density difference is maintained between cells B and C that drives
a geostrophic current along the ice shelf base. This mechanism is
only present where the ice shelf topography cuts through grid layers
and thus lead to singular velocities and thus noise. The horizontal
jumps in density are also present in a experment with large vertical
cell thicknesses and not using partially-filled cells, where the noise
problem, in agreement with Adcroft et al. [1997], is much larger
(not shown).
In summary, both cooling as a function of cell thickness and
the excitation of grid scale waves, produce noise patterns that are
stationary, but that do not jeopardize the numerical stability of the
solution, and can be addressed with the above methods.
3.2. Effects of horizontal mixing and vertical resolution
The ISOMIP values for horizontal mixing lead to rather viscous
solutions. In a z-coordinate model, mixing is strictly horizontal [in
the absence of an eddy mixing scheme as in Gent and McWilliams,
1990], and strong mixing coefficients emphasize this effect. Figure 6 illustrates this bias along with a standard solution. With
the standard horizontal mixing coefficient for ISOMIP (left panel
of Figure 6) the light plume rises along the ice shelf base, but is
diluted laterally by horizontal mixing. Without explicit diffusion
(κh = 0 m2 s−1 ) and reduced numerical diffusion (with the help of
a high-order advection scheme following Daru and Tenaud [2004],
A. Adcroft, pers. comm.) the plume is much thinner (center). Isopycnic mixing according to Gent and McWilliams [1990], but without
the skew flux, makes the plume slightly thicker, but compared to
the horizontal mixing case, the lateral dilution of the plume is much
reduced (right panel). With the current grid resolution, both slope
and plume are resolved. However, if the vertical and horizontal
resolution is not sufficient to resolve the topographic slope—as is
typically the case in overflow regions in global ocean models—zcoordinate models have difficulties representing flow along such a
slope [Legg et al., 2006].
In order to assess the impact of vertical resolution, the experiment is repeated for layer thicknesses of 450 m (2 layers), 225 m
X-6
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
2
2
horizontal κ = 100 m /s
κ = 0 m /s
−200
−200
−400
−400
1
−2.
−600
−600
−800
−80
0
−70
−200
−2
−400
−800
−75
2
= 100 m /s
−2
.1
0
GM
−2
.1
0
−2
isopycnal κ
h
−600
−2
−800
−80
−75
−70
−2
−80
°
°
−75
−70
°
latitude [ N]
latitude [ N]
−2
depth [m]
h
latitude [ N]
Figure 6. Meridional temperature section in the center of the domain along 7.5◦ E for the standard experiment with
strong horizontal mixing (left, κh = 100 m2 s−1 ), without explicit horizontal diffusivity (center, κh = 0), and with
isopycnal mixing (right, κh = 0 and κGM = 100 m2 s−1 ).
relative freqency [%]
0.12
0.1
0.08
0.06
∆z = 450 m
∆z = 225 m
∆z = 100 m
∆z = 75 m
∆z = 45 m
∆z = 30 m
∆z = 20 m
∆z = 10 m
0.04
0.02
0
−2.5
−2.4
−2.3
−2.2
−2.1
−2
temperature [°C]
Figure 7. Histogram of temperatures as a function of vertical resolution. ∆z = 30 m corresponds to the solution in Figure 4.
(4 layers), 100 m (9 layers), 75 m (12 layers), 45 m (20 layers), 30 m
(30 layers, this is the standard case), 20 m (45 layers), and 10 m
(90 layers). Figure 5 shows the mean, minimum, and maximum
melt rates as a function of layer thickness. Note that, because of
the closed domain, the total heat and freshwater flux between ice
and ocean is required to be zero in steady state; the mean melt rates
in Figure 5 refer to the mean beneath the inclined ice-shelf base,
because only this quantity can be affected by the resolution in this
experiment. Mean melt rates are slightly lower for very low resolution, but remain nearly constant for ∆z ≤ 100 m. Maximum
freezing rates become smaller with very high vertical resolution.
Maximum melt rates reduce with smaller ∆z; these maximum melt
rates are found in thick cells next to very thin cells (at topographylayer intersections), in agreement with the discussion of the noise
in Figure 2. Figure 7 gives an impression of mixing and dilution by
means of a histogram of temperature values normalized by the total
number of grid points. For vertical layer thicknesses ≤ 100 m, the
temperature distribution is stable along the ice shelf base (low temperatures). Smaller layer thicknesses allow warmer temperatures
near the bottom of the interior because of a sharper pycnocline underneath the ice shelf and reduced vertical mixing. At least in this
configuration with this particular slope and horizontal resolution,
∆z = 100 m appears to be the minimum vertical resolution that is
required to resolve ice shelf-ocean processes.
4. Global Ocean Circulation Model with Ice Shelf
Cavities
The advantage of an ice shelf cavity model in z-coordinates is its
straightforward applicability to existing global ocean models with
z-coordinates. Up to now, the majority of global models are zcoordinate models and adding an ice shelf cavity model in, say,
σ-coordinates to such a model requires considerable recoding. In
this section results from a comparison of a “standard”, but coarse
ocean general circulation model in z-coordinates with and without
explicit modeling of ice shelf cavities are presented.
4.1. Model configuration
A (nearly) global ocean model (excluding the Arctic Ocean) with
sea ice is set up at a nominal resolution of 2◦ . In the northern
hemisphere the grid spacing is constant at 2◦ and the domain is
closed at 80◦ N in order to avoid any difficulties associated with the
convergence of latitudes and the pole singularity. In the southern
hemisphere, the grid spacing is locally isotropic, that is, the latitude
(φ) spacing is scaled by cos φ, giving a resolution of 100 km near
60◦ S and around 50 km in the ice shelf regions of the Ross Sea and
the Filchner-Ronne Ice Shelf in the Weddell Sea. While 23 vertical
layers allow only limited vertical resolution in the ice shelf cavities,
this situation represents a typical problem of z-coordinate models:
the vertical resolution near the bottom is poor. This problem is partially alleviated by the partial cell treatment of topography [Adcroft
et al., 1997]. The layer thicknesses are 10, 10, 15, 20, 20, 25, 35,
50, 75, 100, 150, 200, 275, 350, 415, 450, and 7×500 m.
Realistic topography is derived from a combination of GEBCO
[British Oceanographic Data Center, 2003] and the Smith and
Sandwell topography [Smith and Sandwell, 1997; Marks and Smith,
2006]; the ice shelf topography is taken from the Bremerhaven Regional Ice Ocean System [BRIOS, Beckmann et al., 1999] and interpolated to the computational grid. Because of the coarse resolution,
no attempt to resolve the small Antarctic ice shelves has been made;
only the large ice shelf in the Ross Sea and the Filchner-Ronne Ice
Shelf in the Weddell Sea are considered here.
The ocean model is initialized with temperature and salinity climatology [Levitus et al., 1994; Levitus and Boyer, 1994] and forced
by daily wind, air temperature and humidity, downward long and
short wave radiation fields, monthly precipitation fields and a constant run-off field. All fields are part of the climatology (“normal
X-7
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
0
4
16
24 6
8
0
26
12 14
0
64
20
0
0
depth [m]
28
−2
0
−20
0
1102
−40
28
0
−60
−4
2 −4
−2
−4
−5000
−80
28
−4
−2
60
82
14
1620 18
22
24
26
24
128022
14
10
12
0
−8
−6
0
−4000
28
14
8
10
0
−3000
8
−2000
6
−2
−1000
−−1−1− −−2222426−30
2 0 104
18828
−18
−−20
8−34
46
0 1 16
230
−3
360−2
84−4
−4 23422 20
6 4 −0812146−−
−6−32−−2
12
12
1
4
10
16
18
12
16
6
10 8
4
6
2
0
42
0
−2
−2
−4
−
6
−6
2
4
8 20 0
10
1186
−2
0
12 14
0
0
o
latitude [ N]
20
40
0
60
Figure 8. Global overturning stream function in Sv. The contour interval is 2 Sv.
0 00
0 0
0
60oN
0
0
0
0
0
0
0
0
0
0
0
0
o
20
0
0
0
0
20
40
060 120
1080
60oW
−20
20
120
0o
−020
40
60
10800
60oE
120
0
0−2
40
201 60
080
0
0
0
0
0
0
−−420
0
−20
−20
20 40
8060
60 S 100
120
120
o
180 W
120oW
0
−2
o
30 S
0
12
0
0
0
−20
0
0
0
0
o
20
0
0
40
0
20
0
0
0
20
20 20
0
30oN
0
0
0
120oE
180oW
Figure 9. Barotropic stream function in Sv. The contour interval is 10 Sv.
year”) of the Common Ocean-ice Reference Experiments (CORE)
data set [Large and Yeager, 2004]. Wind stress and buoyancy fluxes
are computed from bulk formulae [Large and Pond, 1981, 1982;
Large and Yeager, 2004].
Horizontal mixing is parameterized following Gent and
McWilliams [1990] with a variable diffusivity following Visbeck
et al. [1996] with an imposed maximum of 600 m2 s−1 and Large
et al. [1997]’s slope clipping scheme; the horizontal viscosity for
harmonic mixing of momentum is flow dependent according to a
scheme by Leith [1996]. Resulting maximum viscosities are on the
order of 5×104 m2 s−1 in western boundary currents and the Antarctic Circumpolar Current. In most parts of the ocean the viscosities
are much lower. For vertical mixing, the KPP-scheme [Large et al.,
1994] is used. Density is computed from a fully non-linear equation
of state [McDougall et al., 2003]. No surface restoring to climatology is applied.
The dynamic-thermodynamic sea ice model is described in detail by Losch et al., (manuscript in preparation). It is based on the
model used in Menemenlis et al. [2005], but involves a new discretization on a C-grid. Stress and buoyancy flux coupling to the
ocean is standard.
In two experiments the model is spun-up for 100 years with full
ice shelf-ocean interaction following Hellmer and Olbers [1989]
and with an accelerated time step of 12 h for the tracer equations and
20 min for the momentum equations to reach a quasi-equilibrium at
least for the upper waters. In the first case the ice shelf cavities are
kept open, in the second case the ice shelf cavities are closed and
replaced by land.
4.2. General features of the circulation
The general circulation of this ocean model includes all features
expected of a simulation of this resolution and degree of realism.
For a summary, Figure 8 and Figure 9 show the global overturning
stream function and the barotropic stream function of the reference
run. With a Drake Passage transport of 116 Sv below the “canonical”
value of 134 Sv [e.g., Whitworth and Peterson, 1985], the Antarctic
Circumpolar Current is on the weak side, while the Atlantic overturning at 45◦ N of 28 Sv is stronger than generally estimated. The
temperature and salinity bias is similar to that of recent global models of similar resolution [Gnanadesikan et al., 2006].
Guided by our interest in the effects of the ice shelf cavities, we
restrict any detailed analysis to the Southern Ocean, in particular the
Weddell Sea. Here, the observed temperature and salinity structure
[Olbers et al., 1992; Klatt et al., 2005] is mainly preserved after
the spin-up, but a warm bias compared to climatology [Levitus and
Boyer, 1994] is clearly visible in the Warm Deep Water (WDW)
and bottom waters in the Weddell Sea (Weddell Sea Bottom Water,
WSWB), see Figure 10. This warm bias is associated with a haline
bias (not shown); these biases are not unusual for ocean models with
this resolution and time scale [see e.g., Gnanadesikan et al., 2006].
The warming of the WDW does not have local causes but can be
explained by the inflow of overly warm Circumpolar Deep Water
(CDW) into the Weddell gyre east of the Greenwich Meridian. The
X-8
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
o
0 −0.2
−0.5
1
3 2.5
1.5
0 −0.2
−0.5
2 1.5
0.5
Potential temperature along Weddell Sea section
0
−1
−0.5
−0.2
−1.5
100.50.2
1
−200
0.2 1−1.5
0.5−1
−200
0
0
60
65
70
depth [m]
−1
.2
−0.8
−0.6
0 1.81.61.4
−0.2
2 −0.4
1010.8
0. 2.4
55
−200
−400
60
65
−1000
−2000
−3000
−4000
−5000
70
60
70
80
Difference along Weddell Sea section
−1−0.8 −0.6
−0.4
0.4
0.6
1.20.2
1 0.8
1.6
21.4
−0.
1.41
2
0.8
0.6
−1
−0.8−−0.6
0.4
0.6 0 1.4
1.2
0−0.4
1 0.8
1.6
21.4
.2
10.2
0.8
0.
4
.6
.2 0.6
1112.8
.2
0
2
0
−2
50
latitude [°degS]
4
2
0
0
2
1.
−0.
00.4
.4
0 12 1.4
0.2
0.6
−0
1.2
2
1.8
1.6
11.2
0.8
0.8
1 0.6
1
0
6
0.
−400
−1000
−2000
−3000
−4000
−5000
−1
−0.8
−0.6
0 1.6 2 −0.4
−0.2
1.8 1.4 010.8
..2
1 02.4
6
0.
−200
0.4 2
−0.
−0
0 1 1.4
0.2
0.4
0.6
1.2
2
1.8
1.6
0.50.2
0
50
Difference along 0oE
0
−10.2
−0.5
−0.2
100.5
1 −1.5
43.5.5 5
3 22 1.
−1000
−2000
−3000
−4000
−5000
0.2
−0.2
55
−400
0.2 1−1.5
0.5−1
1
depth [m]
−400
−1000
−2000
−3000
−4000
−5000
C
1
o
Potential temperature along 0 E
0
60
70
80
latitude [°S]
Figure 10. Top: Two selected sections of potential temperature in the Weddell Sea. Left: A one year mean of
sections along the 0◦ meridian. Right: A section through the Weddell Sea from (75◦ W,80◦ S) to (15◦ W,50◦ S) [see
also Timmermann and Beckmann, 2004]. Bottom: Difference to climatology (model minus climatology) [Levitus and
Boyer, 1994].
o
0
o
o
E
4.3. Circulation in the Filchner-Ronne Ice Shelf cavity
0
72 oS
W
2.5
oS
0
oS
78
oS
82
60 oS
o
180 W
−0.01
Figure 12. Vertical mean of potential temperature difference between a run with and without Filchner-Ronne and Ross Sea ice
shelf cavities (ice shelf minus no ice shelf, in ◦ C). The magenta
line in the temperature plot marks the location of the section
along the anomaly minimum in Figure 13.
−2.5
oS
80
−0.005
12
o
0
m/y
5
0 oE
84 oS
12
The mean depth-averaged circulation in the Filchner-Ronne Ice
Shelf (FRIS) cavity is shown in Figure 11. In some of the previous model results the circulation underneath the FRIS is determined
largely by geostrophic contours, and hence by water column thick-
76
0.005
temperature [ C]
60 o
W
0.01
60
bottom waters are warming because there is little to no flux of cold
water down the continental slopes of the Weddell Sea, which is a
typical problem of z-coordinate models without a special bottom
layer treatment [e.g., Beckmann and Doescher, 1997; Campin and
Goosse, 1999].
The Weddell Gyre with a mean transport of a little over 26 Sv is
weaker than in other simulations with a σ-coordinate model [Beckmann et al., 1999; Timmermann et al., 2002a], but stronger than, for
example, in the z-coordinate model of Timmermann et al. [2005].
o
54oW 36 o
72 W
W
−5
Figure 11. Depth-averaged mean circulation pattern and freshwater flux into the ocean (in m/y, negative values mean melting)
underneath the Filchner-Ronne Ice Shelf.
ness [Grosfeld et al., 1997; Gerdes et al., 1999]. The circulation in
Figure 11 does not follow geostrophic contours, but it is similar to the
solutions shown in Timmermann et al. [2002b, their 1989 solution]
and Jenkins and Holland [2002]: The water enters in the western
part of the ice shelf, circulates counter-clockwise (anti-cyclonically)
around Berkner Island and leaves the cavity as cold and fresh water through the Filchner Trough in the east. As a side remark, the
X-9
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
o
temperature difference [ C]
−500
−0.4
4
−0.
0
−0.4
−0.8
4
0
−0.8
−0.8
.4
−0
0
depth [m]
0
−0.
−0.4
0 −0.4
−1000
−1500
0
500
1000
1500
2000
o
salinity difference [ C]
−0.
0 06 −0.0−0.08
2 −0.04 0−0.02
0
.02.04
−0−0
−0.06
−0.04
−500
−1000
0
depth [m]
02
0.
0
−1500
0
500
02 0
−0.06
−0.06
−0. −0.08
−0.08
−0.0
4
−0.04
−0.02
1000
1500
distance from ice shelf [km]
2000
the solution in Figure 11, yield mean values that are only 20% of
those of Hellmer and Olbers [1989]. In a sensitivity experiment
with such values, the mean freshwater flux for FRIS can be reduced
by over 50% without changing the circulation and melting pattern
significantly (not shown). Second, poor vertical resolution may be
a problem. The vertical z-layer thickness in the ice shelf cavity is
on the order of 100 to 400 m (the actual cell thickness can be smaller
due to the partial cell treatment of the topography), so that cold and
fresh melt water at the ice-ocean interface is immediately mixed
into a large volume and the temperature at the ice shelf interface
remains well above freezing, leading to more melting. Third, as a
consequence of a bias of the model far away from the ice shelves,
the water entering the ice shelves may be too warm (see Figure 10).
Then, ice shelf processes can not cool the water sufficiently, which
may also explain the absence of a large region of refreezing west
of Berkner Island that is observed in previous simulations [Gerdes
et al., 1999; Jenkins and Holland, 2002]. In the present simulation
this region is characterized by reduced melt rates but no refreezing.
4.4. Effect of ice shelf cavities on the global circulation
Figure 13. Vertical section of potential temperature (in ◦ C)
and salinity difference between a run with and without FilchnerRonne and Ross Sea ice shelf cavities (ice shelf minus no ice
shelf). The location of the section along the anomaly minimum
is shown in the temperature plot of Figure 12.
Figure 12 shows the vertically averaged temperature difference
between a run with and without ice shelf cavities. In the Weddell
Sea, warm and saline water enters the FRIS in the west and emerges
a) Weddell Sea tracer
o
years
80
0
o
15 S
o
30 S
30 oW
o
45 W
oW
60
S
o
180 W
o
120 W
o
60 W
o
o
0
o
60 E
0
o
120 E
180 W
b) Weddell Sea section of Weddell Sea tracer
S
15
o
2
−1000
1.5
S
o
76
3255
25 50
o
72
25
35
0
S
5 5
1 45
68
ppt
0
0
0
depth [m]
o
64
20
o
75 S
5
35
o
40
60oS
15 oW
0o
60
60
45oS
1
−3000
−4000
max = 164 cm
S
−2000
0.5
−5000
−70
o
80
−60
−50
−40
−30
−20
−10
0
0
°
latitude [ N]
c) Ross Sea tracer
S
0o
15oS
o
30 S
years
80
60
45oS
Figure 14. Difference of sea ice volume per unit area in the
Weddell Sea between a run with and without ice shelf cavities
(ice shelf minus no ice shelf, in centimeters).
40
60oS
20
o
75 S
180oW
60oW
0o
60oE
120oE
0
180oW
d) Ross Sea section of Ross Sea tracer
ppt
0
2
−1000
depth [m]
orientation of the flow is sensitive to the circulation in the Weddell
Sea: In more viscous runs (with a constant viscosity coefficient
of 5 × 104 m2 s−1 ), the direction of the circulation reverses to cyclonic and cold and fresh water leaves the cavity in the West, similar
to Timmermann et al. [2002b]’s 1992 solution (not shown). With
Hellmer and Olbers [1989]’s standard parameters for the ice shelf
ocean interaction (in particular the constant exchange coefficients
for heat and fresh water of 1 × 10−4 m s−1 and 5.05 × 10−7 m s−1 ,
respectively), the mean melt rate underneath FRIS is 0.94 m y−1 ,
maximum melt and freezing rates can be as high as 7.8 m y−1 and
1.4 m y−1 , respectively. Melt rates are highest near the grounding line where the ice shelf is deepest (Figure 11). The implied
mean total freshwater flux is 13.9 × 103 m3 s−1 , hence three to four
times higher than in previous studies [Hellmer, 2004; Timmermann
et al., 2001]. There are multiple possible reasons for this overestimation. First, the current implementation with constant exchange
coefficients may be inappropriate. Velocity dependent exchange
coefficients following Holland and Jenkins [1999], diagnosed for
120oW
1.5
−2000
1
−3000
−4000
0.5
−5000
−70
−60
−50
−40
−30
−20
−10
0
0
latitude [°N]
Figure 15. Arrival time (in years) of a threshold concentration
of 1 ppt anywhere in the water column for a passive dye tracer
that is released in a) the Filchner-Ronne Ice Shelf cavity, c) the
Ross Sea Ice Shelf cavity. The vertical distribution of the corresponding tracers (in ppt of the original concentration) after 100
years are averaged over longitude bands of b) 36◦ W to 56◦ W
and d) 160◦ E to 160◦ W.
X - 10
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
from the Filchner Trough as cold and fresh water east of Berkner
Island (Figure 11). The water is buoyant and rises to a lower depth
(parameterized by vertical mixing of the KPP scheme). The cold
and fresh anomaly moves off the shelf and enters the circulation
system of the Weddell Gyre and rises further until it reaches the surface east of the northern tip of the Antarctic peninsula (Figures 12
and 13). Part of the cold and fresh water reaches depths of approximately 1500 m after leaving the shelf (Figure 13), but deep waters
below the exit depth of the Filchner Trough tend to be warmer than
in the run without ice shelf cavities pointing to less bottom water
formation as discussed in Hellmer [2004]. The inclusion of the ice
shelf processes also lead to more sea ice with a pattern (Figure 14)
that is comparable to that found by Hellmer [2004]. Furthermore, in
the run with ice shelves the northward circulation along the Antarctic peninsula is moved off the shelf, allowing a warm anomaly on
the shelf. The origin of this anomaly, which is compensated by a
negative salinity anomaly, is not clear.
Passive tracer simulations, in which the ice shelf cavities are
filled once with passive tracers of concentration 1 at the beginning
of the integration, illustrate that in the Ross Sea, the cold and fresh
glacial melt spreads westwards in the Antarctic coastal current (Figure 15c). Also the water mass modified by melt water tends to be
less buoyant than surrounding waters and sinks to greater depths
where it spreads northwards across the equator (visible as a northward pointing tongue along approximately 170◦ W in Figure 15c and
d). The glacial melt water from the Weddell Sea circulates along
the shelf break. Along its path it is diluted and cannot sink as deep
as in the Ross Sea although small tracer concentrations are found
below 2000 m. Thus, the melt water from the FRIS is confined to
the Antarctic Circumpolar Current (Figure 15a and b); it circulates
around Antarctica before re-entering the South Atlantic through the
Drake Passage along the cold water path [Rintoul, 1991].
The potential influence of ice-shelf melt water on the global circulation was the initial driving force for this work, but the simulation presented here is too short to provide conclusive evidence
about inter-hemispheric or global effects. However, it is observed
that the inclusion of ice shelf cavities has an effect particularly in
the sensitive convection areas in the North Atlantic (not shown). A
similar effect was observed by Hellmer et al. [2005] who introduced
the effect of ice shelf cavities by assimilating data from a regional
model with ice shelves into a global model. The observation in the
present model run should be taken with caution because of the solid
boundary at 80◦ N in this simulation, although at this point it may
be speculated that the anomalies represent small changes in the general circulation, for example a horizontal shift of convection events.
These changes may be triggered by baroclinic anomaly waves originating in the Southern Ocean. How and on what time scales these
shifts occur is beyond the scope of this paper and will be discussed
elsewhere.
5. Conclusions
We implemented an ice shelf cavity model in a z-coordinate
ocean model. Difficulties associated with spurious horizontal mixing in z-coordinate models can be overcome by sufficient grid resolution and parameterized isopycnic mixing. While not essential, the
partial cell treatment of topography makes it possible to represent
the complicated topography of the ice shelves more accurately. The
new model approach compares favorably with established ice shelf
models in the framework of ISOMIP. The pressure gradient error
which is notorious in terrain following models, is not an issue for zcoordinate models by construction. This can be shown in a second
ISOMIP experiment where the thermodynamic ice-ocean interaction is turned off; as expected, the model remains at rest within the
64bit working precision.
Solid (castellated) boundaries and the Coriolis term are a perpetual source of noise on a C-grid. Furthermore, if the vertical
resolution is not sufficient to resolve the ice shelf topography, the
effect of castellated boundaries is amplified at the ice-ocean interface because there, as opposed to lateral or bottom boundaries, the
effect of the local thermodynamic ice-ocean interaction is a function
of varying cell thickness. In principle, the noise emerging from the
thermodynamics can be reduced by introducing a simple boundary
layer scheme, but both types of noise are damped effectively with
general techniques such as bi-harmonic viscosity (not shown) and
a special treatment of the Coriolis term [Adcroft et al., 1997]. In a
realistic global ocean simulation this noise is not observed.
In practice, one has to find a compromise between affordable
resolution in a global model and the necessary resolution beneath
ice shelves. For the ISOMIP experiment, the ice shelf slope is reasonably resolved with a vertical layer thickness of 100 m and less:
The vertical resolution dependence of mean melt rates and internal
mixing becomes small in experiments with a vertical resolution better than 100 m. The prescribed large ISOMIP mixing parameters
emphasize the z-coordinate models’ weakness of spurious diapycnic mixing. Although the experiments with horizontal mixing can
reproduce the rising light melt water plumes, spurious diapycnal
mixing can be reduced dramatically with the help of an isopycnic
mixing scheme [Gent and McWilliams, 1990] and high-order advection schemes, thus leading to a thinner boundary layer along the
ice shelf.
In a numerical experiment simulating the global ocean the principal pattern of the circulation in the FRIS is reproduced. Estimated
melt rates are higher than the observed and previously simulated
values. This can be explained by the fact that the vertical resolution
in the depth range of the ice shelf cavities is poor and more melting
is required to cool a large (grid cell) volume of water. Previous models with terrain-following or isopycnic vertical coordinates have the
advantage of high vertical resolution (in shallow cavities the vertical resolution in σ-models can be so high that it imposes constraints
on the numerical stability, H. Hellmer, personal communication),
and can resolve thin layers of cold water beneath the ice shelf. As
a possible solution for z-coordinate models, the exchange coefficients for heat and fresh water could be made a function of the local
cell thickness or interior mixing (as provided by the oceanic mixing
scheme) in order to reduce the melt rates. Further, parameterizing
the exchange coefficient as a function of friction velocity following
Holland and Jenkins [1999] could improve the melt rate estimates
in a global context. One can also speculate that the melt rate is controlled to a large extent by the properties of the water entering the
ice shelf cavities. This water is too warm in the present simulation
leading to too strong melting.
The approach presented here can be implemented in any zcoordinate model that supports partial cells in the vertical. Because of the numerical effects and the limited resolution near the
ice-ocean interface in z-models it may be desirable to combine the
benefits of a z-coordinate model (no pressure gradient error over
topography) with the high resolution at the ice-ocean interface of
the terrain-following coordinate model of, for example, Beckmann
et al. [1999]. This can be achieved with the help of a surface following coordinate system [Adcroft and Campin, 2004]. Alternatively, a
boundary layer model analogous to bottom boundary layer models
[e.g., Killworth and Edwards, 1999] could be implemented. However, for both approaches, the pressure gradient errors need to be
addressed in the same way as in terrain-following coordinate models.
In the coarse global ocean simulation of this study, the cold and
fresh melt water from the ice shelf cavities leads to additional sea
ice thickness. This points to a reduced production of bottom water
and thus a reduced overturning in the model [Hellmer, 2004]. The
glacial melt of the Ross Sea ice shelf cavity can be traced along the
bottom as far north as 15◦ S on a 100 year timescale implying far
field effects of ice shelf processes. In contrast, glacial melt from the
Weddell Sea is confined to the ACC. Previous circulation models
with ice-shelf cavities were regional models and could not detect
these large scale effects. Both local, regional and global effects
emphasize that for ocean state predictions, the (effects of) ice shelfocean interaction ought to be included in ocean general circulation
models.
Acknowledgements: I thank my colleagues Klaus Grosfeld,
Hartmut Hellmer, Michael Schodlok, Sergey Danilov, Malte
Thoma, and Jill Schwarz for many helpful discussions. MS also
kindly provided the BRIOS topographies used in this work.
LOSCH: ICE SHELF CAVITIES IN A Z-COORDINATES
Appendix A: Thermodynamics
Two different thermodynamic schemes are used in this paper. For
the ISOMIP experiments in Section 3, the upward heat flux Q at the
interface is parameterized in a bulk formulation proportional to the
temperature difference between the ice shelf base and the sea-water
Q = ρcp γT (TW − Tb )
(A1)
where ρ is the density of sea-water, cp = 3974 J kg−1 K−1 is the
specific heat capacity of water, and γT = 10−4 m s−1 the turbulent
exchange coefficient. TW is the temperature of the model cell adjacent to the ice-water interface. The temperature at the interface Tb
is assumed to be the in-situ freezing point of sea-water Tf which is
computed from
3/2
2
Tf =1.710523 × 10−3 SW − 2.154996 × 10−4 SW
− 0.0575 SW − 7.53 × 10−4 pW
(A2)
with the salinity SW and the pressure pW (in dBar) in the cell at the
ice-water interface [Gill, 1982].
Neglecting the heat flux through the ice, all heat is used for basal
melting and freezing. The associated upward freshwater flux (negative melt rate, in units of fresh water mass per time) can be computed
by
Q
(A3)
q=−
L
−1
with the latent heat of fusion L = 334000 J kg . Upward heat
flux implies basal melting therefore a downward freshwater flux,
hence the minus sign. From the freshwater flux a virtual salt flux is
computed using a constant reference salinity of 34.4 according to
the ISOMIP specifications.
The more realistic three-equation-thermodynamics of Hellmer
and Olbers [1989] with modifications following Jenkins et al. [2001]
is sketched for completeness. Instead of Eq. (A3) the total heat flux
in Eq. (A1) is expressed as
Q = cp (ργT − q)(TW − Tb ) = −Lq − ρI cp,I κ
(TS − Tb )
(A4)
h
where ρI = 920 kg m−3 , cp,I = 2000 J kg−1 K−1 , and TS are the
density, heat capacity and the surface temperature of the ice shelf;
κ = 1.54×10−6 m2 s−1 is the heat diffusivity through the ice-shelf
and h is the ice-shelf draft. The second term on the right hand side
describes the heat flux through the ice shelf. A constant surface
temperature TS = −20 ◦ C is imposed. From the salt budget, the
(virtual) salt flux across the shelf ice-ocean interface is equal to the
(virtual) salt flux due to melting and freezing:
(ργS − q)(SW − Sb ) = −Sb q,
(A5)
where γS = 5.05 × 10−3 γT is the turbulent salinity exchange coefficient, and SW and Sb are defined in analogy to temperature as the
salinity of the model cell adjacent to the ice-water interface and at
the interface, respectively. Equations (A4) and (A5), together with
a linear equation for the freezing temperature of sea water can be
solved for Sb or Tb , from which the freshwater flux q and the heat
flux Q can be computed. This formulation yields smaller melt rates
than the simpler formulation of equations (A1) to (A3) because the
freshwater flux due to melting decreases the salinity which raises
the freezing point temperature and thus leads to less melting at the
interface.
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