dansereau etal

dansereau etal
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,
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2
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Simulation of sub-ice shelf melt rates in a general
circulation model: velocity-dependent transfer and
the role of friction
1
1
Véronique Dansereau, Patrick Heimbach, Martin Losch,
2
Corresponding author: Veronique Dansereau, Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, office 54-1421, Cambridge, MA 02139,
USA. (vero [email protected])
1
Department of Earth, Atmospheric and
Planetary Sciences, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA.
2
Alfred Wegener Institute, Bremerhaven
D-27570, Germany.
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Abstract.
Two parameterizations of turbulent boundary layer processes at the in-
6
terface between an ice shelf and the cavity circulation beneath are investi-
7
gated in terms of their impact on simulated melt rates and feedbacks. The
8
parameterizations differ in the transfer coefficients for heat and freshwater
9
fluxes. In their simplest form, they are assumed constant and hence are in-
10
dependent of the velocity of ocean currents at the ice shelf base. An augmented
11
melt rate parameterization accounts for frictional turbulence via transfer co-
12
efficients that do depend on boundary layer current velocities via a drag law.
13
In simulations with both parameterizations for idealized as well as realistic
14
cavity geometries under Pine Island Ice Shelf, West Antarctica, significant
15
differences in melt rate patterns between the velocity-independent and de-
16
pendent formulations are found. Whereas patterns are strongly correlated
17
to those of thermal forcing for velocity-independent transfer coefficients, melt-
18
ing in the case of velocity-dependent coefficients is collocated with regions
19
of high boundary layer currents, in particular where rapid plume outflow oc-
20
curs. Both positive and negative feedbacks between melt rates, boundary layer
21
temperature, velocities and buoyancy fluxes are identified. Melt rates are found
22
to increase with increasing drag coefficient Cd , in agreement with plume model
23
simulations, but optimal values of Cd inferred from plume models are not
24
easily transferable. Uncertainties therefore remain, both regarding simulated
25
melt rate spatial distributions and magnitudes.
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1. Introduction
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Interactions between the ocean circulation and the ice/ocean interface under floating
27
ice shelves have received considerable attention in the context of observed changes in
28
flow speed and thinning of marine ice sheets around Antarctica (e.g., Joughin and Alley
29
[2011] for a review of the fast-growing literature on this subject). Among the most recent
30
studies, Pritchard et al. [2012] deduced maximum overall thinning rates of up to 6.8
31
m/y between 2003 and 2008 for ice shelves along the Amundsen and Bellingshausen Sea
32
coasts, despite thickening of the firn layer and increased influx from glacier tributaries.
33
They concluded that regional thinning is caused by increased basal melt, driven by ice
34
shelf-ocean interactions. Observations by Jacobs et al. [2011] indicated a 6% increase
35
between 1999 and 2004 in the temperature difference between the base of Pine Island
36
Ice Shelf (PIIS) in the Amundsen Sea Embayment and the ocean just below, consistent
37
with an increased volume of warmer Circumpolar Deep Water (CDW) outside the cavity.
38
Although significant, the authors pointed out that this warming is too small to explain
39
the 77% increase in the strength of the circulation under PIIS and the 50% increase
40
in meltwater production observed over the same period. Their results suggest that the
41
internal cavity dynamics is at least as, if not more important, than hydrographic conditions
42
of the far field ocean in controlling the ice shelf mass balance.
43
Deploying instruments at the base of hundreds of meters thick ice shelves is a serious
44
technological challenge, hampering direct measurements of ice shelf-ocean interactions
45
and associated melt rates. The investigation of ice shelf cavities dynamics therefore rely
46
largely on model simulations. In particular, the turbulent mixing that occurs within a
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thin boundary layer at the ice shelf base was identified as the critical process by which
48
the sensible heat and kinetic energy of the ocean impact the melting and refreezing that
49
control both the mass balance of the ice shelf and the buoyancy forcing on the cavity cir-
50
culation [Holland and Jenkins, 1999; Jenkins et al., 2010a]. Current modeling approaches
51
do not resolve the turbulent boundary processes. Hence turbulence closure schemes, i.e.
52
parameterizations of these fluxes, are required to infer melt rates. Since turbulent pro-
53
cesses have not yet been directly measured at the ice shelf-ocean interface [Jenkins et al.,
54
2010a], these parameterizations remain highly uncertain.
55
The turbulence closure employed in most models is based on a standard approach in
56
which fluxes are related to spatial gradients of temperature and salinity via bulk turbulent
57
exchange velocities (or piston velocities) γ. The simplest (and earliest) parameterization
58
with constant heat and freshwater exchange velocities γT and γS [Hellmer and Olbers,
59
1989] implicitly assumes a temporally and spatially uniform ocean velocity at the ice shelf
60
base. In this case, the only direct forcing on melt rates is the gradient in temperature
61
between the ice interface at the local freezing point and the ocean just below. Example
62
models that have adopted this approach are BRIOS and BRIOS-2 [Beckmann et al., 1999;
63
Timmermann et al., 2002a, b], ROMS [Dinniman et al., 2007] and HIM [Little et al., 2008].
64
Ocean currents are the dominant physical driver of turbulent heat and salt transfers
65
at the ice shelf base. Where tidal currents are large, they are thought to be a major
66
source of turbulent kinetic energy in ice shelf cavities [MacAyeal , 1984a, b, 1985a, b;
67
Holland , 2008; Jenkins et al., 2010a; Mueller et al., 2012; Makinson et al., 2012]. In
68
the velocity-independent melt rate parameterizations, the impact of currents or tides on
69
the distribution of sub-ice shelf melting is indirect, hence limited. A more physically
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motivated parameterization of the turbulent heat and salt exchanges therefore accounts
71
for the kinematic stress at the ice-ocean interface and defines transfer coefficients γT and
72
γS in terms of a friction velocity that is directly related to current velocity [Jenkins, 1991;
73
Holland and Jenkins, 1999; Jenkins et al., 2010a]. Such a parameterization is inspired
74
by formulations employed in models of sea ice-ocean interactions [McPhee et al., 1987;
75
McPhee, 1992; McPhee et al., 1999, 2008].
76
Many models employed today to simulate sub-ice shelf melt rates have adopted velocity-
77
dependent parameterizations of turbulent heat and freshwater transfer, e.g., Holland and
78
Jenkins [2001]; Jenkins and Holland [2002]; Holland et al. [2003, 2008]; Makinson et al.
79
[2011] (MICOM), Smedsrud et al. [2006]; Holland et al. [2010] (MICOM/POLAIR), Hol-
80
land and Feltham [2006] (plume model), Little et al. [2009] (HIM), Timmermann et al.
81
[2012] (FESOM), and Dinniman et al. [2011]; Mueller et al. [2012]; Galton-Fenzi et al.
82
[2012] (ROMS). Nevertheless, velocity-independent formulations are also still in use. Ex-
83
amples of the latter that either appeared since the review on the subject by Jenkins et al.
84
[2010a] or were not mentioned in that review are Dinniman et al. [2007] (using ROMS,
85
but later updated to velocity-independent, Dinniman et al. [2011]), Heimbach and Losch
86
[2012] and Schodlok et al. [2012] (using MITgcm) and Kusahara and Hasumi [2013] (using
87
COCO). More importantly, in most cases where models have been updated from velocity-
88
independent to dependent formulations, the impact has not been documented. To our
89
knowledge, the work of Mueller et al. [2012] on Larsen C ice shelf is the only published
90
direct model comparison of the spatial distribution of melt rates and cavity circulation
91
simulated with and without a velocity-dependent melt rate parameterization. The results
92
of our study indicate that further comparisons and sensitivity analyses of the two types of
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parameterizations are warranted to better understand the heat and freshwater transfers
94
simulated in models currently in use.
95
Performing such comparisons for models with different vertical discretization is a further
96
motivation of our study. The ROMS model used by Mueller et al. [2012] is based on
97
terrain-following (or “σ”) vertical coordinates, which may exhibit different behavior to
98
that of isopycnal models (e.g., MICOM, HIM), or z-level or height) models. In this study,
99
we use a z-level model, the Massachusetts Institute of Technology general circulation
100
model [MITgcm, Marshall et al., 1997a; Adcroft et al., 2004].
101
Another important distinction in the context of ice shelf-ocean interactions is that
102
between “cold” and “warm” ice shelves. Larsen C is an example of the former, floating in
103
waters near the surface freezing point. One interest behind the present study is in refining
104
our understanding of simulated melt rates under PIIS. This ice shelf is in contact with
105
CDW nearly 3◦ C above the surface freezing point and hence is an example of the later.
106
It is therefore unclear to which extent results obtained by Mueller et al. [2012] for Larsen
107
C are readily transferrable to PIIS and adjacent warm ice shelves.
108
PIIS is home to the strongest ocean thermal forcing and mass loss in Antarctica [Rignot
109
and Jacobs, 2002; Joughin et al., 2010; Jacobs et al., 2011]. Two recent studies [Heimbach
110
and Losch, 2012; Schodlok et al., 2012] have simulated sub-ice shelf melt rate magnitudes
111
and spatial patterns using the MITgcm, although neither of these have used velocity-
112
dependent transfer coefficients. An in-depth understanding of the dependence of melt
113
rates on the parameterization employed is a crucial step to increase our confidence in
114
simulated melt rates in this important region.
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Finally, the anticipated increased use of the MITgcm, an open-source code, for ice
116
shelf-ocean interaction studies merits a detailed assessment of issues surrounding the for-
117
mulation of turbulent exchange velocities in melt rate parameterizations.
118
The purpose of this study is to provide such an assessment. Main goals here are to
119
identify differences in melt rate patterns associated with the use of velocity-dependent
120
versus velocity-independent parameterizations, and to understand the physical processes
121
responsible for these differences and possible cavity circulation changes. Another goal is
122
to identify potential feedback mechanisms between melting, circulation, meltwater plume
123
velocity and hydrographic properties and transfer coefficients.
124
The paper is structured as follow: The MITgcm and its ice shelf cavity component are
125
briefly reviewed in Section 2, along with a description of the model configurations used
126
in this study. Comparisons of simulations using the velocity-independent and velocity-
127
dependent parameterizations, and drag coefficient sensitivity experiments are presented
128
in Section 3. Simulations are conducted using both an idealized ice shelf cavity and
129
a realistic configuration of the cavity underneath PIIS. A discussion of the results is
130
provided in Section 4, and conclusions are summarized in Section 5.
2. The MITgcm model and experimental setup
131
The MITgcm forms the basis for our study. It is the first z-coordinate ocean model
132
capable of simulating sub-ice shelf cavity circulation and under-ice shelf melting [Losch,
133
2008]. At resolutions above 1 km the three-dimensional flow is hydrostatic [Marshall et al.,
134
1997b]. As in virtually all sub-ice shelf cavity circulation simulations published so far,
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the ice shelf base is maintained fixed regardless of the melting and refreezing. Convective
136
adjustment parameterizes vertical motion in case of unstable stratification.
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2.1. Sub-ice shelf melt rate parameterization in the MITgcm
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The initial velocity-independent formulation implemented in the MITgcm assigns con-
138
stant values to γT,S . Details are described in Appendix A. We note that a previous
139
description (but not the actual implementation in the code) in Losch [2008] contains
140
errors that have been corrected in Appendix A.
141
In the velocity-dependent formulation the piston velocities γT,S are functions of the
142
frictional drag at the ice shelf base via a friction velocity, u∗ , which is related to the velocity
143
of ocean currents through a simple quadratic drag law involving a drag coefficient Cd . A
144
brief outline is given in Appendix B. This implementation mostly follows the approach
145
suggested by Holland and Jenkins [1999]. In the light of their sensitivity analyses of melt
146
rates to the details of the parameterization, several approximations have been adopted
147
here and are summarized in Appendix C.
148
The heat and salt balances and associated sign conventions used in the present model
149
are illustrated in Figure 1. In particular, the melt rate m, as defined in terms of freshwater
150
mass flux in eqns. (A1)–(A2), is negative for melting and positive for refreezing. Variables
151
and constants of the melt rate parameterizations are listed in Table 1.
152
153
Two important aspects, the treatment of the ice-ocean mixed-layer and the choice of
drag coefficients are discussed in more detail in the following.
154
Treatment of ”mixed layer” properties: Although we will adopt the term “mixed
155
layer” used by Holland and Jenkins [1999], we acknowledge that in our z-coordinate
156
model the definition of a mixed layer is ambiguous. We often refer to the first ocean grid
157
cell underneath the ice shelf as the “mixed layer”, because hydrography and momentum
158
are homogenized in this layer (see below). With the no-slip condition at the ice shelf
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base, ocean currents are weaker in the grid cells directly in contact with the ice interface
160
than in the cells further away from the shelf base. Where melt rates are large enough
161
along the path of outflow plumes, the grid cells adjacent to the interface are also filled
162
with buoyant, cold and fresh meltwater. Hence increasing the depth of the model mixed
163
layer, which can be achieved by increasing the number of vertical grid cells over which
164
hydrographic properties and ocean currents are averaged to obtain TM , SM , and UM , is
165
expected to locally increase both the thermal and dynamical forcing and hence the melt
166
rates. Sensitivity experiments in this regard will be presented in Section 3.2.
167
Choice of drag coefficient: The choice of drag coefficient Cd also deserves special
168
attention. Although roughness characteristics underneath ice shelves are likely spatially
169
variable [Nicholls et al., 2006], a constant Cd is usually employed. MacAyeal [1984a, b] first
170
used values suggested by Ramming and Kowalik [1980] for open water (Cd = 2.5 · 10−3 )
171
and ice shelf covered water (Cd = 5.0 · 10−3 ) in a barotropic model of the circulation
172
beneath Ross Ice Shelf, hence attributing the same drag to the seabed and ice shelf base.
173
Holland and Jenkins [1999] and Holland and Feltham [2006] later adopted a lower value
174
of Cd of 1.5 · 10−3 at the ice shelf base to account for smoothing effects by melting and ice
175
pumping. More recently, Jenkins et al. [2010a] tuned Cd in their model, and found the best
176
agreement between melt rates simulated using the velocity-dependent parameterization
177
and measurements of ablation rates underneath Ronne ice shelf for Cd = 6.2 · 10−3 . A
178
conclusion is the recognition that Cd is a highly uncertain parameter. While it might
179
require adjustments, a simple tuning of the drag coefficient might compensate for other
180
deficiencies of the current models [Jenkins et al., 2010a]. This issue will be taken up in
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more detail in Section 3.2.
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2.2. Model configurations
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All model configurations used here have a horizontal resolution of 1/32◦ corresponding
183
to grid cells of roughly 1 × 1 km2 size, and a uniform vertical discretization of 50 vertical
184
levels of 20 meters thickness. Partial cells [Adcroft et al., 1997] are used to accurately
185
represent both sea floor topography and ice shelf geometry. A volume-weighted vertical
186
interpolation between neighboring boundary layer grid points reduces numerical noise that
187
is associated with the partial-cell treatment. Biharmonic viscosity is used to dampen the
188
noise in the velocity fields associated with excitation of grid-scale waves. Very weak and
189
stationary noise patterns remain in the model results, but do not affect the numerical
190
stability of the solution. Details on the sources of noise are discussed by Losch [2008].
191
2.2.1. Realistic simulation configuration
192
The model domain encompassing PIIS is delimited by the 102◦ 200 W and 99◦ 220 W
193
meridians and the 74◦ 300 S and 75◦ 270 S parallels. The portion of the cavity south of about
194
74◦ 480 S is referred to in the following as “PIIS proper” and is more extensively analyzed
195
than the more stagnant area to the north [Payne et al., 2007]. The ice shelf geometry
196
and the sea floor bathymetry are based on the Timmermann et al. [2010] data set, which
197
includes the information about draft and cavity bathymetry from in-situ Autosub data
198
[Jenkins et al., 2010b]. The sea floor reaches a maximum depth of about 1000 m and the
199
ice shelf draft varies between 200 m at the ice shelf front and about 900 m at the grounding
200
line. Another important feature of this data set is the presence of a sill of about 300 m
201
rising above its surroundings, oriented in the southwest-northeast direction approximately
202
half-way between the ice shelf front and the deepest reaches of the grounding line in the
203
southeastern corner of PIIS proper (Figure 2a). The domain has one open boundary to
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the west; all other boundaries are closed. Time-mean vertical profiles of zonal velocity,
205
potential temperature and salinity are prescribed at the western open boundary (solid
206
curves in Figures 3a and 3b respectively). These are the same profiles used by Heimbach
207
and Losch [2012]. They were estimated from in situ data provided by five hydrographic
208
stations located along the ice shelf front and are uniform in the meridional direction.
209
Relatively fresh and cold water leaves the cavity at the surface and warm, salty water
210
enters the cavity at depth.
211
2.2.2. Idealized simulation configuration
212
The idealized configuration serves to examine in more detail the impact of velocity-
213
dependence in the turbulent ice-ocean transfer on the melt rates and ocean circulation
214
underneath the ice shelf. The rectangular domain is delimited by the 105◦ 300 W and
215
99◦ 220 W meridians and by the 74◦ 300 S and 75◦ 270 S parallels. Its eastern half is covered
216
by a meridionally-uniform ice shelf and the western half is an open ocean that exchanges
217
neither heat nor mass with the atmosphere. The westernmost 20 grid cells act as a sponge
218
layer with a relaxation time of 10 days. The cavity geometry is representative of a typical
219
ice shelf, and scaled to be consistent with the specific case of PIIS. The ice shelf base
220
depth increases monotonically from 200 m at the ice shelf front to 900 m depth at the
221
grounding line. The sea floor is flat and at a depth of 1000 m (see Figure 2b).
222
As for the realistic configuration, time-mean, meridionally uniform profiles of zonal
223
velocity, ocean temperature and salinity are prescribed at the western open boundary
224
(dashed curves in Figures 3a and 3b). These profiles were chosen to be consistent in
225
magnitude and shape with the mean profiles used in the realistic experiments, hence
226
to represent the conditions at the mouth of a typical “warm” ice shelf in contact with
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CDW. The sinusoidal profile of zonal velocity ensures a zero net volume flux at the open
228
boundary. The circulation and melt rates are not sensitive to the specific amplitude of this
229
prescribed zonal current profile, as long as it does not significantly exceed the magnitude
230
of the barotropic circulation in the cavity.
231
All simulations are started from rest. The initial hydrographic profiles are meridionally
232
and zonally uniform, and correspond to the western open boundary profiles. A spinup
233
of three years is performed to reach steady-state hydrographic conditions and melt rates.
234
Monthly averaged fields for the last month of the spinups are analyzed. Unless otherwise
235
stated, a default drag coefficient of Cd0 = 1.5 · 10−3 is employed, as in Holland and Jenkins
236
[1999] and Holland and Feltham [2006]. As mentioned in section 2.1, this value lies in the
237
low range of values employed in previous modeling studies. In all simulations, the drag
238
coefficient in the melt rate parameterization is the same as for the frictional drag at the
239
ice-ocean interface in the momentum equations. Table 2 summarizes the characteristics
240
of each set of experiments.
3. Results
241
The experiments conducted fall into two main categories: velocity-dependent versus in-
242
dependent parameterizations (Section 3.1), and sensitivity to the choice of drag coefficient
243
(Section 3.2). For a clear understanding of the results, all simulations were conducted for
244
both the idealized and realistic configurations.
3.1. Velocity-independent versus dependent parameterizations
245
3.1.1. Idealized experiments
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Various authors have investigated the ocean circulation and melt rate distribution un-
247
derneath idealized ice shelves. Their cavity geometries were typically north-south oriented
248
with base depths decreasing monotonically southward from a few hundred meter thick ice
249
shelf front to a 1 to 2 km deep grounding line. Among these are Hellmer and Olbers
250
[1989]; Determann and Gerdes [1994]; Grosfeld et al. [1997]; Holland and Jenkins [2001];
251
Timmermann et al. [2002b]; Holland et al. [2008]; Losch [2008]; Little et al. [2008]. Recur-
252
ring results of these idealized studies were: (1) the set up of a density-driven overturning
253
circulation due to the depression of the freezing point temperature of seawater with pres-
254
sure and resulting temperature differences between the ice interface and ambient ocean at
255
depth; (2) predominantly geostrophic mixed layer currents constrained by the distribution
256
of background potential vorticity, i.e., by the water column thickness gradient; (3) max-
257
imum melt rates occurring along the south eastern boundary of the cavity, where warm
258
waters first reach the ice shelf base at the grounding line; and (4) maximum refreezing
259
rates concentrated at the western boundary, along the path of the buoyant meltwater
260
plume that rises along the ice shelf base. Rotation and cavity geometry, in turn, were
261
identified to exert strong constraints on the spatial distribution of melting and refreezing,
262
in agreement with potential vorticity considerations.
263
Our simulation of sub-ice shelf cavity melt rates and circulation (Figure 4a) using the
264
velocity-independent parameterization is consistent with this picture (but note the dif-
265
ference in cavity orientation, which in the present study is west–east to align with the
266
realistic Pine Island cavity geometry). Maximum melt rates are found near the grounding
267
line over the northeastern corner of the cavity where the warmest waters reach the ice shelf
268
base (see Figures 5a and 5b for the thermal forcing, TM − TB ). The horizontal stream-
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function for the vertically-integrated volume transport (contours in Figure 4a) indicates
270
a cyclonic gyre covering the whole domain. In the eastward branch of the gyre, warm
271
water from the open ocean entering the cavity is diverted along the northern boundary,
272
consistent with a buoyancy-induced cyclonic circulation set up by melting at the ice shelf
273
base. From the northeastern corner of the cavity, where maximum melt rates occur, the
274
water mass formed through mixing of meltwater and ambient water flows southward along
275
the ice shelf base. Melt rates decrease southward as the plume becomes more diluted with
276
meltwater and exhausts its heat potential. The barotropic streamfunction indicates an
277
intensification of the westward flowing branch along the southern boundary, in agree-
278
ment with the intensification of an ageostrophic flow against the topographically-induced
279
background potential vorticity gradient.
280
The fact that the circulation and melt rate patterns are consistent with results of Little
281
et al. [2008] and ISOMIP experiments, which in comparison are representative of large,
282
”cold” ice shelves, suggests that the buoyancy and dynamical constrains discussed above
283
are applicable to a broad range of ice-ocean systems.
284
The main differences between our velocity-independent simulations and that of previous
285
studies is that ice does not accumulate at the ice shelf base and that densified water does
286
not downwell at the ice shelf front. Instead, the plume escapes the cavity and interacts
287
with the open ocean. As pointed out by Holland et al. [2008], such conditions are con-
288
sistent with small, steep ice shelves in contact with CDW with temperatures exceeding
289
1◦ C. Observational support for this behavior can be found in Jacobs et al. [1996]. Con-
290
sistent with the absence of freezing-induced downwelling at the ice shelf edge and with
291
the meltwater plume ”overshooting” out of the cavity, the cyclonic gyre characterizing
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the vertically integrated volume transport is not restricted to the cavity but extends into
293
the open ocean. This suggests greater barotropic exchanges between the open ocean and
294
the ice shelf cavity relative to the typical ”cold” ice shelf circulation [Grosfeld et al., 1997;
295
Losch, 2008].
296
The idealized model run with the velocity-dependent parameterization and the default
297
drag coefficient Cd0 produces a depth-integrated volume transport (contours in Figure 4b)
298
and a meridionally averaged overturning circulation (not shown) that are very similar
299
to those of the run with the velocity-independent parameterization. However, the spatial
300
distribution of melt rates differs substantially between the two simulations. In the velocity-
301
dependent case, the maximum melt rates are found along the exit path of the meltwater
302
plume, that is, over the intensified westward branch of the cyclonic circulation along the
303
southern edge of the cavity, and over an area extending westward from the southern part
304
of the grounding line. There is no melt rate maximum associated with the northeastern
305
inflowing branch of the cavity circulation.
306
The correspondence between the overturning and horizontal circulations simulated in
307
the two experiments implies that hydrographic properties inside the cavity are similarly
308
distributed in both cases. The discrepancies in melting patterns therefore suggest that the
309
melt rate is not as sensitive to ocean temperature in the velocity-dependent than in the
310
velocity-independent simulations. Instead, the frictional effect of the mixed layer currents
311
might dominate over the thermal forcing in setting the heat flux through the ice-ocean
312
boundary layer in the velocity-dependent case.
313
To test this hypothesis, we compare the velocity-independent and dependent melt rate
314
patterns to the patterns of the two main drivers of the diffusive heat flux (QM
T ). These
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are (see equations A4 and B2 in the appendix) the difference in temperature across the
316
boundary layer, TM − TB (Figures 5a,b), and, through the formulation of the friction ve-
317
locity, the magnitude of mixed layer current velocity, UM (Figures 5c,d). As expected, the
318
spatial patterns of both the thermal forcing and mixed layer velocity are very similar in
319
the velocity-independent and dependent simulations. In both cases, the highest tempera-
320
ture gradients across the ice-ocean boundary layer are found over the northeastern corner
321
of the cavity, at depth, where the warmest mixed layer waters are found. The mixed layer
322
water cools as it flows southward. The fastest mixed layer currents are concentrated along
323
the southern cavity wall over the region of plume outflow, and increase southward over
324
the interior part of the cavity.
325
The spatial correlation between the melt rates and either forcing is however very differ-
326
ent between the two simulations: in the velocity-independent case, melt rate maxima are
327
collocated with maxima in thermal forcing and are insensitive to the details of the mixed
328
layer velocity pattern. In the velocity-dependent case, melt rates are not collocated with
329
thermal forcing, but instead are well aligned with the distribution of UM , such that the
330
highest rates are found over the regions of fastest mixed layer currents, i.e., over the path
331
of the outflow plume.
332
This shift of maximum melt rates from areas of high ocean heat to regions of strong
333
currents is consistent with results by Mueller et al. [2012]. They found that between
334
two experiments in which they used the velocity-dependent parameterization of Holland
335
and Jenkins [1999] (modified by adopting the scalar transfer coefficients of McPhee et al.
336
[2008]) and the velocity-independent parameterization of Hellmer and Olbers [1989], max-
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337
imum melt rates shifted from the vicinity of the deep grounding line, where TB is low, i.e,
338
the thermal forcing is high, to regions of strongest time-mean barotropic currents.
339
A similar behavior was simulated by Gladish et al. [2012] who applied the model of
340
Holland and Feltham [2006] to the floating tongue of Petermann Gletscher (Northwest
341
Greenland). They found a somewhat larger spatial correlation between melt rate and
342
mixed layer current forcing than between melt rate and thermal forcing. However, in their
343
model vertical profiles of T and S were prescribed and homogeneous in the horizontal and
344
their thermal forcing was high and approximately uniform in the regions of high melt
345
rates, which is not necessarily the case in the present experiments.
346
Moving from melt rate patterns to magnitudes reveals that melting is overall lower in
347
the velocity-dependent simulation with Cd 0 than in the velocity-independent one. The
348
lower melting explains the difference in the strength of the mixed layer currents and
349
thermal forcing between the two experiments. In the velocity-independent simulation,
350
higher melt rates lead to stronger buoyancy-flux induced density gradients and support
351
faster mixed layer currents over the interior part of the cavity. The production of larger
352
volumes of buoyant melt water overall cools the mixed layer and hence reduces thermal
353
forcing relative to the low melt rates in the velocity-dependent simulation. Section 3.2
354
discusses these effects in detail in the context of the sensitivity of velocity-dependent melt
355
rates to the drag coefficient.
356
3.1.2. Experiments with realistic geometry
357
Melt rates simulated with the realistic PIIS configuration using the velocity-independent
358
and velocity-dependent parameterizations (with the default Cd 0 ) are shown in Figures
359
6a and 6b, respectively. Corresponding patterns of temperature difference across the
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360
boundary layer, TM − TB , and of mixed layer velocities, UM , are illustrated in Figures
361
7a–d.
362
As in the idealized experiments, spatial patterns of TM − TB and of UM are very similar
363
between the two parameterizations, but important differences are seen in the melt rate
364
distributions. Velocity-independent melt rates are highly spatially correlated with the
365
thermal forcing. Figure 6a shows melting to be largest over the southeastern portion
366
of the cavity where the ice shelf base is deepest, i.e. where TB is lowest. Vertical cross
367
sections of temperature and salinity (not shown) confirm that the warmest and most salty
368
waters reach the grounding line in this area.
369
Figure 6c shows the vertically-integrated volume transport along with the water column
370
thickness (black, dashed contours). As in the idealized experiments, the structure of the
371
circulation suggests that the barotropic transport inside the cavity is strongly controlled
372
by the distribution of water column thickness (nearly equivalent to the distribution of
373
background potential vorticity, f /h). Three prominent gyres are labeled in the Figure:
374
(1) a strong cyclonic gyre over the exit of the cavity; (2) a second prominent cyclonic
375
gyre deep inside the cavity, inward of the sill; and (3) a weaker anti-cyclonic gyre also
376
inward of the sill and to the north of cyclonic gyre 2. Transport over the sill is weak, with
377
cross-sill exchanges confined to its northern and southern sides.
378
Figures 7b and 7d indicate that the melt rate pattern simulated with the velocity-
379
dependent parameterization is not correlated with the thermal forcing. Instead, it mimics
380
the distribution of the mixed layer currents. In agreement with the idealized cavity con-
381
figuration, melt rate maxima are collocated with rapid plume outflows. The strongest
382
outflow occurs at the southern flank of cyclonic gyre (1) around 75◦ 060 S, 101◦ 300 W. This
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383
position marks a convergence zone with waters originating from the southern flank of
384
cyclonic gyre (2). The water leaves the cavity at the southern edge of the ice front.
385
The outflow at the northern flank of anti-cyclonic recirculation gyre (3) around 74◦ 550 S,
386
100◦ 300 W, coincides with the convergence of currents against the eastern cavity wall of
387
PIIS proper. A third weaker outflow collects melt water from the more stagnant northern
388
portion of the cavity. Only part of these two outflows leaves the cavity when reaching
389
the ice shelf front. The other part is steered southward along the ice front and the first
390
stronger outflow near the southern boundary. Two patches of relatively higher melt rates
391
are also seen downstream of the deepest portions of the grounding line, corresponding to
392
locally intensified mixed layer currents.
393
As in the idealized experiment, the realistic velocity-dependent simulation with Cd0 pro-
394
duces smaller melt rates than the corresponding velocity-independent simulation. The
395
maximum velocity-dependent integrated volume transport is reduced by about 40% rel-
396
ative to the velocity-independent transport. The overall structure of the transport is
397
similar in both cases.
398
3.1.3. Observational evidence
399
Observational melt rate estimates under PIIS are limited. In the following, we compare
400
our simulated melt rate pattern with recent studies that produced estimates of melt rate
401
distribution under PIIS from available observations and to the plume model simulations of
402
Payne et al. [2007], which to our knowledge produced the only published high-resolution
403
velocity-dependent melt rate distribution for the entire ice shelf.
404
A notable similarity between our realistic velocity-dependent simulations and that of
405
Payne et al. [2007] is that local melt rate maxima are collocated with the paths of two
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406
principal outflow plumes underneath PIIS and with a third weaker outflow collecting
407
meltwater under the northern portion of the ice shelf (see Figures 4 and 6 from [Payne
408
et al., 2007]). Our results also agree with their melt rate estimates from ice flux divergence
409
calculations based on ice velocity and shelf thickness data. These calculations indicated
410
local melt rate maxima near the southernmost part of the ice shelf front and along the
411
northern cavity wall of PIIS proper (see their Figure 10).
412
Payne et al. [2007] pointed out that Advanced Spaceborne Thermal Emission and Re-
413
flection Radiometer (ASTER) images indicate a retreat of sea ice in front of the ice shelf
414
over three isolated areas collocated with their plume outflows, suggesting the presence of
415
warm upwelling plume water there. Bindschadler et al. [2011] analyzed 116 Landsat im-
416
ages spanning 35 years and a few images from Advanced Very High Resolution Radiometer
417
(AVHRR) and Moderate Resolution Imaging Spectroradiometer (MODIS) and observed
418
three recurrent polynyas at the same fixed locations. The largest of these polynyas was
419
positioned at the southern edge of the ice shelf front, where our realistic model and that
420
of Payne et al. [2007] simulate the strongest outflow and where Jacobs et al. [2011] also
421
observed concentrated meltwater outflows. Analyses of temperature, salinity and current
422
profiles from a research cruise in 2009 and of Landsat thermal band and thermal infrared
423
(TIR) images from two austral summers during which the ocean was sea ice-free at the
424
ice shelf front support the presence of warmer waters exiting the ice shelf cavity in the
425
same locations of the three polynyas present during other summers [Bindschadler et al.,
426
2011; Mankoff et al., 2012].
427
A notable difference to the results of Payne et al. [2007] is the structure of the mixed-
428
layer flow. In their simulations it is concentrated mostly in the primary outflow along the
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429
southern boundary and to a lesser degree along the outflow crossing the middle part of
430
PIIS proper. In our experiments, this latter outflow is stronger and concentrated along
431
the northern boundary of PIIS. This discrepancy can be explained by the different nature
432
of the two models and their interaction with cavity geometry. The mixed layer flow in
433
ice shelf cavities is expected to be predominantly geostrophic and constrained by the
434
background potential vorticity, i.e. by the water column thickness gradient which is set
435
by the bed geometry and by the ice shelf base topography [Little et al., 2008]. As the
436
effect of bathymetry is not accounted for in vertically integrated (plume) models such as
437
that of Payne et al. [2007], the mixed layer flow (i.e., the buoyant plume) is steered only
438
by the ice shelf base topography. Important features of the sub-ice shelf topography in the
439
Payne et al. [2007] simulations are two inverted channels collocated with their southerly
440
outflow and with the one roughly in the center of the ice shelf (see their Figure 4).
441
Recent observations [Dutrieux et al., 2013] support the presence of two 3 km-wide chan-
442
nels merging at the southernmost edge of the ice front of PIIS. Landsat images indicate a
443
significant longitudinal surface trough running in the middle of the ice shelf, which, in hy-
444
drostatic equilibrium, suggests the presence of a deep inverted trough in the underside of
445
the ice shelf susceptible of channeling buoyant outflow waters [Bindschadler et al., 2011].
446
These channels are not represented in our shelf base topography (contours in Figures
447
6a,b). Instead, the cavity geometry feature that appears to exert a strong constraint on
448
the circulation and to give rise to the gyres described above, is the pronounced ridge in our
449
bathymetry data [see also Schodlok et al., 2012]. Our two strongest outflows correspond
450
to areas of convergence along the cavity walls of mostly geostrophic currents.
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451
We also notice a difference in the location of the (near) grounding line maximum melt
452
rates between our simulations and both the simulations and flux divergence estimates of
453
Payne et al. [2007], which we attribute to the use of very different PIIS cavity geometries
454
between the two studies. Comparing the present shelf base topography (contours in
455
Figures 6a,b) to the one derived by Payne et al. [2007] (see their Figure 2), we note that
456
the deepest portion of the grounding line is not at the same location in the two models.
457
In our setup, the shelf base is deepest (900 meters) in the southeastern corner of PIIS
458
proper (around 75◦ 18’ S, 99◦ 30’ W). In Payne et al. [2007] it is also about 900 meters
459
deep at that location, but is even deeper (> 1000 meters) in a hollow portion of the cavity
460
to the northeast (around 75◦ 06’ S, 99◦ 45’ W) where the shelf base is only 600 to 400
461
meters deep in our model. The presence of an inverted channel downstream of this deep
462
portion of the grounding line in the shelf topography of Payne et al. [2007] results in
463
a steep gradient of shelf base depth that is not seen in the present ice cavity geometry
464
derived from recent Autosub data [Jenkins et al., 2010b]. This has implications for ice
465
flux divergence calculations.
466
Moreover, we note that melt rate magnitudes in our velocity-dependent simulation
467
with Cd 0 are overall lower than previously published estimates [Payne et al., 2007; Jacobs
468
et al., 2011; Dutrieux et al., 2013]. In order to match previous and their own observational
469
estimates of the cavity-average melt rate under PIIS, Payne et al. [2007] tuned four poorly
470
constrained parameters of their plume model. For example, they varied the drag coefficient
471
between 1 and 6 · 10−3 . In the following, we investigate how our simulated velocity-
472
dependent melt rates are affected when varying this parameter.
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3.2. Melt rate dependence on the drag coefficient
473
Energy conservation at the ice-ocean interface, eqn. (A1), requires that the latent heat
474
flux associated with melting and refreezing be equal to the diffusive heat flux through the
475
boundary layer, QTM , minus the fraction of heat lost to the ice shelf by conduction, QTI .
476
Usually, the conductive heat flux term is one order of magnitude smaller than the diffusive
477
heat flux term [e.g., Holland and Feltham, 2006; Holland and Jenkins, 1999; Determann
478
and Gerdes, 1994], so that we can express the melt rate as
m=−
479
cpM
u∗ ΓT (TM − TB ).
Lf
(1)
480
Because of the dominance of molecular over turbulent diffusion in the viscous sublayer
481
closest to the ice interface, the heat and salt exchange coefficients ΓT,S are only weakly
482
dependent on the friction velocity. Eqn. (1) then predicts to first order a linear dependence
483
of the melt rate on u∗ or
√
Cd .
484
To investigate the dependence of the melt rates on Cd and assess the relative importance
485
of various feedbacks associated with variations of the drag coefficient, we conducted both
486
idealized and realistic PIIS simulations in which Cd was varied between 1/16 and 16 times
487
the default value of Cd0 = 1.5 · 10−3 .
488
3.2.1. Idealized experiments
490
Figure 8a shows the area-averaged melt rate m (black dots) calculated for velocityp
dependent simulations as a function of Cd /Cd 0 . The area-averaged melt rate of the
491
velocity-independent simulation with Cd 0 is also plotted as a reference (dashed black line).
492
As predicted by theory, m in the velocity-dependent simulations increases with
493
order to understand this behavior, we examine the effect of the two direct forcings on the
489
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November 24, 2013, 9:42pm
√
Cd . In
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494
melt rates as Cd is changed by comparing area-averaged values of the friction velocity and
495
of the difference in temperature across the ice shelf boundary layer.
496
Positive feedback – friction velocity: Similar to melt rates, friction velocity in√
√
497
creases with
498
a power law fit of the form u∗ = a Cd
499
(within a 95% confidence interval, b = [1.163, 1.373]). This is because the area-averaged
500
mixed layer velocity in our simulations (Figure 8c, orange dots) also increases with
501
in a sub-linear manner.
Cd (Figure 8c). Fitting the area-averaged friction velocity against
b/2
Cd with
indicates that the dependence of u∗ is above-linear
√
Cd ,
502
On the one hand, the increase of mixed layer currents with Cd is consistent with the
503
strengthening of buoyancy-flux induced density gradients under the shelf that occurs with
504
the intensification of the melting. In turn, stronger mixed layer currents enhance the
505
diffusive heat flux across the boundary layer, thereby amplifying the increase of melt rates
506
with Cd . This positive feedback between melt rates, buoyancy flux-induced gradients and
507
mixed layer currents is not accounted for in a velocity-independent parameterization. On
508
the other hand, the fact that the increase of UM with
509
the enhanced frictional drag.
√
Cd is sub-linear is consistent with
510
Negative feedback – thermal forcing: Figure 8e shows a decrease of the cavity-
511
averaged thermal forcing (purple dots) with increasing drag coefficient. This points to an
512
overall cooling of the mixed layer. It is consistent with the production of a larger volume
513
of cold buoyant meltwater that spreads at the ice shelf base, stratifying the upper water
514
column and forming an insulating film [Gill , 1973; Little et al., 2009]. This reduction in
515
thermal forcing is a negative feedback on the increase of melting with Cd .
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516
Provided that the cooling is due to a larger production of meltwater, the salinity at
517
the ice shelf base, SB , will also decrease with increasing Cd . Through the dependence of
518
the freezing point, Tfreeze , on salinity (eqn. A3), this should raise Tfreeze and reduce the
519
difference of temperature across the boundary layer, thereby slowing the increase in melt
520
rates with Cd . Because the dependence of the freezing point of seawater on salinity is only
521
weak, this effect is expected to be small.
522
To verify whether this salt feedback actually has a non-negligible effect on the ther-
523
mal forcing, TM − TB , with changing Cd , we calculate the area-averaged thermal driving
524
underneath the ice shelf (red dots),
525
T∗ = TM − TB − a (SM − SB )
526
with (SM − SB ), the salinity difference across the boundary layer and a, the (negative)
527
salinity coefficient given in Appendix A. Thermal driving is the thermal forcing obtained
528
when neglecting the effects of salt diffusivity on the temperature gradient at the ice shelf
529
base [Holland and Jenkins, 1999]. In the present experiments, its area average is higher
530
than the area averaged thermal forcing by about 0.3 to 0.8◦ C, indicating that neglecting
531
the effects of salt diffusivity would significantly overestimate the melt rates. Figure 8d
532
shows that the thermal forcing and driving behave very similarly as a function of Cd in
533
the model. This suggests that salinity feedbacks on the simulated melt rates are not
534
significant, as anticipated.
535
(2)
Melt rate versus Cd fit: Returning to Figure 8a, a power law fit of the form m =
b/2
to the area-averaged melt rate against
√
536
a Cd
537
interval (b = [0.579, 0.922]), suggesting that the negative feedback of the decreased thermal
538
forcing on the melt rates exceeds the positive feedback associated with the increased mixed
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November 24, 2013, 9:42pm
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539
layer velocity. The scattering of the calculated melt rates around the fitted curve in Figure
540
8a reveals, however, systematic deviations from the simple power law fit over different
541
ranges of
542
the model, and that are not accounted for in the above considerations.
√
Cd . It suggests other feedbacks or non-linearities, or both, to be at play in
543
Spatial patterns: Over the range of Cd values investigated, the spatial patterns
544
described in the previous section for the velocity-dependent γT,S simulation (see Figures 6b
545
and 5b,d) remain overall unchanged. We therefore only report the results, while omitting
546
supporting figures. Substantial melting near the grounding line is a persistent feature,
547
with a decrease westward towards the ice front. Maximum melting is collocated with
548
the outflow of the meltwater plume along the southern boundary. As Cd is increased,
549
both melting and mixed layer currents increase in these regions, as expected from the
550
strengthening of buoyancy-induced zonal density gradients. Melt rates therefore remain
551
highly spatially correlated with the mixed layer velocity. Slow refreezing occurs over a
552
limited region bordering the northern edge of the plume for Cd > 4 · Cd 0 .
553
The temperature difference across the boundary layer diminishes over the region of
554
largest melt when Cd is increased. For Cd > 2 · Cd 0 , both the temperature and salinity
555
of the mixed layer locally decrease below the lowest surface temperature and salinity
556
prescribed as initial conditions. This confirms that the cooling of the mixed layer is due
557
to an increased production of melt water rather than a redistribution of hydrographic
558
properties in the cavity. Consistent with this picture, zonal sections of temperature and
559
salinity across the westward outflow indicate a cooling, freshening and thickening of the
560
plume as the drag coefficient is increased (not shown). For the case of Cd = 16 · Cd 0 , this
561
negative feedback of thermal forcing on melting seems to have a noticeable impact on the
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562
melt rate pattern. In this case, melt rates near the grounding line become comparable to
563
that along the path of the outflow plume and are highest towards the northern half of the
564
cavity, where thermal forcing is maximal.
565
Both the depth-integrated volume transport and the meridionally-integrated zonal over-
566
turning circulation strengthen with increasing Cd , but again with a spatial pattern vir-
567
tually unchanged compared to that for Cd 0 (Figure 4d). The increase in the barotropic
568
circulation is consistent with increased melting and enhanced buoyancy-induced density
569
gradients [Little et al., 2008]. The strengthening of the overturning circulation agrees with
570
the production of larger volumes of melt water, the increase of vertical density gradients,
571
and the enhanced buoyancy of the plume [Holland et al., 2008].
572
3.2.2. Realistic experiments
573
Thermodynamic forcings, melt rates and circulation in the experiments with realistic ice
574
shelf and sea floor geometries of PIIS behave in a very similar manner as in the idealized
575
experiments when varying Cd , as revealed by comparing the left and right panels of
576
Figure 8. The same holds true for a number of inferences made, including (1) the positive
577
feedback between enhanced melting, strengthened buoyancy-induced density gradients
578
and mixed layer currents, (2) the increased production of meltwater that insulates the ice
579
interface from the warmer waters below, (3) the negligible impact of salinity through the
580
dependence of the freezing point of seawater, (4) the overall conservation of the spatial
581
patterns of melting, thermal and ocean current forcings represented in Figures 6b and 7b,
582
d, and of the structure of the barotropic circulation shown in Figure 6d.
583
The fact that the spatial distribution of melt rates in the velocity-dependent experiments
584
is robust and does not seem to depend on the specific drag coefficient over a a wide range
D R A F T
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
585
of values is a valuable result, since in practice, the appropriate value for Cd underneath
586
ice shelves remains unknown.
587
In both the idealized and realistic experiments, the thermal forcing is higher in the
588
velocity-dependent than in the velocity-independent simulation over the entire range of
589
Cd values investigated. This is a consequence of the regions of rapid melting and of high
590
thermal forcing being spatially decorrelated in the velocity-dependent case. Even if the
591
production of cold meltwater increases with Cd and mixed layer temperatures drops locally
592
over region of strong mixed layer currents and rapid melting, TM remains comparatively
593
high where thermal forcing is strong.
594
In the realistic experiments, the area-averaged mixed layer velocity is lower in the
595
velocity-dependent than in the velocity-independent simulation for all values of Cd . This
596
is not the case in the idealized simulations, for which a drag coefficient about four times the
597
default value matches the mixed layer velocities. Moreover, a drag coefficient about 8 times
598
the default value is required to match the velocity-dependent and velocity-independent
599
melt rates in the realistic case. In the idealized experiments, Cd ≈ 2 · Cd 0 is required.
600
These differences indicate that no value of drag coefficient reconciles the two melt rate
601
parameterizations in all simulations and suggests that the ice shelf cavity system reaches
602
different thermodynamic steady states between our idealized and realistic experiments
603
that are not readily comparable. This might be indicative of additional feedbacks between
604
melt rates, mixed layer velocities, buoyancy fluxes and topographic features that occur in
605
the more realistic case.
606
A drag coefficient about 4 to 8 times our default value would be needed to match
607
our cavity-averaged melt rate under PIIS to the ice flux divergence based estimate of
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608
Payne et al. [2007] of 20.7 m/yr (Figure 8b). Using Cd = 4 · Cd 0 and Cd = 8 · Cd 0 , the
609
spatial average over PIIS proper varies from 23 m/yr to 31 m/yr. These values compare
610
favorably with the 29.7 m/yr PIIS-proper value of Payne et al. [2007], and with the
611
24 ± 4 m/yr estimate of Rignot [1998]. Figure 9a shows the distribution of melt rates for
612
Cd = 6 · 10−3 = 4 · Cd 0 . Maximum melt rates of 60 to almost 100 m/yr are found over
613
the path of the outflow plume that exits at the southern end of the ice shelf front and
614
rates of up to 70 m/yr are collocated with the outflow along the northern boundary of
615
PIIS proper. Melting near the southeastern portion of the grounding line exceeds 50 m/yr
616
and decreases rapidly downstream to 10-20 m/yr, outside the regions associated with the
617
outflows, in agreement with the result of ice flux divergence calculations of Rignot [1998]
618
and the more recent estimates along four airborne survey lines over PIIS by Bindschadler
619
et al. [2011]. In the case of Cd = 8 · Cd 0 , the pattern is virtually the same, and these
620
values become 80 to 113 m/yr and 90 m/yr for the two main outflows, 80 m/yr near the
621
grounding line, and 20 − 30 m/yr downstream of the grounding line melt region.
622
Figure 9b shows the difference between melt rates simulated using Cd 0 and Cd = 4 · Cd 0 .
623
Melting increases more rapidly with Cd over the regions that are already local melt rate
624
maxima for Cd = Cd 0 . The increase is comparable along the outflows and over the
625
regions downstream of the grounding line. Therefore, as melt rates are lower there than
626
along plume paths in the default Cd simulation, this indicates that melting increases
627
more rapidly downstream of the grounding line with enhanced frictional drag. As in the
628
idealized experiments, for Cd = 16 · Cd 0 melt rates near the grounding line slightly exceed
629
those along the outflow plumes. This is again indicative of the decorrelation of melt
630
rates and thermal forcing in the velocity-dependent experiments. It can also be related to
D R A F T
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631
entrainment: as increased frictional drag increases the melt rates, enhanced mixed layer
632
currents underneath the ice shelf result in more entrainment of water from below. As the
633
temperature difference between the ice shelf base and ocean below and the shear related
634
to the steepness of the ice shelf base are both highest near the grounding line, entrainment
635
is expected to have the highest impact on melt rates there [Little et al., 2009].
4. Discussion
636
Despite the higher level of complexity of the velocity-dependent melt rate parame-
637
terization compared to the velocity-independent version, the representation of physical
638
processes involved in ice-ocean interactions, such as frictional drag due to rough surfaces
639
or entrainment still deserves further attention. A number of aspects are discussed below.
4.1. Effects of roughness and frictional drag
640
The drag coefficient Cd in our model serves two purposes: (1) in a general sense, it
641
captures a number of unresolved scales at the ice-ocean interface (and ocean bottom)
642
that give rise to roughness and therefore exert a frictional drag on the flow, an effect
643
represented via a stress term in the momentum equation; (2) in the thermodynamical
644
melt rate parameterization it establishes a relationship between frictional forcing and
645
melt rates.
646
Thermodynamic forcing: Varying Cd may be justified by the fact that its value
647
is unknown and may depend on the material and morphological roughness properties of
648
the interface considered. Increasing Cd by 4 times the default value to Cd = 6 · 10−3
649
in our model to approach published melt rate estimates is in line with Jenkins et al.
650
[2010a], who increased Cd to 6.2 · 10−3 to match their observational estimate of ablation
D R A F T
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X - 31
651
rates underneath Ronne Ice Shelf. Although this number is at the high end of previously
652
published values, melting near the grounding line in our velocity-dependent experiments
653
remains low compared to recent estimates of melt rates under PIIS, locally in excess of
654
100 m/yr [Payne et al., 2007; Bindschadler et al., 2011; Dutrieux et al., 2013]. Obtaining
655
such high melt rates requires increasing Cd to 16 times its default value. Depending on
656
the model, other parameters may be available for tuning observed melt rates. Payne et al.
657
[2007] tuned their simulated melt rates by varying shelf core temperature, horizontal eddy
658
viscosity, entrainment coefficient and drag parameter. Sensitivities of cavity-averaged melt
659
rate were found to be largest with respect to drag and entrainment parameters (see their
660
Figure 13).
661
Momentum forcing and vertical discretization: While the functional dependence
662
of the melt rate on Cd simulated here (melt rates vary sub-linearly with drag coefficient)
663
is in overall agreement with the plume model results of Holland and Feltham [2006] and
664
Payne et al. [2007], an important difference is that we do not encounter a critical Cd value
665
beyond which melt rates would decrease (which may be expected if excessive frictional
666
drag impedes the plume flow). We attribute this to the different treatment of the frictional
667
drag at the ice shelf base. In layer and plume models, the mixed layer (plume) depth and
668
properties evolve in time and space. With increasing melt rates, larger volumes of buoyant
669
meltwater are produced and the plume thickens and accelerates. However, with increasing
670
drag, the impeding effect of friction on the plume dominates and the melting effectively
671
decreases for very large values of drag.
672
In our z-level model, the drag does not act explicitly on the entire plume layer but only
673
on the first grid cell below the ice-ocean interface. Further vertical mixing of momentum
D R A F T
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
674
(i.e. the effect of the drag) and heat supply from below are parameterized by vertical
675
diffusion (in our case even with constant coefficients) that may not be effective enough to
676
form a thick plume. The acceleration by thermal forcing is mostly confined to the first
677
grid cell layer and the counteracting drag is not strong enough for the flow to slow down.
678
The sub-linear behavior of the mixed layer velocity in Figure 8c and Figure 8d shows that
679
the negative feedback of increasing drag starts to act for high values of Cd . However, in
680
the absence of a more sophisticated mixed-layer treatment, the negative feedback of drag
681
onto the melt rates is not expected to be as important in level as in layer models.
682
Drag and geophysical roughness: Recent acoustic (Autosub) survey, laser altime-
683
try, and radar data helped identify a network of basal channels with width on the order
684
of 0.5 m to 3 km and height of up to 200 m on the underside of PIIS [Bindschadler
685
et al., 2011; Vaughan et al., 2012; Dutrieux et al., 2013; Stanton et al., 2013]. These are
686
thought to be formed near the grounding line, enlarged by basal melting downstream of
687
the grounding line, and subsequently smoothed by melting towards the ice shelf front.
688
Dutrieux et al. [2013] showed that the medium-scale (10 km) melt rates under PIIS are
689
strongly modulated by melt variability at the scale of these channels. They reported high
690
melting in channels near the grounding line, on the order of 40 m/yr (i.e., 80% more
691
melting in channels than in keels) and lower channel melting of 15 m/yr in the region
692
downstream. Stanton et al. [2013] also reported melting of approximately 20 m/yr at the
693
apex of a basal channel under PIIS and near-zero melting on its flanks.
694
A number of studies related the formation and deepening of these features to an accel-
695
eration of mixed layer currents within the narrow channels leading to enhanced melting
696
[Vaughan et al., 2012; Gladish et al., 2012; Rignot and Steffen, 2008; Sergienko, 2013].
D R A F T
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X - 33
697
These findings suggest that ice-ocean interactions are strongly modulated by kilometer-
698
scale processes and imply that higher resolution models are required to accurately estimate
699
both the spatial average and distribution of melting in ice shelf cavities.
700
One perhaps crude yet simple way of accounting for the effect of basal channels in large-
701
scale models might be through the frictional drag. The studies mentioned above show that
702
channel features, and hence the large-scale roughness characteristics of the base of PIIS
703
are very heterogeneous. Associated with these channels are narrower surface and basal
704
crevasses [Vaughan et al., 2012], which further enhance the irregularity of the ice-ocean
705
interface. While current velocity-dependent models employ a constant ice shelf basal drag
706
coefficient, the use of a spatially varying value might be more appropriate to account for
707
the distribution of these basal channels and crevasses.
4.2. Role of entrainment
708
Entrainment of warm waters by the buoyant plume as it rises along the ice shelf base can
709
impact the melt rates in at least two ways. First, as the ambient ocean is warmer than the
710
meltwater plume, entrainment raises the temperature of the plume and provides a heat
711
source for melting. Payne et al. [2007] applied the reduced gravity plume model of Holland
712
and Feltham [2006] to a realistic PIIS cavity and showed that buoyant plumes are indeed
713
primarily fed by entrainment of warm waters near the grounding line. Second, Holland
714
and Feltham [2006] identified that the inclusion of entrainment in their plume model
715
decreases the relative importance of drag at the ice shelf base and therefore accelerates
716
the plume. As the highest melt rates in our velocity-dependent model are collocated with
717
the path of meltwater plumes, an increase in the speed of plume outflows would directly
718
increase the maximum ablation rates.
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
719
The representation of entrainment in numerical models is very sensitive to the details
720
of the vertical discretization. At scales typically considered (1 km and larger), the issue
721
of too low entrainment is confined to layer (isopycnal or sigma) models (e.g., Adcroft and
722
Hallberg [2006]), which therefore requires specific attention through adequate parameter-
723
izations. In contrast, Legg et al. [2006] and others have shown that level (z-coordinate)
724
models typically suffer from excessive entrainment due to numerical diffusion, unless non-
725
hydrostatic scales down to 1 to 10 m are resolved (e.g., Sciascia et al. [2013]). While tuning
726
their plume model, Payne et al. [2007] found that entrainment had by far the largest effect
727
on their predicted melt rates. In their model, melt rates increased monotonically with
728
the entrainment coefficient such that any cavity-average target in the range of previously
729
estimated melt rates for PIIS could be matched. In the present model, such tuning is not
730
possible and entrainment by the meltwater plume cannot be easily quantified.
731
However, a shortcoming of plume models compared to three-dimensional baroclinic
732
models is the need to prescribe ocean properties, hence not permitting an evolution of
733
oceanic forcing, and not accounting for the effects of depth-independent flows within the
734
cavity [Holland and Feltham, 2006]. Payne et al. [2007] justified their use of a plume
735
model to simulate melt rates under PIIS by assuming that the control of barotropic flows
736
on the redistribution of melting in “warm” ice shelf cavities might not be as important as
737
in ”cold” and more weakly stratified cavities. The present experiments suggest, however,
738
that the convergence of depth-independent currents along the steep cavity wall sets the
739
location of the outflow plumes under PIIS.
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4.3. Sensitivity to mixed-layer thickness
740
To test whether the spatial distribution and magnitude of melt rates obtained with
741
our z-coordinate model depend strongly on the fixed thickness of the mixed layer, we
742
conducted additional experiments in which we increased the vertical resolution of the
743
model from 20 meters to 10 meters, and varied the thickness of the averaging layer for
744
TM , SM and (UM , VM , WM ) between 10, 20 and 50 meters.
745
The melt rates simulated using Cd 0 are shown in Figure 10a–c for the velocity-
746
independent and Figure 10d–f for the velocity-dependent experiments.
Overall, the
747
ablation pattern is maintained when varying the mixed layer depth. In the velocity-
748
independent experiments, the maximum melt rates are located downstream of the ground-
749
ing zone, while in the velocity-dependent simulations, melt rates are still highest along
750
the path of the plume outflows, where currents underneath the shelf are strong.
751
As expected, the maximum melt rates increase with increasing thickness of the mixed
752
layer. The velocity-dependent mean melt rate is nearly unchanged, while the velocity-
753
independent mean melt rate increases slightly between the 10 and 20 meters cases. Larger
754
changes in magnitudes only occur in the 50 meters velocity-independent case. This last
755
case, however, is not used in the present study and is thought to overestimate the mixed
756
layer thickness [Jenkins et al., 2010a; Stanton et al., 2013]. The decrease in mean and
757
maximum melt rate with increasing vertical resolution was also observed by Losch [2008]
758
(using a velocity-independent parameterization only). It is attributed to the fact that
759
increasing the resolution decreases the total heat content of the grid cells adjacent to the
760
ice shelf. Melting fills these cells with buoyant meltwater near the freezing temperature.
D R A F T
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
761
Therefore, higher vertical resolution at the ice shelf base (i.e., thinner cells) reduces the
762
heat supply to the ice shelf from the ocean layer directly in contact with the ice shelf base.
763
In conclusion, our main result that the spatial distribution of the melting is very differ-
764
ent between the velocity-dependent and velocity-independent melt rate parameterizations
765
is not affected by the specified thickness of the mixed layer. Furthermore, mean melt
766
rate magnitudes remain nearly unchanged (velocity-dependent) or change only slightly
767
(velocity-independent) when changing from 10 m to 20 m mixed layer thickness. There-
768
fore, our results appear to be affected only marginally by the inability of the model to
769
account for the spatial and temporal variability of an evolving mixed layer near the ice
770
shelf base.
5. Summary and conclusion
771
The goal of this study was to assess two parameterizations of turbulent heat and salt
772
transfer at the base of an ice shelf in terms of the simulated sub-ice shelf cavity circu-
773
lation and melt rate patterns in the context of a three-dimensional z-coordinate general
774
circulation model. The first parameterization is based on the work of Hellmer and Olbers
775
[1989]. It assigns constant values to the turbulent exchange velocities, γT,S , and hence im-
776
plies constant ocean current speeds underneath the ice shelf. The second accounts for the
777
turbulence generated by ocean currents at the ice interface and couples the turbulent ex-
778
change velocities with the mixed layer flow [Holland and Jenkins, 1999]. Our simulations
779
exposed important differences between the velocity-dependent and velocity-independent
780
parameterizations, particularly in terms of the distribution of melting. The main findings
781
of our simulations are summarized as follows:
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X - 37
782
• Our velocity-dependent simulations differ significantly from previously-published ice
783
shelf-ocean modeling studies using a velocity-independent melt rate parameterization.
784
The experiments performed here suggest that, under conditions of current velocities and
785
thermal forcing typical of PIIS or other “warm” ice shelves, the effects of parameterized
786
turbulence in the proximity of the fixed ice interface dominate over those of temperature
787
gradients in setting the diffusive heat flux through the ice-ocean boundary layer and,
788
hence, the location of high melt rates. In our velocity-dependent experiments, the regions
789
of largest melting coincide with strong outflow plumes and fast mixed layer currents, in
790
agreement with Payne et al. [2007]. This is true over a range of two orders of magnitudes
791
of drag coefficient values (1/16 to 8 times Cd 0 ), encompassing the values employed in
792
published ice shelf-ocean interactions studies.
793
• Sensitivity experiments in which the drag coefficient is varied over this wide range of
√
794
values indicate that the melt rate increases with
795
on the melt rates. (1) They indicate a negative feedback due to the production of larger
796
volumes of meltwater, which spreads at the shelf base and insulates the ice interface
797
from the warmer water below. (2) They also indicate a positive feedback associated
798
with the acceleration of geostrophic mixed layer currents, by increased buoyancy flux-
799
induced density gradients underneath the ice shelf [Little et al., 2008] and by stronger
800
outflow plumes that feed on enhanced meltwater production. This second feedback is not
801
accounted for in velocity-independent melt rate parameterizations. In the present velocity-
802
dependent model, no critical value of Cd is found beyond which melt rates decrease with
803
increasing drag coefficient because of the negative feedback of increased frictional drag
804
on the mixed layer currents. Possible explanations for this behavior are strong buoyancy
D R A F T
Cd and reveal two important feedbacks
November 24, 2013, 9:42pm
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
805
fluxes in our “warm” ice shelf simulations that dominate over friction in setting mixed
806
layer velocities when increasing Cd , but also the treatment of friction in our z-coordinates,
807
namely that friction is distributed over a fixed depth.
808
• No unique value of Cd reconciles the velocity-independent and dependent melt rates
809
in both the idealized and realistic experiments. This suggests that feedbacks between
810
melt rates, mixed layer velocities, and buoyancy fluxes depend on the details of the cav-
811
ity geometry and restoring hydrographic properties. Similarly, optimal drag coefficients
812
inferred from plume model simulations are not easily transferable to three-dimensional
813
baroclinic models. For example, melt rates simulated using the velocity-dependent plume
814
model of Payne et al. [2007] were in best agreement with an ice flux divergence calculation
815
based on surface mass balance, ice thickness and ice flow data for Cd = 3 · 10−3 . Their
816
ice flux calculation indicated melt rates in excess of 100 m/y in a few localized regions,
817
a PIIS proper average of 29.7 m/y, and a cavity average of about 20.7 m/y. Such melt
818
rates require the use of Cd ≈ 6 · 10−3 to 12 · 10−3 in our realistic PIIS model.
819
A step toward ascertaining the relative contributions of ocean circulation, thermal forc-
820
ing and entrainment in determining the location and strength of melting under PIIS may
821
ultimately require non-hydrostatic simulations down to the scales of meters, and in the
822
presence of tidal currents. The latter issue will be taken up elsewhere. Nevertheless,
823
a robust result at present is the marked differences in melt rate patterns depending on
824
whether velocity-dependent or independent transfer coefficients are used. Given the im-
825
portant implications on where within an ice shelf cavity the maximum melt rates are
826
expected and their potential impact on ice shelf dynamical responses, our results call for
827
more detailed observations that would resolve the spatial distribution of melt rates. First
D R A F T
November 24, 2013, 9:42pm
D R A F T
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
X - 39
828
steps to this end have been made with the recent drilling through PIIS and deployment
829
of a specialized suite of oceanographic instrumentation (“flux package”), measuring ocean
830
velocity, temperature, and salinity at a sufficiently fast rate (4 Hz) so as to enable the
831
inference of vertical turbulent fluxes of momentum, heat, and salt Stanton et al. [2013].
832
Such data hold the prospect of vastly improving constraints on turbulent transfer pro-
833
cesses at the ice-ocean interface and improve melt rate parameterizations used in today’s
834
ocean climate models.
Appendix A: Thermodynamical melt rate parameterizations
835
Typical melt rate parameterizations are based on the assumption that phase changes at
836
the ice-ocean boundary occur in thermodynamic equilibrium. The three-equation model
837
uses two conservation equations for heat and salt, along with a third linearized relation
838
[e.g., Hellmer and Olbers, 1989; Holland and Jenkins, 1999; Jenkins et al., 2010a] that
839
expresses the dependence of seawater freezing point temperature on salinity and pressure
840
using empirical parameters a, b, c:
QTI + QTM = −Lf ρM m
(A1)
QSI + QSM = −ρM mSB
(A2)
845
Tfreeze = TB (pB , SB ) = a SB + b pB + c.
(A3)
846
QTM and QSM are the diffusive heat and salt fluxes across the ice-ocean boundary layer,
847
QTI and QSI are the conductive heat flux and diffusive salt flux through the ice shelf,
848
respectively, Lf is the latent heat of fusion/melting, ρM is the ocean mixed layer density,
849
Tfreeze , is the freezing temperature, TB , SB and pB are the hydrographic properties and
841
842
843
844
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
850
pressure at the ice shelf base, and m is the melt rate, expressed here as a volume flux per
851
unit area (with corresponding mass flux q = ρM m).
852
In the present model, m is defined negative for melting and positive for refreezing
853
(in contrast to Holland and Jenkins [1999]). All simulated melt rates reported here are
854
expressed in terms of equivalent ice thickness. The salt in the ice shelf is neglected so
855
that QSI = 0 [Eicken et al., 1994]. Following Losch [2008], we choose a salinity coefficient
◦
856
857
858
a = −0.0575◦ C, a pressure coefficient b = −7.61 · 10−4 C dBar−1 , and c = 0.0901◦ C.
For a turbulent boundary layer, the turbulence-induced variability of the diffusivities of
tracers X = {T, S} may be represented by a non-dimensional Nusselt number, Nu:
X
QX
M = ρM cpM
859
Nu κX
M
(XM − XB ),
D
(A4)
860
where cTpM is the heat capacity of the mixed layer (and cp SM = 1), κX
M are the thermal and
861
salt diffusivities and D is the thickness of the boundary layer. The factors γX =
862
have dimensions of velocity and are referred to respectively as the turbulent heat and
863
salt exchange or piston velocities (hereinafter, γT,S ). We note that the description of the
864
three-equation model in the Appendix of Losch [2008] contains errors. These have been
865
corrected in the present formulation.
NuκX
M
D
866
Together with these generic expressions for QTM and QSM in terms of γT,S , the set of equa-
867
tions (A1) – (A3) provides solutions for TB , SB and m. They are used to infer boundary
868
conditions for the temperature (T ) and salinity (S) tendency equations, represented here
869
as a generic equation for tracer X:
870
D R A F T
∂X κ
= (γX − m)(XB − XM )
∂z B
(A5)
November 24, 2013, 9:42pm
D R A F T
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
X - 41
871
with vertical diffusion κ [Jenkins et al., 2001]. The heat and salt balances and associ-
872
ated sign conventions in our model are illustrated in Figure 1, the various variables and
873
constants are listed in Table 1.
Appendix B: Accounting for drag at the ice-ocean interface
874
To account for the circulation-driven turbulent exchanges at the ice shelf base the piston
875
velocities γT,S are turned into functions of the frictional drag at the ice shelf base via a
876
friction velocity, u∗ , which is related to the velocity of ocean currents through a simple
877
quadratic drag law of the form:
2
u∗ 2 = Cd UM
,
878
(B1)
p
2
2
u2M + vM
+ wM
, the magnitude of
879
with Cd a dimensionless drag coefficient and UM =
880
the mixed layer current velocity. The piston velocities are expressed as
γT,S = ΓT,S u∗ = ΓT,S
881
p
Cd UM ,
(B2)
882
where ΓT and ΓS (hereinafter, ΓT,S ) are turbulent transfer coefficients for heat and salt,
883
respectively. Holland and Jenkins [1999] formulated expressions for ΓT,S that include the
884
effects of rotation and of melting and refreezing on the stability of the boundary layer:
ΓT,S =
885
886
(B3)
with
ΓT urb
887
888
1
,
ΓT urb + ΓT,S
M ole
1
= ln
k
u∗ ξN η∗2
f hν
+
1
1
− ,
2ξN η∗ k
(B4)
and
889
2/3
ΓT,S
− 6.
M ole = 12.5(Pr, Sc)
890
Here, Pr and Sc are the Prandtl and Schmidt numbers for seawater, k is the von Karman
891
(B5)
constant, f is the Coriolis parameter, ξN is a dimensionless stability constant, hν is the
D R A F T
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
892
thickness of the viscous sublayer, estimated as hν = 5 uν∗ . η∗ is the stability parameter,
893
formulated in terms of a critical flux Richardson number and the Obukhov length. It is
894
negative for a destabilizing and positive for a stabilizing buoyancy flux. Other parameter
895
values adopted from Holland and Jenkins [1999] are listed in Table 1.
896
Two caveats regarding the velocity-dependent parameterization are worth mentioning.
897
First, both Jenkins [1991] and Holland and Jenkins [1999] make the assumption of a
898
hydraulically smooth interface. While this approach may be applicable over ablating por-
899
tions of the ice shelf base, it might not be entirely adequate over regions of refreezing.
900
Support for this assumption comes from the work of McPhee [1992] and McPhee et al.
901
[1999], who measured turbulent transfers underneath sea ice over a wide variety of rough-
902
ness characteristics. They found that turbulent transfers appear to be independent of
903
the roughness of the ice-ocean interface. Uncertainties remain, nevertheless, regarding
904
roughness characteristics of ice shelf-ocean interfaces. Jenkins et al. [2010a] pointed out
905
that little observational evidence exists to date that supports the direct applicability of
906
findings from sea ice studies to the ice shelf problem.
907
Second, using a quadratic drag law introduces an unknown drag coefficient Cd in
908
eqn. (B2). Current observations do not provide enough information to allow estimat-
909
ing the drag and turbulent transfer coefficients independently [Jenkins et al., 2010a].
Appendix C: Approximations to the velocity-dependent melt rate parameterization
910
Based on the sensitivity analyses performed by Holland and Jenkins [1999], some ap-
911
proximations were adopted in the implementation of velocity-dependent melt rate param-
912
eterization in the MITgcm. They are briefly summarized in the following.
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913
914
915
X - 43
• The heat flux through the ice shelf, QTI is only described by vertical diffusion, i.e.,
vertical advection is neglected. In this case the gradient in ice temperature at the shelf
I
B
base is linear and can be estimated as ∂T
= TSh−T
, with TS the (constant) surface
∂z B
I
916
temperature of the ice shelf and hI , the local thickness of the ice shelf. Sensitivity analysis
917
of simulated melt rates to the parameterization of heat flux through the ice shelf by
918
Holland and Jenkins [1999] suggest that for high melt rates, as those obtained in our
919
”warm” idealized and realistic ice shelf experiments, omitting vertical advection increases
920
the simulated melt rates by about 10% (see their Figure 7b and c). However, as this
921
percentage varies very little over a wide range (2◦ C) of thermal driving (see their Figure
922
7c), we do not expect this choice to significantly impact our simulated melt patterns.
923
• As in Holland and Jenkins [1999], it is assumed that all phase changes occur at the
924
ice-ocean boundary. The formation of sea ice in front of the ice shelf is not simulated.
925
The formation of frazil ice through supercooling in the water column is not parametrized
926
either. Neglecting this process is not expected to affect the cavity dynamics greatly,
927
because regions over which the plume refreezes underneath both our “warm” idealized
928
and realistic PIIS ice shelves are very limited.
929
• Following the argument of Holland and Jenkins [1999] that direct freezing onto the ice
930
shelf base is limited, we neglect the effect of a destabilizing buoyancy flux on the freezing
931
rate and set the stability parameter η∗ in equation (B4) to 1 in the case of refreezing.
932
Contrary to Holland and Jenkins [1999], we also neglect the stabilizing effect of melting
933
on the boundary layer, and hence in the present model, η∗ = 1 also in the case of melting.
934
Holland and Jenkins [1999] compared melt rates computed both with and without taking
935
into account the effects of stabilizing/destabilizing buoyancy fluxes. They found that the
D R A F T
November 24, 2013, 9:42pm
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
936
explicit calculation of η∗ in eqn. (B4) changed the melt rates by less than 10% under
937
“moderate” conditions of friction velocity and thermal driving (see eqn. (2)), which they
938
identified as u∗ > 0.001 m s−1 and T∗ < 0.5◦ C.
939
The area-averaged friction velocity underneath both the idealized and realistic PIIS ice
940
shelves is above 0.001 m s−1 for most values of Cd used in this study (Figures 8c and
941
8d). However, for all values of Cd employed here, the thermal driving is larger than 0.5◦ C
942
(Figures 8e and 8f) and is representative of most “warm” ice shelves in contact with
943
CDW [e.g., Jacobs et al., 1996; Payne et al., 2007; Holland , 2008; Holland et al., 2008;
944
Jenkins et al., 2010a]. Hence, parameterizing the stabilizing or destabilizing effect of the
945
melting or refreezing-induced buoyancy fluxes on the boundary layer underneath the ice
946
shelf could impact our simulated melt rates. Furthermore, the relatively large melt rates
947
and associated stabilizing buoyancy fluxes may significantly suppress mixing underneath
948
the ice shelf and inhibit further melting.
949
Holland and Jenkins [1999] pointed out that solving for melt rates and γT,S in the presence
950
of a stability parameter requires a computationally expensive iteration. Whether the
951
addition of this extra level of complexity is necessary to obtain accurate estimates of melt
952
rates underneath “warm” ice shelves such as PIIS requires further studies.
953
Acknowledgments. This work was supported in part by NSF grant # 0934404 and
954
NASA grant # NNX11AQ12G. V.D. gratefully acknowledges a Postgraduate Doctoral
955
Scholarship from the National Sciences and Engineering Research Council of Canada and
956
from the Fonds Québécois de la Recherche sur la Nature et les Technologies. We thank
957
two anonymous reviewers for their detailed comments and suggestions.
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X - 45
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1153
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
Figure 1.
Schematic representation of the heat and salt balances at the base of an idealized ice shelf, as formulated in
the present thee-equation model. The diagram represents an ice shelf of thickness hI (dark grey shaded area), an ice-ocean
boundary layer of thickness D at the ice shelf base and a mixed layer outside the boundary layer with fixed depth hM .
The sign convention is such that a positive (upward) heat flux through the boundary layer leads to melting (downward
flux of freshwater) and a to positive conductive heat flux (upward) into the ice shelf. QTM , QTlatent and QTI have dimensions
of a heat flux per unit volume (J ms−1 m−3 or Wm−2 ). QS
M has dimensions of a flux of mass of salt per unit volume
(kg ms−1 m−3 ).
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0
50
0
100
100
0
400
50
500
−300
200
0
60
48’
−200
50
400
700
−100
50
36’
100
50
100
20
500 03
00
0
−400
800
75 S
800
−500
1000
−600
50
0
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600
500
400
50
200
300
300
0
(a)
0
−700
50
12’
0
10
400
0
24’
0
30
600
0
90
900
o
0
50
100
−800
300
200
10
0
50
200
100
50
0
−900
0
30’
o
102 W
o
101 W
30’
100oW
−1000
30’
0
−100
36’
−200
−300
48’
−400
−500
o
75 S
−700
200
300
400
500
(b)
600
12’
700
800
−600
−800
−900
24’
o
105 W
o
104 W
o
103 W
102oW
101oW
100oW
−1000
Figure 2. (a) Geometry of the ice shelf cavity in the realistic experiments. Shading is used for the bathymetry (m)
and contours show the water column thickness (m). (b) Geometry of the idealized cavity. Shading indicates the depth of
the ice shelf base (m) and contours, the water column thickness (m). The solid black line indicates the ice shelf front in
both cases.
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34.2
0
34.3
34.4
S (psu)
34.5
34.6
34.7
34.8
idealized
realistic
100
200
depth (m)
300
400
500
600
700
800
900
1000
−2
(a)
−1.5
−1
−0.5
0
T (°C)
0.5
1
0
1.5
idealized
realistic
100
200
depth (m)
300
400
500
600
700
800
900
1000
−0.03
(b)
−0.02
−0.01
0
u (ms−1)
0.01
0.02
0.03
Figure 3.
Vertical profiles of (a) temperature and salinity and (b) zonal velocity prescribed as the western open
boundary conditions in the idealized and realistic experiments. Profiles are all uniform in the meridional direction.
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0.25
0.05
0.05
0.15
0.25
0
0.65.45 0
.35
0.95
0.
55
0.
85
(b)
0.0
5
5
0.2
101oW
0.05
0.35
5
0.05 0.2 0.15
102oW
101oW
0.25
0.15
0.3
0.55 5
0.7
5
0.
45
0.25
−20
−30
0.15
5
0.2
0.45 0.65
0.2
0.05 5 0.15
o
o
103 W
104 W
0.05
0.8
5
−10
15
0.
0.5
5
5
0.4
105 W
0
0.35
o
−50
0.35
0.55
0.85
0.95
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95
0.
100oW
0.0
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0.
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25
0.05 0.15 0.
102oW
0.05
0.15
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0.55
0.35
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o
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−40
0.45
75 S
0.15
o
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0.65
0.75
o
0.55
0.15 0.25
0.05
mean =−15 m/yr
max =−78 m/yr
48’
0.15
36’
0.25
105 W
0.35
o
95
0.
−20
−30
5
0.5
24’
−10
0.65
0.95
0.75
0.55
0.35
0.15
0.05
12’
0.45
75 S
0.6
5
5
0.7
0.75
0.85
0.95
o
0.45
0.55
48’
0
0.3
5
25
0.
mean =−21 m/yr
max =−69 m/yr
0.0
5
0.
35
(a)
0.
05
0.
15
36’
0.05
0.1
5
X - 57
100oW
−40
−50
Figure 4.
Melt rate (shading, in m/yr) and barotropic streamfunction for the depth-integrated horizontal volume
transport (black countours, in Sv) in the idealized cavity setup using (a) the velocity-independent and (b) the velocitydependent melt rate parameterization with Cd 0 . The maximum and cavity-averaged melt rates are given in the top left
corner of each panel.
D R A F T
November 24, 2013, 9:42pm
D R A F T
X - 58
101oW
100oW
−15 −20
0.25
48’
0.2
(d)
o
105 W
o
104 W
o
103 W
102oW
101oW
−25 −25
−
30
−30
−30
−
30−20
−
30
−30
−15
0
24’
−25
−25 0
−5
−20 −25
−10
−35
30
−
40
−50
−45
12’
0.05
−25
75 S
2.5
2
1.5
1
0.5
0
−0.5
0.3
0.25
0.2
−15
0.1
o
−25 −25 −30
−30
−30
−
30−20
−
30
−30
−15
−25
mean =0.15 m/s
max =1.41 m/s
102oW
−10
100oW
o
103 W
−25
−25 0
−5
−20 −25
−10
−35
30
−
40
−50
−45
101oW
−15
102oW
−10
o
103 W
36’
o
104 W
−10
o
104 W
0.3
o
105 W
−10
o
105 W
05
−15
−−
20
(c)
−50
−40−45
−35
−30
−25−10
−15
−10
12’
−0.5
0.15
−20
75 S
(b)
24’
−05
o
12’
0
−10
48’
o
75 S
−10
100oW
−40 −35
−45
−50
−30
−10
−25
−30
−35 −25
−20
−15
−10
−0
5
mean =0.22 m/s
max =1.57 m/s
101oW
102 W
103 W
−10
o
104 W
o
0.5
3
48’
1
−40 −50
−30
o
105 W
o
0
−20
24’
mean =1.8 C
°
max =3.5 C
−15 −20
−30
−20
−10
(a)
24’
2
1.5
o
75 S
12’
36’
2.5
48’
36’
3
3.5
°
−05
−20
0
mean =1.5 C
max =3.2°C
3.5
−60
−50 −40
−10
−30
°
−10
36’
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
100oW
0.15
0.1
0.05
0
Figure 5.
(a, b) Thermal forcing (C◦ ) and (c, d) and ocean mixed layer velocity (m/s) in the idealized cavity setup
with Cd . The values of area-averaged and maximum thermal forcing and mixed layer velocity are given in the top left
corner of each figure. Black contours show the spatial distribution of melt rates (m/yr). Vectors indicate the direction
and relative magnitude of the mixed layer currents on figures (c) and (d). Left panels show the the velocity-independent
simulation results and right panels, results from the velocity-dependent simulation.
0
D R A F T
November 24, 2013, 9:42pm
D R A F T
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DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
5
−20
(a)
40
0
400
70
800 0
100oW
30’
0.04
50
0.02
100 50
50
24’
30’
o
101 W
30’
100oW
500
−0.04
30’
−0.06
2
20
0
(d)
o
102 W
30’
−0.02
50
50
200
0
40
o
101 W
300
300
100
12’
100
300
100
100
300
300
3
300
500
300
−0.02
800
600
400
50
0
50
100
1
200
200
o
75 S
3
2
20
0
(c)
o
102 W
0
50
100
0
40
50
24’
max =0.14 Sv
100
600
600
300
12’
−30
800 700
1
800
600
400
50
30’
400
48’
800 700
o
75 S
200
30
0
0
50
0.02
100oW
1050
0
100
50
50
400
48’
100
300
500
50
100
36’
0
60
200
30
0
0.04
0
50
0
60
400
50
0
20
0
20
100
300
500
30’
0.06
50
max =0.23 Sv
1050
0
100
o
101 W
30’
o
102 W
−20
−25
500
(b)
−70
0.06
50
36’
12’
−60
30’
600
400
−50
24’
o
101 W
30’
o
102 W
500
0
50
0
50
400
500
−15
0
50 0
30
−40
600
400
12’
−10
200
40
0
500
−5
0
30
48’
o
75 S
0
50 0
30
24’
300
−30
200
o
75 S
0
20
0
0
0
30
48’
max =−57 m/yr
36’
−10
300
400
mean =−8 m/yr
50
20
0
0
200
70
800 0
max =−71 m/yr
36’
10
0
0
30
0
400
mean =−24 m/yr
30
0
200
400
0 10
0
200
−0.04
100 50
30’
100oW
30’
−0.06
Figure 6.
(a, b) Melt rate (m/yr) simulated using (a) the velocity-independent and (b) the velocity-dependent model
in the realistic PIIS setup, with Cd 0 . Black contours indicate the depth of the ice shelf base (m). The maximum and
area-averaged melt rates are indicated in the top right corner of each panel. Different scales are used to bring out clearly the
spatial distribution of melt rates in both cases. (c, d) Barotropic streamfunction for the depth-integrated horizontal volume
transport (Sv) calculated using (c) the velocity-independent and (d) the velocity-dependent model. Dashed contours show
the distribution of water column thickness (m) and the solid black line, the position of the ice shelf front. The three main
depth-integrated ocean gyres discussed are indicated with numbers.
D R A F T
November 24, 2013, 9:42pm
D R A F T
X - 60
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
−20
36’
−10
−30
−10
3
mean =2.4°C
max =3.6°C
2.5
48’
2
−0.5
(b)
30’
o
102 W
o
101 W
0.3
−2
0
−20
48’
−40
10−2
−−
−4300
0
101 W
o
100 W
−1
0
−1
0
0.05
−10
30’
30’
0.15
0.1
12’
24’
o
−30
o
75 S
0.1
0
30’
0
0
−50
102 W
−6
0
−30
−30
−30
−2
0
−10
0
−2
0.15
−30
o
0.2
0
0
−3
−2
0
(c)
0.3
0
−1
0
−2
−40
0
−2
24’
−0.5
0.25
−2
0
0
0
−1
12’
30’
mean =0.05 m/s
max =0.48 m/s
36’
0.2
−30
−20
100oW
0
−30
o
75 S
30’
−10
0 −10
−1
48’
0
0.25
−3
0
−10
−10 −10
0.5
−10
36’
0
−20
−50 mean =0.1 m/s
−40
−30max =0.83 m/s
−20
30’
−30
0
1
0
100oW
−10
0
30’
1.5
12’
24’
o
101 W
−10
1
0.5
−30
30’
o
102 W
o
75 S
−20
−−
30
−10
−60
−50
−10
−30 0
(a)
1.5
0
−20
−40−30
−30
−20
−20
−20
24’
0
−20
−40
−30
−10
12’
2
−20
0
o
75 S
3
2.5
−20
−10
−10 −30
48’
3.5
0
−20 −30
−40
−50
0
−10
36’
3.5
°
mean =1.6 C
max =3.3°C
0
(d)
o
102 W
0.05
0
30’
o
101 W
30’
100oW
30’
0
Figure 7.
(a, b) Thermal forcing (C◦ ) and (c, d) and ocean mixed layer velocity (m/s) in the realistic PIIS setup
with Cd 0 . Black contours show the spatial distribution of melt rates (m/yr) and the area-averaged and maximum values
of the forcings are given at the top right corner of each panel. Vectors indicate the direction and relative magnitude of
the mixed layer currents on panels (c) and (d). Left and right panels show the results from the velocity-independent and
velocity-dependent simulation respectively.
D R A F T
November 24, 2013, 9:42pm
D R A F T
X - 61
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
45
40
40
35
35
30
30
Mean m (m/yr)
Mean m (m/yr)
45
(a)
25
20
b
a*sqrt(Cd)
15
a =15.442
b =0.750
10
25
20
a*sqrt(Cd)b
15
a =8.514
b =0.960
10
5
5
0
(b)
0.5
1
1.5
2
2.5
3
3.5
0
4
0.5
1
1.5
C1/2
/ C 0 1/2
d
0.25
d
0.25
(c)
U , u (m/s)
d
3
3.5
4
(d)
b
a*sqrt(C )
a =0.0021927
b =1.3765
0.15
*
a =0.006
b =1.267
d
M
UM, u* (m/s)
a*sqrt(C )b
0.15
0.1
0.1
0.05
0.05
0.5
1
1.5
2
2.5
3
3.5
0
4
0.5
1
1.5
3
2
2.5
3
3.5
4
3
3.5
4
/ C 0 1/2
C1/2
d
C1/2 / C 0 1/2
d
d
d
3
(e)
(f)
2.5
TM −TB, T* (°C)
2.5
TM −TB, T* (°C)
2.5
d
0.2
0.2
0
2
C1/2 / C 0 1/2
d
2
1.5
2
1.5
1
1
0.5
0.5
0.5
1
1.5
2
2.5
3
3.5
4
0.5
1
C1/2
/ C 0 1/2
d
1.5
2
2.5
C1/2
/ C 0 1/2
d
d
d
Figure
8.
√
(a, b): Area averaged melt rate (m/yr, black dots) as a function of the square root of the drag coefficient
Cd and power law fit (black curves, with coefficients in the lower right corner of the graph). The dashed black curve
shows area averaged melt rates for the velocity-independent experiments with Cd 0 . √
(c, d): Area averaged mixed layer
velocity (m/s, orange dots) and friction velocity (m/s, blue asterisks), as a function of Cd . The orange dotted line shows
area-averaged mixed layer velocity UM for the velocity-independent experiment with Cd 0 . The solid blue curve is the power
law fit to the area-averaged friction velocity. (e, f): Area averaged
thermal driving (◦ C, red dots) and thermal forcing
√
◦
( C, purple dots) across the boundary layer as a function of Cd . The solid lines of the same colors are the corresponding
power law fits. The red and purple dotted lines show respectively the area-averaged thermal driving and thermal forcing
in the velocity-independent experiment with Cd 0 . Left and right panels show the results of the idealized and realistic PIIS
simulations, respectively.
D R A F T
November 24, 2013, 9:42pm
D R A F T
X - 62
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
20
0
0
−20
40
0
50
0
30
500
400
24’
o
102 W
−50
o
101 W
30’
100oW
30’
500
400
−70
o
102 W
30’
o
101 W
30’
5
0
500
(b)
10
600
12’
24’
30’
0
30
400
−60
500
15
0
50
0
50
12’
(a)
50
−40
600
400
25
0
30
20
o
75 S
70
800 0
75 S
30
300
48’
−30
200
o
min =−5 m/yr
20
0
0
100
100
0
30
400
max =40 m/yr
36’
−10
300
48’
0
200
30
0
max =−97 m/yr
35
10
0
70
800 0
36’
400
mean =−19 m/yr
40
0
30
0
200
200
10
0
100oW
30’
−5
Figure 9.
(a) Melt rate (m/yr) simulated using the velocity-dependent model in the realistic PIIS setup and
Cd = 4 · Cd 0 = 6.0 · 10−3 . The maximum and area-averaged melt rates are indicated in the top right corner of the figure.
For this value of drag coefficient, the area-averaged melt rate is comparable to the ice flux divergence based estimate of
Payne et al. [2007] (20.7 m/yr). (b) Difference between the velocity-dependent melt rate simulated using Cd = 4 · Cd 0
and Cd 0 . Positive differences indicate a higher melt rate for the larger drag coefficient experiment. The maximum and
minimum differences are indicated in the top right corner of the figure. Black contours indicate the depth of the ice shelf
base (m) on both figures.
D R A F T
November 24, 2013, 9:42pm
D R A F T
X - 63
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
o
101 W
30’
200
100oW
30’
−30
500
50
300
200
50
50
300
500
30’
−15
10
0
50
o
102 W
800
600
400
200 300
50
400
−20
300
12’
300
(f)
−25
50
−10
50
200
20
0
−5
700
0
10
100
50
50
600
(e)
24’
5
400
−20
300
300
−70
0
200
30
0
500
−15
10
0
400
12’
−25
50
0
60
20
0
800
600
400
200 300
50
30’
100
50
o
75 S
100oW
mean =−5 m/yr
500
800
−10
30’
50
100
400
100
50
24’
−60
50
100max =−37 m/yr
50
300
48’
50
300
(d)
o
75 S
50
−20
300
12’
−5
700
200
−15
10
0
400
50
0
10
200
800
600
400
200 300
50
200
30
0
600
0
10
600
50
50
10
0
36’
0
100
800
−10
5
o
101 W
30’
o
102 W
400
800
500
48’
700
−70
mean =−5 m/yr
500
−5
400
48’
30’
50
100
0
60
0
60
500
50
100oW
100max =−32 m/yr
50
300
400
400
200
30
0
20
0
200
36’
0
100
50
10
0
mean =−5 m/yr
30’
200
5
50
100
o
101 W
30’
o
102 W
50
−70
100max =−30 m/yr
50
300
500
100
50
24’
300
30’
500
(c)
−60
500
200
100oW
−50
300
300
500
300
50
10
0
30’
400
300
50
−40
10
0
50
20
0
−30
50
200
o
101 W
30’
0
10
100
50
800
600
400
200 300
50
12’
24’
o
102 W
o
75 S
300
(b)
−60
50
−50
300
12’
−20
400
100
50
−40
400
−10
50
500
(a)
24’
o
75 S
10
0
0
700
−30
50
50
50
−50
300
300
20
0
800
600
400
200 300
50
200
30
0
600
−40
400
48’
800
200
200
o
75 S
10
0
−20
0
10
−30
50
−10
700
600
0
10
600
800
600
400
200 300
50
mean =−29 m/yr
100
0
60
800
12’
36’
48’
700
50
100
100max =−69 m/yr
50
500
50
400
400
−20
200
30
0
500
0
60
500
0
60
800
36’
300
400
−10
50
50
10
0
0
100
400
400
200
30
0
mean =−22 m/yr
50
100
48’
o
75 S
36’
50
100
100max =−63 m/yr
50
300
500
500
500
50
10
0
0
200
mean =−19 m/yr
50
36’
50
100
100max =−59 m/yr
50
300
300
50
10
0
100
50
20
0
−25
50
24’
o
102 W
30’
o
101 W
30’
100oW
30’
−30
o
102 W
30’
o
101 W
30’
100oW
30’
−30
Figure 10. Melt rate (m/yr) simulated using (a to c) the velocity-independent and (d to f) the velocity-dependent
model in the realistic PIIS setup with Cd 0 and a (a, d) 10 m, (b, e) 20 m and (c, f) 50 m thick mixed layer for averaging of
TM , SM and UM . Dashed contours show the distribution of water column thickness (m). The maximum and area-averaged
melt rates are indicated in the top right corner of each panel.
D R A F T
November 24, 2013, 9:42pm
D R A F T
X - 64
Table 1.
D R A F T
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
Three-equation model parameters and constants
Parameter
Ice shelf
thickness
surface temperature
bulk salinity
ice density
heat capacity
molecular thermal conductivity
Ice-ocean boundary layer
temperature
salinity
pressure
Ocean mixed layer
thickness
temperature
salinity
water density
specific heat capacity
Latent heat of fusion
Symbol
Latent heat flux
Brine flux
Diffuisve heat flux through the BL
Diffusive salt flux through the BL
Diffusive heat flux through the ice shelf
Diffusive salt flux through the ice shelf
Melt/refreezing rate
Transfer velocities parameterizations
Turbulent transfer velocity for heat
Turbulent transfer velocity for salt
stability parameter
Von Karman’s constant
stability constant
kinematic viscosity of sea water
Coriolis parameter
Prandlt number
Schmidt number
Model parameters
Advection scheme
Vertical advection and diffusion
Equation of state
Vertical viscosity
Laplacian viscosity
Bi-harmonic viscosity
Vertical diffusion
Horizontal diffusion
Quadratic bottom and shelf base drag
Minimum partial cell factor
Reference ocean density,
QTlatent
QS
brine
QTM
QS
M
QTI
QS
I
m
hI
TS
SI
Value
−20.0◦ C
0 psu
917 kg m−3
2000 J kg−1 K−1
1.54 · 10−6 m2 s−1
TB
SB
pB
hM
TM
SM
ρM
cp M
Lf
γT
γS
η∗
κ
ξN
ν
f
Pr
Sc
Cd
ρref
20 m (default)
3998 J kg−1 K−1
334000 J kg−1
0
1.0
0.4
0.052
1.95 · 10−6 m2 s−1
13.8
2432
3rd order direct space-time
Implicit for T and S
Jackett and McDougall (1995)
10−3 m2 s−1
0.2
0.02
5 · 10−5 m2 s−1
10 m2 s−1
Cd 0 = 1.5 · 10−3 (default)
0.1 (1/8◦ ), 0.3 (1/32◦ )
1000 kg m−3
November 24, 2013, 9:42pm
D R A F T
DANSEREAU, HEIMBACH, LOSCH: ICE SHELF-OCEAN INTERACTIONS IN A GCM
Table 2.
Summary of experiments
Section
D R A F T
X - 65
setup
γT,S formulation
Cd
3.1.1
idealized
vel-dep. & indep.
Cd 0
3.1.2
realistic
vel-dep. & indep.
Cd 0
3.2.1
idealized
vel-dep.
1/16 to 16 · Cd 0
3.2.2
realistic
vel-dep.
1/16 to 16 · Cd 0
November 24, 2013, 9:42pm
D R A F T
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