A comment on the equation of state and the freezing point equation with respect to
subglacial lake modelling
Malte Thoma
a,bKlaus Grosfeld
aAndrew M. Smith
cChristoph Mayer
baAlfred Wegener Institute for Polar and Marine Research, Bussestrasse 24, 27570 Bremerhaven, Germany
bBayerische Akademie der Wissenschaften, Kommission f¨ur Glaziologie, Alfons-Goppel-Str. 11, 80539 M¨unchen, Germany
c British Antarctic Survey, High Cross, Madingley Road, Cambridge, CB3 0ET, United Kingdom
1
Abstract
2
The empiricalEquation of State (EoS) allows to calculate the density of water in
3
dependence of salinity, temperature, and pressure. The three parameters have a
4
complex interdependency on the EoS. Hence, whether warmer water parcels sink
5
or raise depends on the surrounding salinity and pressure. The empirical Equation
6
of Freezing Point (EoFP) allows to calculate the pressure and salinity dependent
7
freezing point of water. Both equations are necessary to model the basal mass bal-
8
ance below Antarctic ice shelves or at the ice-water interface of subglacial lakes.
9
This article aims three tasks: First we comment on the most common formulations
10
of theEoSand the EoFP applied in numerical ocean and lake models during the
11
past decades. Then we describe the impact of the recent and self-consistent Gibbs
12
thermodynamic potential-formulation of theEoSand theEoFP on subglacial lake
13
modeling. Finally, we show that the circulation regime of subglacial lakes covered
14
by at least 3000 m of ice, in principle, is independent of the particular formula-
15
tion, in contrast to lakes covered by a shallower ice sheet, like e.g., subglacial Lake
16
Ellsworth. However, as modeled values like the basal mass balance or the distri-
17
bution of accreted ice at the ice-lake interface are sensitive to different EoS and
18
EoFP, we present updated values for subglacial Lake Vostok and subglacial Lake
19
Concordia.
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Key words: Subglacial Lakes, Equation of State, Freezing Point Equation,
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Numerical Modelling, Ice-ocean Interaction, Lake Vostok, Lake Concordia, Lake
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Ellsworth, Antarctica
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Email address: Malte.Thoma@awi.de(Malte Thoma).
1 Introduction
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Water flow within oceans and subglacial lakes is modelled by solving thehydro-
25
static primitive equations numerically (e.g., Haidvogel and Beckmann, 1999;
26
Griffies, 2004). These equations describe the flow of a fluid on the rotating
27
earth by the equation of motion, the conservation laws of temperature and
28
salinity, and an equation of state (EoS). Some fundamental differences be-
29
tween different models relate to the implementation of the vertical coordinate,
30
which may be orientated planar, terrain-following, or along isopycnals. Well
31
known representatives for these type of models are theModular Ocean Model
32
(MOM, e.g. Pacanowski and Griffies, 1998; Griffies et al., 2003), thePrinceton
33
Ocean Model(POM, e.g., Blumberg and Mellor, 1983; Ezer and Mellor, 2004),
34
and theMiami Isopycnic Coordinate Ocean Model (MICOM, e.g., Bleck, 1998;
35
Holland and Jenkins, 2001), respectively. Other approaches to solve the equa-
36
tions on unstructured grids apply spectral formulations (SEOM, e.g., Patera,
37
1984), finite volumes (MITgcm, e.g., Marshall et al., 1997a,b), or finite ele-
38
ments (COM, e.g., Danilov et al., 2004; Timmermann et al., 2009). The num-
39
ber of ocean models originating from these, in particular of those with struc-
40
tured horizontal grids, is high. However, each model has to implement the
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EoS. The empirical EoS is a complex nonlinear function to calculate the
42
density as a function of temperature, salinity, and pressureρ=ρ(T, S, p). For
43
the global ocean, it has to cover a wide parameter range inS (0 to 42 psu),T
44
(−2 to 40◦C), andp(0 to 100 MPa). Subglacial lakes range at the lower bound-
45
aries for T and S and the medium pressure range. In this parameter range,
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the slope of the calculated density is at its vertex, which has implications for
47
the circulation and basal mass balance within subglacial lakes (Thoma et al.,
48
2008b). Models that also include the interaction between ice and water, ad-
49
ditionally apply an equation for the pressure-dependent freezing point of sea
50
water (EoFP)Tf =Tf(S, p).
51
In the following we briefly review different representations of EoSand EoFP
52
used in ocean modelling, before we discuss the relevance of their improved
53
formulations for the modelling of subglacial lakes. Finally we present updated
54
results of subglacial lake modelling studies, with respect to the revised EoS
55
and EoFP.
56
1.1 Equation of State (EoS)
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Early ocean models applied theKnudsen-Ekman equation, which relies on the
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Boussinesq approximation and linearises theEoS around some reference val-
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ues for temperature, salinity and pressure (e.g., Fofonoff, 1962; Bryan and Cox,
60
1972). Although this approach reduces the computational effort significantly,
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it is only appropriate over very narrow ranges ofT and S. A more general ap-
62
proach is the so-called UNESCO-EoS (Fofonoff and Millard, 1983), derived
63
from the fundamental work of Millero et al. (1980) and Millero and Poisson
64
(1981). It consists of a set of 15 coefficients, to calculate the ocean’s sur-
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face density ρ0(T, S) = ρ(T, S, p = 0) and 26 subsequent coefficients for the
66
secant bulk modulus κ to evaluate the pressure dependence: ρ(T, S, p) =
67
ρ0(T, S)/(1−p/κ(T, S, p)). This equation is valid over a large parameter range
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−2◦C< T < 40◦C, 0< S < 42 psu, and 0 < p <108Pa (≈10 000 m depth),
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and could hence be applied to the global ocean as a whole.
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However, a complication arises from the fact, that the ocean models intrinsic variable is not the temperature T, but the potential temperature θ, which excludes temperature changes induced by adiabatic processes. To bypass the time-consuming conversion of different temperature representations in ocean models, Jackett and McDougall (1995) published a modified set of coefficients for the UNESCO-formulation. This allows a straight calculation of the density from the potential temperature
ρ(θ, S, p) = ρ0(θ, S)
1−p/κ(θ, S, p). (1)
The pressure in (1) is calculated from integrating the hydrostatic equation
∂p
∂z =−ρg ⇒ p=g
Z 0
z ρ(θ, S, p)dz (2)
from the surface to the depthz. To improve efficiency in numerical ocean mod-
71
els solving (1) and (2) iteratively, either the density of a former model-timestep
72
has to be used, or another set of coefficients for the UNESCO-formulation of
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the EoS has to be applied, which allows for a depth-dependent density cal-
74
culation instead of pressure ρ = ρ(θ, S, z) (Haidvogel and Beckmann, 1999).
75
However, this set of coefficients is based on a homogeneously stratifiedstandard
76
ocean and has significant limits as soon as deviations from this standard strati-
77
fication arise. Figure 1 indicates the deviation of the Haidvogel and Beckmann
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(1999) formulation from the Jackett and McDougall (1995) formulation as
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soon as the temperature, salinity and/or depth diverges from the assumed
80
reference values, which refer to the mean oceanic properties.
81
The most up-to-date approach for calculating the density of seawater depends
82
on the Gibbs thermodynamic potential (e.g., Feistel, 1993; Feistel and Hagen,
83
1995; Feistel, 2003; Jackett et al., 2006). Thermodynamic properties, like den-
84
sity, freezing point, heat capacity, and many more, are calculated in a self-
85
consistent way by derivatives from this Gibbs potential. The improved density
86
algorithm provided by Jackett et al. (2006) shows only minimal adjustments
87
with respect to Jackett and McDougall (1995). However, because of the consis-
88
tency of the derived thermodynamic properties and the significantly reduced
89
S = 35 psu
0 1000 2000 3000 4000 5000 6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
0 1000 2000 3000 4000 5000 6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
1028 1032 1032
1036 1036
1040 1040
1044 1048
S = 0 psu
0 1000 2000 3000 4000 5000 6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
0 1000 2000 3000 4000 5000 6000
Depth (m)
−5 0 5 10 15 20 25 30 35 40
Temperature (°C)
1000 1000
1004 1004
1008 1008
1012 1012
1016 1016
1020 1024
Jackett & McDougall (1995) Haidvogel & Beckmann (1999)
Jackett et al. (2006) 0 1 2 3
Difference (kg/m3)
Fig. 1. Density (kg/m3) as a function of depth and potential temperature for oceanic water masses (left) and fresh water (right). The blue and green lines, which are quite close together, refer to Jackett and McDougall (1995) and Jackett et al.
(2006), respectively, while the red lines refers to Haidvogel and Beckmann (1999).
The background color indicates the increasing difference between the pressure- and depth-dependent density according to Jackett and McDougall (1995) and Haidvogel and Beckmann (1999).
computational effort, the implementation of the Gibbs-potential algorithms in
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ocean models is the preferred formulation.
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1.2 Equation of freezing point (EoFP)
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For an adequate treatment of the ice-water interaction the equations for the conservation of temperature and salinity are complemented by an equation to calculate the pressure- and salinity-dependent freezing point of water (EoFP, e.g., Holland and Jenkins, 1999)
Tf =Tf(S, p)≈αS+β+γp, (3) where α = 0.057◦C/psu, β = 0.0939◦C, and γ = 7.64·10−4◦C/dbar. For
93
an analytic solution of the complete set of the three equations a linearized
94
version of theEoFPis needed as indicated on the right hand side of (3). This
95
set of coefficients dating back to Foldvik and Kvinge (1974) is still in use in
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models dealing with ice-water interaction and has not always been replaced
97
by a linearised version of the more precise (but higher order) formulation
98
of Fofonoff and Millard (1983). One drawback of (3) is the need for regular
99
temperature conversions betweenT and the models intrinsic variableθ. Also,
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the EoFP (3) was not designed for the high-pressure, low-salinity environ-
101
ment within subglacial lakes, which are covered by several thousand meters
102
of ice (Feistel, 2003, 2008). Jackett et al. (2006) present an algorithm to cal-
103
culate the freezing point in terms of the potential temperature θf =θf(S, p),
104
based on the Gibbs-potential considerations of Feistel (2003). This formu-
105
lation of the EoFP is also valid for high-pressure environments found in
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subglacial lakes. To make this formulation applicable with the analytic so-
107
lution of the three-equation formulation, it has to be linearised with respect
108
to the specific environmental needs (S ∼mean-salinity-at-ice-water-interface,
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p ∼ mean-interface-depth). For subglacial Lake Vostok (with S = 0 psu and
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p ≈3700 m) the adjusted linearized equation (3) is indicated by the red line
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in Figure 2, while the original freezing point line (according to Jackett et al.,
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2006) is drawn in black.
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2 Relevance for subglacial lake modelling
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In former studies of subglacial lake circulation, different formulations of the
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EoShave been applied. In the first three-dimensional numerical model studies
116
of Lake Vostok, the simplistic Knudsen-Ekman equation was used (Williams,
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2001; Mayer et al., 2003). Later studies dealing with Lake Vostok and Lake
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Concordia (Thoma et al., 2007, 2008a,b, 2009) applied the improved depth-
119
dependent EoSafter Haidvogel and Beckmann (1999). However, Figure 1 in-
120
dicates that in the fresh-water regime of subglacial lakes the application of this
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convenient approach is questionable. Although the absolute densities are quite
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similar (Figure 1), the different vertical gradient and in particular the resulting
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significantly different isopycnal-vertices determine the characteristics of flow
124
and basal mass balance within subglacial lakes. The line of maximum density
125
(LoMD) connects the vertices of the isopycnals, indicated as a dashed line in
126
Figure 2. The LoMD determines if warming leads to rising of water masses
127
or sinking. By using the improved Gibbs-potential formulation, theLoMD is
128
moved to a greater depth compared to the Haidvogel and Beckmann (1999)
129
approach. However, as long as the a lake’s depth below the ice surface remains
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well below the LoMD in Figure 2, the principle circulation regime doesn’t
131
change (Thoma et al., 2008b).
132
3 Updated subglacial lake model results
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The most up-to date model to simulate the three-dimensional flow regime and
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the basal mass balance within subglacial lakes isRombax(Thoma et al., 2007,
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0
500
1000
1500
2000
2500
3000
3500
4000
4500
Water Depth (m)
−3 −2 −1 0
Temperature (°C) solid
fluid
Lake Vostok Lake Ellsworth Lake Concordia
1000 kg/m3
1002.5 kg/m3
1005 kg/m3
1007.5 kg/m3
1010 kg/m3
1012.5 kg/m3
1015 kg/m3
1017.5 kg/m3
1020 kg/m3
1022.5 kg/m3
warming
➼ sinking
warming
➼ raising 0
500
1000
1500
2000
2500
3000
3500
4000
4500
Water Depth (m)
−3 −2 −1 0
Temperature (°C)
Fig. 2. Water depth and potential temperature dependence of isopycnals (Feistel, 2003; Jackett et al., 2006). The black solidus line shows the depth-dependent freez- ing point of fresh water (Feistel, 2003; Jackett et al., 2006), the red solidus line in- dicates the linearized form of the freezing point equation adjusted for Lake Vostok.
The dashed line connects the isopycnal’s vertices and indicates the line of maximum density (LoMD). The dotted gray line indicates the former LoMD according to Haidvogel and Beckmann (1999) as published in Thoma et al. (2008b). Coloured dots show the captured space of potential temperatures and equivalent water depth for Lake Vostok, Lake Concordia, and Lake Ellsworth, respectively. Dots within the grey shaded area above the solidus line represent supercooled water masses with freezing capability.
2008a,b, 2009). In order to investigate the impact of the improved formulations
136
of theEoSand theEoFP, we repeated the most important model runs of our
137
former studies and reanalyse the results. The model set-up for Lake Vostok
138
uses the bathymetry model of Filina et al. (2008). The corresponding bound-
139
ary conditions are described in detail in Thoma et al. (2007, 2008a,b). In addi-
140
tion to the previously applied geothermal heat flux of 54 mW/m2 Maule et al.
141
(2005), which is based on the interpretation of satellite magnetic data, we
142
also apply a value of 48 mW/m2, from the interpretation of seismic data
143
(Shapiro and Ritzwoller, 2004). This allows us to estimate an uncertainty for
144
Lake Vostok, with respect to this parameter, as specified in Table 1.
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The model set-up for Lake Concordia is fully described in Thoma et al. (2009).
146
Here we only present the updated results with respect to the revisedEoSand
147
EoFP with otherwise identical configurations. Since Lake Vostok and Lake
148
Concordia are still located well below the line of maximum density (LoMD,
149
Figure 2), no fundamental regime shifts are observed. However, the absolute
150
values of the modelled flow, the basal mass balance, as well as the derived
151
distributions of the accreted ice at the ice-lake interface, and the lake water
152
residence times do change slightly. In Table 1 we present updates of the most
153
relevant results and their uncertainties for Lake Vostok and Lake Concordia
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published in the aforementioned studies. A complete set of Figures indicating
155
the circulation, temperature regime, basal mass balance, and the distribu-
156
tion and thickness of accreted ice for Lake Vostok and for Lake Concordia is
157
presented in the supplemental material.
158
4 Summary and implications for future subglacial lake studies
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The general circulation regime within subglacial lakes is generated by buoy-
160
ancy forces, originating from the geothermal heat flux and the thermodynamic
161
interactions at the ice-lake interface. However, the specific flow as well as the
162
basal mass balance of any lake is determined by its complex bathymetry and
163
the steepness of the ice-lake interface slope. This makes reliable generalized
164
predictions of any specific values for an individual lake impossible; each lake
165
must be considered individually.
166
The buoyancy force, which drives the flow within subglacial lakes, depends
167
very much on theEoS. According to W¨uest and Carmack (2000) and Thoma et al.
168
(2008b), a fundamental regime shift is observed when the LoMD is ap-
169
proached or crossed. With respect to this, the previous results on subglacial
170
Lake Vostok and subglacial Lake Concordia do not change in their general
171
aspects, but in their specific quantities. In contrast, the recently investigated
172
Lake Ellsworth (Woodward et al., 2009) provides a rather different situation
173
Compared to many other subglacial lakes, Lake Ellsworth is covered by a
174
Table 1
Revised values for important modelled results within subglacial lakes with respect to improved versions of theEoSand theEoFP. The uncertainties are derived from model runs with varying boundary conditions.
Lake Vostok Lake Concordia
Min. stream func. (mSv) −11.6±0.1 −0.10±0.01
Max. stream func. (mSv) +22.5±0.1 +0.11±0.01
Merid. overturning (µSv) (±1.8±0.1)·103 −14.7±0.1 Zonal overturning (µSv) (−11.6±0.1)·103 +55.6±0.4
Velocity (horizontal) (mm/s) O1 O0.1
(vertical) (µm/s) O10 O1
Turb. kin. energy (10−2cm2/s2) 1.9±0.1 (3.52±0.05)·10−2
Freezing area (km2) 5212±85 115±55
Mean melt rate (mm/a) 16.8±0.3 3.8±1.2
Mean freeze rate (mm/a) 24.7±0.3 1.3±0.2
Fresh water gain (10−1m3/s) 15.7±1.6 0.57±0.27 Basal ice loss (10−2km3/a) 5.0±0.5 0.18±0.09
Accreted ice area (km2) 11 000±500 125±55
volume (km3) 855±20 2.0±1.6
average thickness (m) 70±10 12±7
Melting rate in meteoric area (mm/a) 17.0±0.4 3.8±1.1 Lake water residence time (ka) 51.7±5.6 18.9±7.4
thinner ice sheet, moving it towards the LoMD (Figure 2). Additionally, the
175
slope of the ice-lake interface is significantly larger (about 1.9%) compared to
176
Lake Vostok or Lake Concordia (about 0.4%), which will have its impact on
177
the basal mass balance. A future detailed modelling study of subglacial Lake
178
Ellsworth will show this in detail.
179
Acknowledgements: This work was funded by the DFG through grant
180
MA33471-2. The authors wish to thank Aike Beckmann, Rainer Feistel, R¨udiger
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Gerdes, Kate Hedstrom, Adrian Jenkins, Martin Losch, Trevor McDougall,
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and Ralph Timmermann for helpful comments and discussions.
183
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Corr, H. F. J., King, E. C., Grosfeld, K., 2009. Bathymetry of Subglacial
280
Lake Ellsworth, West Antarctica and implications for lake access. Geophys.
281
Res. Lett. in preparation.
282
W¨uest, A., Carmack, E., 2000. A priori estimates of mixing and circulation
283
in the hard-to-reach water body of Lake Vostok. Ocean Modelling 2 (1),
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29–43.
285
Supplemental material for A comment on the
1
equation of state and the freezing point
2
equation with respect to subglacial lake
3
modelling
4
Malte Thoma
a,bKlaus Grosfeld
aAndrew M. Smith
c5
Christoph Mayer
b6
aAlfred Wegener Institute for Polar and Marine Research,
7
Bussestrasse 24, 27570 Bremerhaven, Germany
8
bBayerische Akademie der Wissenschaften, Kommission f¨ur Glaziologie,
9
Alfons-Goppel-Str. 11, 80539 M¨unchen, Germany
10
c British Antarctic Survey, High Cross, Madingley Road, Cambridge, CB3 0ET,
11
United Kingdom
12
1 Introduction
13
The application of the Gibbs thermodynamic potential for the formulation
14
of the Equation of State (EoS) and the Freezing Point Equation (EoFP)
15
enables a consistent description for their application to ocean and/or sub-
16
glacial lake flow models. As already discussed in the corresponding article, the
17
general pattern of subglacial lake circulation, melting and freezing, and the
18
thermal regime remains unchanged, but their quantitative structure adapts
19
to the new formulations. While these revised quantities are published in the
20
corresponding paper, we supply a new set of figures for subglacial Lake Vos-
21
tok as well as for subglacial Lake Concordia in order to update the results
22
shown in Thoma et al. (2007), Thoma et al. (2008b), Thoma et al. (2008a),
23
and Thoma et al. (2009).
24
Email address: Malte.Thoma@awi.de(Malte Thoma).
2 Lake Vostok
25
For the figures in this section the most up-to date bathymetry model of
26
Filina et al. (2008) as well as a geothermal heat flux of 48 mW/m2 is applied.
27
All other model parameters as well as boundary conditions are fully described
28
in the corresponding publications.
29
−258˚
−258˚
−256˚
−256˚
−254˚
−254˚
−252˚
−252˚
−78.5˚ −78.5˚
−78˚ −78˚
−77.5˚ −77.5˚
−77˚ −77˚
−76.5˚ −76.5˚
−76˚ −76˚
3900
3900 3900 3900
3900 3900 4100 4100
4100
4100
4100
4100 4100
4100 4300
4300 4300 4300
4500
4500
0 20 40
3500 4000 4500 5000 Bedrock
m
Vostok Station
a)
−258˚
−258˚
−256˚
−256˚
−254˚
−254˚
−252˚
−252˚
−78.5˚ −78.5˚
−78˚ −78˚
−77.5˚ −77.5˚
−77˚ −77˚
−76.5˚ −76.5˚
−76˚ −76˚
30 30
70 90
200
200 200
200
200 200
200
400
400 400
400 600
600 600
800
800
0 20 40
4995 9990
Water Column m
Vostok Station
b)
Fig. 1. Bedrock topography (a) and water column thickness (b) of Lake Vostok.
The solid red line indicates the track along the cross sections shown in Figures 4b.
−258˚
−258˚
−256˚
−256˚
−254˚
−254˚
−252˚
−252˚
−78.5˚ −78.5˚
−78˚ −78˚
−77.5˚ −77.5˚
−77˚ −77˚
−76.5˚ −76.5˚
−76˚ −76˚
−5
−5 5
5
5 5 5
5
15
15 15
0 20 40
−25 0 25
Stream Function mSv
Vostok Station
a)
−258˚
−258˚
−256˚
−256˚
−254˚
−254˚
−252˚
−252˚
−78.5˚ −78.5˚
−78˚ −78˚
−77.5˚ −77.5˚
−77˚ −77˚
−76.5˚ −76.5˚
−76˚ −76˚
−25−20−15−10
−5
−5 0
0
0 0
0
0
0
0 0
0
5
5 5 1.0 mm/s
0 20 40
−25 0 25
Vertical Velocity µm/s
Vostok Station
b)
Fig. 2. a) Vertically integrated mass transport stream function (1 mSv = 103m3/s).
b) Integrated vertical velocity, arrows indicate the flow in the lake’s bottom layer.
−258˚
−258˚
−256˚
−256˚
−254˚
−254˚
−252˚
−252˚
−78.5˚ −78.5˚
−78˚ −78˚
−77.5˚ −77.5˚
−77˚ −77˚
−76.5˚ −76.5˚
−76˚ −76˚
−6
−6
−4
−4
−4
−4 −4
−2
−2
−2−2
−2
−2
−2
−2
−2
−2
−2 −2
−2
−2
−2
−2
−2 0
0
0
0 0 0
2 2
4
4
4 6 108
1.0 mm/s
0 20 40
−10 0 10
Basal Mass Balance cm/a
Vostok Station
a)
−258˚
−258˚
−256˚
−256˚
−254˚
−254˚
−252˚
−252˚
−78.5˚ −78.5˚
−78˚ −78˚
−77.5˚ −77.5˚
−77˚ −77˚
−76.5˚ −76.5˚
−76˚ −76˚
−2.89
−2.87
−2.85
−2.85
−2.83
−2.81
0 20 40
−2.9 −2.8 −2.7 Pot. Temperature
°C
Vostok Station
b)
Fig. 3. a) Modelled basal mass balance at the ice–lake interface. Negative values (blue/green) indicate melting, positive (yellow/red) values freezing. Velocities in the ice–lake boundary layer are indicated by arrows. b) Modelled temperatures at the ice–lake interface. The solid red line indicates the track along the cross sections shown in Figures 4b.
3500
4000
4500
Depth (m)
−78 −77 −76
Latitude
−1
−1
0 01
1
−2 0 2
Meridional Overturning
mSv 3500
4000
4500
Depth (m)
102 104 106 108
Longitude
−11−9
−7
−5
−5
−1−3
−15 0 15
Zonal Overturning
mSv
a)
3500
4000
4500
Depth (m)
0 100 200 300
Distance (km)
−2.85
−2.85
−2.83
−2.9 −2.8 −2.7 Pot. Temperature
°C
b)
Fig. 4. a) Zonal and meridional overturning stream functions (1 mSv = 103m3/s). b) South-north temperature cross section across Lake Vostok along the track indicated in Figure 1 and 3b.
−258˚
−258˚
−256˚
−256˚
−254˚
−254˚
−252˚
−252˚
−78.5˚ −78.5˚
−78˚ −78˚
−77.5˚ −77.5˚
−77˚ −77˚
−76.5˚ −76.5˚
−76˚ −76˚
1 1
1 1
5 5
5
20 20
20
20 100
100
100 100 100
0 20 40
100 101 102 103 Accreted Ice
m
Vostok Station
Fig. 5. Modelled accreted ice thickness (in meter) at the ice–lake interface The corresponding ice flow direction is indicated. The horizontal flow velocity is assumed to be 3.7 m/a, which results in 210 m of accreted ice, as measured at Vostok Station.
This value is within the proposed measured velocities of about 1.9 and 4.2 m/a (e.g., Kwok et al., 2000; Bell et al., 2002; Tikku et al., 2004; Wendt, 2005).
3 Lake Concordia
30
For the model output in this section the bathymetry model presented in
31
Thoma et al. (2009) as well as a geothermal heat flux of 57 mW/m2(Maule et al.,
32
2005; Tikku et al., 2005) and a heat flux into the ice sheet of 28.6 mW/m2
33
(Thoma et al., 2009) is applied.
34
−235.4˚ −235.2˚−235.2˚ −235˚−235˚ −234.8˚−234.8˚ −234.6˚−234.6˚
−74.2˚ −74.2˚
−74.1˚ −74.1˚
−74˚ −74˚
−73.9˚ 3700 −73.9˚
3750 3750
3750 3750
3750
3750
3750
3800
3800
3800
3800 3800
3800 3800
3800
0 4 8
3600 3750 3900 Bedrock
m
a)
−235.4˚ −235.2˚−235.2˚ −235˚−235˚ −234.8˚−234.8˚ −234.6˚−234.6˚
−74.2˚ −74.2˚
−74.1˚ −74.1˚
−74˚ −74˚
−73.9˚ −73.9˚
50
50
50
50
50 50
50 50
50
100
100
100 100
100
0 4 8
0 75 150
Water Column m
b)
Fig. 6. Bedrock topography (a) and water column thickness (b) of Lake Concordia.
The solid red line indicates the track along the cross sections shown in Figures 9b.
−235.4˚ −235.2˚−235.2˚ −235˚−235˚ −234.8˚−234.8˚ −234.6˚−234.6˚
−74.2˚ −74.2˚
−74.1˚ −74.1˚
−74˚ −74˚
−73.9˚ −73.9˚
−80 −80
−60
−60
−60
−40
−40
−40
−20
−20
−20
−20 0
0
0
0
0
0 0
0 0
20
20
20
20 20
20 40
40
40 40
40 60
60
60 60
60
80
80 80
100 100
0 4 8
−120 0 120
Stream Function µSv
a)
−235.4˚ −235.2˚−235.2˚ −235˚−235˚ −234.8˚−234.8˚ −234.6˚−234.6˚
−74.2˚ −74.2˚
−74.1˚ −74.1˚
−74˚ −74˚
−73.9˚ −73.9˚
−0.8
−0.8
−0.6
−0.6
−0.6
−0.4
−0.4
−0.4
−0.4
−0.4
−0.2
−0.2
−0.2
−0.2
−0.2 0
0
0
0
0 0.2
0.2
0.2 0.2
0.2 0.2 0.2 0.2
0.4
0.4 0.4 0.4
0.4
0.4
0.2 mm/s
0 4 8
−1 0 1
Vertical Velocity µm/s
b)
Fig. 7. a) Vertically integrated mass transport stream function (1 Sv = 10 m3/s).
b) Integrated vertical velocity, arrows indicate the flow in the lake’s bottom layer.
−235.4˚ −235.2˚−235.2˚ −235˚−235˚ −234.8˚−234.8˚ −234.6˚−234.6˚
−74.2˚ −74.2˚
−74.1˚ −74.1˚
−74˚ −74˚
−73.9˚ −73.9˚
−9
−9
−9
−9 −6
−6
−6
−6
−6
−6
−3
−3 0
0.2 mm/s
0 4 8
−15 0 15
Basal Mass Balance mm/a
a)
−235.4˚ −235.2˚−235.2˚ −235˚−235˚ −234.8˚−234.8˚ −234.6˚−234.6˚
−74.2˚ −74.2˚
−74.1˚ −74.1˚
−74˚ −74˚
−73.9˚ −73.9˚
−2.834
−2.832
−2.83
−2.828
−2.826
−2.824
−2.822
−2.82
−2.818
−2.816
0 4 8
−2.84 −2.83 −2.82 −2.81 Pot. Temperature
°C
b)
Fig. 8. a) Modelled basal mass balance at the ice–lake interface. Negative values (blue/green) indicate melting, positive (yellow/red) values freezing. Velocities in the ice–lake boundary layer are indicated by arrows. b) Modelled temperatures at the ice–lake interface. The solid red line indicates the track along the cross sections shown in Figures 9b.
3650 3700 3750 3800 3850
Depth (m)
−74.3 −74.2 −74.1 −74.0 −73.9
Latitude
−12
−8
−8
−8
−4 −4
−20 0 20
Meridional Overturning
µSv 3650
3700 3750 3800 3850
Depth (m)
124.6 124.8 125.0 125.2 125.4
Longitude
10
20 20
20 30
30 40 5040
−60 0 60
Zonal Overturning
µSv
a)
3650
3700
3750
3800
Depth (m)
0 10 20 30 40
Distance (km)
−2.83 −2.824
−2.84 −2.83 −2.82 −2.81 Pot. Temperature
b) °C
Fig. 9. a) Zonal and meridional overturning stream functions (1 Sv = 10 m3/s).
b) South-north temperature cross section across Lake Concordia along the track indicated in Figure 6 and 8b.
−235.4˚ −235.2˚−235.2˚ −235˚−235˚ −234.8˚−234.8˚ −234.6˚−234.6˚
−74.2˚ −74.2˚
−74.1˚ −74.1˚
−74˚ −74˚
−73.9˚ −73.9˚
1 5
20
0 4 8
100 101 102
Accreted Ice m
Fig. 10. Modelled accreted ice thickness (in meter) at the ice–lake interface The corresponding ice flow line direction is east-northeastward. The horizontal ice flow velocity is assumed to be 25 cm/a (Tikku et al., 2005).
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