The Cryosphere, 4, 1–12, 2010
www.the-cryosphere.net/4/1/2010/
© Author(s) 2010. This work is distributed under
the Creative Commons Attribution 3.0 License.
The Cryosphere
Interaction between ice sheet dynamics and subglacial lake
circulation: a coupled modelling approach
M. Thoma1,2 , K. Grosfeld2 , C. Mayer1 , and F. Pattyn3
1 Bavarian
Academy and Sciences, Commission for Glaciology, Alfons-Goppel-Str. 11, 80539 Munich, Germany
Wegener Institute for Polar and Marine Research, Bussestrasse 24, 27570 Bremerhaven, Germany
3 Laboratoire de Glaciologie, Département des Sciences de la Terre et de l’Environnement (DSTE), Université Libre de
Bruxelles (ULB), CP 160/03, Avenue F.D. Roosevelt, 1050 Bruxelles, Belgium
2 Alfred
Received: 15 September 2009 – Published in The Cryosphere Discuss.: 29 September 2009
Revised: 15 December 2009 – Accepted: 15 December 2009 – Published: 8 January 2010
Abstract. Subglacial lakes in Antarctica influence to a large
extent the flow of the ice sheet. In this study we use an idealised lake geometry to study this impact. We employ a) an
improved three-dimensional full-Stokes ice flow model with
a nonlinear rheology, b) a three-dimensional fluid dynamics
model with eddy diffusion to simulate the basal mass balance at the lake-ice interface, and c) a newly developed coupler to exchange boundary conditions between the two individual models. Different boundary conditions are applied
over grounded ice and floating ice. This results in significantly increased temperatures within the ice on top of the
lake, compared to ice at the same depth outside the lake area.
Basal melting of the ice sheet increases this lateral temperature gradient. Upstream the ice flow converges towards the
lake and accelerates by about 10% whenever basal melting at
the ice-lake boundary is present. Above and downstream of
the lake, where the ice flow diverges, a velocity decrease of
about 10% is simulated.
1
Introduction
During the last decades our knowledge on subglacial lake
systems has greatly increased. Since the discovery of the
largest subglacial lake, Lake Vostok (Oswald and Robin,
1973; Robin et al., 1977), more than 270 other lakes have
been identified so far (Siegert et al., 2005; Carter et al.,
2007; Bell, 2008; Smith et al., 2009). Plenty of efforts have
been undertaken to reveal partial secrets subglacial lakes may
hold, mostly referring to Lake Vostok. For instance the speculation that extremophiles in subglacial lakes may be encountered (e.g., Duxbury et al., 2001; Siegert et al., 2003) has
Correspondence to: M. Thoma
([email protected])
been nurtured by microorganisms discovered in ice core samples (Karl et al., 1999; Lavire et al., 2006). These samples
originate from the at least 200 m thick accreted ice, drilled
at the Russian research station Vostok (Jouzel et al., 1999).
Discussions about the origin and history of the lake (Duxbury
et al., 2001; Siegert, 2004; Pattyn, 2004; Siegert, 2005) are
still ongoing. It is still unknown whether Subglacial Lake
Vostok existed prior to the Antarctic glaciation and if it could
have survived glaciation, or whether it formed subglacially
after the onset of glaciation. The impact of subglacial lakes
on the flow of the overlying ice sheet has been analysed
(Kwok et al., 2000; Tikku et al., 2004, e.g.,) and modelled
(Mayer and Siegert, 2000; Pattyn, 2003; Pattyn et al., 2004;
Pattyn, 2008), but self-consistent numerical models, which
include the modelling of the basal mass balance, are lacking
at present. Several numerical estimates and models (West
and Carmack, 2000; Williams, 2001; Walsh, 2002; Mayer
et al., 2003; Thoma et al., 2007, 2008b; Filina et al., 2008)
as well as laboratory analogues (Wells and Wettlaufer, 2008)
of water flow within the lake have been carried out, permitting insights into the water circulation, energy budget and
basal processes at the lake-ice interface. Finally, observations (Bell et al., 2002; Tikku et al., 2004) and numerical
modelling (Thoma et al., 2008a) of accreted ice at the lakeice interface gave insights about its thickness and distribution
over subglacial lakes.
Early research assumed that subglacial lakes were isolated systems, while only recently Gray et al. (2005), Wingham et al. (2006) and Fricker et al. (2007) found evidence
that Antarctic subglacial lakes are connected, hence forming an extensive subglacial hydrological network. Several
plans to unlock subglacial lakes exist (Siegert et al., 2004;
Inman, 2005; Siegert et al., 2007; Schiermeier, 2008; Woodward et al., 2009) and valuable knowledge will be available
as soon as direct samples of subglacial water and sediments
are taken.
Published by Copernicus Publications on behalf of the European Geosciences Union.
2
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
However, current knowledge about the interaction between subglacial lakes and the overlying ice sheet is lacking. The most important parameter exchanged between ice
and water is heat. The exchange of latent heat associated
with melting and freezing dominates heat conduction, but
the latter process has also be accounted for in subglacial lake
modelling (Thoma et al., 2008b). Melting and freezing is
closely related to the ice draft which varies spatially over
subglacial lakes (Siegert et al., 2000; Studinger et al., 2004;
Tikku et al., 2005; Thoma et al., 2007, 2008a, 2009a). The
ice draft, and hence the water circulation within the lake, is
maintained by the ice flow across lakes. Without this flow,
lake surfaces would even out (Lewis and Perkin, 1986). On
the other hand, a spatially varying melting/freezing pattern
will have an impact on the overlying ice sheet as well. In order to get an insight in the complex interaction processes between the Antarctic ice sheet and subglacial lakes, this study
for the first time couples a numerical ice-flow and a lake-flow
model with a simple asynchronous time-stepping scheme to
overcome the problem of different adjustment time scales.
We describe the applied ice-flow model R IMBAY and lakeflow model ROMBAX in the first two Sections, respectively.
Each section starts with a general description of the particular model, describes the applied boundary conditions, and
the results of the uncoupled model runs. The newly developed R IMBAY–ROMBAX–coupler R IROCO is introduced in
Sect. 4, where we also discuss the impact of a coupled icelake system on the ice flow and lake geometry.
2
2.1
Ice model R IMBAY
General description
The ice sheet model R IMBAY (Revised Ice sheet Model
Based on frAnk pattYn) is based on the work of Pattyn
(2003), Pattyn et al. (2004) and Pattyn (2008). Within this
thermomechanically coupled, three-dimensional, full-Stokes
ice model, a subglacial lake is represented numerically by a
vanishing bottom-friction coefficient of β 2 = 0, while high
friction in grounded areas is represented by a large coefficient; in this study we use β 2 = 106 . The choice of the latter
parameter has no influence on the model results as long as it
is much larger than zero.
The constitutive equation, governing the creep of polycrystalline ice and relating the deviatoric stresses τ to the
strain rates ε̇, is given by Glen’s flow law: ε̇ = Aτ n (e.g.,
Pattyn, 2003), with a temperature dependent rate factor A =
A(T ). Here we apply the so-called Hooke’s rate factor
(Hooke, 1981). Experimental values of the exponent n in
Glen’s flow law vary from 1.5 to 4.2 with a mean of about 3
(Weertman, 1973; Paterson, 1994); ice models traditionally
assume n = 3. However, a simplified viscous rheology with
n = 1 stabilises and accelerates the convergence behaviour
The Cryosphere, 4, 1–12, 2010
of the implemented numerical solvers. Therefore, previous
subglacial lake simulations within ice models were limited
to viscous rheologies (Pattyn, 2008).
2.2
Model setup and boundary conditions
The model domain used in this study is a slightly enlarged
version of the one presented in Pattyn (2003): It consists of a
rectangular 168 100 km2 large domain with a model resolution of 5 km (resolutions of 2.5 km and 10 km are also used
for comparison) and 41 terrain-following vertical layers. The
surface of the initially 4000 m thick ice sheet has an initial
slope of 2% (similar to Lake Vostok, Tikku et al., 2004) from
left (upstream) to right (downstream). An idealized circular lake with a radius of about 48 km and an area of about
7200 km2 is located in the center of the domain, where a
1000 m deep cavity modulates the otherwise smoothly sloped
bedrock (similar to Pattyn, 2008). The lake’s maximum water depth of about 600 m, resulting in a volume of about
1840 km3 (the lake’s geometry is also indicated in Figs. 3, 4).
We apply a constant surface temperature of −50 ◦ C at the
ice’s surface, a typical value for central Antarctica (Comiso,
2000). The bottom layer boundary temperature depends on
the basal condition: above bedrock a Neumann boundary
condition is applied, based on a geothermal heat flux of
54 mW/m2 (a value suggested for the Lake Vostok region
by Maule et al., 2005). Assuming isostasy above the subglacial lake, the bottom layer temperature is at the pressuredepended freezing point. This temperature, prescribing a
Dirichlet boundary condition, depends on the local ice sheet
thickness (Tb = −H ×8.7×10−4 ◦ C/m ,e.g., Paterson, 1994).
Accumulation and basal melting/freezing are ignored during
the initial experiments in this Sect., where no coupling is applied. In the subsequent coupling-experiments, basal melting
and freezing at the lake’s interface, as modelled by the lake
flow model, is accounted for (Sect. 4).
The lateral boundary conditions are periodic, hence values
of ice thicknesses, velocities, and stresses are copied from
the upstream (left) side to the downstream (right) side of the
model domain and vice versa. The same applies to the lateral
borders along the flow. The initial integration starts from
an isothermal state. The basal melt rate is set to zero over
bedrock as well as over the lake.
2.3
Model improvements
Compared to Pattyn (2008) we improved the model R IMBAY
in two significant ways to allow a numerically stable and fast
representation of the more realistic non-linear flow law with
an exponent of n = 3: First, a gradual increase of the friction
coefficient β 2 at the lake’s boundaries is considered (Fig. 1a).
Physically, this smoothing can be interpreted as a lubrication
of the ice sheet base on the grounded side and as stiffening
due to debris on the lake side of the lake’s edge. Our experiments have shown, that a slight prescribed β 2 -smoothing
www.the-cryosphere.net/4/1/2010/
0.6
M. Thoma
et al.: Coupled
and lake-flow
M. Thoma
et al.: Coupled
ice- andicelake-flow
Weight factor
Scaled friction coefficient
0.8
3
3
M. Thoma
et al.: Modelled interaction between ice sheet and subglacial lakes
0.6
0.8
0.6
0
0.4
50
0.4
100
0.2
150
200
250
300
350
0.2
0.2
0.4
0.4
0.0
400
X−distance (km)
b)
0.6
0.6
(3/2) Smoothed β2
(1/0) Smoothed β2
(0/0) Unsmoothed β2
0.0
0.8
b)
Weight factor
0.6
a)
Weight factor
0.2
0.8
Scaled friction coefficient β2
Scaled friction coefficient β2
0.4
0.4
0.8
1.0
a)
1.0
3
−5
−4
−3
−2
−1
0
1
2
3
4
5
Node distance
0.2
0.2
(3/2) Smoothed
β2
(3/2) Smoothed
β2
2
6
Smoothed
β2 the maximum of 10 ) along the central x-axis for an unsmoothed and two test-smoothed cases.
Fig. 1. a) Friction coefficient
β(1/0)
(scaled
to
(1/0) Smoothed
β2
(0/0) Unsmoothed
β2
(0/0) Unsmoothed
β2
The smoothing coefficient
outside
and inside of the lake’s border, respectively.
0.0 (x/y) represents the number of nodes smoothed
0.0
0.0
0.0
50 150
100
150
200
250 350
300 400
350
b) Weight factors for 0a Gaussian-type
filter
with250
different
width
from 400
one−5(red)
to −3
five
(cyan)
500 100
200
300
−4 −2
−3 −1−2depending
1 the
5 central node
−4−5
0 −1 1 0 2on
3 2 distance
4 3 5 4 to the
X−distance
located at zero.
X−distance
(km) (km)
distance
Node Node
distance
2
6 of 106 ) along the central x-axis for an unsmoothed and two test-smoothed cases.
Fig. 1. a) Friction coefficient
β 2 (scaled to the maximum
Fig. 1. a) Friction coefficient
β (scaled to the maximum of 10 ) along
the central x-axis for an unsmoothed and two test-smoothed cases.
2 (scaled
6 ) along
Fig. 1. (a) Friction
coefficient β
to the maximum
of 10nodes
the central
x-axis for
anlake’s
unsmoothed
and two test-smoothed cases.
The smoothing
coefficient
(x/y) represents
the of
number
smoothed
outside
andofinside
of theborder,
border, respectively.
The smoothing
coefficient
(x/y) represents
the number
nodesofsmoothed
outside and
inside
the lake’s
respectively.
The smoothing
coefficient
(x/y)
represents
thewith
number
nodes
smoothed
and
inside
of the
thedistance
lake’s
respectively.
b) factors
Weightfor
factors
for
a Gaussian-type
filter
withof
different
width
(red)
to five
(cyan)
depending
on
the distance
to the
central node (b) Weight
b) Weight
a Gaussian-type
filter
different
width
from
onefrom
(red)one
tooutside
five (cyan)
depending
on
toborder,
the central
node
located
factors for alocated
Gaussian-type
filter with different width from one (red) to five (cyan) depending on the distance to the central node located at
at
zero. at zero.
zero.
a)
b)
Smooth β2:
(0/0)
Gaussian filter: 0
c)
Smooth β2:
(1/0)
Gaussian filter: 2
a)
a)
b)
b)
2:
Smooth
(0/0)
Smooth β2:
(0/0) β filter:
0
Gaussian filter:Gaussian
0
Smooth β2:
(3/2)
Gaussian filter: 5
c)
c)
2:
(1/0)
Smooth β2: Smooth
(1/0) β filter:
2
Gaussian filter:Gaussian
2
2:
(3/2)
Smooth β2: Smooth
(3/2) β filter:
5
Gaussian filter:Gaussian
5
0
0
0
0
100
150
20 )
m
Y (k 250
300
400
350
2019
200
250
m)
100
X (k
150
350
300
200
2120
2221
2322
2423
100
24
250
400
300
m)
X (k
50
log(η)
log(η)
0
400
19
150
50
350
0
100
0
400
50
50
β10
300
400
m)
X (k
250
400
0
350
350
150
350
m)
100
300
300
0
5
10810
200
0
400
300
105β
350
300
250
250
200
200
0
m)0
m) 15 X (k25
150
100
X (k
50
200
250
400
400
86
Y (k
350
400
400
64
m)
300
350
100
50
0
150
200
250
m)
400
400
350
105β
42
Y (k
200
400
350
300300
350
350
300
0 m)
X (k
50
20
350
150
150
Y (k
300
100
100
100
2
25050
300
300
400
0
0
0
20
200 m)
Y (k m)
250
m)
250
250
m)
m)
Y (k
150
Y (k
200
200
150
200
Y (k
Y (k
150
100 150
100
100
150
15
0
10 0
20
00
35
40
250
50
50
50
1
50 00
50
50
100
0
50
0
200
m)
X (k
50
0
0
0
50
0
0
50
100
0
300
35
250
0
200 30 ) 150
0 0
m
1525
100
X (k
50
150
350
200
400
250
300
350
400
m)
X (k
log(η)
2 24
4 of viscosity
8of the layer
10 andlayer
19
20an
22 model.
2. 2Logarithm
surface
and bedrock-friction
parameter
β idealized
for21
an idealized
No β 2 –smoothing
Fig. 2.0Fig.
Logarithm
of viscosity
η6of theηsurface
bedrock-friction
parameter
β 2 for
model.
a)23
No βa)
–smoothing
2
2
or Gaussian
β 2 –smoothing
and Gaussian
with width
a filterofwidth
c) βStrong
β 2 –smoothing
(3/2) and
or Gaussian
filtering.filtering.
b) Slightb)βSlight
–smoothing
(1/0) and(1/0)
Gaussian
filtering filtering
with a filter
two. of
c) two.
Strong
–smoothing
(3/2) and
filtering
withwidth
a filter
width
of five.
that the represent
Figures represent
theconditions
initial conditions
for a FS-model
the filter
applied
filter and22
22afor
GaussianGaussian
filtering
with a filter
of
five. Note
thatNote
the
Figures
the initial
for
FS-model
with thewith
applied
and
Fig.
of
viscosity
η
of
the
surface
layer
and
bedrock-friction
parameter
β
for
an
idealized
model.
a)
No ββ -smoothing
–smoothing
Fig.2.2. Logarithm
Logarithm
of
viscosity
η
of
the
surface
bedrock-friction
parameter
β
an
idealized
model.
(a)
No
β 2 –smoothing.
β 2 –smoothing.
22
22
2
Gaussianfiltering.
filtering. b)
(b)Slight
Slightββ–smoothing
-smoothing(1/0)
(1/0) and
and Gaussian
Gaussian filtering with a filter
ororGaussian
filter width
width of
of two.
two. (c)
c) Strong
Strong ββ -smoothing
–smoothing(3/2)
(3/2)and
and
Gaussianfiltering
filteringwith
witha afilter
filterwidth
widthofoffive.
five.Note
Notethat
thatthe
theFigures
Figs. represent
Gaussian
representthe
theinitial
initialconditions
conditionsfor
foraaFS-model
FS-modelwith
withthe
theapplied
appliedfilter
filterand
and
22
-smoothing.
ββ
–smoothing.
coefficient of (1/0) is suited to decrease the integration time
significantly and stabilises the numerical results. In this notation, (1) indicates one lubricated node on the bedrock side
of the boundary and (0) indicates no stiffed (debris) node on
the lake side. If the number of lubricated nodes is reduced
to zero, the model’s integration time increases without a significant impact on the velocity field. If the number of stiffed
nodes over the lake is increased, lower velocities over the
lake are achieved. However, for higher model resolutions
than used in this study, higher β-values should be considered
to make the transition more realistic.
www.the-cryosphere.net/4/1/2010/
Second, we implement a three-dimensional Gaussian-type
filter to smooth the viscosity as well as the vertical resistive stress with a variable filter width. Without this filter, the numerics becomes unstable. Figure 1b shows a
one-dimensional equivalent to the implemented three dimensional filter. Figure 2 compares three different smoothing
parameter results on the friction coefficient β 2 as well as the
Gaussian-type filter impact on the viscosity for three different filter widths. Sensitivity experiments have shown, that
with a slight transient β 2 smoothing at the lake edges combined with a gentle (quadratic) Gaussian filter, as shown in
Fig. 2b, numerical stability is achieved.
The Cryosphere, 4, 1–12, 2010
4
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
(a)
(b)
1.2m/a
0
1.2m/a
300
600
0
300
Lake Depth (m)
Lake Depth (m)
300
−80
−20
100
200
−80
200
−20
y (km)
300
−80
−80
−20
400
−20
400
y (km)
600
100
0
0
0
100
200
300
400
0
x (km)
100
200
300
400
x (km)
Fig. 3. Results for a full-Stokes ice dynamic model with a horizontal resolution of 5 km. (a) Lake depth (color), ice sheet surface elevation
(dashed contours), and ice-flow velocity (black arrows). (b) Vertically averaged horizontal velocity of the standard full-Stokes experiment.
2.4
Results
The standard experiment is a thermomechanically coupled
full-Stokes (FS) model with a horizontal resolution of 5 km.
A quasi-steady state is reached after 300 000 years. Figure 3a
shows the lake’s position and depth in the center of the model
domain. The frictionless boundary condition over the lake
results in an increased velocity, not only above the lake, but
also in it’s vicinity (Fig. 3b). From mass conservation it follows, that the (vertically averaged) horizontal velocity converges towards the lake from upstream and diverges downstream. The flattened ice sheet surface over the lake is a consequence of the isostatic adjustment. This effect is also visible in the geometry of the profiles in Fig. 4 and in particular
in Fig. 5. The inclined lake-ice interface is maintained by the
constant ice flow over the lake and hence, independent of a
possible basal mass exchange.
Figure 3b shows the vertically averaged horizontal velocity and Fig. 4a a vertical cross section of the velocity along
the flow at the lake’s center at y = 200 km. Over the lake,
the ice sheet behaves like an ice shelf, featuring a vertically constant velocity. Towards the lake, the surface velocity increases by more than 70% from about 0.7 m/a to about
1.2 m/a. The largest velocity gradients occur in the vicinity
of the grounding line (Figs. 3b, 4a, 5). Convergence of ice
towards the lake results in an accelerated ice flow which undulates at the lake’s edges because of significant changes in
The Cryosphere, 4, 1–12, 2010
the ice thickness and ice sheet surface gradients (yellow and
red line in Fig. 5, respectively). The surface velocity (green
line in Fig. 5) accelerates towards the lake until the ice sheet
surface flattens. The velocity increase in deeper layers close
to the bedrock (blue line in Fig. 5) is suspended just before
the frictionless lake interface is reached, because of a strong
increase of the ice thickness filling the trough. The maximum
velocity is reached across the lake (Figs. 3b, 4a, 5), where
the basal friction is zero. The vertical temperature profile
(Fig. 4b) is nearly linear, as accumulation and basal melting
are neglected. Geothermal heat flux is not sufficient to melt
the bottom of the grounded ice, but over the lake the freezing
point is maintained by the boundary condition. This results
in submerging isotherms. At the downstream grounding line
a slight overshoot of the upwelling isotherm is observed.
2.5
Robustness of the results
To investigate the impact of the horizontal resolution on the
model results, simulations with a coarser 10 km as well as
a finer 2.5 km grid resolution have been performed. Within
the coarser resolution most features are reproduced well, but
there is a significant impact on ice velocities, in particular
along the grounding line where the differences reaches about
10%. In general, the model with the higher spatial resolution has increased velocities along the flowlines across the
lake and decreased velocities outside. In addition, the local
www.the-cryosphere.net/4/1/2010/
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
(a)
5
(b)
0.0
0.5
−50
1.0
−25
0
Temperature (˚C)
Horizontal u−Velocity (m/a)
0
0
−500
−500
−1000
−1000
−1500
−1500
0
−40
−4
−40
−30
−30
z (m)
z (m)
−2000
0.8
1
0.8
−2000
1
−30
−2
−20
0
−2500
−2500
−20
0.6
0
0.6
0.4
0.2
−3500
−10
−1
−3000
−3000
0.4
0.2
−10
−3500
−4000
−4000
−4500
−4500
0
100
200
300
0
400
100
200
300
400
x (km)
x (km)
Fig. 4. Results for a full-Stokes ice dynamic model with a horizontal resolution of 5 km. (a) Lake depth (color), ice sheet surface elevation
M. Thoma et al.: Coupled ice- and lake-flow
7
(dashed contours), and ice-flow
velocity (black arrows). (b) Horizontal downstream velocity (in x-direction). (c) Temperature
profile.
1.0
Surface Velocity
Bedrock Velocity
0.5
0.0
1.0
Ice Sheet Surface
Ice Thickness
Bedrock Friction
0.5
0.0
0
50
100
150
200
250
300
350
400
x (km)
Fig. 5. Cross sections along y = 200 km across the lake’s center. Shown are the normalized variables bedrock friction, ice sheet surface,
and ice
panal) as well
the corresponding
at the
close to the bedrock
(upper panal).
Fig. 5. Cross sections along
ythickness
= 200(lower
km across
theaslake’s
center.velocities
Shown
aresurface
the and
normalized
variables
bedrock friction, ice sheet surface, and
ice thickness (lower panal) as well as the corresponding velocities at the surface and close to the bedrock (upper panal).
40
80
20
25
30
mSv
35
−1
0
Stream Function
27
Heat Cond.
1
27
2
−1
Zonal Overturning
1
−0.3
−0.6
.9
−0
.2
S
−1
.2
−0
6
N
300
−4
.6
−0
−0
.
N
0
−10
26
0.2
0
−300
−8
26
25
0.6
80
12
µSv
−1
www.the-cryosphere.net/4/1/2010/
0
mm/a
Meridional Overturning
−1
40
Basal Mass Balance
mSv
1
0
−12
60
120
−6
25
We also investigated,
whether a higher order model, ned)
glecting resistive stress and vertical derivatives of the vertical velocity (see Pattyn 2003, Saito et al. 2003, Marshall
W
2005 for furtherE details), is able to reproduce the results obtained with the full-Stokes model. All other aspects of the
c)
60
0
.9
−0.6
−0
−0.3
80
mW/m2
180
40
12 18 2
0 0 40
0
0
−6
velocity maximum at the
grounding line (Fig.
5) cannot be
a)
b)
resolved with the 10 km resolution. The finer 2.5 km model
resolution needs a much longer integration time, without producing significant differences in the results compared to the
intermediate 5 km grid resolution.
The Cryosphere, 4, 1–12, 2010
N
N
Fig. 6. a) Heat conduction into the ice (QIce ). b) Vertically integrated mass transport stream function, positive values indicate a clockwise
circulation, negative values an anticlockwise circulation. c) Zonal (from west to east) and meridional (from south to north) overturning. d)
6
M. Thoma et al.: Modelled interaction between
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
(b)
0
40
80
(c)
0
40
80
mW/m2
20
25
30
mSv
35
−1
0
Heat Cond.
1
60 0
12
Stream Function
27
12 18 2
0 0 40
0
−6
60
(a)
180
6
26
−1
Zonal Overturning
.6
25
−0
−0.3
−1
.2
26
−0
M. Thoma et al.: Modelled
interaction between iceN sheet and subglacial lakes
N
mSv
−1
0
Stream Function
1
.9
−0.6
−0
0
40
−12
0
80
mm/a
12
0
−6
60
80
.6
(c)
(d)
Fig. 6. (a) Heat conduction into the ice (QIce ). (b) Vertically integrated mass transport stream function, po
circulation, negative values an anticlockwise circulation. (c) Zonal (from west to east) and meridional (fro
Basal mass balance.
60 0
Basal Mass Balance
12
12 18 2
0 0 40
40
−0
180
0
−0.6
.9
.2
−0
0.2
(b)
300
3
1
1
0
Meridional Overturning
−1
0.6
0
−300
−0.
27
25
μSv
mSv
−6
2005 for further details), is able to reproduce the results obfurther details). The horizonta
2
tained with the full-Stokes μSv
model. −1All other aspects of the
about 0.7 × 1.4 km), the num
mSv
0
300
geometry as well−300as the
parametrization
are kept identical.
well as the horizontal and ver
−1
0
1
Zonal Overturning
Our
experiments
indicate
a
moderate
impact
on
the
ice
flow:
and 0.025 cm2 /s, respectively
Meridional Overturning
−1
1
.6
calculated velocities
across the subglacial lake are about 5%
of subglacial Lake Concordia
−0
−0.3 −0.6
9
0.6
.2
0.
lower
compared
to
the
full-Stokes
model.
In
the
vicinity
of
model domain of about 170 ×
0
−
−
0.2
the lake, the− impact
of the full-Stokes terms decreased, but
tion within the lake as well as
.9
−0.6
0.
6
−0
is still enhanced
along
the flowlines.N Although the FS-model
at the lake–ice interface are ca
N
implements more mathematics in the numercial code it conlake a geothermal heat flux o
verges faster than the higher order model, hence all further
the ice-flow model’s boundar
e ice (QIce ).Fig.
(b)6. Vertically
integrated
mass
transport
stream
function,
positive
values
indicate
a
clockwise
(a) Heat conduction
into
the
ice
(Q
).
(b)
Vertically
integrated
mass
transport
stream
function,
positive
values
indicate
a clockwise
studies areIceperformed with the full-Stokes model.
vious
subglacial
lake simulat
circulation,
negative
values
an
anticlockwise
circulation.
(c)
Zonal
(from
west
to
east)
and
meridional
(from
south
to
north)
overturning.
clockwise circulation. (c) Zonal (from west to east) and meridional (from south to north) overturning. (d) et al., 2007, 2008a; Filina e
(d) Basal mass balance.
(Thoma et al., 2009a), or Lak
3 Lake floware
model
“ROMBAX”
2009) used a prescribed avera
geometry as well as the parametrization
kept identical.
3 Lake flow model ROMBAX
Our experiments indicate a moderate impact on the ice flow:
◦ ×0.0125◦ , (QIce = dT /dz × 2.1 W/(K m)
3.1
General
description
e to reproduce
the results
obfurther
details).
The
horizontal
resolution
(0.025
General
description
calculated
velocities
across3.1
the
subglacial
lake
are
about
5%
ture measurements and thickn
odel. All other
aboutmodel.
0.7 × In
1.4thekm),
the ofnumber
of vertical
layers
(16),
lower aspects
comparedof
to the
the full-Stokes
vicinity
To simulate
the water
flow
in theasprescribed
lake
mates. subglacial
The availability
of the
metrizationthe
arelake,
kepttheidentical.
well
as the
horizontal
eddy
diffusivities
(5 m2 /s
impact of theTo
full-Stokes
terms
decreased,
but vertical
ROMBAX
,subglacial
a terrain-following,
primitive equations,
simulate
the
water and
flow
in we
theapply
prescribed
ical
ice-sheet
model
“RIMBA
2 /s,
is still
flowlines.
Although
the FS-model
three-dimensional,
fluid
derate impact
onenhanced
the ice along
flow:the lake
andwe0.025
cm“ROMBAX”,
respectively)
are adopted from
a dynamics
model model (e.g., Griffies,
apply
a terrain-following,
primitive
QIce (Figure
6a).andAcross the
implements
more
mathematics
in
the
numercial
codeConcordia
it con2004).
R
OMBAX
simulates
the
between ice
subglacial lake are about 5% equations,
of subglacial
Lake
(Thoma
et
al.,
2009a).
Ininteraction
a
three-dimensional, fluid dynamics model (e.g.,
following
gradient
verges faster than the higher order model, hence all further
subjacent water in terms of melting and freezing, according from abou
okes model.studies
In the
vicinity of the
model
domain
of about 170 ×
88
× 16 grid
cells the circula2004).
“ROMBAX”
simulates
the interaction
beare performed withGriffies,
full-Stokes
model.
to heat and salinity conservation and the
sidepressure
of ice dependent
flow to about 24 m
l-Stokes terms decreased, but tween
tion ice
within
the
lake
as
well
as
the
melting
and
freezing
rates
and subjacent water in terms
melting
freezfreezingofpoint
at theand
interface
(Holland
and Jenkins,
1999).
results
from the
modelled tem
lines. Although the FS-model ing,
at according
the lake–ice
interface
are calculated.
theand
bottom
of the and has been applied
to heat
and salinity
conservation
the presThe
modelAt
uses
spherical
coordinates
(Fig. 4b) and is used as input
2 , ice-shelf
in the numercial code it con- sure
lake
a geothermal
heatpoint
flux atofsuccessfully
54 interface
mW/mto
consistent
with
cavities
(e.g., Grosfeld et al., 1997;
and
dependent
freezing
the
(Holland
et al.,
Thoma
et al., 2006) as well as to subrder model, hence all further Jenkins,
the ice-flow
boundary
condition,
is2001;
applied.
Pre1999).model’s
The model
usesWilliams
spherical
coordinates
and
glacial
lakes
(Williams,
2001;
Thoma3.3
et al.,Results
2007, 2008a,b;
full-Stokes model.
lake simulations
of Lake
Vostok
(Thoma
hasvious
been subglacial
applied successfully
to ice-shelf
cavities
(e.g., GrosFilina et al., 2008; Thoma et al., 2009a,b,c; Woodward et al.,
et et
al.,al.,2007,
Filina
et 2009).
al., 2008),
feld
1997;2008a;
Williams
et al.,
2001;
ThomaLake
et al.,Concordia
2006)
et subglacial
al., 2009a),lakes
or Lake
Ellsworth
(Woodward
et al., The initial model run starts w
as (Thoma
well as to
(Williams,
2001;
Thoma et al.,
AX”
2009)
used a prescribed
average
conduction
into the ice 200 years a quasi-steady stat
2007,
2008a,b;
Filina et al.,
2008; heat
Thoma
et al., 2009a,b,c;
within the lake is shown in
Woodward
et /dz
al., 2009).
(QIce = dT
× 2.1 W/(K m)), based on borehole temperaThe Cryosphere, 4, 1–12, 2010
www.the-cryosphere.net/4/1/2010/
ture measurements and thickness temperature-gradient esti- tegrated mass transport stream
3.2mates.
Model
and boundary
conditions
Thesetup
availability
of the results
of the thermomechan- about 1.3 mSv, 1 mSv=1000 m
in the prescribed subglacial
turning (about 0.3 mSv) show
−4
−8
−1
−0.
.2
3
−10
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
8
7
M. Thoma et al.: Coupled ice- and lake-flow
R IMBAY
R IMBAY
(restart)
extrap.
melt/freeze
rates
Restart
Water temperature
Output/Input
Ice thickness
Water column thickness
Temperature gradient
extrap.
tracer
ROMBAX
(restart)
ROMBAX
ROMBAX
R I RO C O
Out-/Input
Freezing rate
Melting rate
Output/Input
Ice thickness
Water column thickness
Temperature gradient
Initial
Geothermal heat
Restart
Ice geometry
Ice temperature
Ice flow
R IMBAY
Initial
Ice thickness
Bedrock height
Lake Geometry
Fig. 7. Couple scheme schematics. The upper part represents the ice-flow model R IMBAY, the lower part the lake-flow model ROMBAX. In the
middle
parameters
byThe
the coupler
R I Rrepresents
O C O are shown.
The gray
linesRinIMBAY
the left
partlower
of the
figure
represent the
initial
start-up. In
Fig. 7. the
Couple
schemeexchanged
schematics.
upper part
the ice-flow
model
, the
part
the lake-flow
model
ROMBAX
sequence
initial model
runs), by
dashed
loops indicate
(successive
model
that
apriori
unspecified
number
the middle(the
thetwo
parameters
exchanged
the coupler
R IROCOcycles
are shown.
The gray
linesruns)
in the
leftmay
partrepeat
of thean
Fig.
represent
the initial
start-up
of
times. Two
additional
ovals indicate
the individual
regarding
successive
restarts/coupling:
The iceunspecified
flow calculated
sequence
(the two
initial yellow
model runs),
dashed loops
indicate model
cycles needs
(successive
model
runs) that
may repeat an apriori
number
by
R IMBAY Two
may additional
result in new
lakeovals
nodes.indicate
For these
the basal
balance
is calculated
by restarts/coupling:
averaging adjacentThe
nodes.
For Rcalculated
OMBAX
of times.
yellow
the nodes
individual
modelmass
needs
regarding
successive
ice flow
abymodified
desires
thelake
extrapolation
temperatures
a previous
run to
skip the
time-consuming
R IMBAYgeometry
may result
in new
nodes. Forofthese
nodes the(and
basalother
masstracers)
balancefrom
is calculated
bymodel
averaging
adjacent
nodes.
For ROMBAX
spin-up
process.
a modified geometry desires the extrapolation of temperatures (and other tracers) from a previous model run to skip the time-consuming
spin-up process.
370
375
380
385
390
only parameter exchange beween both models was the uni3.2 Model
setupof
and
lateral
initilisation
theboundary
lake flow conditions
model ROMBAX with the
modelled geometry and temperature gradient of the ice flow 395
The bedrock
draft,
needed7).forThe
the
model
R IMBAY topography
(indicated byand
thethe
grayicelines
in Figure
lake-flow
model
ROMBAX
, iswhen
obtained
from oftheROMBAX
output
real
coupling
procedure
starts,
the results
(Sect.
2.4) of into
the ice-flow
model R IMBAY
for
are
reinserted
R IMBAY (indicated
by the (see
blackSect.
lines4 in
further7).
details). The horizontal resolution (0.025◦ ×0.0125◦ ,
Figure
about 0.7 × 1.4 km), the number of vertical layers (16), as400
4.2
Results
well as
the horizontal and vertical eddy diffusivities (5 m2 /s
and 0.025 cm2 /s, respectively) are adopted from a model
From
ROMBAX the
basal
mass balance
at the
in- a
of subglacial
Lake
Concordia
(Thoma
etice
al.,sheet–lake
2009a). In
terface
considered
for subsequent
IMBAY .
model is
domain
of about
170 × 88 ×initialisations
16 grid cells of
theRcirculaMelting
dominates
(which
is negligible)
and hence
tion within
the lakefreezing
as well as
the melting
and freezing
rates
the
icelake-ice
sheet is interface
loosing mass.
Note that the
molten
ice does
at the
are calculated.
At the
bottom
of the405
not
the lake’sheat
volume,
in the2 ,ice-sheet
model
lakeaffect
a geothermal
flux neither
of 54 mW/m
consistent
with
nor
in
the
lake-flow
model,
as
this
is
constant
per
definithe ice-flow model’s boundary condition, is applied. Pretion.
mass imbalance
can be of
interpreted
as a (Thoma
virtual
vious This
subglacial
lake simulations
Lake Vostok
constant
water
flow
out
of
the
lake
without
any
feedback
et al., 2007, 2008a; Filina et al., 2008), Lake Concordia
to
the iceetsheet.
The coupler
RO C O applies restarts from 410
(Thoma
al., 2009a),
or LakeR IEllsworth
(Woodward et al.,
previous model runs. Consequently the models reach their
2009) used a prescribed average heat conduction into the ice
new quasi-steady state after a significant shorter integration
(QIce = dT /dz × 2.1 W/(K m)), based on borehole temperatime. Ice sheet volume in the model domain is decreasing by
ture measurements
and thickness temperature-gradient estiabout 600 km3 per 100 000 years, equivalent to about 3.5 m
mates. The availability of the results of the thermomechanithickness. Ice draft reduction in the lake-flow model within 415
cal ice-sheet model R IMBAY permits a spatially varying Q
100 000 years is shown in Figure 8a. Most ice is lost in theIce
(Fig. 6a). Across the lake’s center, a general draft-following
center of the lake (up to 8 m), and the area of maximum mass
www.the-cryosphere.net/4/1/2010/
loss is slightly shifted to the upstream
side of the lake. Con2 on the upstream side of ice
gradient from
about
27 mW/m
sequently,
the water
column
thickness
increases where most
2 on the downstream side results from
flow
to aboutand
24 mW/m
ice
is molten
decreases
where less ice is molten (Figthe8b).
modelled
temperature
distribution
in the ice
ure
However,
the ice-thickness
reduction
and(Fig.
slope4b)
ad-and
is
used
as
input
for
the
lake-flow
model.
justment is too small to change the surface pressure on the
lake water, and hence the water flow, significantly. Just a few
iteration
cycles are needed to bring this coupled lake-water
3.3 Results
system into a quasi-steady state.
The initial model run starts with a lake at rest. After about
Several impacts of bottom melting on ice on top of the lake
200 years a quasi-steady state is reached. The circulation
are observed: First, the temperature gradient at the ice sheet’s
within the lake is shown in Fig. 6b–c. The vertically inbottom is increased. This results in an increase of heat contegrated
mass
transport
stream
function8d),
(with
a strength
duction
into
the ice
by about
22%3(Figures
compared
to a of
about
1.3
mSv,
1
mSv=1000
m
/s)
as
well
as
the
zonal
overmodel run without basal melting (Figures 6a). Consequently,
turning
(about
0.3
mSv)
show
a
two-gyre
structure,
while
more heat is extracted from the lake and the modelled aver-the
meridional
overturning
1.3 (Figures
mSv) indicates
just one
age
melting decreases
by (about
about 7%
8c). Second,
iceanticyclonic
gyre.
The
strength
of
the
mass
transport
is
sheet thickness above the lake is reduced. This increases thebetween gradient
those modelled
for upstream
Lake Vostok
and decreases
those for the
Lake
surface
towards the
part and
Concordia
(Thoma
et
al.,
2009a),
and
hence
reasonable
surface gradient towards the downstream part. Hence, the icefor
subglacial
lakes.
There is
only
a slight ice
draft slope
from
flow
upstream
accelerates
and
decelerates
downstream
(Figabout
4006
m
to
3927
m
across
the
90
km
of
the
lake
(Figs.
ure 9a). The magnitude of the ice velocity change is about 5,
4). This
results
in a decrease
of melting
fromand
about
12 mm/a
10%.
Third,
bottom
melting removes
mass
a vertical
in
the
West
to
a
negligible
freezing
along
the
eastern
shoredownward velocity follows from mass conservation. This
line
(Fig.
6d).
A
significant
amount
of
freezing
would
only
advects colder ice from the surface towards the bottom, re◦ a steeper slope.
be
modelled
if
the
ice
draft
would
have
sulting in a relative cooling of up to –2.1 C (Figure 9b) above
the lake. This negative temperature (compared to the model
The Cryosphere, 4, 1–12, 2010
8
4
4.1
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
Coupling
General description
The bedrock topography and the ice draft, necessary to set up
the geometry for the lake-flow model ROMBAX (Sect. 3.2),
was obtained from the modelled output geometry of the iceflow model R IMBAY (Sect. 2.4). In addition, the thermodynamic boundary condition of the heat conduction into the
ice was calculated from the temperature gradient at the ice
sheet’s bottom. The left part of Fig. 7 (gray lines) shows
schematically the performed operations. A coordinate conversion is necessary as R IMBAY uses Cartesian coordinates
while ROMBAX is based on spherical coordinates.
The modelled melting and freezing rates (Fig. 6d) are
considered to replace the previously zero-constrained lower
boundary condition in the ice-flow model R IMBAY (indicated by the central triangle within the yellow area in Fig. 7).
Again a coordinate transformation is necessary. To speed up
the integration time, ice geometry, ice flow, and ice temperature from the initial model run are reused (indicated by the
black-lined triangle within the blue area within Fig. 7). An
additional process has to be considered since the lake’s area
may change during an ice-model run. As the basal mass balance (melting/freezing) depends on the former lake-model
results and cannot be calculated during a specific model run,
extrapolation of neighbouring values may be necessary (indicated by the embedded yellow oval in the blue area of Fig. 7).
In each coupling cycle, successive initialisations of the
lake-flow model ROMBAX are performed with the slightly
changed ice draft and water column thickness (indicated by
the right triangle in the yellow area of Fig. 7), as well as the
temperature field of the previous model-run (indicated by the
central triangle in the orange area of Fig. 7). Because the lake
flow model does not permit dynamically changing geometry,
temperatures of emerging nodes have to be extrapolated from
neighbouring nodes (indicated by the embedded yellow oval
in the orange area of Fig. 7). It is not necessary to reuse
(and possibly extrapolate) the water circulation, as this value
is based on the tracer (density) distribution and converges
quickly.
The coupling mechanism, including initial starts of the
individual models, parameter exchanges, coordinate transformations, and restart-initialisations are embedded into and
controlled by the R IMBAY–ROMBAX–Coupler R IROCO.
In Sect. 2.4 and Sect. 3.3 the results of the initial runs
of the individual models have been described. So far the
only parameter exchange beween both models was the unilateral initilisation of the lake flow model ROMBAX with the
modelled geometry and temperature gradient of the ice flow
model R IMBAY (indicated by the gray lines in Fig. 7). The
real coupling procedure starts, when the results of ROMBAX
are reinserted into R IMBAY (indicated by the black lines in
Fig. 7).
The Cryosphere, 4, 1–12, 2010
4.2
Results
From ROMBAX the basal mass balance at the ice sheet-lake
interface is considered for subsequent initialisations of R IM BAY . Melting dominates freezing (which is negligible) and
hence the ice sheet is loosing mass. Note that the molten
ice does not affect the lake’s volume, neither in the ice-sheet
model nor in the lake-flow model, as this is constant per definition. This mass imbalance can be interpreted as a virtual
constant water flow out of the lake without any feedback
to the ice sheet. The coupler R IROCO applies restarts from
previous model runs. Consequently the models reach their
new quasi-steady state after a significant shorter integration
time. Ice sheet volume in the model domain is decreasing by
about 600 km3 per 100 000 years, equivalent to about 3.5 m
thickness. Ice draft reduction in the lake-flow model within
100 000 years is shown in Fig. 8a. Most ice is lost in the center of the lake (up to 8 m), and the area of maximum mass
loss is slightly shifted to the upstream side of the lake. Consequently, the water column thickness increases where most
ice is molten and decreases where less ice is molten (Fig. 8b).
However, the ice-thickness reduction and slope adjustment is
too small to change the surface pressure on the lake water,
and hence the water flow, significantly. Just a few iteration
cycles are needed to bring this coupled lake-water system
into a quasi-steady state.
Several impacts of bottom melting on ice on top of the
lake are observed: First, the temperature gradient at the ice
sheet’s bottom is increased. This results in an increase of heat
conduction into the ice by about 22% (Fig. 8d), compared to
a model run without basal melting (Fig. 6a). Consequently,
more heat is extracted from the lake and the modelled average melting decreases by about 7% (Fig. 8c). Second, ice
sheet thickness above the lake is reduced. This increases the
surface gradient towards the upstream part and decreases the
surface gradient towards the downstream part. Hence, the
ice flow upstream accelerates and decelerates downstream
(Fig. 9a). The magnitude of the ice velocity change is about
10%. Third, bottom melting removes mass and a vertical
downward velocity follows from mass conservation. This advects colder ice from the surface towards the bottom, resulting in a relative cooling of up to −2.1 ◦ C (Fig. 9b) above the
lake. This negative temperature (compared to the model run
without melting, Fig. 4b) is advected downstream by the horizontal velocity. The lake-ice interface itself is cooled by less
than 0.007 ◦ C, as it is maintained at the pressure-dependent
freezing point. The cooling, visible at the lateral boundaries
in Fig. 9b, result from the applied periodic boundary condition. It is an artifact of the numerical representation of the
setup and not discussed here.
www.the-cryosphere.net/4/1/2010/
M.interaction
Thoma etbetween
al.: Modelled
interaction
between
M. Thoma et al.: Modelled
ice sheet and
subglacial lakes
(a)
ice sheet and subglacial lakes
(b)
0
40
(c)
80
0
40
0
80
m
−10
−5
9
40
80
mm/a
m
0
−4
−2
Ice Thickness
0
−1.0
2
−0.5
0.0
0.5
1.0
Basal Mass Balance
Water Column
0.2
−0.3
.3
−0
−2.8
−1.2
−1.2
−3
−7
ction between ice sheet and subglacial lakes
b)
0
40
m
−4
−2
0
40
80
0
40
.3
−0
80
mW/m2
mm/a
−0.5
0.0
0.5
1.0
20
25
Basal Mass Balance
Water Column
.3
N
(c)
(d)
Fig. 8. Impact of melting after 100 000 years on (a) the ice draft, (b) the water column thickness, and
the geometry difference between the lake-flow model restart and the initial geometry. (d) Indicates the
−1.0
2
1
N
0
80
−0
9
N
30
35
Heat Cond.
0.2
The ice flow converges
where an ice-shelf type flow
28
servation requires a downs
of the deceleration when th
5 Summary
Melting at the lake–ice inte
32
31
eration upstream and decre
Observations from space show that large subglacial lakes
as well as downstream, bec
have a significant
impact on the shape and dynamics of the
−0
.3
This impacts on the veloci
1
.3
N
N
Antarctic
Ice Sheet
as they flattenN the ice sheets surface
−0
and downstream of the la
(Siegert et al., 2000; Kwok et al., 2000, 2004; Leonard et al.,
corresponds to about 20 tim
2004; Tikku et al., 2004; Siegert, 2005). These regions depict
at the
Fig. 8. Impact of melting after 100 000 years on (a) the ice draft, (b) the water column thickness, and (c) the basal mass
balance. Shown
is ice sheet
00 years on (a) the ice draft, (b) the
water column
thickness,
and
(c)tothe
basal
mass balance.
Shown is temperature
a change
in surface
slopeand
due
the
isostatic
adjustment
the geometry difference between
the lake-flow
model restart
the initial
geometry.
(d) Indicates
the heatof
conduction
into
the
ice
(Q
).
Ice
pressure melting
point, and
lake-flow model restart and the initial
geometry.
Indicates
the heat
conduction
into bottom
the ice (Q
Ice ).
the ice
sheet. (d)
This,
combined
with
the lacking
fricbased ice in the vicinity. A
tion, results in an observable redirection of ice flow. In this
ing with
at the
lake–ice
5 Summary
a three dimensional fluid dynamics model
eddy
diffu- interface
study, we apply a newly coupled sivity
ice sheet–lake
flow
model
and simulates
the lake
the mass
balthe as
lake.
Advection
transpo
erical representation of the on anThe
ice flow
accelerates
the flow
lake,as well
idealized
ice converges
sheet–lake and
configuration
totowards
investigate
ance at the lake-ice
interface.the
This model,
previously
only
Observations from space show
that
large
subglacial
lakes
where anbetween
ice-shelfthe
type flow structure
establishes.
Mass
con- stream, and hence the tem
applied
to lake-exclusive
studies with a prescribed ice thicksystems.
The full-Stokes
have a significant impactfeedbacks
on the shape
and dynamicsindividual
of the
have impacts on the ice flo
servation
requires a downstream
divergence
ofnow
flow,
and bedrock,
receives
2008) is ness
improved
to handle
a because
more its geometry directly from
Antarctic Ice Sheet as ice
theysheet
flattenmodel
the ice(Pattyn,
sheets surface
Our the
idealised
the ice
flow model.
In addition,
heat flux from
lake intoconfigurati
of
the2000;
deceleration
theThe
flow
reaches
the
bedrock
again.
(Siegert et al., 2000; Kwok
et al.,
Leonard rheology.
et when
al., 2004;
realistic
non-linear
lake-flow
model
is
based
theincreases
ice sheet isthe
an exchange
Besides
other exterimportant
impact of the
at theregions
lake–ice
interface
iceeddy
flow parameter.
accel- an
Tikku et al., 2004; Siegert,Melting
depict
a
on a2005).
three These
dimensional
fluid
dynamics
model
with
difnal forcing
fields,
the ice-flow
model receives
the
basal
mass
change in surface slope dueeration
to the isostatic
adjustment
of
the
upstream
and decreases
theasicewell
flowinterface
on
of the
lake on the ice sheet’s dynamic
fusivity
simulates
lake flow
thetop
mass
balance
at theas
frombalthe dynamic lake-flow model.
hat large ice
subglacial
sheet. This,lakes
combined
with theand
lacking
bottom the
friction,
coupled model will focus o
as well
aslake–ice
downstream,
because of the
reduced
surfaceonly
slope.model
order to previously
stabilise the ice-flow
numerically with
at the
in an observable
redirection
of ice
flow. In interface.
this study, ThisInmodel,
shape andresults
dynamics
of the ance
Lake
Vostok
This
impacts
on
the
velocity
can
be
traced
about
75
km
upa
nonlinear
rheology,
a
slight
Gaussian
filtering
of theand
ice its glacia
we
apply
a
newly
coupled
ice
sheet-lake
flow
model
on
an
ten the ice sheets surface applied to lake-exclusive studies with a prescribed ice thickis necessary.
and downstream
ofthe
the
lake inviscosity
specific
setting,from
which vestigate the flow dynamic
idealized ice sheet-lake configuration
to investigate
feedbedrock,
now
receives
itsthis
geometry
directly
2000, 2004;
Leonard et al., ness and
observations.
backs between the individual
systems.
The
full-Stokes
ice
The
ice
flow
converges
and
accelerates
towards
the lake,
to about
20 timesheat
the flux
ice sheet
The with
thecorresponds
ice
flow model.
In
addition,
from thickness.
the
lake
into
2005). These
regions
depict2008)
is
improved
to
handle
a
more
resheet
model (Pattyn,
where
an
ice-shelf
type
flow
structure
establishes.
Mass
contemperature
atanthe
ice sheet bottom on Besides
top of the lakeexteris at the
icelake-flow
sheet ismodel
exchange
the isostatic
adjustment
of theThe
alistic
non-linear rheology.
is based onparameter.
servation requires aother
downstream divergence of flow, because
pressure melting point, and hence warmer than the bedrockth the lacking bottom fric- nal forcing fields, the ice-flow model receives the basal mass
based ice in the vicinity. According to our simulation, meltrection ofwww.the-cryosphere.net/4/1/2010/
ice flow. In this balance at the interface from the dynamic lake-flow model.
The Cryosphere, 4, 1–12, 2010
ingorder
at theto
lake–ice
interface
reduces
the temperature
top of
In
stabilise
the ice-flow
model
numericallyonwith
ice sheet–lake flow model
the
lake.
Advection
transports
the
relatively
colder
ice
downfiguration to investigate the a nonlinear rheology, a slight Gaussian filtering of the ice
stream, and hence the temperature dependent rheology will
29
−2.8
30
31
30
33
29
33
33
−1.2
−1.2
225267
2822
276
32 5
tion. It is−0.3an artifact of the numerical representation of the
setup and not discussed here.
.3
−0
33
10
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
(a)
(b)
−0.10
−0.05
0.00
0.05
0.10
−2
−1
Horizontal u−Velocity (m/a)
0
1
2
Temperature (˚C)
0
0
−0.01
−0.22
−0.05
−2500
−0.05
−3000
−1.1
−0
.22
−1.1
−0.22
−0.01
−3000
−3500
−0.22
−0.22
z (m)
−0.05
−0.22
−2500
−1.1
−2000
−1.1
−0.22
−0.01
−1500
−0.01
z (m)
−0.22
−2000
−0.22
−1500
−1.1
−0.22
−1000
−1.1
−0.01
−1000
−500
−0.22
−500
−3500
−0.01
−0.22
−4000
−4000
−4500
−4500
0
100
200
300
400
0
x (km)
100
200
300
400
x (km)
Fig. 9. Differences between coupled-model result and initial model (Fig. 4) for (a) horizontal downstream velocity and (b) temperature.
of the deceleration when the flow reaches the bedrock again.
Melting at the lake-ice interface increases the ice flow acceleration upstream and decreases the ice flow on top of the lake
as well as downstream, because of the reduced surface slope.
This impacts on the velocity can be traced about 75 km upand downstream of the lake in this specific setting, which
corresponds to about 20 times the ice sheet thickness. The
temperature at the ice sheet bottom on top of the lake is at the
pressure melting point, and hence warmer than the bedrockbased ice in the vicinity. According to our simulation, melting at the lake-ice interface reduces the temperature on top of
the lake. Advection transports the relatively colder ice downstream, and hence the temperature dependent rheology will
have impacts on the ice flow beyond the lake.
Our idealised configuration of an ice-lake system indicates
an important impact of the interaction between both systems
on the ice sheet’s dynamics. Next applications of this new
coupled model will focus on realistic configurations, such as
Lake Vostok and its glacial drainage system. In order to investigate the flow dynamical impact which can be compared
with observations.
Acknowledgements. This work was funded by the DFG through
grant MA3347/2-1. The authors wish to thank Saito Fuyuki,
Povl Christoffersen, and an anonymous reviewer for helpful
comments as well as Andrea Bleyer for proof reading.
Edited by: I. M. Howat
The Cryosphere, 4, 1–12, 2010
References
Bell, R. E.: The role of subglacial water in ice-sheet mass balance,
Nat. Geosci., 1, 297–304, doi:10.1038/ngeo186, 2008.
Bell, R. E., Studinger, M., Tikku, A. A., Clarke, G. K. C., Gutner,
M. M., and Meertens, C.: Origin and fate of Lake Vostok water
frozen to the base of the East Antarctic ice sheet, Nature, 416,
307–310, 2002.
Carter, S. P., Blankenship, D. D., Peters, M. E., Young, D. A.,
Holt, J. W., and Morse, D. L.: Radar-based subglacial lake
classification in Antarctica, Geochem. Geophys. Geosyst., 8,
doi:10.1029/2006GC001408, 2007.
Comiso, J. C.: Variability and trends in Antarctic surface temperatures from in situ and satellite infrared measurements, J. Climate,
13, 1674–1696, 2000.
Duxbury, N. S., Zotikov, I. A., Nealson, K. H., Romanovsky, V. E.,
and Carsey, F. D.: A numerical model for an alternative origin
of Lake Vostok and its exobiological implications for Mars, J.
Geophys. Res., 106, 1453–1462, 2001.
Filina, I. Y., Blankenship, D. D., Thoma, M., Lukin, V. V., Masolov,
V. N., and Sen, M. K.: New 3D bathymetry and sediment distribution in Lake Vostok: Implication for pre-glacial origin and numerical modeling of the internal processes within the lake, Earth
Pla. Sci. Let., 276, 106–114, doi:10.1016/j.epsl.2008.09.012,
2008.
Fricker, H. A., Scambos, T., Bindschadler, R., and Padman, L.: An
active subglacial water system in west Antarctica mapped from
space, Science, 315, 1544–1548, doi:10.1126/science.1136897,
2007.
Gray, L., Joughin, I., Tulaczyk, S., Spikes, V. B., Bindschadler,
www.the-cryosphere.net/4/1/2010/
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
R., and Jezek, K.: Evidence for subglacial water transport
in the West Antarctic Ice Sheet through three-dimensional
satellite radar interferometry, Geophys. Res. Lett., 32, 3,
doi:10.1029/2004GL021387, 2005.
Griffies, S. M.: Fundamentals of ocean climate models, Princeton
University Press, Princeton, 2004.
Grosfeld, K., Gerdes, R., and Determann, J.: Thermohaline circulation and interaction beneath ice shelf cavities and the adjacent
open ocean, J. Geophys. Res., 102, 15595–15610, 1997.
Holland, M. D. and Jenkins, A.: Modeling Thermodynamic IceOcean Interaction at the Base of an Ice Shelf, J. Phys. Oceanogr.,
29, 1787–1800, 1999.
Hooke, R. L.: Flow law for polycrystalline ice in glaciers: Comparison of theoretical predictions, laboratory data, and field measurements, Rev. Geophys. Space Phys., 19, 664–672, 1981.
Inman, M.: Antarctic drilling: The plan to unlock Lake Vostok, Science, 310, 611–612, doi:10.1126/science.310.5748.611, 2005.
Jouzel, J., Petit, J. R., Souchez, R., Barkov, N. I., Lipenkov,
V. Y., Raynaud, D., Stievenard, M., Vassiliev, N. I., Verbeke,
V., and Vimeux, F.: More than 200 meters of lake ice above
subglacial Lake Vostok, Antarctica, Science, 286, 2138–2141,
doi:10.1126/science.286.5447.2138, 1999.
Karl, D. M., Bird, D. F., Bjrkman, K., Houlihan, T., Shackelford, R., and Tupas, L.: Microorganisms in the accreted ice
of Lake Vostok, Antarctica, Science, 286, 5447, 2144–2147,
doi:10.1126/science.286.5447.2144, 1999.
Kwok, R., Siegert, M. J., and Carsey, F. D.: Ice motion over Lake
Vostok, Antarctica: constraints on inferences regarding the accreted ice, J. Glaciol., 46, 689–694, 2000.
Lavire, C., Normand, P., Alekhina, I., Bulat, S., Prieur, D., Birrien, J., Fournier, P., Hnni, C., and Petit, J.: Presence of Hydrogenophilus thermoluteolus DNA in accretion ice in the subglacial Lake Vostok, Antarctica, assessed using rrs, cbb and
hox., Environ. Microbiol., 8, 2106–2114, doi:10.1111/j.14622920.2006.01087.x, 2006.
Leonard, K., Bell, R. E., Studinger, M., and Tremblay, B.:
Anomalous accumulation rates in the Vostok ice-core resulting
from ice flow over Lake Vostok, Geophys. Res. Lett., 31, 24,
doi:10.1029/2004GL021102, 2004.
Lewis, E. L. and Perkin, R. G.: Ice pumps and their rates, J. Geophys. Res., 91, 11756–11762, 1986.
Marshall, S. J.: Recent advances in understanding ice sheet dynamics, Earth Pla. Sci. Let., 240, 191–204, 2005.
Maule, C. F., Purucker, M. E., Olsen, N., and Mosegaard, K.: Heat
Flux Anomalies in Antarctica Revealed by Satellite Magnetic
Data, Science, 309, 464–467, doi:10.1126/science.1106888,
2005.
Mayer, C. and Siegert, M. J.: Numerical modelling of ice-sheet dynamics across the Vostok subglacial lake, central East Antarctica,
J. Glaciol., 46, 197–205, doi:10.3189/172756500781832981,
2000.
Mayer, C., Grosfeld, K., and Siegert, M.: Salinity impact on water flow and lake ice in Lake Vostok, Antarctica, Geophys. Res.
Lett., 30, 1767, doi:10.1029/2003GL017380, 2003.
Oswald, G. K. A. and Robin, G. d. Q.: Lakes Beneath the Antarctic
Ice Sheet, Nature, 245, 251–254, doi:10.1038/245251a0, 1973.
Paterson, W. S. B.: The Physics of Glaciers, Butterworth Heinemann, Oxford, 3 Edn., 1994.
Pattyn, F.: A new three-dimensional higher-order thermomechan-
www.the-cryosphere.net/4/1/2010/
11
ical ice sheet model: Basic sensitivity, ice stream development,
and ice flow across subglacial lakes, J. Geophys. Res., 108, 1–15,
doi:10.1029/2002JB002329, 2003.
Pattyn, F.: Comment on the comment by M. J. Siegert on “A numerical model for an alternative origin of Lake Vostok and its
exobiological implications for Mars” by N. S. Duxbury et al., J.
Geophys. Res., 109, E11004, doi:10.1029/2004JE002329, 2004.
Pattyn, F.: Investigating the stability of subglacial lakes with
a full Stokes ice-sheet model, J. Glaciol., 54, 353–361,
doi:10.3189/002214308784886171, 2008.
Pattyn, F., de Smedt, B., and Souchez, R.: Influence of subglacial
Vostok lake on the regional ice dynamics of the Antarctic ice
sheet: a model study, J. Glaciol., 50, 583–589, 2004.
Robin, G. D. Q., Drewry, D. J., and Meldrum, D. T.: International
Studies of Ice Sheet and Bedrock, Philos. Tr. R. Soc., 279, 185–
196, 1977.
Saito, F., Abe-Ouchi, A., and Blatter, H.: Effects of first-order
stress gradients in an ice sheet evaluated by a three-dimensional
thermomechanical coupled model, Ann. Glaciol., 37, 166–172,
doi:10.3189/172756403781815645, 2003.
Schiermeier, Q.: Russia delays Lake Vostok drill, Antarctica’s hidden water will keep its secrets for another year, Nature, 454, 258–
259, doi:10.1038/454258a, 2008.
Siegert, M. J.: Comment on “A numerical model for an alternative origin of Lake Vostok and its exobiological implications for
Mars” edited by: Duxbury, N. S., Zotikov, I. A., Nealson, K. H.,
Romanovsky, V. E., and Carsey, F. D., J. Geophys. Res., 109,
1–3, doi:10.1029/2003JE002176, 2004.
Siegert, M. J.: Reviewing the origin of subglacial Lake Vostok and
its sensitivity to ice sheet changes, Prog. Phys. Geog., 29, 156–
170, doi:10.1191/0309133305pp441ra, 2005.
Siegert, M. J., Kwok, R., Mayer, C., and Hubbard, B.: Water exchange between the subglacial Lake Vostok and the overlying
ice sheet, Nature, 403, 643–646, doi:10.1038/35001049, 2000.
Siegert, M. J., Tranter, M., Ellis-Evans, J. C., Priscu, J. C., and
Lyons, W. B.: The hydrochemistry of Lake Vostok and the potential for life in Antarctic subglacial lakes, Hydr. Proc., 17, 795–
814, 2003.
Siegert, M. J., Hindmarsh, R., Corr, H., Smith, A., Woodward,
J., King, E. C., Payne, A. J., and Joughin, I.: Subglacial Lake
Ellsworth: A candidate for in situ exploration in West Antarctica, Geophys. Res. Lett., 31, 1–4, doi:10.1029/2004GL021477,
2004.
Siegert, M. J., Carter, S., Tabacco, I. E., Popov, S., and Blankenship, D. D.: A revised inventory of Antarctic subglacial lakes,
Anatarct. Sci., 17, 453–460, doi:10.1017/S0954102005002889,
2005.
Siegert, M. J., Behar, A., Bentley, M., Blake, D., Bowden, S.,
Christoffersen, P., Cockell, C., Corr, H., Cullen, D. C., Edwards, H., Ellery, A., Ellis-Evans, C., Griffiths, G., Hindmarsh,
R., Hodgson, D. A., King, E., Lamb, H., Lane, L., Makinson,
K., Mowlem, M., Parnell, J., Pearce, D. A., Priscu, J., Rivera,
A., Sephton, M. A., Sims, M. R., Smith, A. M., Tranter, M.,
Wadham, J. L., Wilson, G., and Woodward, J.: Exploration of
Ellsworth Subglacial Lake: a concept paper on the development,
organisation and execution of an experiment to explore, measure
and sample the environment of a West Antarctic subglacial lake,
Rev. Env. Sci. Bio., 6, 161–179, doi:10.1007/s11157-006-91099, 2007.
The Cryosphere, 4, 1–12, 2010
12
M. Thoma et al.: Modelled interaction between ice sheet and subglacial lakes
Smith, B. E., Fricher, H. A., Joughin, I. R., and Tulaczyk, S.: An inventory of active subglacial lakes in Antarctica detected by ICESat (2003–2008), J. Glaciol., 55, 573–595, 2009.
Studinger, M., Bell, R. E., and Tikku, A. A.: Estimating the depth and shape of subglacial Lake Vostok’s water
cavity from aerogravity data, Geophys. Res. Lett., 31, 12,
doi:10.1029/2004GL019801, 2004.
Thoma, M., Grosfeld, K., and Lange, M. A.: Impact of the Eastern
Weddell Ice Shelves on water masses in the eastern Weddell Sea,
J. Geophys. Res., 111, doi:10.1029/2005JC003212, 2006.
Thoma, M., Grosfeld, K., and Mayer, C.: Modelling mixing and circulation in subglacial Lake Vostok, Antarctica, Ocean Dynamics,
57, 531–540, doi:10.1007/s10236-007-0110-9, 2007.
Thoma, M., Grosfeld, K., and Mayer, C.: Modelling accreted ice
in subglacial Lake Vostok, Antarctica, Geophys. Res. Lett., 35,
1–6, doi:10.1029/2008GL033607, 2008a.
Thoma, M., Mayer, C., and Grosfeld, K.: Sensitivity of Lake Vostok’s flow regime on environmental parameters, Earth Pla. Sci.
Let., 269, 242–247, doi:10.1016/j.epsl.2008.02.023, 2008b.
Thoma, M., Filina, I., Grosfeld, K., and Mayer, C.: Modelling flow
and accreted ice in subglacial Lake Concordia, Antarctica, Earth
Pla. Sci. Let., 286, 278–284, doi:10.1016/j.epsl.2009.06.037,
2009a.
Thoma, M., Grosfeld, K., and Lange, M. A.: Modelling the impact of ocean warming on Eastern Weddell Ice Shelf melting and
water masses in their vicinity, Ocean Dynamics, re-submitted,
2009b.
Thoma, M., Grosfeld, K., Smith, A. M., and Mayer, C.: A comment on the equation of state and the freezing point equation
with respect to subglacial lake modelling, Earth Pla. Sci. Let.,
submitted, 2009c.
Tikku, A. A., Bell, R. E., Studinger, M., and Clarke, G.
K. C.: Ice flow field over Lake Vostok, East Antarctica inferred by structure tracking, Earth Pla. Sci. Let., 227, 249–261,
doi:10.1016/j.epsl.2004.09.021, 2004.
The Cryosphere, 4, 1–12, 2010
Tikku, A. A., Bell, R. E., Studinger, M., Clarke, G. K. C., Tabacco,
I., and Ferraccioli, F.: Influx of meltwater to subglacial Lake
Concordia, east Antarctica, J. Glaciol., 51, 96–104, 2005.
Walsh, D.: A note on eastern-boundary intensification of flow in
Lake Vostok, Ocean Modelling, 4, 207–218, 2002.
Weertman, J.: Creep of ice, in: Physics and Chemistry of Ice, edited
by Whalley, E., Jones, S. J., and Gold, L. W., pp. 320–337, Royal
Society of Canada, Ottawa, 1973.
Wells, M. G. and Wettlaufer, J. S.: Circulation in Lake Vostok:
A laboratory analogue study, Geophys. Res. Lett., 35, 1–5,
doi:10.1029/2007GL032162, 2008.
Williams, M. J. M.: Application of a three-dimensional numerical
model to Lake Vostok: An Antarctic subglacial lake, Geophys.
Res. Lett., 28, 531–534, 2001.
Williams, M. J. M., Grosfeld, K., Warner, R. C., Gerdes, R., and
Determann, J.: Ocean circulation and ice-ocean interaction beneath the Amery Ice Shelf, Antarctica, J. Geophys. Res., 106,
22383–22400, 2001.
Wingham, D. J., Siegert, M. J., Shepherd, A., and Muir, A. S.:
Rapid discharge connects Antarctic subglacial lakes, Nature,
440, 1033–1036, doi:10.1038/nature04660, 2006.
Woodward, J., Smith, A. M., Ross, N., Thoma, M., Siegert, M. J.,
King, M. A., Corr, H. F. J., King, E. C., and Grosfeld, K.:
Bathymetry of Subglacial Lake Ellsworth, West Antarctica and
implications for lake access, Geophys. Res. Lett., in preparation,
2009.
West, A. and Carmack, E.: A priori estimates of mixing and circulation in the hard-to-reach water body of Lake Vostok, Ocean
Modelling, 2, 29–43, 2000.
www.the-cryosphere.net/4/1/2010/