1.
Measurements of ocean temperature, salinity and velocity
During August 16 – 18, 2010, we collected ocean temperature, salinity and velocity data at 8
stations spanning the width of the glacial fjord in front of Store Glacier, Greenland, using an
InterOcean S4 conductivity, temperature, depth/current profiler (S4, hereafter). The distance
interval between stations varied from 0.5 to 0.9 km (Fig. 1c-e). The instrument accuracy is
±0.02 °C for temperature, ±0.02 psu for salinity and ±0.01 m/s for velocity. The instrument
collects data at 2 Hz, or every 0.5 second. We programmed the instrument to output data
averaged every 30 seconds. At each station, the instrument was lowered in the ocean and
stationed at discrete depth intervals (5-m steps above 50 m, 10-m steps between 50 m and 100 m,
and 25-m steps below 100 m) for 2-3 minutes to obtain 3-5 average velocity readings at each
depth. These 3-5 average velocity data at each depth were then averaged to obtain an average
value. According to the manufacturer specifications
(http://www.interoceansystems.com/s4specs.htm), a 2-3 min averaging guarantees that the noise
level will be less than the resolution of the instrument, which is 0.03-0.35 cm/s at 2 Hz.
It took about 1.5 hours to measure one cast down to 400 m depth. The total time of survey
was three days: three casts were made on August 16 between 4:00 pm and 10:30 pm; four casts
on August 17 between 10:00 am and 7:00 pm; and one on August 18 between 11:40 am to 1:20
pm.
An additional cast was made at a station 4 km from the glacier front using a Seabird SBE19plus conductivity, temperature and depth profiler down to 500 m depth (Fig. S1). The profile
shows that the ocean temperature and salinity are nearly constant below 450 m depth. This
information justifies our assumption that temperature and salinity may be assumed to be constant
below 480-m depth of the S4 casts.
1
Figure S1. a) LandSat-7 image of Store Glacier with a red dot at the location of the Seabird
SBE19plus CTD cast and yellow dots at the locations of S4 casts; b) temperature (T) and salinity
(S) profiles measured by the Seabird SBE19plus CTD.
2. Calculation of the melt rate from oceanographic data
We calculate the subaqueous melt rate of Store Glacier by applying the conservation of mass,
heat, and salinity to the hydrographic section [Motyka et al., 2003; Rignot et al., 2010].
First, we omit the upper 20 m of the water column from the calculation. The reasons are that
the upper column is strongly influenced by solar heating and melting debris that do not
participate in the subsurface melting. Fig. S2 shows that the upper 20 m are characterized by
high temperature and low salinity, hence stable. During the survey, few icebergs were present in
front of the glacier, most icebergs break off in a myriad of debris that slowly melt near the
surface, not at depth. By removing the top 20 m from our calculation, we neglect the impact of
2
calved ice in our calculation of the melt rate, i.e. we concentrate on the melt rate at depth,
typically several hundred meters below the surface.
Figure S2. Temperature (T) and salinity (S) of the top 100 m at 8 stations of the hydrographic
section of Store Glacier, Greenland on Aug. 16-18, 2010. High temperature and low salinity in
the surface layer is caused, respectively, by solar heating and melting of ice debris.
To fill data gaps in the hydrographic section, we linearly interpolate the measured ocean
temperature (T), salinity (S) and velocity (V) data (Fig. 1c,d,e). At the sides and bottom of the
hydrographic section, where no data are available, we linearly relax the measured T and S to the
mean profile shown in Fig. 1b, which is obtained by averaging all 8 observed profiles. T and S
are assumed to remain constant with depth below the bottom end of the cast, as discussed earlier
(Fig. S1), and also consistent with data collected in other glacial fjords where Atlantic waters are
present [Straneo et al., 2012]. We assume the ocean velocity at the sidewalls is zero (no-slip).
The measured velocity data are linearly interpolated to the sidewalls (Fig. 1e).
To replace missing velocity data at depth, we close the salinity budget by assuming zero net
salinity flux through the hydrographic section, i.e., we assume the total ocean salinity between
3
the hydrographic section and the glacier face to be constant). The zero net salinity flux is
expressed by the equation
 (S
a
ua   a )  Sb ub   b  0
(S1)
where subscript “a” stand for upper layer properties, and “b” stands for properties below the
depth of our measurements; u is the water velocity and δ is the area of each grid cell; Sb and u b
are, respectively, the mean salinity and velocity of the entire bottom section with missing data.
From Eq. S1, we calculate u b = –0.4 cm/s for the bottom part of the casts (Fig. 1e).
The T, S, V of the entire section is then applied to deduce the melt rate from the heat budget.
The net heat flux through the entire section, H, is
H   C pw   (T  T f )u   
(S2)
where C pw is the heat capacity of water,  is the water density, and T f is the in-situ freezing
point, which is pressure and salinity dependent. The subglacial freshwater discharge is at the insitu freezing point and therefore contributes no heat flux. We assume the entire available heat
flux melts ice on the calving face (i.e., ocean temperatures between the section and glacier are
steady), so the volume flux of melt water (Qm) is
Qm  H / (L   )
(S3)
where L is latent heat of fusion of ice. Here, we use water density ( r ) so Qm is the volume flux
of melt water rather than melted ice. The melt rate (qm) per unit area equals
qm 
Qm

 86400 s/day = 3.0 m/d
(S4)
which is measured in meters of water per day.
4
Last, the conservation of water volume flux is applied to balance the subglacial water
discharge, Qsg, with the water mass flux from the ocean and from ice melting, Qm, i.e.,
Qsg  Qm   u    0
(S5)
From Eq. S5, we deduce the subglacial water discharge, Qsg = 246 m3/s.
3. Error estimation of the melt rate calculated from the oceanographic data
A Monte-Carlo method is used to estimate the errors on the melt rate due to 1) velocity
measurement errors and 2) errors from interpolation and extrapolation of the velocity data.
We use the published S4 velocity accuracy of 1 cm/s as the RMS error for each horizontal
velocity component. To quantify the melt rate uncertainty due to instrument accuracy, we add
random noise with a Gaussian distribution, zero mean and sigma =1 cm/s to all measured
velocity (204 points), recalculate the melt rate, and repeat the procedure until the standard
deviation of the recalculated melt rates converges to a stable value. The standard deviation of
these Monte-Carlo experiments represents the error from limited instrument accuracy. We find
that the error of 1 cm/s in the velocity data results in a melt rate error of 1.0 m/d and an error in
Qsg of 10 m3/s.
For estimating the error from interpolation and extrapolation, we first calculate the standard
deviation of all 204 velocity data, which is 12.8 cm/s. We then assume that the error of
interpolated values in each grid cells does not exceed this number. Gaussian noise with standard
deviation of 12.8 cm/s is applied to each grid cell of interpolated and extrapolated velocity data.
Using a Monte-Carlo method, we find an uncertainty in melt rate of 0.3 m/d and in Qsg of 6 m3/s.
4. Formulation used in the MITgcm to model melt rates
The three-equation formulation is widely used for sea ice and ice shelf basal melting [e.g.,
Holland et al., 2008; Walker and Holland, 2007]. Xu et al. [2012] applied this formulation to
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subaqueous melting on a near-vertical glacier face in two dimensions (2D). Here, we modify this
formulation for a three-dimensional (3D) simulation.
The three-equation formulation expresses: 1) the dependence of the freezing point of
seawater as a function of salinity and pressure, 2) the conservation of heat at the ice-ocean
boundary, and 3) the conservation of salinity at the ice-ocean boundary. Equation S6 is a
standard equation, identical in all model formulations. In Equations S7-S8, the heat and salinity
transfer coefficients, γT and γS, respectively, are taken equal to Cd1/2ГT and Cd1/2ГS, as in Jenkins
et al. [2010]. The speed in Eq. S7-S8 is the norm |u| of the velocity vector of water at the
boundary. Equations S7-S8 include background heat and salinity transfer rates, ГT0 and Гs0,
which reflect that melting occurs even in the absence of water currents. The values of ГT0 and Гs0
are taken from Hellmer and Olbers [1989] and Losch [2008].
TB = a SB + b + cpB
(S6)
Cpw  (γT |u| + ГT0)(T − TB) = −q [Lf + CpI (Tice − TB)]
(S7)
 (γs |u| + ГS0)(S − SB) = −q (SB – Sice)]
(S8)
6
5. Table S1. Parameters and variables used in Eq. S6-S8
Symbol
a
b
c
γT
γS
ГT0
ГS0
Lf
CpI
CpW

u
pB
T
TB
Tice
S
SB
Sice
q
Description
Parameter for freezing point
Parameter for freezing point
Parameter for freezing point
Heat transfer coefficient
Salinity transfer coefficient
Background heat transfer rate
Background salinity transfer rate
Latent heat of water
Heat capacity of ice
Heat capacity of water
Seawater density
Velocity of water along the ice face
In-situ pressure
Seawater temperature
Boundary layer temperature
Ice temperature
Seawater salinity
Boundary layer salinity
Ice salinity
Melt rate of ice
Value
-0.0575
0.0901
-7.61×10-4
1.1×10-3
3.1×10-5
1.0×10-4
5.05×10-7
334000
2000
3994
0
Unit
C psu-1
C
C db1
m/s
m/s
J kg-1
J kg-1 C-1
J kg-1 C-1
kg m-3
m/s
Db
C
C
C
Psu
Psu
Psu
kg m-2 s-1
6. Subglacial freshwater discharge and channel sizes
We use surface runoff outputs from the Regional Atmospheric Climate Model (RACMO) to
constrain the subglacial freshwater discharge, Qsg, of Store Glacier in 2010. Qsg is near zero (<
0.02 m3/s) in winter and peaks at 1200 m3/s in mid-July across the 5-km wide ice front
[van Angelen et al., 2012]. We do not know, however, the size and spatial distribution of
subglacial water channels. Evidence from the surface suggests the presence of at least three
major channels on the left and right sides of the glacier and at the center, but the possible
existence of other channels cannot be excluded. Röthlisberger [1972] suggested that the crosssectional area of the subglacial channels is controlled by the rate of closure caused by ice
deformation and the rate of opening due to melting of the inner face. Store Glacier has a draft of
about 500 m below sea level and a freeboard height of 70 m above sea level [Howat et al., 2010].
7
This configuration makes Store Glacier near hydrostatic equilibrium at the terminus, and
therefore the pressure difference between the water in the channel and ice above will be small,
which should yield low channel closure rates. Subglacial channels are therefore expected to have
a large cross-sectional area and be broad and low in shape [Hooke et al., 1990]. In our
simulations, we only consider the case of large, low and wide channels.
7. Dependence of the melt rate on subglacial water distribution.
To evaluate the impact of channel size, we keep TF constant as in Fig. 1b, and we compare
the results obtained with four different channel configurations that contribute to a total Qsg of 10
m3/s: 1) a 1-m high by 20-m wide central channel with a freshwater flow speed of 0.5 m/s; 2) a
1-m high by 50-m wide central channel with a flow speed of 0.2 m/s; 3) a 1-m high by 150-m
wide channel below the entire ice face with a flow speed of 0.067 m/s; and 4) 25 channels of 1 m
in height and 2 m in width, evenly distributed at the grounding line, with a freshwater speed of
0.2 m/s in each channel.
Figure S3 shows that for channel type 1, the plume is turbulent and occupies the entire model
domain above 300 m depth. The maximum melt rate is 6 m/d about 80 m above the channel and
the average melt rate over the entire ice face is 2.2 m/d. For channel type 2, the plume is initially
laminar; it becomes turbulent above 380 m depth, at which point it spreads laterally. The highest
melt rate is 8 m/d about 200 m above the channel and the average melt rate is 2.4 m/d. For
channel type 3, the plume remains laminar and rises all the way to the surface. The average melt
rate is 5.0 m/d. Finally, for channel type 4, the individual plumes merge into a larger plume that
melts the entire ice face; the maximum melt rate is 6 m/d and the average melt rate is 2.8 m/d. If
we omit the case of channel type 3, where the plume never reaches turbulence because of
relatively low Reynolds number of the simulation, we conclude that uncertainties in the shape of
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the subglacial channel introduce an uncertainty in melt rate of ±15 %. This is the level of
uncertainty that we use in Fig. 3.
Figure S3. Simulations of the melt rate
of a submerged calving face with
different configuration of subglacial
water channels corresponding to a
constant subglacial discharge Qsg = 10
m3/s. Thermal forcing, TF = 4.34oC
as in Fig. 1b. a-d) side view of
salinity (S), with water velocity
represented as black arrows; e-h) face
view of water speed (V) adjacent to
the ice face; and i-l) face view of the
time-averaged melt rate (qm).The 4
channel configurations are: a,e,i) a
central channel of 1 m in height × 20
m in width; b,f,j) a central channel of
1 m × 50 m; c,g,k) a wide channel of 1
m × 150 m; and d,h,l) 25 channels of 1
m × 2 m, evenly distributed along the
grounding line.
9
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