1. Modeling of Subaqueous Melting of Greenland Tidewater glaciers
Yun Xu1, Eric Rignot1,2, Dimitris Menemenlis2, Ian Fenty2, Mar Flexas2, François Primeau1
1 Earth System Science, University of California Irvine, Irvine, CA, United States 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, United States

2. Subaqueous Melting Warm water
Melting occurs on the vertical calving face of tidewater glaciers
large subaqueous melt rate [m/day]
a potential trigger of calving
causes glacier un-grounding and retreat, trigger glacier acceleration
Warm water
Subglacial runoff (fresh, cold)
(Motyka, 2003; Rignot et al., 2010)
First, let me explain the term ‘subaqueous melting’ Greenland tidewater glaciers are usually grounded on the sea floor, and subaqueous melting here refers to the melting on the vertical calving face of tidewater glaciers.
Subaqueous melting is important because
Subaqueous melt rate is fast [m/d], it is an important term for the mass balance of glacier.
It might be a trigger of calving
Cause glaciers un-grounding and retreat.
Two factors can influence the subaqueous melting, one is ocean temperature.If ocean temperature increases, melt rate could increase. Another one is subglacial runoff from the inland surface melting. surface melt water drains to the bottom of the glacier and flows into the ocean under the ice. It’s fresh and buoyant, so it upwells in front of the ice, causes turbulent mixing... We include both processes in the model.

3. Modeling of subaqueous melting with MITgcm
Melt water
In model, we have ocean and ice. Ocean transport heat and salinity to the ice, melt the ice, and the melt water is added to the ocean and change ocean T/S. Here in the model, ice is fixed, doesn’t change position after melting. The heat transfer from the ocean to the ice is proportional to the temperature difference between ocean and ice boundary, also to a transfer coefficient gammaT, gammaS. GammaT gammaS, are proportional to the velocity near the ice face. we add small constant term for the background metling because if we put a piece of ice in the water, it would melt even if there is no large scale motion. There terms has much smaller effects compare to velocity term, when subglacial runoff is added.
ΓT, ΓT follow Jenkins et al. (2010)
γT0, γS0 follow Losch (2008) x z
subglacial runoff
γT / γS
Heat /salinity transfer coefficient W
Vertical velocity
TB/SB
Temperature/salinity at the Boundary
T / S
Temperature/salinity of ocean water

4. 2-D Sensitivity Experiments Results
Melt rate (q) vs subglacial flux (Qsg)
 q depends on Qsg: q is very small when there is no Qsg, and increases sub-linearly with Qsg.
q is not sensitive to channel height
q ~ m/d
Melt rate (q) vs thermal forcing (TF)
 TF = ocean temperature – freezing point
 q increases linearly with TF
Melt rate is not sensitive the channel height unless at high subglacial flux, when the velocity of the subglacial runoff is high and kind of detach from the ice.

5. 2-D Sensitivity Experiments – 20 m(h) x 5 m(v) resolution
Subglacial plume is not resolved
— 3D high-resolution
In the simulations to get previous resutls, the resolution is 20m x 5m.
The color here shows the subglacial freshwater. We see that subglacial plume is confined in the first one or two grids from the ice, it’s not resolved.
This make me think about 3D high-resolution simulations.

6. Simplified Store Glacier
3-D Experiments
Simplified Store Glacier
Resolution (ΔX × ΔY × ΔZ) : 2m × 2m × 1-2m
Domain size (L × W × H) : 1000m × 300m × 500m
Subglacial discharge* : ~30 m3/s
5 km
In 3D experiments, we simulate the simplified case of store glacier.
Store glacier is 5 km wide at the ice front, and 500m deep under sea surface. Freshwater is flowing in the subglacial channels, and enter the ocean through discrete locations.
In this 3D experiment, I want to see a lot of details of the subglacial plume behavior, the ocean circulation in front of the ice… so I use 2m resolution here.
Because of the high resolution and because I am an impatient person (I don’t want to wait for result for more than one week). I can only simulate a small domain size shown in the black box. Its 1km from the ice, 300m along the ice front, and 500m deep.
RACMO date shows the subglacial discharge is about 500m3/s in summer. If it’s evenly distributed, there is 30m3/s of freshwater discharge in our 300m wide domain.
500 m
* Subglacial freshwater discharge in summer is ~500 m3/s (RACMO data), assumed to enter the ocean through discrete subglacial channels over 5-km ice front.

7. Subglacial Channel Size
Assume one channel in the middle of the domain.
Store glacier is close to hydrostatic equilibrium or afloat at the terminus. The channel could be very big.
Tank experiments: In big channels, fresh water could flow only through the top of the channel to ensure Fr > 1. Fr = inertia / buoyancy =
So we have the total amount of discharge, we need to determine the channel size.
Assume there is only one channel in the middle of the domain at the sea floor.
We know Store glacier is very close to hydrostatic equilibrium at the glacier terminus, and our ocean survey even indicate it’s afloat at some locations. Therefore the subglacial channel could be very big at the floating case or calculated based on R-channel theory.
After doing some simple fishtank experiments, I realized that in big channels ,freshwater could flow through the top of channel to ensure Fr >1. i.e. inertial has to overcome buoyancy to flow out of the channel.
Which means, if there is a big channel shown in the black box, and only a relative small amount of freshwater flowing in the channel, it will altomatically ocupy only the top portion of the channel. This is similar the case the channels at the ice shelf.
So in the experiment, we determine the freshwater is flowing in the rectangular shap of 3m high and 10m width at the speed of 1m/s. in this case, Fr =
In this designed channel,
Fr = 1.13,
Re = 5000, (108 in reality)
10 m
3 m
1 m/s

8. 3-D Model Configuration Free surface, non-hydrostatic regime
Domain Size (L × W × H)
1000m × 300m × 500m
Resolution (ΔX × ΔY × ΔZ)
2 m × 2 m × 1-2 m
Ocean Forcing
T, S from field observation
Subglacial channel size (W × H)
10 × 3 m
plume speed
1.0 m/s
viscA4
0.1
diffusivity
To sum up the 3D modle configuration, I use free surface non-hydrostatic regime. Very high resolution in a small domain. ocean initial condition and boundary condition, I use a T/S profile from observation.
Viscosity is small to ensure large Reynolds number, and diffusivity is 0.

9. Results - Salinity View from side View from ice Depth (m)
The high-resolution run clearly show the turbulent plume in the salinity map. Two figures, one is view from the side, we have ice of the right, and one is view from the ice.
The freshwater jet is entering the ocean at the speed of 1m/s, but it’s so buoyant, and therefore, it doesn’t move more that 10m at the floor, and turn to a upwelling plume. The turbulent plume grow in width at all directions along the ice face and away from the ice. It mixed with a lot of ambient water and become … at 100 – 150m depth, flow away from the ice. Ocean water flows to the plume below 200m.
Distance to the ice
(m)
Distance to the plume
(m)

10. Temperature Distance to the ice (m) Depth (m) Distance to the plume
View from side
View from ice
Temperature also show the turbulent plume pattern. From the temperature we also see that, the subglacial plume is colder than is sea water at the beginning, but it mixed with a lot of ambient water, and quickly become warm, and it bring the warm water rises along the ice face. only a small portion of heat carried by the plume is used to melt the ice, it’s barely noticed from the plume temperature, so the plume is still warm when it rises to the ??? neutrient layer. This can explain why In the ocean CTD data, cold layer at the mid-layer becomes less cold when getting close to the ice. It’s because of the turbulent mixing at the ice front.

11. Melt Rete Average melt rate = 1.7 m/d View from ice Depth (m)
This is the averaged melt rate over 10 minutes. We see the maximum melt in the plume middle is close higher than 5m/d, and melt rate is very low away from the plume. The averaged melt rate over the entire ice face is 1.7 m/d
Distance to the plume
(m)

12. Melt Rate Derived From Observations
Based on mass, heat and salinity conservation,
melt rate = 1.6 ± 1 m/d
Errors mainly from
1. velocity measurement,
2. omission of top 10m,
3. constant current velocity assumption where there is no measurement, and
4. tidal currents
We also derived melt rate from ocean data. We collect ocean CTD at 9 stations across the glacial fjord 1km from the ice. There are temperature, salinity and velocity data. Based on mass, heat and salinity conservations, we estimate the melt rate is about m/d. The error is large. But we find the melt rate is close to the value from the model, 1.7m/d

13. Conclusions
The MITgcm shows that subglacial runoff is dependent on the subglacial runoff and the ocean thermal forcing in 2-D sensitivity experiments.
The growth of turbulent plume is resolved in the 3-D 2 m-resolution experiment. This upwelling plume brings heat from ocean bottom to the ice face to melt the ice.
The summer melt rate of Store Glacier is about 1.7 m/d in the model, close to the observation derived melt rate of 1.6 ± 1 m/d.
Several aspects of the model require more work, e.g.
sensitivity on subglacial runoff configuration;
apply to larger domain.

14. Thank You!
I am working on the numerical modeling of subaqeuous melt rate of Greenland tidewater glaciers on the vertical ice-ocean interface using a ocean GCM.
Previous 2D sensitivity numerical experiments indicate that melt rate increases sub-linearly with subglacial discharge, and linearly with ocean thermal forcing. And the melt rate could reach a few meters per day in summer.
We now apply 3D 1m-resolution using NASA super computer in order to better compare with ocean observations