To investigate ice sheet evolution over the timescale of a glacial
cycle, 3-D ice sheet models (ISMs) are typically run at “coarse” grid
resolutions (10–50
We develop a new flow line SG model for embedding in coarse resolution
models.
A 1
The resolution used in any model of complex environmental systems (e.g. ice
sheet models (ISMs), general circulation models or hydrological models)
limits the processes that can be represented. For continental-scale glacial
cycle contexts, ISMs are currently run at resolutions of about 10–50 km
Any model of complex environmental systems will have sub-grid (SG) processes
that are, by definition, not dynamically resolved. Accurate modelling of such
systems must therefore determine whether the SG processes variability is relevant
for the given context. If it is, some of the impact of this SG variability
may be captured in a parameterized form
In this paper, “
In this paper, we develop a new SG model extending the approach of
The sub-grid model is a finite difference flow line model composed of a
diagnostic equation for the ice velocities and a prognostic equation for ice
thickness evolution. The surface mass balance is calculated using a positive
degree day (PDD) method. The elevation of a 3-D region is parameterized using
a hypsometric curve. Differences between the new SG model and the
Differences between our new SG model and the
We use 1
At any time step,
To compute the slope at the lowest bin we assume an ice cliff boundary
condition. The surface elevation
The flow line model requires an effective width,
Schematic representation of the effective
width of the 7th hypsometric bin for a region of 10
We use the positive degree day method described in
The GSM and ISSM compute the surface mass balance using the same PDD method.
The prognostic equation for the ice thickness (
In their most recent experiments,
Equation (
At the highest bin, we assume that no ice flows into the region. At the lowest bin ice is allowed to flow out of the region.
The shallow ice approximation, used to compute fluxes, is formally invalid
for high surface slopes such as present in mountain ranges like the Rockies.
Simulating ice evolution over a 3-D terrain using a flow line model limits
the ice flow representation. Ice flows from one SG bin to another using an
average slope. Our model configuration does not allow for ice at high
elevations to flow into an adjacent coarse grid cell. Nor does it allow for
ice present at low elevations, in isolated regions having a closed drainage
basin, to stay in a coarse grid cell. Moreover, the Arrhenius coefficient is
computed with a constant ice temperature of 0
The hypsometric length parameterization inferred from the surface slopes are correct for ice free regions, but it is only an approximation once the ice starts building up. At the lowest hypsometric bin, slopes are computed assuming ice cliff boundary conditions.
For the comparison against ISSM results, the surface temperature is
downscaled with a lapse rate of 6.5
The core of the GSM is a 3-D thermomechanically coupled ice sheet model. The
model incorporates sub-glacial temperatures, basal dynamics, a visco-elastic
bedrock response, climate forcing, surface mass balance, a surface drainage
solver, ice calving and margin forcing. The grid resolution used for this
study is 1.0
The thermomechanically coupled ice sheet model, described in detail in
The visco-elastic bedrock response is asynchronously coupled to the GSM with
a 100-year interval. This module is based on the complete linear
visco-elastic field theory for a Maxwell model of the Earth
At the surface, the parameterized climate forcing
The calving module, described in detail in
In this section, we describe how the SG model is embedded in the
GSM and the conditions applied to activate or deactivate the SG model in each
CG cell. The GSM is run, at all times, over all the CG cells and the ice
thickness is updated in cases where the SG model is activated.
Figure
Communication between the GSM and the SG model for one CG cell.
There is two-way communication between the GSM and SG models to exchange
information about ice thickness, surface mass balance, and surface
temperature. Ice in a CG cell is added to the SG level SG level
represents the hypsometric curve while CG level corresponds to a GSM cell.
Once the SG model is reactivated in a CG cell during deglaciation, the ice volume present at the CG level is distributed over the different hypsometric bins. To account for the higher volume of ice in valleys, represented by the lowest hypsometric bins, the average of the following two mass-conserving distributions is used for SG initialization. The first is even distribution across every bin. The second keeps equal surface elevation for the lowest bins, starting from the lowest bin and using as many bins as necessary.
The SG model flux module is coupled asynchronously and runs at half the SG mass balance time step. Glacial isostatic adjustment from the CG level is imposed on the SG basal topography.
Unlike
As a detailed description of the ISSM is given in
The SG model computation time for a 3000-year simulation, using 10 hypsometric
bins, is about 0.02 s. At a resolution of 1
We compare 2
The bed topography for this test is an inclined plane topography with a
constant slope of 0.014 and a maximum basal elevation of 800
SG model vs. ISSM differences over
an idealized inclined plane terrain. Average ice thickness differences
(SG model
The SG model is tested on 21 regions from the Canadian Rockies, representing
a wide range of topographic complexity (e.g. Fig.
Topography characteristics for six regions
over the Canadian Rockies.
In contradiction with the simplified inclined plane configuration, increasing
the number of hypsometric bins does not reduce the misfits with ISSM
simulations
(Fig.
Ratio of the SG model over ISSM
total ice volume for six different regions in the Rockies as a function of
hypsometric bins. The simulations were run until steady state with a
constant sea level temperature of 0
We examine the impact of including more topographic characteristics in the velocity parameterization. Characteristics considered include the flow direction, the terrain ruggedness (measured as the variation in three-dimensional orientation using a radius of 5 grid cells around the grid cell of interest), the sum of the squared slopes, the variance in the slopes, the number of local maxima (tested with radius sizes of 2, 6 and 10 grid cells) and the standard deviation of the surface elevation topography.
The ISSM and the sub-grid model were run until steady state (2
Average ice thickness in metres for different topographic regions in the Rockies. Results are shown for the ISSM, the regression model (generated by the stepwise regression fit including only the standard deviation of the topography) and the SG model using 10 hypsometric bins.
To explore potential improvement from accounting for the standard deviation
of the high resolution topography,
This equation is used in a simulation initialized with the ice thickness,
velocities and slopes of ISSM values at steady state. The parameters
The lowest hypsometric bin has the most significant misfits (e.g. Fig. S4 in
the Supplement). This is likely related to the margin ice cliff slope
parameterization. To try to correct this, we test the following
parameterization for the lowest hypsometric bin velocity:
Using the same least-squares approach as above, the parameters
Average ice thickness
root mean square error (RMSE)
between the ISSM and the SG model for different topographic regions.
Simulations are run over 2
The following modifications of the current version of the SG model have been
explored but did not improve the model. The central difference
discretization of the ice thickness in the effective diffusivity coefficient
was replaced by an upwind scheme. Simulations with different values of the
Arrhenius coefficient, the power of the ice thickness and the slope, in
Eq. (
Surface
elevation generated by the ISSM (solid blue line), the SG model with
no flux term, using 5 and 10 hypsometric bins, (dotted lines)
and the SG model including the flux term (solid thin red
line). These simulations use a constant sea level temperature of
0
We present results of simulations over the last glacial cycle. The 39
“ensemble parameters” of the GSM (attempting to capture the largest
uncertainties in climate forcing, ice calving, and ice dynamics) have been
subject to a Bayesian calibration against a large set of palaeoconstraints
for the deglaciation of North America, as detailed in
Using
a conversion factor of 2.519
The SG model can significantly alter the pattern of ice accumulation and
loss. Figure
Elevation comparisons when the
SG model is turned on (blue) or off (red) at different
time steps using the parameter vector nn9894.
The ensemble of simulations of the last glacial cycle over North America with
the SG model activated generates, on average, between 0 and 1
Ensemble mean (solid red line) and standard deviation (dotted blue line) eustatic sea level equivalent of the total ice volume differences when the SG model is turned on and off, for an ensemble run over the last glacial cycle.
The impact of the SG model depends, however, on the climate forcing and the
ice sheet extent and elevations. During inception, when the SG model is
turned on, ice accumulating in higher regions flows downhill and accumulates
in regions close to the ELA and in valleys (Fig.
Ice field during
inception at 115
Ice field at 50
The accounting of SG fluxes has varying impacts over a glacial cycle
simulation (Fig.
To better understand the range of responses to CG ice flow between grid cells
that have SG activated, three case scenarios can be considered. Case 1: ice
flows out of the lowest SG bins located above the ELA into the lowest SG bins
located above the ELA of another CG cell. There is limited impact of not
allowing ice to flow out of the CG cell as in both cases ice accumulates.
Case 2: ice flows out of the lowest SG bins located above the ELA into the
lowest SG bins located below the ELA of another CG cell. In that case,
turning off the fluxes between CG cells tends to reduce the total melt. Case
3: ice flows out of the lowest SG bins located below the ELA into the lowest
SG bins located below the ELA of another CG cell. Ice flowing into lower SG
bins generates higher melting rates, so permitting fluxes between CG cells
will in this case tend to increase ice mass loss. In cases 2 and 3, the
combination of ice flowing below the ELA from the adjacent CG cell and from
the bins above the ELA can raise the surface elevation of lower bins above
the ELA and reduce the melt. Depending on the proportion of each of these
cases, not allowing ice fluxes out of coarse grid cells with SG activated
generates higher or lower ice volumes
(Fig.
Total ice volume evolution for a simulation using parameter vector nn9894. “flux on” and “flux off” both include the SG surface mass balance calculations but the latter has no SG ice fluxes. “NofluxOut” has SG on, but no SG ice flux between coarse grid cells. The “SG OFF” line is most of the time hidden under the “flux off” line.
With
Ice volume evolution for a
simulation over North America (parameter vector nn9894) with the SG model turned on
during inception. “our flux” represents the flux code used in our SG model and “Marshall flux” the
flux code used in
As described in Sect.
Total ice volume evolution for a simulation over North America during inception with the SG model turned on (SG on) using the parameter vector of run nn9927. Different methods of ice redistribution at the CG level are compared. VC is for ice volume conservation, SC for surface elevation conservation and MC uses the maximum of the previous two methods. “SG off” represents a run where the SG model has been turned off.
Figure
Total ice volume evolution for a simulation using parameter vector nn9894. Different curves represent simulations with different minimum altitude variation thresholds used for the SG activation.
Our new sub-grid surface mass balance and flux model extends the
initial work of Depending on the regional topographic characteristics, the new SG
model simulates ice volumes 45 % lower to 15 % higher than simulated by
the ISSM (using 10 hypsometric bins). Increasing the number of hypsometric
bins to more than 10 did not reduce misfits for simulations over rough
topographic regions extracted from the Canadian Rockies. Turning off the SG internal fluxes increases the ice volume misfits
with ISSM simulations by a minimum of 100 %. Increasing the number of topography characteristics used in the SG
model, as suggested by
An ensemble of simulations over the last glacial cycle of the North American
ice complex shows, on average, an increase of ice generated with inclusion of
the SG model. The ensemble mean for each time step is between 0 and
1
Simulated ice evolution is sensitive to the treatment of ice fluxes
within the SG model and between the SG and CG levels.
The flux term has an important impact on the SG model. Not allowing
ice to flow between hypsometric bins increases the total ice volume with a
maximum increase of 50 % at 50 The flux term used in the Not allowing ice to flow out of a CG cell where SG is activated
increases or decreases the total ice volume depending of the ice
configuration. At 50 The ice configuration from simulations over the last glacial cycle of North
America is sensitive to the choice of SG to CG ice redistribution scheme.
We have identified the representation of SG fluxes between CG cells to be a challenging issue that can significantly impact modelling ice sheet evolution.
We have shown that the above geometric and ice dynamics factors can have
significant impacts on modelled ice sheet evolution (with up to a
35
The sub-grid code is available upon request from the first two authors.
Kevin Le Morzadec and Lev Tarasov designed the experiments. Kevin Le Morzadec developed the SG model code and performed the simulations. Kevin Le Morzadec and Lev Tarasov coupled the SG model into the GSM. Mathieu Morlighem and Helene Seroussi supported ISSM installation and helped build a new surface mass balance module for the ISSM. Kevin Le Morzadec prepared the manuscript with contributions from Lev Tarasov and the other co-authors. Lev Tarasov heavily edited the manuscript.
We thank Vincent Lecours and Rodolphe Devillers for extracting some of the topographic characteristics. Support provided by the Canadian Foundation for Innovation, the National Science and Engineering Research Council, and ACEnet. Tarasov holds a Canada Research Chair. We finally thank Philippe Huybrechts, as well as Fuyuki Saito and an anonymous reviewer, whose comments helped significantly improve the clarity of the manuscript.Edited by: D. Roche