We describe a program that produces paleo-ice sheet reconstructions using an assumption of steady-state, perfectly plastic ice flow behaviour. It incorporates three input parameters: ice margin, basal shear stress and basal topography. Though it is unlikely that paleo-ice sheets were ever in complete steady-state conditions, this method can produce an ice sheet without relying on complicated and unconstrained parameters such as climate and ice dynamics. This makes it advantageous to use in glacial-isostatic adjustment ice sheet modelling, which are often used as input parameters in global climate modelling simulations. We test this program by applying it to the modern Greenland Ice Sheet and Last Glacial Maximum Barents Sea Ice Sheet and demonstrate the optimal parameters that balance computational time and accuracy.
Reconstructing past ice sheets is a complex task, due to the large number of
parameters that can affect their growth and retreat. For example,
Since past climatic parameters are generally only well characterized in areas
outside of where paleo-ice sheets existed, ice sheet reconstructions that are
independently determined using evidence of glacial-isostatic adjustment (GIA)
are often used in paleo-climate simulations
The program presented in this paper produces a physically realistic ice sheet
reconstructions taking into account changes in basal shear stress and
topography, while being simple enough that it does not depend on numerous
parameters with large uncertainties. The goal of this program is to provide
an compromise between the GIA-only ice sheet reconstructions that have
limited or no physics applied to their construction, and the full glacial
systems models that demand considerable computational resources. The
reconstructions are based on the assumption of perfectly plastic,
steady-state ice conditions. It allows for the rapid determination of
paleo-ice sheet configurations, which is desirable when matching observations
of GIA. We present an example application of this program to the Barents Sea
Ice Sheet, a relatively short-lived portion of the Eurasian Ice Sheet
complex, by trying to match an existing GIA based model. We also apply the
model to the contemporary Greenland ice sheet to provide an indication of how
well the model is capable of reconstructing a known ice sheet geometry.
Ultimately, the goal would be to reconstruct, in a timestepped fashion, the
entire history of an ice sheet complex. In this case, the basal topography is
relatively well determined (since there is no existing ice), and the basal
shear stress can be established to a certain extent by the surficial geology
and geomorphology. The ice topography and basal shear stress are determined
through time using external evidence, such as the nature of GIA. An example
of this is presented for the western Laurentide Ice Sheet by
The reconstructions produced by the ICESHEET program are based on the
assumption that ice rheology adheres to perfectly plastic, steady-state
conditions (i.e. ignoring lateral shear stresses, and assuming that the ice
surface is not dynamically changing). The two-dimensional form of this theory
was derived by
In order to overcome problems with spatial changes in basal topography and
shear stress, in addition to the uncertainties in the location of the ice
sheet centre,
Equation (
It is important to note that assuming perfectly plastic, steady-state conditions for the ice sheet is not accurate in areas where the ice sheet was highly dynamic, or where lateral shear stress was an important factor. Due to this, the output basal shear stress is unlikely to reflect the true basal shear stress in those areas.
Schematic illustrating the steps in calculating the ice sheet, illustrating steps 7, 8, 10, 11, 12 and 14 in Sect. 2.2. The black lines indicate the initial elevation contour, blue lines indicate calculated flowlines, red lines indicate the next elevation contour, black circles indicate flowline initiation points, unfilled circles indicate added initiation points for the next elevation contour, crosses indicate flagged points that are not included in the next elevation contour.
In order to solve Eqs. (
The ice sheet reconstruction is calculated in a piece-wise manner (see
Fig.
All parameters (ice sheet margin, shear stress map, topography map)
are converted from geographical coordinates to a Cartesian coordinate system
prior to the execution of the program. Estimates of the basal shear stress for the area of interest are
read into the program. The shear stress values must be adjusted for each time
epoch to produce an appropriate ice sheet configuration. The basal topography data for the area of interest are read in.
For the first iteration of ice sheet model development, it uses modern
topography or topography adjusted for changes in global mean sea level (in
practice, it has limited impact on the final reconstruction, i.e. The program reads in the margin, and defines locations along the perimeter
where the flowline calculation initiates. The minimum distance along the
margin between where flowline calculation is initiated is user-defined. The
program defines the initial direction of flow to be perpendicular to the
margin, away from the centre of the ice sheet. The margin is set to have an initial ice thickness of 1 m.
If the margin is located where the topography is below sea level, it is
assumed that the margin corresponds to the grounding line of the ice sheet. A
conservative estimate of the thickness of ice at this point is set to
The calculation of ice elevation contours is a recursive process.
If the contour crosses over itself (signifying a saddle on the surface of the
ice sheet), the contour polygon is split, and the calculation is continued as
separate polygons (see step 12). The program searches for points on the contour that are below the next
contour elevation. The elevation may be above the next contour elevation
along the margin, or if a point coincides with a nunatuk (see step 14). It
then calculates the flowline by numerical integration of
Eqs. ( If the flowline calculation cannot reach the next contour elevation,
which happens when the topography is too high ( If the flowline direction changes sufficiently so that If the calculated flowline goes outside the last calculated contour
polygon, it is flagged and the point is not included in the next contour.
This happens when the ice surface is near its peak height. This can also
happen in areas where there is a sudden change in topography or basal shear
stress, which causes a deflection in the flowline direction
(Fig. After the flowlines are calculated for each applicable point along the
polygon, the program checks to see if any of the calculated flowlines cross
over. Offending crossovers are eliminated using a motorcycle algorithm
At this point, an initial polygon of the next elevation contour can be
constructed. This is checked to ensure that it is a simple polygon (i.e. a
polygon that does not cross over itself). If it is not, then the program
breaks it into several polygons, and determines whether they represent domes
(ice gradient is increasing towards the centre of the polygon) or saddles
(the ice gradient is decreasing towards the centre of the polygon). Where a
saddle is identified, it is determined to have reached its peak elevation and
is eliminated from subsequent calculations (Fig. The ice elevation and thickness for all points on a valid polygon
(including flagged points) are written to file. The polygon is resampled using the user-defined distance interval. There
is also a check using Eq. (
This process is repeated for each time interval of interest. After
calculation of the ice reconstruction, the calculated elevation values are
averaged into a grid to be used as input for a GIA calculation program. The
grid is created using a continuous smoothing algorithm, which is part of
Generic Mapping Tools
The Barents Sea Ice Sheet was predominantly marine based, and likely formed
by the merging of isolated ice caps over Svalbard, Franz Josef Land, Novaya
Zemlya and the Scandinavia Ice Sheet
In this sample problem, the ice sheet extent is taken as the “most likely”
configuration at 20 ka from the DATED project
Basal topography used in the resolution test, which is modern
topography minus the 133 m drop in global mean sea level at 20 ka. Also shown
in brown is the 20 ka ice margin
This purpose of this test is to demonstrate that GIA has an impact on the ice
sheet reconstruction. This test only includes the Barents Sea Ice Sheet for
the calculation of GIA. In a full glacial reconstruction
Results of the resolution test
In order to test the optimal parameters for producing ice sheet
reconstructions, a series of tests with different distance and contour
intervals were performed, the results can be found in
Table
The program execution time largely depends on the chosen sampling interval
along the contour polygons (Table.
Example from central–western Svalbard of how spatial changes in basal topography and basal shear stress affect the reconstructed ice surface topography of the ice sheet (see text). The contour interval is 100 m in the figure, though this sample was calculated with a 5 km spacing and 20 m contour interval. The dark black lines are the modern-day coastlines.
When an ice sheet grows, the basal topography is modified by GIA, which will
significantly impact the Barents Sea Ice Sheet example. Therefore, in order
to obtain an accurate characterization of the ice sheet surface topography
and thickness, it is necessary to re-run the program with the modified basal
topography. The Earth model used in this sample problem is spherically
symmetric and includes a 90 km thick elastic lithosphere,
The results show that one iteration of GIA has a significant effect on ice
sheet reconstruction, and in this case increases the total volume by about
5.8 % (Fig.
Additional tests by
Ice sheet reconstruction after one iteration of GIA.
The Greenland Ice Sheet serves as a good example of the capabilities of the
ICESHEET program. The basal topography under the ice sheet is an
observationally constrained, mass continuity based inversion of the
contemporary ice thickness
The goal of this example is to determine the misfit between the ICESHEET
reconstructed ice surface topography and the contemporary ice sheet using a
methodology analogous to the reconstruction of a paleo-ice sheet. The input
grounded ice margin and basal topography data come from the IceBridge
BedMachine Greenland, Version 2 data set
Sample reconstruction of the contemporary Greenland Ice Sheet.
The resulting reconstruction is shown in Fig.
The resolution test was also performed with the Greenland simulation
(Table
ICESHEET 1.0 is a program that can quickly create reconstructions of
paleo-ice sheets, with a given margin configuration and estimated basal shear
stress. We have provided two proof of concept examples showing
reconstructions of the modern Greenland Ice Sheet and the Barents Sea Ice
Sheet at the LGM. It is recommended that at least one iteration of GIA is
included to best characterize the thickness and ice surface topography. It is
also recommended (if a 5 km basal topography grid is used) to use a flowline
spacing interval of 5 km and contour interval of 20 m for optimal
calculation speed. This program has been used to create a full late glacial
GIA based ice sheet reconstruction of the western Laurentide ice sheet
The source code, licensed under GPL version 3, and Greenland Ice Sheet
example are available in the Supplement. Software updates will be available
on EJG's website (
The ICESHEET program was developed as part of a PHD project by EJG, and was
funded by an ANU Postgraduate Research Scholarship. This study is funded as
part of a Swedish Research Council FORMAS grant (grant 2013-1600) to Nina
Kirchner. We thank Nina Kirchner for comments that improved this paper.
Computing resources used for the development of ICESHEET were provided by
Terrawulf