Earth deformation models

The option -bed_def [iso, lc] turns one of the two available bed deformation models.

The first model -bed_def iso, is instantaneous pointwise isostasy. This model assumes that the bed at the starting time is in equilibrium with the load. Then, as the ice geometry evolves, the bed elevation is equal to the starting bed elevation minus a multiple of the increase in ice thickness from the starting time:

\[b(t,x,y) = b(0,x,y) - f \left[H(t,x,y) - H(0,x,y)\right].\]

Here \(f\) is the density of ice divided by the density of the mantle, so its value is determined by setting the values of bed_deformation.mantle_density and in the configuration file; see PISM’s configuration parameters and how to change them. For an example and verification, see Test H in Verification.

The second model -bed_def lc is much more physical. It is based on papers by Lingle and Clark [88] and Bueler and others [89]. It generalizes and improves the most widely-used earth deformation model in ice sheet modeling, the flat earth Elastic Lithosphere Relaxing Asthenosphere (ELRA) model [90]. It imposes essentially no computational burden because the Fast Fourier Transform is used to solve the linear differential equation [89]. When using this model in PISM, the rate of bed movement (uplift) and the viscous plate displacement are stored in the PISM output file and then used to initialize the next part of the run. In fact, if gridded “observed” uplift data is available, for instance from a combination of actual point observations and/or paleo ice load modeling, and if that uplift field is put in a NetCDF variable with standard name tendency_of_bedrock_altitude in the input file, then this model will initialize so that it starts with the given uplift rate.

Here are minimal example runs to compare these models:

mpiexec -n 4 pisms -eisII A -y 8000 -o
mpiexec -n 4 pisms -eisII A -bed_def iso -y 8000 -o
mpiexec -n 4 pisms -eisII A -bed_def lc -y 8000 -o

Compare the topg, usurf, and dbdt variables in the resulting output files. See also the comparison done in [89].

To include “measured” uplift rates during initialization, use the option -uplift_file to specify the name of the file containing the field dbdt (CF standard name: tendency_of_bedrock_altitude).

Use the -topg_delta_file option to apply a correction to the bed topography field read in from an input file. This sets the bed topography \(b\) at the beginning of a run as follows:

(16)\[b = b_{0} + \Delta b.\]

Here \(b_{0}\) is the bed topography (topg) read in from an input file and \(\Delta b\) is the topg_delta field read in from the file specified using this option.

A correction like this can be used to get a bed topography field at the end of a paleo-climate run that is closer to observed present day topography. The correction is computed by performing a “preliminary” run and subtracting modeled bed topography from present day observations. A subsequent run with this correction should produce bed elevations that are closer to observed values.


The variable viscous_bed_displacement does not correspond to any measured physical quantity. Do not even attempt to analyze it without a careful reading of [89].

Trying to provide a “hand-crafted” viscous_bed_displacement field to PISM is not a good idea.

Keep in mind that zero viscous_bed_displacement does not mean that the bed deformation model is in equilibrium.

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