The Table 24 summarizes options controlling calving parameterizations implemented in PISM.

Table 24 Options for the calving models in PISM.
Option Description
-calving_cfl Apply CFL-type criterion to reduce (limit) PISM’s time step using the horizontal calving rate computed by eigen_calving or vonmises_calving.
-calving eigen_calving Physically-based calving parameterization [75], [30]. Whereever the product of principal strain rates is positive, the calving rate is proportional to this product.
-eigen_calving_K (\(m s\)) Sets the proportionality parameter \(K\) in \(\text{m}\,\text{s}\).
-calving vonmises_calving Physically-based calving parameterization [76] that uses the tensile von Mises stresses.
-vonmises_calving_sigma_max (\(Pa\)) Sets the maximum tensile stress \(\tilde{\sigma}\) in \(\text{Pa}\).
-calving thickness_calving Calve all near-terminus ice which is thinner than ice threshold thickness \(H_{\textrm{cr}}\).
-thickness_calving_threshold (m) Sets the thickness threshold \(H_{\textrm{cr}}\) in meters.
-thickness_calving_threshold_file Specifies the file containing the variable calving_threshold to be used as the spetially-variable thickness threshold
-calving float_kill All floating ice is calved off immediately.
-float_kill_margin_only At each time step, calve cells at the ice margin only instead of removing all floating ice.
-float_kill_calve_near_grounding_line Calve floating ice near the grounding line (this is the default). Disable using -float_kill_calve_near_grounding_line off.
-calving ocean_kill All ice flowing into grid cells marked as “ice free ocean”, according to the ice thickness in the provided file, is calved.
-ocean_kill_file Sets the file with the thk field used to compute maximum ice extent.

To select several calving mechanisms, use a comma-separated list of keywords mentioned in Table 24:

-calving eigen_calving,thickness_calving,ocean_kill,vonmises_calving

Eigen calving

PISM-PIK introduced a physically-based 2D-calving parameterization [75]. This calving parameterization is turned on in PISM by option -calving eigen_calving. Average calving rates, \(c\), are proportional to the product of principal components of the horizontal strain rates, \(\dot{\epsilon}_{_\pm}\), derived from SSA-velocities

(18)\[c = K\; \dot{\epsilon}_{_+}\; \dot{\epsilon}_{_-}\quad\text{and}\quad\dot{\epsilon}_{_\pm}>0\:.\]

The rate \(c\) is in \(\text{m}\,\text{s}^{-1}\), and the principal strain rates \(\dot\epsilon_\pm\) have units \(\text{s}^{-1}\), so \(K\) has units \(\text{m}\,\text{s}\). The constant \(K\) incorporates material properties of the ice at the front. It can be set using the -eigen_calving_K option or a configuration parameter (calving.eigen_calving.K in src/pism_config.cdl).

The actual strain rate pattern strongly depends on the geometry and boundary conditions along the confinements of an ice shelf (coast, ice rises, front position). The strain rate pattern provides information in which regions preexisting fractures are likely to propagate, forming rifts (in two directions). These rifts may ultimately intersect, leading to the release of icebergs. This (and other) ice shelf calving models are not intended to resolve individual rifts or calving events, but it produces structurally-stable calving front positions which agree well with observations. Calving rates balance calving-front ice flow velocities on average.

The partially-filled grid cell formulation (section Partially-filled cells at the boundaries of ice shelves) provides a framework suitable to relate the calving rate produced by eigen_calving to the mass transport scheme at the ice shelf terminus. Ice shelf front advance and retreat due to calving are limited to a maximum of one grid cell length per (adaptive) time step. The calving rate (velocity) from eigen_calving can be used to limit the overall timestep of PISM–thus slowing down all of PISM–by using -calving_cfl. This “CFL”-type time-step limitation is definitely recommended in high-resolution runs which attempt to model calving position accurately. Without this option, under certain conditions where PISM’s adaptive time step happens to be long enough, dendritic structures can appear at the calving front because the calving mechanism cannot “keep up” with the computed calving rate.

Von Mises stress calving


This code is experimental and has not yet been thoroughly tested, use at your own risk.

While eigen-calving (section Eigen calving) is appropriate for Antartic ice shelves, it does not work for outlet glaciers that flow in narrow fjords. Along valleys with nearly parallel walls, the transverse component of the velocity is close to zero, and the transversal strain rate is therefore also close to zero and noisy.

Instead of the product of the eigen strain rates, [76] proposes a calving law where the calving rate \(c\) is a functionally related to tensile stresses:

(19)\[c = |\mathbf{u}| \frac{\tilde{\sigma}}{\sigma_{max}},\]

where \(\tilde{\sigma}\) is the tensile von Mises stress and \(\sigma_{max}\) is a threshold that has units \(Pa\). It can be set as a configuration parameter (calving.vonmises.sigma_max in src/pism_config.cdl). As the tensile fracture strength is much smaller than the compressive fracture strenth, the effective tensile strain rate is defined as

(20)\[\tilde{\dot{\epsilon}}_e = \left(\frac{1}{2}\left(\max(0,\dot{\epsilon}_{_+})^2 + \max(0,\dot{\epsilon}_{_-})^2\right)\right)^{1/2}.\]

Following [76] \(\tilde{\sigma}\) is given by

(21)\[\tilde{\sigma} = \sqrt{3} B \tilde{\dot{{\epsilon}}}_e^{1/n},\]

where \(B\) is the ice hardness.

Similar to eigen_calving, the calving rate from vonmises_calving can be used to limit the overall timestep of PISM — thus slowing down all of PISM — by using -calving_cfl.

Additional calving methods

PISM also includes three more basic calving mechanisms (Table 24). The option -calving thickness_calving is based on the observation that ice shelf calving fronts are commonly thicker than about 150–250 m (even though the physical reasons are not clear yet). Accordingly, any floating ice thinner than \(H_{\textrm{cr}}\) is removed along the front, at a rate at most one grid cell per time step. The value of \(H_{\mathrm{cr}}\) can be set using the -thickness_calving_threshold option or the calving.thickness_calving.threshold configuration parameter.

To set a spatially-variable ice thickness threshold, use the option -thickness_calving_threshold_file or the parameter calving.thickness_calving.threshold_file. This file should contain the variable calving_threshold in meters (or other compatible units).

Option -calving float_kill removes (calves), at each time step of the run, any ice that satisfies the flotation criterion. Use of this option implies that there are no ice shelves in the model at all.

Use the option -float_kill_margin_only to restrict this to cells at the ice margin.

Sometimes it is useful to preserve a one-cell-wide shelf near the grounding line. To do this, set calving.float_kill.calve_near_grounding_line to false.

Option -calving ocean_kill chooses the calving mechanism removing ice in the “open ocean”. It requires the option -ocean_kill_file, which specifies the file containing the ice thickness field thk. (This can be the input file specified using -i.) Any locations which were ice-free (thk == 0) and which had bedrock elevation below sea level (topg < 0), in the provided data set, are marked as ice-free ocean. The resulting mask is not altered during the run, and is available as diagnostic field ocean_kill_mask. At these places any floating ice is removed at each step of the run. Ice shelves can exist in locations where a positive thickness was supplied in the provided data set.

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