PIK options for marine ice sheets

References [78], [75], [30] by the research group of Prof. Anders Levermann at the Potsdam Institute for Climate Impact Research (“PIK”), Germany, describe most of the mechanisms covered in this section. These are all improvements to the grounded, SSA-as-a-sliding law model of [22]. These improvements make PISM an effective Antarctic model, as demonstrated by [79], [3], [80], among other publications. These improvements had a separate existence as the “PISM-PIK” model from 2009–2010, but since PISM stable0.4 are part of PISM itself.

A summary of options to turn on most of these “PIK” mechanisms is in Table 22. More information on the particular mechanisms is given in sub-sections Stress condition at calving fronts through Sub-grid treatment of the grounding line position that follow the Table.

Table 22 Options which turn on PIK ice shelf front and grounding line mechanisms. A calving law choice is needed in addition to these options.
Option Description
-cfbc apply the stress boundary condition along the ice shelf calving front [30]
-kill_icebergs identify and eliminate free-floating icebergs, which cause well-posedness problems for the SSA stress balance solver [30]
-part_grid allow the ice shelf front to advance by a part of a grid cell, avoiding the development of unphysically-thinned ice shelves [78]
-subgl apply interpolation to compute basal shear stress and basal melt near the grounding line [81]
-no_subgl_basal_melt don’t apply interpolation to compute basal melt near the grounding line if -subgl is set [81]
-pik equivalent to option combination -cfbc -kill_icebergs -part_grid -subgl

Note

When in doubt, PISM users should set option -pik to turn on all of mechanisms in Table 22. The user should also choose a calving model from Table 24. However, the -pik mechanisms will not be effective if the non-default FEM stress balance -ssa_method fem is chosen.

Stress condition at calving fronts

The vertically integrated force balance at floating calving fronts has been formulated by [36] as

(17)\[\int_{z_s-\frac{\rho}{\rho_w}H}^{z_s+(1-\frac{\rho}{\rho_w})H}\mathbf{\sigma}\cdot\mathbf{n}\;dz = \int_{z_s-\frac{\rho}{\rho_w}H}^{z_s}\rho_w g (z-z_s) \;\mathbf{n}\;dz.\]

with \(\mathbf{n}\) being the horizontal normal vector pointing from the ice boundary oceanward, \(\mathbf{\sigma}\) the Cauchy stress tensor, \(H\) the ice thickness and \(\rho\) and \(\rho_{w}\) the densities of ice and seawater, respectively, for a sea level of \(z_s\). The integration limits on the right hand side of equation (17) account for the pressure exerted by the ocean on that part of the shelf, which is below sea level (bending and torque neglected). The limits on the left hand side change for water-terminating outlet glacier or glacier fronts above sea level according to the bed topography. By applying the ice flow law (section Ice rheology), equation (17) can be rewritten in terms of strain rates (velocity derivatives), as one does with the SSA stress balance itself.

Note that the discretized SSA stress balance, in the default finite difference discretization chosen by -ssa_method fd, is solved with an iterative matrix scheme. If option -cfbc is set then, during matrix assembly, those equations which are for fully-filled grid cells along the ice domain boundary have terms replaced according to equation (17), so as to apply the correct stresses [78], [30].

Partially-filled cells at the boundaries of ice shelves

Albrecht et al [78] argue that the correct movement of the ice shelf calving front on a finite-difference grid, assuming for the moment that ice velocities are correctly determined (see below), requires tracking some cells as being partially-filled (option -part_grid). If the calving front is moving forward, for example, then the neighboring cell gets a little ice at the next time step. It is not correct to add that little mass as a thin layer of ice which fills the cell’s horizontal extent, as that would smooth the steep ice front after a few time steps. Instead the cell must be regarded as having ice which is comparably thick to the upstream cells, but where the ice only partially fills the cell.

Specifically, the PIK mechanism turned on by -part_grid adds mass to the partially-filled cell which the advancing front enters, and it determines the coverage ratio according to the ice thickness of neighboring fully-filled ice shelf cells. If option -part_grid is used then the PISM output file will have field ice_area_specific_volume which tracks the amount of ice in the partially-filled cells as a “thickness”, or, more appropriately, “volume per unit area”. When a cell becomes fully-filled, in the sense that the ice_area_specific_volume reaches the average of the ice thickness in neighboring ice-filled cells, then the residual mass is redistributed to neighboring partially-filled or empty grid cells.

The stress balance equations determining the velocities are only sensitive to “fully-filled” cells. Similarly, advection is controlled only by values of velocity in fully-filled cells. Adaptive time stepping (specifically: the CFL criterion) limits the speed of ice front propagation so that at most one empty cell is filled, or one full cell emptied, per time step by the advance or retreat, respectively, of the calving front.

Iceberg removal

Any calving mechanism (see section Calving) removes ice along the seaward front of the ice shelf domain. This can lead to isolated cells either filled or partially-filled with floating ice, or to patches of floating ice (icebergs) fully surrounded by ice free ocean neighbors. This ice is detached from the flowing and partly-grounded ice sheet. That is, calving can lead to icebergs.

In terms of our basic model of ice as a viscous fluid, however, the stress balance for an iceberg is not well-posed because the ocean applies no resistance to balance the driving stress. (See [38].) In this situation the numerical SSA stress balance solver will fail.

Option -kill_icebergs turns on the mechanism which cleans this up. This option is therefore generally needed if there is nontrivial calving or significant variations in sea level during a simulation. The mechanism identifies free-floating icebergs by using a 2-scan connected-component labeling algorithm. It then eliminates such icebergs, with the corresponding mass loss reported as a part of the 2D discharge flux diagnostic (see section Saving time series of spatially-varying diagnostic quantities).

Sub-grid treatment of the grounding line position

The command-line option -subgl turns on a parameterization of the grounding line position based on the “LI” parameterization described in [82] and [81]. With this option PISM computes an extra flotation mask, available as the cell_grounded_fraction output variable, which corresponds to the fraction of the cell that is grounded. Cells that are ice-free or fully floating are assigned the value of \(0\) while fully-grounded icy cells get the value of \(1\). Partially grounded cells, the ones which contain the grounding line, get a value between \(0\) and \(1\). The resulting field has two uses:

  • It is used to scale the basal friction in cells containing the grounding line in order to avoid an abrupt change in the basal friction from the “last” grounded cell to the “first” floating cell. See the source code browser for the detailed description and section MISMIP3d for an application.
  • It is used to adjust the basal melt rate in cells containing the grounding line: in such cells the basal melt rate is set to \(M_{b,\text{adjusted}} = \lambda M_{b,\text{grounded}} + (1 - \lambda)M_{b,\text{shelf-base}}\), where \(\lambda\) is the value of the flotation mask. Use -no_subgl_basal_melt to disable this.

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