Grounding Line Migration
Presently CISM uses the same grounding line retreat and advance mechanism used in Glimmer. This is based up the flotation criteria combined with a calving rate. Algorithmically
At a cell at the grounding line, if the continuity equation predicts that at time
then the cell becomes occupied with ice, thickness , otherwise the cell has zero ice thickness. Here is the fraction of the ice that is assumed to calve away.
While this is a fairly typical treatment, we hope to do better.
van der Veen's Method
Following Fundamentals of Glacier Dynamics (2002) page 396, from continuity, the evolution of thickness at the grounding line obeys
The term is our primary concern, as it is the stretching of the ice as it moves onto an ice shelf. It is simpler to consider the one dimensional problem of the ice spreading outwards, normal to the shelf. The one dimensional continuity equation is
is the longitudinal stretching of the ice shelf, . A one dimensional analytical expression of this can be derived from the constituative relation for ice
and by noting that hydrostatic equilibrium at the ice/ocean interface implies
The balance of forces in an ice shelf is
where the lateral drag on the ice shelf is and the ice shelf's half width is .
Substitution of previous expressions for and recalling that an ice shelf at floatation has driving stress gives
With the 'back pressure' equal to
This derivation neglects the back pressure that may arise from ice rises. As we will ultimately seek a parametrization of , rather than doing an actual integral, it is alright to lump that contribution into .
With out being able to evaluate by direct means, we hypothesis that it is a function of 1) geometry of the embayment, 2)temperature of the ocean and atmosphere, and 3) position of the calving front. Each is now treated in turn.
geometry of embayment
The geometry can be crudely measured in terms of the aspect ratio of the embayment. If the ratio is length in transverse divided by length in longitudinal flow direction, then larger ratios produce larger resistances.
temperatures of ocean and atmosphere
The temperature and position dependence would lead to a time delay for the temperature sensitivity that is asymmetric. The delay between cold temperatures and ice shelf advance is longer than the delay between warm temperatures and ice shelf retreat.