ASIS:Solver thickness

Thickness solver overview
(Tony P initial description for distribution to interested parties)

Ice geometry evolves according to

$$\frac{\partial H}{\partial t}=b-\nabla .H\bar{u}$$

where is H ice thickness, b the local mass balance (the difference between snowfall and snow and ice melt, including submarine melt in the case of floating ice shelves),

$$\nabla .$$ is the 2d-horizontal divergence operator and

$$\bar{u}$$ vertically-averaged horizontal velocity.

This equation is solved on a regular, staggered horizontal grid using a variety of techniques including leap frog. Recently, we have started using the piece-wise parabolic method. We use an explicit third-order scheme. It solves 2d advection by using operator splitting, i.e. solving 1d problems in each direction. Each 1d problem evolves the flow in Lagrangian coordinates using known $$\bar{u}$$ (i.e. calculated separately, see below), evolves mass by applying Lagrangian mass conservation and then remaps mass into Eulerian coordinates. The advection term used in remapping is calculated using parabolic interpolation functions that allow good representation of steep gradients.

The lateral boundary condition for an ice sheet that is losing mass via melt (either into the ocean or simply subaerial melt) is simply where the horizontal ice flux goes to zero. In practice, a fringe of non-ice covered cells is maintained around the ice sheet so that it can advance. At the end of each time step, this fringe is checked for negative ice thickness (which is set to zero). Some test cases prescribe the location of the margin.

The elevation of upper and lower surfaces is calculated from flotation, if the ice is in contact with the ocean, or simply by adding thickness to bedrock elevation, if the ice is grounded.

Depending on the type of ice flow (i.e., pattern of velocity, see below), two types of flow are found: parabolic in the slow flowing areas (flow is strongly related to gradients in ice-surface elevation, i.e. gravitation driving stress) or hyperbolic in fast flowing ice streams and ice shelves.