Execute Halfar on Greenland, and couple with Navier-Stokes

[lukem12345]

Halfar-NS

We have demonstrated the Halfar glacier dynamics on a sphere and on a scientific dataset of the Grigoriev ice cap. We have also demonstrated the Mohamed et al. formulation of the incompressible Navier-Stokes equations running on the sphere.

However, we have not demonstrated the Halfar dynamics running on a glacier "in context".

We should create a demo of the Halfar dynamics running on the Greenland ice sheet which is on the Earth. For the remainder of the Earth, we can simulate flow according to the Mohamed et al. formulation of the incompressible Navier-Stokes equations.

Coupling

To couple these physics, we need to specify what components they should share, and how.

The simplest interaction between Halfar's ice height, $h$, and Navier-Stokes' flow of water, $u$, is to have ice "block" the flow of water. That is, we want to couple:

graph LR
    A[Halfar] --> B((h))
    D[Blocking] --> B
    D[Blocking] --> E 
    C[NS] --> E((u))

Loading

$Halfar$ will be the result of the usual composition of Halfar's equation and Glen's law, and $NS$ will be the usual Decapode.

Ice Blocking Flow

To encode $Blocking$, we should say that $u$ is set to 0 when the wall of ice reaches it. We can "detect" the wall of ice by taking the gradient of ice height, and noting where it is non-zero: $dh$. If an edge is not crossing the wall of ice, we can set $u$ to the value it already is. Mathematically, we can state this like:
$$u = (1-\sigma(dh))u$$
, where $\sigma$ is a function which is 1 when $dh$ is nonzero, and 0 when $dh$ is 0. To ease the computational burden, we can have $\sigma$ do this smoothly, a la a sigmoid function.