Hydrology Solution - Shreve Approximation

Physical basis

This model is the one described in [LeBrocq2009]. Here we present only the main equations.

Water column

The model applied here is the most simplistic form of the water-film model, as described by the Weertman theory [Weertman1957]. The model solves for the thickness $Equation 1$ of the water-film as follows:

$Equation 2$

where:

$Equation 4$ is the source term $Equation 3$
$Equation 6$ is the water velocity vector $Equation 5$

The water velocity vector $Equation 7$ is a depth-averaged two dimensional horizontal vector, which is computed using a theoretical treatment of laminar flow between two parallel plates:

$Equation 8$

$Equation 10$ is the hydraulic potential $Equation 9$
$Equation 12$ is the water viscosity $Equation 11$

In this model, the hydraulic potential $Equation 13$ is defined following the Shreve approximation [Shreve1972], which hypothesizes a null effective pressure. Assuming this null effective pressure gives the hydraulic potential gradient as follows:

$Equation 14$

where:

$Equation 16$ is the density of the ice $Equation 15$
$Equation 18$ is the density of fresh water $Equation 17$
$Equation 20$ is the surface elevation $Equation 19$
$Equation 22$ is the gravitational acceleration $Equation 21$
$Equation 24$ is the bedrock elevation $Equation 23$

Numerical implementation

To stabilize the equation, artificial diffusion might be added to the left hand side:

$Equation 25$

where $Equation 26$ is the artificial diffusivity. We take:

$Equation 27$

Model parameters

The parameters relevant to the water column solution can be displayed by running:

>> md.hydrology

md.hydrology.spcwatercolumn: water thickness constraints (NaN means no constraint) $Equation 28$
md.hydrology.stabilization: artificial diffusivity (default is 1).

Running a simulation

To run a simulation, use the following command:

>> md = solve(md, 'Hydrology');