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 \(w\) of the water-film as follows:

\(\frac{\partial w}{\partial t}=S - \nabla \cdot {\bf u}_{w} w\) where:

  • \(S\) is the source term \([m\,s^{-1}]\)
  • \({\bf u}_w\) is the water velocity vector \([m\,s^{-1}]\)

The water velocity vector \({\bf u}_w\) is a depth-averaged two dimensional horizontal vector, which is computed using a theoretical treatment of laminar flow between two parallel plates:

\[{\bf u}_w = \frac{w^2}{12 \mu}\nabla \phi\]
  • \(\phi\) is the hydraulic potential \([Pa]\)
  • \(\mu\) is the water viscosity \([Pa\,s]\)

In this model, the hydraulic potential \(\phi\) 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:

\(\nabla \phi=\rho_{ice} g \nabla s + \left(\rho_w - \rho_{ice}\right) g \nabla h\) where:

  • \(\rho_{ice}\) is the density of the ice \([kg\,m^{-3}]\)
  • \(\rho_w\) is the density of fresh water \([kg\,m^{-3}]\)
  • \(s\) is the surface elevation \([m]\)
  • \(g\) is the gravitational acceleration \([m\,s^{-2}]\)
  • \(h\) is the bedrock elevation \([m]\)

Numerical implementation

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

\(\frac{\partial w}{\partial t} +{\color{red} \nabla \left(\mathfrak{D} \nabla w\right)} =S - \nabla \cdot {\bf u}_w w\) where \(\mathfrak{D}\) is the artificial diffusivity. We take:

\[\mathfrak{D} = \frac{h}{2}\left(\begin{array}{cc}\left|vx\right| & 0 \\\\0 & \left|vy\right|\end{array}\right)\]

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) \([m]\)
  • md.hydrology.stabilization: artificial diffusivity (default is 1).

Running a simulation

To run a simulation, use the following command:

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

References

  • A. M. Le Brocq, A. J. Payne, M. J. Siegert, and R. B. Alley. A subglacial water-flow model for West Antarctica. J. Glaciol., 55(193):879-888, 2009.

  • R. L. Shreve. Movement of water in glaciers. J. Glaciol., 11(62):205-214, 1972.

  • J. Weertman. On the sliding of glaciers. J. Glaciol., 3:33-38, March 1957.