# Shallow-shelf approxmation

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The shallow shelf equations are given by

\begin{align} & \frac{\partial }{\partial x}\left( 2\bar{\eta }H\left( 2\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y} \right) \right)+\frac{\partial }{\partial y}\left( \bar{\eta }H\left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right) \right)=\rho gH\frac{\partial s}{\partial x} \\ & \frac{\partial }{\partial y}\left( 2\bar{\eta }H\left( 2\frac{\partial v}{\partial y}+\frac{\partial u}{\partial x} \right) \right)+\frac{\partial }{\partial x}\left( \bar{\eta }H\left( \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x} \right) \right)=\rho gH\frac{\partial s}{\partial y} \\ \end{align},

where u and v are the depth-independent x and y components of velocity, $\bar{\eta }$ is the depth-averaged effective viscosity, H is the ice thickness, ρ is the ice density, g is the acceleration due to gravity, and s=s(x,y) is the ice surface elevation.

Notice the symmetry in the equations. This means that, computationally, many of the same subroutines can be used for discretization.