Blatter-Pattyn model

The starting point for the Blatter-Pattyn model is the full Stokes equations

$\begin{align} & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=0 \\ & y:\quad \frac{\partial \tau _{yy}}{\partial y}-\frac{\partial P}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=0 \\ & z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}+\frac{\partial \tau _{zy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial x}=\rho g \\ \end{align}$,

where P is the pressure and τ is the deviatoric stress tensor. The latter is given by

$\tau _{ij}=\sigma _{ij}+P\delta _{ij}$,

where σ is the full stress tensor.

There are a number of ways to argue that because of the "shallowness" of ice sheets - that is because the ratio of H/L, where H is the thickness and L is a relevant horizontal length scale, is small - the equations above can be reduced to the following "first-order approximation"

$\begin{align} & \frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\frac{\partial P}{\partial x} \\ & \frac{\partial \tau _{yy}}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=\frac{\partial P}{\partial y} \\ & \frac{\partial \tau _{zz}}{\partial z}=\rho g+\frac{\partial P}{\partial z} \\ \end{align}$