Solving the equation for thickness evolution

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A generic conservation equation

Evolution of the ice sheet geometry is simply the result of mass conservation. Consider a generic conservation equation for some scalar value φ

\frac{\partial \varphi }{\partial t}=-\nabla \cdot \vec{Q}+S_{\varphi }

On the left-hand side we have the change in φ per unit time, which is given by the negative of the divergence of the flux, Q (a vector), plus any contribution from sources, S_{\varphi }. The negative sign in front of the divergence operator (the \nabla) insures that, for example in the case of heat flow, a negative temperature gradient causes an increase in temperature over time in the absence of source. Intuitively, we expect heat to flow from warmer to colder regions rather than vice versa. This is, perhaps, more obvious if we expand the equation above in terms of partial derivatives and the vector-valued flux components,

\frac{\partial \varphi }{\partial t}=-\left( \frac{\partial Q_{x}}{\partial x}+\frac{\partial Q_{y}}{\partial y} \right)+S_{\varphi }.

Conservation of mass for an ice sheet

For an ice sheet (or any depth-integrated mass flow for that matter), the equation becomes

  & \varphi =H,\quad \vec{Q}=\vec{U}H,\quad \vec{U}=\left( U,V \right),\quad S_{H}=\dot{b} \\ 
 &  \\ 
 & \frac{\partial H}{\partial t}=-\left( \frac{\partial \left( UH \right)}{\partial x}+\frac{\partial \left( VH \right)}{\partial y} \right)+\dot{b} \\ 

where {\vec{U}} is the depth-averaged velocity vector (in map plane), and U and V are the depth integrated x and y velocity fields, H is the ice thickness, and {\dot{b}} is the source term, which is the surface mass balance (>0 for accumulation and <0 for ablation).