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 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 (the \nabla \cdot operation) of the flux, Q, plus any contribution from sources, S_{\varphi }.