# 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, $S_{\varphi }$.