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


\begin{align}
  & \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} \\ 
\end{align}


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). Note that, as above, a negative sign in front of the divergence insures sensible behavior. Consider a section of the ice sheet where the thickness is nearly constant and there is no accumulation or ablation. If the velocity gradient along-flow is negative (the ice is slowing down), we expect that to lead to thickening locally (left-hand side of the equation > 0) and vice versa (if the ice is speeding up along flow, that should lead to thinning locally).

This is the equation that needs to be solved to calculate ice sheet evolution. For the case of a SIA model, the values of U and V are recast in terms of ice thickness and elevation gradients, in which case the whole problem can be recast as a diffusion equation in ice thickness. In 1d, the equation becomes


\frac{\partial H}{\partial t}=\frac{\partial }{\partial x}\left( D\frac{\partial h}{\partial x} \right)+\dot{b},\quad D=\frac{2A}{n=2}\left( \rho g \right)^{n}H^{n+2}\left| \frac{\partial h}{\partial x} \right|^{n-1}


where D is the non-linear diffusivity (because it depends on the solution to the equation, H), A is the rate factor in Glen's flow lay and n is the power-law exponent, h is the ice surface elevation, and ρ and g are the ice density and the acceleration due to gravity.