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.

Importantly, we need only local information in order to solve the above equation. If our velocity solution can not be solved locally, as in the case of higher-order models, we cannot easily use the above formulation to solve ice sheet evolution. In attempt to use this form and retain a diffusion-solution approach to the problem (diffusion problems generally have nice numerical properties), we could try the following approach (again, in 1d only),


\frac{\partial H}{\partial t}=\frac{\partial }{\partial x}\left( D\frac{\partial h}{\partial x} \right)+\dot{b},\quad D=UH\left( \frac{\partial h}{\partial x} \right)^{-1},


where the U in the expression for D is the depth-integrated velocity field from the higher-order model. This is the approach initially taken by Pattyn (2003). Notice, however, that ice sheet surface slope is in the denominator of the diffusivity here and, as the slopes get smaller and smaller, as they tend to do in regions of fast flow like ice streams and ice shelves, the diffusivity will get larger and larger (approaching infinity as the slope goes to zero). This is a severe restriction on this approach because the diffusive CFL condition says that


\Delta t<\frac{\left( \Delta x \right)^{2}}{2D},


where Δt is the time step required for stability and Δx is the grid spacing. As the diffusivity goes to infinity, the stable time step goes to zero. In practice, this approach has proven very difficult to use for calculating ice sheet evolution in most of the areas we care about (fast flowing areas with shallow slopes). Thus, it seems that we need an alternate approach.


Advection schemes

The alternate approach is to solve the evolution equation using an advection scheme. Numerically, advection schemes are more problematic than diffusion schemes, but in some cases like this one, they are difficult to get around. The most general advection scheme, and one that we will implement in a related exercise, is a first-order, upwind advection scheme (first-order refers to first-order accurate, as opposed to second-order accurate, which we might prefer). For the 1d version of our ice sheet evolution equation


\frac{\partial H}{\partial t}=-\frac{\partial \left( UH \right)}{\partial x}+\dot{b}.


Most ice sheet models (and fluid dynamic models in general) perform calculations on a "staggered" grid of the type shown below, where velocity components live on one grid and scalar components (e.g. temperatures, pressures, thickness, etc.) live on a grid that is staggered 1/2 grid space in the horizontal dimensions (this leads to numerical advantages - like stability - that we won't get in to much here).


Figure 1: Staggered grid in two dimensions, showing scalars (pressure, P, and temperature, φ) at grid cell centers and velocity components, u and w, at grid cell faces. This is a "C" grid. Another staggered-grid possibility is a "B" grid, for which both velocity components live at the grid cell corners.

Stag grid.png


A "control volume" (or finite volume) approach to solving the problem would be to integrate our equation over the control volume centered on the scalar values (for us the scalar is thickness, H). Ignoring source terms for now, and in 1d only, we have


\frac{\partial H}{\partial t}=-\frac{1}{\Delta x}\int_{w}^{e}{\frac{\partial \left( UH \right)}{\partial x}}dx=-\frac{1}{\Delta x}\left( HU_{e}-HU_{w} \right).


The "east" and "west" (subscripts e and w) faces of the control volume are shown in the figure below.


Figure 2: Staggered grid in two dimensions, showing locations of interfaces and control volume dimensions.

Stag gridb.png

Reference

  • Pattyn, F.