Solution of the Blatter-Pattyn model

The final form of the equations we'd like to solve is:

$\begin{align} & x:\quad 4\frac{\partial }{\partial x}\left( \eta \frac{\partial u}{\partial x} \right)+\frac{\partial }{\partial y}\left( \eta \frac{\partial u}{\partial y} \right)+\frac{\partial }{\partial z}\left( \eta \frac{\partial u}{\partial z} \right)=-2\frac{\partial }{\partial x}\left( \eta \frac{\partial v}{\partial y} \right)-\frac{\partial }{\partial y}\left( \eta \frac{\partial v}{\partial x} \right)+\rho g\frac{\partial s}{\partial x} \\ & y:\quad 4\frac{\partial }{\partial y}\left( \eta \frac{\partial v}{\partial y} \right)+\frac{\partial }{\partial x}\left( \eta \frac{\partial v}{\partial x} \right)+\frac{\partial }{\partial z}\left( \eta \frac{\partial v}{\partial z} \right)=-2\frac{\partial }{\partial y}\left( \eta \frac{\partial u}{\partial x} \right)-\frac{\partial }{\partial x}\left( \eta \frac{\partial u}{\partial y} \right)+\rho g\frac{\partial s}{\partial y} \\ \end{align}$

Operating Splitting

Again, note that for the x equation we've moved all the terms containing gradients in v to the right-hand side (RHS).

We've set it up this way in order to solve the equations using an operator splitting approach; for the x equation, we treat v as known (where we take the values of v from the previous iteration) and solve for u, and vice versa when we solve they y equation for v. The "splitting" refers to the fact that we are breaking the multi-dimensional divergence operation into multiple steps. Rather than solving one big matrix equation for u and v simultaneously we solve two smaller matrix equations in sequence with one of the unknowns treated as a known "source" term.

Coordinate Transform

As with the 0-order model, we need to change from Cartesian to sigma coordinates. The first normal-stress term first term on the left-hand side becomes

Pattyn introduces a dimensionless vertical coordinate so that variations in ice thickness do not complicate the numerical treatment considerably (Pattyn 2003). This dimensionless vertical coordinate is referred to as $\zeta$ by Pattyn. I will refer to it as $\sigma$ for consistency with CISM's existing notation. The rescaled coordinate is defined as:

$\sigma = \frac{(s - z)}{H}$

This means that at the surface of the ice sheet $\sigma = 0$, and at the base $\sigma = 1$ regardless of the ice thickness (Pattyn 2003). As a result of this transformation, a coordinate $(x,y,z)$ is mapped to $(x',y',\sigma)$ (Pattyn 2003). This means that function derivatives must be re-written (using $\frac{\partial f}{\partial x}$ as an example) as:

$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x'} \frac{\partial x'}{\partial x} + \frac{\partial f}{\partial y'} \frac{\partial y'}{\partial x} + \frac{\partial f}{\partial \sigma} \frac{\partial \sigma}{\partial x}$

Similarly for $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$. Pattyn simplifies this by assuming that

$\frac{\partial x'}{\partial x}, \frac{\partial y'}{\partial y} = 1$

and

$\frac{\partial x'}{\partial y}, \frac{\partial x'}{\partial z}, \frac{\partial y'}{\partial x}, \frac{\partial y'}{\partial z} = 0$.

This assumption is valid if the bed and surface gradients are not too large (Pattyn 2003). This simplifies the above to:

$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x'} + \frac{\partial f}{\partial \sigma}\frac{\partial \sigma}{\partial x}$

$\frac{\partial f}{\partial y} = \frac{\partial f}{\partial y'} + \frac{\partial f}{\partial \sigma}\frac{\partial \sigma}{\partial y}$

$\frac{\partial f}{\partial z} = \frac{\partial f}{\partial \sigma}\frac{\partial \sigma}{\partial z}$

Rescaling parameters $a_x$, $a_y$, $b_x$, $b_y$, and $c_{xy}$ are defined. Presenting the x derivative case, as the y derivative case is analogous,

$a_x = \frac{1}{H}(\frac{\partial s}{\partial x'} - \sigma \frac{\partial H}{\partial x'})$

$b_x = \frac{\partial a_x}{\partial x'} + a_x \frac{\partial a_x}{\partial \sigma} = \frac{1}{H} (\frac{\partial^2 s}{\partial x'^2} - \sigma \frac{\partial^2 H}{\partial x'^2} - 2a_x \frac{\partial H}{\partial x'})$

$c_{xy} = \frac{\partial a_y}{\partial x'} + a_x \frac{\partial a_y}{\partial \sigma} = \frac{\partial a_x}{\partial y'} + a_y \frac{\partial a_x}{\partial \sigma}$

Using these, expressions for the x derivatives become:

$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial x'} + a_x \frac{\partial f}{\partial \sigma}$

$\frac{\partial }{\partial x}\left( \eta \frac{\partial u}{\partial x} \right)=\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial u}{\partial \hat{x}} \right)+\frac{\partial \sigma }{\partial \hat{x}}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial u}{\partial \hat{x}} \right)+\frac{\partial \sigma }{\partial \hat{x}}\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial u}{\partial \sigma } \right)+\left( \frac{\partial \sigma }{\partial \hat{x}} \right)^{2}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial u}{\partial \sigma } \right)+\left( \frac{\partial _{{}}^{2}\sigma }{\partial \hat{x}_{{}}^{2}} \right)\eta \frac{\partial u}{\partial \sigma }$

where hatted values refer to the coordinate directions in sigma coordinates. Similarly, the first cross-stress term on the RHS is given by

$\frac{\partial }{\partial x}\left( \eta \frac{\partial v}{\partial y} \right)=\underset{{}}{\mathop{\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial v}{\partial \hat{y}} \right)}}\,+\underset{{}}{\mathop \frac{\partial \sigma }{\partial \hat{x}}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial v}{\partial \hat{y}} \right)}\,+\underset{{}}{\mathop \frac{\partial \sigma }{\partial \hat{y}}\frac{\partial }{\partial \hat{x}}\left( \eta \frac{\partial v}{\partial \sigma } \right)}\,+\underset{{}}{\mathop \frac{\partial \sigma }{\partial \hat{x}}\frac{\partial \sigma }{\partial \hat{y}}\frac{\partial }{\partial \sigma }\left( \eta \frac{\partial v}{\partial \sigma } \right)}\,+\underset{{}}{\mathop \frac{\partial _{{}}^{2}\sigma }{\partial \hat{x}\partial \hat{y}}\eta \frac{\partial v}{\partial \sigma }}\,$

One term has become five terms and each one of those is pretty scary looking on its own. Luckily, there is also a lot of symmetry here. Notice that if we wanted to design subroutines to discretize the terms on the RHS, we could re-use a lot of them by either applying them to the correct velocity component (either to the u OR the v discretization) or by passing the appropriate arguments (pass either the grid spacing in the x direction OR the y direction, where appropriate). There is still a lot of work here ...

Continue with the Blatter-Pattyn Boundary Conditions.

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