Scaling

In this section, subscript 0 is used to denote scales; primed symbols denote scaled quantities; unadorned symbols denote unscaled quantities; and finite difference approximations are denoted using [math]\displaystyle{ \Delta\, }[/math].

Scaling the equation for ice thickness evolution,

[math]\displaystyle{ \frac{\delta\,s} {\delta\,t}= b + \frac{\delta\,} {\delta\,x} D \frac{\delta\,s} {\delta\,x} + \frac {D} {w} \frac{\delta\,w} {\delta\,x}\frac{\delta\,s} {\delta\,x} }[/math]

[math]\displaystyle{ \left [\frac{H_{0}}{T_{0}}\right ]\frac{\delta\,s'} {\delta\,t'} = \left [B_{0}\right]b' + \left [\frac{H_{0}D_{0}}{X_{0} ^2}\right ]\frac{\delta\,} {\delta\,x'}D' \frac{\delta\,s'} {\delta\,x'} + \left [\frac{H_{0}D_{0}W_{0}}{X_{0} ^2W_{0}}\right ]\frac {D'} {w'} \frac{\delta\,w'} {\delta\,x'}\frac{\delta\,s'} {\delta\,x'} }[/math]

and choosing

[math]\displaystyle{ B_{0}= \frac{H_{0}}{T_{0}} }[/math]

[math]\displaystyle{ D_{0}= \frac{X_{0} ^2}{T_{0}} = \frac{B_{0} X_{0} ^2}{H_{0}} }[/math]

we obtain

[math]\displaystyle{ \left [\frac{H_{0}}{T_{0}}\right ]\frac{\delta\,s'} {\delta\,t'} = \left [\frac{H_{0}}{T_{0}}\right ]b' + \left [\frac{H_{0}X_{0}^2}{T_{0}X_{0} ^2}\right ]\frac{\delta\,} {\delta\,x'}D' \frac{\delta\,s'} {\delta\,x'} + \left [\frac{H_{0}X_{0}^2W_{0}}{X_{0} ^2 T_{0} W_{0}}\right ]\frac {D'} {w'} \frac{\delta\,w'} {\delta\,x'}\frac{\delta\,s'} {\delta\,x'} }[/math]

In the code, diffusivity is found as (sigma refers to averaging needed to move between normal and staggered grid)

[math]\displaystyle{ diff = - \frac{2fA}{n+2} (\rho\,g)^n \left ( \frac{\sum H}{2} \right ) ^{n+2} \left | \frac{\Delta s}{\Delta x} \right |^{n-1} }[/math]

Note that diff is not simply the scaled diffusivity but carries the appropriate time and grid steps (final bracketed quantity). To find ice flux,

however, in the code, need to remember to reset grid steps

Finally, we need to look at the convergence term. In this case, nothing needs to be done because the various grid steps are needed however some averaging between grids is required