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 $$\Delta\,$$.

Scaling the equation for ice thickness evolution,

$$\frac{\delta\,s} {\delta\,t}= b + \frac{\delta\,} {\delta\,x} D \frac{\delta\,s} {\delta\,x} + \frac {D} {w} \frac{\delta\,w} {\delta\,x}\frac{\delta\,s} {\delta\,x} $$

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

and choosing

$$              B_{0}= \frac{H_{0}}{T_{0}} $$

$$              D_{0}= \frac{X_{0} ^2}{T_{0}} = \frac{B_{0} X_{0} ^2}{H_{0}} $$

we obtain

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

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

$$ 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}\left [\frac{H_{0}^{2n+1}}{X_{0} ^{n+1} D_{0}}\right ]\left [\frac{\Delta\, t} {\left (\Delta\,x\right )^2}\right ]

$$.

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

$$ Q' = - D'w' \frac{\Delta\, s'} {\Delta\,x'}

$$

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

$$ Q' = - diff w' \frac{\Delta\, s'} {\Delta\,x'} \left [\frac{\left (\Delta\,x'\right )^2}{\Delta\,t'}\right ] = - diff w'\Delta\, s'\left [\frac{\Delta\,x'}{\Delta\,t'}\right ]

$$.

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,

$$ conv = \frac{\sum diff} {2w'} \frac {\Delta\,w'} {2\Delta\,x'}

$$

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