Nondimensionalization

In writing computer code for finite-difference schemes, it should be evident from Equations (11.4) to (11.11) that the code will be much simpler if the units of spatial discretization (subdividing the domain into many small discrete units) are of equal size. Thus, Az should not change with depth. If the problem at hand involves a substantial part of an ice sheet, say along a flowline that is broken into columns of width Ax, each of which is then subdivided into depth increments, Az (Figure 11.3), it is clearly impossible to keep both the number of depth increments and their size, Az, constant from one column to the next. To avoid problems of this type, modelers commonly normalize the depth by dividing by the thickness. Thus a point at a depth, z, of 600 m in an ice sheet that is H = 1000 m thick will be at a normalized depth, z*, of 0.6. The columns then all have a non-dimensional thickness, H*, of 1.0, and if they are each subdivided into 20 equal depth increments, all increments will have a non-dimensional thickness of H*/20. Non-dimensionalization or scaling of lengths in this way generally requires scaling of the other parameters in the equations.

Figure 11.3. Illustration of problems encountered with a finite-difference discretization of an ice sheet along a flowline.

Figure 11.3. Illustration of problems encountered with a finite-difference discretization of an ice sheet along a flowline.

Such scaling greatly simplifies the mathematics in many situations. However, results from such computations have to be restated in dimensional form before most of us can derive useful physical insights from them. As our goal is to develop an appreciation for physical concepts, we will not delve further into this subject.

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