16.2.3.1 Accuracy of numerics

When resolution is constant, the finite difference numerics are second order accurate. In this case, grid cells and grid points are at the same locations regardless of whether they are constructed using method 1 or method 2. However, contrary to widespread belief, when resolution is non-uniform, numerics are still second order accurate if the stretching is based on a smooth analytic function. See Treguier, Dukowicz, and Bryan (1995).

Even though methods 1 and 2 are second order accurate, is one slightly better than the other? In particular, does the horizontal staggering of grid cells implied by method 2 give slightly better horizontal advection^16.3 of tracers while the staggering implied by method 1 give more accurate horizontal advection of momentum? Also, since T cells and U cells are not staggered in the vertical, does method 2 gives more accurate vertical advection of tracers and momentum than method 1?

The reasoning behind these questions can be seen by referring to Figure 16.5 and noting the placement of T grid points within T cells^16.4. Advective fluxes are constructed as the product of advective velocities and averages of quantities to be advected.

$$advective\;\;flux = W_k\cdot\frac{T_k + T_{k+1}}{2}$$ (16.5)

When resolution is constant, both advective velocity and averaged quantities lie on cell faces, but when resolution is non-uniform, averaged quantities may lie off the cell faces. Whether or not this happens depends on the placement of grid points within grid cells. Method 2 defines tracer points off center in such a way that the averaged tracer given by Equation (16.4) is placed squarely on the cell faces. Recall from Chapter A that defining the average operator differently than in Equation (16.4) will not conserve second moments.

Simple one dimensional tests using a constant advection velocity of 5 cm/sec to advect a gaussian shaped waveform through a non-uniform resolution varying from 2 to 4 degrees and back to 2 degrees suggests that method 2 is better than method 1. Also, a one dimensional thermocline model employing a stretched vertical coordinate indicates again that method 2 is better than method 1. In both cases, better means that the variance of the solution was closer to the variance of the analytic solution by a few percent. In short integrations using MOM, the effect showed up as less spurious creation of tracer extrema in grids constructed with method 2 as compared to method 1. Whether this difference is robust in all integrations has not been demonstrated. Nevertheless, in light of these results, the default grid construction in MOM is method 2. Method 1 can be implemented by using option centered_t.

It should be stressed, that regardless of which method of grid construction is used, the equations don't change and first and second moments are conserved. Of primary importance when constructing a grid is whether the physical scales are adequately resolved by the number of grid points. Beyond this, grid construction by method 1 or method 2 is of secondary importance.