33.2.4.1 Richardson number

In previous discretizations of this scheme, the Richardson number was first computed at the base of U-cells, then averaged onto T-cells after which the three dimensional viscosity coefficient $visc\_cbu_{i,k,j}$ was computed using Richardson numbers at the base of U-cells and the three dimensional diffusivity coefficient $diff\_cbt_{i,k,j}$ was computed using Richardson numbers at the base of T-cells.

Actually, the above is only one of many possible ways to discretize the scheme. Instead of averaging Richardson numbers, another way is to compute Richardson numbers, viscosity, and diffusivity coefficients at the base of U-cells and then average the diffusivity coefficients onto T-cells. A third way is to compute separate Richardson numbers at the base of T-cells and U-cells separately after which the mixing coefficients are then computed.

Two other approaches are as follows: The first involves computing Richardson numbers at the base of T-cells, averaging to the base of U-cells, then computing mixing coefficients at the base of T-cells and U-cells. The second involves computing Richardson numbers at the base of T-cells, computing both mixing coefficients at the base of T-cells, and then averaging the viscosity coefficients onto U-cells.

Based on a combination of analytic and numerical explorations at GFDL by Anand Gnanadesikan, the last way turns out to be the most accurate of the above five schemes. This last scheme was used in an intermediate version^33.1 of MOM 2 for about 8 months. However, a variety of three dimensional simulations indicated that a weak instability exists which under certain conditions manifests as a two grid point structure (numerical noise) in the tracer fields. Based on these results, a change was made back to the original scheme which is currently being used and is described below. All vertical mixing schemes compute the Richardson number the following way.

The Richardson number is first computed at the base of U-cells using

$$\frac{-grav\cdot \overline{\delta_z(\rho_{i,k,j,\tau-1})}^{\lambd... ...u-1}))^2 + (\delta_z(u_{i,k,j,2,\tau-1}))^2} \;\;\;\;\;\;\;\;(k < kmu_{i,jrow})$$ riu_i,k,j = (33.7)

$$0\;\;\;\;\;\;\;\;(k \ge kmu_{i,jrow})$$ = (33.8)

where grav=980.6 cm/sec² and the local index j is related to the global index jrow by the memory window offset given in Equation 11.4. Note that, because of the averaging in latitude and longitude, riu_i,k,j is defined at the base of U-cells and can only be computed for rows 1 through jmw-1 in the memory window (Refer to Section 22.2.4 for a discussion). Note also that riu_i,k,j is zero on the top and bottom faces of land cells. The Richardson number at the base of U-cells is then averaged onto the base of T-cells by

$$rit_{i,k,j} = \overline{riu_{i-1,k,j-1}}^{\lambda \phi}$$ (33.9)

However, with the aid of masking (not shown) only non-zero values of riu are considered in the average. This prevents generating low values of rit next to topography for the wrong reason^33.2. Again, because of the averaging in latitude and longitude, rit_i,k,j can only be computed for rows 2 through jmw-1 in the memory window.