29.2.2 The coefficient matrices

Equation (29.14) involves nine values of $\Delta \psi$centered at $\Delta\psi_{i,jrow}$ which may be written as

$$\sum_{i'=-1}^{1}\sum_{j'=-1}^{1} \coeff_{i,jrow,i',j'} \Delta\psi_{i+i',jrow+j'} = ztd_{i,jrow}$$ (29.18)

where the 3rd and 4th subscripts on the coefficient matrix refer to coefficients on neighboring cells. For example, i'=-1 and j'=0 refers to the coefficient of $\Delta\psi_{i-1,jrow}$ (the value on the western neighboring cell). The coefficient of $\Delta\psi_{i+1,jrow+1}$ on the northeast neighboring cell is given by i'=1 and j'=1. When option sf_9_point is enabled, MOM calculates the coefficient matrix $\coeff_{i,jrow,i',j'}$ for Equation (29.18) using summation formulas given in Section 30.2. This nine point coefficient matrix differs slightly from the one in MOM 1 and is more accurate. The elliptic solvers also converge in fewer iterations using this coefficient matrix. The $\bar{u}_{i,j,1,\tau+1}$ and $\bar{u}_{i,j,2,\tau+1}$ derived from the solution of Equation (29.14) are exact solutions^29.7 of the finite difference vertically averaged momentum Equations (29.3) and (29.4).

When option sf_5_point is enabled, the nine point coefficient matrix is approximated by a five point coefficient matrix involving five non-zero coefficients $\coeff_{i,jrow,i',j'}$ for i'=0 or j'=0as in Bryan (1969). It is arrived at by averaging terms used to construct the nine point coefficient matrix in a different way resulting in a coefficient matrix that differs from the one used by Bryan (1969). The five point coefficient matrix is not as accurate as the nine point matrix although the nine point matrix has a checkerboard null mode. Refer to Appendix E for a discussion on null modes. However, in the stream function, this null mode is largely suppressed because $\psi_{i,jrow}$ is constant along boundaries. Also, spatial derivatives of $\psi_{i,jrow}$ remove the this null mode and so it is of no dynamical consequence. The five point matrix does not have this checkerboard null mode. However, both five and nine point operators have an arbitrary unspecified constant null mode. Therefore, the value of $\psi$ on one land mass can be arbitrarily specified (usually set to zero) and all stream function values referenced to this land mass value. Either the five point or nine point operator must be chosen by enabling options sf_5_point or sf_9_point.