21.2.3 Other rules

There is no ambiguity in manipulating finite difference objects. As noted in Bryan (1969), there are formal rules and some are given below. They can be verified by substituting the basic finite difference derivative and averaging operators and expanding the terms. For illustrative purposes, consider one dimensional quantities $\alpha_{i}$ and $\gamma_i$ (defined on longitudes of T cell grid points) and $\beta_i$ (defined on longitudes of U cell grid points).

  

$$\overline{\alpha_i \gamma_i}^\lambda \overline{\alpha_i}^\lambda\overline{\gamma_i}^\lambda + \frac{1}{4} dxu^2_i\delx(\alpha_i)\delx(\gamma_i)$$ = (21.16)

$$\delx(\alpha_i \gamma_i) \overline{\alpha_i}^\lambda\delx(\gamma_i) + \overline{\gamma_i}^\lambda\delx(\alpha_i)$$ = (21.17)

$$dxt_{i+1}\cdot \delx(\overline{\alpha_i}^\lambda \beta_i) \overline{\beta_i\cdot dxu_i\cdot\delx(\alpha_i)}^\lambda + \alpha_{i+1}\cdot dxt_{i+1}\cdot \delx(\beta_i)$$ = (21.18)

$$\overline{\overline{\alpha_i}^\lambda \beta_i}^\lambda \alpha_{i+1}\cdot\overline{\beta_i}^\lambda + \frac{1}{4}dxt_{i+1}\delx(\beta_i\cdot dxu_i\cdot\delx(\alpha_i))$$ = (21.19)

These expressions also hold along other dimensions. In particular if $\phi$ is substituted^21.2 for $\lambda$ or if zis substituted for $\lambda$. Consider further the two dimensional case where $\alpha_{i}$ is defined at T cell grid points, and $\beta_i$ is defined at U cell grid points. The following rule may be derived by combining Equations (21.18) and (21.19)

$$\alpha_{i,k,j}\cdot\dxti\cdot\delx(\overline{\beta_{i-1,k,j-1}}^\... ...1,k,j-1}dxu_{i-1}\cdot\overline{\delx(\alpha_{i-1,k,j-1})}^\phi}^{\phi \lambda}$$

$$dxt_i\cdot\delx(\overline{\alpha_{i-1,k,j}}^\lambda\overline{\beta{i-1,k,j-1}}^\phi)$$ =

$$\frac{1}{4}\dytj\dely(\overline{\beta_{i-1,k,j-1}\cdot\dyujm\dely(dxu_{i-1}\delx(\alpha_{i-1,k,j-1})))}^\lambda)$$ + (21.20)