21.1.2 Derivative operators

Simple derivative operators in space and time are defined as follows:

    

$$\delta_\lambda(\alpha_{i,k,j}) \frac{\alpha_{i+1,k,j} - \alpha_{i,k,j}}{a\Delta\lambda_i}$$ = (21.4)

$$\delta_\phi(\alpha_{i,k,j}) \frac{\alpha_{i,k,j+1} - \alpha_{i,k,j}}{a\Delta\phi_{jrow}}$$ = (21.5)

$$\delta_z(\alpha_{i,k,j}) -\frac{\alpha_{i,k+1,j} - \alpha_{i,k,j}}{\Delta z_k}$$ = (21.6)

$$\delta_\tau(\beta_{\tau}) \frac{\beta_{\tau+1} - \beta_{\tau-1}}{2\Delta\tau}$$ = (21.7)

where grid distances (measured in cm) are determined by the distance between variables as indicated in Figures 16.2, 16.3, and 16.4 and discussed in Chapter 14. These operators are second order accurate with non-uniform resolution as long as the grid is constructed so that the stretching is based on a smooth analytic function. See Treguier, Dukowicz, and Bryan (1995). When $\alpha_{i,k,j}$ is defined at grid points, then $a\Delta\lambda_i = dxu_i$ (a=6370 x 10⁵ cm) and when $\alpha_{i,k,j}$ is defined at grid points, then $a\Delta\lambda_i = dxt_{i+1}$. Similarly, $a\Delta\phi_{jrow} = \dyuj$ when $\alpha_{i,k,j}$ is defined at grid points and $a\Delta\phi_{jrow} = dyt_{jrow+1}$ when $\alpha_{i,k,j}$ is defined at grid points. Note the negative sign in the vertical derivative. This is because z increases upwards while k increases downwards. The negative sign in Equation (21.6) is usually absorbed by reversing the indexing to give

 

$$\delta_z(\alpha_{i,k,j}) = \frac{\alpha_{i,k,j} - \alpha_{i,k+1,j}}{\Delta z_k}$$ (21.8)

In the finite difference approximation to the continuous time derivative, Equation (21.7) is appropriate for the normal leapfrog time steps where $2\Delta\tau$ is in seconds. As indicated in Section 21.4, on mixing time steps, the denominator is replaced by $\Delta\tau$.