7.1 Initial Data

In this section, we describe how the time integration is started from data consistent with the spectral truncation. The land surface model requires its own initial data, as described by Bonan [22]. The basic initial data for the model consist of values of $u, v, T, q, \Pi,$ and $\Phi_s$ on the Gaussian grid at time $t=0.$ From these, $U, V, T'$, and $\Pi$ are computed on the grid using (3.11), and (3.49). The Fourier coefficients of these variables $U^m, V^m, T'^m, \Pi^m,$ and $\Phi^m_s$ are determined via an FFT subroutine (3.149), and the spherical harmonic coefficients $T'^m_n, \Pi^m_n$, and $\left(\Phi_s \right)^m_n$ are determined by Gaussian quadrature (3.150). The relative vorticity $\zeta$ and divergence $\delta$ spherical harmonic coefficients are determined directly from the Fourier coefficients $U^m$ and $V^m$ using the relations,

$$\zeta = \frac{1}{a(1 - \mu^2)} \frac{\partial V}{\partial \lambda} - \frac{1}{a} \frac{\partial U}{\partial \mu} ,$$ (7.1)

$$\delta = \frac{1}{a(1 - \mu^2)} \frac{\partial U}{\partial \lambda} + \frac{1}{a} \frac{\partial V}{\partial \mu} .$$ (7.2)

The relative vorticity and divergence coefficients are obtained by Gaussian quadrature directly, using (3.154) for the $\lambda$-derivative terms and (3.157) for the $\mu$-derivatives.

Once the spectral coefficients of the prognostic variables are available, the grid-point values of $\zeta, \delta, T', \Pi,$ and $\Phi_s$ may be calculated from (3.180), the gradient $\nabla \Pi$ from (3.183) and (3.184), and $U$ and $V$ from (3.189) and (3.190). The absolute vorticity $\eta$ is determined from the relative vorticity $\zeta$ by adding the appropriate associated Legendre function for $f$ (3.117). This process gives grid-point fields for all variables, including the surface geopotential, that are consistent with the spectral truncation even if the original grid-point data were not. These grid-point values are then convectively adjusted (including the mass and negative moisture corrections).

The first time step of the model is forward semi-implicit rather than centered semi-implicit, so only variables at $t=0$ are needed. The model performs this forward step by setting the variables at time $t = -\Delta t$ equal to those at $t=0$ and by temporarily dividing $2 \Delta t$ by 2 for this time step only. This is done so that formally the code and the centered prognostic equations of chapter 3 also describe this first forward step and no additional code is needed for this special step. The model loops through as indicated sequentially in chapter 3. The time step $2 \Delta t$ is set to its original value before beginning the second time step.