Chapter 6

GFDL will provide the additional documentation by the end of May 2021.

(This information is a reproduction of Section 6 from HCZC20)

An equation for \(z\) can be derived from the definition of \(w\):

\[ w = \frac{Dz}{Dt} = D_Lz + \vec{U} \cdot \nabla z \\ \tag {6.1} \]

The time-tendency of geopotential height is then the sum of the advective height flux along the Lagrangian interfaces and the vertical distortion of the surfaces by the gradient of \(z\). Discretizing:

\[ z^{n + 1} = z^n + F [\tilde{u^*}, \delta z_y] + G[\tilde{v^*}, \delta z_x] + w^{n + 1} \triangle t \\ \tag {6.2} \]

Since \(w\) is solved for on the interfaces, we can then simply take the vertical difference to get \(\delta z\).

Recalling that the Lagrangian dynamics in (7) only performs the forward advection of the vertical velocity, yielding \(w^*\), we then need to evaluate the vertical pressure-gradient force:

\[ w^{n + 1} = w^* -g \delta z^{n + 1} \delta_z p'^{n + 1} \\ \tag {6.3} \]

The pressure perturbation \(p'\) can be evaluated from the ideal gas law,

\[ p' = p - p^* = \frac{\delta p}{g \delta z} R_d T_v - p^* \\ \tag {6.4} \]

requiring simultaneous solution of \(w\), \(p'\), and \(\delta z\) using a tridiagonal solver.

There is an option to off-center the semi-implicit solver toreduce implicit diffusion. The parameter \(\alpha\) can be varied between 0.5 and 1 to control the amount of off-centering, with \(\alpha = 1\) being fully-implicit. As discussed in Chapter 4 this off-centering parameter should be set to \(\alpha = \beta - 1\), consistent with that used for the horizontal pressure-gradient force.

The boundary conditions used are \(p' = 0\) at the model top, and \(\vec{U} \widehat{\cdot} n_s = 0\) at the lower boundary of \(z = z_s\). This is the “free-slip” boundary condition, that the lower boundary is a streamline. The surface vertical velocity \(w_s\) can be computed from (11) by advecting the surface height \(z_s\):

\[ w_s = \frac{z^*_s - z_s}{\triangle t} \\ \tag {6.5} \]

where \(z_s^*\) is the advected value and \(z_s\) the height of the topography.