Moist Held Suarez with Topography

I’ll describe the process of using topography with idealized physics. In this document I’ll be explaining how to use the landfrac variable in the TJ16 compset to only apply surface fluxes over the ocean.

The implicit update reads as \(\pder{q_a}{t} = \frac{C_E |\boldsymbol{v}_a| (q_{\textrm{sat}, a} - q_a)}{z_a}\) (with an analogous expression for \(T_a\)). It is updated with one step of a fixed-point iterationpatterned on backwards Euler. This reads as

\[\begin{align*} &\phantom{\implies} \frac{q_a^{t+1} - q_a^t}{\Delta t} = \frac{C_E |\boldsymbol{v}_a| (q_{\textrm{sat}, a} - q_a^{t+1})}{z_a} \\ &\implies q_a^{t+1} - q_a^t = \frac{\Delta t C_E |\boldsymbol{v}_a| q_{\textrm{sat}, a}}{z_a} - \frac{\Delta t C_E |\boldsymbol{v}_a| q_a^{t+1}}{z_a} \\ &\implies \left( \frac{\Delta t C_E |\boldsymbol{v}_a|}{z_a} + 1 \right)q_a^{t+1} = \frac{\Delta t C_E |\boldsymbol{v}_a| q_{\textrm{sat}, a}}{z_a} + q_a^t \\ &\implies q_a^{t+1} = \frac{q_{\textrm{sat}, a} + \frac{\Delta t C_E |\boldsymbol{v}_a| }{z_a} }{ 1 + \frac{\Delta t C_E |\boldsymbol{v}_a|}{z_a}} \\ \end{align*}\]

So if we assume that the RHS of the continuous-time equation is scaled by \(C_{\textrm{landfrac}}\), then anywhere that a \(C_E\) appears, you also apply \(C_{\textrm{landfrac}}\).