4.6 Dry Adiabatic Adjustment

If a layer is unstable with respect to the dry adiabatic lapse rate, dry adiabatic adjustment is performed. The layer is stable if

$$\frac{\partial T}{\partial p} < \frac{\kappa T}{p}.$$ (4.160)

In finite-difference form, this becomes

$$T_{k+1} - T_k < C1_{k+1} (T_{k+1} + T_k) + \delta ,$$ (4.161)
where

$$C1_{k+1} = \frac{\kappa (p_{k+1} - p_k)}{2 p_{k+1/2}} .$$ (4.162)

If there are any unstable layers in the top three model layers, the temperature is adjusted so that (4.161) is satisfied everywhere in the column. The variable $\delta$ represents a convergence criterion. The adjustment is done so that sensible heat is conserved,

$$c_p(\hat{T}_k \Delta p_k + \hat{T}_{k+1} \Delta p_{k+1}) = c_p (T_k \Delta p_k + T_{k+1} \Delta p_{k+1}) ,$$ (4.163)

and so that the layer has neutral stability:

$$\hat{T}_{k+1} - \hat{T}_k = C1_{k+1} (\hat{T}_{k+1} + \hat{T}_k) .$$ (4.164)

As mentioned above, the hats denote the variables after adjustment. Thus, the adjusted temperatures are given by

$$\hat{T}_{k+1} = \frac{\Delta p_k}{\Delta p_{k+1} + \Delta p_k C2_{k+1}} T_k + \frac{\Delta p_{k+1}}{\Delta p_{k+1} + \Delta p_k C2_{k+1}} T_{k+1},$$ (4.165)
and

$$\hat{T}_k = C2_{k+1} \hat{T}_{k+1} ,$$ (4.166)
where

$$C2_{k+1} = \frac{1 - C1_{k+1}}{1 + C1_{k+1}} .$$ (4.167)

Whenever the two layers undergo dry adjustment, the moisture is assumed to be completely mixed by the process as well. Thus, the specific humidity is changed in the two layers in a conserving manner to be the average value of the original values,

$$\hat{q}_{k+1} = \hat{q}_k = (q_{k+1} \Delta p_{k+1} + q_k \Delta p_k)/(\Delta p_{k+1} + \Delta p_k) .$$ (4.168)

The layers are adjusted iteratively. Initially, $\delta = 0.01$ in the stability check (4.161). The column is passed through from $k=1$ to a user-specifiable lower level (set to 3 in the standard model configuration) up to 15 times; each time unstable layers are adjusted until the entire column is stable. If convergence is not reached by the 15th pass, the convergence criterion is doubled, a message is printed, and the entire process is repeated. If $\delta$ exceeds 0.1 and the column is still not stable, the model stops.

As indicated above, the dry convective adjustment is only applied to the top three levels of the standard model. The vertical diffusion provides the stabilizing vertical mixing at other levels. Thus, in practice, momentum is mixed as well as moisture and potential temperature in the unstable case.