Analytical Exact Solution of the Cloud Thickness and Inversion Height Time Evolution for the Mixed Layer Model

Abstract:

Various schemes based on numerical and simulation methods exist in the literature that describe the evolution of stratocumulus clouds over land under a mixed layer model (MLM) assumption. Although it is possible to obtain sensitivity and time evolution results through multiple simulations, the obtained dependency is only an approximation to the underlying physical phenomena. In order to mitigate the disadvantages of these methods, we have derived the exact analytical solution of the MLM differential equations, in closed-form. An analytical, closed-form expression has many advantages: exact expression for the cloud dissipation time and how it relates to the climate conditions, sensitivity and time evolution of any parameter, time constant of the climate and the effect of the initial conditions on the time evolution; are some of the advantages of an analytical exact solution.

We first couple the MLM, which solves the boundary layer mass, heat and moisture budget equations, with an equation that relates the cloud thickness tendency to the budget equations. We utilize the total moisture and liquid potential temperature within the boundary layer as the conserved variables. For the radiative effects, we used a temperature dependent longwave radiation scheme and a solar zenith angle dependent, Delta-Eddington approximation based shortwave scheme. Entrainment velocity is parametrized as a function of the buoyancy flux within the cloud layer, whereas the land surface flux is parametrized based on the net surface radiation. Utilizing the described physical phenomena, we obtain closed-form analytical expression for the time evolution of the cloud thickness. To solve the various differential equations, we only use mathematical approximations, not physical approximations.

Furthermore, using the cloud thickness evolution solution, we trace our derivation steps back to investigate the evolution of the other variables involved in the process such as the conserved variables, entrainment velocity and inversion height structure. We believe that the analytical solution will help us understand, in a broader perspective, the sensitivity of the cloud thickness towards all parameters governed within the process.