A Simple Dynamical Model of a Stratocumulus-Topped Boundary Layer

Abstract

Some latent heating effects on horizontal wavenumber selection and cell circulation patterns in stratocumulus- topped boundary layers are investigated via a study of two-dimensional shallow moist Boussinesq convection. No-entrainment and zero moisture flux assumptions are used to develop a simple latent heating representation; accordingly, upward and downward motions below a uniform cloud base are assumed to be dry adiabatic and those above cloud base are moist adiabatic. A nonlinear nine-coefficient low-order spectral model is developed in which representations of all nonlinear and linear terms of the original system of partial differential equations are retained. In dry Rayleigh-Bénard convection, the first convective steady solution branching from the conductive state is assumed to occur at the smallest value of the critical Rayleigh number and the resulting flow field is dominated by one horizontal wavenumber. In this case, the circulation patterns are symmetric because the horizontal wavelengths of the updraft and the downdraft within a single cell are equal; these preferred wavelengths are given by the above minimization calculation. In moist convection, owing to the vertically asymmetric effects of latent heating, the minimum values of the critical Rayleigh number are smaller than those for dry convection. Moreover, these latent heating effects cause the activation of two new additional linear terms in the moist spectral model. Because the branching convective solution is dominated now by two horizontal wavenumbers. the resulting circulation patterns are asymmetric. The horizontal widths of the updraft and the downdraft of the convective cell are not equal, but the preferred wavelengths of the cell couplets are given by the above minimization procedure. A linear stability analysis of the conductive solution to the low-order model is compared with that for the original partial differential system. and the low-order model is shown to produce excellent approximations of both the critical Rayleigh number and the preferred wavelengths. Cell circulation patterns for a few selected cases are shown, and qualitative differences in the steady convective solutions for deep and shallow clouds are demonstrated. As the amount of latent heating is increased when the cloud is deepened. the horizontal asymmetry of the circulation within a convective couplet increases correspondingly. When stratocumulus clouds fill ? to ½of the domain, the initial circulation is elevated from the ground and occurs primarily within the cloud itself., in a sense, the circulation within the cloud layer is detached from that in the subcloud layer. As the value of the Rayleigh number increases, the circulation gradually fills the domain, thereby linking the two layers; only circulation patterns that fill the domain occur when shallow clouds are present.