HIERARCHICAL TROPICAL CLOUD SYSTEMS IN AN ANALOG SHALLOW-WATER MODEL

Recent observations have revealed an hierarchy of cloud clusters and s uperclusters within the Madden-Julian oscillation of the equatorial tr oposphere. The authors here report on the results of simulations with a model based on a simple nonlinear analog of the shallow-water equati ons. The model consists of a troposphere represented by single values of vertical velocity and temperature and in which the horizontal motio ns are assumed always to represent the first baroclinic mode. The trop osphere overlies a subcloud layer of fixed depth in which the evolutio n of moist entropy is predicted. The model is driven by specified valu es of radiative cooling and sea surface temperature, and a Newtonian r elaxation of the surface wind toward a specified value. The system is horizontally homogeneous except for an anisotropy owing to the equator ial beta effect. An eastward-propagating low-wavenumber disturbance co ntaining an hierarchy of superclusters and cloud clusters is spontaneo usly generated from a random initial state using each of three cumulus parameterizations: a Kuo-like scheme and two prognostic schemes. A fo urth scheme, which assumes an instantaneous adjustment to convective n eutrality, fails to produce an hierarchical structure. Experiments wit h fixed wind speed in the surface flux formulas demonstrate that the W ISHE (wind-induced surface heat exchange) mechanism is responsible for the organized structures in the model fields, except when the Kuo-lik e scheme is used; even in this case the modes ate strongly affected by WISHE. The supercluster resembles a Matsuno-Gill pattern in all three cases, but the horizontal cloud distribution within the superclusters differs substantially among the three schemes. The Kuo-like scheme pr oduces grid-column convection aligned along the convergence zones as a result of its direct coupling of convection with the large-scale conv ergence. This scheme always produces grid-scale motions. The prognosti c schemes, which allow for the finite timescale of convection, are les s prone to gridpoint structure, but the degree of such structure depen ds sensitively on the parameters of the schemes and the presence or ab sence of time-lagged downdrafts. The authors find that the wavenumber spectrum of convective updrafts is nearly flat, while the zonal wind s pectrum is strongly peaked at low wavenumbers. This behavior exists ev en if the nonlinear advection terms are switched off, showing that the se play little or no role in the final wavenumber selection. Even turn ing off all of the model nonlinearity except for the ''up-only'' natur e of convection preserves the essential structure of the full solution , although it does weakly flatten the zonal wind spectrum. The depende nce of the behavior of the system on the magnitude and direction of th e background surface wind is also explored. A weaker easterly wind for cing leads to a modulation of the superclusters into a yet lower-waven umber structure; this modulation propagates faster than the superclust ers. Westerly wind forcing suppresses the Kelvin-type mode and generat es a mixed mode resembling a mixed Rossby-gravity wave. A further incr ease of the westerly wind forcing induces westward-moving disturbances as well. This model is considered as a framework for interpreting mor e complicated tropical models.