On Cellular Cloud Patterns. Part 1: Mathematical Model

Abstract

The relationship of “open” or “closed” cellular cloud patterns to large-scale sinking or rising motion is investigated. In particular, it is shown that the open cell patterns typically found behind cold fronts can be determined by a large-scale sinking motion of a convectively unstable layer. The mathematical model treated is one in which a layer of Boussinesq fluid between two conducting porous boundaries is given a uniform vertical velocity w0. The linear stability problem for small γ=w0/κ, where κ is the thermal diffusivity and d the depth of the layer, is solved for a critical Rayleigh number Rc. The solutions for the flow field for this linear problem are infinitely degenerate. Steady finite-amplitude solutions of the nonlinear Boussinesq equations are obtained by a double expansion of the fields in powers of γ and an amplitude ε. The stability of the nonlinear solutions is investigated and it is shown that for a certain range of Prandtl numbers, (i) for γ>0, only hexagonal cells with upward flow in their centers are stable, (ii) for γlt;0, only hexagonal cells with downward flow in their centers are stable, and (iii) for γ=0, only rolls are stable. In the earth's atmosphere (i) corresponds to closed cells, while (ii) corresponds to open cells, and (iii) may correspond to cloud streets.