A Theoretical Investigation of Atmospheric Convective Modes as a Function of Rayleigh Number, Prandtl Number and Eddy Anisotropy

[摘要] A mathematical model consisting of a complete set of the linearized Boussinesq equations governing atmospheric convection has been solved. The significant new features of the model include (1) anisotropic eddy heat and momentum diffusion coefficients with the eddy viscosity for momentum varying with height through the convecting layer; (2) convection cells of general horizontal geometry as determined by parameter values appearing in a modified form of the generalized Christopherson shape function; (3) a non-zero complex stability factor allowing for steady and time-dependent convective modes; (4) a diabatic heating term in the conservation of energy equation allowing for the development of a moist atmospheric Rayleigh number (Ra); and (5) an atmospheric Prandtl number (Pγ) where molecular viscosity and conductivity have been replaced by their eddy counterparts. The important physical parameters in specifying the convective state are Ra, Pγ, and the eddy anisotropy coefficients for momentum and heat diffusion.Results obtained show that: (1) aspect ratio, cell geometry, and stability of the convective mode are very sensitive to variations in Ra, Pγ, and the anisotropy coefficients; (2) Regime-Stability (R-S) diagram reveals a convective order which closely corresponds to stability diagrams obtained in laboratory studies using different species of fluids; (3) cell flatness, as determined by values of the aspect ratio, is found to be inversely proportional to the square of the convective depth, with the proportionality constant being an increasing function of the momentum diffusion anisotropy coefficient; (4) a family of curves is found to exist where the diameter to depth ratio increases with increasing eddy anisotropy, supporting the observational evidence that such atmospheric convective phenomena as open and closed mesoscale cellular convection (MCC), radar cells in clear air, isolated tropical rings and thunderstorm cells are ordered according to their degree of anisotropy on a cell flatness versus depth diagram.Using an assumed periodic form of the perturbation solutions, the eigenvalue problem is reduced to one of solving a sixth-order differential equation for the vertical wind field with variable coefficients and accompanying boundary conditions. By applying the two-variable small perturbation technique, a Regime-Stability diagram was constructed yielding information concerning: (1) the geometry of the convective mode; (2) the stability of the convective mode; and (3) the anisotropic character of the eddy diffusion coefficients.

[学科分类] 大气科学