A two-dimensional, second-order model for turbulent flow in the upper atmosphere

A second order closure model for the atmospheric, two-dimensional, steady, turbulent flow in a zonal channel for the β-plane is developed, and the non-linear system of equations resulting from the model is solved using a standard iterative technique. An important feature of the model is that it attempts to simulate thermal effects in the energy- and enstrophy budgets by introducing a fictitious buoyancy-like forcing term, which corresponds to a baroclinic force field of intensity Γ[T] acting southwards in the β-plane. From the original governing equations a full set of relations for the Reynolds stresses is developed, in addition to one equation each for the mean momentum- and the energy balance (in the energy equation, a Boussinesq approximation is employed). The governing equations are properly scaled from synoptic observations to closely resemble those of the large eddy dynamics, Then, an invariant modelling technique is used, to obtain a non-linear system for the Reynolds'- and the thermal stresses. Since the source of energy required to maintain the flow is in the mean temperature difference between two limiting latitude circles, the [negative] temperature gradient is assumed from empirical data corresponding to a stable environment. The model parameters are reduced to standard Reynolds-, Rossby-, Richardson- and Prandtl numbers. A two-step double iterative procedure is employed to achieve convergence while maintaining physically sound profiles. The model shows a very high sensitivity to the turbulent kinetic energy profile, and the predicted energy conversion mechanisms, in particular the meridional eddy momentum flux, the eddy potential energy, and the temperature variances, are in good agreement with the available experimental data. In particular, the model reproduces reasonably well the general dynamics of synoptic flows, including the reverse energy cascade and the direct enstrophy cascade. Further possible developments and feasible research topics in this area are briefly discussed.