A numerical spectral model for the mean zonal circulation and the tides in the middle and upper atmosphere

A three-dimensional, time-dependent, spectral model has been developed that solves the fully non-linear Navier Stokes equations without any of the commonly adopted hydrostatic or anelastic approximations. The marching in time is implicit, except for the terms that are non-linear and involve the Coriolis force. Horizontal variations of the dependent variables are expressed in terms of spherical harmonics and, through spectral transforms, the non-linearities are evaluated in physical space. The equations are formulated such that the total mass and total angular momentum are conserved to round-off error. Simplified, linear versions of the model have been derived to describe: (i) the stationary, axisymmetric mean circulation with zonal wave number m = 0, and (ii) stationary thermal tides by expressing the time derivatives as multiples of iω, where ω is the frequency with the period of a day. In both cases the same eddy diffusion and Rayleigh friction parameterizations are used for internal consistency. These solutions are obtained implicitly in one step by inverting the coupling matrices for a sufficiently large number of spherical harmonics that assure convergence. The models simulate the essential features of the observed mean circulation and tides in the mesosphere and thermosphere under solstice and equinox conditions. When the model is used in the time marching mode, the results obtained reproduce the stationary solutions, albeit with relatively long integration times. Non-linear processes are then also accounted for and some of the effects are discussed here.