Asymptotic models for planetary scale atmospheric motions

Observations indicate the existence of a large number of low-frequency (periods longer than 10 days) atmospheric regimes with planetary spatial scales (of the order of the earth's radius, ca. 6300 km) that have an important influence on the variability of the atmosphere. Further studies show that the interactions between such planetary scale flows and the synoptic eddies (characteristic length and time scales : 1000 km and 2-6 days) play a crucial role for the atmospheric dynamics. In this theses we derive reduced model equations for three planetary regimes by applying a multiple scales asymptotic method. This method allows us to take into account in a systematic way the interactions with the synoptic scales. The numerical experiments with a primitive equations model showed that two of the asymptotic regimes reproduce basic properties of the planetary scale dynamics. The Planetary Regime (PR) is characterized by isotropic planetary horizontal scales and by a corresponding advective time scale of about one week. The variations of the background potential temperature in this regime are comparable in magnitude with those adopted in the classical quasi-geostrophic (QG) theory, larger variations are assumed in the Planetary Regime with Background Flow (PRBF). In the PR we obtain as leading order model the planetary geostrophic equations (PGEs). We derive in a systematic way from the asymptotic analysis a closure for the PGEs in the form of an evolution equation for the vertically averaged (barotropic) component of the pressure. Relative to the prognostic closures adopted in existing reduced-complexity planetary models, this new dynamical closure may provide for a more realistic large scale and long term variability in future implementations. Using a two scale asymptotic ansatz, we extended the region of validity of the PR to the synoptic spatial and temporal scales. We derive modified QG equations for the dynamics on the synoptic scale as well as terms describing new interactions between the synoptic and planetary scales. In the Anisotropic Planetary Regime (APR) we investigate motions with planetary modulation in zonal direction but with a meridional extent confined to the synoptic scale, the same assumption for the background temperature as in the PR is made. As leading order model we obtain the QG model, describing the synoptic evolution of the leading order synoptic potential vorticity (PV). The second order asymptotic model describes a coupling between the planetary evolution of this leading order synoptic PV, the synoptic evolution of the planetary scale vorticity field and the synoptic dynamics of higher order PV corrections. By applying a primitive equations model, we studied the balances in the vorticity transport on the planetary and synoptic scale. The numerical experiments showed that only the PR and the APR are relevant for the earth's atmosphere. These two models helped us to understand different aspects of the dynamics on the planetary scale and they can be further employed for the construction of intermediate complexity models for long term climate simulations