Geography Reference
In-Depth Information
Fig. 8.9
Energy flow in an amplifying baroclinic wave.
An analysis of the structure of baroclinic modes for realistic mean zonal wind
profiles is quite complex, and indeed can only be done by numerical methods.
However, without obtaining specific normal mode solutions, it is possible to obtain
necessary conditions for baroclinic or barotropic instability from an integral theo-
rem first developed by Rayleigh. This theorem, which is discussed in Section 8.4.2,
also shows how baroclinic instability is intimately related to the mean meridional
gradient of potential vorticity and the mean meridional temperature gradient at the
surface.
If a number of simplifying assumptions are made, it is possible to pose the
stability problem for a continuously stratified atmosphere in a fashion that leads to
a second-order differential equation for the vertical structure that can be solved by
standard methods. This problem was originally studied by the British meteorologist
Eady (1949) and, although mathematically similar to the two-layer model, provides
additional insights. It is developed in Section 8.4.3.
8.4.1
Log-Pressure Coordinates
Derivation of the Rayleigh theorem and the Eady stability model is facilitated if we
transform from the standard isobaric coordinates to a vertical coordinate based on
