& Tornado Alley
the Baroclinic Instability Model
of the Atmosphere
This is a development (i.e., derivation) of the baroclinic instability model
of a continuously stratified atmosphere. This is the general model. When a point is reached in which the equations become mathematically intractible the two-layer model will be presented.
The coordinate system used here is one involving log-pressure as the vertical coordinate. The mathematics for this coordinate system is present elsewhere (log-pressure coordinates) The vertical coordinate is
where p is pressure and ps is pressure at the surface. H is the standard scale height given by H=RT/g where R is the gas constant, T is absolute temperature and g is the acceleration due to gravity. The governing equations for momenta are given by:
where V is the wind velocity vector, f is the Coriolis paramenter and Φ is the geopotential height. The parcel-following derivative is D/Dt=(∂/∂t + V·∇ + ω*(∂/∂z*), where ω=Dz*/Dt.
The continuity equation in the log-pressure coordinate system is
The thermodynamic equation in the absence of heat sources or sinks is, in the isobaric coordinate system:
where the static stability parameter
Absolute temperature T may be expressed in term of a gradient of the geopotential height throught the following derivation:
Now the thermodynamic equation can be expressed as
The vertical pressure velocity ω needs to be replaced by the corresponding variable for z*; i.e.,
With this replacement the thermodynamic equation becomes
The coefficient of w* in this equation may be defined as N2. This definition is
which, as it happens, is the same as the Brunt-Väisällä bouyancy frequency. N varies little with height and can be assumed to be constant.
The thermodynamic eqation in final form is thus
The quasi-geostrophic potential vorticity q is defined as the sum of relative vorticity, planetary vorticity and a stretching vorticity; i.e.,
For geostrophic (and quasi-geostrophic) flow the divergence is zero so there exists a quasi-geostrophic streamfunction ψ such that the wind velocities are given by V = k×∇ψ and relative vorticity by ζ=∇2ψ. This quasi-geostrophic stream function is related to the geopotential height by
The quasi-geostrophic potential vorticity q is preserved by stream flow; i.e.,
(To be continued.)
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