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 The Development of 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 Basic Governing Equations

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

#### z* = -Hln(p/ps)

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:

#### DV/Dt + fk×V = -∇Φ

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

#### ∇V + (1/ρ)(∂(ρω*)/∂z*) = 0

The thermodynamic equation in the absence of heat sources or sinks is, in the isobaric coordinate system:

#### (∂/∂t + V·∇)T - Spω = 0

where the static stability parameter

#### Sp= -(T/θ)(∂θ/∂p) = RT/(cpp) - ∂θ/∂p

Absolute temperature T may be expressed in term of a gradient of the geopotential height throught the following derivation:

• From the hydrostatic balance equation

#### (∂z/∂p) = -1/gρ so (∂(gz)/∂p) = -1/ρ and for an ideal gas for which ρ = p/RT, (∂(gz)/∂p) = -RT/p;

• Geopotential height Φ is gz and therefore

#### (∂Φ/∂p) = -RT/p p(∂Φ/∂p) = -RT (∂Φ/∂(ln p) = -RT and, since dz*=-Hd(ln p) (∂Φ/∂z*) = RT/H and hence T = (H/R)(∂Φ/∂z*)

Now the thermodynamic equation can be expressed as

#### (∂/∂t + V·∇)(∂Φ/∂z*) - (R/H)Spω = 0

The vertical pressure velocity ω needs to be replaced by the corresponding variable for z*; i.e.,

#### w* = -Hω/p = -H[D(ln p)/Dt]

With this replacement the thermodynamic equation becomes

#### (∂/∂t + V·∇)(∂Φ/∂z*) + (RSp/p)w* = 0

The coefficient of w* in this equation may be defined as N2. This definition is

#### N2 = (RSp/p) = R[-(T/θ)(∂θ/∂p) = -RT(∂( ln θ)/∂(ln p))

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

## Quasi-Geostrophic Potential Vorticity

The quasi-geostrophic potential vorticity q is defined as the sum of relative vorticity, planetary vorticity and a stretching vorticity; i.e.,

#### q = ζ + f + (∂/∂z*)(ε(∂ψ/∂z*))

where ε=(f0/N)2.

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

#### ψ = Φ/f0

The quasi-geostrophic potential vorticity q is preserved by stream flow; i.e.,

#### Dq/Dt = 0 where D/Dt = (∂/∂t + V·∇)

(To be continued.)