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/p_s)
 

where p is pressure and p_s 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 - S_pω = 0
 

where the static stability parameter

S_p= -(T/θ)(∂θ/∂p)
= RT/(c_pp) - ∂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)S_pω = 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*) + (RS_p/p)w* = 0
 

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

N² = (RS_p/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

(∂/∂t + V·∇)(∂Φ/∂z*) + N²w* = 0
 

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.)