The Basics

A baroclinic model of the atmosphere is one that does not use the assumptions of the barotropic model; i.e., that density depends only upon pressure and thus isobaric surfaces are also isothermal. A baroclinic model is more general than a barotropic model but it is not fully general. Winds are given by the geostrophic approximation. Geostrophic winds are nondivergent; i.e., their divergence is everywhere equal to zero. Because geostrophic winds are nondivergent there exists a stream function ψ such that

V = k×∇ψ = (-(∂ψ/∂y), (∂ψ/∂x))
ζ = ∇²ψ
 

where V and ζ are the geostrophic wind velocity vector and vorticity, respectively. ∇ represents the horizontal Laplacian.

Because the quasi-geostrophic wind is given by

V = (1/f₀)∇Φ
where Φ is geopotential height
and therefore it follows that
ψ = (1/f₀)Φ
 

Thus ψ may be considered a quasi-geostrophic stream function. Note for later use that

(∂Φ/∂p) = f₀((∂&pst;/∂p)
 

The Dynamics

The preservation of absolute vorticity leads to the geostrophic vorticity tendency equation, which in pressure coordinates is:

∂ζ/∂t = -V·∇(ζ+f)+f(∂ω/∂p)
 

where f is the Coriolis parameter (planetary vorticity) and ω is dp/dt, the vertical velocity.

This equation says that the local rate of change in vorticity is the sum of advection of absolute vorticity, (ζ+f), and a component for the stretching/shrinking of the fluid column due to divergence. The advection of planetary vorticity is βv, where β=∂f/∂y and v is the meridian wind velocity. The above vorticity tendency equation can be expressed in terms of the geostrophic stream function as:

(∂/∂t + V·∇)∇²ψ + β(∂ψ/∂x) - f(∂ω/∂p) = 0
 

In the quasi-geostrophic formulation β is taken to be a nonzero constant but f in the term f(∂ω/∂p) is also taken to be a constant f₀.

The quasi-geostrophic version of the thermodynamic equation in the absence of heat sources or sink is:

(∂/∂t + V·∇)T - (σp/R)ω = 0
 

where σ, the stability parameter, is taken to be constant.

The Elimination of Temperature as a Model Variable

As the model now stands there are two fields to be determined; that of ψ and that of T. Surprisingly there is a relationship between the ψ field and the T field that can be used to eliminate temperature as a separate variable. To find this relationship first note that the hydrostatic equilibrium equation (∂p/∂z)=-gρ could be expressed as

(∂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
and since, as noted earlier, Φ=f₀ψ so
T = -(f₀p/R)(∂ψ/∂p)

The thermodynamic equation can now be expressed as

(∂/∂t + V·∇)[-(f₀p/R)(∂ψ/∂p)] - (σp/R)ω = 0
which upon multiplying through by -R
and dividing by f₀ becomes
(∂/∂t + V·∇)(p∂ψ/∂p) + (σp/f₀)ω = 0
 

The local tendency operator ∂/∂t is defined for x, y and p constant and the horizontal gradient operator ∇ operator is defined for p and t constant so p is a constant in both differentiation expressions and can be taken outside the differentiation operations and factored out of the equation. The result is strictly in terms of the stream function ψ

(∂/∂t + V·∇)(∂ψ/∂p) + (σ/f₀)ω = 0
 

Although the model is called a two-layer model that apellation is misleading. The layers are not uniform over height so it is actually a five-level model as shown in the diagram below.

The gradients of vertical motion, ∂ω/∂p, at levels 1 and 3 are approximated by finite differences; i.e.,

(∂ω/∂p)₁ = (ω₂-ω₀)/(p₂-p₀)
(∂ω/∂p)₃ = (ω₄-ω₂)/(p₄-p₂)
 

The boundary conditions are that ω₀=0 and ω₄=0. The differences in pressure levels are taken as being of equal to Δp so p₂-p₀=2Δp and p₄-p₂=2Δp. (J.R. Holton in An Introduction to Dynamic Meteorology idiosyncratically defines 2Δp to be δp.)

The vorticity equations for level 1 and 3 are:

(∂ + V₁·∇)∇²ψ₁ + β(∂ψ₁/∂x) - (f₀/2Δp)ω₂ = 0
(∂ + V₃·∇)∇²ψ₃ + β(∂ψ₃/∂x) + (f₀/2Δp)ω₂ = 0
 

In evaluating the thermodynamic equation at level 2 it is necessary to make use of the finite difference approximation

(∂ψ/∂p)₂ = (ψ₃-ψ₁)/(p₃-p₁)
= (ψ₃-ψ₁)/(2Δp)
 

With this approximation, the thermodynamic equation is, after multiplying through by 2Δp,

(∂/∂t + V₂·∇)((ψ₃-ψ₁) - (σ2Δp/f₀)ω₂
 

The vectors of wind velocities V₁ and V₃ are given by:

V_i = (-(∂ψ/∂y)_i, (∂ψ/∂x)_i)) = (-(∂ψ_i/∂y), (∂ψ_i/∂x))
for i=1,3
 

but this formula cannot be applied for V₂ directly because ψ₂ is not known or computed in the model. But ψ₂ can be approximated as the mean of ψ₁ and ψ₃. With this approximation the equation for V₂ is:

V₂ = (1/2)(-(∂(ψ₁+ψ₃)/∂y), (∂(ψ₁+ψ₃)/∂x))
 

The model is now complete and determines ψ₁, ψ₃, V₁, V₃ and ω₂.

The Analysis of the Two-Layer Baroclinic Model