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 The Derivation of the Two-Layer Model of Baroclinic Instability

## 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

## 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:

#### Vi = (-(∂ψ/∂y)i, (∂ψ/∂x)i)) = (-(∂ψi/∂y), (∂ψi/∂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.