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of Baroclinic Instability |
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
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
Thus ψ may be considered a quasi-geostrophic stream function. Note for later use that
The preservation of absolute vorticity leads to the geostrophic vorticity tendency equation, which in pressure coordinates is:
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:
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 f_{0}.
The quasi-geostrophic version of the thermodynamic equation in the absence of heat sources or sink is:
where σ, the stability parameter, is taken to be constant.
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
Geopotential height Φ is gz and therefore
The thermodynamic equation can now be expressed as
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
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.,
The boundary conditions are that ω_{0}=0 and ω_{4}=0. The differences in pressure levels are taken as being of equal to Δp so p_{2}-p_{0}=2Δp and p_{4}-p_{2}=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:
In evaluating the thermodynamic equation at level 2 it is necessary to make use of the finite difference approximation
With this approximation, the thermodynamic equation is, after multiplying through by 2Δp,
The vectors of wind velocities V_{1} and V_{3} are given by:
but this formula cannot be applied for V_{2} directly because ψ_{2} is not known or computed in the model. But ψ_{2} can be approximated as the mean of ψ_{1} and ψ_{3}. With this approximation the equation for V_{2} is:
The model is now complete and determines ψ_{1}, ψ_{3}, V_{1}, V_{3} and ω_{2}.
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