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Two-Layer (Five-Level) Model of the Atmosphere |
The governing equations for the model, which are derived elsewhere, are
where V_{1}, V_{2} and V_{3} are the wind velocity vectors at levels 1, 2 and 3, respectively, and are determined from the stream function ψ at those two levels. ω_{2} is the vertical pressure velocity at level 2. The relationships to the levels is shown below.
It is assumed that V_{2}=(V_{1}+V_{2})/2. Rather than replacing V_{2} by this relationship it is more convenient to express the 1 and 3 levels of the variables in terms of the level 2 variables and the difference between level 1 and level 3 values. But first it is necessary to obtain the linearized form of the equations.
It is assumed that the stream function ψ is of the form
This notation will be later replaced by a notation in which ψ_{i}(x,y,t) means ψ(x,y,p_{i},t).
Under the assumption concerning the stream function then
In the linearization process the only parts of the wind velocity vectors that can survive in the momentum equations are the background velocities; i.e., (U(p), 0). Thus the linearized momentum equations are:
Let
where the Th in U_{Th} stands for thermal.
With these definitions the momentum equations become
Or, upon rearrangement,
Now if these two equations are added and the result is:
Likewise if the second equation is subtracted from the first the result is:
Using the defintions
the previous two equations can be expressed as
The linearization of the equation which comes from the thermodynamic equation is only a bit more complex. First, we must go back to the original stream functions
When this last expression is substituted into the thermodynamic equation the result is, after division by 2,
If the variables are to have wave-like solutions
then the complex constants A, B and C must be such that
C may be eliminated by dividing the third equation by (σΔp/f_{0}) and multiplying by (f_{0}/2Δp) and subtracting from the second equation. This is equivalent to multiplying the coefficients of A and B in the third equation by [(f_{0}/2Δp)]/[(σΔp/f_{0})] = f_{0}^{2}/(4σ(Δp)^{2}) and subtracting from the second equation. This factor of f_{0}^{2}/(4σ(Δp)^{2}) will be denoted as Λ^{2}. The result is
After elimination of common factors of ik the set of equations to be satisfied by A and B is
For a nontrivial solution the determinant of the coefficient matrix for these equations must be zero.
The value of the determinant is
When this is set equal to zero the result is a quadratic equation in (c-U_{2}); i.e.,
Its solution is
The dispersion relation for the wave solutions is
The dividing line between stable and unstable waves is then
The latter equation can be put into the convenient form
which is a quadratic equation in (k/Λ)^{4} and the solution is
Thus as U_{Th} → ∞ k goes asymptotically to 0 or 1.
The relationship of U_{Th} to (k/Λ) is as is shown below:
The minimum value of U_{Th} is where the two roots for (k/Λ) are equal; i.e.,
The value of (k/Λ)^{4} where that minimum of U_{Th} is achieved is (1/2) so the value of k is (1/2)^{1/4}Λ.
For the following values of the parameters, all in SI units,
the value of Λ is 6.32×10^{-6} and the corresponding wavelength is 0.9935×10^{6} m = 993.5 km. The minimal value of an unstable thermal wind U_{th} is 12.1 m/s, which occurs for a wave of length equal to 1181 km.
(To be continued.)
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