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Then, the linearized momentum, mass continuity and thermodynamic equations have the form: 46 ∂u ' ∂φ ' = + f v', ∂t ∂x ∂φ ' ∂v' = - f u', ∂y ∂t (2) ∂u ' ∂v' ∂ω ' + + = 0, ∂y ∂p ∂x ∂ ∂φ ' ( ) + σ ω' = 0, ∂t ∂p where ω = dp/dt is the vertical velocity in p-coordinates; φ = gz is geopotential height and z is the height of a constant pressure surface; σ = − α d ln θ/dp is the static stability parameter which is assumed here as a function of p only. We can eliminate ω by differentiating (2d) with respect to p and then using (2c).

T ∂t ∂z The symbols "δh" and "δc" are used to trace the effects of the vertical acceleration in the 3rd momentum equation, and the compressibility in the continuity equation, respectively. If we set δh = 0, the hydrostatic conditions are implied, while if we set δc = 0, which is equivalent to dρ/dt = 0, it means that the fluid is incompressible. Eqs. (6a) - (6d) are composed of four linear equations in four unknowns: u', w', p' and α'. Again, the effect of mean flow has been omitted since it merely adds to the propagation in the x-direction.

Now we can assume solutions of the form (u*, w*, p*, α*) = ( uˆ , wˆ , pˆ , αˆ ) ei(kx + mz -ωt), (7b) where uˆ , wˆ , pˆ and αˆ are constants. Substituting into (8) gives -iω uˆ + ik pˆ = 0, 1 -δh iω wˆ + (im - 2H ) pˆ - g αˆ = 0, 1 -δciω αˆ + δc wˆ /H - ik uˆ - (im + 2H ) wˆ = 0, -iω pˆ - g wˆ + ca2 (-iω αˆ + wˆ /H) = 0. The above equations can be organized into matrix form as ⎡−ω ⎤ ⎡ uˆ ⎤ 0 0 k ⎢ ⎥ −1⎥ ⎢ −δhω −g im − (2H) ⎥ ⎢iwˆ ⎥ ⎢ 0 = 0. ⎢−k im + (2H)−1 − δc /H −δcω ⎥ ⎢ αˆ ⎥ 0 ⎢ ⎥ ⎢ ˆ ⎥ g(γ −1) c 02ω ω ⎣ 0 ⎦ ⎣ p ⎦ (9) In order for a solution to exist, the determinant of coefficients must vanish, that is, € ⎡ € −δhω −g im − (2H)−1⎤ ⎡ 0 −δhω −g ⎤ ⎢ ⎥ ⎢ −1 ⎥ −1 0 ω ⎢im + (2H) − δc /H −δcω ⎥+k ⎢−k im + (2H) − δc /H −δcω⎥ = 0.

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A short course on atmospheric and oceanic waves by Derome J., Zhang D.L.


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