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It should be noted that Eq. (9') contains only low-frequency Rossby waves because of the use of geostrophic assumption (8b) to replace (1a). This assumption also eliminates other types of waves. Specifically, without making the quasi-geostrophic assumption, the linearized equations (1a), (9) and (10) form a closed set of equations describing the propagation of the types of waves in a divergent flow. Substituting the harmonic form of the solution into (1a), (9) and (10) leads to δ(-iω + ikU) uˆ - f0 vˆ + ikg hˆ = 0, (12a) (-iω + ikU) ik vˆ + β vˆ + f0 ik uˆ = 0, (12b) f0U (-iω + ikU) hˆ - g vˆ + ikH uˆ = 0, (12c) where δ is a tracer to evaluate the significance of the quasi-geostrophic assumption (8b).

From the previous discussions, we can see that the positive root is for high-frequency acoustic waves, while the negative root is for low-frequency gravity waves. Fig. 1 shows the solution for the case of a basic state at rest. 1) includes all possible solutions. Now let us examine the wave properties for some special cases. In particular, the propagation of acoustic waves is not of interest in meteorological and oceanic problems. , small error in the initial conditions) in a forecast model. In fact, their presence has significant effects on the efficiency of a numerical model.

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. Because of the terms containing α (z), Eqs. (6a) - (6d) are a set of partial differential equations with variable coefficients for which we could not simply assume the harmonic form of solutions. , ρu'2 = const. Thus, we may assume: (u', w', p', α') = (u* α 1/2, w* α 1/2, p* α -1/2, α* α 3/2) into (6), we obtain 53 (7a) ∂u * ∂p * + = 0, ∂t ∂x δh ∂w * ∂p * 1 dα + p* - gα* = 0, ∂t ∂z 2α dz ∂α * 1 d α ∂u * ∂w * 1 dα + w*) - ( + + w*) = 0, ∂t α dz ∂x ∂z 2α dz € ∂p * ∂α * 1 dα - gw*+ ca2 ( + w*) = 0.

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

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