On the propagation of mountain waves: linear theory

TL;DR

This work casts the linear mountain-wave problem in two dimensions as a Dirichlet boundary-value problem for a Helmholtz-type equation with an altitude-dependent Scorer parameter $F(z)$. Using a Weyl–Titchmarsh transform and Lyra's radiation condition, the authors construct a physically correct solution $w$ via a Green's function $K$, decomposed into evanescent, radiated, and trapped components, revealing when vertically propagating or trapped lee waves occur. The analysis generalizes beyond common Boussinesq simplifications, provides explicit solution formulas in representative cases (including $F=0$, constant $F$, and Morse-type $F_0-F$), and offers rigorous justification of existence, uniqueness, and monotonicity upstream. The results greatly enhance the theoretical understanding of mountain-wave propagation, enabling precise prediction of vertical and horizontal wave structures in stratified atmospheres and clarifying conditions for trapped lee waves to form.

Abstract

We derive and establish a solution concept for the linear mountain wave problem in two dimensions. After linearizing the governing equations and a change of variables, the problem can be stated as a Dirichlet boundary value problem for a Helmholtz equation in terms of the vertical wind profile in the upper half-plane, with altitude-dependent potential (the Scorer parameter). To single out the correct solution, we have to make use of a radiation condition which is, due to the different physical situation, different from the classical Sommerfeld radiation condition for electromagnetic or acoustic waves. We rigorously develop a transform method and construct the physically correct solution, following Lyra's monotonicity criterion for mountain waves. In this procedure, we clearly recognize the two typical types of mountain waves: vertically propagating waves and trapped lee waves. This paper is the first rigorous work on Helmholtz-like equations in the upper half-plane subject to such a non-classical radiation condition.

Figures (9)

• Figure 1: Breaking mountain waves over Mount Duval in Australia (Source: $\copyright$GRAHAMUK, CC BY-SA 3.0): the wave slopes steepens until the top of the wave overruns the lower part.
• Figure 2: Water vapor satellite images showing vertically propagating mountain waves over the Rocky Mountains in Colorado by visualizing atmospheric moisture patterns, with relatively dry regions rendered as dark and high-humidity regions shown in white (Source: COMET$\circledR$ https://learn.meted.ucar.edu $\copyright$ 1997--2025). By converting radiation measurements of the effective layer (the highest altitude with appreciable water vapor) into temperature, this imagery can trace upper-level atmospheric flows when cloud condensation does not occur and therefore visible imagery is powerless.
• Figure 3: Left: photograph of nacreous clouds taken from an aircraft flying in the lower stratosphere, above the highest clouds in the troposphere. The intensely bright wavelike nacreous clouds are higher still (about 20-30 km in the stratosphere). (Source: $\copyright$ Paul A. Newman/NASA). Right: aerial photograph of lee wave clouds (Reproduced with permission from Comet Program/NOAA.). The lee wave clouds are not carried along with the air flow: they are standing clouds, being continuously formed at the leading edge of each wave and continuously eroded at the trailing edge.
• Figure 4: Left: cap cloud covering the 2462 m high summit crater of Mayon Volcano, Philippines, on 23 April 2019 (Credit: Patryk Reba CC BY-SA 4.0). Right: lenticular clouds over the 4322 m high Mount Shasta in California, on 25 October 2013 (Credit: NOAA).
• Figure 5: Sketch of the two main types of mountain waves.

Theorems & Definitions (9)

• Proposition 1
• Remark 2
• Lemma 3
• Lemma 4
• proof
• Theorem 5
• Remark 6
• proof : Proof of Theorem \ref{['thm:main']}
• proof