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A multilayer Fourier method is developed for modeling gravity waves above some altitude at which the vertical velocity is given as a boundary condition. The method is described in a general context in which the altitude and time variation of Fourier components for fixed horizontal wavenumbers is specified by a linear homogeneous partial differential equation (PDE) of any order, with coefficients varying with altitude. The coefficients are required to meet certain conditions so that causality is preserved and upgoing and downgoing modes can be defined. It is shown that causality is not preserved unless dissipative modes are included. An imaginary frequency shifting technique is introduced to allow upgoing and downgoing modes to be defined in certain situations that would otherwise be problematic. The pervasive problem of numerical swamping, endemic to multilayer approaches to viscous and thermally diffusive gravity-wave problems, is solved using a scattering matrix method, related to a method from seismology, which differs fundamentally from earlier gravity-wave solution methods cited herein. The method is applied to gravity waves in the linear, nonhydrostatic, anelastic, viscous, and thermally diffusive case with vertically varying background winds, in addition to certain limiting cases. The PDE in altitude and time is obtained from a dispersion relation that includes odd powers of the vertical wavenumber, making necessary the aforementioned imaginary frequency shifting technique.

In previous work (Knight et al., 2019) a multilayer Fourier method was introduced for obtaining solutions for a partial differential equation (PDE) associated with a dispersion relation for linear gravity waves in a vertically varying, anelastic, nonhydrostatic, viscous, and thermally diffusive atmosphere over some altitude range, given boundary conditions that allow waves to enter at the lower boundary and exit at the top. The previous work described an exact solution method for the dispersion-relation PDE using all wave modes. The current work describes a type of approximate solution, based on a single upgoing gravity-wave mode for each Fourier component, which is computationally cheaper and simpler to implement. Two different approaches to defining the single-mode solution are presented, one based on the Wentzel–Kramers–Brillouin (WKB) method and the other based on a simplified version of the multilayer approximation. The two approaches are shown to be mathematically equivalent. The accuracy of the single-mode approximation is investigated through comparison with the all-mode solution for a realistic example in which gravity waves generated by a convective storm in the troposphere propagate into the upper thermosphere and for which the single-mode approximation gives reasonably accurate results. It is necessary to include imaginary frequency shifts in some Fourier components in order for the different types of modes (e.g., upgoing vs. downgoing and gravity-wave vs. dissipative) to be well-defined in the presence of molecular viscosity. A systematic approach to determining the smallest possible imaginary frequency shift for any given Fourier component is introduced. The new method efficiently suppresses a specific type of numerical artifact associated with large imaginary frequency shifts.

An approximate solution for the steady linear internal wave field generated by a localized, horizontally moving source in a thermocline is derived using ray and WKB (Wentzel–Kramers–Brillouin) theory. The waves are assumed to be steady in a reference frame moving with the source velocity. This solution is shown to agree in the limit of propagating waves with an existing solution that is expressed in terms of an eigenfunction expansion. The WKB method is shown to reproduce all the factors in the eigenfunction solution, not just the eigenfunctions themselves. It also reveals the physical significance of the terms in the eigenfunction solution. Furthermore, the WKB solution suggests an alternate way to compute the wave field numerically that is computationally more efficient.

Expressions are derived for the wave steepness η z of mountain waves, defined as the altitude ( z ) derivative of the vertical displacement η . The derivation begins with a known Fourier integral for η . The altitude derivative of the Fourier integral for η introduces a singularity in the integrand that is not absolutely integrable at any altitude. It is seen, nonetheless, that the altitude derivative can be moved under the integral sign for altitudes above the ground, but not at the ground, for which a special expression is found relating wave steepness to the mountain height function. It is shown that the upwind limit of wave steepness at the ground is zero, but the downwind limit is nonzero in general, and an expression for the downwind limit in terms of the mountain height function is given. It is also found that the imaginary part of complex wave steepness diverges approaching the ground, and an asymptotic formula is given. The results are applied to three mountain topography shapes: elliptical Gaussian, circular with algebraic decay, and infinite ridge.