A causality-preserving Fourier method for gravity waves in a viscous, thermally diffusive, and vertically varying atmosphere

Abstract 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.