An application of WKBJ theory for triad interactions of internal gravity waves in varying background flows

Abstract

Motivated by the question of whether and how wave–wave interactions should be implemented into atmospheric gravity‐wave parametrizations, the modulation of triadic gravity‐wave interactions by a slowly varying and vertically sheared mean flow is considered for a non‐rotating Boussinesq fluid with constant stratification. An analysis using a multiple‐scale WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) expansion identifies two distinct scaling regimes, a linear off‐resonance regime, and a nonlinear near‐resonance regime. Simplifying the near‐resonance interaction equations allows for the construction of a parametrization for the triadic energy exchange which has been implemented into a one‐dimensional WKBJ ray‐tracing code. Theory and numerical implementation are validated for test cases where two wave trains generate a third wave train while spectrally passing through resonance. In various settings, of interacting vertical wavenumbers, mean‐flow shear, and initial wave amplitudes, the WKBJ simulations are generally in good agreement with wave‐resolving simulations. Both stronger mean‐flow shear and smaller wave amplitudes suppress the energy exchange among a resonantly interacting triad. Experiments with mean‐flow shear as strong as in the vicinity of atmospheric jets suggest that internal gravity‐wave dynamics are dominated in such regions by wave modulation. However, triadic gravity‐wave interactions are likely to be relevant in weakly sheared regions of the atmosphere.

The Interaction Between Rossby Waves and the Mean Flow

“…It would be interesting to obtain a corresponding extension of the theory, e.g. following the route indicated by [16]. Such work would close a conceptual gap that we still seem to have in the weakly nonlinear regime between the quasilinear regime (allowing a spectral approach) and the large-amplitude regime (where only locally monochromatic GW fields are possible).…”

“…In the weakly nonlinear case n = 1 the wave-action equation is to be supplemented by the effect of wave-wave interactions, of either GWs with GWs or GWs with GMs. To the best of our knowledge, the corresponding scattering integrals have only been worked out fully for Boussinesq dynamics without mean flow [14,15], and first steps for Boussinesq dynamics with non-vanishing mean flows have been taken by [16]. In the atmospheric context they have so far been simply ignored.…”

“…A numerical implementation of spectral wave-action dynamics could either be done in a Eulerian finite-volume formulation, based on (71), or in a Lagrangian approach starting from (72). In the latter [16][17][18][19] one sub-divides the part of phase space with non-zero wave-action density into rectangular so-called ray volumes. Wave-action density is conserved along trajectories (called rays) satisfying…”

Resonant and near-resonant internal wave triads for non-uniform stratifications. Part 2. Vertically bounded domain with mild-slope bathymetry

“…Analytical methods have also been developed for studying wave-wave interactions in non-resonant regimes in the presence of a slowly varying background shear flow (e.g. Grimshaw 1988Grimshaw , 1994Voelker, Akylas & Achatz 2021), where the wave train can pass through non-resonant regimes and resonant regimes. However, this is not within the scope of this paper.…”