Weakly non-linear analysis of small-amplitude internal gravity waves forced by isolated topography
Geophysical and Astrophysical Fluid Dynamics , Volume 108 - Issue 5 p. 503- 535
Asymptotic methods and numerical simulations are used to examine the evolution of an internal gravity wave packet comprising a continuous spectrum of horizontal wavenumbers and propagating upwards in a continuously stratified shear flow. In the multiple-scale framework for a horizontally localized wave packet generated by stratified flow over a localized mountain range with multiple peaks, there are in general two horizontal scales: the "fast" scale which is defined by the oscillations within the packet, i.e. the number of peaks, and the "slow scale" which is defined by the horizontal extent of the packet, i.e. the width of the mountain range. The focus here is on the specific case of an isolated mountain where the spectrum of horizontal wavenumbers is centred at zero and the multiple-scaling procedure is thus simplified by the absence of the fast spatial scale. The background flow is vertically sheared and critical-level interactions occur. The time frame within which non-linear critical-level effects become significant is determined by the magnitude of the non-linear terms in the governing equations. With the isolated mountain forcing this time frame is significantly longer than in the case of a multiple-peak mountain range forcing and it depends on the horizontal scale of the forcing, as well as on the amplitude. At leading-order, the non-linear asymptotic solution approaches a steady state in the outer region at late time, but the zero-wavenumber component of the solution continues to evolve with time in the vicinity of the critical level.
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Campbell, L, & Nikitina, L.V. (L. V.). (2014). Weakly non-linear analysis of small-amplitude internal gravity waves forced by isolated topography. Geophysical and Astrophysical Fluid Dynamics, 108(5), 503–535. doi:10.1080/03091929.2014.903944