Absorption of inertia-gravity waves in vertically sheared rotating stratified flows
Abstract
It is well established that gravity waves have a substantial role on the large-scale atmospheric circulation, particularly in the middle atmosphere. In the present work, we re-examine the reflection and transmission of gravity waves through a critical layer surrounded by two inertial levels for the case of a constant vertically sheared flow. In this configuration, the vertical structure of the disturbance can be described as quasi-geostrophic from the critical layer up to the inertial levels, at which the Doppler-shifted frequency is equal to the Coriolis parameter. Near and beyond these levels, the balanced approximations do not apply and there is a transition from the quasi-geostrophic solution to propagating gravity waves. The three-dimensional disturbance solution is obtained analytically using both an exact method, in terms of hypergeometric functions, and a WKB approximation valid for large Richardson numbers; the latter includes an exponentially small term which captures the radiation feedback in the region between the inertial levels. We first focused on the homogeneous part of the disturbance equations, under the assumption of an unbounded domain. In contrast with past studies which show that there is a finite reflection and did not analyze the transmission (Yamanaka and Tanaka, 1984), we find that the reflection coefficient is too small to be significant and that the transmission coefficient is exactly like in the much simpler non-rotating case analyzed by Booker and Bretherton (1966). Our theoretical predictions are found to be in very good agreement with those obtained by numerically integrating the complete hydrostatic-Boussinesq equations with a small Rayleigh damping. The discrepancies between our results and those in Yamanaka and Tanaka (1984) are related to the fact that the solutions are given in term of multivalued functions and the values of the reflection and transmission coefficients are exponentially small, e.g. quite difficult to cross check numerically. More specifically, we suspect that the differences come from their treatment of the analytic continuations in the matching regions (e.g. the inertial layers). Our results are useful to study the evolution of initial disturbances. As an illustration, we consider the problem of gravity waves generated by potential-vorticity anomalies, a problem that was recently studied in Lott et al. (2013) for an unbounded atmosphere. The vertical structure of the potential-vorticity anomaly is represented by a Dirac distribution localized at the critical level. The disturbance field can be deduced from the homogeneous solutions above and below the critical level, by using suitable jump conditions. It is shown how the inclusion of a boundary condition within the problem, below the potential-vorticity anomaly, changes the amplitude of the radiated gravity wave, especially when the Richardson number is not too large. This process may be related to the occurrence of radiative instability waves in sheared rotating stratified flows.
- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2012
- Bibcode:
- 2012AGUFM.A31C0042M
- Keywords:
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- 3285 MATHEMATICAL GEOPHYSICS / Wave propagation;
- 3367 ATMOSPHERIC PROCESSES / Theoretical modeling;
- 3363 ATMOSPHERIC PROCESSES / Stratospheric dynamics