Interaction between Meso- and Sub-Mesoscale Gravity Waves

Present-day high-resolution numerical weather prediction (NWP) models are resolving the mesoscale part of the gravity wave (GW) spectrum, while the effect of potentially relevant sub-mesoscale GWs is not taken into account. It seems worthwhile examining how the effect of such waves can be described efficiently in NWP codes. In contrast to the classic situation where GW parameterisations have been developed for interactions of mesoscale GWs with a synoptic-scale balanced flow, these unresolved GWs now propagate in a completely different background that cannot be assumed to be balanced to leading order. Consequently, it is necessary to reconsider the basic theory and study the interaction between meso- and sub-mesoscale GWs theoretically and numerically. A multi-scale asymptotic analysis is applied in Boussinesq dynamics in order to identify regimes for this interaction, characterised by the amplitude and aspect ratio of small-scale waves, and the ratio of Coriolis parameter and Brunt-Väisälä frequency, where powers of the latter are acting as the scale-separation parameter. It is found that mesoscale waves are mainly influenced by the vertical flux of horizontal momentum associated with the sub-mesoscale waves. Moreover, the sub-mesoscale GW field is able to produce mesoscale wind patterns, connected to a resonance phenomenon (Tabei and Akylas 2007). As variations of background stratification and mesoscale wind patterns also impact the characteristics of the submesoscale wave field, a two-way coupling occurs that can be studied by a WKB model. The numerical implementation uses a Lagrangian-WKB ray-tracer approach with a phase-space wave-action density (Muraschko et al 2015), thereby avoiding instabilities due to caustics. Fully nonlinear Large Eddy Simulations, resolving also sub-mesoscale wave structures, serve in idealized settings as a reference, confirming our theoretical findings, and validating the phase space approach of the Lagrangian WKB model for vertically as well as horizontally confined wave fields. The theory and the corresponding ray tracer are described, and validating numerical simulations are shown. Several cases are discussed, exhibiting e.g. the radiation of a mesoscale GW by a sub-mesoscale GW packet.

References:

Muraschko, J., Fruman, M.D., Achatz, U., Hickel, S., and Y. Toledo, 2015. On the application of Wentzel-Kramer- Brioullin theory for the simulation of the weakly nonlinear dynamics of gravity waves. Quart. J. Roy. Met. Soc.,
141, 676-697.

Tabaei, A., and T.R. Akylas, 2007. Resonant Long-Short Wave Interactions in an Unbounded Rotating Stratified Fluid. Stud. Appl. Math., 119, 271-296.