Do transient gravity waves in a shear flow break?

Abstract

ABSTRACT:The propagation of transient gravity waves in a shear flow towards their critical levels is examined using a ray tracing approximation and a higher-degree (quasi-optic) approximation. Because of its transient forcing, the amplitude of transient waves decays to zero in the neighbourhood of the critical region so that it is not clear whether transient gravity waves will reach the convective instability threshold or not. The analysis shows that the horizontal perturbation decays asymptotically as the inverse of the square root of time, while the vertical wavenumber depends linearly on time, thus transient gravity waves attain convective instability for long times. The theoretical results are compared with numerical simulations. The ray path approximation is not able to reproduce the maximum amplitude, but the quasi-optic approximation gives a reasonable agreement at short and long times. There are three breaking regimes for transient gravity waves. For wave packets with a narrow frequency spectrum (quasi-steady waves) and large enough initial wave amplitude, the wave breaking is similar to the abrupt monochromatic wave overturning. On the other hand, highly transient wave packets will dissipate near the critical region for very long times with small wave amplitudes and high vertical wavenumber. The third regime is a transition between the two extremes; in this case both wave amplitude and vertical wavenumber are important to produce the convective threshold. The dependencies of the convective instability height (a quantity that may be useful for gravity wave parametrizations) on the Richardson number and the frequency spectral width are obtained.

Review of wave‐turbulence interactions in the stable atmospheric boundary layer

“…The T-G equation (1) can become singular at a critical level, z cl , where U(z cl ) = c and c is the wave phase speed in the direction of wind velocity. Most investigations of wave critical levels in the literature focus on their impact on wave generation and property changes such as wave dissipation/breaking [e.g., Geller et al, 1975;Nappo and Chimonas, 1992;Moustaoui et al, 2004;Lane and Sharman, 2008;Pulido and Rodas, 2008], mean flow acceleration through wave momentum flux deposition when IGWs break near critical levels [e.g., Jones and Houghton, 1971;Hirt, 1981;Weinstock, 1982;Nappo and Chimonas, 1992], IGW reflection from critical levels [e.g., Jones, 1968;Hirt, 1981], wave energy leakage across critical levels [e.g., Booker and Bretherton, 1967;Jones and Houghton, 1971;Teixeira et al, 2008], and three-dimensional wave instability near critical levels [Winters and D'Asaro, 1994]. A background shear flow can supply energy to waves through instabilities and can extract energy from waves through wave momentum deposition at wave critical levels [e.g., West, 1981].…”

The breaking of transient inertio-gravity waves in a shear flow using the Gaussian beam approximation

“…Note that (2.15) is similar to the expression for the wave energy density obtained in Pulido & Rodas (2008) for internal gravity waves, but in that case the derivation is based on energy equipartition while for inertio-gravity waves that principle is not valid.…”

“…where the c subindex represents evaluation at the central absolute frequency, Ω c is the central intrinsic frequency and Ω ic is the initial central intrinsic frequency. The transient inertio-gravity wave solution obtained in (2.19) differs from the solution obtained by Pulido & Rodas (2008) for transient internal waves in the amplitude and phase terms due to the dispersion relation. The part of the exponential with imaginary argument in (2.19) represents the phase that defines the central ray given by the stationary phase points ∂ ω ψ c = 0.…”

“…To determine the wave amplitude and height of the breaking for transient waves, the usual spatial ray theory (Lighthill 1978) is not useful, and nor are the stationary The breaking of transient inertio-gravity waves 679 phase methods, because they only give the asymptotic tendency for long times, while transient waves may reach the instability threshold before attaining the asymptotic regime. Indeed, transient gravity wavepackets reach the maximum amplitude during propagation well below the critical level, a feature that is not captured with the first-order asymptotic approaches (Pulido & Rodas 2008). The Gaussian beam formulation, which is extensively used in other contexts such as optics (Heyman & Felsen 2001), quantum mechanics (Hagedorn 1984) and seismic waves (Cerveny, Popov & Psencik 1982;Cerveny 1983), is able to represent the wavepacket from the early times -this formulation is free of the caustic artifacts that appear in the spatial ray path, in particular at the initial time -up to long times when it reaches the asymptotic behaviour.…”

“…In the case of a transient wave in a shear flow, there is a continuous frequency spectrum associated with the wave so that the different components of the frequency spectrum reach critical levels at different heights. Pulido & Rodas (2008) examined a transient internal gravity wave that propagates in a shear flow (in a system without rotation) towards the critical level. They found that the wave amplitude of the horizontal wind reaches a maximum and then decays asymptotically as the inverse root square of time, contrary to steady-state waves (i.e.…”