Dissipative wave-mean interactions and the transport of vorticity or potential vorticity

Abstract

Wave-mean interactions of the classical type, in which the effect of the waves on the mean motion depends on wave breaking or other types of wave dissipation, are to be sharply distinguished from other types of wave-mean interaction that have no such dependence on dissipation. Important cases arise both for unstratified (homentropic) flow and for stably stratified flow under gravity. A very general way of characterizing what is meant by the classical, dissipative type of wave-induced mean motion is to say that the wave-induced mean motions are balanced motions, in a sense to be discussed, and that the effective mean force corresponds to the wave-induced vorticity or potential vorticity transport that results from wave dissipation. For a stratified fluid, ‘potential vorticity’ is to be understood in the sense of Rossby and Ertel. ‘Balanced’ is to be understood in whatever sense is needed to imply the invertibility of the vorticity or potential vorticity field to give the other fields describing the mean motion. At first sight this appears to require that an appropriate Mach, Froude and/or Rossby number for the mean motion should be much smaller than unity, but the fundamental, and in practice less stringent, principal requirement appears to be that the spontaneous emission, or aerodynamic generation, of sound, gravity and/or inertio-gravity waves by the mean flow should be weak.Three basic examples of dissipative wave-induced mean flow generation are presented and discussed. The first is the transport of vorticity by dissipating sound waves, which gives rise to classical acoustic streaming of the quartz-wind type. The transport or flux of vorticity can always be taken to be an exactly antisymmetric tensor; and in the case of a plane sound wave this tensor fluctuates about a mean value equal to and E are the mean mass and wave-energy densities, ω the intrinsic frequency, and k the wavenumber. This is a succinct way of making evident why it is only the contribution q to the radiation stress convergence per unit mass that is significant for the generation of mean streaming. The second example is the transport of Rossby–Ertel potential vorticity (PV) by internal gravity waves that are either dissipating laminarly, or ‘breaking’ to produce inhomogeneous three-dimensional turbulence. This PV transport gives rise to mean streaming in much the same way as the vorticity transport in the acoustic example. The transport or flux of PV can always be taken to be directed exactly along the isentropic surfaces θ = constant of the stable stratification, where θ is potential temperature or potential density as appropriate; and in the case of a plane internal gravity wave the wave-induced PV transport fluctuates about a mean value G × q, where G is the basic gradient of θ associated with the stable stratification. This is a succinct way of making evident why it is only the projection of q onto the basic stratification surfaces that is significant. In both the acoustic and the internal-gravity examples the transport is non-advective, and often upgradient. The third example is the corresponding problem for Rossby waves, in which the typical effect of wave dissipation is a downgradient PV transport. This is brought about in an entirely different way, namely through advection of PV anomalies by the fluctuating velocity field of the wave motion, whether the dissipation be laminar or by breaking.Processes of the sort idealized in the second and third examples are ubiquitous in the Earth's atmosphere and, for instance, largely control the strength of the global-scale middle atmospheric circulation and hence, for instance, the e-folding residence times (∼ 102 y) of man-made chlorofluorocarbons in the lower atmosphere.