Nonlinear Evolution of an Upward Propagating Gravity Wave: Overturning, Convection, Transience and Turbulence

Abstract

A numerical simulation of the nonlinear evolution of an upward propagating gravity wave shows that over-turning (the turning over of isopotential temperature surfaces) is the mechanism responsible for limiting the growth of the wave. Wave saturation (the state in which wave amplitude is constant with height) in the mesosphere results in turbulence (random, subkilometer-scale motions), but turbulence is not responsible for limiting wave amplitude. Therefore, parameterizations of wave drag and wave-associated eddy diffusivity that derive from the turbulence model of wave saturation have no rigorous justification and could give erroneous results if employed in studies of middle atmosphere circulation and minor constituent mixing. The immediate consequence of wave overturning is small-scale convection (regular, cellular structures with length scales of several to tens of kilometers) in the moving unstable phases of the wave. Cellular convection grows at the expense of the wave and provides a stabilization of the gross stratification. Convection ultimately decays into turbulence. Strong residual gradients in stratification are removed by the turbulence. Turbulence is primarily an end product of the overturning process that limits wave amplitude. Strong overturning is possible before the onset of convection, and, as a result, the transport of heat and constituents is up, rather than down, the mean gradient over the unstable phase of the wave. Net heat and constituent transports over the entire phase of the wave are reduced by the counter-mean gradient contributions to the transports. Overturning is confined to only a limited region of the wave (the unstable phase); it leads to localization of intense convection and turbulence. Extremely large values of mixing would be required to halt wave growth without a significant degree of overturning. The effects of localization on eddy diffusivity parameterizations may be incorporated through multiplication of the Lindzen eddy diffusivity by the factor [1 ? (δ/2π) sin(2&pi/δ)]?1 where 1/δ is the fraction of the wave involved in turbulence. Wave kinetic energy rapidly reaches a level of slow growth during the nonlinear growth of the wave and then rapidly decreases after the onset of convection. In contrast, wave available potential energy increases monotonically until convection begins, at which time it exceeds wave kinetic energy by a large amount. Self-acceleration of the wave retards self-destruction through critical level formation in the accelerated mean flow. The Doppler-shifted phase speed is relatively constant with height as a consequence of self-acceleration and the tendency for the prevalence of faster waves, excited by the finite duration of forcing, to increase with altitude. Wave transience is responsible for generating a substantial mean wind which persists in the region of wave breakdown. The normal growth with height of transient nonlinear gravity waves is substantially less than the inverse square root of density growth because breakdown occurs before a steady state can be established through a deep layer, and because wave energy flux is diminished to supply the conversion of wave kinetic energy to wave available potential energy. As a consequence, wave momentum fluxes vary with height approximately as if the wave had reached constant amplitude, even prior to breakdown by overturning. Therefore, observations of limited wave growth with height may reflect the natural and gradual nonlinear evolution of upward propagating gravity waves rather than their breakdown and saturation.