Multiscale Large Eddy States in Weakly Stratified Planetary Boundary Layers

Abstract

We first discuss observations of two classes of two-dimensional large eddy states in weakly stratified atmospheric boundary layers. One class is characterized by large eddies with a single horizontal scale. The other contains multiscale large eddies, with horizontal wavelengths on the order of the boundary layer depth (as generally supported by spectra), coexisting with scales that are several times this depth (as generally seen in cloud street patterns). We then present a theory which describes these different large eddy states as the result of wave–wave interactions between growing disturbances in the planetary boundary layer. We develop nonlinear, coupled evolution equations that govern the temporal evolution of a set of three isolated disturbances of initial eigenmode form in a general unstable flow. These equations model both wave/wave and wave/mean flow interactions. They imply that in unstable flow regimes, modes with relatively large growth rates as predicted by linear theory act as catalysts for the transfer of mean flow energy into modes for which linear theory would predict slow growth or even decay. This energy transfer results in the acceleration of the growth of these initially slow modes. In essence, relatively fast growing instabilities to an unstable mean state are themselves unstable. Because of this indiscriminate catalytic property, a wide range of waves can compete for prominence other than the primary instabilities to a given mean state. We therefore explore four broad classes of wave–wave interactions. This produces predicted large eddy velocity fields which we relate to two classes of observed large eddy states. The predicted multiscale states may be described as a long wavelength modulation of short wavelength roll vortices. Finally, we suggest that the determining factor in selecting a large eddy state in the boundary layer for given synoptic conditions is the presence or absence of sources of waves to the boundary layer that complement the dynamically or convectively driven boundary layer roll vortices. These additional waves could be due to other instability mechanisms, or topographic effects. They would project onto the eigenstates of the boundary layer, introducing relatively well defined scales in the flow field along with the dominant boundary layer disturbances, thereby selecting a set of waves for interaction.