Spatio-temporal Spectra and Spectral Transfers in Fluid Dynamics.

Abstract

Motivated in part by interest in low-frequency variability in complicated flows such as the ocean or atmosphere but also by a general interest in the spatio-temporal structure of turbulent flows, we investigate two-dimensional and quasi-two-dimensional turbulence in the wavenumber-frequency domain.

First, given a general equation of motion we derive spatio-temporal spectral transfers and corresponding fluxes in terms of a general bilinear time-frequency representation. Such transfers generalize the spatial spectral transfers used in the well-known theories for the cascade of energy or enstrophy in two- and three-dimensional turbulence. Specifically for transfers based on the short-time Fourier transform, we also develop a theoretical model that quantifies the effects of either a mean flow or isotropic sweeping on the spatio-temporal spectral transfers.

Second, we use spatio-temporal spectral transfers as a diagnostic in simulations of forced-dissipated two-dimensional homogeneous isotropic turbulence, where the forcing is narrowband in both wavenumber and frequency. We use the simulations both to illustrate the physical meaning of the spectral transfers and to investigate the robustness of the diagnostic when applied to imperfect data. We find that temporal spectral fluxes remain quantitatively reliable for a range of dataset limitations, such as low temporal resolution, limited record duration, and the presence of a trend. The theory and numerical investigations outlined above provide a foundation for the interpretation of spatio-temporal transfers in more complex systems.

Third, we conduct a wavenumber-frequency analysis of a quasi-two-dimensional system: the single-layer shallow-water quasi-geostrophic equation on the beta plane, one of the simplest models for large scale oceanic and atmospheric dynamics. We report that the "nondispersive line" that sometimes appears in zonal wavenumber-frequency spectra is not just a signature of westward propagating vortices. The nondispersive line can also be a signature of westward propagating meandering jets, although the propagation speed of jets is slightly slower than that of westward propagating vortices. We also report the discovery of new spectral features, such as a "nonlinear dispersive curve" that also appears in simulations with meandering jets, and quasi-sinusoidal dispersive curves that appear in simulations with nearly zonal jets.