Spectral Properties of One-Dimensional Diffusive Systems Subject to Stochastic Forcing

Abstract

The vertical wavenumber and frequency spectra of horizontal wind and temperature in stochastically driven systems with diffusion, either due to uniform background eddy and molecular transport, or due to adjustment processes associated with shear or convective instability, are studied. Because of the dominating role of vertical transport in a stratified fluid, one-dimensional Langevin-type equations could be ascribed to such systems in the vertical direction. The linear equation with uniform diffusion is solved explicitly, and the spectra follow power-law distributions if the stochastic force is Gaussian. The nonlinear equations with gradient (either shear or lapse rate) dependent diffusion coefficients are shown to support scale invariance, and the power-law indices of the spectra are determined from dynamic renormalization group (DRG) analysis under rather general conditions. The exact power-law indices vary with the spectrum of the stochastic force and the nonlinearity of the systems. If the wavenumber spectrum of the force is moderately red (between k0 and k?2), the spectral indices of horizontal wind and temperature and the range of their variability are in general agreement with those inferred from wind and temperature measurements. The indices in both linear and nonlinear cases are confirmed by numerical simulations. This theory may suggest an alternative explanation to the universal vertical wavenumber and frequency spectra and their variability. By relating the universal spectra to systems characterized by stochastic forcing and background diffusion or diffusive adjustment due to shear or convective instability, which are ubiquitous in a stratified fluid, the difficulty to associate the time- and location-independent spectral features directly with the highly time- and location-dependent gravity waves or wave-breaking events is avoided. If such systems are suggestive of the real atmosphere, there is a need to be cautious in making assumptions regarding gravity waves solely based on the universal spectra when analyzing and interpreting wind and temperature observations.