Quasi-Wavelet Models of Turbulent Temperature Fluctuations

Abstract

Here, we contribute to the continuing development of the quasi-wavelet (QW) model of turbulence that is currently being used in simulations of sound propagation and scattering in the turbulent atmosphere. We show that a QW model of temperature fluctuations exists for any physically reasonable temperature spectrum of isotropic homogeneous turbulence, including the widely used von Kármán spectrum. We derive a simple formula for the QW shape that reproduces a given spectrum exactly in the energy, transition, and inertial subranges. We also show that simple QW shapes can be normalized to yield an analytic expression for a temperature spectrum that is fairly close to any given spectrum. As an example, we match the Gaussian QW model to the von Kármán spectrum as closely as possible, and find remarkably good agreement in all subranges including the dissipation subrange. We also derive formulae for the variance and kurtosis associated with the QW model, and show how the latter depends on the QW packing fraction and size distribution. We also illustrate how the visual appearance of several QW-simulated temperature fluctuation fields depends on the QW packing fraction, size distribution, and kurtosis.

Sound Propagation and Scattering in Nighttime Atmospheric Boundary Layers

“…In particular, we found the parent functions and, hence, QW models, which exactly result in the von Kármán spectra of temperature and wind velocity fluctuations, see Refs. [3,16,22,23]. This is an important result which can be considered as a phenomenological motivation for the von Kármán spectra.…”

“…2 and 3. Finally, in [7], nonspherical QWs were used to model anisotropic, shear-driven velocity fluctuations. Note that wind shear is one of the mechanisms of turbulence in NBL.…”

“…QWs are localized, self-similar temperature or wind velocity eddies. The ensemble of QWs with many different sizes and properly chosen scaling laws resembles actual turbulence[3,7,16,18,23].…”

A quasi-wavelet model for the turbulent refractive index

“…A random field based on quasi-wavelets is created inspired by the Frisch's β model [11]: fluctuations of different sizes are randomly arranged over the available space. That is, following [9], we define the fluctuations of the atmospheric refractive index as…”

“…Over the past several years, a new model has been developed by Goedecke et al [8,9] describing the statistical properties of the velocity and temperature fluctuations of atmospheric turbulence: the quasi-wavelet (QW) model. Specifically, this model has successfully been employed in numerical simulations of atmospheric acoustics predicting wave scattering, outdoors sound propagation, and the forward problem of acoustic tomography [10].…”