Generalized turbulence space-correlation and wave-number spectrum-function pairs

Abstract

A new generalized pair of space-correlation and wave-number spectrum functions, which has properties indicated by fluid dynamics and by intuition, is proposed. In contrast with previously used combinations, both are analytic functions and smooth and cover the complete range of their arguments. For example, the spectrum form can apply to all wave numbers rather than to any limited portion, such as the inertial range. Whenever convolution is necessary, it becomes a routine computer calculation. Furthermore, the correlation function has a zero first derivative and a negative second derivative at the origin, giving both an integral scale and a microscale for the turbulence. The spectrum function can have an inertial region with some power-law decay for intermediate wave numbers and has an exponential decay for very large wave numbers. There are three adjustable parameters, determined by experiment, namely, the spectral power-law decay index, the turnover point where this power law starts, and the turnover point where the power dependence changes to an exponential decay. A library of such spectrum functions is not required, since the single proposed function, with its adjustable parameters, can be made to fit most data. One can also allow any even power of wave number for the spectral dependence in the limit of zero wave number. Simple anisotropic functional forms can readily be incorporated. It is indicated that two inertial falloffs with different power laws can be provided and the resultant correlation function is derivable by convolution. The results are applied to simplified versions of plasma electron density fluctuations and to velocity fluctuations of the background fluid. Various scales of turbulence and the related pressure-correlation function are derived. Expressions of correlation functions of amplitude and phase over parallel-line-of-sight propagation paths are also deduced.The proposed pair of functions should only be considered as a generalized model of Tatarski's form, which allows easy conversion from the correlation to the spectrum function and vice versa. Perhaps even more realistic forms can be invented.