The Statistics of Oceanic Turbulence Measurements. Part II: Shear Spectra and a New Spectral Model

Abstract

Abstract This manuscript provides (i) the statistical uncertainty of a shear spectrum and (ii) a new universal shear spectrum, and (iii) shows how these are combined to quantify the quality of a shear spectrum. The data from four collocated shear probes, described in Part I, are used to estimate the spectra of shear, Ψ(k), for wavenumbers k ≥ 2 cpm, from data lengths of 1.0 to 50.5 m, using Fourier transform (FT) segments of 0.5 m length. The differences of the logarithm of pairs of simultaneous shear spectra are stationary, distributed normally, independent of the rate of dissipation, and only weakly dependent on wavenumber. The variance of the logarithm of an individual spectrum, , equals one-half of the variance of these differences and is , where Nf is the number of FT segments used to estimate the spectrum. This term σlnΨ provides the statistical basis for constructing the confidence interval of the logarithm of a spectrum, and thus, the spectrum itself. A universal spectrum of turbulence shear is derived from the nondimensionalization of 14 600 spectra estimated from 5 m segments of data. This spectrum differs from the Nasmyth spectrum and from the spectrum of Panchev and Kesich by 8% near its peak, and is approximated to within 1% by a new analytic equation. The difference between the logarithms of a measured and a universal spectrum, together with the confidence interval of a spectrum, provides the statistical basis for quantifying the quality of a measured shear (and velocity) spectrum, and the quality of a dissipation estimate that is derived from the spectrum. Significance Statement The results reported here can be used to estimate the statistical uncertainty of a spectrum of turbulent shear or velocity that is derived from a finite number of discrete Fourier transform segments, and they can be used to quantify the quality of a spectrum.