M. C. Gregg, H. E. Seim and D. B. Percival (1993), `Statistics of Shear and Turbulent Dissipation Profiles in Random Internal Wave Fields,' Journal of Physical Oceanography, 23, no. 8, pp. 1777-99.
Because breaking internal waves produces most of the turbulence in the thermocline, the statistics of epsilon, the rate of turbulent dissipation, cannot be understood apart from the statistics of internal wave shear. The statistics of epsilon and shear are compared for two sets of profiles from the northeast Pacific. One set, PATCHEX, has internal wave shear close to the Garrett and Munk model, but the other set, PATCHEX north, has average 10-m shear squared S^2_10 about four times larger than the model.
The 10-m shear components, S_x and S_y, were measured between 1 and 9 MPa and referenced to a common stratification by WKB scaling. The scaled components, (hat S)_x and (hat S)_y, are found to be independent and normally distributed with zero means, as assumed by Garrett and Munk. This readily leads to analytic forms for the probability densities of (hat S)^2_10 and (hat S)^4_10. The observed probability densities of (hat S)^2_10 and (hat S)^4_10 are close to the predicted forms, and both are strongly skewed. Moreover, the standard deviations of the logs of (hat S)^2_10 and (hat S)^4_10 are constants, independent of the standard deviations of (hat S)_x and (hat S)_y. The probability density of the inverse Richardson number is a scaled version of the probability density of S^2_10. The PATCHEX distribution cuts off near a Richardson number of 4, as found by Eriksen, but the PATCHEX north distribution extends to higher values, as predicted analytically. Consequently, a cutoff at a Richardson number of 4 is not a universal constraint.
Over depths where the mean value of N^2 is nearly uniform, the probability density of 0.5-m epsilon can be approximated, to varying degrees of accuracy, as the sum of a noise variate with an empirically determined distribution and a lognormally distributed variate whose parameters can be estimated using a minimum chi-square fitting procedure. The 0.5-m epsilon samples, however, are far from being uncorrelated, a circumstance not considered by Baker and Gibson in their analysis of microstructure statistics. Obtaining approximately uncorrelated samples requires averaging over 10 m for PATCHEX and 15 m for PATCHEX north. These lengths correspond approximately to reciprocals of the wavenumbers at which the respective shear spectra roll off. After correcting the uncorrelated epsilon samples for noise and then scaling to remove the dependence on stratification, the scaled dissipation rates are lognormally distributed. (Without noise correction and scaling the data with the mean of N^2, the data are not lognormal; e.g., noise correction and scaling with either the mean of N or the mean of N^{3/2} do not produce lognormality.)
It is hypothesized that the approximate lognormality of the scaled epsilon data results from generation of turbulence in proportion to (hat S)^4_10. Lognormality is well established for isotropic homogeneous turbulence (Gurvich and Yaglom), and Yamazaki and Lueck show that it also occurs within individual turbulent patches. Bulk ensembles from the thermocline, however, include samples from many sections lacking turbulence as well as from a wide range of uncorrelated turbulent events at different evolutionary stages. Consequently, the bulk data do not meet the criteria used to demonstrate lognormality under more restricted conditions. If the authors are correct, the high-amplitude portion of scaled bulk ensembles scaled by the mean of N^2 is lognormal or nearly so owing to generation of the turbulence of a highly skewed shear moment. As another consequence, the standard deviation of the log of (hat S)^4_10 should provide an upper bound of 2.57 for the standard deviation of 10 meter log scaled epsilon data when turbulence is produced by the breaking of random internal waves. Because many parts of the profile lack turbulence, sensor noise limits the epsilon distribution to smaller spreads than those of (hat S)^4_10. In practice we observe a standard deviation of log epsilon hat approximately equal to 1.2 when (hat S)^2_10 equals GM76 (and 1.5 when (hat S)^2_10 is about three times GM76). For the larger spread, 95% confidence limits require that n be approximately equal to 60 for accuracies of plus/minus 100%; to 140 for for accuracies of plus/minus 50%; and to 2000 for for accuracies of plus/minus 10%. Owing to instrumentation uncertainties in epsilon estimates, the authors suggest accepting less restrictive confidence limits at one site and sampling at multiple sites to estimate average dissipation rates in the thermocline.
Log normal distribution
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