Statistics of Shear and Turbulent Dissipation Profiles in Random Internal Wave Fields

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  • 1 Applied Physics Laboratory, College of Ocean and Fishery Sciences, University of Washington, Seattle, Washington
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Abstract

Because breaking internal waves produces most of the turbulence in the thermocline, the statistics of ε, the rate of turbulent dissipation, cannot be understood apart from the statistics of internal wave shear. The statistics of ε 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, 〈S210〉, about four times larger than the model.

The 10-m, shear components, Sx and Sy, were measured between 1 and 9 MPa and referenced to a common stratification by WKB scaling. The scaled components, S^x and 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 S^210 and S^410. The observed probability densities of S^210 and S^410 are close to the predicted forms, and both are strongly skewed. Moreover, σlnS^210 and σlnS^410 are constants, independent of the standard deviations of S^x and Sy. The probability density of the inverse Richardson number, Ri−110S210/〈N2〉, is a scaled version of the probability density of S210. The PATCHEX distribution cuts off near Ri−110 = 4, as found by Eriksen, but the PATCHEX north distribution extends to higher values, as predicted analytically. Consequently, a cutoff at Ri−110 = 4 is not a universal constraint.

Over depths where 〈N2〉 is nearly uniform, the probability density of 0.5-m ε 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 ε, 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 ε samples for noise and then scaling to remove the dependence on stratification, the scaled dissipation rates, ε^ ≡ ε(N20/〈N2〉), are lognormally distributed. (Without noise correction and 〈N2〉 scaling the data are not lognormal; e.g., noise correction and scaling with 〈N1〉 and 〈N3/2〉 do not produce lognormality.)

It is hypothesized that the approximate lognormality of bulk ensembles of ε^ results from generation of turbulence in proportion to S410. 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 〈N2〉-scaled bulk ensembles is lognormal or nearly so owing to generation of the turbulence by a highly skewed shear moment. As another consequence, σlnS^410 = 2.57 should be an upper bound for 10 m σ˜lnε^ when the turbulence is produced by the breaking of random internal waves. Because many parts of the profile lack turbulence, sensor noise limits the ε distribution to smaller spreads than those of S410. In practice we observe σ˜lnε^ ≈ 1.2 when S^210 equals GM76, and σ˜lnε^ ≈ 1.5 when S^210 is about three times GM76. For the larger spread, 95% confidence limits require n ≈ 60 for accuracies of ±100%, n ≈ 140 for ±50%, and n ≈ 2000 for ±10%. Owing to instrumental uncertainties in ε 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.

Abstract

Because breaking internal waves produces most of the turbulence in the thermocline, the statistics of ε, the rate of turbulent dissipation, cannot be understood apart from the statistics of internal wave shear. The statistics of ε 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, 〈S210〉, about four times larger than the model.

The 10-m, shear components, Sx and Sy, were measured between 1 and 9 MPa and referenced to a common stratification by WKB scaling. The scaled components, S^x and 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 S^210 and S^410. The observed probability densities of S^210 and S^410 are close to the predicted forms, and both are strongly skewed. Moreover, σlnS^210 and σlnS^410 are constants, independent of the standard deviations of S^x and Sy. The probability density of the inverse Richardson number, Ri−110S210/〈N2〉, is a scaled version of the probability density of S210. The PATCHEX distribution cuts off near Ri−110 = 4, as found by Eriksen, but the PATCHEX north distribution extends to higher values, as predicted analytically. Consequently, a cutoff at Ri−110 = 4 is not a universal constraint.

Over depths where 〈N2〉 is nearly uniform, the probability density of 0.5-m ε 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 ε, 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 ε samples for noise and then scaling to remove the dependence on stratification, the scaled dissipation rates, ε^ ≡ ε(N20/〈N2〉), are lognormally distributed. (Without noise correction and 〈N2〉 scaling the data are not lognormal; e.g., noise correction and scaling with 〈N1〉 and 〈N3/2〉 do not produce lognormality.)

It is hypothesized that the approximate lognormality of bulk ensembles of ε^ results from generation of turbulence in proportion to S410. 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 〈N2〉-scaled bulk ensembles is lognormal or nearly so owing to generation of the turbulence by a highly skewed shear moment. As another consequence, σlnS^410 = 2.57 should be an upper bound for 10 m σ˜lnε^ when the turbulence is produced by the breaking of random internal waves. Because many parts of the profile lack turbulence, sensor noise limits the ε distribution to smaller spreads than those of S410. In practice we observe σ˜lnε^ ≈ 1.2 when S^210 equals GM76, and σ˜lnε^ ≈ 1.5 when S^210 is about three times GM76. For the larger spread, 95% confidence limits require n ≈ 60 for accuracies of ±100%, n ≈ 140 for ±50%, and n ≈ 2000 for ±10%. Owing to instrumental uncertainties in ε 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.

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