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Rolf G. Lueck

Abstract

The temperature anomaly of a fluid moving through circular and rectangular cylinders induced by the heat stored in the walls of these hollow cylinders is derived under the assumption of quasi-steady heat transfer. These geometries correspond closely to the SBE-4 conductivity cell manufactured by Sea-Bird Electronics (SBE) and the NBIS Mark III cell made by EG&G Ocean Products (NBIS). For a step change of ambient temperature, the initial volume-weighted thermal anomalies are 4.3 and 12%, and the relaxation times are 4.3 and 0.23 s, for the SBE and NBIS cells, respectively, under typical operating conditions. The error in the measured conductivity is significant both in magnitude and longevity. Ale magnitude and the relaxation time of the anomaly can be considerably reduced by increasing the rate of flow through a cell, by forcing the flow to be turbulent, and by appropriate changes in the wall thickness and material. The wall is not a source or sink of salt, consequently no analogous effect is produced by changes in the ambient salinity. The effect of the thermal inertia of the wall has not been previously reported because frequency response calibrations have been made in isothermal salt-stratified tanks.

The signal reported by a conductivity cell is governed by: (i) the initial flushing by the free stream in the cell, (ii) the thermal and saline boundary layers on the wall of the cell and (iii) the heat stored in the wall of the cell. The bulk of the response is determined by the flushing of the cell, which has a time scale of order L/u≈0.05 s and should be nearly the same for conductivity changes imposed by either temperature or salinity. The boundary layer is not the same for temperature and salinity because the diffusivities of heat and salt differ by a factor of 100. The time scale of boundary layer diffusion is approximately 0.1 s for heat and 0.4 s for salt. Finally, the heat stored in the wall influences the temperature in the boundary layer. The time scale of this influence is determined by the dimensions and geometry of the cell, the thermal properties of the wall, and the flow through the cell.

It is impossible, in principle, to remove transient salinity errors by merely matching the response of a conductivity cell to the response of a thermometer because the temperature and salinity responses of a cell are different.

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Rolf G. Lueck

Abstract

Some advection of water across the North Pacific subtropical front occurs by the subduction of surface mixed layers from the north side of the front underneath surface waters on the south side. Cross-frontal advection in the thermocline is obscure because waters from both sides of the front follow a single trajectory in θ−S space. When winds are less than 10 m s−1, turbulence between these layers is too small to generate significant vertical diffusion. However, typical winter storms could mix these layers, in less than 20 days, to form a single homogeneous surface layer up to 145 m thick. When surface winds are too weak to maintain mixing over the entire depth of a surface mixed layer, turbulence associated with internal waves in the top of the thermocline contributes to the restratification of the surface layers. On a sampling grid of 37 km, there is no evidence for a systematic geographic variation of the rate of dissipation of kinetic energy. The rate of dissipation near the front is larger than in low energetic regions like the Sargasso Sea or of Vancouver Island, but smaller than in highly energetic ones such as the Equatorial Undercurrent or warm-core rings.

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Rolf G. Lueck

Abstract

An empirically derived statistic is used to estimate the confidence interval of a dissipation estimate that uses a finite amount of shear data. Four collocated shear probes, mounted on a bottom anchored float, are used to measure the rate of dissipation of turbulence kinetic energy ϵ at a height of 15 m above the bottom in a 55 m deep tidal channel. One pair of probes measures ∂w/∂x while the other measures ∂υ/∂x, where w and υ are the vertical and lateral velocity. The shear-probe signals are converted into a regularly resampled space series to permit the rate of dissipation to be estimated directly from the variance of the shear using ϵ¯L=7.5ν(w/x)2¯L (and similarly for the υ component), for averaging lengths, L ranging from 1 to 104 Kolmogorov lengths. While the rate of dissipation fluctuates by more than a factor of 100, the fluctuations of the differences of ln(ϵ¯L) between pairs of probes are stationary, zero mean, and distributed normally for averaging lengths of L = ∼30 to 104 Kolmogorov lengths. The variance of the differences, σlnϵ2, scales as L −7/9, independent of stratification for buoyancy Reynolds numbers larger than ∼600, and for dissipation rates from ∼10−10 to ∼10−5 W kg−1. The variance decreases more slowly than L −1 because the averaging is done in linear space while the variance is evaluated in logarithmic space. This statistic provides the confidence interval of an ϵ estimate such as the 95% interval CI95(ϵ)=ϵexp(±1.96σlnϵ). This result also applies to the traditional ϵ estimates that are made by way of spectral integration, after L is adjusted for the truncation of the shear spectrum.

Significance Statement

The results reported here can be used to estimate the statistical uncertainty of a dissipation estimate that is derived from a finite length of turbulence shear data.

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Rolf G. Lueck

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, σlnΨ2, equals one-half of the variance of these differences and is σlnΨ2=1.25Nf7/9, where Nf is the number of FT segments used to estimate the spectrum. This term σ lnΨ 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.

Open access
Rolf G. Lueck and Fabian Wolk

Abstract

A technique is presented for determining if an estimate of covariance (or correlation) of two time series is statistically significantly different from zero. The technique makes no assumptions about the spectrum or the probability distribution of either time series. It is more efficient than the method of Yamazaki and Osborn and Fleury and Lueck by several factors of 10.

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Youyu Lu and Rolf G. Lueck

Abstract

A four-transducer, 600-kHz, broadband acoustic Dopple current profiler (ADCP) was rigidly mounted to the bottom of a fully turbulent tidal channel with peak flows of 1 m s−1. Rapid samples of velocity data are used to estimate various parameters of turbulence with the covariance technique. The questions of bias and error sources, statistical uncertainty, and spectra are addressed. Estimates of the Reynolds stress are biased by the misalignment of the instrument axis with respect to vertical. This bias can be eliminated by a fifth transducer directed along the instrument axis. The estimates of turbulent kinetic energy (TKE) density have a systematic bias of 5 × 10−4 m2 s−2 due to Doppler noise, and the relative statistical uncertainty of the 20-min averages is usually less than 20%–95% confidence. The bias in the Reynolds stress due to Doppler noise is less than ±4 × 10−5 m2. The band of zero significance is never less than 1.5 × 10−5 m2 s−2 due to Doppler noise, and this band increases with increasing TKE density. Velocity fluctuations with periods longer than 20 min contribute little to either the stress or the TKE density. The rate of production of TKE density and the vertical eddy viscosity are derived and in agreement with expectations for a tidal channel.

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Youyu Lu and Rolf G. Lueck

Abstract

This paper discusses the principles of measuring the mean velocity and its vertical shear in a turbulent flow using an acoustic Doppler current profiler (ADCP), and presents an analysis of data gathered in a tidal channel. The assumption of horizontal homogeneity of the first moments is fundamental to the derivation of the mean velocity vector because the velocity is never homogeneous over the span of the beams in a turbulent flow. Two tests of this assumption are developed—a comparison of the mean error velocity against its standard deviation and against the mean speed. The fraction of the samples that pass these tests increases with increasing spatial averaging and exceeds 95% for distances longer than 55 beam separations. The statistical uncertainty of the velocity and shear vector, averaged over 10 min and longer, stems from turbulent fluctuations rather than Doppler noise. Estimation of the vertical velocity requires a correction for the bias in the measured tilt.

The mean velocity and shear estimates from this natural tidal channel show more complex depth–time variations than found in idealized one-dimensional channel flow, which seldom occurs in nature. The ADCP measurements reveal the secondary circulation, bursts of up- and downwelling, shear reversals, and transverse velocity shear.

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Thomas R. Osborn and Rolf G. Lueck

Abstract

A towed body suitable for measuring oceanic velocity and temperature microstructure is described. The development was motivated by i) a requirement for long times series to produce statistically reliable estimates of dissipation rates, ii) the desire to observe salt fingers directly, and iii) the need to map the horizontal distribution of turbulence. The major problem with towed measurements is the contamination of the velocity signal by body vibrations which has been reduced by making the body almost neutrally buoyant and by minimizing aerodynamic forces. System noise in terms of ε, the rate of dissipation of kinetic enemy, is 2 × 10−6 W m−3, less than “typical” values of 10−4 W m−3 in wind mixing layers and 10−5 W m−3 in the seasonal thermocline.

Observations over the continental slope off Monterey Bay show a 1700 meter long turbulent layer in the seasonal thermocline. Values of log10ε are normally distributed with a standard deviation of 0.3. There is a significant excess of small dissipation values due to intermittency and a small deficit of large values due to a finite Reynolds number. The uniformity of the dissipation values and the responsiveness of dissipation rates to changes in the production of turbulent kinetic energy indicate that the mean vertical shear must be fairly uniform over the 1700 m long turbulent layer.

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Edward R. Levine and Rolf G. Lueck

Abstract

Horizontal profiles of the microstructure of velocity and temperature were obtained with a large autonomous underwater vehicle (AUV) using two piezoelectric shear probes, an FP07 thermistor, and three orthogonal accelerometers mounted on a sting at the forward end of the vehicle. A winter field trial in Narragansett Bay provided a run in the midwater pycnocline at 8-m depth that contained a thermal front, an ascending profile to 3-m depth, and a run in the weakly stratified surface layer at this depth. Although shear spectra were strongly contaminated by narrowband vibrations produced by the motor and actuators, this contamination was highly coherent with the measured acceleration and was removed with standard signal processing techniques. The corrected spectra agreed well with the Nasmyth universal spectrum for wavenumbers up to 40 cpm. The estimated rate of dissipation of kinetic energy varied from 0.8 to 250 × 10−8 W kg−1 and was consistent with the rate of production of turbulence by surface wind forcing and bottom stress.

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Hidekatsu Yamazaki, Rolf G. Lueck, and Thomas Osborn

Abstract

The dissipation rate of kinetic energy, ε, was estimated from adjacent and simultaneous measurements with a submarine and a vertical profiler. The submarine cycled up and down through the water column measuring both a vertical and a horizontal component of the turbulent velocity. The free-fall profiler measured two perpendicular, horizontal components, The mean value of the dissipations from the two systems, between 50 and 120 m depth along a 25 km transect, differed by a factor of 1.8. This difference was statistically significant and led us to examine the statistical distribution of the dissipation estimates.

For each profile from the submarine and the vertical profiler, the probability density function of the dissipation estimates is bimodal and well represented by a mixture of two lognormal distribution. This division of the data is a combination of an active mode and a relatively quiescent mode. The mean values of ε for the more dissipative modes are comparable for the submarine and the vertical profiler, often exceeding 20ρνN 2. The mean rates of the quiescent mode are not all the same. Those from the vertical velocity components are one decade smaller than the other three estimates which are all based on horizontal velocity components. Thus, this lower mode of the turbulence is anisotropic. All the means from the lower mode are smaller than 20ρνN 2.

The effect of anisotropy on the dissipation estimates is not sufficient to explain the differences in average dissipation between the two platforms. The difference was due to heterogeneity of the turbulence, on kilometer scales, which was revealed by the denser spatial sampling of the submarine but not resolved by the vertical profiler.

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