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## 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.

## 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.

## Abstract

In their derivation of the lognormal probability density function for volume-averaged dissipation rates, Gurvich and Yaglom assumed explicitly that these dissipation rates are statistically homogeneous and that the averaging scale is small compared to the domain scale of the turbulent flow and large compared to the Kolmogorov scale. Estimates of dissipation rates in the oceanic thermocline reported by various researchers do not, in general, distribute lognormally because these datasets are often not homogeneous, nor is the averaging scale small compared to the scale of the turbulent patches. The conventional method of computing dissipation rates, a spectral technique, is incompatible with the assumptions for a lognormal distribution. Dissipation rates do distribute lognormally when they are computed with an alternative method that is consistent with the assumptions made by Gurvich and Yaglom. The shortest averaging scale that produced a lognormal distribution is three Kolmogorov length scales.

## Abstract

In their derivation of the lognormal probability density function for volume-averaged dissipation rates, Gurvich and Yaglom assumed explicitly that these dissipation rates are statistically homogeneous and that the averaging scale is small compared to the domain scale of the turbulent flow and large compared to the Kolmogorov scale. Estimates of dissipation rates in the oceanic thermocline reported by various researchers do not, in general, distribute lognormally because these datasets are often not homogeneous, nor is the averaging scale small compared to the scale of the turbulent patches. The conventional method of computing dissipation rates, a spectral technique, is incompatible with the assumptions for a lognormal distribution. Dissipation rates do distribute lognormally when they are computed with an alternative method that is consistent with the assumptions made by Gurvich and Yaglom. The shortest averaging scale that produced a lognormal distribution is three Kolmogorov length scales.

## Abstract

Vertical profiles of microstructure velocity over the San Diego Trough showed enhanced levels of kinetic energy dissipation in the intrusive region between the California Undercurrent and the surface California Current. If the observed rate of dissipation is typical, then the kinetic energy of the undercurrent is extracted with a minimum time scale of 11 days. The time scale for the dissipation of total mechanical energy (kinetic plus potential) and the transit time from southern California to Vancouver Island are comparable. The vertical eddy diffusivity is less than 1.9 × 10^{−5} m^{−2} s^{−1} and is not a factor in the mixing of the undercurrent.

The most frequently observed thickness of a turbulent layer is 1–2 m. Layers thinner than 6 m contribute the most to the total dissipation, while thicker and less frequent layers are noticeable contributors.

## Abstract

Vertical profiles of microstructure velocity over the San Diego Trough showed enhanced levels of kinetic energy dissipation in the intrusive region between the California Undercurrent and the surface California Current. If the observed rate of dissipation is typical, then the kinetic energy of the undercurrent is extracted with a minimum time scale of 11 days. The time scale for the dissipation of total mechanical energy (kinetic plus potential) and the transit time from southern California to Vancouver Island are comparable. The vertical eddy diffusivity is less than 1.9 × 10^{−5} m^{−2} s^{−1} and is not a factor in the mixing of the undercurrent.

The most frequently observed thickness of a turbulent layer is 1–2 m. Layers thinner than 6 m contribute the most to the total dissipation, while thicker and less frequent layers are noticeable contributors.

## Abstract

A transect of velocity profiles across Gulf Stream warm-core ring 82-B shows that the thermostad contained virtually no vertical shear. Enhanced downward-propagating near-inertial wave shear was present at the base of the core, consistent with these waves encountering critical layers as they try to leave the region of negative vorticity. Wave amplification and a shrinking vertical wavelength at the critical layer should lead to instability and shear production of turbulence. High dissipation rates have been observed at the base of ring 81-D's core.

## Abstract

A transect of velocity profiles across Gulf Stream warm-core ring 82-B shows that the thermostad contained virtually no vertical shear. Enhanced downward-propagating near-inertial wave shear was present at the base of the core, consistent with these waves encountering critical layers as they try to leave the region of negative vorticity. Wave amplification and a shrinking vertical wavelength at the critical layer should lead to instability and shear production of turbulence. High dissipation rates have been observed at the base of ring 81-D's core.

## Abstract

The airfoil shear probe is the only robust sensor currently available for measuring the rate of dissipation of kinetic energy in the ocean. The wavenumber (or spatial) resolution of the shear probe is determined by its physical dimensions, while the bandwidth of shear fluctuations is determined by the rate of dissipation. For most oceanic work, the conventional airfoil probe resolves the shear spectrum adequately. However, measurements taken in regions of larger dissipation rates, such as boundary layers, require a resolution beyond that of currently available shear probes. A newly designed probe, with dimensions approximately one-half of those of the conventional probe, was tested side by side with the conventional probe in a vigorously turbulent tidal channel. The relative response of the two types of probes indicates that both probes are characterized by a single-pole low-pass filter, with half-power wavenumbers of 49 and 88 cpm for the larger and smaller probes, respectively. After correction for this response, the spectra from both probes agree closely for the dissipation range 10^{−7} to 10^{−4} W kg^{−1}. Variance estimates from corrected spectra only agree with the Nasmyth empirical spectrum over a limited range in dissipation rate.

## Abstract

The airfoil shear probe is the only robust sensor currently available for measuring the rate of dissipation of kinetic energy in the ocean. The wavenumber (or spatial) resolution of the shear probe is determined by its physical dimensions, while the bandwidth of shear fluctuations is determined by the rate of dissipation. For most oceanic work, the conventional airfoil probe resolves the shear spectrum adequately. However, measurements taken in regions of larger dissipation rates, such as boundary layers, require a resolution beyond that of currently available shear probes. A newly designed probe, with dimensions approximately one-half of those of the conventional probe, was tested side by side with the conventional probe in a vigorously turbulent tidal channel. The relative response of the two types of probes indicates that both probes are characterized by a single-pole low-pass filter, with half-power wavenumbers of 49 and 88 cpm for the larger and smaller probes, respectively. After correction for this response, the spectra from both probes agree closely for the dissipation range 10^{−7} to 10^{−4} W kg^{−1}. Variance estimates from corrected spectra only agree with the Nasmyth empirical spectrum over a limited range in dissipation rate.

## Abstract

A moored and autonomous instrument that measures velocity and temperature fluctuations in the inertial subrange using shear probes and FP07 thermistors has been deployed in a swift [*O*(1 m s^{−1})] tidal channel for eight days. The measured velocity signals are free from body vibrations for frequencies below 16 Hz in flows faster than 0.5 m s^{−1} and below 8 Hz for slower flows. At lower frequencies, fluctuations of torque on the instrument, due mainly to fluctuations of the ambient current, produce large pitching and rolling motions (≈4° peak) that can easily be reduced by minor mechanical changes. Vibrations at higher frequencies do not scale with flow speed and stem mainly from mechanical structures. The velocity spectrum is free from contamination by body motions between 0.2 and 20 cpm (the inertial subrange), and consistent estimates of the rate of dissipation of kinetic energy are obtained from the spectral levels of vertical and lateral velocity fluctuations within this range.

## Abstract

A moored and autonomous instrument that measures velocity and temperature fluctuations in the inertial subrange using shear probes and FP07 thermistors has been deployed in a swift [*O*(1 m s^{−1})] tidal channel for eight days. The measured velocity signals are free from body vibrations for frequencies below 16 Hz in flows faster than 0.5 m s^{−1} and below 8 Hz for slower flows. At lower frequencies, fluctuations of torque on the instrument, due mainly to fluctuations of the ambient current, produce large pitching and rolling motions (≈4° peak) that can easily be reduced by minor mechanical changes. Vibrations at higher frequencies do not scale with flow speed and stem mainly from mechanical structures. The velocity spectrum is free from contamination by body motions between 0.2 and 20 cpm (the inertial subrange), and consistent estimates of the rate of dissipation of kinetic energy are obtained from the spectral levels of vertical and lateral velocity fluctuations within this range.

## 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.

## 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.

## 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 *υ* component), for averaging lengths, *L* ranging from 1 to 10^{4} Kolmogorov lengths. While the rate of dissipation fluctuates by more than a factor of 100, the fluctuations of the differences of *L* = ∼30 to 10^{4} Kolmogorov lengths. The variance of the differences, *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 *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.

## 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 *υ* component), for averaging lengths, *L* ranging from 1 to 10^{4} Kolmogorov lengths. While the rate of dissipation fluctuates by more than a factor of 100, the fluctuations of the differences of *L* = ∼30 to 10^{4} Kolmogorov lengths. The variance of the differences, *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 *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.

## 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, *N _{f}* 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.

## 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, *N _{f}* 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.