Abstract

Simultaneous measurements of temperature, salinity, their vertical gradients, and the vertical gradient of velocity across a 1.4-m-long Lagrangian float were used to investigate the accuracy with which the dissipation of scalar variance χ can be computed using inertial subrange methods from such a neutrally buoyant float. The float was deployed in a variety of environments in Puget Sound; χ varied by about 3.5 orders of magnitude. A previous study used an inertial subrange method to yield accurate measurements of ɛ, the rate of dissipation of kinetic energy, from this data. Kolmogorov scaling predicts a Lagrangian frequency spectrum for the rate of change of a scalar as ΦDσ/Dt(ω) = βsχ, where βs is a universal Kolmogorov constant. Measured spectra of the rate of change of potential density σ were nearly white at frequencies above N, the buoyancy frequency. Deviations at higher frequency could be modeled quantitatively using the measured deviations of the float from perfect Lagrangian behavior, yielding an empirical nondimensional form ΦDσ/Dt = βsχH(ω/ωL) for the measured spectra, where L is half the float length, ω3L = ɛ/L2, and H is a function describing the deviations of the spectrum from Kolmogorov scaling. Using this empirical form, estimates of χ were computed and compared with estimates derived from ɛ. The required mixing efficiency was computed from the turbulent Froude number ω0/N, where ω0 is the large-eddy frequency. The results are consistent over a range of ɛ from 10−8 to 3 × 10−5 W kg−1 implying that χ can be estimated from float data to an accuracy of least a factor of 2. These methods for estimating ɛ, χ, and the Froude number from Lagrangian floats appear to be unbiased and self-consistent for ɛ > 10−8 W kg−1. They are expected to fail in less energetic turbulence both for instrumental reasons and because the Reynolds number typically becomes too small to support an inertial subrange. The value of βs is estimated at 0.6 to within an uncertainty of less than a factor of 2.

1. Introduction

Measurements of mixing rates in the ocean have typically been made using free-fall or loosely tethered instruments deployed from research vessels (Lueck et al. 2002). As our understanding of mixing physics has increased, there is an increasing need to make long-term mixing measurements in a variety of locations and therefore develop techniques for making such measurements from autonomous platforms. The most commonly used methods to measure mixing from profilers involve the measurement of the turbulent kinetic energy dissipation rate ɛ and the temperature variance dissipation rate χT. D’Asaro et al. (1996) suggest that ɛ could be measured from the high-frequency vertical motion of a neutrally buoyant float. Lien and D’Asaro (2006) make a direct test of this method using data from a Lagrangian float equipped with a microstructure velocity sensor and find it to be accurate to within experimental error. Lien et al. (1998) and Lien and D’Asaro (2004) suggest that χx, the dissipation of the variance of scalar x, could be measured similarly from the high-frequency fluctuations of x measured on a float. Lien et al. (1998) report reasonable agreement in a small dataset.

Here, we develop and evaluate the accuracy of an inertial subrange method to measure scalar dissipation χ from neutrally buoyant floats using the same float deployments analyzed by Lien and D’Asaro (2006). The spectral forms for acceleration, velocity, and scalars measured on floats are reviewed and extended to include instrumental effects. This yields a practical method to estimate χ. The theory relating χ and ɛ measurements through the mixing efficiency Γ is reviewed and a practical, Froude number–based method to estimate Γ from float data using the analysis of Ivey and Imberger (1991) is developed. These methods are combined to verify the new measurements of χ with estimates of ɛ from Lien and D’Asaro (2006). Finally, the advantages and limitations of float-based estimates ɛ, χ, and Γ are discussed.

2. Instrumentation and data

Data were obtained using a special Lagrangian float configured for these measurements. The details, development, and technology of the Lagrangian float are described in D’Asaro (2003b). Lagrangian floats are designed to follow the three-dimensional, high-frequency motion of the surrounding water. The combination of high drag, a hull with a compressibility close to that of seawater, and active control of buoyancy based on precise temperature, salinity, and pressure measurements acts to make the float Lagrangian at high frequencies and isopycnal at low frequencies. The transition frequency is about N/30, less than N, the buoyancy frequency, due to the drag of the drogue.

The float used here is a version of the second-generation mixed layer float (MLFII) described in D’Asaro (2003b) (see Fig. 1). The hull is about 0.9 m long and 0.27 m in diameter. A circular red cloth drogue is 1.16 m in diameter. Two Seabird SBE-43 conductivity–temperature–depth (CTD) sensors are mounted on the two ends of the float separated by about 1.44 m; they sample every 30 s during the periods of Lagrangian drift and every 15 s during active profiling. A three-beam 5-MHz SonTek acoustic Doppler velocimeter (ADV) mounted on the bottom of the float measures three components of velocity 0.93 m below the center of the float at 25 Hz. Data were sampled for 80 s followed by a 20-s gap. Additional float details are found in Lien and D’Asaro (2006).

Fig. 1.

(a) Spectra of vertical velocity from all deployments. Spectra are grouped by ɛ and the average of each group is plotted. Different groups are plotted with different colors. The legend indicates the range of ɛ in each group and the number of spectra averaged. Spectra are normalized by ɛ and ωL. The two groups of curves show spectra of the float’s vertical velocity and the vertical velocity relative to the float measured by the ADV. The thick gray curves show theoretical float velocity spectra computed from (4); the thick green curves show the theoretical relative velocity spectra (5). (b) Acceleration spectra. Thin lines show theoretical spectra of float acceleration (gray) and relative acceleration [i.e., the time derivative of relative velocity (green) and their sum (black)]. Thick colored lines show the sum of the measured acceleration and relative acceleration. Color code is the same as in (a). Acceleration spectra are scaled by ɛ. (c) A picture of the float.

Fig. 1.

(a) Spectra of vertical velocity from all deployments. Spectra are grouped by ɛ and the average of each group is plotted. Different groups are plotted with different colors. The legend indicates the range of ɛ in each group and the number of spectra averaged. Spectra are normalized by ɛ and ωL. The two groups of curves show spectra of the float’s vertical velocity and the vertical velocity relative to the float measured by the ADV. The thick gray curves show theoretical float velocity spectra computed from (4); the thick green curves show the theoretical relative velocity spectra (5). (b) Acceleration spectra. Thin lines show theoretical spectra of float acceleration (gray) and relative acceleration [i.e., the time derivative of relative velocity (green) and their sum (black)]. Thick colored lines show the sum of the measured acceleration and relative acceleration. Color code is the same as in (a). Acceleration spectra are scaled by ɛ. (c) A picture of the float.

The measurements reported here were made during five deployments (December 2003 to February 2004) in Puget Sound as detailed in Lien and D’Asaro (2006). These are labeled ADV1–ADV5. In each, the float alternated between “profiling” to the surface every 12 h and “drifting” during which the float was operated to be Lagrangian. As in Lien and D’Asaro (2006), periods when the float was very close to, or on the bottom or surface, were excluded from the analysis. Data from all five missions were used unlike in Lien and D’Asaro (2006), where ADV2 was not used because the ADV data were not recorded properly.

A key measurement in this analysis is the stratification N. This was computed from the difference in potential density measured at the two CTD sensors. Examination of this difference shows an offset of about 0.005 psu between the two salinities using the manufacturer’s calibration. This does not necessarily represent a calibration error; it could have resulted from our installation of the CTD boards. The relative calibration of the two CTD sensors was established by comparing the vertical salinity gradient computed from their difference with that computed from the difference in subsequent samples from both during vertical profiling. The best estimate was obtained from data in a bottom boundary layer measured during ADV2. Two vertical profiles through the layer show a potential density gradient of about 0.001 kg m−3 over 10 m, equivalent to a difference between the two sensors of less than 0.2 × 10−3 kg m−3. Using a salinity offset between the two sensors of 5.3 × 10−3 psu, the mean density difference between the two sensors in the layer is less than 0.1 × 10−3 kg m−3 with a standard deviation of 0.4 × 10−3 kg m−3 over 26 data points. Using the same salinity offset on a similar uniform layer in ADV5, the mean density difference is about 0.2 × 10−3 kg m−3 with a standard deviation of 0.7 × 10−3 kg m−3 over 95 data points. Based on these data, the calibration error between the two CTD sensors is estimated as less than 0.3 × 10−3 kg m−3, resulting in a bias in N2 of less than 0.00142 s−2.

3. Theory and analysis

a. Velocity spectra and kinetic energy dissipation ɛ

For a turbulent flow at sufficiently high Reynolds number, the one-dimensional wavenumber spectrum of velocity has the form

 
formula

where k is the wavenumber and α = 0.53 ± 0.055 is a Kolmogorov constant (Sreenivasan 1995) appropriate for longitudinal spectra, that is, the velocity u in the same direction as k. This spectral form is valid over a wavenumber range from the “large-eddy” wavenumber k0 to the Kolmogorov wavenumber kν = (ɛ/ν3)1/4. This expression often applies for the Reynolds numbers Rλ based on the Taylor microscale as low as 50 and becomes universally true for Rλ > 1000. Under these conditions, it is an accurate method to estimate ɛ.

Velocity measured along a Lagrangian trajectory also exhibits an inertial subrange (Tennekes and Lumley 1972). This is most simply expressed in terms of the frequency ω spectrum of acceleration a as

 
formula

and is valid between a large-eddy frequency ω0 and a Kolmogorov frequency ων = (ɛ/ν)1/2. The Kolmogorov constant, β = 1.9 ± 0.1 (Lien and D’Asaro 2002), is based on a small number of numerical, laboratory, and field observations.

Lagrangian floats make highly accurate measurements of their own vertical velocity along a nearly Lagrangian trajectory. However, their size, here characterized by a half-length L, is much larger than 1 kν−1, implying dimensionally that the frequency

 
formula

rather than ων will control the high-frequency limit of (2). Lien et al. (1998) investigate this problem in detail and derive a modified expression for the spectrum measured from a float of length 2L:

 
formula

where B and F are nondimensional functions that describe the deviations of the spectrum from (2) due to the influence of large eddies and the finite float size, respectively.

Lien and D’Asaro (2006) fit (4) to observed spectra of vertical float acceleration to estimate a value of ɛ, here denoted by ɛL (L for Lagrangian). They also fit velocity spectra measured by the ADV to a slightly modified version of (1) to estimate a value ɛE (E for Eulerian). The values of ɛE and ɛL agree to within a factor of 2 over nearly four orders of magnitude in ɛ using data from ADV1, ADV3, ADV4, and ADV5. Because the estimation of ɛE from (1) is a well-established technique, they conclude that estimation of ɛL (4) is also accurate.

A more precise test of the accuracy of (4) can be made by examining both the spectrum of float velocity, computed from its measured pressure, and the spectrum of velocity relative to the float measured by the ADV (Fig. 1a). The upper “velocity of float” curves (colored) plot average spectra of vertical velocity binned by the ɛL. Not surprisingly, these closely match the velocity spectrum Φw(ω) = Φa(ω)/ω2 computed from (4) (thick gray line) because the parameters ω0 and ɛ in (4) were derived by fitting it to these data. Here the spectra are plotted against ω/ωL, so that a nondimensional form showing the function F in (4) is obtained for ωωL. For ωωL the curves for each ɛ bin diverge because the ratio of ω0 to ωL is not constant.

The curves labeled “velocity relative to float” show the average spectra of vertical velocity measured by the ADV scaled in the same way. These are measured independently of the float velocity and therefore provide new information. At frequencies less than ωL the float is smaller than the typical eddy size. It is therefore accurately Lagrangian and the velocity relative to the float should be much less than the float velocity (Fig. 1a). At frequencies much larger than ωL the float is larger than the typical eddy size and therefore averages over many eddies. The float’s velocity spectrum should therefore fall below the inertial subrange value, that is, the function F in (4), and the velocity relative to the float should increase toward the inertial subrange level. The relative velocity is modeled as

 
formula

and plotted in Fig. 1a as a thick green curve for each ɛ group. This function fits the relative velocity data surprisingly well.

A test of this formulation is shown in Fig. 1b, where the sum of the float and relative vertical accelerations [Φwrel(ω) + Φw(ω)]ω2 are plotted both for the data and from (4) and (5). If the model were perfect, this sum should be constant for ω > ωL as shown by the black lines. The sum is indeed flatter than either of its components but varies by almost a factor of 3, as emphasized by the linear scale. This highlights the imperfections in (5). First, the relative velocity is taken only at one location, while (5) is an average over all locations near the float. More importantly, the rate of change of relative velocity is not a Lagrangian acceleration, because unlike the rate of change of float velocity, it is not taken along a Lagrangian trajectory of the water parcels. Despite this, however, relative velocity spectra from a wide range of ɛ nearly collapse to a single curve, albeit only approximately given by (5), for ω > ωL. Similar analysis, not shown here, yields the same result for the other two components of relative velocity.

Both the velocity of the float and the velocity of the water relative to the float are accurately given by nondimensional forms that scale with ɛ, ω, and ωL. For ω < ωL the float is smaller than the turbulent eddies and is accurately Lagrangian; the motion relative to the float is much smaller than the motion of the float. For ω > ωL it is larger than the typical eddies. Its motion averages over the effects of many eddies and is therefore reduced; the motion relative to the float becomes larger than the motion of the float itself. Because the relative velocity can be scaled on the local ɛ, other sources of shear, for example, internal waves or the mean estuarine flow, are not significant at these frequencies (see D’Asaro and Lien 2000b for further explanation).

b. Scalar spectra and scalar variance dissipation χ

Spectra of conserved scalars, potential temperature, and salinity also exhibit universal spectral forms. For high Reynolds number flow (Rλ > 1000) an inertial subrange, one-dimensional wavenumber spectrum (Obukhov–Corrsin spectrum) for scalars exists with a form

 
formula

where χ is the rate of dissipation of the variance of the scalar and αs is a universal Kolmogorov constant equal to 0.4 with an uncertainty of about 25% (Sreenivasan 1996). Here

 
formula

is the Reynolds number based on the Taylor microscale λ and 0.5 q2 is the turbulent kinetic energy (Tennekes and Lumley 1972),

 
formula

where ν is the viscosity.

For water the scalar spectra of temperature and salinity extend to higher wavenumbers than the spectra of velocity because their scalar diffusivities D are smaller than the viscosity v, that is, the Prandtl and Schmidt numbers are greater than one. The spectrum at wavenumbers above the Kolmogorov wavenumber kν flattens to a slope of −1 before decreasing rapidly near k = kB = (ɛ/νD)1/4, the Batchelor (1959) wavenumber. The resulting “Batchelor spectrum”

 
formula

accurately describes the gradient spectrum. The value of q ≈ 3.7, although there is some uncertainty. Following Klymak and Moum (2007), we model the total scalar spectrum through the inertial subrange to well past the Batchelor scale as the sum

 
formula

where T(k/kB) tapers Φis(k) to small values much faster than ΦBatch(k). Its exact form is unimportant. Our results are insensitive to the details of the spectra near the Batchelor scale because we do not resolve at such a fine scale.

The Lagrangian inertial subrange spectrum of a scalar is most simply expressed in terms of the rate of change of the scalar

 
formula

It is white, as in the case of acceleration, and proportional to χ for that scalar. The value of the Kolmogorov constant is not well known. Tennekes and Lumley (1972) suggest 0.75, Lien et al. (2002) use 1.13, and Lien and D’Asaro (2004) derive 1/π. A major goal here is to improve these estimates so that χ can be estimated from scalar spectra. A value βs = 0.6 is used and will be shown to be consistent with our data.

Figure 2 shows measured Lagrangian frequency spectra of the rate of change of potential density referenced to the surface. Using σ, rather than temperature or salinity, enables a direct connection to diapycnal mixing rate. The spectra are averaged into bins of χ, here used to denote the dissipation rate of potential density variance. The value of χ is estimated from the low-frequency spectral level [see (14) below]. These spectra are subject to noise, which appears as isolated spikes or bursts of high variance in the measured salinity. These may be the result of plankton ingestion into the conductivity cell. Noisy χ estimates are eliminated by computing χ from both the upper and lower CTD sensors, computing the ratio Rχ = χupper/χlower, and rejecting all estimates for which Rχ + 1/Rχ is larger than 50. The subsequent analysis uses the lower CTD because it is close to the ADV measurement volume.

Fig. 2.

Average spectra of rate of change of potential density grouped by χ. The average value of χ in each group and number of spectra averaged are annotated. The value of χ is computed from the spectral level. Bars give 95% confidence limits based on number of spectra and chi-squared statistics. Circles show value of ωL for each group.

Fig. 2.

Average spectra of rate of change of potential density grouped by χ. The average value of χ in each group and number of spectra averaged are annotated. The value of χ is computed from the spectral level. Bars give 95% confidence limits based on number of spectra and chi-squared statistics. Circles show value of ωL for each group.

Each of the average spectra in Fig. 2 is characterized by a white region for ω < ωL (circle) and an increasingly steep upward slope for ω > ωL. The white region is easily explained as the Lagrangian inertial subrange (11) with a level proportional to χ. The level of this spectrum averaged over a range of frequencies will be used to estimate χ. However, it is necessary to first understand why the spectrum deviates from a constant level in order to pick an appropriate range of frequencies. Because the inertial subrange itself extends far past ωL, the deviation of the measured spectrum from the constant level predicted by (11) for ωωL must result from some non-Lagrangian aspect of the measurement. Consider a float embedded in a uniform shear. The center of the float moves with the average velocity and a steady velocity U is present at the lower CTD. Assuming a wavenumber spectrum (6) and Taylor’s hypothesis, the frequency spectrum of /Dt would have the form ΦDσ/Dt(ω) = αsχ(ɛU2)−1/3ω1/3. This would rise above the white Lagrangian spectrum with a slope of 1/3. This example shows that advection of spatial density variations past the CTD sensors could cause the measured spectra to deviate from the Lagrangian form. This effect should become increasingly important for ωωL, as the turbulent eddies become smaller than the float. However, a uniform shear is clearly too simple a model; the observed spectra clearly have slopes steeper than 1/3. The actual velocity advecting the scalar field at frequency ω will be approximately given by U2 ≈ ∫ω0 ΦU(ω) dω, where ΦU(ω) is the spectrum of velocity measured by the ADV near the lower CTD. Modeling this by (5), U2 increases roughly as ω for ω > ωL implying a large contribution of high frequencies to the advective velocity. Taylor’s hypothesis needs to be modified to include unsteady advection resulting from the time-varying turbulent eddies surrounding the float.

The observed spectra nearly collapse to a single nondimensional form when scaled with ωL and χ:

 
formula

where H is an empirical nondimensional function. Figures 3a and 3c plot the averaged spectra from Fig. 2 in nondimensional form. The function H is plotted as a black line.

Fig. 3.

Scaled spectra of /Dt. (a) Scaled spectra from Fig. 2. Black line is H(ω/ωL). (b) Scaled spectra simulated by advecting (10) with measured ADV velocity. (c) Deviation of scaled spectra from H(ω/ωL). (d) Deviation of scaled simulated spectra from H(ω/ωL). Colors and legend indicate average χ for each plotted spectrum and number of spectra averaged. Data from ADV2 not used in this plot because no velocity data are available.

Fig. 3.

Scaled spectra of /Dt. (a) Scaled spectra from Fig. 2. Black line is H(ω/ωL). (b) Scaled spectra simulated by advecting (10) with measured ADV velocity. (c) Deviation of scaled spectra from H(ω/ωL). (d) Deviation of scaled simulated spectra from H(ω/ωL). Colors and legend indicate average χ for each plotted spectrum and number of spectra averaged. Data from ADV2 not used in this plot because no velocity data are available.

These spectra are modeled (Figs. 3b, 3d) assuming Taylor’s hypothesis and the scalar spectrum (10) with a D appropriate for temperature. Because σ includes contributions from both temperature and salinity, this spectrum is inaccurate at high wavenumbers. However, our results are not sensitive to the spectral levels at these small scales; using only the inertial subrange part of (10) yields similar results.

For each data segment σ(t) from which a real spectrum has been computed, a random Gaussian realization of (10) σ̂(x) is generated assuming the values of ɛ and χ are estimated from the real data for this segment. This spatial series is converted to a time series σ̂[x(t)] using

 
formula

where U is the velocity measured by the ADV. A frequency spectrum is computed from σ̂[x(t)] and substituted for the real spectrum in the analysis. This numerically implements the Lumley (1965) analysis of the unsteady Taylor hypothesis. The same approach was used by Lien and D’Asaro (2006). The resulting normalized spectra (Figs. 3b, 3d) compare well with the real spectra (Figs. 3a, 3c), approximately reproducing the spectral form H(ω/ωL). On average, the scatter in the simulated spectra is somewhat larger than that in the real spectra for ω > ωL. The simulated spectra also fall below the real spectra for ω < ωL because they do not include the Lagrangian density fluctuations that dominate at low frequency. The measured frequency spectra of Dσ/Dt therefore reflect the Lagrangian density fluctuations for ωωL and the unsteadily advected Eulerian density fluctuations for ωωL. The overall level of the spectrum scales with χ and the spectral shape is reasonably described by H(ω/ωL). Given these results, estimates of χ are computed from spectra of /Dt using

 
formula

An upper cutoff of 2ωL is used because H increases rapidly for higher frequencies, potentially biasing the χ estimates due to uncertainties in and scatter around its assumed functional form.

c. Diapycnal mixing rates

Two techniques to estimate diapycnal mixing rates are used widely in oceanography. The Osborn and Cox (1972) method computes the downgradient flux of any scalar from its dissipation rate and its gradient. Winters and D’Asaro (1996) show that this formula is exact if the correct definition of diapycnal gradient is used. Applying this to σ yields the diapycnal diffusivity of potential density

 
formula

The Osborn (1980) formula

 
formula

relates the diapycnal diffusivity to ɛ in terms of a “mixing efficiency” Γ. Equating these yields either an expression for ɛ in terms of χ:

 
formula

or an expression for Γ:

 
formula

Ocean mixing usually occurs at near-maximum efficiency, so that using a constant Γ = 0.2 is often a good approximation (Gregg 1987). More detailed studies (Ivey and Imberger 1991; Smyth et al. 2001), however, find that Γ varies as a function of the Froude number. One definition is

 
formula

where LO and LT are the Ozmidov and Thorpe scales, respectively. For float data it is more convenient to use a definition based on the time scales of stratification N−1 and overturning ω−10:

 
formula

where the constant F0 will need to be chosen to convert between (20) and other definitions of Froude number. A prediction of Γ can now be made using the Ivey and Imberger (1991) formulas for flux Richardson number Rf:

 
formula

and its relationship to Γ:

 
formula

For our purposes, the important property of this formulation is that Γ decreases with increasing Fr from a maximum value Γ = 0.3 near Fr = 1.

4. Results

a. Froude number

Figure 4 shows the average acceleration spectra grouped by ɛ. The value of ω0 for each spectrum is found by fitting (4) and is shown by the vertical dashed line. The fits are good at all but the lowest energy (Fig. 4a), which appears to be below the wave–turbulence transition (D’Asaro and Lien 2000b) so that the spectrum does not include a turbulent component. The error in these estimates of ω0 is less than 50%. For each spectrum N is computed from the average difference between the top and bottom CTD sensors. Spectra with N < 10−3 s−1 are not used. For each group of spectra an average N is found by averaging N2 for each spectrum in the group.

Fig. 4.

Average acceleration spectra grouped by ɛ and fit with the model from (4). Spectra are shown with 95% bootstrap confidence limits. Fit is shown by dashed curve. Vertical lines show fit value of ω0 (dashed) and the measured N (gray). Circle shows upper-frequency limit of fit. Number of points and average value of ɛ for each average spectra are annotated.

Fig. 4.

Average acceleration spectra grouped by ɛ and fit with the model from (4). Spectra are shown with 95% bootstrap confidence limits. Fit is shown by dashed curve. Vertical lines show fit value of ω0 (dashed) and the measured N (gray). Circle shows upper-frequency limit of fit. Number of points and average value of ɛ for each average spectra are annotated.

At lower energies (Figs. 4a–d) ω0 is less than N, while at higher energies (Figs. 4e,f) it is larger than N, implying an increase in Fr with energy (see also Fig. 5a). The change in Fr is about a factor of 6, well above the estimated error in ω0.

Fig. 5.

(a) Buoyancy N and large-eddy ω0 frequencies for each group in Fig. 4. (b) Froude number (20) (divided by 10), flux Richardson number Rf , and mixing efficiency Γ (22) computed from these using (21) assuming F0 = 2.

Fig. 5.

(a) Buoyancy N and large-eddy ω0 frequencies for each group in Fig. 4. (b) Froude number (20) (divided by 10), flux Richardson number Rf , and mixing efficiency Γ (22) computed from these using (21) assuming F0 = 2.

The increase in Fr can be explained by the changing mechanisms through which the turbulence is forced. This can be seen by examining the gradient Richardson number Ri = 1.5 〈N2〉/〈S2〉. The shear S is estimated from the time-averaged velocity measured by the ADV divided by the distance to the center of the float, 0.93 m. Values were computed for time averages ranging from 100 to 5000 s and averaged into bins of ɛ. The value of N2 was computed from the difference between the density at the two CTDs divided by their separation, about 1.44 m. The difference in separations between density and velocity biases Ri low by a factor of 1.5 assuming constant gradients, so the factor 1.5 is included to compensate. The results are insensitive to the amount of time averaging. For ɛ < 10−7 W kg−1, Ri = 0.5–0.75. The value steadily decreases with increasing ɛ, because of both increasing shear and decreasing stratification, reaching values of 0.001 for ɛ > 10−5 W kg−1. This indicates that stratification is much more important to the turbulence at low ɛ.

At low ɛ the internal waves are the only mechanism to supply the energy for local mixing. Mixing is driven by local internal wave breaking as in the open ocean. Small-scale internal wave properties will scale with ɛ and N. The value of ω0 should scale with N implying a constant value of Fr. The values 2ω0 = N, that is, Fr = 2 and Fr = 1, reported by D’Asaro and Lien (2000a), appear accurate here also. The mixing efficiency from (21) and (22) is about 0.3, close to its maximal value.

The highest energy data come from the tidal channels of Tacoma Narrows and Colvos Passage in Washington. These are highly turbulent from top to bottom (Seim and Gregg 1997). The Ozmidov length, 23 m in the highest energy group, approaches the water depth, 60 m in Tacoma Narrows. Extrapolating our Ri to 100-m scale, it is still below critical. Thus, stratification plays a small role in these flows; the shear S rather than N will scale the turbulence. In an unstratified log layer, for example, the shear is Su*/z, where u* is the friction velocity, ɛu3*/z, and Ku*zɛ/S2. Buoyancy plays no role in the energetics and Γ = 0. As the shear becomes dominant, we therefore expect ω0 to be roughly equal to S rather than N implying 2ω0N, Fr ≫ 1. The diffusivity will be given by ɛ/S2 rather than (16) and Γ will be much less than its maximal value.

This brief analysis merely suggests the nature of the dynamic changes and suggests that changes in Γ are entirely plausible. An expansive analysis of these effects is beyond the scope of this paper.

b. Can χ be computed from float data?

Figure 6 tests the accuracy with which χ can be computed from the level of the inertial subrange of ΦDσ/Dt(ω) by comparing ɛL computed from the float’s motion (Lien and D’Asaro 2006) with ɛχ (17) computed from χ. This requires estimates of ρ and N, both of which are accurately measured on the float, and assumptions about the mixing efficiency Γ. Figures 6a and 6b assume a constant Γ = 0.2 in accord with the usual oceanographic practice. Figures 6c and 6d use the functional form of Γ(ɛ) from Fig. 5b to compute ɛχ.

Fig. 6.

Comparison of ɛ computed from float’s vertical acceleration (horizontal axes) with ɛχ computed from χ using (17). (a), (c) Scatterplots; (b), (d) the same data plotted as a ratio. The mixing efficiency is assumed constant in (a), (b) but allowed to vary as shown in Fig. 5b in (c), (d). The blue lines plot the mean of the data in ɛ bins with 95% bootstrap confidence limits. The yellow region in (d) spans 0.5 to 2.

Fig. 6.

Comparison of ɛ computed from float’s vertical acceleration (horizontal axes) with ɛχ computed from χ using (17). (a), (c) Scatterplots; (b), (d) the same data plotted as a ratio. The mixing efficiency is assumed constant in (a), (b) but allowed to vary as shown in Fig. 5b in (c), (d). The blue lines plot the mean of the data in ɛ bins with 95% bootstrap confidence limits. The yellow region in (d) spans 0.5 to 2.

Figures 6c and 6d show a strong correlation between ɛχ and ɛ. They are equal to within a factor of 2 over about 3.5 orders of magnitude change in ɛ. This implies that χ can be estimated from ΦDσ/Dt(ω) to better than a factor of 2. The values of χ are computed assuming βs = 0.6. This is therefore the best current estimate for this constant. Its uncertainty is less than a factor of 2.

5. Summary and discussion

Simultaneous measurements of temperature, salinity, their vertical gradients, and the vertical velocity gradient across a 1.4-m-long Lagrangian float were used to investigate the accuracy with which the dissipation of scalar variance χ can be computed using inertial subrange methods from such a neutrally buoyant float. The float was deployed in a variety of environments in Puget Sound chosen to span a wide range of turbulent intensities. Lien and D’Asaro (2006) show that an inertial subrange range method yields accurate measurements of ɛ, the rate of dissipation of kinetic energy, from this data. Here, a similar method to estimate χ is investigated. Major points include the following.

  • Kolmogorov scaling predicts a white Lagrangian frequency spectrum (11) for the rate of change of a scalar. The spectral level is βsχ, where βs is a universal Kolmogorov constant.

  • Measured spectra of potential density σ are approximately white but rise increasingly fast at frequencies much above N (Fig. 2).

  • The deviation of the spectra from (11) can be explained by the advection of spatial density gradients past the sensor at frequencies for which the turbulent eddies are smaller than the float, resulting in a nondimensional empirical form for the measured spectra (12).

  • This form is used to derive a method to estimate χ (14), which accounts for these instrumental imperfections.

  • The accuracy of these estimates is evaluated by comparing diapycnal diffusivities computed from χ (15) with those computed from ɛ (16). A mixing efficiency is needed for this comparison, which is estimated from a turbulent Froude number ω0/N (20) using results from laboratory experiments (21) and (22). The large-eddy frequency ω0 is estimated from the float’s acceleration spectrum.

  • The diffusivities based on χ and ɛ are highly correlated over a range of ɛ = 10−8 to 3 × 10−5 W kg−1. The proposed method to estimate χ is accurate to better than a factor of 2.

  • The Kolmogorov constant βs = 0.6 with an uncertainty of less than a factor of 2.

This paper, combined with Lien and D’Asaro (2006), validates techniques for estimating Ri = N2/S2, ɛ, χ, and Fr from Lagrangian floats. Taken together, these parameters describe both the local rate of mixing and much of its turbulent physics. Current floats can operate autonomously with lifetimes of months to a year anywhere in the open ocean. These capabilities should allow detailed measurements of mixing rates and physics in a much wider range of environments than is possible with ship-based instrumentation.

These methods are limited, however, to relatively energetic turbulence. The most fundamental limitation is imposed by the use of inertial subrange methods. These are only accurate for sufficiently high Reynolds number. The existing evidence (Sreenivasan and Antonia 1991; Sreenivasan 1995, 1996), however, suggests that the inertial subrange spectra scale with ɛ and χ for Rλ > 102, but that full isotropy and universality may occur only for Rλ > 103. The spectral forms in Fig. 1a imply ɛ = w2 N/3, where w is the vertical velocity (D’Asaro and Lien 2000a). Turbulence is confined to frequencies ω > N so that only half of w2 is turbulence, the rest being internal waves. Isotropy of the turbulence implies

 
formula

Note that this has the simple interpretation that the time scale of the turbulent eddies is N−1, consistent with Ozmidov scaling. Combining (7), (8), and (23)

 
formula

which relates Rλ, commonly used by turbulence researchers, to ɛ/νN2, commonly used by the ocean microstructure research community. Thus Rλ > 103 corresponds to ɛ/νN2 > 3300 and Rλ > 102 corresponds to ɛ/νN2 > 33. We take these as thresholds for “universality” and “usefulness” of inertial subrange methods. They are somewhat lower than those found by Gargett et al. (1984). A common criteria for “active turbulence” (Gregg 1987) is ɛ/νN2 > 25 or Rλ > 87. At typical “Garrett–Munk” levels of ocean mixing K = 10−5, ɛ/νN2 is about 38 using (16) and inertial subrange methods seem unlikely to work well. For our data, with N = 3 × 10−3, ɛ > 10−8 W kg−1, and ɛ/νN2 > 854, the inertial subrange methods should work well for all but the least energetic cases.

Several practical limits currently set higher bounds on the accuracy of these methods. The float size, L = 0.42 m for the floats used here, is a crucial parameter, expressed here in terms of the frequency ωL (3). Floats make unbiased measurements of the Lagrangian inertial subrange for ω0 ≈ 0.5 N < ω < ωL. The upper and lower limits are equal for ɛ ≈ 0.125N3 L2 = 2 × 10−8 W kg−1 for N = 0.01 s−1. The spectral forms compensated for float size effects, (4) and (12), allow useful measurements to be made at smaller ɛ but at the expense of a rapidly diminishing signal. Thus both the Eulerian and Lagrangian velocity inertial subranges measured by Lien and D’Asaro (2006) were limited by the combination of sensor noise and reduced signal to ɛ > 10−8 W kg−1. Scalar Lagrangian spectra, shown here, appear to have a lower noise floor as there is little evidence of noise in even the lowest spectrum (Fig. 2). However, the estimation of χ becomes increasingly dependent on the accuracy of the modeled deviations of the spectrum from its Lagrangian form.

Inertial subrange methods require that the flow be sufficiently steady that the inertial subrange comes to equilibration with the forcing. This requires that the forcing not change faster than the overturning times of the largest eddies in the inertial subrange. Unlike in, for example, turbomachinery, interior mixing in the ocean is generally in equilibrium. Interior mixing is driven by internal wave breaking, which necessarily has time scales not much faster than N−1, this being the maximum frequency for the waves. Most of the energy in the waves is at lower frequencies so the time scales of forcing are likely often much longer than N−1. Because the frequency of the upper end of the inertial subrange is about N, the forcing should generally be slower than the equilibration time of the inertial subrange. The same may not be true very near the ocean surface, where wave breaking can cause more rapid forcing, and very near the bottom, where sharp topographic features could cause strong separation.

Floats are particularly good at measuring vertical velocity and acceleration spectra because these can be computed from pressure. These estimates are nearly immune to surface wave contamination (D’Asaro 2003b). However, there is currently no simple way to measure horizontal velocity and acceleration. There may be concerns that variations in the isotropy of the turbulence would bias the estimates of dissipation rates. Our existing data on isotropy (D’Asaro and Lien 2000a), derived from acoustically tracked floats, suggest that anisotropy is limited to frequencies ω < N, within the internal wave frequency band, and that the turbulent and inertial subrange frequencies, ω < N, are isotropic. This apparent ability to separate waves and turbulence in a stratified flow, and sample only the turbulence, suggests that these measurement methods are insensitive to anisotropy caused by stratification. On the other hand, measurements made near the ocean surface do not show the same universal spectral forms (D’Asaro 2003a), perhaps due to anisotropy caused by the surface or near-surface shear.

There may also be concerns that the float itself could generate turbulence and thereby affect its measurements (particularly the ADV and CTD measurements). It is crucial to realize that because the float is neutrally buoyant it has the same inertia as the water that it displaces. In rapidly accelerating flows with scales larger than the float, there is no delay between the motion of the water and that of the float, because both are subject to the same pressure forces and respond to these forces with the same inertia. Drag, lift, and therefore float-generated turbulence, are only important because the water has scales of motion smaller than the float. These act to create shears across the float, which could cause it to move relative to the surrounding water. Perhaps some of the deviations from our simple models in Fig. 1 are due to such effects. Lien and D’Asaro (2006) analyze the deviations in ɛ measured by the ADV compared to those from acceleration as a function of the direction of the flow in an attempt to see if the turbulent wake of the float could be detected. No significant signal was found. Small-scale velocities also cause mixing of density and velocity, which act on the water but not on the float. The main purpose of the drogue is to help make the float stay with the water when such effects cause the water to accelerate away from the float or change its local density. Detailed discussions on both of these effects can be found in D’Asaro et al. (1996) and D’Asaro (2003b).

The accuracy of these techniques is limited statistically. Lagrangian floats gather statistics slowly compared to profilers because they must wait for the fluid to overturn, in a time of roughly 1/N, to measure a new eddy realization. A float will typically gather only 1000 degrees of freedom per day for turbulence parameters. Thus, short-lived events may not be well sampled, although useful information on the time variations in mixing may be obtained (Lien et al. 2002). In addition, because the inertial subrange spectra used to compute both ɛ and χ are white with levels proportional to the dissipation rates and upper and lower bounds, which do not depend strongly on the levels, spectra averaged over many events with different dissipation rates tend to have the same form as individual spectra. Thus, average values of the dissipations can be computed from averaged spectra. This, and their longevity, makes floats particularly appropriate tools for measuring average mixing rates. Like most things in the ocean, the variability in mixing is most likely to be larger at longer times.

Acknowledgments

This work was supported by NSF Grant OCE 0241244.

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Footnotes

Corresponding author address: Eric D’Asaro, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA 98115. Email: dasaro@apl.washington.edu