Abstract

This study tests the ability of a neutrally buoyant float to estimate the dissipation rate of turbulent kinetic energy ɛ from its vertical acceleration spectrum using an inertial subrange method. A Lagrangian float was equipped with a SonTek acoustic Doppler velocimeter (ADV), which measured the vector velocity 1 m below the float's center, and a pressure sensor, which measured the float's depth. Measurements were taken in flows where estimates of ɛ varied from 10−8 to 10−3 W kg−1. Previous observational and theoretical studies conclude that the Lagrangian acceleration spectrum is white within the inertial subrange with a level proportional to ɛ. The size of the Lagrangian float introduces a highly reproducible spectral attenuation at high frequencies. Estimates of the dissipation rate of turbulent kinetic energy using float measurements ɛfloat were obtained by fitting the observed spectra to a model spectrum that included the attenuation effect. The ADV velocity measurements were converted to a wavenumber spectrum using a variant of Taylor's hypothesis. The spectrum exhibited the expected −5/3 slope within an inertial subrange. The turbulent kinetic energy dissipation rate ɛADV was computed from the level of this spectrum. These two independent estimates, ɛADV and ɛfloat, were highly correlated. The ratio ɛfloat/ɛADV deviated from one by less than a factor of 2 over the five decades of ɛ measured. This analysis confirms that ɛ can be estimated reliably from Lagrangian float acceleration spectra in turbulent flows. For the meter-sized floats used here, the size of the float and the noise level of the pressure measurements sets a lower limit of ɛfloat > 10−8 W kg−1.

1. Introduction

Measurements of mixing rates in the ocean have typically been made using free-fall or wire-lowered instruments deployed from research vessels (Lueck et al. 2002). Given the practical limitations on ship time, long-term measurements of ocean mixing will need to be made from autonomous platforms. D'Asaro et al. (1996) suggest that the turbulent kinetic energy dissipation rate ɛ could be measured from the high-frequency vertical motion of a neutrally buoyant float. Mixing rates could thus be measured using the Osborn (1980) relationship. Here, we compare the estimates of ɛ measured in this way with those measured from the level of the “−5/3” wavenumber spectrum from the same instrument; excellent agreement over a wide range of ɛ is reported.

A common method for measuring ɛ in a turbulent flow is based on the existence of an inertial subrange. For a sufficiently high Reynolds number, the one-dimensional wavenumber spectrum of velocity has the form

 
formula

where k is the wavenumber and α is a Kolmogorov constant (Tennekes and Lumley 1972). Sreenivasan (1995) reviews a large quantity of experiment data and finds α = 0.53, with a variability of about 0.055, for longitudinal spectra, that is, the velocity u in the same direction as k, when the Reynolds numbers Rλ based on the Taylor microscale are >50. For isotropic turbulence, the Kolmogorov constant for transverse spectra is expected to be 4/3 times that for longitudinal spectra; that is, α = 0.71. This is confirmed when the Reynolds numbers of the turbulent flow are higher than 1000 (Sreenivasan 1995). Thus, the experimental evidence clearly indicates that values of ɛ estimated from (1) are accurate as long as the Reynolds number is sufficiently high.

Velocity measured along a Lagrangian trajectory is also expected to exhibit an inertial subrange between the large-eddy frequency and the Kolmogorov frequency (Tennekes and Lumley 1972). This is most simply expressed as the frequency spectrum of acceleration

 
formula

where ω is the frequency and β is a different Kolomogorov constant. Note that (2) applies to any component of acceleration. However, only the vertical component of acceleration is measured by the float and is considered in this analysis. The value of β is related to the constant C0 in the structure function of turbulence velocity as C0 = πβ (Monin and Yaglom 1975). Lien and D'Asaro (2002) compiled results of numerical models and laboratory and field observations for both C0 and β and conclude that β = 1.9, with an uncertainty of 0.1 for Rλ > 100. Although the quantity of Lagrangian data is far less, the uncertainty in β appears comparable to that of α. Based on this, it should be possible to estimate ɛ from Φa(ω) with the same accuracy and reliability with which it is commonly estimated from Φu(k).

The practical use of (2) is motivated by the development and use of highly Lagrangian autonomous floats (D'Asaro et al. 1996; D'Asaro 2003). These Lagrangian floats can measure their own depth and the temperature and salinity of the surrounding water at sampling rates much faster than the local buoyancy frequency, typically every 20–30 s, thus sampling the frequencies of turbulence and internal waves (D'Asaro and Lien 2000). Lien et al. (1998) used float data from turbulent upper ocean mixed layers to evaluate the validity of (2). They report that significant corrections for the averaging effect of the float's size (about 1 m) were needed, but because these were highly reproducible, reliable estimates of ɛ were possible. The resulting estimates were consistent with the surface forcing. Lien et al. (2002) applied this method to the equatorial deep-cycle layer and found patterns and magnitudes of ɛ consistent with previous observations. Nevertheless, estimates of ɛ from float measurements using (2) have never been compared with other direct measurements of ɛ.

The intrinsic intermittency and inhomogeneity of turbulence makes direct comparisons of ɛ measured on different platforms difficult, even if they are only a short distance away (Moum et al. 1995). Accordingly, an accuracy test of (2) for a Lagrangian float requires an independent measurement of ɛ on the same float. This was done by mounting a high-accuracy rapid-sampling microstructure velocity sensor, a SonTek acoustic Doppler velocimeter (ADV) (Voulgaris and Trowbridge 1998), on a float. The ADV measured velocity approximately 1 m below the center of the float. Relative velocity between the float and this point allowed Φu(k) to be evaluated using Taylor's hypothesis, and thus ɛADV to be computed from (1). The vertical motion of the float was used to compute ɛfloat using (2). This paper describes the measurement, computation, and comparison of these two quantities.

The instrument and experiment are described in section 2. The ADV measurements and the estimation of ɛADV through a variation of Taylor's frozen turbulence hypothesis are discussed in section 3. The estimation of ɛfloat via a Lagrangian model spectrum is described in section 4. These are compared in section 5. Errors in ɛADV resulting from the failure of Taylor's hypothesis and the float's wake effect are discussed in section 6. A summary section follows.

2. Instrument and experiment

a. ADV Lagrangian float

A special Lagrangian float (ADV float) was configured to compare ɛADV and ɛfloat (Fig. 1). The details, development, and technology of the Lagrangian float are described in D'Asaro (2003). Lagrangian floats are designed to follow the three-dimensional high-frequency motion of the surrounding water. This is done through a combination of matching the density of the water and high drag provided by a cloth drogue. The float's compressibility matches closely that of seawater so that it settles naturally toward a surface of constant potential density. This tendency is reinforced at low frequencies (less than 10−4 s−1) by active control of the float's volume based on CTD measurements. At high frequencies, the float's neutral buoyancy and drag cause it to follow water motions. Thus, the float is Lagrangian at high frequencies and isopycnal at low frequencies. The transition frequency depends on the float drag. D'Asaro (2003) shows that in a stratified fluid the drag is primarily due to internal wave generation and that the transition from Lagrangian to isopycnal behavior occurs at a frequency of about N/30, where N is the buoyancy frequency. The inertial subrange occurs at frequencies larger than N (D'Asaro and Lien 2000). The float is Lagrangian at these frequencies.

Fig. 1.

ADV Lagrangian float. The float is equipped with two CTD sensors on both ends; a 5-MHz SonTek acoustic Doppler velocimeter at the bottom; two pressure sensors and a high-drag drogue in the middle of the float; and an Iridium antenna, a GPS antenna, and an Argos antenna on the top.

Fig. 1.

ADV Lagrangian float. The float is equipped with two CTD sensors on both ends; a 5-MHz SonTek acoustic Doppler velocimeter at the bottom; two pressure sensors and a high-drag drogue in the middle of the float; and an Iridium antenna, a GPS antenna, and an Argos antenna on the top.

The float used here is a version of the second-generation mixed layer float (MLFII) described in D'Asaro (2003). The hull of the float is about 0.9 m long and 0.27 m in diameter. The drogue is 1.16 m in diameter. The float is equipped with Seabird SBE-43 CTD sensors mounted on the two ends of the float separated by about 1.44 m; a buoyancy control piston at the bottom end of the float; two pressure sensors and a high-drag drogue at the middle of the float; and an Iridium antenna, a GPS antenna, and an Argos antenna on the top of the float. Only one of the two pressure sensors was used for data analysis; it sampled at 1 Hz. CTD data were sampled every 30 s during periods of Lagrangian drift, and 15 s during profiling. A three-beam 5-MHz SonTek ADV was mounted on the bottom of the float. The ADV sensor protruded 0.3 m from the bottom of the float and the sensing volume of the ADV measurements was about 0.18 m below the transducers. Measurements of ADV were thus about 0.93 m below the center of the float. The ADV measured three velocity components at 25 Hz.

b. Experiments

The measurements reported here were made during five ADV float deployments between December 2003 and January 2004 (Fig. 2). For each mission the float was deployed by a small boat. Using preprogrammed knowledge of its mass, volume, and compressibility, the float used the measured CTD values to adjust its buoyancy to that of the water and thus achieve nearly neutral buoyancy. After a predetermined time, usually about 1 day, the float made itself buoyant, surfaced, and determined its position using GPS. Rapid-sampling profile data were taken during the initial descent, and the ascent and descent associated with each surfacing. The float position plus a subset of the data were telemetered via Iridium. Based on this information, the float was either recovered or allowed to operate for another day.

Fig. 2.

GPS fixes of the Lagrangian float during five missions in Puget Sound, WA. These fixes were obtained when the float surfaced. The beginning and ending of missions are labeled. Open circles indicate the deployment locations and solid black dots indicate the recovery locations. Gray dots represent locations of floats in the middle of the missions. Curves illustrate the possible float trajectories. The possible float trajectory for the third mission is represented by a dashed curve to better distinguish if from those of missions 1 and 2.

Fig. 2.

GPS fixes of the Lagrangian float during five missions in Puget Sound, WA. These fixes were obtained when the float surfaced. The beginning and ending of missions are labeled. Open circles indicate the deployment locations and solid black dots indicate the recovery locations. Gray dots represent locations of floats in the middle of the missions. Curves illustrate the possible float trajectories. The possible float trajectory for the third mission is represented by a dashed curve to better distinguish if from those of missions 1 and 2.

All deployments were made in the Puget Sound between Seattle, Washington, and the Tacoma Narrows. The first mission (Fig. 2) began just south of Colvos Passage in Washington. The float drifted northward into the passage immediately after the deployment on the strong ebb tide. In the second mission, the ADV float was deployed within Tacoma Narrows during the ebb. The float drifted into the Colvos Passage. Unfortunately, the ADV failed to record data. In the third mission (Fig. 2, dashed curve) the float was again deployed into the Tacoma Narrows and drifted into Colvos Passage, eventually beaching near its northern end. In the fourth mission the float was deployed west of Alki Point during the flood tide. The float drifted southward and was caught in the back eddy north of Point Beals. In the fifth mission the float was deployed north of Three-Tree Point during the flood and drifted south.

In stratified turbulent flows the existence of the inertial subrange and the validity of the local isotropy assumption depend on the value of the turbulence activity parameter I, defined as I = ksk−1b = (ɛν−1N−2)4/3, where ks = ɛ1/4ν−3/4 is the Kolmogorov wavenumber, ν is the molecular kinematic viscosity, and kb = ɛ−1/2N3/2 is the Ozmidov wavenumber (Gargett et al. 1984). Here I represents the ratio between the large-eddy scale and the viscous dissipation scale. In stratified turbulence the large-eddy scale corresponds to the scale of instability. For I ≥ 3000, more than one decade of inertial subrange exists and the local isotropy is confirmed. For I = O(1000) and I = O(300), inertial subrange wavenumbers are still distinguishable, especially for the longitudinal component; the ratios of transverse spectra to longitudinal spectra, however, are about one in the inertial subrange, smaller than the expected value of 4/3 for the isotropic turbulence. For IO(100), the inertial subrange does not exist (Gargett et al. 1984).

Locations of five deployments were chosen to sample a variety of environments, including (i) a weakly stratified, highly turbulent flow, where I = O(6000) in mission 1; (ii) a change from unstratified to weakly stratified strong turbulent flow, where I > 105 in missions 2 and 3; (iii) weak turbulence in a weakly stratified flow, where I = O(1000) in mission 4; and (iv) intermittent turbulence in a stratified flow, where I = O(300) in mission 5. These measurements provided nearly five decades of variation in ɛ, from 10−8 to 10−3 W kg−1. The stratification varied from N less than 0.002 to about 0.01 s−1. We expected a clear inertial subrange in missions 1–3 and a distinguishable inertial subrange in missions 4–5.

Time series of the float's pressure, that is, depth, during the five missions are shown in Fig. 3. The first mission (Fig. 3a) lasted about 1 day. Errors in the buoyancy computation caused it to be slightly heavy. Accordingly, the float trajectory alternates between periods when it bounces along the bottom (near day 350.4) and periods when turbulence picks it off of the bottom and it passes into the interior (day 350.2–350.32). The vertical profile of stratification was taken at around day 349.9 when the float was at the south end of the Colvos Passage. The second mission (Fig. 3b) was about 2 days long. The float encountered strong turbulence in Tacoma Narrows, that is, large high-frequency vertical displacements. Again, the float was somewhat heavy and spent significant periods on the bottom in Colvos Passage. The third mission (Fig. 3c) was about 0.5 day long. The float was deployed in Tacoma Narrows and was rapidly carried into Colvos Passage. The float was somewhat light and thus stayed at the surface after day 20.3. The fourth mission (Fig. 3d) was about 3 days long. The float was deployed in the thermocline west of Alki Point in the main basin of the Puget Sound. It remained neutrally buoyant within the water column at depths varying from 100 to 150 db. In the fifth mission (Fig. 3e), the float was deployed close to Three-Tree Point. Nearly 2 days of data were taken. Again, the float remained in the water column, but mostly at shallower depths.

Fig. 3.

Time series of (a)–(e) pressure fluctuations of the Lagrangian float and (f)–(j) vertical profiles of buoyancy frequency N during the five missions. Periods when the float surfaces are labeled. Black dots in the right column are values of N computed by CTD sensors at two ends of the float during the profile mode (labeled surfacing in the plot). Thick gray curves are the depth-bin-averaged profiles of N. Shadings in the (a), (c), (d), and (e) mark periods when the float was Lagrangian and when ɛ was estimated using float measurements.

Fig. 3.

Time series of (a)–(e) pressure fluctuations of the Lagrangian float and (f)–(j) vertical profiles of buoyancy frequency N during the five missions. Periods when the float surfaces are labeled. Black dots in the right column are values of N computed by CTD sensors at two ends of the float during the profile mode (labeled surfacing in the plot). Thick gray curves are the depth-bin-averaged profiles of N. Shadings in the (a), (c), (d), and (e) mark periods when the float was Lagrangian and when ɛ was estimated using float measurements.

In Colvos Passage N is nearly constant in the whole water column, with a value near 3 × 10−3 s−1 (Fig. 3g). In Tacoma Narrows, N reaches to 1–2 × 10−2 s−1 in the upper layer and decreases to 5 × 10−3 s−1 below 20 m (Fig. 3h). East of Vashon Island, N is 5–10 × 10−3 s−1 in the upper 75 m and shows a minimum of 10−3 s−1 at 110 m (Figs. 3i and 3j).

3. ADV energy spectra and estimates of ɛADV

a. Measurements

The ability to calculate ɛ using the velocity measured by the ADV is based on the averaging effect of the finite float size and on the location of the ADV measurement. Lagrangian spectra of the float's motion (section 4) are accurately modeled by assuming that the float moves with the average velocity of the water in a region extending a distance L up and down from the center of the float. Here L = 0.7 m, the float's half-length. Lien et al. (1998) shows that the float does not respond accurately to velocities with a Lagrangian frequency larger than ωL = ɛ1/3L−2/3. The float's motion is thus a low-passed filtering of the total water motion with a low-pass frequency of ωL. Accordingly, a velocity measured by the ADV sensor mounted on the float is the residual velocity of the total water motion relative to the float's motion, that is, a high-passed filtering of the total velocity. However, for the values of ɛ sampled here, the value of ωL is often smaller than the frequencies resolved by our analysis; the measured velocity fluctuations at frequencies greater than ωL are the same as if the float were stationary and are therefore Eulerian. In addition, it is the center of the float that follows an approximate Lagrangian trajectory. The ADV samples velocity about = 1 m below the center of the float. In a stratified flow the typical shear is N, resulting in a typical velocity difference of Nℓ = 1–10 mm s−1. Turbulence will cause additional velocity differences with a magnitude (ɛℓ)1/3 or 2–100 mm s−1. It is these relative velocities that advect smaller-scale turbulence past the ADV and allow a wavenumber spectrum to be computed.

The ADV measured vector velocity at 25 Hz. Eighty-second bursts of measurements were taken, followed by 20 s of processing and recording. Velocity spectra from each 80-s segment were computed using a multitaper spectral analysis with two tapers (Percival and Walden 1993). Frequency spectra of horizontal kinetic energy, vertical velocity, and total kinetic energy during energetic turbulence, ɛ ≈ 10−6 W kg−1, are shown in Fig. 4a. At low frequencies, all spectra exhibit a clear −5/3 inertial subrange spectral slope. White spectral noise exists at high frequencies. Vertical velocity has a noise of 0.8 mm s−1; horizontal velocity has a noise of 5.3 mm s−1. The observed noises are consistent with those reported by Voulgaris and Trowbridge (1998). The noise shown in Fig. 4a is the same as that found in a quiet test tank, indicating that the flow's small shear and Doppler spreading are not important noise sources in these measurements. The vertical velocity spectrum shows a roll-off deviation from the −5/3 slope at frequencies between 1 s−1 and where the spectrum hits the white-noise floor. This roll-off is presumably because of the sampling volume averaging, which is larger in the vertical than in the horizontal (Voulgaris and Trowbridge 1998).

Fig. 4.

(a) Examples of horizontal kinetic energy frequency spectrum (black curve), vertical velocity spectrum (red curve), and total kinetic energy spectrum (blue curve) computed from ADV measurements for ɛ ≈ 10−6 W kg−1. (b) Wavenumber spectra of total kinetic energy for ɛ ≈ 10−8 (black), 10−7 (red), 10−6 (blue), 10−5 (green), and 10−4 (magenta) W kg−1. (c) Average of spectra shown in (b) at different ɛ levels and (d) normalized spectra (dots) and the average of normalized spectra (solid curve). Shading in (a) and (c) are the 95% confidence interval of spectra. The dashed lines in (c) and (d) are Kolmogorov scaling of kinetic energy spectra in inertial subrange.

Fig. 4.

(a) Examples of horizontal kinetic energy frequency spectrum (black curve), vertical velocity spectrum (red curve), and total kinetic energy spectrum (blue curve) computed from ADV measurements for ɛ ≈ 10−6 W kg−1. (b) Wavenumber spectra of total kinetic energy for ɛ ≈ 10−8 (black), 10−7 (red), 10−6 (blue), 10−5 (green), and 10−4 (magenta) W kg−1. (c) Average of spectra shown in (b) at different ɛ levels and (d) normalized spectra (dots) and the average of normalized spectra (solid curve). Shading in (a) and (c) are the 95% confidence interval of spectra. The dashed lines in (c) and (d) are Kolmogorov scaling of kinetic energy spectra in inertial subrange.

The calibration of the ADV was confirmed by measurements in a steady-state laminar flow flume. Water velocity was measured by timing the motion of dye dots as they traveled past the ADV sensors. The ADV sensor measured the mean speed of the flow to be 2.52 cm s−1, with a standard deviation of 0.04 cm s−1. The mean horizontal speed of the dye was 2.57 cm s−1, with a standard deviation of 0.05 cm s−1. The ADV measurements agreed with the dye velocity within their standard deviations.

b. Kolmogorov scaling

Estimation of ɛADV requires conversion of the velocity time series to a spatial record so that wavenumber spectra can be computed. Conventionally, one would assume Taylor's frozen turbulence hypothesis (Lumley and Terray 1983), and convert time t to space x by x = ut, where u is the mean velocity. However, this requires that the mean velocity is at least 10 times greater than the perturbation velocity (Soloviev and Lukas 2003). This assumption is not justified in our measurements. Instead of using the mean velocity, we use instantaneous velocity measurements to convert from time to space, as suggested by Lumley (1965) for this situation. The wavenumber spectra are then taken along the path defined by this integrated velocity. This approach is analogous to Taylor's conversion, but avoids the assumption of a large ratio of the mean to perturbation velocities. When the mean current is much stronger than the fluctuations, our approach reduces to the conventional Taylor's theory. The accuracy of this approach is assessed in section 6a.

The position of each ADV measurement is taken as X = ∫u dt, where u is the instantaneous velocity. The distance along this path is x = ∫|u| dt, where |u| is the instantaneous speed. Distances are linearly interpolated to a fixed interval, chosen to be the mean interval of x. Velocities at these fixed positions are obtained by the linear interpolation of observed velocity. Wavenumber spectra of three velocity components are computed from u(x).

Within the inertial subrange, the longitudinal velocity spectra Φu||(k||) and transverse velocity spectra Φu(k||) are expressed as

 
formula
 
formula

where u|| and u are velocity components parallel and perpendicular to the mean flow, and k|| is the wavenumber in the mean flow. The values α|| = 0.53 and α = 0.71 are adopted from Sreenivasan (1995). The total kinetic energy spectrum ΦKE(k||) is expressed as

 
formula
 
formula

where 𝒜|| = (1/2)(α|| + 2α) = 0.98. Here, we have assumed isotropic turbulence. Turbulence kinetic energy dissipation rate is estimated by solving (6),

 
formula

Here, the angle brackets 〈〉 represent averaging in the inertial subrange. In a convoluted path, it is nontrivial to decompose the velocity vector to its lateral and longitudinal components. However, in this analysis only the total kinetic energy spectrum ΦKE(k||) = (1/2){Φu(k||) + Φυ(k||) + Φw(k||)} is necessary to compute ɛADV; the longitudinal and lateral components need not be computed individually. The accuracy of the present conversion scheme is discussed in section 6a.

c. Observed kinetic energy spectrum and estimates of ɛADV

A value of ɛADV was estimated for each 80-s burst of ADV data as follows. Each individual kinetic energy wavenumber spectrum was examined. The k−5/3 inertial subrange was identified. The value of 1.04 [ΦKE(k||)k5/3||]3/2 was averaged in the inertial subrange. Not all bursts yielded valid estimates. Bursts were discarded if the average correlation of any of three ADV beams was lower than 0.9. Spectra with unclear inertial subranges, mostly resulting from low energy, were also discarded. Among those discarded were measurements taken when the float was at the bottom, at the surface, or during the profile mode. Overall only 19% of the bursts yielded good estimates of ɛADV.

Wavenumber spectra of the total kinetic energy ΦKE(k||) are shown in Fig. 4b; the different colors indicate different levels of ɛADV, 10−8–10−4 W kg−1. Average spectra at these different ɛ levels exhibit a clear −5/3 spectral slope (Fig. 4c). Wavenumber spectra normalized by ɛADV and the average of normalized spectra are shown in Fig. 4d. The normalized spectra are in excellent agreement with Kolmogorov scaling. This agreement is expected because we have chosen those spectra of a clear inertial subrange and derived ɛ from the selected inertial subrange.

The ADV velocity spectra resolve turbulence dissipation rate as low as 10−8 W kg−1. Nearly one decade of the inertial subrange is found for ɛ ≈ 10−8 W kg−1, and nearly two decades for ɛ ≈ 10−4 W kg−1. There is no hint of a spectral peak at low wavenumbers, indicating that the large-eddy scale is almost always greater than the lowest resolved wavenumber in the 80-s segment. During the 80-s time interval, the water at the ADV sensing position travels 0.5–3 m relative to the float.

4. Estimation of ɛfloat from pressure measurements

a. Model spectra

Lien et al. (1998) proposed a universal form of the Lagrangian velocity spectra measured by Lagrangian floats. The form was derived analytically based on a kinematic turbulence model relating wavenumber to frequency (Fung et al. 1992). It was found to be in good agreement with a compilation of observed float spectra. The universal spectrum includes (i) an inertial subrange where the acceleration spectrum is white, that is, Eq. (2); (ii) a low-frequency roll-off function B(ω/ω0) below the large-eddy frequency ω0; and (iii) a high-frequency roll-off function F(ω/ωL), which models the averaging effect of the float's finite size. The universal Lagrangian spectrum of acceleration is expressed as

 
formula
 
formula
 
formula

Here ω0 is the large-eddy frequency, ωL = ɛ1/3L−2/3, and L ≈ 0.7 m is half of the float's length. The Kolmogorov constant β = 1.9. In stratified flows, D'Asaro and Lien (2000) found that the above spectral form fits to observed spectra very well with ω0 = N/2.

b. Estimation of ɛfloat

Measurements of float pressure, that is, depth, sampled at 1 Hz and taken during the Lagrangian mode, were used to compute Φa(ω). Periods when the float was in the Lagrangian mode are shown in Fig. 3. The float operated in a non-Lagrangian mode when it either was deployed, was ballasted to its target density, surfaced or rested on the bottom because of a poor ballasting, was profiled for taking vertical stratification, or surfaced for communications. Measurements taken during these periods were not used to compute Lagrangian spectra in the following analysis. Spectra were computed on 600-s-long data segments. The large-eddy frequency ω0 = N/2 lies between 10−3 and 10−2 s−1. The 600-s segment length was chosen to be sufficiently short to resolve the variation of ɛ and sufficiently long to resolve the Lagrangian inertial subrange. Vertical velocity was computed from the central differencing of pressure. Vertical acceleration was computed from the central differencing of vertical velocity and the acceleration spectrum computed using a multitaper spectral analysis with two tapers (Percival and Walden 1993).

All Lagrangian frequency spectra of acceleration at different levels of ɛ, from 10−8 to 10−4 W kg−1, are shown in Fig. 5a. At the highest frequencies a spectral roll-off resulting from the central differencing used to derive acceleration from pressure is seen. Below this, an ω4 spectral slope is seen at all but the highest ɛ. This results from the white-noise floor of the pressure sensor, and represents an rms error of 1.8 mm. This pressure noise limits estimates of ɛfloat to greater than about 10−8 W kg−1, depending on the large-eddy frequency. The spectra rise above the noise floor to a flat spectrum at low frequency. The roll-off is because of the float's finite size, that is, F(ω/ωL), so that the true white inertial subrange is achieved only at the lowest frequencies and for larger ɛ. In other words, the range of frequencies lies mostly above ω0, and often above ωL. The quality of the ɛfloat estimates thus depends on an accurate form for F(ω/ωL), especially near ωL where most of the variance exists.

Fig. 5.

(a) All Lagrangian acceleration spectra for ɛ = 10−8 (black), 10−7 (red), 10−6 (blue), 10−5 (green), and 10−4 (magenta) W kg−1, respectively; (b) the average of observed spectra at different ɛ levels (solid curves) and their model spectral fit (dashed curves), and the shadings are 95% confidence intervals; and (c) the normalized spectrum versus normalized frequency (dots). The blue curve in (c) is the average of the observed normalized spectrum; the red curve is the model spectrum [Eq. (8)]; the green curve is the polynomial fit to the observed average spectrum.

Fig. 5.

(a) All Lagrangian acceleration spectra for ɛ = 10−8 (black), 10−7 (red), 10−6 (blue), 10−5 (green), and 10−4 (magenta) W kg−1, respectively; (b) the average of observed spectra at different ɛ levels (solid curves) and their model spectral fit (dashed curves), and the shadings are 95% confidence intervals; and (c) the normalized spectrum versus normalized frequency (dots). The blue curve in (c) is the average of the observed normalized spectrum; the red curve is the model spectrum [Eq. (8)]; the green curve is the polynomial fit to the observed average spectrum.

Each measured spectrum was fit with the model spectrum (8). Its magnitude and shape depend on ɛ, the large-eddy frequency ω0, and the float response frequency ωL. We have assumed ω0 = N/2 (D'Asaro and Lien 2000), and therefore ɛ is the only fitting parameter. Fitting was done iteratively by minimizing the sum of the square of the difference between the model spectrum and the observed spectrum. We compared observed spectra with the model noise spectra of a prescribed rms pressure noise of 2.7 mm, that is, 1.5 times the observed rms value of pressure noise. Observed spectra with spectral levels lower than the model noise spectra were excluded from the fitting process to obtain estimates of ɛ.

Averaged spectra at different levels of ɛ show a good fit to the model spectra (Fig. 5b). The normalized acceleration spectrum for all observed spectra is shown in Fig. 5c. The observed spectra indeed fit the model spectrum well, except near the high-frequency portion of the response function, ω/ωL > 5. Such disagreement also appears in Fig. 5b at two extreme values of ɛ (magenta and black curves). The difference at high ɛ may be statistical because of so few spectra. The difference at low ɛ may represent unmodelled physical aspects of the float response. The response may, for example, not be perfectly characterized by a single length scale L. These discrepancies, however, do not significantly affect the estimates of ɛ because most of the spectral variance is at frequencies near the roll-off frequency, that is, ω/ωL = O(1). A modified version of F(ω/ωL) was created from a polynomial fit to the data in Fig. 5c (green curve). Use of this function reduces the computed values of ɛ by nearly 10%, However, observed spectra at intermediate ɛ, 10−7–10−5 W kg−1, do not fit well to the modified model spectrum using the modified response function. Thus, although there is some uncertainty in the form of the model spectrum, this does not significantly affect the accuracy of the estimates of ɛfloat.

5. Comparison of ɛADV and ɛfloat

Time series of ɛfloat (Fig. 6, dots) computed from the Lagrangian acceleration spectra and ɛADV (Fig. 6, crosses) computed from the Eulerian kinetic energy spectra fluctuate in unison. No values of either were computed from periods when the float was profiling or at the bottom or surface (see Fig. 3 and section 2b). Values of ɛADV were not computed if the spectra either were of poor quality, had a low correlation for measuring ADV beam velocity, or had an unclear inertial subrange.

Fig. 6.

Time series of ɛ estimates using the Lagrangian frequency spectra method (dots) and ADV wavenumber spectra method (crosses). Estimates of ɛ from Lagrangian measurements are computed in 600-s segments. Estimates of ɛ from ADV measurements were computed in 80-s bursts.

Fig. 6.

Time series of ɛ estimates using the Lagrangian frequency spectra method (dots) and ADV wavenumber spectra method (crosses). Estimates of ɛ from Lagrangian measurements are computed in 600-s segments. Estimates of ɛ from ADV measurements were computed in 80-s bursts.

During the first mission in the Colvos Passage, the float was in a strong tidal channel and ɛ varied between 10−7 and 10−5 W kg−1. During the third mission in Tacoma Narrows, the float was in an even stronger tidal channel and ɛ varied between 10−7 and 10−3 W kg−1. During the fourth and fifth missions, ɛ varied between 10−8 and 10−6 W kg−1. Intermittent bursts of ɛ > 10−6 W kg−1 last several hours. The two bursts in mission 5 at day 41.25 and around 41.6 are likely because of the hydraulics across the ridge associated with Three-Tree Point.

Figure 7 shows a direct comparison of ɛfloat and ɛADV. Estimates of ɛADV were averaged in 600-s segments corresponding to the intervals used to estimate ɛfloat. On average, three ɛADV values, each computed from an 80-s burst, were used in every 600-s segment. There were 262 pairs of ɛfloat and ɛADV. The two independent estimates of ɛ are strongly correlated (correlation coefficient = 0.93 and 95% significance level = 0.12). The agreement spans nearly five decades of ɛ from 10−8 to 10−3 W kg−1. The ratio ɛfloat/ɛADV has a mean value of 0.92, with the 95% confidence interval of 0.14 and 3.66. This ratio does not vary significantly with ɛADV, except for a deviation of marginal significance near 10−7 W kg−1 (Fig. 8a). Overall, average values of ɛfloat vary with ɛADV to within a factor of 2.

Fig. 7.

Scatter comparison of ɛfloat estimates from Lagrangian measurements and average of ɛADV from ADV measurements in 600-s segments. The thick solid curve represents the mean ratio between ɛfloat and ɛADV, i.e., ɛfloat = 0.92ɛADV. The shading represents the 95% confidence interval of the ratio.

Fig. 7.

Scatter comparison of ɛfloat estimates from Lagrangian measurements and average of ɛADV from ADV measurements in 600-s segments. The thick solid curve represents the mean ratio between ɛfloat and ɛADV, i.e., ɛfloat = 0.92ɛADV. The shading represents the 95% confidence interval of the ratio.

Fig. 8.

Ratios of ɛfloat/ɛADV computed from ADV measurements as a function of (a) ɛADV, (b) the ratio between the mean speed and the perturbation speed, (c) the variation of flow direction; (d) orientation of flow from horizontal, and (e) the minimum distance from the drogue of the float. The thick curves are averaged ratios and the thin dashed curves are their 95% confidence intervals. The averages and confidence intervals are computed using the bootstrap method. (f)–(j) The probability density functions of (a)–(e) are shown in insets, respectively. Color shadings and sizes of dots represent the magnitude of ɛADV, shown in (a), e.g., largest red dots for 10−4 W kg−1 < ɛADV < 10−3 W kg−1 and smallest cyan dots for 10−8 W kg−1 < ɛADV < 10−7 W kg−1.

Fig. 8.

Ratios of ɛfloat/ɛADV computed from ADV measurements as a function of (a) ɛADV, (b) the ratio between the mean speed and the perturbation speed, (c) the variation of flow direction; (d) orientation of flow from horizontal, and (e) the minimum distance from the drogue of the float. The thick curves are averaged ratios and the thin dashed curves are their 95% confidence intervals. The averages and confidence intervals are computed using the bootstrap method. (f)–(j) The probability density functions of (a)–(e) are shown in insets, respectively. Color shadings and sizes of dots represent the magnitude of ɛADV, shown in (a), e.g., largest red dots for 10−4 W kg−1 < ɛADV < 10−3 W kg−1 and smallest cyan dots for 10−8 W kg−1 < ɛADV < 10−7 W kg−1.

6. Discussion

a. Accuracy of Taylor's hypothesis

The estimation of ɛADV uses a modified version of Taylor's hypothesis x = ∫T0 |u| dt to convert the ADV time series of velocity to a scalar distance x (see section 3c). Although, unlike the traditional Taylor's hypothesis, this technique does not require the turbulent velocity to be small compared to the mean advection velocity, it is still expected to fail for low values of the mean flow relative to turbulence intensity (Figs. 8b–c).

Figure 8b shows the variation of the ratio ɛfloat/ɛADV as a function of the flow speed variability 〈|u|〉/〈σ|U|〉, where σ|U| is the standard deviation of speed and the angle brackets 〈〉 indicate an average over the 80-s burst. Figure 8c shows the variation of the same ratio with σθ = (σθυ + σθh)/2, where σθh is the standard deviation of the horizontal flow direction and σθυ is the standard deviation of the vertical flow direction.

For both plots (Figs. 8b–c), ɛfloat/ɛADV shows weak trends that are not statistically significant. There is little evidence that failures in Taylor's hypothesis significantly bias the estimates of ɛADV.

b. Float wake effects

The ADV sensor was mounted at the bottom of the float and measurements were taken 0.95 m from the drogue. If the flow is downward relative to the float, the ADV could be within the turbulent wake of the float or drogue and the estimates of ɛADV may be contaminated.

Figure 8d shows the ratio ɛfloat/ɛADV as a function of the vertical angle of attack θ = tan−1(w/), where w is the mean vertical velocity and is the mean horizontal speed. Positive (negative) θ corresponds to upward (downward) relative vertical velocity. Contamination by the float wake is possible for negative θ. No significant effect is seen in Fig. 8d, because θ is rarely sufficiently negative for the float wake to reach the ADV. A value of θ < −60° is needed to bring fluid from the edge of the drogue to the ADV sampling volume. The values of θ are mostly larger than −40°. The flow at the ADV sampling volume is thus mostly horizontal relative to the float, possibly reflecting the effect of stratification or the effect of the drogue in limiting vertical flow relative to the float. In either case wake effects are minimal.

Figure 8e compares ɛfloat/ɛADV with another indicator of wake contamination. We computed the water displacement by integrating velocity measured by the ADV. If the computed vertical displacement is greater than the distance between the ADV sensing volume and the drogue of the float, the wake contamination may be important. The vertical displacement of water parcels arriving at the ADV is computed assuming a spatially uniform flow. The minimal vertical distance from the drogue D is estimated. Because the ADV measurements were taken about 0.95 m below the drogue, D varies between 0 and 0.95 m. If the particle encounters the drogue, D = 0 m. Less than 1% of the observations show the particle encountering the drogue. Most of time, particles stay at least 0.5 m away from the drogue. The ratio ɛfloat/ɛADV does not show significant dependence on D.

7. Summary

Estimates of the turbulent kinetic energy dissipation rate ɛ were obtained using a Lagrangian float deployed in a variety of flow environments. Estimates of ɛ varied from 10−8 to 10−3 W kg−1, with the lower bound set by our instrumentation. These estimates were computed assuming a Lagrangian inertial subrange of turbulence modified by a float response function as derived by Lien et al. (1998). Observed Lagrangian acceleration spectra were fit to the Lien et al. (1998) model spectrum to obtain an estimate of ɛfloat.

Independent estimates of ɛ were obtained from velocity measured below the float using an acoustic velocity sensor. Time series of velocity were converted to space series using an extension of Taylor's hypothesis and computed wavenumber spectra. Estimates of ɛADV were obtained by fitting the observed kinetic energy wavenumber spectra to the Kolmogorov scaling within the “−5/3” inertial subrange.

Average values of ɛADV and ɛfloat show an agreement to within a factor of 2 over the entire five decades of ɛ. The ratio ɛfloat/ɛADV is nearly independent of ɛ, the flow direction, turbulent intensity, and indicators of contamination by the float's wake. Weak trends with each of these variables suggest that the general deviations of ɛfloat/ɛADV could be the result of errors in either or both methods.

The comparison of ɛ estimates from different sensors is often difficult because of the intrinsic intermittency and inhomogeneity of turbulence (Moum et al. 1995). The success of this comparison occurs because the two measurements were taken simultaneously and were less than 1 m apart, and because of the large dynamic range of turbulence encountered in the experiment.

These results validate the ability of Lagrangian floats to measure ɛ from variations in their depth using the inertial subrange method originally proposed by Lien et al. (1998). They suggest a Lagrangian acceleration Kolmogorov constant value β = 1.9 ± 0.1, consistent with the results of the literature survey by Lien and D'Asaro (2002). For the meter-sized floats used here, this method has a lower limit of ɛfloat > 10−8 W kg−1 (Fig. 5b). This is set by the noise level of the pressure sensor and the rapid decrease in spectral level at high frequencies caused by the float's finite size. This limit could be decreased if N were very small, thus extending the inertial subrange to a lower frequency, if the float size were decreased, thus reducing the attenuation resulting from the float size, or if the noise level for acceleration measurements were decreased. At energy levels well below those considered here, the Reynolds number of the flow may become sufficiently small that additional corrections may be required.

Acknowledgments

This work and analysis was supported by the National Science Foundation under Grant OCE 0241244. Mike Ohmart (APL-UW) braved the winter storms of Puget Sound; thanks are given to the entire APL-UW engineering team. Chris Garrett provided helpful comments that improved the presentation of this paper.

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Footnotes

Corresponding author address: Dr. Ren-Chieh Lien, Applied Physics Laboratory and School of Oceanography, College of Ocean and Fishery Sciences, University of Washington, Seattle, WA 98105. Email: lien@apl.washington.edu