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).
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 N 2 of less than 0.00142 s−2.
3. Theory and analysis
a. Velocity spectra and kinetic energy dissipation ɛ
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 ω ≫ ωL. For ω ≪ ωL the curves for each ɛ bin diverge because the ratio of ω0 to ωL is not constant.
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
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.
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 ω ≫ ωL 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 Dσ/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 ω ≫ ωL, 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.
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.
c. Diapycnal mixing rates
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 N 2 for each spectrum in the group.
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.
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 〈N 2〉/〈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 N 2 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 S ≈ u*/z, where u* is the friction velocity, ɛ ≈ u3*/z, and K ≈ u*z ≈ ɛ/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 2ω0 ≫ N, 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.
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 = N 2/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.
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 ≈ 0.125N 3 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.
REFERENCES
Batchelor, G. K., 1959: Small-scale variation of convected quantities like temperature in turbulent fluid. J. Fluid Mech., 5 , 113–139.
D’Asaro, E. A., 2003a: The ocean boundary layer beneath Hurricane Dennis. J. Phys. Oceanogr., 33 , 561–579.
D’Asaro, E. A., 2003b: Performance of autonomous Lagrangian floats. J. Atmos. Oceanic Technol., 20 , 896–911.
D’Asaro, E. A., and Lien R-C. , 2000a: Lagrangian measurements of waves and turbulence in stratified flows. J. Phys. Oceanogr., 30 , 641–655.
D’Asaro, E. A., and Lien R-C. , 2000b: The wave–turbulence transition for stratified flows. J. Phys. Oceanogr., 30 , 1669–1678.
D’Asaro, E. A., Farmer D. M. , Osse J. T. , and Dairiki G. T. , 1996: A Lagrangian float. J. Atmos. Oceanic Technol., 13 , 1230–1246.
Gargett, A. E., Osborn T. R. , and Nasmyth P. W. , 1984: Local isotropy and the decay of turbulence in a stratified fluid. J. Fluid Mech., 144 , 231–280.
Gregg, M., 1987: Diapycnal mixing in the thermocline: A review. J. Geophys. Res., 92 , 5249–5286.
Ivey, G. M., and Imberger J. , 1991: On the nature of turbulence in a stratified fluid. Part I: The energetics of mixing. J. Phys. Oceanogr., 21 , 650–658.
Klymak, J. M., and Moum J. , 2007: Oceanic isopycnal slope spectra. Part II: Turbulence. J. Phys. Oceanogr., 37 , 1232–1245.
Lien, R-C., and D’Asaro E. A. , 2002: The Kolmogorov constant for the Lagrangian velocity spectrum and structure function. Phys. Fluids, 14 , 4456–4459.
Lien, R-C., and D’Asaro E. A. , 2004: Lagrangian spectra and diapycnal mixing in stratified flow. J. Phys. Oceanogr., 34 , 978–984.
Lien, R-C., and D’Asaro E. A. , 2006: Measurement of turbulent kinetic energy dissipation rate with a Lagrangian float. J. Atmos. Oceanic Technol., 23 , 964–976.
Lien, R-C., D’Asaro E. A. , and Dairiki G. T. , 1998: Lagrangian frequency spectra of vertical velocity and vorticity in high-Reynolds-number oceanic turbulence. J. Fluid Mech., 362 , 177–198.
Lien, R-C., D’Asaro E. A. , and McPhaden M. J. , 2002: Internal waves and turbulence in the upper central equatorial Pacific: Lagrangian and Eulerian observations. J. Phys. Oceanogr., 32 , 581–600.
Lueck, R., Wolk F. , and Yamazaki H. , 2002: Oceanic velocity microstructure measurements in the 20th century. J. Oceanogr., 58 , 153–174.
Lumley, J. L., 1965: Interpretation of time spectra measured on high-intensity shear flows. Phys. Fluids, 8 , 1056–1062.
Osborn, T. R., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr., 10 , 83–89.
Osborn, T. R., and Cox C. S. , 1972: Oceanic fine structure. Geophys. Fluid Dyn., 3 , 321–345.
Seim, H. E., and Gregg M. C. , 1997: The importance of aspiration and channel curvature in producing strong vertical mixing over a sill. J. Geophys. Res., 102 , 3451–3471.
Smyth, W. D., Moum J. N. , and Caldwell D. R. , 2001: The efficiency of mixing in turbulent patches: Inferences from direct simulations and microstructure observations. J. Phys. Oceanogr., 31 , 1969–1992.
Sreenivasan, K. R., 1995: On the universality of the Kolmogorov constant. Phys. Fluids, 7 , 2778–2784.
Sreenivasan, K. R., 1996: The passive scalar spectrum and the Obukhov-Corrsin constant. Phys. Fluids, 8 , 189–196.
Sreenivasan, K. R., and Antonia R. A. , 1991: The phenomenology of small scale turbulence. Annu. Rev. Fluid Mech., 29 , 435–472.
Tennekes, H., and Lumley J. L. , 1972: A First Course in Turbulence. MIT Press, 300 pp.
Winters, K. B., and D’Asaro E. A. , 1996: Diapycnal fluxes in density stratified flows. J. Fluid Mech., 317 , 179–193.
(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.
Citation: Journal of Atmospheric and Oceanic Technology 24, 6; 10.1175/JTECH2031.1
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.
Citation: Journal of Atmospheric and Oceanic Technology 24, 6; 10.1175/JTECH2031.1
Scaled spectra of Dσ/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.
Citation: Journal of Atmospheric and Oceanic Technology 24, 6; 10.1175/JTECH2031.1
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.
Citation: Journal of Atmospheric and Oceanic Technology 24, 6; 10.1175/JTECH2031.1
(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.
Citation: Journal of Atmospheric and Oceanic Technology 24, 6; 10.1175/JTECH2031.1
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.
Citation: Journal of Atmospheric and Oceanic Technology 24, 6; 10.1175/JTECH2031.1