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Experimental Evidence of the Kraichnan Scalar Spectrum at High Reynolds Numbers

D. J. Bogucki Department of Physical and Environmental Sciences, Texas A&M University-Corpus Christi, Corpus Christi, Texas

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H. Luo Department of Physical and Environmental Sciences, Texas A&M University-Corpus Christi, Corpus Christi, Texas

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J. A. Domaradzki Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, California

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Print Publication:
01 Oct 2012
DOI:
https://doi.org/10.1175/JPO-D-11-0214.1
Page(s):
1717–1728
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Abstract

Measurements of horizontal temperature gradient spectra were carried out in a 25 m × 12 m × 3 m freshwater well mixed swimming pool. The associated flow Reynolds number was found to vary from 4 × 103 to 6 × 103. In all analyzed cases, at large wavenumbers k, the temperature gradient spectra exhibited behavior consistent with the Kraichnan spectral form: ~exp[−(6qk)1/2k] with constant . The measurements are in agreement with direct numerical simulations of a passive scalar flow with of Bogucki et al. and with recent experimental observations of Sanchez et al.

Corresponding author address: Darek Bogucki, Department of Physical & Environmental Sciences, Texas A&M University-Corpus Christi, 6300 Ocean Drive, Unit# 5892, Corpus Christi, TX 78412-5869. E-mail: darek.bogucki@tamucc.edu

Abstract

Measurements of horizontal temperature gradient spectra were carried out in a 25 m × 12 m × 3 m freshwater well mixed swimming pool. The associated flow Reynolds number was found to vary from 4 × 103 to 6 × 103. In all analyzed cases, at large wavenumbers k, the temperature gradient spectra exhibited behavior consistent with the Kraichnan spectral form: ~exp[−(6qk)1/2k] with constant . The measurements are in agreement with direct numerical simulations of a passive scalar flow with of Bogucki et al. and with recent experimental observations of Sanchez et al.

Corresponding author address: Darek Bogucki, Department of Physical & Environmental Sciences, Texas A&M University-Corpus Christi, 6300 Ocean Drive, Unit# 5892, Corpus Christi, TX 78412-5869. E-mail: darek.bogucki@tamucc.edu
Keywords: Diffusion; Turbulence

1. Introduction

The turbulent state of the upper ocean is of paramount importance when trying to understand transfers that take place between the ocean and atmosphere. The air–water exchange of heat, momentum, and gases are controlled by turbulent fluxes within the marine boundary layer (Anis and Moum 1995; Kantha and Clayson 2003; Bogucki et al. 2010). To determine turbulent flux, the simultaneous values of turbulent kinetic energy (TKE) dissipation rate ɛ and the temperature variance dissipation rate χ within the same part of the water column are needed. When both ɛ and χ are known, their combination can be used to infer the vertical mixing efficiency as well as the local vertical density diffusivity (Oakey 1982). The simultaneous measurements of the pair (ɛ, χ) can be performed by employing two separate sensors: one to measure the flow shear and the other to obtain temperature gradients. However, it is also possible to obtain both ɛ and χ from measurements of only one quantity, the temperature gradient spectra (Dillon and Caldwell 1980), which in principle should be an easier experimental task. For a detailed in depth review of oceanic microstructure measurements see- Lueck et al. (2002).

Historically, in situ measurements of small-scale temperature fluctuations were initially carried out by a thin film resistance thermometer (Grant et al. 1968), a thermocouple (Nash et al. 1999), or most frequently with fast response thermistors (Lueck et al. 1977; Dillon and Caldwell 1980; Gregg and Maegher 1980; Hill 1987). Small-scale shear is usually recorded using airfoils, an approach pioneered by Osborn (1980). Such direct measurements of TKE ɛ with airfoils within the upper ocean are frequently employed in modern microstructure research (Anis and Moum 1995). Yet, this approach is more involved than using temperature gradient spectra to infer the temperature variance spectra, the dissipation rate χ, and the TKE dissipation rates (Dillon and Caldwell 1980; Oakey 1982; Ruddick et al. 2000). However, the latter approach can be employed only if a proper universal function describing the temperature spectra is available (Dillon and Caldwell 1980). Usually, the universal function is assumed to be that derived by Batchelor (1959).

The purpose of experiments reported in this paper is to demonstrate that in nonstratified, high Reynolds number flows the temperature gradient spectra are best described by the Kraichnan spectral form (Kraichnan 1968) rather than by the Batchelor form. This assertion has been known from analyzes of the direct numerical simulation (DNS) results of Bogucki et al. (1997) and recently also observed in the experiments of Sanchez et al. (2011).

The paper is organized as follows. Section 2 will include discussion of the relation between ɛ and the location of the temperature dissipation peak using both Batchelor and Kraichnan one-dimensional temperature dissipation spectra D1T. The experimental setup is described in section 3 along with the method for calculating ɛ and χ using measured temperature spectra. The experimental results are presented in sections 3 and 4, with conclusions in section 5.

2. Review of temperature spectra in water: Batchelor and Kraichnan forms

When investigating incompressible, homogenous, and isotropic flows, the TKE dissipation rate and the temperature variance dissipation rate are expressed as follows (Monin and Yaglom 1971): , and . The apostrophe denotes fluctuating components of velocity and temperature and an overbar signifies averaging, for example, over an ensemble of similar flows.

More detailed small-scale temperature information can be obtained using spectral turbulence theories (Monin and Yaglom 1971). In particular, Batchelor (1959) was the first to deduce the spectrum for scalar characterized by a Prandtl number Pr ≫ 1. The Prandtl number is the ratio of the kinematic viscosity to the thermal diffusivity, Pr = ν/D. Since water temperature is characterized by a Prandtl number Pr ≫ 1, the Batchelor theory has often been applied to describe temperature fluctuations in water. Later Mjolsness (1975) supplied a theoretical scalar spectrum formula based on an alternate theory of Kraichnan (1968). The Kraichnan spectra were observed in a number of DNS simulations: for nonstratified homogenous and isotropic turbulent flows (Bogucki et al. 1997; Yeung et al. 2002, 2004) and for sheared and stratified turbulence (Smyth 1999). The Kraichnan scalar spectra were also observed in oceanic experiments performed by Nash and Moum (2002), and in the recent lake measurements of Sanchez et al. (2011). This evidence points to the Kraichnan spectrum and accompanying theory as the correct description of temperature spectra in water. Despite this evidence many, researchers routinely use the spectrum proposed by Batchelor (Stevens and Smith 2004; Luketina and Imberger 2001; Ruddick et al. 2000) which may lead to erroneous results when computing TKE dissipation rates from temperature dissipation spectra.

It should be noted that the theoretical results normally use wavenumber k expressed in units of radian/m while the experimental spectra are typically expressed as a function of the cyclical wavenumber , bearing the units of cycle per meter (cpm). Both are related through the formula (cpm). To be consistent with other experiments, the results in this paper are reported using wavenumbers in cyclic units, signified by a hat, that is, (cpm).

In case of homogeneous, isotropic turbulence eddies smaller than the associated Kolmogorov length scales are strongly damped by the viscosity. In dimensional units (meter) is
e1
where is the Kolmogorov wavenumber (cpm).
To derive the expression for scalar spectra for Pr ≫ 1 at high wavenumbers Batchelor (1959) assumed that the dissipative, small scalar scales are subject to deformations by a steady compressive principal strain rate, γ, of Kolmogorov eddies, which is locally uniform in space and time at length scales much smaller than . This assumption resulted in the form of three-dimensional temperature spectra in the viscous range:
e2
where is the mean value of the compressive strain rate γ. Batchelor estimated the mean principal strain rate as , where is a universal constant, and subscript B signifies that it is used for the Batchelor temperature spectrum. The universal, nondimensional temperature spectrum form is then obtained by first normalizing wavenumbers by the Batchelor length scale :
e3
and then normalizing the temperature spectra by . Under this scaling the nondimensional form of Eq. (2) becomes
e4
Kraichnan (1968) has extended Batchelor’s analysis by including intermittency of the strain rate, which resulted in the following expression for the scalar spectra in the dissipation range:
e5
where is a universal constant, and the subscript K signifies that it is used for the Kraichnan temperature spectrum. The behavior in the viscous–convective range is preserved by Kraichnan’s theory but the resultant spectrum differs from the Batchelor form in the viscous–diffusive range, having slower fall off in . Kraichnan also obtained the same result using his Lagrangian-history-direct-interaction (LDHI) closure theory (Kraichnan 1968).
Experimental temperature spectra are typically reported as a one-dimensional spectrum E1T (Monin and Yaglom 1971), which is related to its three-dimensional counterpart ET by the following formula:
e6
with normalization
e7
Note that applied normalization, that is, Eq. (7), implies that . The expressions for one-dimensional temperature spectra can be derived from Eq. (4) and Eq. (5) by applying Eq. (6), giving their respective forms:
e8
e9
The corresponding one-dimensional isotropic temperature dissipation spectra become
e10
e11
The location of the dissipation spectrum peak is obtained by setting derivatives of expressions (10) and (11) to zero
e12
Solutions of the resulting equations for lead to expressions for the Batchelor length scale
e13
for the Batchelor spectral form (10) and
e14
for the Kraichnan form (11). A detailed derivation is given in appendix A. Upon substitution of Eq. (13) and Eq. (14) into Eq. (3), the respective turbulent kinetic energy dissipation rates as a function of the dissipation peak location and of a nondimensional constant are
e15
e16
Therefore, the above formulas allow the determination the TKE dissipation rate from the measured location of the peak of the temperature variance dissipation spectrum if the functional form of the spectrum and the associated constant are known.

A number of values for and have been reported in the literature on the subject. Batchelor (1959) suggested ≈ 2 based on available experimental data. Gibson (1968) provided a range of values deduced from the incompressibility and kinematic constraints. Grant et al. (1968) supplied a measurement of = 3.9 ± 1.5, while Oakey (1982) found = 3.7 ± 1.5 based on measured oceanic data. Using a thermocouple Nash et al. (1999) measured the ocean temperature fluctuation spectra and found them to be consistent with the Batchelor form for . Similarly, Stevens and Smith (2004) obtained from oceanic observations.

Kraichnan (1968) estimated < 0.9 using the LHDIA. Newman and Herring (1979) derived = 1.68 using the test field model. Bogucki et al. (1997) studied passive scalar mixing in forced homogenous and isotropic turbulence using DNS results. In that study the best fit to the numerical data was provided by the Kraichnan form with . When the Batchelor form was assumed the constant qB was obtained as 3.9 ± 0.25. The recent lake measurements of Sanchez et al. (2011) yielded . Note that the Bogucki et al. (1997) DNS results are consistent with observations of Sanchez et al. (2011) when taking into account the associated error. Using DNS data, Yeung et al. (2002, 2004) studied a passive scalar mixing in statistically stationary homogenous and isotropic turbulence with a large range of Prandtl numbers from Pr = ⅛ to Pr = 1024 and to . They found that in the dissipation range for Pr ≫ 1, the scalar spectrum was in agreement with the Kraichnan form. Donzis et al. (2010) using an extensive DNS simulation database found that when Pr ≫ 1, was about 4.93 and that it was weakly dependent on the scalar Prandtl number and the flow Reynolds number. For decaying nonstratified turbulence Antonia and Orlandi (2003) DNS data yielded and , respectively. They also found that in the viscous–convection range the Batchelor form approximates the DNS data well, while in high wavenumber range the Kraichnan form provides a better fit. For decaying stratified turbulence Smyth (1999) also found the Kraichnan form with is better than the Batchelor form with . Nash and Moum (2002) used this value of when comparing their measured spectra with the Kraichnan theoretical spectrum.

While no single values for qB and qK emerge, the majority of the reported results are in the range between 3 and 5 for the former, and between 3.4 and 7.9 for the latter quantity. Throughout the rest of the paper fixed values of = 3.9 and = 5.26 will be used. These values were obtained by Bogucki et al. (1997), are consistent with the most recent results of Sanchez et al. (2011), and fall in the middle of the respective ranges.

3. Experimental procedure and results

Near surface turbulence measurements were carried out in the Texas A&M University-Corpus Christi (TAMUCC) freshwater swimming pool having dimensions of 25 m × 12 m × 3 m deep, over the duration of one hour. The data were collected at a depth of around 25 cm. The pool was thermally nonstratified with uniform mean rms temperature fluctuations in both vertical and horizontal directions of around 0.01°C over a 1-m distance. The mean water temperature during the experiment was about 21°C with corresponding Pr of around 7. On the day of the experiment the wind speed was ~5 m s−1, while the pool surface was populated with quasi-randomly distributed surface waves a few centimers in amplitude.

The temperature data in the present work were obtained using the Vertical Microstructure Profiler (VMP)-200 (Rockland Scientific International), which was deployed for taking horizontal measurements. The VMP-200 measured concurrently and —for a detailed description of the system see Lueck et al. (2002). At the time of the experiment, the VMP carried one shear probe (SPM-38), one fast response thermistor FP07, two accelerometers to measure tilt and vibrations, and a pressure gauge. The turbulent velocity fluctuations were measured with the shear probe but are not presented here because of problems with a large angle of attack (around 10 degrees) experienced by the probe during the experiment. Microstructure temperature fluctuations were measured with the FP07 thermistor mounted on the VMP200. The temperature resolution of this probe is 0.0001°C. Temperature was sampled at 512 Hz. The temperature temporal gradient was measured by the FP07 and was converted to spatial gradients via Taylor’s ‘frozen field’ hypothesis:
e17
where w is the mean speed of the oncoming flow (the velocity of the VMP). The electromagnetic (EM) current meter (JFE Advantech-Infinity current meter with a resolution of 0.02 cm/s) was used to measure velocity. Prior to the experiment the current meter was tested in the pool to verify whether the pool water possessed an ionic content high enough for the current meter to perform well. Temperature variance dissipation rates were obtained using the isotropic formula following Monin and Yaglom (1971):
e18
The VMP-200 and the current meter were attached to the remotely operated vehicle (ROV) at 25 cm below the water surface. The current meter was 30 cm downstream from the VMP-200. A sketch of the instruments setup on the ROV is shown in Fig. 1.
Fig. 1.
Fig. 1.

Experiment instrument sketch: ROV with attached VMP200.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

The data were collected over the course of 15 runs, each lasting about 1–3 min. The collected data are labeled as runs B through P. In the selected runs, to test for possible contamination of vibrations the electric motor was switched off and the ROV continued its motion propelled only by its inertia. We have not observed any significant noise in the thermistor data that could be related to the propelling motor. This is attributed to the relatively large inertia of the whole system (>200 kg) when propelled by a relatively weak electric motor (<25 W).

Only subsets of selected runs (C, E, I, and J)—see Fig. 2—were chosen to maximize the resolved temperature variance. These required finding segments with low horizontal velocity w of the ROV and the minimal rms of mean velocity (Δw/w) (see appendix A). The time series of the speed of the ROV for runs C, E, I, and J are presented in Fig. 3. The sub segments selected for analysis marked in Fig. 3 are as follows: 15–16 s for run C, 61.5–64.5 s for run E, 24–29 s for run I, and 113–116 s for run J. The measured flow Reynolds numbers, flow speed, temperature, and energy dissipation rates and uncertainties (i.e., Δw/w, Δχ/χ) for the selected runs are collected in Table 1. The Reynolds number is defined here as , where H is the distance between the VMP and the water’s surface (H = 0.25 m), and represents local mean current speed. The flow Reynolds numbers in our experiments corresponds to upper Re values in the experiments of Sanchez et al. (2011).

Fig. 2.
Fig. 2.

Water temperature time series for the runs C, E, I, and J.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

Fig. 3.
Fig. 3.

Velocity time series for the runs C, E, I, and J.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

Table 1.

Parameters of different runs.

Table 1.

The FP07 thermistor thermal inertia and the VMP electronics limit the observed thermistor response. The VMP-200 carries out a very sophisticated thermistor signal processing which minimizes the electronics noise so that the measured temperature spectra are only limited by the thermistor inertia—see Lueck et al. (2002).

The thermistor thermal inertia is related to the rate of heat transfer from the outside fluid to the thermistor bead through the combined fluid boundary layer and the FP07’s glass coating (Lueck et al. 1977; Gregg and Maegher 1980; Hill 1987; Nash et al. 1999). In general, the response of the FP07 depends on fluid velocity—see Fig. 4—and varies between thermistors. In this experiment to correct the FP07 data for the thermal inertia, a single pole response function was used with parameter = 15 Hz, being consistent with our observations. For more detailed discussion see appendix B.

Fig. 4.
Fig. 4.

FP07 thermistor single pole-correction function, , for three selected ROV speeds. The data presented in this paper were collected at approximately 0.1 m s−1 current speed.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

The spectral correction to the measured FP07 signal in terms of temperature dissipation spectra amounts to multiplying the measured dissipation spectrum by the inverse of the filter function seen here:
e19
where is the true dissipation spectrum. The thermistor correction extends the resolved variance up to wavenumber ~ 2300 cpm at which the measured and corrected temperature dissipation spectrum begins to markedly deviate from the theoretical spectrum as seen in Fig. 5.
Fig. 5.
Fig. 5.

Shows the measured one-dimensional temperature dissipation spectra : (a) run C, 15–18 s, (b) run E, 61.5–64.5 s, (c) run I, 24–29 s, and (d) run J, 113–116 s. The measured one dimensional temperature dissipation spectra for (e) run C, 15–18 s, (f) run E, 61.5–64.5 s, (g) run I, 24–29 s, and (h) run J, 113–116 s. The numbers after each run name correspond to the lengths of the time series used for analysis.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

To estimate the contribution of the unresolved part of the temperature spectrum to the temperature dissipation rate χ, following Gregg (1999), the measured part of the dissipation spectrum was corrected by a factor 1/a (see Table 1), that is,
e20

To find the temperature dissipation peak location a polynomial curve was used to approximate around the dissipation peak. Differentiating this polynomial with respect to and setting the derivative to zero, provides the equation for the dissipation peak location of the corrected dissipation spectrum D1T−corrected. Using this approach was found to be 193, 196, 139, and 110 cpm for runs C, E, I, and J, respectively. The peak location was then converted via Eqs. (15) and (16) to the corresponding turbulent kinetic energy dissipation values—see Table 1.

As seen in Fig. 5, at low wavenumbers both spectral forms agree fairly well with the experimental data. The differences appear at higher wavenumbers, with the Kraichnan form describing the measured spectrum very well over the entire range of experimental wavenumbers. To see the match more clearly, the y axis value was changed to and plotted in linear–linear scale in Figs. 5e–h. These figures show that, neglecting contributions from noise, the experimental values are aligned almost linearly. This implies that , which in turn implies that (where b and c are constant). This shows that the far dissipation part of the temperature spectrum behaves as , consistent with the Kraichnan formula, Eq. (11).

Overall, the energy and temperature dissipation rates exhibit a relatively small scatter seen clearly in Table 1. This is attributed to the experimental conditions, specifically the fact that the measurements were taken as the ROV traversed horizontally in the swimming pool under essentially constant conditions including steady wind stress and the absence of any significant mean currents. This spatial homogeneity results in the dissipation spectra being measured under a background flow with a relatively constant Reynolds number, a situation not frequently experienced in oceanic conditions.

4. Parameters of the temperature variance spectra in the dissipation range

In the spectral forms proposed by Batchelor [Eq. (10)] and by Kraichnan [Eq. (11)] the TKE dissipation rate and are treated as unknowns while all other parameters are either prescribed (ν, D, , and ) or can be computed directly from the measured data such as the temperature variance dissipation rate. In this paper we use fixed values of parameters = 3.9 and = 5.26.

Values of and for the four analyzed experimental runs, that is, C, E, I, and J were obtained by applying the weighted least squares fit of Eq. (10) and Eq. (11) to the measured data. The lower frequency limit was set to 1 Hz and the upper limit set to —for the results see Table 1.

The results of fitting the theoretical spectral forms to the measured data are presented in Fig. 6. The Kraichnan form is consistent with the observations over all experimental wavenumbers. In the case of the Batchelor form, the fit shows large systematic deviation from the measured data at large wavenumbers.

Fig. 6.
Fig. 6.

Measured temperature variance dissipation spectra and their best least squares fit to the Kraichnan and Batchelor spectral forms: (a) run C, 15–18 s; (b) run E, 61.5–64.5 s; (c) run I, 24–29 s; and (d) run J, 113–116 s.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

Figures 7 and 8 show the normalized temperature dissipation spectra in either linear–log or log–log scale. The data are clearly consistent with the Kraichnan spectral form and exhibits large deviations from the Batchelor form at high wavenumbers.

Fig. 7.
Fig. 7.

Normalized, compensated temperature spectra in analyzed runs in linlog scale, where kmax is the largest resolved wavenumber; note that the Kraichnan form behaves as ~k while Batchelor form has parabolic behavior in the far dissipation region.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

Fig. 8.
Fig. 8.

Normalized, compensated temperature spectra in analyzed runs; the k−1 range corresponds to the flat part of the plot at small wavenumbers.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

5. Conclusions

Temperature dissipation spectra were measured in a swimming pool using a fast thermistor FP07. In all analyzed cases the spectra were self-similar and followed closely the Kraichnan spectral form while displaying systematic deviations from the Batchelor form.

We found that the Kraichnan formula with the constant = 5.26 determined previously from numerical simulations represent the temperature variance dissipation data accurately for nonstratified flow in experiments as well.

As noted by Donzis et al. (2010) the constant may weakly depend on the Reynolds number of the flow, such that the somewhat larger flow Re number results in increased value of . In our experiments Re was around 4 × 103–6 × 103 and we used the value of = 5.26. The data of Sanchez et al. (2011) were acquired from experiments where the Reynolds number was larger, Re ~ O(104) (our estimate), and likely varies by at least an order of magnitude as indicated by the variability in their Reynolds buoyancy number of 50 to few 103. One may speculate that the differences between values of in both experiments can be attributed to the differences in the Reynolds numbers. Clearly more small-scale temperature measurements or DNS simulations are needed to establish the details of scalar spectra at high wavenumbers and determine the level of dependence on the Reynolds number, if any.

Based on the results in Table 1, for the kinetic energy dissipation rates, ɛ, obtained from the measured temperature dissipation spectra, the application of the Kraichnan functional form yields estimates of the kinetic energy dissipation rates by a factor of 2 to 3 smaller than the estimates inferred using the Batchelor form. We believe that the estimate based on the Kraichnan formula is more accurate.

Acknowledgments

DB acknowledges TAMUCC TRDF, NASA Physical Oceanography, and BP/The Gulf of Mexico Research Initiative support. We are grateful to Dr. Hua Li (JFE), Dr. John Gonzales (TAMUCC), and Mr. Chris Trombley (TAMUCC) for help with experiment.

APPENDIX A

The Temperature Dissipation Spectra

a. The location of the temperature dissipation spectra maximum

The one dimensional Batchelor temperature dissipation spectrum is
eq1
Introducing the variable in the above formula yields the following:
eq2
To find the value of m that maximizes we find the solution of the equation , which in its explicit form is
eq3
Numerical solution is obtained as , which gives .
Similarly, for the Kraichnan temperature dissipation spectrum
eq4
the value of that maximizes obeys the equations , leading to the following equation for :
eq5
Solving for yields or, alternatively, .

b. Estimate of the measured temperature variance dissipation error

The “frozen field” hypothesis is applied to convert between spatial and temporal gradient: .

The temperature dissipation rate and its relative error can be expressed as
eq6
eq7
Therefore, the relationship between the relative errors of the temperature dissipation rate and the current velocity is
eq8

APPENDIX B

Experimental Estimate of Thermistor Dynamic Response Function

A sensor such as the FP07 thermistor relies on contact with water to provide an indirect measure of water temperature. In frequency domain the true water temperature spectrum T0(f) is related to the observed water temperature spectrum Tm(f) via the filtering (low-pass filter) or the response function H(f):
eq9
The relationship between the true and thermistor measured spectrum is then
eb1
The spatial temperature spectra can be obtained by substituting into Eq. (B1), where w is the current speed.
There are two empirical H(f) filter functions used for the FP07 thermistor (Lueck et al. 1977; Gregg and Maegher 1980; Hill 1987; Nash et al. 1999), a single-pole filter function
eq10
and a double-pole filter function
eq11
where the filter cutoff frequency fc represents 3-dB/6-dB drop for either single-pole or double-pole filter functions. The cutoff frequency fc has been observed to depend on current speed −w: fc = f0wp where f0 and p are thermistor dependent.

In principle, the parameters of a thermistor transfer function can be inferred from experimental temperature spectra provided repeated measurements are taken of the same turbulent structure [that is a structure characterized by the same temperature spectra ] but utilizing a different thermistor speed.

To estimate the thermistor transfer function in the experiment two neighboring time intervals were selected such that the thermistor speed was significantly different between the measurements. The chosen intervals are at the beginning of Run J: from the 2nd to 4th second and from 5th to 7th second—see Fig. 3 with corresponding ROV speeds: w1 = 0.06 m s−1 and w2 = 0.14 m s−1.

To estimate the thermistor transfer function, the function R() is generated from observations of spectra at two time segments and and at two different speeds, w2 > w1 such that
eb2
The thermistor spatial filter functions are then
eq12
or
eq13
for the single-pole or the double-pole filter. The speed dependent constant C(w) is then .
If the measured temperature spectra do not vary between the measured intervals [i.e., does not change between intervals] then for the single pole filter R1() should be
eb3
while for the double pole R2():
eb4
Additionally, to estimate the influence of the spectral variability it is assumed that the temperature dissipation for each analyzed segment varies by a factor of 0.3. The results of this spectral variability can be seen in Fig. B1. The blue lines in the graph denote the R1() wavenumber response (single pole) when the temperature dissipation for the two time segments is allowed to vary (shown by a dotted black line representing the undisturbed response) by a factor 0.3 between the black dotted line and either blue line. The corresponding double-pole filter response (similarly the temperature dissipation for each analyzed segment is allowed to vary by a factor of 0.3) is indicated by black lines while the undisturbed double-pole filter function is represented by a dotted black line in the center.
Fig. B1.
Fig. B1.

The measured high wavenumber portion of function R(k) is shown in red. The blue lines denote the theoretical single-pole filter response when temperature dissipation is allowed to vary by a factor 0.3 between the two time segments. The corresponding double-filter response is marked by black lines. The undisturbed double pole filter function is represented by a black-dotted line for either double- or single-pole filter, respectively.

Citation: Journal of Physical Oceanography 42, 10; 10.1175/JPO-D-11-0214.1

The results show that the high wavenumber part ( > 200 cpm) of R() is less sensitive to variability and is appropriate to use when determining the thermistor filter function from an observed temperature time series.

By fitting the modeled R() function to the observed data following Eq. (B3) or Eq. (B4), it is clearly evident that the data is best approximated by a single-pole filter with f0 =30 (Hz) and p = 0.3 if w is expressed in meters per second (see Fig. B1).

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    • Export Citation

  • Luketina, D. A., and J. Imberger, 2001: Determining turbulent kinetic energy dissipation from Batchelor curve fitting. J. Atmos. Oceanic Technol., 18, 100–113.

    • Search Google Scholar
    • Export Citation

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    • Search Google Scholar
    • Export Citation

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    • Search Google Scholar
    • Export Citation

  • Nash, J. D., D. R. Caldwell, M. J. Zelman, and J. N. Moum, 1999: A thermocouple probe for high-speed temperature measurement in the ocean. J. Atmos. Oceanic Technol., 16, 1474–1482.

    • Search Google Scholar
    • Export Citation

  • Newman, G. R., and J. R. Herring, 1979: A test field study of a passive scalar in isotropic turbulence. J. Fluid Mech., 94, 163–194.

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    • Search Google Scholar
    • Export Citation

  • Osborn, T. R., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr., 10, 83–89.

  • Ruddick, B., A. Anis, and K. Thompson, 2000: Maximum likelihood spectral fitting: The Batchelor spectrum. J. Atmos. Oceanic Technol., 17, 1541–1555.

    • Search Google Scholar
    • Export Citation

  • Sanchez, X., E. Roget, J. Planella, and F. Forcat, 2011: Small-scale spectrum of a scalar field in water: The Batchelor and Kraichnan models. J. Phys. Oceanogr., 41, 2155–2167.

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    • Export Citation

  • Smyth, W. D., 1999: Dissipation-range geometry and scalar spectra in sheared stratified turbulence. J. Fluid Mech., 401, 209–242.

  • Stevens, C. L., and M. J. Smith, 2004: Temperature microstructure beneath surface gravity waves. J. Atmos. Oceanic Technol., 21, 1747–1757.

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  • Yeung, P. K., S. Xu, and K. R. Sreenivasan, 2002: Schmidt number effects on turbulent transport with uniform mean scalar gradient. Phys. Fluids, 14, 4178–4191.

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    • Export Citation

  • Yeung, P. K., S. Xu, D. A. Donzis, and K. R. Sreenivasan, 2004: Simulations of three-dimensional turbulent mixing for Schmidt numbers of the order 1000. Flow Turbul. Combust., 72, 333–347.

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  • Donzis, D. A., K. R. Sreenivasan, and P. K. Yeung, 2010: The Batchelor spectrum for mixing of passive scalars in isotropic turbulence. Flow Turbul. Combust., 85, 549–566.

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  • Gibson, C. H., 1968: Fine structure of scalar fields mixed by turbulence. II. Spectral theory. Phys. Fluids, 11, 2316–2327.

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  • Gregg, M. C., 1999: Uncertainties and limitations in measuring ɛ and χT. J. Atmos. Oceanic Technol., 16, 1483–1490.

  • Gregg, M. C., and T. B. Maegher, 1980: The dynamic response of glass rod thermistor. J. Geophys. Res., 85 (C5), 2779–2786.

  • Hill, K. D., 1987: Observations on the velocity scaling of thermistor dynamic response functions. Rev. Sci. Instrum., 58, 1235–1238.

  • Kantha, L. H., and C. A. Clayson, 2003: Ocean Mixed Layer in Encyclopedia of Atmospheric Sciences. Academic Press, 2779 pp.

  • Kraichnan, R., 1968: Small-scale structure of a scalar field convected by turbulence. Phys. Fluids, 11, 945–953.

  • Lueck, R. G., O. Hertzman, and T. R. Osborn, 1977: The spectral response of thermistors. Deep-Sea Res., 24, 951–970.

  • Lueck, R. G., F. Wolk, and H. Yamazaki, 2002: Oceanic velocity microstructure measurements in the 20th century. J. Oceanogr., 58, 153–174.

    • Search Google Scholar
    • Export Citation

  • Luketina, D. A., and J. Imberger, 2001: Determining turbulent kinetic energy dissipation from Batchelor curve fitting. J. Atmos. Oceanic Technol., 18, 100–113.

    • Search Google Scholar
    • Export Citation

  • Mjolsness, R. C., 1975: Diffusion of a passive scalar at large Prandtl number according to the abridged Lagrangian interaction theory. Phys. Fluids, 18, 1393–1394.

    • Search Google Scholar
    • Export Citation

  • Monin, A. S., and A. M. Yaglom, 1971: Statistical Fluid Mechanics: Volume 2. The MIT Press, 886 pp.

  • Nash, J. D., and J. N. Moum, 2002: Microstructure estimates of turbulent salinity flux and the dissipation spectrum of salinity. J. Phys. Oceanogr., 32, 2313–2333.

    • Search Google Scholar
    • Export Citation

  • Nash, J. D., D. R. Caldwell, M. J. Zelman, and J. N. Moum, 1999: A thermocouple probe for high-speed temperature measurement in the ocean. J. Atmos. Oceanic Technol., 16, 1474–1482.

    • Search Google Scholar
    • Export Citation

  • Newman, G. R., and J. R. Herring, 1979: A test field study of a passive scalar in isotropic turbulence. J. Fluid Mech., 94, 163–194.

  • Oakey, N. S., 1982: Determination of the rate of dissipation of turbulent energy from simultaneous temperature and velocity shear microstructure measurements. J. Phys. Oceanogr., 12, 256–271.

    • Search Google Scholar
    • Export Citation

  • Osborn, T. R., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr., 10, 83–89.

  • Ruddick, B., A. Anis, and K. Thompson, 2000: Maximum likelihood spectral fitting: The Batchelor spectrum. J. Atmos. Oceanic Technol., 17, 1541–1555.

    • Search Google Scholar
    • Export Citation

  • Sanchez, X., E. Roget, J. Planella, and F. Forcat, 2011: Small-scale spectrum of a scalar field in water: The Batchelor and Kraichnan models. J. Phys. Oceanogr., 41, 2155–2167.

    • Search Google Scholar
    • Export Citation

  • Smyth, W. D., 1999: Dissipation-range geometry and scalar spectra in sheared stratified turbulence. J. Fluid Mech., 401, 209–242.

  • Stevens, C. L., and M. J. Smith, 2004: Temperature microstructure beneath surface gravity waves. J. Atmos. Oceanic Technol., 21, 1747–1757.

    • Search Google Scholar
    • Export Citation

  • Yeung, P. K., S. Xu, and K. R. Sreenivasan, 2002: Schmidt number effects on turbulent transport with uniform mean scalar gradient. Phys. Fluids, 14, 4178–4191.

    • Search Google Scholar
    • Export Citation

  • Yeung, P. K., S. Xu, D. A. Donzis, and K. R. Sreenivasan, 2004: Simulations of three-dimensional turbulent mixing for Schmidt numbers of the order 1000. Flow Turbul. Combust., 72, 333–347.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Experiment instrument sketch: ROV with attached VMP200.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Water temperature time series for the runs C, E, I, and J.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Velocity time series for the runs C, E, I, and J.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    FP07 thermistor single pole-correction function, , for three selected ROV speeds. The data presented in this paper were collected at approximately 0.1 m s−1 current speed.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Shows the measured one-dimensional temperature dissipation spectra : (a) run C, 15–18 s, (b) run E, 61.5–64.5 s, (c) run I, 24–29 s, and (d) run J, 113–116 s. The measured one dimensional temperature dissipation spectra for (e) run C, 15–18 s, (f) run E, 61.5–64.5 s, (g) run I, 24–29 s, and (h) run J, 113–116 s. The numbers after each run name correspond to the lengths of the time series used for analysis.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Measured temperature variance dissipation spectra and their best least squares fit to the Kraichnan and Batchelor spectral forms: (a) run C, 15–18 s; (b) run E, 61.5–64.5 s; (c) run I, 24–29 s; and (d) run J, 113–116 s.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Normalized, compensated temperature spectra in analyzed runs in linlog scale, where kmax is the largest resolved wavenumber; note that the Kraichnan form behaves as ~k while Batchelor form has parabolic behavior in the far dissipation region.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Normalized, compensated temperature spectra in analyzed runs; the k−1 range corresponds to the flat part of the plot at small wavenumbers.

  • <sc>Fig</sc>. B1.
    View in gallery
    Fig. B1.

    The measured high wavenumber portion of function R(k) is shown in red. The blue lines denote the theoretical single-pole filter response when temperature dissipation is allowed to vary by a factor 0.3 between the two time segments. The corresponding double-filter response is marked by black lines. The undisturbed double pole filter function is represented by a black-dotted line for either double- or single-pole filter, respectively.

Fig. 1.

Experiment instrument sketch: ROV with attached VMP200.
Fig. 2.

Water temperature time series for the runs C, E, I, and J.
Fig. 3.

Velocity time series for the runs C, E, I, and J.
Fig. 4.

FP07 thermistor single pole-correction function, , for three selected ROV speeds. The data presented in this paper were collected at approximately 0.1 m s−1 current speed.
Fig. 5.

Shows the measured one-dimensional temperature dissipation spectra : (a) run C, 15–18 s, (b) run E, 61.5–64.5 s, (c) run I, 24–29 s, and (d) run J, 113–116 s. The measured one dimensional temperature dissipation spectra for (e) run C, 15–18 s, (f) run E, 61.5–64.5 s, (g) run I, 24–29 s, and (h) run J, 113–116 s. The numbers after each run name correspond to the lengths of the time series used for analysis.
Fig. 6.

Measured temperature variance dissipation spectra and their best least squares fit to the Kraichnan and Batchelor spectral forms: (a) run C, 15–18 s; (b) run E, 61.5–64.5 s; (c) run I, 24–29 s; and (d) run J, 113–116 s.
Fig. 7.

Normalized, compensated temperature spectra in analyzed runs in linlog scale, where kmax is the largest resolved wavenumber; note that the Kraichnan form behaves as ~k while Batchelor form has parabolic behavior in the far dissipation region.
Fig. 8.

Normalized, compensated temperature spectra in analyzed runs; the k−1 range corresponds to the flat part of the plot at small wavenumbers.
Fig. B1.

The measured high wavenumber portion of function R(k) is shown in red. The blue lines denote the theoretical single-pole filter response when temperature dissipation is allowed to vary by a factor 0.3 between the two time segments. The corresponding double-filter response is marked by black lines. The undisturbed double pole filter function is represented by a black-dotted line for either double- or single-pole filter, respectively.
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