a. Finestructure

The most commonly observed finestructure occurs when a probe passes vertically through a stationary density stratified fluid. For the case where heat is the stratifying agent, this results in a spectrum with a negative slope, which is quantified as follows: the vertical temperature profile in many regions of a stable temperature stratified fluid can be represented by

TT_oβz^γzγβ(1)

where γ and β are constants, z′ is the vertical ordinate whose origin is at the center of the region of interest and is positive upward, and T_o is the temperature at the center of the region. The temperature gradient is thus given by

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Examples of stable 1-m-long temperature profiles with a 1°C change from top to bottom, which satisfy (1), are shown in Fig. 2a. By examining Fig. 2a it is clear that (2) describes a wide range of stable temperature profiles. In practice, it is expected that γ would most likely be in the range 0 ⩽ γ ⩽ 2 (β then depends on the temperature change from the top to the bottom of the profile).

The one-sided finestructure power spectrum S_f(k) of (2) is given by S_f(k) = 2F(k)F*(k), where F denotes the Fourier transform of dT/dz′, the asterisk denotes the complex conjugate, and k is the wavenumber (rad m⁻¹). The Fourier transform of (2) is

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where Γ(γ + 1) = ∫^∞₀ x^γe^−x dx. Thus S_f(k) is given by

S_fkc_fk^−2γγ(4)

where c_f = 8[(2π)^−γγβ cos(γπ/2)Γ(γ)]². For γ in the expected range of 0 ⩽ γ ⩽ 2 the slope of the spectrum, as given by (4), will be between 0 and −4 in log–log space.

Band-averaged spectra S_f(k) of the gradients of the temperature profiles of Fig. 2a are shown in Fig. 2b. The temperature gradients were determined by applying a Gaussian gradient filter to the temperature profiles—this removed the problem of the temperature gradient as given by (2) becoming infinite at z′ = 0. The temperature gradient data record was then windowed (see Press et al. 1989, p. 423 for details) prior to applying a discrete Fourier transform to yield F(k). The filtering is responsible for the spectra with γ = 0 and γ = 0.5 deviating from the theoretical spectra given by (4) at wavenumbers above approximately 300 cpm. The spectrum with γ = 2 deviates because the high frequency content of the temperature gradient profile is located at the ends of the data record, which are subsequently modified by windowing. In any case, it is expected that the finestructure spectrum S_f(k) derived via discrete Fourier transforms will generally have a slope of between 0 and −4 at low wavenumbers in log–log space.

b. Internal waves

Internal waves with a wavelength smaller than the Rossby radius (Pedlosky 1987, p. 9) will have a temperature gradient spectrum of the form (Gregg et al. 1991; also see Fig. 1)

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where the subscript w denotes internal waves and, based on the data of Gregg et al. (1973), c_w is an O(1) constant. Internal wave effects can extend up to a maximum frequency of N, where N² = −(g/ρ)(∂ρ/∂z) and ρ is the fluid density (in wavenumber space this will correspond to a maximum wavenumber of 2πN/u, where u is the sensor velocity relative to the water).

c. Inertial convective subrange

For scales smaller than 2πN/u, but large enough to be influenced by viscosity, we have the inertial convective subrange. The temperature gradient spectrum in the inertial convective subrange is given by (Monin and Yaglom 1975, 479–485)

S_ICSkc_ICSχ_T^−1/3k^1/3(6)

where c_ICS is an O(1) constant and χ_T is the dissipation of temperature variance due to turbulence and is defined for the case of isotropic turbulence by

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where the overbar denotes an ensemble average, D_T is the diffusivity of heat, and T′ is the temperature fluctuation. The spectrum given by (6) is valid to a wavenumber (c_ICS/c_VC)^3/2k_k where the Kolmogorov wavenumber is k_k = (ɛ/ν³)^1/4 and c_VC is a constant related to the spectrum in the viscous convective subrange (see below).

d. Batchelor spectrum

In his paper, Batchelor (1959) developed the form of the temperature gradient wavenumber spectrum for high Reynolds number, isotropic, homogeneous turbulence. Gibson and Schwartz (1963) used Batchelor’s results to derive the one-dimensional Batchelor spectrum S(k) of the temperature gradient

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where k has units of radians per meter, q is a universal constant, Batchelor wavenumber k_B is defined as the inverse of the Batchelor length scale L_B = (D²_Tν/ɛ)^1/4, and α is a dimensionless wavenumber given by (2q)^1/2kk⁻¹_B. The normalized spectrum S_N(α) is given by

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The one-dimensional form S(k) has been confirmed in laboratory experiments involving turbulence generated behind a grid (Gibson and Schwartz 1963), in the upper ocean (Dillon and Caldwell 1980), and in lakes (Imberger 1985).

The initial part of the Batchelor spectrum for Pr = ν/D_T > 1 is the viscous convective subrange and is given by (Batchelor 1959; Gibson and Schwartz 1963)

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where c_VC is an O(1) constant (Dillon and Caldwell 1980; Oakey 1982). Sherman and Davis (1995) have shown that the viscous convective subrange is only found for events with large values of the Cox number C_T = χ_T/2D_TT²_z, where T_z is the mean temperature gradient. We shall refer to the portion of the Batchelor spectrum at wavenumbers higher than that of the spectral maximum as the roll-off region (see Fig. 1a).

The value of the universal constant q was estimated to be 3^1/2 < q < 2(3^1/2) from theoretical arguments advanced by Gibson (1968a,b). Grant et al. (1968) estimated q = 3.9 ± 1.5, and Dillon and Caldwell (1980) used q = 2(3^1/2) = 3.46. Oakey (1982) determined q = 3.67 ± 1.52 by comparing direct measurements of dissipation with those obtained from temperature gradient spectra. In this paper it is assumed that q = 2(3^1/2).

It should be noted that the Batchelor spectrum can be applied to any resolvable tracer that has a Prandtl reasonably greater than unity.

e. Noise spectra

Noise is generated either at the sensor or by the processing circuitry. Commonly, the noise spectrum will have a slope of around 2 (this corresponds to white noise). In any case, the approach used in this paper is not sensitive to the exact form of the noise spectrum. The wavenumber at the intersection of the Batchelor and noise spectra is denoted k_n.

There have been various attempts to reduce the effects of noise. For example, Dillon and Caldwell (1980) used the in-phase power between redundantly amplified channels to reduce electronic noise. However, this technique is not necessarily useful for newer microstructure probes where electronic noise levels have been reduced resulting in sensor noise dominating electronic noise.

a. Background

Before proceeding, it should be noted that (8) applies to a spectrum based on radian wavenumbers k having units of radians per meter. If the spectra being fitted are based on cyclical wavenumbers κ having units of inverse meters, then the nondimensional wavenumber α is defined by α = (2q)^1/2κκ⁻¹_B, where κ_B = (2πL_B)⁻¹. For the remainder of this paper we shall only deal with cyclical wavenumbers. The measured spectrum will be denoted D(κ) to distinguish it from the theoretical or fitted spectrum S(κ). The universal constant q in (7) is assumed to be 2(3^1/2) and the nondimensional theoretical spectrum S_N(α) as defined in (9) is approximated by

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where

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(see Abramowitz and Stegun 1970, section 26.2.17). The dissipation of turbulent kinetic energy is then found by rearranging the definition of the Batchelor wavenumber to give

π⁴νD²_Tκ⁴_B(14)

b. Overview of method

Once a value of the variance χ_T is estimated, the theoretical spectrum S(κ) can be fitted to an appropriate portion of the measured spectrum D(κ) with the Batchelor wavenumber κ_B being the free variable. Since we do not know a priori what the “appropriate portion” of the measured spectrum is, we use a trial and error approach of fitting the theoretical Batchelor spectrum to all possible portions (the method for determining possible portions is discussed later in this section). This entire process is repeated up to six times with the estimates of the noise wavenumber κ_n and the variance χ_T being progressively refined (see the following section). The dissipation is then found by substituting the wavenumber κ_B, associated with the best global fit into (14).

A nonlinear curve fitting algorithm could be used to try to find the best fit (i.e., that with the least error) of the theoretical temperature gradient spectrum to the measured data. However, experience has shown that these fitting algorithms often find only the best local rather than global fit. Further, they are computationally demanding. For this reason, a trial and error approach is used to find the best fit of the theoretical Batchelor spectrum to each of the possible portions of the measured spectrum. In this approach, we trial 200 geometrically spaced values of the Batchelor wavenumber κ_B such that ɛ ranges from 10⁻¹² to 10⁻⁴ m² s⁻³ (however, if κ_B exceeds κ_n, the noise wavenumber, then the upper limit for κ_B is taken to be κ_n). Of these 200 values, the wavenumber κ_B resulting in the least fitting error is assumed to be the Batchelor wavenumber.

The fitting procedure is done in log–log space as spectra are commonly presented in this manner and the roll-off region is well defined compared with a linear–linear format. There is no reason why the procedures that are presented here cannot be adapted for use with variance preserving spectra in a linear–linear format or the error function modified. In any case, the focus here is on developing an automated procedure for determining the Batchelor portion of a temperature gradient spectrum and an appropriate estimate of the variance. Until issues such as these are resolved, use of a statistically optimum error function is somewhat academic.

c. Estimating variance

If we know the Batchelor like part of the measured temperature gradient spectrum (bounded by wavenumbers, κ_L and κ_n) and have fitted a theoretical spectrum to this part, we can estimate the temperature variance dissipation χ_T using

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where the molecular diffusivity of heat D_T for water can be estimated from formulas given in Fofonoff and Millard (1983). However, we have a chicken and egg problem as the theoretical spectrum cannot be fitted until we have estimated the temperature variance dissipation χ_T which, in turn, requires the limiting wavenumbers, κ_L and κ_n, to be estimated. We get around this problem by an iterative procedure.

Prior to the first iteration, κ_L is set to the lowest wavenumber of the measured spectra D(κ). For each subsequent set of trials, κ_L is the lowest wavenumber at which the measured spectra D(κ) and the best-fit theoretical spectra S(κ) (from the set of fits done within the preceding iteration) intersect (see Fig. 3; if there is no intersection, κ_L does not change).

Prior to the first iteration, the noise wavenumber κ_n is set to the wavenumber corresponding to the minimum of the measured spectrum at wavenumbers greater than 30 m⁻¹. This generally ensures that the appropriate minima are selected. However, in some instances, this estimate of κ_n may be too high. By comparing the best-fit spectrum from the first iteration with the measured spectrum it is possible to reevaluate whether κ_n should be reduced. First, the highest wavenumber intersection of the theoretical (i.e., fitted) and measured spectra is found (see Fig. 3b). The measured spectrum is then scanned in the direction of higher wavenumbers until a positive gradient, indicating noise, is encountered. If the wavenumber at which this positive gradient is encountered is less than the previous value of the noise wavenumber κ_n, κ_n is replaced (see Fig. 3c). This is done only at the end of the first iteration and prior to the second iteration.

d. Which portion of the measured spectrum should be used

Once wavenumbers higher than the noise wavenumber k_n have been excluded, the only basis for determining the Batchelor portion of the measured spectrum is that it is shaped like a Batchelor spectrum. It is assumed that some portion of the data at lower wavenumbers than k_n is Batchelor shaped. Accordingly, different portions of the data between the lowest wavenumber of the measured spectrum and the noise wavenumber κ_n are fitted. The portion of the data giving the best fit to the theoretical spectrum is deemed to be the Batchelor portion of the data.

When selecting the portions of the measured spectrum to fit, it is necessary to ensure that at least the peak of the spectrum and the initial part of the roll-off region are included in the fitting. Further, to ensure that a sufficient amount of data is used, the wavenumbers of the data being fitted must span not much less than a decade. If we define the portions as extending from wavenumber κ₁ to κ₂, then the preceding restrictions can be written as

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where the peak of the theoretical spectrum is S(κ_peak) and κ_peak is equal to 0.176κ_B. Potential values of κ₁ and κ₂ satisfying the first two criteria are shown in Fig. 3.

e. Error function

One standard curve fitting approach would be to specify the fitting error based on the difference between the theoretical and measured spectrum at each wavenumber. This approach was only found to be successful using a relatively complicated error function (Luketina 1987). The key to a more robust error function is to recognize, as Stewart and Grant (1962) did, that once χ_T is estimated, the theoretical curve as given by (8) is constrained to move along a slope of −1 in log–log space. For this reason, when working in log–log space, as is done here, the error should be the difference between the theoretical and measured spectrum in the direction of slope −1 (see Fig. 4). This error for the ith point is given by

λ_iDκ_iSκ_Si²κ_iκ_Si^21/2(17)

where S(κ_Si) is the point on the theoretical spectrum that lies on the −1 slope passing through D(κ_i) (see Fig. 4). The error function that we have found to be effective involves a normalization of the error given by (17) and is

View Expanded

where w_i is a weight based on the number of raw spectral data points band averaged to produce D(κ_i), m is the number of data points (after band averaging) between wavenumbers κ₁ and κ₂, m − 1 is the number of degrees of freedom, f_r is the fraction of the maximum possible number of data points used (i.e., all those at wavenumbers lower than κ_n), and c is a constant.

The use of (f_r)^c in the error function biases the fitting procedure toward accepting a larger portion of the spectrum. Setting c to 1.5 has been found to give the best fits to a wide range of measured spectra (the procedure is not sensitive to the value of c if the measured spectrum is clearly Batchelor shaped). Dividing by the degrees of freedom (m − 1) is a standard normalization. The weights w_i are based on the standard deviation of n_i points band averaged to a single point being reduced by a factor of (n_i)^−1/2. More specifically, the weights are calculated using w_i = (n_i)^−1/2 and are then multiplied by a factor so that the average value of the weights is unity.

In order to evaluate (17) and (18) we must find find S(κ_Si), or more specifically κ_Si. The method is illustrated in Fig. 4 and is presented in terms of the normalized wavenumber α_Si = (2q)^1/2κ_Si/κ_B. Figure 4 shows a data point α_i, D_N(α_i) on the normalized spectrum through which passes a line L_N(α) of slope −1. The normalized spectrum is given by

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Also shown is the normalized theoretical spectrum S_N(α) with the −1 tangent point α_tan, S_N(α_tan) indicated. Note that α_tan = 1.036. A parameter Δ is defined as

₁₀S_Nα₁₀L_Nα(20)

We can now approximate the points α_left and α_right for Δ > 0 using the following series expansion:

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where ζ = lnΔ. The wavenumber α_Si is then taken to be the closer of α_left and α_right (in the example of Fig. 4, α_Si = α_left) or, if Δ < 0 (i.e., the line and fitted curve do not intersect), α_Si is set equal to α_tan.

Before moving onto the next section, we wish to make clear that we do not claim that the error function given by (18) is the statistically optimum error function (see Ruddick et al. 2000 for a discussion of the merits of different forms of the error function). However, the error function used here is convenient and is sufficient for our purposes as the error in the estimate of the log dissipation for synthetic spectra is within a few percent of the actual dissipation for the majority of cases (see section 5). Further, Ruddick et al. (2000) shows that a variety of error functions can give good results but that all error functions have some bias.

a. Examples of fitting to data

Figures 5 and 6 show the results of applying the curve fitting algorithm to a range of temperature gradient microstructure data. Figure 5 shows relatively good fits, while Fig. 6 shows poorer fits. These poorer fits are due to the measured spectrum lacking a clear Batchelor-shaped region. The value of the fitting error δ_e as a function of dissipation is also shown. It can be seen that the error function has a unique minimum for a measured spectrum that is clearly Batchelor shaped. As the measured spectrum becomes less Batchelor like, the minimum in the error tends to be flatter, indicating that the uncertainty in the value of the dissipation is increased. In general, poor Batchelor fits result in higher values of the minimum error and the error curve shape deviating from the shapes shown in Fig. 5—either by being flatter in the region of the minimum or more irregularly shaped.

The algorithm has been applied to many hundreds of sets of temperature gradient data and has proven itself to be robust in selecting the Batchelor portion of the temperature gradient spectrum. In measured spectra where there is a clear Batchelor portion, the placement of the theoretical Batchelor spectrum is always close to the best fit “by eye.”

b. Accuracy

The most accurate curve fitting, and hence dissipation estimates, will be when the measured spectrum is free of noise, finestructure, internal waves, and calibration errors. Nonetheless, there will still be inaccuracies due to the fitting procedure, because each spectrum is only a sample of a larger population, and because the Nyquist wavenumber limits the highest wavenumber that can be resolved. These inaccuracies were assessed by using the Batchelor fitting technique on spectra derived from an artificial time series. A discrete artificial time series η(t_i) was calculated using

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where θ_j is a uniformly distributed random variable in the range from 0 to 2π; p is the number of wavenumber (or Fourier) components being used to simulate η(t); and κ_j = (j/p)Δκ, where Δκ = (2pΔtu) and Δt is the sampling interval of the time series. The magnitude of the component associated with wavenumber κ_j is

A_jSκ_jκ^1/2(23)

where S(κ) is given by (8). The above relationships ensure that spectra based upon the time series will not be aliased as there is no energy in the time series above the Nyquist frequency.

Values of D_T, p, Δt and u were fixed as 1.4 × 10⁻⁷ m² s⁻¹, 300, 0.01 s, and 0.1 m s⁻¹, respectively. This left ɛ and χ_T to be specified so that a time series could be generated. Spectra were then derived by using adjacent segments of data of duration τ (or record length L_r = uτ) from a time series consisting of 51 200 points. Statistical quantities calculated for the log of the dissipation determined from Batchelor fitting were the mean μ_logɛ and the standard deviation s_logɛ. The originally specified dissipation was denoted ɛ.

Examples of artificially generated spectra and curve fits are shown in Fig. 7 for a range of dissipations. The segment length (in this instance 0.512 m) limits the amount of the low wavenumber data that can be captured, while the chosen sampling interval (Δt = 0.01 s, which corresponds to 1 mm in space in this instance) limits the highest wavenumber to 500 m⁻¹. The viscous convective subrange is absent from the low dissipation spectrum due to the low wavenumber restriction, while the roll-off region is only just discernable for the high dissipation spectrum due to the high wavenumber limit.

Figure 8 shows the relationship between the actual ɛ and mean of the dissipation estimates μ_logɛ in the form of a percentage error 100(logɛ − μ_logɛ)/logɛ. It is clear that the estimated and actual values are in good agreement in the range from 10⁻⁹ to 10⁻⁵ m² s⁻³. Below 10⁻⁹ m² s⁻³, dissipation tends to be underestimated due to a lack of data in the viscous convective subrange, while above 10⁻⁵ m² s⁻³, dissipation tends to be overestimated, due to little data being in the roll-off region. Figure 8 shows that the systematic error in the mean of the low (i.e., less than 10⁻⁹) dissipation estimates is more than 5% when κ_B/κ_r < 3.2, where κ_r = L⁻¹_r. At high values of dissipation (i.e., more than 10⁻⁵) the systematic error is generally larger than 5%. There is a slight reduction in the error as the (segment) record length is increased. Systematic errors can be corrected using Fig. 8 for a probe having sampling characteristics similar to the one used in this paper. For other probes, the preceding method can be applied to estimate errors.

In the field, there is rarely an opportunity to average large numbers of dissipation estimates. In this case, we are concerned with the likely error or standard deviation of a single dissipation estimate. Figure 9 shows the highest random error occurs at high values of dissipation due to the roll-off region not being resolved. However, the error reduces as the record length increases. More specifically, the random error associated with the log dissipation estimates is more than 5% for κ_B/κ^−1/3_r > 600.

c. Statistical stationarity

Dillon and Caldwell (1980) claimed that dissipation estimates in nonstationary regions would be biased toward the higher dissipation values in the sample. In other words, the estimated dissipation would be greater than the volume averaged dissipation of the sample. Here we examine this assertion using artificially generated nonstationary datasets. More specifically, 2048 sample time (or depth) series was generated such that each half of the dataset had different values of dissipation. Figure 10 shows spectra of two of these artificial time series. From examining Fig. 10 it is evident that the spectrum for the entire profile (i.e., 2048 sample dataset) basically follows the spectrum of the lower dissipation portion of the profile at low wavenumbers and vice versa at high wavenumbers. In general, the spectrum of a temperature gradient profile with several statistically stationary regions is equivalent to taking, at each wavenumber, the average of the spectra for each stationary region. This is one reason why measured temperature gradient spectra may not be close in shape to the theoretical Batchelor spectrum.

The ratio of the estimated dissipation ɛ_B and the volume averaged dissipation ɛ_vol was calculated for a number of the artificial nonstationary datasets. The results are plotted in Fig. 11 where it can be seen that dissipation is overestimated at low values of ɛ_vol and underestimated at high values of ɛ_vol. This occurs because the low ɛ_vol spectra are well resolved (i.e., the roll-off region is well defined) as in Fig. 10b, while the high ɛ_vol spectra are dominated by the viscous convective region (e.g., Fig. 10a). These results are consistent with Dillon and Caldwell’s (1980) assertion that dissipation estimates in nonstationary regions would be biased toward the higher dissipation values in the sample because they were only using well-resolved spectra.

The dissipation value at which we switch from underestimating to overestimating dissipation will depend upon the bandwidth of the measured spectra. In turn, the bandwidth will depend upon the sampling frequency and the record length L_r. The presence of finestructure and internal waves will affect the effective bandwidth due to the fitting algorithm rejecting the lower wavenumber portion of the spectrum; this will tend to result in dissipation being overestimated rather than underestimated.

The biasing of dissipation estimates in nonstationary regions highlights the importance of being able to divide or segment data into stationary regions prior to estimating dissipation. To demonstrate that such segmentation is viable we have applied the segmentation algorithm of Imberger and Ivey (1991) to one of the artificially generated nonstationary temperature gradient datasets (see Fig. 12). The segmentation parameter shown in Fig. 12b has a large value wherever there is an appreciable change in the frequency content of the signal (the artificially generated dataset changes from ɛ = 10⁻⁶ to ɛ = 10⁻⁷ m² s⁻³ at its midpoint). In particular, that the largest value of the segmentation parameter occurs at the midpoint transition. Thus, selecting a threshold of, say, 100 for the segmentation parameter would result in a segmentation boundary being placed at the transition, thereby dividing the data into stationary regions.