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  • Townsend, A. A., 1976: The Structure of Turbulent Shear Flow. Cambridge University Press, 429 pp.

  • View in gallery

    A schematic showing the various processes contributing to the temperature gradient spectrum as a function of cyclic wavenumber κ. (a) Contributing processes shown are internal waves (dashed line at low wavenumbers), finestructure (dashed and dotted line), the inertial convective subrange (dotted line at low wavenumbers), the Batchelor spectrum (thin line), and thermal/electronic noise (dashed line at high wavenumbers). (b) An example is shown of a spectrum where internal wave, finestructure, and ICS contributions are negligible. Such a spectrum may be found where there is energetic turbulence and weak stratification such as in a wind mixed layer. (c) A spectrum is sketched for the case where the finestructure contains a significant amount of energy at the lower wavenumbers. This type of spectrum may be found, for example, in a stratified thermocline. Note that the thick lines in (a), (b), and (c) represent the composite (i.e., measured) temperature gradient spectrum.

  • View in gallery

    (a) Statically stable temperature profiles 1 m long with a 1°C change from top to bottom that satisfy Eq. (1) with γ = 0 (thin line), γ = 0.5 (thick line), and γ = 2 (dashed line), and (b) the corresponding spectra of the temperature gradients.

  • View in gallery

    A schematic showing the various stages in fitting a Batchelor spectrum (dark line) to a measured spectrum (gray line) to determine the kinetic energy dissipation ɛ. The dashed vertical line indicates the noise wavenumber κn. (a) The potential range of the wavenumbers κ1 and κ2 are indicated for a Batchelor spectrum being fitted during the first iteration. Following the first iteration, the wavenumber κL is determined and the noise wavenumber κn modified as shown in (b). (c) The potential range of the wavenumbers κ1 and κ2 are indicated for a Batchelor spectrum being fitted during the second iteration. Following the second iteration, the wavenumber κL is modified as shown in (d). Subsequent iterations repeat the procedure shown in (c) and (d), resulting in the wavenumber κL being progressively modified.

  • View in gallery

    Finding the deviation between a data point on the normalized measured spectrum and the normalized theoretical spectrum along a slope of −1.

  • View in gallery

    Measured temperature gradient spectra D(κ) (solid line) and fitted theoretical Batchelor spectra S(κ) (dashed line) for data from a single profile in Lake Kinneret, Israel. The portion of the measured spectra that was used for fitting is between the two gray vertical lines. (a) The measured spectrum follows the theoretical form fairly well. (c) The curve fitting algorithm has ignored finescale/internal wave contamination at the low wavenumbers. In both cases the algorithm has successfully fitted the Batchelor spectrum despite the presence of high frequency noise. Note that the noise decreases at the very high wavenumbers due to the use of a brickwall filter on the time series prior to the spectra being calculated. The fitting error δe as a function of dissipation ɛ for cases (a) and (c) is shown in (b) and (d), respectively. In both cases there are clear minima in the error function.

  • View in gallery

    (a) The measured spectrum does not follow the theoretical form very well. Consequently, the corresponding error function shown in (b) is flatter in the region of the minima, and the values of the minima are larger than that for the more Batchelor shaped measured spectra of Fig. 5. (c) There is no discernable Batchelor shape. As a result, the the error function shown in (d) is lacking the smoother shape shown in error graphs of Fig. 5. Further, the minimum error is relatively high. See Fig. 5 for explanation of the different lines and symbols.

  • View in gallery

    Band-averaged spectra corresponding to (a) artificially generated dataset 2048 samples long, the first half of the dataset has ɛ = 10−4 m2 s−3, while the second half of the dataset has ɛ = 10−5 m2 s−3; (b) is similar except that ɛ = 10−10 m2 s−3 and 10−11 m2 s−3 for the first and second parts of the dataset. The different lines and symbols are explained in Fig. 5.

  • View in gallery

    Systematic error in estimates of log dissipation as a function of (a) Batchelor wavenumber κB (based on actual dissipation) and record wavenumber κr, which is the reciprocal of the record length and, (b) actual dissipation and record length Lr.

  • View in gallery

    Absolute value of the random error (standard deviation) in estimates of log dissipation as a function of (a) Batchelor wavenumber κB (based on actual dissipation); record wavenumber κr, which is the reciprocal of the record length; and (b) actual dissipation and record length Lr.

  • View in gallery

    Band-averaged spectra corresponding to (a) artificially generated dataset 2048 samples long (solid line), the first half of the dataset (line with short dashes), which has ɛ = 10−4 m2 s−3 and the second half of the dataset (line with long dashes), which ɛ = 10−5 m2 s−3; (b) is similar except that ɛ = 10−10 m2 s−3 (line with short dashes) and ɛ = 10−11 m2 s−3 (line with long dashes).

  • View in gallery

    Ratio of estimated dissipation ɛB to actual (volume averaged) ɛvol dissipation as a function of ɛvol when the data are nonstationary.

  • View in gallery

    Segmentation of a nonstationary dataset: (a) an artificially generated temperature gradient dataset 2048 samples long where the upper and lower halves have ɛ = 10−6 m2 s−3 and ɛ = 10−7 m2 s−3, respectively; and (b) the corresponding segmentation parameter calculated using the method of Imberger and Ivey (1991). The dashed line in (b) shows an arbitrary theshold of 100, which would result in a segmentation boundary being placed at the transition from ɛ = 10−6 to 10−7 m2 s−3.

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Determining Turbulent Kinetic Energy Dissipation from Batchelor Curve Fitting

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  • 1 School of Civil and Environmental Engineering, University of New South Wales, Sydney, New South Wales, Australia
  • | 2 Centre for Water Research, University of Western Australia, Perth, Western Australia, Australia
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Abstract

An algorithm is presented for obtaining the rate of turbulent kinetic energy dissipation by fitting the theoretical Batchelor spectrum to the temperature gradient spectrum at high wavenumbers. The algorithm is relatively robust in selecting the turbulent Batchelor component from temperature gradient spectra, which have finestructure, internal wave, and noise contributions. The theoretical curve is fitted using an error function that takes into account many of the characteristics of the Batchelor spectrum. Overall, the use of the algorithm to determine dissipation of the turbulent kinetic energy is considerably more time efficient than manual methods. Some limits on the accuracy of the method are also discussed.

Corresponding author address: Dr. David Luketina, Civil Engineering, Asian Institute of Technology, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand.

Email: davidl@ait.ac.th

Abstract

An algorithm is presented for obtaining the rate of turbulent kinetic energy dissipation by fitting the theoretical Batchelor spectrum to the temperature gradient spectrum at high wavenumbers. The algorithm is relatively robust in selecting the turbulent Batchelor component from temperature gradient spectra, which have finestructure, internal wave, and noise contributions. The theoretical curve is fitted using an error function that takes into account many of the characteristics of the Batchelor spectrum. Overall, the use of the algorithm to determine dissipation of the turbulent kinetic energy is considerably more time efficient than manual methods. Some limits on the accuracy of the method are also discussed.

Corresponding author address: Dr. David Luketina, Civil Engineering, Asian Institute of Technology, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand.

Email: davidl@ait.ac.th

1. Introduction

The rate of turbulent kinetic energy dissipation ɛ is an important parameter in turbulent mixing theory. Knowledge of the magnitude of ɛ allows the estimation of the rate of vertical mixing (Gregg et al. 1977b; Ivey and Imberger 1991; and Saggio and Imberger 2000, hereafter SI).

It is well known that the higher wavenumber part of the one-dimensional temperature gradient spectrum due to turbulence (the Batchelor spectrum) is a function of the dissipation ɛ and the dissipation of temperature variance χT. Stewart and Grant (1962) describe a graphical method for fitting a nondimensionalized Batchelor spectrum to temperature gradient data to obtain the dissipation ɛ. Dillon and Caldwell (1980) and Dillon (1982) used a nonlinear least squares method to fit the Batchelor spectrum to well-resolved temperature gradient spectra with high signal-to-noise levels. Some temperature gradient spectra are difficult to fit in this way because of instrument noise at the high wavenumber end of the spectra (Gregg et al. 1986), internal wave and finestructure “contamination” at the low-wavenumber end of the spectra (Gregg 1977a,b), and the lack of a clearly defined viscous convective subrange (Sherman and Davis 1995). Figure 1 depicts typical spectra with these additional spectral components. Note that although temperature gradient spectra may be corrected for sensor response, uncertainty in the response is a source of distortion in measured spectra (Gregg 1999).

In this paper, we present an algorithm for obtaining ɛ by fitting the theoretical Batchelor spectrum to the measured spectrum. Key issues in fitting the theoretical Batchelor curve that are discussed in this paper are 1) what is the appropriate portion of the measured spectrum to use, 2) how should the dissipation of temperature variance χT be determined from the measured spectrum, and 3) what is an appropriate error function to use in determining the optimum fit? This paper focuses on the first two of these items and describes a fitting algorithm that has been developed and refined over a period of several years. This algorithm has been successfully applied to temperature gradient spectra from a wide variety of turbulent forcing in the field (e.g., Luketina and Imberger 1989; Imberger and Ivey 1991; SI) and in the laboratory (e.g. Teoh et al. 1997; and De Silva et al. 1997).

The data discussed in this paper were generally collected at 100 Hz using an FP07 thermistor mounted on a probe rising or falling at a nominal velocity of 0.1 m s−1. However, the algorithm can be applied to a probe traveling in any direction and at velocities other than of 0.1 m s−1.

The paper is divided into six sections. In section 2 the features of temperature gradient spectra are discussed. Data requirements, including preprocessing of data, are covered in section 3. The method for obtaining the dissipation ɛ by fitting the theoretical Batchelor spectrum to the measured temperature gradient spectrum (hereafter referred to as the measured spectrum) is outlined in section 4. Results from the application of the fitting method to measured and artificially generated spectra are discussed in section 5. Conclusions are presented in section 6.

Finally, it should be noted that wavenumbers are expressed as both cycles per meter (cpm) and radians per meter in this paper. The symbols κ and k are consistently used for wavenumbers in units of cpm and rad m−1, respectively. Wavenumbers used in figures always have units of cpm.

2. Temperature gradient spectra

The typical temperature gradient spectrum shown in Fig. 1 has five different components as shown. These components are due to finestructure, internal waves, inertial convective turbulence (the inertial convective subrange), turbulence affected by viscosity and diffusivity (the Batchelor spectrum), and noise.

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
TToβzγzγβ
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 To is the temperature at the center of the region. The temperature gradient is thus given by
i1520-0426-18-1-100-e2

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 Sf(k) of (2) is given by Sf(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
i1520-0426-18-1-100-e3
where Γ(γ + 1) = 0xγex dx. Thus Sf(k) is given by
Sfkcfk−2γγ
where cf = 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 Sf(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 Sf(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)
i1520-0426-18-1-100-e5
where the subscript w denotes internal waves and, based on the data of Gregg et al. (1973), cw 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)
SICSkcICSχT−1/3k1/3
where cICS is an O(1) constant and χT is the dissipation of temperature variance due to turbulence and is defined for the case of isotropic turbulence by
i1520-0426-18-1-100-e7
where the overbar denotes an ensemble average, DT is the diffusivity of heat, and T′ is the temperature fluctuation. The spectrum given by (6) is valid to a wavenumber (cICS/cVC)3/2kk where the Kolmogorov wavenumber is kk = (ɛ/ν3)1/4 and cVC 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
i1520-0426-18-1-100-e8
where k has units of radians per meter, q is a universal constant, Batchelor wavenumber kB is defined as the inverse of the Batchelor length scale LB = (D2Tν/ɛ)1/4, and α is a dimensionless wavenumber given by (2q)1/2kk−1B. The normalized spectrum SN(α) is given by
i1520-0426-18-1-100-e9

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/DT > 1 is the viscous convective subrange and is given by (Batchelor 1959; Gibson and Schwartz 1963)
i1520-0426-18-1-100-e10
where cVC 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 CT = χT/2DTT2z, where Tz 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 31/2 < q < 2(31/2) 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(31/2) = 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(31/2).

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 kn.

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.

3. Data requirements

Before Batchelor curve fitting can be used to determine dissipation it is usually necessary that 1) the frozen field assumption be valid, 2) the temperature signal is corrected for the thermistor response, and 3) stationary data segments are selected. The frozen field assumption, or Taylor hypothesis (Taylor 1938), is used to convert the response of a thermistor from frequency ω to wavenumber k space using k = ω/u, where the frequency ω is measured in radians per second and wavenumbers given in radians per meter. The Taylor hypothesis is generally considered to be valid if variations in the fluctuating velocities are small compared with u, the probe velocity relative to the fluid (see Townsend 1976, p. 67). Ideally, the probe speed will be reasonably greater than the largest turbulent velocity fluctuations but not so great that sensor roll-off becomes a problem. Given that u ∼ (ɛl)1/3 where u and l are the rms velocity and overturn size of the turbulence (Luketina and Imberger 1989), and l is typically less than 0.1 m and ɛ around 10−5 m2 s−3 in an energetic environment, results in u ∼ 10−2 m s−1. Thus a nominal probe velocity of 0.1 m s−1 is a good compromise between satisfying the frozen field requirement and minimizing sensor roll-off. Higher probe velocities can be used in less energetic environments.

Because much of the spectral energy encountered by a thermistor is near the highest frequencies the thermistor can resolve, it is usually necessary to correct for the thermistor response. Generally, the response of a thermistor to changes in temperature is limited by the time for heat to diffuse through the boundary layer surrounding the sensor (Gregg and Meagher 1980). Response functions and correction methods for fast response thermistors have been published by Gregg and Meagher (1980), Dillon and Caldwell (1980), Fozdar et al. (1985), and Sherman and Davis (1995). Here the method of Fozdar et al. (1985) is used.

It is necessary to divide the data into statistically stationary segments or intervals if the measured spectrum D(k), or a portion of it, is to behave as a Batchelor spectrum (Imberger and Boashash 1986). An example of this is provided in section 4; arbitrarily dividing the data into short segments is not a guarantee that the turbulence is homogeneous within a segment.

Note that comments relating to measured spectra in subsequent sections of this paper refer to (wavenumber) band-averaged spectra. The number of spectral points ni averaged to form the ith point in the band-averaged spectrum is a geometric series with ni increasing from low to high wavenumbers. Thus confidence in the accuracy of the spectral estimate increases with increasing wavenumber (see Bendat and Piersol 1986, p. 285).

4. Method

When a theoretical Batchelor spectrum as given by (8) is fitted to a spectrum for which the variance has been estimated, the only free variable is the Batchelor wavenumber, which in turn, defines the dissipation. As mentioned previously, key issues in fitting the theoretical Batchelor curve are 1) what is the appropriate portion of the measured spectrum to use, 2) how should the variance χT be estimated from the measured spectrum, and 3) what is an appropriate error function to use in determining the optimum fit? These issues and other aspects of the method are discussed in this section.

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−1B, where κB = (2πLB)−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(31/2) and the nondimensional theoretical spectrum SN(α) as defined in (9) is approximated by
i1520-0426-18-1-100-e11
where
i1520-0426-18-1-100-e12
(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
π4νD2Tκ4B

b. Overview of method

Once a value of the variance χT is estimated, the theoretical spectrum S(κ) can be fitted to an appropriate portion of the measured spectrum D(κ) with the Batchelor wavenumber κB 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 κn and the variance χT being progressively refined (see the following section). The dissipation is then found by substituting the wavenumber κB, 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 κB such that ɛ ranges from 10−12 to 10−4 m2 s−3 (however, if κB exceeds κn, the noise wavenumber, then the upper limit for κB is taken to be κn). Of these 200 values, the wavenumber κB 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, κL and κn) and have fitted a theoretical spectrum to this part, we can estimate the temperature variance dissipation χT using
i1520-0426-18-1-100-e15
where the molecular diffusivity of heat DT 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 χT which, in turn, requires the limiting wavenumbers, κL and κn, to be estimated. We get around this problem by an iterative procedure.

Prior to the first iteration, κL is set to the lowest wavenumber of the measured spectra D(κ). For each subsequent set of trials, κL 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, κL does not change).

Prior to the first iteration, the noise wavenumber κn 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 κn may be too high. By comparing the best-fit spectrum from the first iteration with the measured spectrum it is possible to reevaluate whether κn 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 κn, κn 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 kn 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 kn is Batchelor shaped. Accordingly, different portions of the data between the lowest wavenumber of the measured spectrum and the noise wavenumber κn 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
i1520-0426-18-1-100-e16
where the peak of the theoretical spectrum is S(κpeak) and κpeak is equal to 0.176κB. 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 χT 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
λiDκiSκSi2κiκSi21/2
where S(κSi) is the point on the theoretical spectrum that lies on the −1 slope passing through D(κi) (see Fig. 4). The error function that we have found to be effective involves a normalization of the error given by (17) and is
i1520-0426-18-1-100-e18
where wi is a weight based on the number of raw spectral data points band averaged to produce D(κi), m is the number of data points (after band averaging) between wavenumbers κ1 and κ2, m − 1 is the number of degrees of freedom, fr is the fraction of the maximum possible number of data points used (i.e., all those at wavenumbers lower than κn), and c is a constant.

The use of (fr)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 wi are based on the standard deviation of ni points band averaged to a single point being reduced by a factor of (ni)−1/2. More specifically, the weights are calculated using wi = (ni)−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(κSi), or more specifically κSi. The method is illustrated in Fig. 4 and is presented in terms of the normalized wavenumber αSi = (2q)1/2κSi/κB. Figure 4 shows a data point αi, DN(αi) on the normalized spectrum through which passes a line LN(α) of slope −1. The normalized spectrum is given by
i1520-0426-18-1-100-e19
Also shown is the normalized theoretical spectrum SN(α) with the −1 tangent point αtan, SN(αtan) indicated. Note that αtan = 1.036. A parameter Δ is defined as
10SNα10LNα
We can now approximate the points αleft and αright for Δ > 0 using the following series expansion:
i1520-0426-18-1-100-e21
where ζ = lnΔ. The wavenumber αSi is then taken to be the closer of αleft and αright (in the example of Fig. 4, αSi = αleft) or, if Δ < 0 (i.e., the line and fitted curve do not intersect), αSi is set equal to αtan.

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.

5. Discussion

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 δe 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(ti) was calculated using
i1520-0426-18-1-100-e22
where θj 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 κj = (j/pκ, where Δκ = (2pΔtu) and Δt is the sampling interval of the time series. The magnitude of the component associated with wavenumber κj is
AjSκjκ1/2
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 DT, 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 χT 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 Lr = ) 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 μlogɛ and the standard deviation slogɛ. 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 μlogɛ in the form of a percentage error 100(logɛ − μlogɛ)/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 κB/κr < 3.2, where κr = L−1r. 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 κB/κ−1/3r > 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 ɛB and the volume averaged dissipation ɛvol 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 ɛvol and underestimated at high values of ɛvol. This occurs because the low ɛvol spectra are well resolved (i.e., the roll-off region is well defined) as in Fig. 10b, while the high ɛvol 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 Lr. 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.

6. Conclusions

A relatively robust method for determining dissipation from Batchelor curve fitting has been described. The method is relatively robust with regard to determining and making use of the Batchelor portion of a temperature gradient spectrum and determining the variance associated with that portion. Overall, the use of the method to determine dissipation results in considerable time savings when compared to manual methods. The systematic and random errors associated with determining the dissipation have been shown to be a function of the dissipation and the record length. Of course, these errors are the best case, as the presence of internal waves, finestructure, noise and calibration errors will result in increased errors.

Acknowledgments

We would like to thank Greg Ivey, Mike Barry, and the reviewers for providing constructive comments on this paper. Angelo Saggio provided the data shown and Carol Lam checked the coding of the algorithm. This work forms WRL Reference 198 and CWR Reference ED648DL.

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Fig. 1.
Fig. 1.

A schematic showing the various processes contributing to the temperature gradient spectrum as a function of cyclic wavenumber κ. (a) Contributing processes shown are internal waves (dashed line at low wavenumbers), finestructure (dashed and dotted line), the inertial convective subrange (dotted line at low wavenumbers), the Batchelor spectrum (thin line), and thermal/electronic noise (dashed line at high wavenumbers). (b) An example is shown of a spectrum where internal wave, finestructure, and ICS contributions are negligible. Such a spectrum may be found where there is energetic turbulence and weak stratification such as in a wind mixed layer. (c) A spectrum is sketched for the case where the finestructure contains a significant amount of energy at the lower wavenumbers. This type of spectrum may be found, for example, in a stratified thermocline. Note that the thick lines in (a), (b), and (c) represent the composite (i.e., measured) temperature gradient spectrum.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 2.
Fig. 2.

(a) Statically stable temperature profiles 1 m long with a 1°C change from top to bottom that satisfy Eq. (1) with γ = 0 (thin line), γ = 0.5 (thick line), and γ = 2 (dashed line), and (b) the corresponding spectra of the temperature gradients.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 3.
Fig. 3.

A schematic showing the various stages in fitting a Batchelor spectrum (dark line) to a measured spectrum (gray line) to determine the kinetic energy dissipation ɛ. The dashed vertical line indicates the noise wavenumber κn. (a) The potential range of the wavenumbers κ1 and κ2 are indicated for a Batchelor spectrum being fitted during the first iteration. Following the first iteration, the wavenumber κL is determined and the noise wavenumber κn modified as shown in (b). (c) The potential range of the wavenumbers κ1 and κ2 are indicated for a Batchelor spectrum being fitted during the second iteration. Following the second iteration, the wavenumber κL is modified as shown in (d). Subsequent iterations repeat the procedure shown in (c) and (d), resulting in the wavenumber κL being progressively modified.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 4.
Fig. 4.

Finding the deviation between a data point on the normalized measured spectrum and the normalized theoretical spectrum along a slope of −1.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 5.
Fig. 5.

Measured temperature gradient spectra D(κ) (solid line) and fitted theoretical Batchelor spectra S(κ) (dashed line) for data from a single profile in Lake Kinneret, Israel. The portion of the measured spectra that was used for fitting is between the two gray vertical lines. (a) The measured spectrum follows the theoretical form fairly well. (c) The curve fitting algorithm has ignored finescale/internal wave contamination at the low wavenumbers. In both cases the algorithm has successfully fitted the Batchelor spectrum despite the presence of high frequency noise. Note that the noise decreases at the very high wavenumbers due to the use of a brickwall filter on the time series prior to the spectra being calculated. The fitting error δe as a function of dissipation ɛ for cases (a) and (c) is shown in (b) and (d), respectively. In both cases there are clear minima in the error function.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 6.
Fig. 6.

(a) The measured spectrum does not follow the theoretical form very well. Consequently, the corresponding error function shown in (b) is flatter in the region of the minima, and the values of the minima are larger than that for the more Batchelor shaped measured spectra of Fig. 5. (c) There is no discernable Batchelor shape. As a result, the the error function shown in (d) is lacking the smoother shape shown in error graphs of Fig. 5. Further, the minimum error is relatively high. See Fig. 5 for explanation of the different lines and symbols.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 7.
Fig. 7.

Band-averaged spectra corresponding to (a) artificially generated dataset 2048 samples long, the first half of the dataset has ɛ = 10−4 m2 s−3, while the second half of the dataset has ɛ = 10−5 m2 s−3; (b) is similar except that ɛ = 10−10 m2 s−3 and 10−11 m2 s−3 for the first and second parts of the dataset. The different lines and symbols are explained in Fig. 5.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 8.
Fig. 8.

Systematic error in estimates of log dissipation as a function of (a) Batchelor wavenumber κB (based on actual dissipation) and record wavenumber κr, which is the reciprocal of the record length and, (b) actual dissipation and record length Lr.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 9.
Fig. 9.

Absolute value of the random error (standard deviation) in estimates of log dissipation as a function of (a) Batchelor wavenumber κB (based on actual dissipation); record wavenumber κr, which is the reciprocal of the record length; and (b) actual dissipation and record length Lr.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 10.
Fig. 10.

Band-averaged spectra corresponding to (a) artificially generated dataset 2048 samples long (solid line), the first half of the dataset (line with short dashes), which has ɛ = 10−4 m2 s−3 and the second half of the dataset (line with long dashes), which ɛ = 10−5 m2 s−3; (b) is similar except that ɛ = 10−10 m2 s−3 (line with short dashes) and ɛ = 10−11 m2 s−3 (line with long dashes).

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 11.
Fig. 11.

Ratio of estimated dissipation ɛB to actual (volume averaged) ɛvol dissipation as a function of ɛvol when the data are nonstationary.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

Fig. 12.
Fig. 12.

Segmentation of a nonstationary dataset: (a) an artificially generated temperature gradient dataset 2048 samples long where the upper and lower halves have ɛ = 10−6 m2 s−3 and ɛ = 10−7 m2 s−3, respectively; and (b) the corresponding segmentation parameter calculated using the method of Imberger and Ivey (1991). The dashed line in (b) shows an arbitrary theshold of 100, which would result in a segmentation boundary being placed at the transition from ɛ = 10−6 to 10−7 m2 s−3.

Citation: Journal of Atmospheric and Oceanic Technology 18, 1; 10.1175/1520-0426(2001)018<0100:DTKEDF>2.0.CO;2

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