Let us consider a signal u1(t) resolved in a certain range of frequencies f0<f<f1. Converting the wavenumber spectrum to the frequency spectrum, we obtain from Eq. () the following relation for the number of signal crossings per unit time u1′2N12=4∫f0f1f2S(f)df. Similarly as in the velocity variance method described in Sect. , let us now filter the signal using a band-pass filter characterized by a different cutoff frequency f2<f1. In such a case we obtain a different signal u2(t) with a reduced number of zero crossings N2<N1: u2′2N22=4∫f0f2f2S(f)df. If we subtract Eq. () from Eq. () we obtain u1′2N12-u2′2N22=4∫f2f1f2S(f)df. In the inertial range, S(f) is described by Eq. (); hence, if both f1 and f2 belong to the inertial range, u1′2N12-u2′2N22=4C1U2π2/3ϵ2/3∫f2f1f1/3df=3C1U2π2/3ϵ2/3f14/3-f24/3. If we proceed further and filter the signal using a series of cutoff frequencies fi<f2, we can estimate ϵ from Eq. () using a linear-least-squares fitting method.

In the above derivation we assumed that the applied filter is rectangular in the frequency space. The issue of frequency response characteristics of a filter will be discussed further in Sect. .

In this method we attempt to account for the impact of the missing part of the dissipation spectrum by introducing a correcting factor to the number of zero crossings per unit length NL. The number of crossings per unit length is calculated from the measured signal where the fine-scale fluctuations having the highest wavenumber kcut will be denoted by Ncut and the variance of this signal will be denoted by ucut′2. From Eq. () it follows that Ncut is related to NL by the formula u′2NL2=ucut′2Ncut2∫0∞k12E11dk1∫0kcutk12E11dk1=ucut′2Ncut21+∫kcut∞k12E11dk1∫0kcutk12E11dk1. We then assume a certain form of the energy spectrum, Eq. (). For simplicity we take fL=1, i.e., we neglect the contribution of the largest scales to the value of the dissipation rate based on zero crossings and we consider two different forms of fη, as proposed in . The first is the simple exponential form fη=e-βkη, with β=2.1. The second is the more complex formula fη=e-βkη4+βcη41/4+βcη, where β=5.2 and cη=0.4. With this, the energy spectrum reads E(k)=Cϵ2/3k-5/3fη(βkη), where C=1.5. The integral from 0 to ∞ of the dissipation spectrum 2νk2E(k) should be equal to ϵ, which results in β=2.1 in Eq. () and provides a relation between β and cη in Eq. (). The latter case, due to the additional degree of freedom in fη fits the experimental data better in the dissipative range .

The corresponding one-dimensional spectrum E11 can be calculated from Eq. (): E11(k1)=Cϵ2/3∫k1∞k-8/3fη(βkη)1-k12k2dk. Next we change the variables in the integral Eq. () to ξ=βkη, introduce Eq. () into Eq. () and once again change the variables to ξ1=βk1η. As a result we obtain u′2NL2≈ucut′2Ncut21+∫kcutβη∞ξ12∫ξ1∞ξ-8/3fη(ξ)1-ξ12ξ2dξdξ1∫0kcutβηξ12∫ξ1∞ξ-8/3fη(ξ)1-ξ12ξ2dξdξ1︸CF=ucut′2Ncut2CF, here CF is the correcting factor.

The value of ϵ can be calculated numerically using an iterative procedure.

As a starting point for this procedure, a first guess for the Kolmogorov length η=(ν3/ϵ)1/4 should be given. With this, we calculate the correcting factor CF from Eq. () taking either the form of Eq. () or () for fη. Next, from Eq. () the value of dissipation can be estimated as ϵ=15π2νucut′2Ncut2CF. We start the next iteration by calculating again the Kolmogorov length η=(ν3/ϵ)1/4, the corrected value of CF from Eq. () and the new value of ϵ from Eq. (). After several iterations the procedure converges to the final values of the dissipation rate and Kolmogorov's length η with an error defined by a prescribed norm Δη=|ηn+1-ηn|≤dη. The successive steps are summarized in the form of Algorithm 1.

It should be noted that in this approach we do not have the empirical inertial range constant C, and we calculate the dissipation rate directly from the formula with viscosity, Eq. (), as in the original zero-crossing method (see Eq.  and ).

In order to present the more detailed properties of the procedure, we used the velocity signal from one of the horizontal flight segments that took place within the turbulent atmospheric boundary layer. This segment was a part of flight 13 of the POST airborne research campaign . The data were provided in the east, north, up (ENU) coordinate system. For further study we have calculated time series of the longitudinal velocity component along the track. The signals' sampling frequency was fs=40Hz and the duration was T=438.75s. The magnitude of the vector difference between the aircraft velocity and the wind velocity U, averaged over the track vector, was about 55ms-1 and the standard deviation u′=0.28ms-1.

We have estimated the dissipation rate based on the number of zero crossings, according to the methods outlined in Sect. . The average dissipation rate calculated from the frequency spectrum and the structure function for the whole flight fragment Eqs. () and () was close to equal, with ϵPSD=2.48×10-4 and ϵSF=2.52×10-4m2s-3, respectively. These values were obtained from the linear-least-squares fit procedure in the range f= 0.3–5 Hz for the frequency spectrum and r= 11–183 m for the structure function (see Fig. ).

Before applying the threshold-crossing procedures the signal had to be filtered in order to eliminate errors due to large-scale tendencies as well as small-scale measurements noise. For this purpose we used the sixth-order low-pass Butterworth filter implemented in Matlab^®. Figure  presents the velocity signal over t=50s before filtering (top graph) and the same signal after filtering with fcut=5 and fcut=1Hz.

The original formula of was derived for the case when both the signal and its derivative have Gaussian probability density functions (PDFs) and are statistically independent. In general, such assumptions do not hold for turbulent signals. However, as discussed by , experimental observations and further theoretical studies indicate that the formula of has a more general applicability than for what it was mathematically proven for originally and is satisfied with a fair accuracy even for the case of strongly non-Gaussian signals. Figure a presents PDFs of the normalized original signal and the filtered signals compared with the normalized Gaussian distribution. As it is seen, filtering does not lead to significant changes in the investigated PDFs.

It is worth noting that the spectra (f2S(f), Fig. b) display a peak at f=10 Hz. This phenomenon has been indicated in the previous analyses of POST and appears due to measurement errors. However, as the highest cutoff frequencies used in the present study are 5 Hz, it should not affect our results.

In order to use the method based on successive signal filtering we filtered the signal with different values of fcut in the range fcut=0.1-19Hz. For each fcut=fi we calculated the number of zero crossings Ni based on the filtered signal. The zero-crossing event was detected when the product of two consecutive values of velocity fluctuation is less than zero: v(t)v(t+Δt)<0, here Δt=1/fs=0.025s. In order to estimate the value of dissipation rate we used Eq. (). The values ui′2 were calculated from filtered time series. Results for f1=0.3Hz and fi in the range (0.3,5)Hz are presented in Fig. .

Using Eq. () we have used linear fitting of the differences ui′2Ni2-u1′2N12 against fi4/3-f14/3. The resulting value for the analyzed flight section was ϵNCF=2.54×10-4m2s-3, where the subscript NCF denotes variables estimated from the number-of-crossings method with successive filtering of a signal. This value is comparable with the estimations performed using classic methods based on the power spectra and structure functions.

Even if the local isotropy assumption of is satisfied with a good accuracy, the TKE dissipation rate estimates are subject to errors that can result from a finite sampling frequency of a signal, a finite time window, sensor bias and noise. The last of those three causes was investigated in , where it was shown that both the variance of the noise 〈n2〉 and the variance of its derivative 〈n˙2〉 influence the measured number of crossings. A possible remedy was proposed by , who suggested to use the threshold crossings, i.e., counting the number of times a signal crosses a given threshold T≠0, instead of the zero crossings in the case of signals with low signal-to-noise ratios. As for the signal considered in the previous section the signal-to-noise ratio becomes significant at higher frequencies (above 5 Hz), see Fig. , which are removed by the low-pass filter used in the proposed number-of-crossings method. We also applied the method of ; however, as it did not lead to any systematic change in our estimates, we further present results for the zero crossings only.

In order to quantify the errors resulting from the finite sampling frequency and finite time window and to test the performance of the proposed method, we performed the simulation analysis . Results are compared with the standard spectral retrieval estimates without any additional corrections. However, we note in passing that spectral methods can be improved to account for the bias errors. The example is the maximum likelihood approach where, instead of the von Kármán model, a model power spectral density Sfmodel is used, which takes into account the procedure for generating the empirical spectrum from discrete time series of finite length. Analogous approaches could also be formulated for the methods based on the number of crossings, which is a perspective for a further study.

To test the performance of the new proposals we generated a number of artificial velocity signals with frequency spectra and two point correlation functions prescribed by the model. The equations resulting from the application of this model to the one-sided spectra considered in this paper are written below. R11(r1e1)≈0.592548u′2rL01/3K1/3rL0,S(f)≈0.4754482πUu′2L01+L022πfU25/6, where K1/3 is the modified Bessel function of order 1/3. Coefficients of the Fourier series expansion of velocity signal were calculated as wj=Wj(a+ib) where i=-1, a and b are random numbers from the standard Gaussian distribution with zero mean and unit variance, and Wj=S(fj)Δf, j=1,…,N. Alternatively, the coefficients Wj can be calculated as the discrete Fourier transform of R11, as described in . The artificial velocity signal is finally constructed as the discrete inverse Fourier transform of wj (see ).

We used artificial signals with U=55ms-1 and the standard deviation u′=0.28ms-1. Those characteristics correspond to the ones of the signal considered in the previous Sect. . We set L0=83.9 m in Eq. () to also obtain a comparable dissipation rate estimate ϵ=2.5×10-4m2s-3. Our first aim was to test how a finite sampling rate influences the number of crossings. For this purpose in each run we created an artificial signal of length N=217 points and with the sampling frequency 200Hz (5 times larger than the sampling of the signal considered in Sect. ), which resulted in signal duration t≈650s. We treated this velocity series as a reference. Next, we took every fifth sample of this signal to create a 40Hz velocity time series. We then calculated the number of crossings, as described in Sect. , and the power spectral density. We repeated the procedure 500 times and calculated the average of the obtained profiles (see Fig. ). Due to the finite sampling frequency we observe the effect of aliasing – spectral densities for f higher than the Nyquist frequency are added to the spectral densities at f<20 Hz. Distortions are visible for higher frequencies both in the power spectrum, Fig. a, and the Ni2ui2 profiles, Fig. b. We estimated the TKE dissipation rate from the averaged profiles using the method described in Sect. , Eq. (), keeping the lower bound of the fitting range f1=0.3Hz constant and changing the upper bound f2 from 1 to 19Hz. Results are presented in Fig.  and compared with the corresponding ϵPSD values, using the von Kármán model as the reference model spectrum.

We observe an increase in ϵPSD estimates with increasing f2 and a moderate increase in ϵNCF over the input ϵ=2.5×10-4m2s-3, which shows that the number of crossings respond to finite sampling effects differently than the power spectrum. We note here that ϵNCF values calculated from the averaged profiles of the 200Hz reference signal (black line in Fig. ) seem to be slightly overpredicted in comparison to the input ϵ, especially for smaller f2. A possible reason is the response of the filter used in the number-of-crossings method. We attempted to estimate this error using relation () between the number of crossings and the dissipation spectrum. We first integrated f2Sf of an unfiltered signal from f0 to fi. Next, we calculated a spectrum Sffiltered of a band-pass-filtered signal taking f1 and fi as the lower and upper bounds of a filter. We integrated f2Sffiltered over the whole available range of f. The difference between the two integrals should represent a correction due to filter response. The ϵ values estimated from the corrected Ni2ui′2 are presented in Fig.  as a black dot-dashed line. As it is seen that the estimations for the lower f2 are improved as f2 increases, ϵNCF seems to be underpredicted. A possible reason for this might be that the filter influences the number-of-crossings statistics somewhat differently than the spectrum alone.

Next, we tested the influence of the finite temporal window on the calculated statistics. We generated 1000 artificial signals, each time changing slightly the u′ value in Eq. (), which led to a change in input ϵ (see ); the value of L0 remained unchanged. For each signal we estimated ϵPSD from the standard power spectral density using the Welch overlapped segment averaging estimator implemented in Matlab^® with a 28 window and ϵNCF from the number of crossings, Eq. (). We performed these tests for the 40Hz signals and the fitting range 0.3–5 Hz.

We first decreased the time window, taking each time only 1/8 of the created artificial signal for the analysis, which, in terms of L0 from Eq. (), resulted in the signal length L≈50L0. Results of ϵPSD and ϵNCF estimates as functions of the corresponding input ϵ from the theoretical profile Eq. () are presented in Fig.  (upper plots). It can be seen that the bias error is larger for ϵPSD; however, the scatter of ϵNCF is larger. The linear fits and the correlation coefficients are ϵPSD=0.9104ϵ-2.32×10-5,r=0.9898,ϵNCF=0.9878ϵ+6.80×10-5,r=0.9343. We repeated the simulation analysis for signals with 217 points, i.e., with L≈400L0, obtaining ϵPSD=1.0377ϵ+4.56×10-6,r=0.9898,ϵNCF=1.0379ϵ+2.25×10-5,r=0.9989. Hence, for the signal length comparable to the lengths from the POST campaign we can expect a small underprediction of ϵPSD estimates due to bias error and some overprediction due to aliasing (see Fig. ). Both result in a small overprediction of ϵPSD (Fig. , left column, lower plot). As far as ϵNCF is concerned, the simulation analysis shows that it is less sensitive to the bias error (Fig. , right column); however, it has a larger scatter than ϵPSD, at least for the generated artificial velocity fields. Results for the 40Hz signal are slightly overpredicted (Fig. , right column, lower plot) due to aliasing and the fact that the number-of-crossings method gives somewhat larger ϵ estimates in this fitting range (see Fig. ).

Finally, we would like to address the issue of larger scatter observed for ϵNCF. It follows from our study that the scatter in ϵNCF depends on the value of filter cutoffs in the fitting range. In the final test, we set u′=0.28ms-1 and input epsilon ϵ=2.5×10-4m2s-3 and repeated the simulation 500 times for consecutively, 1/512, 1/265, 1/128, 1/65 of the original signal of 217 points. This corresponds to approximately 1L0, 2L0, 4L0, 6L0 for L0 in Eq. (). We band-pass filtered the signal and consider small fitting range of 16–18 Hz. We normalized the results obtained by the input ϵ and calculated their standard deviations. Results presented in Fig.  show that at least for this case the standard deviations of ϵNCF+ is comparable with the standard deviation of ϵPSD+

The measurement signal used in Sect.  was also analyzed using the second method proposed in Sect. , Eqs. () and (). We consider both formulas for the function fη, Eqs. () and (). The advantage of the simpler, exponential formula () is that the one-dimensional spectrum function E11(k1), Eq. (), can be written in terms of the incomplete Γ function as follows E11(k1)=Cϵ2/3βη5/3Γ(-5/3,k1βη)-βη2k12Γ(-11/3,k1βη), where Γ(a,x)=∫x∞e-tta-1dt. The correcting factor () in terms of the Γ functions reads CF=1+∫kcutβη∞ξ12Γ(-5/3,ξ1)-ξ12Γ(-11/3,ξ1)dξ1∫0kcutβηξ12Γ(-5/3,ξ1)-ξ12Γ(-11/3,ξ1)dξ1. If Eq. () is used as a model for fη, both integrals in Eq. () must be calculated numerically. On the other hand, as discussed in , Eq. () provides a better fit of experimental data in the dissipative range.

With such preparation we applied the iterative procedure, as described in Sect. . In the POST experiment the effective cutoff frequency was estimated at fcut=5Hz, which corresponds to kcut=(2πf)/U=0.57m-1. Using the sixth-order Butterworth filter, this resulted in u′2Ncut2=0.0000719⋅ 1 s-2 for this signal. Accordingly we used Algorithm 1 with ν=1.5×10-5m2s-1 and dη=10-6m. We approximated the integrals in Eq. () using the trapezoid rule. The results of successive approximations of CF and ϵ converge fast to a fixed value, independently of the initial guess of ϵ=ϵ0 (Fig. a). The increment dk1 in Eq. () was approximated by Δk1=5⋅10-6m-1. For such choice we obtained ϵNCR=2.61×10-4m2s-3, where the subscript NCR denotes variables calculated with the number-of-crossings method based on the recovered part of the spectrum. We used this as a reference value. In order to estimate the numerical accuracy of the proposed algorithm, we calculated the error Δϵ=|ϵ-ϵNCR| for different values of Δk1 (see Fig. b). We obtain Δϵ∼Δk11.3.

Next we considered Eq. () as a model for fη and calculated the double integral in Eq. () using the trapezoid rule. We obtained the corresponding value ϵNCR=2.58×10-4m2s-3, which is very close to the estimate from the simple exponential form Eq. () and ().

It is worth noting that the proposed method is accounting for a dominant (and not directly measured) part of the spectrum based on the theoretical knowledge about its shape. This knowledge is simply reduced to the form of the correcting factor CF, Eq. (), which contains the integral of k12E11(k1). Fig.  illustrates the relation between the measured and the estimated part of the spectrum for the analyzed case with both forms of the function fη, Eqs. () and (). The spectral cutoff of the data considered here (5Hz) is in the inertial range, where k12E11(k1) with both forms of fη functions is almost indistinguishable (see Fig. ). At the same time integrals of the remaining (recovered) parts of k12E11(k1) are almost equal, as independently of the choice of fη both dissipative spectra 2νk2E(k) must integrate to ϵ. As a result, for the given spectral cutoff, ϵNCR estimates with the simple exponential Eq. () and () forms of fη are very close. This might change for larger cutoff frequencies. We expect that, in the case where the cutoff frequency is placed in a region influenced by the form of the fη function, the spectrum with Eq. () will provide better estimates of the TKE dissipation rate.

The result of application of this method ϵNCR=2.58×10-4m2s-3 with fη described by Eq. () and ϵNCR=2.61×10-4m2s-3 with fη from Eq. () is comparable with the dissipation rates obtained using other methods, as discussed in Sect. , ϵPSD=2.48×10-4m2s-3, ϵSF=2.52×10-4 and ϵNCF=2.54×10-4m2s-3. The relative differences between those estimations are less than 5%.

We finally checked the estimates of the second method using synthetic signals as described in Sect. . For the cutoff fcut=5Hz, we generated 500 artificial signals of length L≈400L0 and with input ϵ=2.5×10-4m2s-3. We obtained the mean 〈ϵNCR〉=2.55×10-4m2s-3 and a standard deviation equal to 9% of the input ϵ value.