# Persistence analysis of velocity and temperature fluctuations in convective surface layer turbulence

## Summary (4 min read)

### II. DATASET AND METHODOLOGY

- The dataset being used is from the Surface Layer Turbulence and Environmental Science Test experiment.
- Note that the authors rotated the coordinate systems of all nine sonic anemometers in the streamwise direction by applying the double-rotation method of Kaimal and Finnigan33 for each 30-min period.
- The associated probability that the u′ signal stays positive or negative for tp amount of time can be evaluated by constructing the persistence PDFs using standard statistical procedures .

### III. RESULTS AND DISCUSSION

- Before describing the features of the persistence PDFs, it is worthwhile to discuss about the presentation of the results.
- In the convective surface layer turbulence, it is a common practice to normalize the spatial length scales in the streamwise direction by the height above the surface.
- Similar to Bershadskii et al.18 and Chamecki,30 the persistence PDFs are computed separately for the positive and negative fluctuations (shown as blue and red markers in Fig. 2) for u′, w′, and T′ signals and compared with the persistence PDFs for the total fluctuations (combining both positive and negative) shown as gray markers.
- It could be noted that for the highly convective stability class [−ζ > 2, Fig. 2(a)], the persistence PDFs show a slight disparity between the positive and negative However, this difference gradually disappears with the change in stability from highly convective to near neutral [Figs. 2(a)–2(f)].

### 1. Linkage between the persistence PDFs and asymmetric distribution

- From phenomenological arguments, the authors will show that such a form of the persistence PDF is equivalent to a timefraction distribution associated with (tpu)/z by directly connecting the premultiplied persistence PDF with the PDF of the corresponding signal itself.
- The authors emphasize that such a connection is not possible to establish from the persistence PDFs alone (shown in Fig. 2) without considering its premultiplied form.

### 2. Premultiplied or logarithmic persistence PDFs of velocity and temperature fluctuations

- Figure 3 shows the logarithmic PDFs of (tpu)/z for the same six different stability classes shown in Fig.
- These PDFs are computed after taking the logarithm of (tpu)/z in base 10 and dividing the fraction of samples by the logarithmic bin-width d log10[(tpu)/z].
- This clear disparity between the premultiplied PDFs of positive and negative.
- This is congruous with the close to Gaussian characteristics of the T′ signal in the near-neutral stability.

### B. Scrutiny of persistence PDFs through surrogate data

- The persistence PDFs are related to the inter-arrival time between the successive zero-crossings of a time series.
- If these zerocrossings were randomly located being independent of each other, the authors would have expected the persistence PDFs to follow a Poisson distribution, which is exponential in nature.
- 1,67 However, in a convective surface layer, the persistence PDFs of the turbulent velocity and temperature fluctuations show a power-law structure.
- Published under license by AIP Publishing 68,69.
- Since the Fourier amplitudes of the surrogate dataset are kept intact during the process of Fourier phase-randomization (PR), this procedure preserves the Fourier spectrum and hence the auto-correlation function of the time series.

### 1. Phase-randomization and randomization experiments

- For their purpose, the authors performed the phase-randomization and randomization experiments by varying the strength of the randomization to investigate their gradual effects on the behavior of the persistence PDFs.
- Figure 4 shows the typical results from these two experiments for the highly convective stability class.
- This is related to the fact that in phase-randomization experiments, the surrogate time series of temperature fluctuations approach an almost Gaussian distribution at even 20% randomization strength [see Fig. S4(a) of the supplementary material].
- On the contrary, from Fig. 4(b), the authors note that the power-law structure of the T′ persistence PDFs gradually disappears as the strength of the randomization is increased to 100%.
- In a turbulent time series, the temporal coherence can be described by the integral time scale, defined as the time up to which the signal remains auto-correlated with itself.

### 2. Auto-correlation functions and integral scales

- Figure 5 shows the auto-correlation functions [Rxx(τ), where x can be u′, w′, or T′ signals and τ is the time lag] of u′, w′, and T′ signals, plotted against the lags for the six different stability classes.
- The auto-correlation functions are plotted against the normalized time lags (τu/z) for u′ (red circles), w′ (blue squares), and T′ (pink inverted triangles) signals for the six different stability classes.
- As the stability changes from highly convective to near neutral, the coefficients K of T′ signals become closer to u′ signals, implying that the ΛT values approach Λu [Figs. 5(a)–5(f)].
- Therefore, it indicates that the power-law behavior of the persistence PDFs is connected to the eddies from the inertial subrange of the turbulence spectra (sizes smaller than the integral scales).

### C. Scaling the persistence time by the integral scales

- The authors begin with discussing the logarithmic persistence PDFs, since in that representation, the disparity between the PDFs corresponding to the positive and negative fluctuations at larger Phys.
- Published under license by AIP Publishing persistence scales is highlighted more clearly (see Fig. 3 in Sec. III A).
- The persistence time tp of u′, w′, or T′ signals are scaled with the integral time scale (Γ) before computing the logarithmic persistence PDFs.
- Note that from the application of Taylor’s frozen turbulence hypothesis, this is equivalent to scaling the persistence length with integral scales Λ.

### 1. Logarithmic persistence PDFs

- Figure 6 shows the logarithmic persistence PDFs with the persistence times normalized by Γ corresponding to u′, w′, or T′ signals for all six stability classes (Table I).
- This discrepancy gradually disappears in near-neutral stability [see Figs. 6(a)–6(f)], implying that the non-Gaussian characteristics of the T′ signal are definitely related to the energy containing scales of motions.
- The exponential decay of the CDFs, F(tp/Γ)∝ exp[−λ(tp/Γ)], (13) in such plots would appear as a straight line with the slope of λ.
- This indicates that the mean time scales of the long persistent events of negative temperature fluctuations become closer to the integral scale of temperature as the near-neutral stability is approached, while the same remains unchanged for the positive fluctuation patterns.
- This explains the closeness of the integral scales of w′ and T′ in highly convective stability [Fig. 5(a)].

### 2. Persistence PDFs

- So far in Fig. 6, the authors have focused on the characteristics of the long persistent events larger than the integral scales (associated with energy containing eddies) by investigating the logarithmic representation of the persistence PDFs.
- Since the power-law behavior is best represented in the original PDFs, Fig. 7 shows the persistence PDFs of u′, w′, and T′ signals with tp scaled with Γ. From Fig. 7, one can note the cut-off scale, where the deviation from the power-law behavior begins, is located almost exactly at the integral scale Γ for all three signals.
- Therefore, in Fig. 7(a) (highly convective stability), the exponents of the power-law functions are determined by performing a linear regression for the range 0.01 ≤ tp/Γ ≤ 1 on the log–log plots.
- Note that the exponent 1.4 for the T′ signal is very close to 1.37 as reported by Bershadskii et al.18 from their turbulent convection experiments.
- For the w′ signal, the extent of the power-law behavior gradually decreases with stability.

### 3. Physical explanation of persistence exponents

- In general, the exponents of the power-laws in persistence PDFs are non-trivial and difficult to compute analytically, except for simple systems such as fractional Brownian motions.
- From Eq. (15), one can infer that the difference in the power-law exponents (γ) for u′, w′, and T′ signals is directly related to the different values of the intermittency exponents (μ).
- Katul, Parlange, and Chu83 and Katul et al.84 demonstrated that in a convective surface layer, the temperature fluctuations are more intermittent compared to the velocity fluctuations, given the presence of sharp drops associated with the cliffs of the ramp-cliff patterns.
- The reason for this difference is not clear at present.
- Nonetheless, at present, both of these frameworks based on SOC and fractals seem plausible to connect these power-law exponents with the physical nature of small-scale turbulence, but further research is required to assess their viability.

### IV. CONCLUSION

- The authors report the statistical scaling properties of the persistence PDFs of turbulent fluctuations in streamwise and vertical velocity (u′ and w′) and temperature (T′) from the SLTEST experimental dataset in a convective surface layer.
- On the other hand, for the long positive T′ events, the mean time scales remain roughly equal to the integral scales, irrespective of stability.
- T′ events is interpreted to be associated with the change in the topology of the coherent structures from cellular convection patterns in highly convective conditions to horizontal streaks in near-neutral stability.
- Subsequently, by scaling the persistence time scales with the integral scales, this statistical property of the persistence PDFs has been associated with the turbulent structures in a convective flow.
- All the authors contributed equally to this work.

Did you find this useful? Give us your feedback

##### Citations

8 citations

7 citations

7 citations

### Cites background or methods from "Persistence analysis of velocity an..."

...Note that, in the parlance of non-equilibrium statistical mechanics, the distribution of zero-crossing time intervals in a stochastic signal is equivalent to the persistence PDFs, where persistence is the probability P (t) that the signal does not change its sign up to the time t (Majumdar, 1999; Chowdhuri et al., 2020a)....

[...]

...Note that, the persistence PDFs are computed via logarithmic binning and subsequent transformation to the linear space using a change of variable, as illustrated by Chowdhuri et al. (2020a)....

[...]

...Since the measurements from the near-neutral stability class belong to the lowest three SLTEST levels, the shrinkage in the power-law regime is related to insufficient sampling of the small scale eddies at 20-Hz sampling frequency (Chowdhuri et al., 2020a)....

[...]

...Since Λw is of the same order as z (Chowdhuri et al., 2020a), such discrepancy reflects distinct attributes of turbulent transport associated with the detached and attached eddies, characterized by length scales smaller and larger than z (Marusic and Monty, 2019)....

[...]

...Besides, Chowdhuri et al. (2020a) reported the respective power-law exponents to be equal to −1.6, −1.25, and −1.4 for the u′, w′, and T ′ signals....

[...]

5 citations

4 citations

##### References

8,276 citations

6,486 citations

### "Persistence analysis of velocity an..." refers background in this paper

...Recently, to explain these power-law exponents, Cava and Katul (2009) and Cava et al. (2012) have proposed an ambitious connection with the self-organized criticality (SOC) observed in the sandpile model of Bak et al. (1987, 1988)....

[...]

5,806 citations

### "Persistence analysis of velocity an..." refers background in this paper

...…ph ys ic s. fl udy n] 1 1 M In turbulent flows, the interest in the concept of persistence or zero-crossings grew with the analytical result from Rice (1945), through which it was possible to show that the frequency of the zero-crossings in a turbulent signal was related to the Taylor…...

[...]

4,734 citations

### "Persistence analysis of velocity an..." refers background or methods in this paper

...…the PDFs of such variables is to take the logarithmic transformation and then binning the transformed variables in the logarithmic space (Christensen and Moloney, 2005; Newman, 2005; Pueyo, 2006; Sims et al., 2007; Benhamou, 2007; White et al., 2008; Dorval, 2011; Newberry and Savage, 2019)....

[...]

...…of the avalanches associated with sandpile models (Christensen and Moloney, 2005), identifying the lévy flight patterns in animal displacements (Benhamou, 2007; Sims et al., 2007), and in many other practical cases (see Newman (2005) and the references therein for a detailed review on this topic)....

[...]

...Additionally, the CDFs also have an inherent advantage of being bin-independent with a smooth convergence towards 1 (Newman, 2005; White et al., 2008)....

[...]

...…to the hypothesis of Yee et al. (1995) and Cava et al. (2012) where they connected this power law behaviour in the persistence PDFs with self-similar Richardson cascading mechanism, given the implied scale invariance associated with power-law distributions (Newman, 2005; Verma et al., 2006)....

[...]

...(2012) where they connected this power law behaviour in the persistence PDFs with self-similar Richardson cascading mechanism, given the implied scale invariance associated with power-law distributions (Newman, 2005; Verma et al., 2006)....

[...]

3,828 citations

### "Persistence analysis of velocity an..." refers background in this paper

...Recently, to explain these power-law exponents, Cava and Katul (2009) and Cava et al. (2012) have proposed an ambitious connection with the self-organized criticality (SOC) observed in the sandpile model of Bak et al. (1987, 1988)....

[...]

##### Related Papers (5)

##### Frequently Asked Questions (2)

###### Q2. What have the authors stated for future works in "Persistence analysis of velocity and temperature fluctuations in convective surface layer turbulence" ?

Since the persistence PDFs are related to the time spent between the zero-crossings in a signal, the future research questions are as follows: 1. How the persistence PDFs of the heat and momentum fluxes are related to the persistence PDFs of the velocity and temperature fluctuations ? 2. How much of the flux variation can be described by the persistence properties of the component signals ?