# Comparison of Different Techniques to Calculate Properties of Atmospheric Turbulence from Low-Resolution Data

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## Abstract

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## 1. Introduction

## 2. Description of Methods

#### 2.1. Characteristic Scales of Turbulence

#### 2.2. Estimation of EDR from 1D Intersections of the Turbulent Velocity Field

#### 2.3. External Intermittency

## 3. Error Analysis of EDR Estimates

## 4. EDR Retrieval from Artificial Signals

#### 4.1. Different Averaging Windows

#### 4.2. Different Averaging Windows and Different Sampling Frequencies

#### 4.2.1. Power Spectra

#### 4.2.2. Structure Functions

#### 4.2.3. Number of Zero-Crossings

#### 4.2.4. Iterative Method

#### 4.3. Deviations from the Kolmogorov’s Scaling

## 5. EDR Retrieval from POST Signals

## 6. Intermittency in Atmospheric Turbulence

## 7. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

EDR | energy dissipation rate |

PS | power spectra |

SF | structure function |

NCF | number of crossings scaling in the inertial range |

VAR | variance of velocity derivative |

NCR | number of crossings with spectrum-reconstruction |

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**Figure 1.**

**Left**: Exemplary frequency spectra of the transverse velocity component from experimental POST data [15],

**Right**: Exemplary second-order structure function. Vertical lines indicate bounds of the fitting range, magenta dashed line is a curve-fit within this range.

**Figure 2.**

**Left**: Exemplary number of zero-crossing scaling calculated from experimental POST data, for cut-off within the inertial range, see Equation (9), dashed magenta line indicates the best fit-line.

**Right**: Power spectra of a signal with spectral cut-off, magenta line indicates reconstructed part of the spectrum.

**Figure 3.**Top plots: Part of an artificial signal, middle plots: EDR estimates from the power spectrum ${\u03f5}_{PS}$, Equation (6), structure function ${\u03f5}_{SF}$, Equation (7) and number of crossings scaling in the inertial range ${\u03f5}_{NCF}$, Equation (9), bottom plots: EDR estimates from the iterative method based on the velocity variance ${\u03f5}_{VAR}$, Equation (10) and the number of zero-crossings per unit length, Equation (12). Left column: averaging window ${\mathrm{T}}_{ch}/4$, right column: averaging window ${\mathrm{T}}_{ch}$.

**Figure 4.**Statistics of EDR estimates for the fitting range $f=[10\xf720]$ Hz, or ${f}_{cut}=20$ Hz from artificial signals with ${f}_{s}=200$ Hz.

**Left panel**: EDR averaged along the signal,

**right panel**: standard deviation of the estimates. EDR estimates based on the power spectrum ${\u03f5}_{PS}$, Equation (6), structure function ${\u03f5}_{SF}$, Equation (7), number of zero-crossings scaling in the inertial range ${\u03f5}_{NCF}$, Equation (9), iterative method based on the velocity variance ${\u03f5}_{VAR}$, Equation (10) and the number of zero-crossings per unit length, Equation (12).

**Figure 7.**Statistics of ${\u03f5}_{PS}$ estimates based on the power spectra, Equation (6), for the fitting range $f=[10\xf720]$ Hz, from artificial signals with ${f}_{s}=200$ Hz, 100 Hz and 50 Hz.

**Left panel**: EDR averaged along the signal,

**right panel**: standard deviation of the estimates.

**Figure 10.**Statistics of ${\u03f5}_{SF}$ estimates based on the second-order structure function, Equation (7), for the fitting range $f=[10\xf720]$ Hz, from artificial signals with ${f}_{s}=200$ Hz, 100 Hz and 50 Hz.

**Left panel**: EDR averaged along the signal,

**right panel**: standard deviation of the estimates.

**Figure 13.**Statistics of ${\u03f5}_{NCF}$ estimates based on the number of zero-crossing scaling, Equation (9), for the fitting range $f=[10\xf720]$ Hz, from artificial signals with ${f}_{s}=200$ Hz, 100 Hz and 50 Hz.

**Left panel**: EDR averaged along the signal,

**right panel**: standard deviation of the estimates.

**Figure 16.**Statistics of ${\u03f5}_{VAR}$ estimates based on the variance of velocity derivative, Equation (10), for the fitting range $f=[10\xf720]$ Hz, from artificial signals with ${f}_{s}=200$ Hz, 100 Hz and 50 Hz.

**Left panel**: EDR averaged along the signal,

**right panel**: standard deviation of the estimates.

**Figure 19.**Mean of EDR estimates for artificial signals which deviate from the $-5/3$ scaling (left panel) or with deviations from the standard value of the Kolmogorov’s constant ${C}_{K}\approx 0.49$.

**Figure 27.**Top plots: Part of an artificial signal, middle plots: $\u03f5$ estimates, bottom plots: Taylor-to Liepmann scale ratio. Left column: signal with the intermittency parameter 0.69, right column: signal with the intermittency parameter 0.84.

**Figure 28.**Top plots: Part of the POST signals, middle plots: $\u03f5$ estimates, bottom plots: Taylor-to Liepmann scale ratio. Left panel: horizontal segment of the flight 13, right panel: flight 3, signal recorded during vertical cloud penetrations.

**Figure 29.**EDR estimates for signals measured during POST flight 3 [15] with ${f}_{s}=10$ Hz.

**Left panel**: fitting range $f=[0.5\xf73.5]$ Hz and ${f}_{cut}=3.5$ Hz,

**right panel**: fitting range corresponding to $f=[2\xf75]$ Hz for ${\u03f5}_{SF}$ and $f=[0.5\xf72.5]$ Hz for ${\u03f5}_{PS}$ and ${\u03f5}_{NCF}$, ${f}_{cut}=2.5$ Hz for ${\u03f5}_{VAR}$.

Signal Number | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Intermittency parameter | 0.84 | 0.69 | 0.60 | 0.30 |

${\lambda}_{n}/\Lambda $ | 0.84 | 0.72 | 0.57 | 0.35 |

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**MDPI and ACS Style**

Wacławczyk, M.; Gozingan, A.S.; Nzotungishaka, J.; Mohammadi, M.; P. Malinowski, S.
Comparison of Different Techniques to Calculate Properties of Atmospheric Turbulence from Low-Resolution Data. *Atmosphere* **2020**, *11*, 199.
https://doi.org/10.3390/atmos11020199

**AMA Style**

Wacławczyk M, Gozingan AS, Nzotungishaka J, Mohammadi M, P. Malinowski S.
Comparison of Different Techniques to Calculate Properties of Atmospheric Turbulence from Low-Resolution Data. *Atmosphere*. 2020; 11(2):199.
https://doi.org/10.3390/atmos11020199

**Chicago/Turabian Style**

Wacławczyk, Marta, Amoussou S. Gozingan, Jackson Nzotungishaka, Moein Mohammadi, and Szymon P. Malinowski.
2020. "Comparison of Different Techniques to Calculate Properties of Atmospheric Turbulence from Low-Resolution Data" *Atmosphere* 11, no. 2: 199.
https://doi.org/10.3390/atmos11020199