# Scale-Dependent Turbulent Dynamics and Phase-Space Behavior of the Stable Atmospheric Boundary Layer

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## Abstract

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## 1. Introduction

## 2. The CASES-99 Data-Set

## 3. Empirical Mode Decomposition of SBL Turbulent Fluctuations

## 4. Phase-Space Reconstruction and Time Delay of Local SBL Fluctuations

## 5. Local Correlation Dimension for Turbulent SBL

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Comparison of Fourier power spectral density $E\left(\omega \right)$ (black line) for sample $16/10/1999$ (from 00:00:00 to 02:00:00) with the Fourier power spectra of different intrinsic mode functions (IMFs) ${\psi}_{j}\left(\omega \right)$ (odd only for figure clarity), as a function of frequency $\omega $ (the curves have been shifted for clarity). The band-like structure of each IMF shows the dyadic nature of the decomposition. The dashed black line indicates the teoretical Kolmogorov spectrum ${\omega}^{-5/3}$ [23,51].

**Figure 2.**Average IMF period of each IMF as a function of the mode j, for each component u, v and w, calculated for the sample dated $16/10/1999$. Error bars show $95\%$ confidence bounds obtained from the inverse instantaneous frequency of the j-th IMF ${\omega}_{i,j}\left(t\right)$. The dashed lines represent the relation ${\tau}_{j}=\alpha \times {\gamma}^{j}$ [44,45,48], with $\gamma =1.89\pm 0.03$, $\gamma =1.96\pm 0.01$ and $\gamma =1.81\pm 0.03$ respectively. The value of $\gamma $ close to 2 indicates the quasi-dyadic filter bank property of the EMD algorithm for these series.

**Figure 3.**FNN percentage as a function of the embedding dimension m, estimated for the ${\psi}_{j}\left(t\right)$ for one sub-interval of sample $16/10/1999$. All curves relative to IMFs with timescale ${\tau}_{j}$ enclosed in the inertial sub-range drop to 0 at $m=3$. IMFs describing the large scale flow tends to a value $m=2$. The last two IMFs are 0 for $m=1$.

**Figure 4.**The phase-space dynamics in the different scaling regimes. Markers of different colors (green, red, black) represent different ranges of scales (large scale, inertial range, dissipative scales, respectively). The three panels are three two-dimensional projections of the three-dimensional phase space.

**Figure 5.**Correlation integral for embedding dimension $m=3$, for the IMF ${\psi}_{5}\left(t\right)$ extracted from the sub-sample dated $15/10/99$$02:00\to 04:00$.

**Figure 6.**Evolution of the local correlation dimension ${D}_{2}^{j}$ as a function of the IMF characteristic timescale ${\tau}_{j}$ for all component u, v, and w of the velocity field $U$ for sub-samples: $15/10/1999$—$00:00\to 02:00$, $16/10/1999$—$02:00\to 04:00$, and $17/10/1999$—$04:00\to 06:00$, respectively. A clear transition from a lower dimensional $d=1$ to an higher dimensional $d=3$ phase space is observed. Vertical dashed lines represent the range of scales relative to the power-law in the PSD reported in Figure 1.

**Table 1.**Time delay ℓ and embedding dimension m for different possible choices of data dimension d and components for the IMF with $j=5$, ${X}_{5}\equiv \left(\right)open="["\; close="]">{\varphi}_{5}\left(t\right),{\psi}_{5}\left(t\right),{\theta}_{5}\left(t\right)$, obtained from the SBL sub-sample dated $15/10/99$$02:00\to 04:00$.

SBL $\mathit{j}=5$ | d | $\mathit{\ell}\Delta \mathit{t}$[s] | m | $\mathit{dm}$ |
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${\varphi}_{5}$ | 1 | 0.35 | 3 | 3 |

${\psi}_{5}$ | 1 | 0.40 | 3 | 3 |

${\theta}_{5}$ | 1 | 0.35 | 3 | 3 |

${\varphi}_{5},{\psi}_{5}$ | 2 | 0.38 | 2 | 4 |

${\varphi}_{5},{\theta}_{5}$ | 2 | 0.35 | 2 | 4 |

${\psi}_{5},{\theta}_{5}$ | 2 | 0.38 | 2 | 4 |

${\varphi}_{5},{\psi}_{5},{\theta}_{5}$ | 3 | 0.37 | 1 | 3 |

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**MDPI and ACS Style**

Carbone, F.; Alberti, T.; Sorriso-Valvo, L.; Telloni, D.; Sprovieri, F.; Pirrone, N.
Scale-Dependent Turbulent Dynamics and Phase-Space Behavior of the Stable Atmospheric Boundary Layer. *Atmosphere* **2020**, *11*, 428.
https://doi.org/10.3390/atmos11040428

**AMA Style**

Carbone F, Alberti T, Sorriso-Valvo L, Telloni D, Sprovieri F, Pirrone N.
Scale-Dependent Turbulent Dynamics and Phase-Space Behavior of the Stable Atmospheric Boundary Layer. *Atmosphere*. 2020; 11(4):428.
https://doi.org/10.3390/atmos11040428

**Chicago/Turabian Style**

Carbone, Francesco, Tommaso Alberti, Luca Sorriso-Valvo, Daniele Telloni, Francesca Sprovieri, and Nicola Pirrone.
2020. "Scale-Dependent Turbulent Dynamics and Phase-Space Behavior of the Stable Atmospheric Boundary Layer" *Atmosphere* 11, no. 4: 428.
https://doi.org/10.3390/atmos11040428