# Boundary Layer via Multifractal Mass Conductivity through Remote Sensing Data in Atmospheric Dynamics

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## Abstract

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## 1. Introduction

- (i)
- (ii)

## 2. Hydrodynamic Multifractal Scenario Conservation Laws

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- t is nonmultifractal time, an affine parameter of movement curves of the entities found in the complex system;
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- x
^{l}is the multifractal spatial coordinate; - -
- v
^{i}is the velocity field at a differentiable scale resolution; - -
- u
^{i}is the velocity field at a non-differentiable scale resolution; - -
- dt is the scale resolution;
- -
- λ is a constant coefficient associated with the multifractal-nonmultifractal scale transition;
- -
- ρ is the state density;
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- ψ is the state function with the amplitude $\sqrt{\mathsf{\rho}}$ and phase s;
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- Q is the scalar specific multifractal potential which quantifies the multifractalization degree of the movement curves in the complex system;
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- f(α) is the singularity spectrum of order α = α(D
_{F}) where D_{F}is the fractal dimension of movement curves of the complex system entities. This spectrum allows the identification of universality classes in the complex system dynamics, even when attractors have different aspects, and it also allows the identification of areas in which the dynamics can be characterized by a specific fractal dimension.

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- The velocity field at a non-differentiable scale:

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- The specific multifractal force field:

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- Conductivity at differentiable scale resolutions:

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- Conductivity at non-differentiable scale resolutions:

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- Conductivity at global scale resolutions:

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- Conductivity at differentiable scale resolutions:

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- Conductivity at non-differentiable scale resolutions:

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- Conductivity at global scale resolutions:

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- Conduction in complex systems is performed through specific mechanisms dependent on the scale resolution. As a consequence, we make the distinction between differentiable conduction $\overline{{\mathsf{\sigma}}_{\mathrm{D}}}$, non-differentiable conduction $\overline{{\mathsf{\sigma}}_{\mathrm{F}}}$ and global conduction $\overline{\mathsf{\sigma}}$;
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- Conduction mechanisms at the two types of scale resolutions are simultaneous and reciprocally conditional. Thus, the values of $\overline{{\mathsf{\sigma}}_{\mathrm{D}}}$ and $\overline{{\mathsf{\sigma}}_{\mathrm{F}}}$ increase along with the increase of the ordering degree (synchronous type conductions) and with the increase of the multifractalization degree $\overline{{\mathsf{\sigma}}_{\mathrm{D}}}$ values increase and $\overline{{\mathsf{\sigma}}_{\mathrm{F}}}$ values decrease (asynchronous type conductions).

## 3. Non-Manifest Dynamic States through Harmonic Mappings

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 0.5; μ = 1.

**Figure 6.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 3; μ = 1.

**Figure 7.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 6; μ = 1.

**Figure 8.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 9; μ = 1.

**Figure 9.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 3; μ = 0.5.

**Figure 10.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 3; μ = 3.

**Figure 11.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 3; μ = 6.

**Figure 12.**$\langle {\left|\overline{{\mathsf{\sigma}}_{\mathrm{F}}}\right|}^{{}^{\prime}2}\rangle $ example plot with θ as control parameter; ξ = 3; μ = 9.

**Figure 14.**Zoomed-in (region of interest) RCS time series, λ = 1064 nm, Galați, Romania, 23 December 2021.

**Figure 16.**Zoomed-in (region of interest) RCS time series, λ = 1064 nm, Galați, Romania, 22 December 2021.

**Figure 17.**${\mathrm{C}}_{\mathrm{N}}^{2}$ time series, λ = 1064 nm, Galați, Romania, 23 December 2021.

**Figure 18.**Zoomed-in (region of interest) ${\mathrm{C}}_{\mathrm{N}}^{2}$ time series, λ = 1064 nm, Galați, Romania, 23 December 2021.

**Figure 19.**${\mathrm{C}}_{\mathrm{N}}^{2}$ time series, λ = 1064 nm, Galați, Romania, 22 December 2021.

**Figure 20.**Zoomed-in (region of interest) ${\mathrm{C}}_{\mathrm{N}}^{2}$ time series, λ = 1064 nm, Galați, Romania, 22 December 2021.

**Figure 22.**Zoomed-in (region of interest) D

_{f}time series, λ = 1064 nm, Galați, Romania, 23 December 2021.

**Figure 24.**Zoomed-in (region of interest) D

_{f}time series, λ = 1064 nm, Galați, Romania, 22 December 2021.

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**MDPI and ACS Style**

Nica, D.-C.; Cazacu, M.-M.; Constantin, D.-E.; Nedeff, V.; Nedeff, F.; Vasincu, D.; Roșu, I.-A.; Agop, M.
Boundary Layer via Multifractal Mass Conductivity through Remote Sensing Data in Atmospheric Dynamics. *Fractal Fract.* **2022**, *6*, 250.
https://doi.org/10.3390/fractalfract6050250

**AMA Style**

Nica D-C, Cazacu M-M, Constantin D-E, Nedeff V, Nedeff F, Vasincu D, Roșu I-A, Agop M.
Boundary Layer via Multifractal Mass Conductivity through Remote Sensing Data in Atmospheric Dynamics. *Fractal and Fractional*. 2022; 6(5):250.
https://doi.org/10.3390/fractalfract6050250

**Chicago/Turabian Style**

Nica, Dragos-Constantin, Marius-Mihai Cazacu, Daniel-Eduard Constantin, Valentin Nedeff, Florin Nedeff, Decebal Vasincu, Iulian-Alin Roșu, and Maricel Agop.
2022. "Boundary Layer via Multifractal Mass Conductivity through Remote Sensing Data in Atmospheric Dynamics" *Fractal and Fractional* 6, no. 5: 250.
https://doi.org/10.3390/fractalfract6050250