# Theoretical and Experimental Designs of the Planetary Boundary Layer Dynamics through a Multifractal Theory of Motion

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## Abstract

**:**

## 1. General Considerations: From Differentiability to Non-Differentiability in Atmospheric Process Dynamics

## 2. Theoretical Design: Non-Differentiability Calibrated on PBL Dynamics in the Form of the Multifractal Hydrodynamic Model

- I.
- The existence of a multifractal specific force implies that all PBL structure units must be considered through a multifractal medium;
- II.
- This medium can be considered a multifractal fluid whose dynamics are characterized by the hydrodynamic model presented previously;
- III.
- Since the velocity field, ${V}_{F}^{i}$, is absent from the multifractal states density conservation laws, it induces the possibility of non-manifest PBL dynamics, meaning that it facilitates the transmission of multifractal specific momentum and multifractal energy;
- IV.
- All potential issues regarding reversibility and existence of the eigenstates are solved by the conservation of multifractal energy and multifractal momentum;
- V.
- When using the tensor:

## 3. PBL Dynamics Mimed as a Multifractal Atmospheric Tunnel Effect

- I.
- The PBL, as a complex system both in a structural and functional perspective, can be assimilated with a mathematical object of multifractal type;
- II.
- PBL dynamics can be described through the scale relativity theory in the form of multifractal hydrodynamic equations;
- III.
- The PBL works as a multifractal atmospheric tunnel effect described through the external scalar potential (see Figure 1):

- (1).
- the multifractal atmospheric incidence zone;
- (2).
- the multifractal atmospheric barrier;
- (3).
- the multifractal atmospheric emergence zone.

- I.
- ${e}^{ikx}$ corresponds to the multifractal incident atmospheric states density (from $-\infty $) in the multifractal zone (1) and to the multifractal emergent atmospheric states density (to $+\infty $) in the multifractal zone (3);
- II.
- ${e}^{-ikx}$ corresponds to the multifractal reflected atmospheric states density, which exists only in the multifractal zone (1), passing from $x=0$ to $x=-\infty $ since in the multifractal zone (3), the external scalar potential is uniformly null.

- The multifractal atmospheric current density of the multifractal atmospheric incident states density in zone (1):$${J}_{i}=2\lambda {(dt)}^{\left(\frac{2}{f\left(\alpha \right)}\right)-1}k{\left|{A}_{1}\right|}^{2}$$
- The multifractal atmospheric current density of the multifractal atmospheric emergent states density in zone (3):$${J}_{e}=2\lambda {(dt)}^{\left(2/f\left(\alpha \right)\right)-1}k{\left|{A}_{3}\right|}^{2}$$
- The multifractal atmospheric current density of the multifractal reflected atmospheric states density:

## 4. Experimental Design

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**External scalar potential configuration (multifractal atmospheric barrier—PBL) for the tunnel effect of the multifractal (atmospheric) type.

**Figure 2.**The 3D variations of the multifractal atmospheric transparency, T, of the dimensionless coordinates, X and Y: (

**a**,

**b**) the dependence T = T (X, Y).

**Figure 3.**The 2D variations of the multifractal atmospheric transparency, T, of the dimensionless coordinates, X and Y: (

**a**) the dependence T = T (X, Y = constant); (

**b**) the dependence T = T (X = constant, Y).

**Figure 4.**The variation of the multifractal atmospheric reflectance, R, of the dimensionless coordinates, X and Y: (

**a**,

**b**) the dependence R = R (X, Y).

**Figure 5.**The 2D variations of the multifractal atmospheric reflectance, R, of the dimensionless coordinates, X and Y: (

**a**) the dependence R = R (X, Y = constant); (

**b**) the dependence R = R (X = constant, Y).

**Figure 8.**Time series of atmospheric temperature profiles; radiometer data; Galati, Romania, 5 May 2022.

**Figure 9.**Time series of atmospheric temperature profiles; theoretical model data; Galati, Romania, 5 May 2022.

**Figure 11.**Time series of atmospheric temperature profiles; radiometer data; Galati, Romania, 6 May 2022.

**Figure 12.**Time series of atmospheric temperature profiles; theoretical model data; Galati, Romania, 6 May 2022.

**Figure 14.**Time series of atmospheric temperature profiles; radiometer data; Galati, Romania, 7 May 2022.

**Figure 15.**Time series of atmospheric temperature profiles; theoretical model data; Galati, Romania, 7 May 2022.

**Figure 17.**Time series of atmospheric temperature profiles; radiometer data; Galati, Romania, 8 May 2022.

**Figure 18.**Time series of atmospheric temperature profiles; theoretical model data; Galati, Romania, 8 May 2022.

**Figure 19.**Profile of atmospheric temperature; radiometer data and theoretical model data; Galati, Romania, 5 May 2022; straight line: radiometer temperature; dotted line: theoretical temperature.

**Figure 20.**Profile of atmospheric temperature; radiometer data and theoretical model data; Galati, Romania, 6 May 2022; straight line: radiometer temperature; dotted line: theoretical temperature.

**Figure 21.**Profile of atmospheric temperature; radiometer data and theoretical model data; Galati, Romania, 7 May 2022; straight line: radiometer temperature; dotted line: theoretical temperature.

**Figure 22.**Profile of atmospheric temperature; radiometer data and theoretical model data; Galati, Romania, 8 May 2022; straight line: radiometer temperature; dotted line: theoretical temperature.

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**MDPI and ACS Style**

Cazacu, M.M.; Roșu, I.-A.; Bibire, L.; Vasincu, D.; Rotundu, A.M.; Agop, M.
Theoretical and Experimental Designs of the Planetary Boundary Layer Dynamics through a Multifractal Theory of Motion. *Fractal Fract.* **2022**, *6*, 747.
https://doi.org/10.3390/fractalfract6120747

**AMA Style**

Cazacu MM, Roșu I-A, Bibire L, Vasincu D, Rotundu AM, Agop M.
Theoretical and Experimental Designs of the Planetary Boundary Layer Dynamics through a Multifractal Theory of Motion. *Fractal and Fractional*. 2022; 6(12):747.
https://doi.org/10.3390/fractalfract6120747

**Chicago/Turabian Style**

Cazacu, Marius Mihai, Iulian-Alin Roșu, Luminița Bibire, Decebal Vasincu, Ana Maria Rotundu, and Maricel Agop.
2022. "Theoretical and Experimental Designs of the Planetary Boundary Layer Dynamics through a Multifractal Theory of Motion" *Fractal and Fractional* 6, no. 12: 747.
https://doi.org/10.3390/fractalfract6120747