In this paper, developments are made towards simulating complex atmospheric behavior using turbulent energy cascade staging models developed through scale relativity theories. Such theoretical considerations imply gauges that describe atmospheric parameters as multifractal functions undertaking scale symmetry breaking at each stage of the turbulent energy cascade. It is found that gauges of higher complexity (in this case, a Riccati-type gauge) can exhibit more complex behavior accordingly, such as both dilation and contraction, but properly parameterizing the solutions formed by these gauges in terms of turbulent staging can be challenging given the multiple constants and parameters. However, it is found that a logistic-type approximation of the multifractal equations of motion that describe turbulent atmospheric entities can be coupled with a model produced by a simpler gauge, and this combination can reveal instances of laminar, or otherwise non-chaotic, behavior in a given turbulent flow at certain scales. Employing the theory with elastic lidar data, quasi-laminar behavior is found in the vicinity of the planetary boundary layer height, and laminar channels are revealed throughout an atmospheric column—these might be used to reveal complex vertical transport behavior in the atmospheric column.

Assimilating the atmosphere with multifractal entities, nonlinear behaviors in the framework of scale relativity theory regarding its hydrodynamic functionality are established at various scale resolutions. From this perspective, revealing a “hidden” symmetry of the specific multifractal force with the SL(2,R) group leads to synchronization of atmospheric entities on the basis of operational procedures (differential and integral Riemann-type geometries, harmonic mappings from Euclidian to hyperbolic space, variational principles, and others) that imply cellular self-structuring, laminar channels and singularities as turbulence generators. These behaviors can then be assimilated and compared with recent discoveries regarding laminar channels found in atmospheric turbulence through lidar data processing.

In this paper, a practical application of theoretical developments found in our previous works is explored in relation to atmospheric lidar data. Multifractal structures, previously named “laminar channels”, have been identified in atmospheric profiles—these exhibit cellular and self-structuring properties, and are spatially ordered across the atmospheric profile. Furthermore, these structures have been connected to the spontaneous emergence of turbulent behavior in the calm atmospheric flow. Calculating the location and occurrence of these channels can help identify features of atmospheric evolution, such as the development of the planetary boundary layer (PBL). Employing this theoretical background to atmospheric lidar data, attempts are made to confirm this suggestion and extract information about atmospheric structure and evolution by analyzing turbulent vortex scale dynamics and scale-corresponding Lyapunov exponents that form the basis of identifying the laminar channels in atmospheric lidar profiles. A parameter named “scale laminarity index” is then introduced, which quantifies the relation between vortex scale and chaoticity throughout the profile. Finally, the algorithmic methods employed in this study are described and distributed for future use.

In this manuscript, multifractal theories of motion based on scale relativity theory are considered in the description of atmospheric dynamics. It is shown that these theories have the potential to highlight nondimensional mass conduction laws that describe the propagation of atmospheric entities. Then, using special operational procedures and harmonic mappings, these equations can be rewritten and simplified for their plotting and analysis to be performed. The inhomogeneity of these conduction phenomena is analyzed, and it is found that it can fluctuate and increase at certain fractal dimensions, leading to the conclusion that certain atmospheric structures and phenomena of either atmospheric transmission or stability can be explained by atmospheric fractal dimension inversions. Finally, this hypothesis is verified using ceilometer data throughout the atmospheric profiles.

In the present paper, correlations between multifractality and stochasticity in atmospheric dynamics are investigated. Starting with two descriptions of atmospheric scenarios, one based on scale relativity theory and another based on stochastic theory, correspondences between parameters and variables belonging to both scenarios are found. In such a context, by replacing an atmospheric conservative passive additive with a non-differentiable component of the atmospheric multifractal velocity, stochastic evolution equations are found for this component, which reveal the multifractal variational transport coefficient and the multifractal molecular diffusion coefficient, along with the multifractal inhomogeneity variation. Furthermore, equations which describe a multifractal Reynolds number and singularity spectrum are also found. Finally, these theoretical results are validated through remote sensing data obtained with the aid of a ceilometer platform.

This paper investigates the nonlinear dynamics of atmospheric aerosols during the transition from laminar to turbulent flows using the framework of Scale Relativity Theory. It is proposed that the transition from multifractal to non-multifractal scales (in the dynamics of the atmospheric aerosols) can be assimilated to the transition between laminar and turbulent states. These transitions are determined by the multifractal diffusion and deposition processes. The methodology used involves the application of the principle of scale covariance, which states that the laws of atmospheric physics remain invariant with respect to spatial and temporal transformations as well as scale transformations. Based on this principle, several conservation laws are constructed. In such context, the conservation law of the density of states associated with the multifractal-non-multifractal scale transition in a one-dimensional case is then considered. The model describes the non-linear behaviour of atmospheric aerosols undergoing diffusion and deposition processes. The theoretical approach was correlated using experimental data from a ceilometer and radar reflectivity factor data.