Multifractal Model of Atmospheric Turbulence Applied to Elastic Lidar Data

Abstract

This paper shall present a multifractal interpretation of turbulent atmospheric entities, considering them a complex system whose dynamics are manifested on continuous yet non-differentiable multifractal curves. By bringing forth theoretical considerations regarding multifractal structures through non-differentiable functions in the form of an adaptation of scale relativity theory, the minimal vortex of an instance of turbulent flow is considered. In this manner, the spontaneous breaking of scale invariance becomes a mechanism for atmospheric turbulence generation. This then leads to a general equation for the non-differentiable vortex itself, with its component velocity fields, and to a vortex turbulent energy dissipation—all of which are plotted and studied. Once the structure of the non-differentiable multifractal structure is mathematically described, an improved phenomenological turbulence model and relations between turbulent energy dissipation and the minimal vortex are employed together, exemplifying the codependency of such models. Using turbulent medium wave propagation theory, certain relations are then extrapolated which allow the obtaining of the inner and outer length scales of the turbulent flow using lidar data. Finally, these altitude profiles are compiled and assembled into timeseries to exemplify the theory and to compare the results with known literature. This model is a generalization of our recent results published under the title “On a Multifractal Approach of Turbulent Atmosphere Dynamics”.

Towards Possible Laminar Channels through Turbulent Atmospheres in a Multifractal Paradigm

“…The following lateral plots show the dependency on n and s separately for a better representation of the function (Figure 2). The main advantage of this model is its simplicity in formulation; despite a lack of complexity, this model has been shown to produce accurate and interesting results regarding atmospheric dynamics [14]. However, certain gauges present a much more variable behavior than the linear one used so far in our studies, and this variability can reveal more complex turbulent phenomena.…”

“…As discussed in the previous section, the notions of scale transition, scale symmetry breaking, and scale symmetry invariance become fundamental to understanding atmospheric dynamics in the context of multifractality. Because of this, a scale transition relation shall be used to construct gauges for turbulent models-for this purpose, some results obtained in a previous work shall be presented [14]. First, let us consider a multifractal function F(x) with x ∈ [a, b] which can be associated with any multifractal variable that describes atmospheric dynamics.…”

“…For now, let us consider the conservation law of the specific impulse associated with the multifractal-nonmultifractal (see Equation ( 16) from [14]) under the specific form:…”

“…As a consequence, any variable purposed to describe the atmosphere dynamics will perform as the limit of a family of mathematical functions, this being non-differentiable for null scale resolutions and differentiable otherwise [8,9]. Using these notions regarding the multifractality of turbulent flow, coupled with correlations between turbulent energy dissipation and the initial and minimal turbulent length scales, it is possible to employ a modified version of the standard turbulent β model to calculate various atmospheric parameters-this model has been validated using lidar data and comparison [13,14]. In the present work, different gauging types of β-constant models and their characteristics are explored, and the various difficulties in applying them are explained.…”

“…Furthermore, in attempting to employ a relatively simple linear gauge β-constant model to represent more complex atmospheric phenomena, the multifractal equations of motion of multifractal entities are approximated by a logistic-type equation, which is then used to identify possible laminar channels crossing the atmosphere both scale-wise and altitude-wise. The model is initialized by extracting the inner and outer turbulent length scale through elastic lidar data as per our previous studies [13,14].…”