Towards Possible Laminar Channels through Turbulent Atmospheres in a Multifractal Paradigm

Abstract

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.

The Search for Atmospheric Laminar Channels: Experimental Results and Method Dissemination

“…In fact, this model is produced by presupposing , , and . Recently, however, we have expanded upon this model in previous studies and successfully implemented it in a practical framework through lidar data [ 11 , 13 ]. It has been found that this model belongs to a class of gauges obtained from scale transition laws in the context of multifractality and that a similarly simple, yet slightly different, model containing a scale resolution parameter can give improved results [ 13 ].…”

“…Recently, however, we have expanded upon this model in previous studies and successfully implemented it in a practical framework through lidar data [ 11 , 13 ]. It has been found that this model belongs to a class of gauges obtained from scale transition laws in the context of multifractality and that a similarly simple, yet slightly different, model containing a scale resolution parameter can give improved results [ 13 ]. By maintaining a constant , but by adding the scale resolution parameter, the following equation: leads to multiple vortices that have their own differing fractal dimensions, i.e., leads to multifractal flow [ 13 ].…”

“…shows how rapidly the scales of the turbulent cascade decrease in size. Multiple other parameters pertaining to turbulent flow can then be calculated through this model, and these results have been previously published [ 11 , 13 ]. Furthermore, this theory has been applied before to lidar data, obtaining time series of these various parameters, and it has been found that these results are in accord with existing scientific literature, especially with regards to the known behavior of the PBL [ 11 , 13 ].…”

“…The manner in which the connection to the logistic map is done shall be shortly detailed in the following segment: following a Galerkin projection of multifractal type of the nondimensional Navier-Stokes equations of multifractal type, it is possible to write an arbitrary velocity Fourier mode of these equations with the iterative equation: which is equivalent to the logistic map, where is Reynold’s number, is a time step parameter, and . is a wavenumber associated with the velocity field decomposition [ 13 ]. Other studies have also been written which prove equivalence between the logistic map and the Navier-Stokes equations [ 14 , 15 ].…”

“…This shows that laminar channels exist and propagate throughout the atmosphere, and they do so in a cellular, self-structuring manner according to equations developed from the multifractal theory of atmospheric motion. In fact, such structures have already been found to exist through experimental lidar data, and in the following segment of the paper, their existence and characteristics can be analyzed again [ 13 , 18 ].…”