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Edited by: Valéry Masson, Météo-France, France

Reviewed by: Eugen Radu, University of Aveiro, Portugal; Sergei Strijhak, Institute for System Programming (RAS), Russia; Iuliana Oprea, Colorado State University, United States

This article was submitted to Atmospheric Science, a section of the journal Frontiers in Earth Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This paper aims to present a multifractal approach of the turbulent atmosphere, by proposing that it can be considered a complex system whose structural units support dynamics on continuous but non-differentiable multifractal curves. Implementing the theoretical framework of multifractality through non-differentiable functions in the form of scale relativity theory with arbitrary and constant fractal dimension, the minimal vortex of an instance of turbulent flow is considered. The results of this assumption lead to an equation that describes the minimal vortex itself, and the velocity fields that compose it, the vortex and turbulent energy dissipation derived from the vortex being plotted and studied. With its structure mathematically described, while employing a classical dynamical turbulence model and relations between turbulent energy dissipation and the minimal vortex, relations are then extrapolated to allow for the solving of multiple turbulent parameters using the inner and outer length scales of the turbulent flow. These equations are then solved as altitude profiles with the necessary length scales obtained from processing lidar data. Finally, profiles are taken periodically and assembled into timeseries, in order to exemplify the method and to compare the results with known literature.

Determinism does not necessarily imply regulated behavior or predictability in atmosphere dynamics. In the standard (linear) analysis focused on atmosphere, unlimited predictability was a fundamental quality of atmosphere dynamics. Development of non-linear analysis and the discovery of laws regarding chaotic behavior demonstrated that not only does the reductionist analysis method, on which the entirety of atmosphere was grounded so far, has limited applicability, but also that unlimited predictability is not an attribute of the atmosphere, but an expected consequence of simplifying its description through linear analysis (Badii,

The chaotic and non-linear nature of the atmosphere is both structural and functional, and interactions between entities of the atmosphere determine reciprocal conditionings of the types microscopic-macroscopic, local-global, individual-collective, and others. Within this theoretical framework, the universality of the laws describing atmosphere dynamics becomes obvious and must be seen in the used mathematical procedures. There is increasing discussion regarding non-differentiable implementations in the description of atmosphere dynamics (Badii,

Usually, models used to describe atmosphere dynamics are based on the uncertain hypothesis that the variables describing it are differentiable (Deville and Gatski,

To describe atmosphere dynamics, while remaining faithful to the differentiable and integrable mathematical procedures, it is necessary to explicitly introduce scale resolution in the expression of the variables, and in the fundamental equations which govern atmospheric dynamics (Nottale,

This mode of describing atmosphere dynamics obviously implies the development of a new geometrical structure and a theory compatible with these structures for which dynamics laws, invariant to spatial, and temporal transformations, are integrated with scale laws invariant to the transformations of scale resolutions. In our opinion, such a geometrical structure can be obtained through the concept of fractal, the physical model associated being the fractal atmospheric dynamics either in the form of scale relativity theory with arbitrary and constant fractal dimension (Merches and Agop,

In the present paper a multifractal model describes the atmosphere dynamics and correspondences of this model with experimental data are proposed.

The atmosphere, both functionally and structurally, is a multifractal; such a hypothesis is sustained by the following typical example: between two successive collisions the trajectory of any atmosphere particle is a straight line that becomes non-differentiable at the impact point. Considering that all collisions impact points form an uncountable set of points, it results that the trajectories of the atmospheric particles becomes continuous and non-differentiable curves. Now, considering both the diversity of atmospheric particles and the variety of the collision processes between its particles, the atmosphere becomes a multifractal. Therefore, the fundamental hypothesis of our mathematical model is that the dynamics of the atmospheric entities (particles) will be described by continuous but non-differentiable curves (multifractal curves). In such context, the dynamics of the atmosphere entities are described through the operator [see

where

The meanings of the variables and of the parameters from (1) and (2) are extensively given in

The operator (1) plays the role of the scale covariant derivative (see

This means that the local multifractal acceleration

If the multifractalities are achieved by Markov—type stochastic processes which involve Lévy type movements (Mitchell,

where λ is a coefficient associated to the differentiable—non-differentiable transition and δ^{il} Kronecker's pseudo-tensor.

Under these conditions, the equation of geodesics takes the simpler form [see

or more, by separating motion on differential and non-differential scale resolutions, hydrodynamic type equation results [see

In this case we discuss about “holographic implementation” of the dynamics of the atmosphere through hydrodynamic fractal “regimes” (i.e., describing dynamics atmosphere by using hydrodynamic equations at various scale resolutions).

From the relations (6) it results that at differentiable scale resolutions “operates” a specific fractal force:

For irrotational motions of the atmosphere dynamics, the complex velocity field

where

Then substituting (8) in (5) the geodesics equation (for details see method from Nottale,

In this case we discuss about “holographic implementations” of the dynamics of atmosphere through Schrödinger fractal “regimes” (i.e., describing dynamics atmosphere by using Schrödinger type equations at various scale solutions).

The explicit form of the velocity field at non-differentiable scale can be shown through the functionality of “evolution” equations (i.e., hydrodynamic equations at non-differentiable scale):

The first of these equations corresponds to the canceling of the specific multifractal force while the second equation corresponds to the incompressibility of the atmosphere.

Generally, it is difficult to obtain an analytical solution for our previous equations system taking into account its non-linear nature [induced both by means of non-differentiable convection

We can still obtain an analytic solution in the case of a plane symmetry (in

where we substituted

First of all, one needs to consider the situation at hand given by the complexity and difficulty of the equations of the atmospheric multifractal: ideally, a three-axis solution would have been reached, but as we have mentioned, this is an exceedingly difficult task. In any case, because the model produces realistic results, as we shall see, a physical interpretation of the phenomena is that our plane-symmetrical multifractal velocity field might be a projection of a true, complete multifractal velocity field.

Using the similarities method (Schlichting and Gersten,

and a constant flux moment per unit of depth,

we obtain the velocity fields as solutions of the Equations (12) and (13) in the form:

The above equations are simplified greatly if we introduce both non-dimensional variables:

and non-dimensional parameters:

where _{0}, _{0}, _{0}, and σ_{0} are specific lengths, specific velocity, and “fractality degree” of the atmosphere. The normalized velocity field is obtained:

In

Normalized velocity field

Normalized velocity field

The above dependences specify the non-linearity behaviors of the velocity fields: a multifractal soliton for the velocity field across the Ox axis, respectively, “mixtures” of multifractal soliton—multifractal kink of the velocity fields across the Oy axis. The multifractality of the atmosphere dynamics is “explained” through its dependence from scale resolutions (

The velocity fields (21) and (22) induces the multifractal minimal vortex

In

Multifractal minimal vortex field Ω, for θ = 180°;

The above dependences specify the non-linear behaviors (through fractal degrees) of the minimal vortex field.

In turbulence, energy is injected in _{0} units, through a “cascade” of intermediary _{n} scales, toward the dissipation scale _{d}. We admit that this process can be described mathematically through the series of discrete scale lengths:

given the associated wavenumbers:

We follow with an analysis neglecting numerical factors, with the exception of those resulted from successive multiplications. Thus, if the specific kinetic energy of the “turbulent fluctuations” associated to the scale _{n} is _{n}, then it is possible to define by means of _{n} as the average value of the velocity difference for _{n} in the form:

Then, the corresponding time interval can be defined as:

We then admit that a certain fraction of the specific energy corresponding to the scale _{n} is transferred to the scale _{n+1} during a period _{n}. As a consequence, the specific energy transfer rate for

In the case of “stationary turbulence” the energy conservation law implies:

where 〈ε〉 is the average dissipation rate. We remind that 〈ε〉 found in (29) can be interpreted both as the rate of energy injection and the rate of energy transfer; the last is a relevant “measure” for the dynamics of the inertial domain. As regards the previous idea, we agree with the addressed Kraichnan critics (Kraichnan,

and from (26) we obtain:

an expression which, after Fourier transformation, is identical to its counterpart in Kolmogorov theory (Kolmogorov,

In the following let us consider the hypotheses that the average number of “vortex fragmentation” is _{n} it is supposed to induce _{n+1} for each value of

in which the second equality is explained through (24). Furthermore, if we admit that the largest units “fill” all the space they have at their disposal, then the

which will be occupied with units of scale _{n}.

Now, we return to the previous argument, but limit ourselves to the “matter” volumes of _{n} is supposed to be:

expressions which, by using the relations (30), (32), and (33) takes the form:

Also, introducing:

and the relation (24) to eliminate

where we introduce the notation:

For an atmosphere with a mono-fractal behavior our model is reduced to the standard β model (Benzi et al.,

Returning to the definition of the average turbulent energy dissipation rate, Tatarski finds the following definition for stationary atmospheric turbulence (Tatarski,

Now, through (24), we can establish a number

In this manner, _{l0} is the number of instances of vortices of _{n} scales that are fractionated into an average of _{n+1} scales in the energy cascade, starting from energy injection to dissipation. Using the relations (34) and (37) in this manner, combined with (40), we find:

where, _{ld} is the velocity difference between two points separated by _{d}. Then, solving for

With (40) and (41) it is now possible to construct the expressions for the kinetic energy of the first and last “stages” of turbulent fluctuations, taking into consideration (35):

The fraction _{ld} is necessarily smaller than _{l0}. In any case, between the liminal stages of the energy cascade, there are _{n}, and all of them can be calculated in this manner, using only length scales and the average of the multifractal minimal vortex obtained by averaging obtained values in the vortex (Equation 23) in order to first obtain

The introduction of the dissipation and injection scales is performed to verify equations with experimental data, and, by using experimental data to solve a number of these equations, to investigate the results. In the following segment of this paper, we present simulations of these various equations with a varying ξ non-dimensional parameter (_{l0} and

The technical specifications of the main components of the lidar platform utilized in the study are as follows: the laser component is a Nd:YAG, producing pulses of laser at a frequency of 30

As mentioned, the length scale profiles used in this study are obtained by first calculating scintillation profiles in order to compile the structure coefficient of the refraction index profile

with

is given, with _{l0}

while the outer scale is linked to the

With turbulent eddies within the inertial subrange, the refraction index profile can be approximated from the definition of the respective structure coefficient (Tatarski,

which then gives us the means to extract the outer scale profile (49).

Regarding the potential influence of noise-related errors to the calculation of the scintillation profile used to obtain the length scales, the overall signal uncertainty added by noise is:

where _{b} is background lidar signal, and ^{−7} or lower (Liu et al., ^{−5} or lower; thus, this uncertainty represents variations hundreds of times smaller than the actual values of the profile. Since the model subtracts the “dark” signal [generated by photocathode thermionic emission, which is collected before the measurements begin (Hamamatsu,

First of all, velocity fields, the multifractal minimal vortex and its dissipation field are exemplified (

Turbulent dissipation rate field ε, for θ = 180°;

The following figure is a timeseries of the RCS lidar signal used to calculate the length scales according to the presented theory (

Lidar-obtained RCS timeseries on the 28th of May 2017; temporal resolution: 1 min; spatial resolution: 3.75 m.

ε(

Regarding the timeseries, 〈ε〉 values appear to be lowest in the region of the PBL (_{l0} shown by the timeseries seems low, roughly between 14 and 17 (

ε(

It must be highlighted that _{l0} values seem highest in the PBL, with dissipation values being lowest. It is possible that this difference can be interpreted such that the turbulent behavior of the atmosphere in the PBL is more active, and more “resilient” to dissipative effects, which does indeed justify why this layer is such a stable and easily-recognizable feature of the atmosphere in most meteorological scenarios. The horizontal stripes of either very high or very low values in the time series, starting from certain intervals above the PBL, are produced due to signal noise present in the RCS lidar data used to calculate the length scales; this noise appears because of the influence of clouds in the vicinity of the PBL.

A simple analysis using a more discrete variation of the non-dimensional parameter ξ is also performed (_{0} = 260 _{d} = 0.0035 _{ld} produces a lower

Calculation of Ω, ε, _{ld}, and

^{∧}2/s^{∧}3) |
||||
---|---|---|---|---|

0.2 | 0.1116 | 0.001839 | 0.229 | 19.6844 |

0.3 | 0.1447 | 0.002068 | 0.797 | 6.015 |

0.4 | 0.1578 | 0.001832 | 1.04 | 4.8521 |

0.5 | 0.1622 | 0.001529 | 1.127 | 4.5548 |

0.6 | 0.1627 | 0.001258 | 1.139 | 4.5162 |

0.7 | 0.1616 | 0.00103 | 1.115 | 4.5942 |

0.8 | 0.1598 | 0.000841 | 1.075 | 4.7372 |

0.9 | 0.1578 | 0.000687 | 1.028 | 4.9206 |

1 | 0.1557 | 0.000561 | 0.979 | 5.1313 |

1.1 | 0.1537 | 0.000458 | 0.932 | 5.361 |

1.2 | 0.1519 | 0.000374 | 0.887 | 5.6043 |

1.3 | 0.1501 | 0.000306 | 0.846 | 5.8573 |

1.4 | 0.1485 | 0.000251 | 0.807 | 6.1172 |

1.5 | 0.1471 | 0.000207 | 0.771 | 6.3818 |

1.6 | 0.1458 | 0.000171 | 0.739 | 6.6495 |

1.7 | 0.1446 | 0.000142 | 0.709 | 6.9188 |

1.8 | 0.1435 | 0.000119 | 0.681 | 7.1887 |

1.9 | 0.1425 | 0.0001 | 0.656 | 7.4583 |

2 | 0.1415 | 0.000085 | 0.633 | 7.7271 |

2.1 | 0.1407 | 0.000073 | 0.612 | 7.9943 |

2.2 | 0.1399 | 0.000064 | 0.592 | 8.2596 |

In this work a holographic (multifractal) approach has been used to describe the non-linear behavior of the atmosphere, by considering that it can be assimilated to a complex system whose structural units support dynamics on continuous and non-differentiable curves. We have named this system the atmospheric multifractal, and the formulation of the motion operator of this atmospheric multifractal has allowed an analytic solution in the case of a plane symmetry for the dynamics of this system. The rotor of the obtained normalized velocity fields is then interpreted as the multifractal minimal vortex, which corresponds to the dissipation scale in the turbulent energy cascade. The velocity fields and the vortex are then plotted using multiple instances of a non-dimensional parameter that is linked to their fractal dimensions, and these results are discussed. Using this new formulation of the minimal vortex, it is then possible to extract an expression of turbulent energy dissipation in the atmospheric flow, and to construct an equation system that can describe the behavior of the cascade of vortices in terms of their scales, the average number of vortex fragmentations per “stage,” and the total number of stages of fragmentation, from injection to dissipation. In this equation system, from an atmospheric modeling and forecast point of view, the remaining unknown quantities are the injection and dissipation scales themselves; however, these quantities can be obtained as an atmospheric profile through a method detailed in one of our previous studies with a lidar platform. Using these profiles, time series of the turbulence parameters detailed in this study have been compiled, and it has been found that, especially regarding the known behavior of the PBL, they are in accord with the presented theory and existing scientific literature. Also, an altitude profile comparison has been made between turbulent energy dissipation calculated with the theory presented in this study and turbulent energy dissipation calculated with theory from one of our previous studies.

The success of these results leads us to believe that a future study might implement these theories, coupled with theoretical means of determining the inner and outer length scales, in order to produce forecasts of turbulent parameters; this, of course, using measured, ground-level, initial parameters. A future study might also include multiple other methods of experimentation and validation using platforms that would contain both traditional measurement instruments and remote sensing instruments. Finally, if these theories, along with many others, would be implemented into a fully functional model, a comparison with other well-known models could be performed in a potential new study.

The datasets generated for this study are available on request to the corresponding author.

IR, MC, MA, AG, and LB: conceptualization, formal analysis, original draft preparation, and visualization. IR, MC, and MA: methodology, validation, resources, data curation, writing—review and editing, supervision, and project administration. IR and MC: software and investigation and funding acquisition. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at: