An Erratum to this paper has been published: https://doi.org/10.1134/S1028334X23050239

The 1934 paper by A N Kolmogorov [1], "Random Motions,'' hereinafter ANK34, uses a Fokker -Plancktype equation for a 6 -dimensional vector with a total rather than a partial derivative with respect to time, and with a Laplacian in the space of velocities. The diffusion coefficient in this case is E, the rate of energy generation/dissipation. The equation is obtained by specifying the accelerations of the particles of the ensemble by Markov processes, i.e., random processes 6 -correlated in time and with each other. The fundamental solution of this equation was already indicated in [1] and was used by A M Obukhov [2] in 1958 to describe a turbulent flow in the inertial interval [3]. It was only recently [4, 5] noticed that the Fokker -Planck -type equation written by Kolmogorov in [1] contains a description of the statistics of other random natural processes, earthquakes, sea waves, and others [5]. This equation, by a change of variables with scales for velocities and for coordinates, is reduced to a self -similar form that does not explicitly contain the diffusion coefficient [6]. Numerical calculations confirm the presence of such scales in systems with the number N of events, in ensembles starting from N = 10. For N = 100, these scales almost exactly coincide with the ANK34 theory. This theory, in principle, containing the results of 1941, paved the way for more complex random systems with enough parameters to form an external similarity parameter. This leads to a change in the characteristics of a random process, for example, to a change in the slope of the time spectrum, as in the case of earthquakes and in a number of other processes (sea waves, cosmic ray energy spectrum, inundation zones during floods, etc.). A review of specific random processes studied experimentally provides a methodology for how to proceed when comparing experimental data with the ANK34 theory. Thus, empirical data illustrate the validity of the fundamental laws of probability theory. The article is an abridged version of the author's monograph [5], where for the first time the ideas of ANK34 were used to explain in a probabilistic sense many experimental patterns that have been considered by pure empiricism for decades.

Using the statistical properties of the solution of the Fokker–Planck–Kolmogorov equation (FPK) for velocities and coordinates and using the vortex identification method, quantitative estimates of the distributions of various characteristics of cyclones and anticyclones (including lifetime, wind speed, size, characteristic forcing, and kinetic energy) depending on their intensity have been obtained. The calculations are based on ERA5 reanalysis data for the period 2010–2021. The vortex lifetimes estimated using the FPK correspond to the lifetimes obtained using the vortex identification method, and the theoretical distribution of cyclones by intensity practically coincides with those observed. The characteristics of vortices during their life cycle are also investigated. In general, all the characteristics of cyclones analyzed increase with their intensification. However, for intense anticyclones, the increase is not as pronounced as for cyclones.

An Erratum to this paper has been published: https://doi.org/10.1134/S1028334X23060120

Since the 1990's, it has been recognized that the full explanation of cosmic rays (CR) and their spectrum may require some new physics. The debate on the origin of CR has led to the conclusion that while most CR come from supernova explosions in the Galaxy, CR with very high energies are likely of extragalactic origin. However, a response to several open questions, still unanswered, concerning CR above 1013 13 eV is required. We herewith study the temporal evolution of the observational CR using data collected by several stations of the ground-based network. The obtained result states that the power spectral density of the CR temporal evolution, especially with a frequency less than 0.1 Hz, exhibits the Kolmogorov-Obukhov 5/3 law that exhibits the energy spectrum of many geophysical quantities. Any small difference found from the 5/3 exponent can be attributed to intermittency corrections and the stations' characteristics. Moreover, natural time analysis applied to the CR time series showed the critical role of the quasi-biennial oscillation to the entropy maximization which occurs following the 5/3 Kolmogorov-Obukhov power law. These findings can be used to more reliably predict extreme CR events that could have an impact even at the molecular level.

In 1982 Lovejoy has published an illustration to Mandelbrot proposal how to characterize the area-perimeter ratio of complicated planar forms and it was found that exponent \(\beta \) for the satellite- and radar-determined cloud and rain areas of such a fractal is 1.35 close to 4/3. Later on it was notified that the same exponent was found also for noctilucent clouds. Such a value might be related to classic turbulence theory of 1941. This text demonstrates this relation using two basic papers by Kolmogorov and Obukhov. The role of prefractal multipliers is revealed, they form a couple of the peculiar invariants for cloud fields and a non-dimensional self-similarity numbers for these fields of sizes \(1 - {{10}^{6}}\,\,{\text{k}}{{{\text{m}}}^{2}}.\) The peculiarity is in their dimensional dependence and in the presence of few invariants, not usual invariants in cloud forms. Further research on random walk of a fluid particle in the 6D phase-space may lead to new discoveries.

The paper by A.N. Kolmogorov 1934 "Random Moves", hereinafter ANK34, uses a Fokker-Planck-type equation for a 6-dimensional vector with a total rather than a partial derivative with respect to time, and with a Laplacian in the space of velocities. This equation is obtained by specifying the accelerations of the particles of the ensemble by Markov processes. The fundamental solution was used by A M Obukhov in 1958 to describe a turbulent flow in the inertial interval. Already recently it was noticed that the Fokker-Planck-type equation written by Kolmogorov contains a description of the statistics of other random natural processes, earthquakes, sea waves, and others. This theory, containing the results of 1941, paved the way for more complex random systems containing enough parameters to form an external similarity parameter. This leads to a change in the characteristics of a random process, for example, to a change in the slope of the time spectrum, as in the case of earthquakes and in a number of other processes (sea waves, cosmic ray energy spectrum, flood zones during floods, etc.). A review of specific random processes studied experimentally provides a methodology for how to proceed when comparing experimental data with the ANK34 theory. Thus, empirical data illustrate the validity of the fundamental laws of probability theory.

We investigate the correlation between precipitation, clouds and cosmic rays based on field observations to develop a new short-term forecasting tool for their extremely high values. The results show that the cosmic ray intensity appears to be highly correlated with the one-year time-lagged rainfall amount with the cosmic rays leading. A plausible explanation is given for this, based on the modulation of cosmic rays by the solar cycle and an earlier finding of a strong correlation between the stratospheric - lower mesospheric temperature fluctuations and the sunspot number cycle with a time lag of one year (sun leading). This suggests that cosmic rays are not the trigger, but rather signals sunspot activity that with a delay of one year may affect cloud formation and therefore rainfall. In addition, high rates of rainfall exhibit a power-law behavior, such as the Gutenberg-Richter law characterizing the large area and perimeter size of rainfall and clouds, which have been proposed recently. Finally, the new tool for the nowcasting the extremely high cosmic ray intensity values can be used for nowcasting the extremely high rainfall values. Their combination can be used to study the modulation of cosmic ray properties by climate change parameters.

Remote sensing plays an increasingly important role in climate research, and as remote sensing observation files span longer periods of time, we are receiving more and more data on climate parameters such as clouds. Four decades ago, it was proposed that the area of planar shapes change with their perimeter following the power law. The exponent of this relation for the areas of clouds (typical and noctilucent) and rain determined by satellites and radar is 1.35 close to 4/3 of the classical turbulence theory of 1941. In this paper, we propose a new relationship between the area and the perimeter of rain and clouds using remote sensing observations. This relationship leads to the development of a new model for the detection of extreme natural phenomena related to rain and clouds.

According to the data on the sizes and intensity of tornadoes and landspouts, their characteristic times and forcings are found. Depending on the intensity, the average times vary by a factor of three and the forcings by a factor of six. The energy of the vortices is estimated and is expressed, among other things, in TNT equivalent mass units.

In the inertial subrange of scales, an exact compressible turbulence universal spectrum law −8/3 for the density fluctuations of cosmic rays (CRs) in the frame of the known two-fluid model of CR dynamics is obtained. It is shown that the origin of this scaling law may be due to the arising of shocks at the breaking of the nonlinear simple waves of CRs near the scale of their Larmor’s radii, as it is well known for the solar wind with the same turbulent spectrum law −8/3. The consistency of the turbulence spectrum −8/3 of CRs with the observed nonthermal differential energy distribution of CRs with a similar index −8/3 due to the possibility of self-reacceleration of the CRs on the self-arising shocks is stated. The turbulent diffusion mechanism for the observed CRs energy spectrum breaks is considered.

An Erratum to this paper has been published: https://doi.org/10.1134/S1028334X23060120

The daily observations of the cosmic ray intensity carried out by the Athens Neutron Monitoring Station in Greece during the period 1 January 2010 to 14 January 2022 are analysed for the study of the observed extreme events. The high values of this parameter were found to obey the Gutenberg-Richter scaling, which has already been identified in several meteorological parameters. Based on this finding a new nowcasting method has been developed herewith for extreme cosmic rays' events that may cause interference to telecommunications and problems with transportation and water supply, as well as several disasters. For the development of this nowcasting model, the innovative domain of 'natural time' has been employed. Moreover, using the data provided by the Cosmic Rays Database, a power-law relation was found that connects the cosmic ray flux and the particle total energy corresponding to a future cosmic ray event with high intensity. This model will make a major contribution to the further development of ways to prevent, prepare for and deal with potential damages following extreme cosmic rays' events.

The profiles of dust aerosol mass concentration obtained with multilevel (0.2, 0.4, 0.8, 1.6, and 3.2 m) daytime measurements in arid conditions in 2020–2022 show a power dependence on height. We distinguish three main types of changes in concentration with height: (a) in low wind (degrees are close to –0.5), (b) burst changes in concentration when wind increases (degrees reach and exceed –1), and (c) inversions (concentration increases with height at two or three lower levels of measurements): weak (about –20 mkg/cm3) and significant (more than 50 mkg/cm3). The power dependence of –0.5 is explained by the collective effect of the rise of the ensemble of closely located bubbles of air warmed around the dust particles. In weak and moderate winds, this mode is more common. Burst changes in concentration are associated with the emergence of turbulent structures.

An exact turbulence universal scaling law-8/3 for the density fluctuations of cosmic ray (CR) is obtained on the basis of a new analytical compressible turbulence theory and known two-fluid model of the CR dynamics. It is shown that the origin of this scaling law may be due to the breaking of the nonlinear simple waves in CR medium near the scale of their Larmor radii as for the space plasma of solar wind and magnetosheath.

Results obtained in the course of unique observations (as part of the TROICA project) of the composition and state of the atmosphere over Russia have been summarized. Scientists and engineers from different countries took part in these observations. The main task of these observations was to measure atmospheric contents of trace gases and aerosols and both meteorological and radiative characteristics of the atmosphere using a mobile laboratory designed and brought into operation specially for transcontinental observations along the network of Russian electrified railroads. A large database obtained in the TROICA experiments using the mobile laboratory, in field experiments, and at scientific stations is still under analysis. In this work, the most important results published earlier and obtained recently are discussed, which makes it possible to form a comprehensive picture of the spatial distribution and temporal variability of the atmospheric composition over northern Eurasia.

The relief spectra of the Moon, Mars and Earth with a very high resolution are discussed (Rexer and Hirt, 2015). According to the Kaula rule (Kaula, 1966), these spectra decay as k(-2). This fact has been recently explained (Gledzer and Golitsyn, 2019) based on the probabilistic laws by Kolmogorov and his school (Kolmogorov, 1934; Obukhov, 1959; Monin and Yaglom, 1967; Golitsyn, 2018; Gledzer and Golitsyn, 2010; Yaglom, 1955). However, the authors (Gledzer and Golitsyn, 2019) have not given a detailed explanation why, for the smallest scales, the relief spectrum of the Moon becomes steeper and behaves like k(-4). The same can be said for Mars and Earth on even smaller spatial scales (Rexer and Hirt, 2015). The explanation has been given by replacing the Markovian character of the probability distribution of accelerations by an internal exponential correlation. Similarity and dimensionality considerations involving the physical properties of the crust make it possible to estimate the scale of the features of the spectra observed.