Numerical and laboratory experiments have been carried out to study the intrusion of anomalies into the axisymmetric distribution of the velocity field generated by sink sources and the MHD method in a rotating circular channel with a sloping bottom. A sectoral decrease in the intensity of the external force in a certain range of values has a decelerating effect on the velocity of propagation of anticyclones through the channel, while it has almost no influence on the dynamics of cyclones. At the same time, a significant part of moving anticyclones can disappear or almost stop, or new quasi-stationary anticyclones appear, despite there being no notable changes in the visible pattern of eddy propagation in the channel in the sector in which external intervention had been performed. However, changes are notable over the entire channel or in its individual parts in the mean characteristics of the eddy field. These anomalies can be interpreted as a decrease in the intensity of the subtropical Hadley cell, which is accompanied by a weakening of the trade winds in a sector of the equatorial atmospheric circulation and a decrease in the westerly transport at midlatitudes. The state of the mixture of standing and moving eddies is considered on the basis of a simple analytical model of the resonant interaction of transient (with an intermediate velocity maximum) modes in a shear flow. In this case, the amplitude of the stationary background state has the same dependence on the β effect as in the well-known Sverdrup relation for the stream function of the surface current in the ocean basin in the studies of the western boundary current intensification.

The model balance equation of turbulence is represented in the form of the first law of thermodynamics. The forms for entropy are derived with an integrating factor. For the Kolmogorov–Obukhov law, the entropy has the same expression as for an ideal gas. A possible formula for the turbulence temperature is proposed on the basis of hydrodynamical instability processes.

The idea of the multiplicity of equilibrium states of the atmospheric circulation in geophysical hydrodynamics goes back to Charne and DeVore 1979, where, for a model with a small number of variables, solutions with significantly different values of the zonal and wave velocity components were obtained (see also Laurie, Bouchet 2015, Herbert et al. 2020). The results of similar studies for low-parameter approximations were given by Kallen 1980, Gluhovsky 2001, Koo, Ghil 2002 ... The circulation modes differed in the magnitude of the zonal component of the flow. At weak transport, the role of almost stationary atmospheric eddies is enhanced, which corresponds to circulation blocking modes. Laboratory confirmation of the effect was obtained from Weeks et al. 1997, Tian et al. 2001.In the same years, in the experiments of A.M. Obukhov and coworkers, modes with differently directed axes of large-scale fluid rotation were observed in closed vessels at the same value of external generation - Obukhov et al. 1976, Gledzer et al. 1981.In the present study, supported by Russian Science Foundation (Project 19-17-00248), the above types of multi-mode are considered based on laboratory and numerical experiments in circular rotating channels. It is known that the permanent magnet location configurations (source-sinks) could create an almost stationary vortex distribution pattern Gledzer et al. 2013,2014. The transition between different states is provided by a change in the value of the main parameter of electric current generation with subsequent restoration of its initial value.The experimental results presented below are obtained for a rotating annular channel (rotation periods up to 1 minute) filled with an electrically conductive 10% copper sulfate solution. The bottoms of circular channels with inner and outer radii of 1) 5.5 cm and 18 cm 2) 5 cm and 36 cm have an axisymmetric conical shape with a height of 1 cm.Depending on channel rotation periods or source configurations, it is possible: 1) Initial and final modes differ quantitatively in the number of generated vortices. 2) The number of vortex formations does not change, but differ in their spatial localization. 3) After changing and restoring the value of the defining parameter, the flow returns to the mode which is practically the same as the initial one.Numerical experiments with the shallow water model confirmed the results obtained in laboratory experiments on the possibility of transition to new modes when the parameter determining the external force is changed for some time. For the source-sink method, a change in the number of large vortices (cyclones) is observed. At MHD generation it is possible to detect a change in the finite spatial position of vortices with preservation of their number.Experiments support the conclusion that different modes of barotropic dynamics may exist. And it is unlikely to be associated with any low-parameter approximation of the velocity field in the model.In our and earlier experiments and models, multi-mode is a property of dynamics in general. The mechanism of multi-mode may be an alternative to the traditional scenario of transition to other modes when external conditions change.

Results of field measurements taken in the near-Caspian (2002, 2003, 2007, 2009, 2010, 2011, 2013, 2014, 2016 years) and near-Aral sea (1998) deserts under the conditions of weak winds (almost in the absence of saltation processes) and strong heating of the land surface are given. These results show that the fine mineral dust aerosol (0.1-1 mu m) considerably contributes to the total aerosol content of the atmospheric surface layer under such conditions. The scaling of daytime mean size d distribution at a height of 2 m is approximated by dependence d(-mu), where mu changes from 4 to 6 for different years of measurements, in contrary to the law d-2 for fraction d > 1 mu m. Different compositions of aerosol particles at 0.1 < d < 1 mu m and d > 1 mu m, including multicomponent fractions (less than 1 mu m) may result in different probabilities of their integration and disintegration. It is hypothesized that the equilibrium distribution of submicron aerosol is a result of sequential disintegration of aerosol particles with a possibility for their reintegration. The simplest distribution approximations are given based on the Kolmogorov direct differential equation which is usually used in considering the Poisson distributions in equations for queuing systems.

The possible existence of distinct regimes of barotropic circulation in closed annular channels at the same external parameters governing the flow dynamics is investigated both experimentally and numerically. Transitions between the regimes are realized by means of varying the value of the main parameter determining the velocity field energy (for example, the current controlling the Ampere force in the case of MHD generation of a velocity field) with subsequent reconstruction of the former parameter value. Depending on the channel rotation period or the configurations of magnet locations in the case of MHD generation or sources and sinks in numerical experiments the following results are possible. (1) The initial and final regimes differ quantitavely in the number of cyclonic or anticyclonic vortices generatted. (2) The number of vortex formations does not change but their localization in space, for example, the angular coordinates of their centers, varies. (3) After the change and reconstruction of the original value of the governing parameter the flow returns to the regime almost undistinguishable from the original regime. The flow patterns and the corresponding diagrams for laboratory experiments and numerical simulations based on shallow water equations are presented.

The particle size distribution function is one of the characteristics reflecting the composition of aerosol during sand lifting and removal in desert regions. This characteristic, in addition to known practical applications, is important in describing radiation processes during the exchange of heat fluxes and in forming cloud systems in the models of atmospheric dynamics. Fine dust-aerosol fractions (less than 2 µm in diameter) are especially important for the atmospheric radiation budget, because such fractions (having a significant lifetime) most efficiently interact with short-wave solar radiation. One of the central regularities in considering the size distributions of simulated dust-aerosol particles is the following formula based on the so-called fragmentation process and verified using a large amount of empirical data N (d) ~ d -2. Similar dependence for particles with size d > 1 µm is associated with the consideration of the fragmentation process as a particle splitting according to the log-normal distribution. Results of field measurements taken in the near–Caspian (2002, 2003, 2007, 2009, 2010, 2011, 2013, 2014, 2016 years) and near–Aral-sea (1998) deserts under the conditions of weak winds (almost in the absence of saltation processes) and strong heating of the land surface are given. These results show that the fine mineral dust aerosol (0.1-1 µm) considerably contributes to the total aerosol content of the atmospheric surface layer under such conditions. The scaling of daytime mean size d distribution at a height of 2 m is close to d -5 in contrast to the law d -2 for fraction d >1 µm. Different compositions of aerosol particles at 0.1 < d < 1 µm, and d >1 µm, including multicomponent fractions (less than 1 µm) may result in different probabilities of their integration and disintegration, which, finally, determine equilibrium particle size distributions. The simplest distribution approximations based on the Kolmogorov direct differential equation are given. This study was supported by the RFBR (19-05-50110) and the Presidium of the Russian Academy of Sciences (programs 12 and 20).

Kaula’s empirical rule has been known for more than 50 years: the coefficients of expansion over spherical harmonics for the fluctuations of the gravitational field and terrain of the planets decrease as the number of the harmonic squared. This was found for Venus, the Moon, Mars, the asteroid Vesta, and very small celestial bodies. The inverse-square line spectra were also found for various types of the Earth’s surface on a scale of up to a hundred kilometers. From this it follows that the spectra of the terrain slope angles are constant, i.e., “white noise.” Thus, they are δ-correlated horizontally. These are the assumptions under which the random walk laws were derived by A.N. Kolmogorov in 1934. Using them, the equation of the horizontal probability diffusion of the terrain with the linear coefficient of diffusion D is derived. Based on the empirical data, D = 1.3 ± 0.3 m for the Earth, while for Venus it is almost an order of magnitude less. The slopes resist the wind; the rock crumbles, and the water flows down the slopes as well. This consideration turns Kuala’s rule into the random walk laws (over terrain) developed by Kolmogorov in 1934.

At the beginning of 1930-s A. N. Kolmogorov has published three papers on analytical methods for the probability theory. The two-page work had the essence of the approach started by A. Einstein and developed further by Fokker and Planck. He proposed a fundamental solution for evolution description of the 6D vector of the probability distribution at the Markov character of action which in modern terms is usually called as time delta-correlated acceleration. This is an approximation of processes when correlation time of random forces is much smaller than the reaction time of the system in consideration.

The results of the laboratory and numerical experiments in circular rotating trays with thin layers of a conductive fluid under the MHD generation of small-scale velocity fields are presented. The configurations of constant magnets for MHD generation were determined based on the numerical calculations with shallow water equations. Both the laboratory and numerical experiments with rotating trays demonstrate the emergence of nonaxisymmetric structures and large-scale near-circular vortices caused by the energy transfer from the system of the externally generated small-scale vortices to the large-scale velocity fields under the action of the Coriolis force. The near-circular vortex has areas with differential rotation when the angular velocity of rotation decreases with the radius. The single large-scale vortices and wide jet flows arise in the regimes of subrotation and superrotation relative to the external rotation depending on its angular velocity. The emergence of the flow structures with the azimuthal wave number m = 2 is demonstrated, and their probable relation to the anomalies of the geomagnetic field observed on the Earth’s surface is considered.

Interactions violating the symmetry of positive and negative total helicity components are considered. In the ideal case where one of the components is zero, the system have two sign-definite integrals of motion, which lead to an inverse energy cascade, as occurs in two-dimensional turbulence. The generation of large-scale modes is considered in the quasi-normal approximation and is manifested as the instability of second moments, a mechanism of which was discussed at the end of previous century. A crucial point in this mechanism is the presence of mean turbulence with large-scale helical disturbances and small-scale sources of energy and helicity. In the case of both helicity components being nonzero, the possibility of the large-scale generation is studied by applying numerical experiments with a shell model and by analyzing special cases of interactions between different shells of the model. In all the approaches used, it is shown that an inverse energy flux (from small to large scales) can exist at a certain level of external helical noises in large-scale modes, which depends on the degree of “mixing” oppositely signed helicity components.

This paper presents the results of experiments in a circular stationary and rotating channels with thin layers of conductive fluid for configurations consisting of a large number of permanent magnets and providing the MHD generation of small-scale velocity fields. The alternating radial configurations of magnets were chosen in such a way as to ensure the conservation of a discrete symmetry of their mutual arrangement relative to rotations of the circular channel around a central axis and were formed on the basis of numerical calculations with the shallow-water equations. Both in numerical and laboratory experiments, large-scale nearly circular vortices were obtained as a result of the energy transfer from the system of externally generated small-scale vortices to large-scale velocity fields (inverse cascade) under the influence of the Coriolis force in the rotating case. Single large-scale vortices and wide jet streams appear in subrotation and superrotation modes relative to external rotation, depending on its angular velocity. Rotation in a nearly circular vortex has a differential character with a decrease in the angular velocity of rotation with the radius in most area of the channel.

The interactions breaking the symmetry of the positive and negative components of the total helicity have been considered with quasinormal approximations and the cascade model of helical turbulence. In the ideal case of the absence of one of them, the equations have two sign-definite integrals of motion; as a result, an inverse energy cascade occurs, as for two-dimensional turbulence. Owing to the instability of the second moments, whose mechanism was proposed in A. Belian, O. Chkhetiani, E. Golbraikh, and S. Moiseev, Physica A 258, 55 (1998), the generation of large-scale modes has been considered within the Fourier equations of hydrodynamics under the assumption of a quasinormal velocity field. The presence of background turbulence with large-scale helical perturbations and small-scale sources of the energy and helicity is a determining circumstance in this mechanism. The possibility of the inverse cascade in the case of incomplete degeneracy of one of the helicity components has been studied in the numerical experiments with the cascade model. It has been shown that the existence of the inverse (toward large scales) energy flux from small-scale perturbations requires a certain external-action-generated level of helical noise in large modes, which depends on the degree of "mixing" of different-sign helical components of the velocity field.

Results of experiments are considered for flows generated by different sources-sinks of mass in the rotating annular channel with beta-effect simulation using the inclined bottom. Diagrams of regimes are presented in parameters of the dimensionless angular velocity of the zonal flow averaged over the channel width and the dimensionless angular velocity of transport of vortex perturbations of cyclonic and anticyclonic types. In experiments and the simplest linear theories, most attention is paid to diagram regions with a slow motion of vortices relative to the rotating coordinate system near the parameters for stationary Rossby waves.

Experiments on the excitation of counterpropagating zonal flows by the magnetohydrodynamic (MHD) method in a rotating cylindrical vessel with a conic bottom have been performed. Flows appear in a conducting fluid layer in the field of ring magnets under the action of a radial electric field. The velocity fields have been reconstructed by the particle image velocimetry (PIV) method. In the fast rotation regimes with a thin fluid layer, where the Rossby-Obukhov scale does not exceed the characteristic sizes of the vessel, the system of perturbations appears with almost immobile blocked anticyclones in the outer part of the flow and rapidly moving cyclones in the main stream. The diagram of regimes is plotted in the variables of the relative angular velocities of the averaged zonal flow and transfer of vortices about the system rotation axis. Attention is focused on the results for the regions of the diagram with slow motion of vortices with respect to the rotating coordinate system near the parameters for stationary Rossby waves (blocking of circulation). The results are compared to the results previously obtained in similar experiments using the source-sink method.