The atmospheric convective boundary layer (CBL) consists of three basic parts: (1) the surface layer unstably stratified and dominated by small-scale turbulence of very complex nature; (2) the CBL core dominated by the energy-, momentum-, and mass-transport of semi-organized structures (large-scale circulations), with a small contribution from small-scale turbulence produced by local structural shears; and (3) turbulent entrainment layer at the upper boundary, characterized by essentially stable stratification with negative (downward) turbulent flux of potential temperature. The energy- and flux budget theory developed previously for atmospheric stably-stratified turbulence and the surface layer in atmospheric convective turbulence is extended to the CBL core using budget equations for turbulent energies and turbulent fluxes of buoyancy and momentum. For the CBL core, we determine global turbulent characteristics (averaged over the entire volume of the semi-organized structure) as well as kinetic and thermal energies of the semi-organized structures as the functions of the aspect ratio of the semi-organized structure, the scale separation parameter between the vertical size of the structures and the integral scale of turbulence and the degree of thermal anisotropy characterized the form of plumes. The obtained theoretical relationships are potentially useful in modeling applications in the atmospheric convective boundary-layer and analysis of laboratory and field experiments, direct numerical simulations, and large-eddy simulations of convective turbulence with large-scale semi-organized structures.

The energy- and flux-budget (EFB) theory developed previously for atmospheric stably stratified turbulence is extended to the surface layer in atmospheric convective turbulence. This theory is based on budget equations for turbulent energies and fluxes in the Boussinesq approximation. In the lower part of the surface layer in the atmospheric convective boundary layer, the rate of turbulence production of the turbulent kinetic energy (TKE) caused by the surface shear is much larger than that caused by the buoyancy, which results in three-dimensional turbulence of very complex nature. In the upper part of the surface layer, the rate of turbulence production of TKE due to the shear is much smaller than that caused by the buoyancy, which causes unusual strongly anisotropic buoyancy-driven turbulence. Considering the applications of the obtained results to the atmospheric convective boundary-layer turbulence, the theoretical relationships potentially useful in modeling applications have been derived. The developed EFB theory allows us to obtain a smooth transition between a stably stratified turbulence to a convective turbulence. The EFB theory for the surface layer in a convective turbulence provides an analytical expression for the entire surface layer including the transition range between the lower and upper parts of the surface layer, and it allows us to determine the vertical profiles for all turbulent characteristics, including TKE, the intensity of turbulent potential temperature fluctuations, the vertical turbulent fluxes of momentum and buoyancy (proportional to potential temperature), the integral turbulence scale, the turbulence anisotropy, the turbulent Prandtl number, and the flux Richardson number.

Compressibility effects in a turbulent transport of temperature field are investigated by applying the quasilinear approach for small Peclet numbers and the spectral tau approach for large Peclet numbers. The compressibility of a fluid flow reduces the turbulent diffusivity of the mean temperature field similarly to that for the particle number density and magnetic field. However, expressions for the turbulent diffusion coefficient for the mean temperature field in a compressible turbulence are different from those for the mean particle number density and the mean magnetic field. The combined effect of compressibility and inhomogeneity of turbulence causes an increase of the mean temperature in the regions with more intense velocity fluctuations due to a turbulent pumping. Formally, this effect is similar to a phenomenon of compressible turbophoresis found previously [J Plasma Phys. 84, 735840502 (2018)] for noninertial particles or gaseous admixtures. The gradient of the mean fluid pressure results in an additional turbulent pumping of the mean temperature field. The latter effect is similar to the turbulent barodiffusion of particles and gaseous admixtures. The compressibility of a fluid flow also causes a turbulent cooling of the surrounding fluid due to an additional sink term in the equation for the mean temperature field. There is no analog of this effect for particles.

The energy and flux budget (EFB) closure theory for a passive scalar (non-buoyant and non-inertial particles or gaseous admixtures) is developed for stably stratified turbulence. The physical background of the EFB turbulence closures is based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy as well as the turbulent flux of particles. The EFB turbulence closure is designed for stratified geophysical flows from neutral to very stable stratification, and it implies that turbulence is maintained by the velocity shear at any stratification. In a steady-state, expressions for the turbulent flux of the passive scalar and the anisotropic non-symmetric turbulent diffusion tensor are derived, and universal flux Richardson number dependencies of the components of this tensor are obtained. The diagonal component in the vertical direction of the turbulent diffusion tensor is suppressed by strong stratification, while the diagonal components in the horizontal directions are not suppressed, but they are dominant in comparison with the other components of the turbulent diffusion tensor. This implies that any initially created strongly inhomogeneous particle cloud is evolved into a thin pancake in a horizontal plane with very slow increase in its thickness in the vertical direction. The turbulent Schmidt number (the ratio of the eddy viscosity and the vertical turbulent diffusivity of the passive scalar) linearly increases with the gradient Richardson number. The physics of such a behavior is related to the buoyancy force that causes a correlation between fluctuations of the potential temperature and the particle number density. This correlation that is proportional to the product of the vertical turbulent particle flux and the vertical gradient of the mean potential temperature reduces the vertical turbulent particle flux. Considering the applications of these results to the atmospheric boundary-layer turbulence, the theoretical relationships are derived, which allows us to determine the turbulent diffusion tensor as a function of the vertical coordinate measured in the units of the local Obukhov length scale. The obtained relations are potentially useful in modeling applications of particle dispersion in the atmospheric boundary-layer turbulence and free atmosphere turbulence.

A turbulent transport of radiation in the solar convective zone is investigated. The mean-field equation for the irradiation intensity is derived. It is shown that due to the turbulent effects, the effective penetration length of radiation can be increased several times in comparison with the mean penetration length of radiation (defined as an inverse mean absorption coefficient). Using the model of the solar convective zone based on mixing length theory, where the mean penetration length of radiation is usually much smaller than the turbulent correlation length, it is demonstrated that the ratio of the effective penetration length to the mean penetration length of radiation increases 2.5 times in the vicinity of the solar surface. The main reasons for this are the compressibility effects that become important in the vicinity of the solar surface where temperature and density fluctuations increase towards the solar surface, enhancing fluctuations of the radiation absorption coefficient and increasing the effective penetration length of radiation.

A review of analytical and numerical results on effects of developed magnetohydrodynamic (MHD) turbulence on mean magnetic pressure and formation of magnetic structures is presented. Suppression of turbulent hydromagnetic pressure (the isotropic part of combined Reynolds and Maxwell stresses) by the mean large-scale magnetic field is related to an effective mechanism for the formation of magnetic inhomogeneous structures in MHD turbulence. At large Reynolds numbers and for sub-equipartition mean magnetic fields, the resulting negative turbulent contribution can be enough large so that the effective mean magnetic pressure (the sum of turbulent and non-turbulent contributions) appears negative. We also investigated the effect of mean current density on the turbulent hydromagnetic pressure reduction, and demonstrated that an enhanced mean current density increases the suppression of the turbulent pressure. Such currents are associated with sharp gradients of the growing magnetic structures. The negative effective mean magnetic pressure was found in direct numerical simulation (DNS) in both, stably stratified forced turbulence and turbulent convection. This phenomenon causes the excitation of the negative effective magnetic pressure instability (NEMPI). By the action of this instability, an initially uniform magnetic field forms flux concentrations whose scale is large compared to the turbulent scale. This instability has been recently detected in DNS of forced stratified MHD turbulence that requires enough large scale separation between the forcing scale and the size of the box (e.g., the number of turbulent eddies in the computational domain is about 30). Strong spontaneous formation of large-scale magnetic structures caused by NEMPI, is seen even without performing any spatial averaging. The characteristic time of the instability is comparable to the turbulent diffusion time. We also demonstrated that the magnetic energy of the forming large-scale inhomogeneous magnetic structures is only weakly dependent on the magnetic Reynolds number, provided its value is large enough for the excitation of NEMPI. Our DNS results support mean-field calculations and analytical results which identified this instability. For example, for an isothermal layer the onset of the instability occurs at the same depth that increases with increasing field strength, the growth rate of NEMPI is independent of the field strength, provided the magnetic structures are fully contained within the domain. NEMPI may play a crucial role in the formation of sunspots and active regions in the upper part of convective zones of Sun and stars. 

The tilt angle, current helicity and twist of solar magnetic fields can be observed in solar active regions. We carried out estimates of these parameters by two ways. Firstly, we consider the model of turbulent convective cells (super-granules) which have a loop floating structure towards the surface of the Sun. Their helical properties are attained during the rising process in the rotating stratified convective zone. The other estimate is obtained from a simple mean-field dynamo model that accounts magnetic helicity conservation. The both values are shown to be capable to give important contributions to the observable tilt, helicity and twist.

A theory of the mean tilt of sunspot bipolar regions (the angle between a line connecting the leading and following sunspots and the solar equator) is developed. Amechanism of formation of the mean tilt is related to the effect of the Coriolis force on meso-scale motions of supergranular convection and large-scale meridional circulation. The balance between the Coriolis force and the Lorentz force (the magnetic tension) determines an additional contribution caused by the large-scale magnetic field to the mean tilt of the sunspot bipolar regions at low latitudes. The latitudinal dependence of the solar differential rotation affects the mean tilt, which can explain deviations from Joy's law for the sunspot bipolar regions at high latitudes. The theoretical results obtained and the results from numerical simulations based on the non-linear mean-field dynamo theory, which takes into account conservation of the total magnetic helicity and the budget equation for the evolution of the Wolf number density, are in agreement with observational data of the mean tilt of sunspot bipolar regions over individual solar cycles 15-24.

An asymmetry in the number density of left- and right-handed fermions is known to give rise to a new term in the induction equation that can result in a dynamo instability. At high temperatures, when a chiral asymmetry can survive for long enough, this chiral dynamo instability can amplify magnetic fields efficiently, which in turn drive turbulence via the Lorentz force. While it has been demonstrated in numerical simulations that this chiral magnetically driven turbulence exists and strongly affects the dynamics of the magnetic field, the details of this process remain unclear. The goal of this paper is to analyse the energetics of chiral magnetically driven turbulence and its effect on the generation and dynamics of the magnetic field using direct numerical simulations. We study these effects for different initial conditions, including a variation of the initial chiral chemical potential and the magnetic Prandtl number, . In particular, we determine the ratio of kinetic to magnetic energy, , in chiral magnetically driven turbulence. Within the parameter space explored in this study, reaches a value of approximately 0.064-0.074-independently of the initial chiral asymmetry and for . Our simulations suggest, that decreases as a power law when increasing by decreasing the viscosity. While the exact scaling depends on the details of the fitting criteria and the Reynolds number regime, an approximate result of is reported. Using the findings from our numerical simulations, we analyse the energetics of chiral magnetically driven turbulence in the early Universe.

We find an instability resulting in generation of large-scale vorticity in a fast-rotating small-scale turbulence or turbulent convection with inhomogeneous fluid density along the rotational axis in anelastic approximation. The large-scale instability causes excitation of two modes: (i) the mode with dominant vertical vorticity and with the mean velocity being independent of the vertical coordinate; (ii) the mode with dominant horizontal vorticity and with the mean momentum being independent of the vertical coordinate. The mode with the dominant vertical vorticity can be excited in a fast-rotating density-stratified hydrodynamic turbulence or turbulent convection. For this mode, the mean entropy is depleted inside the cyclonic vortices, while it is enhanced inside the anticyclonic vortices. The mode with the dominant horizontal vorticity can be excited only in a fast-rotating density-stratified turbulent convection. The developed theory may be relevant for explanation of an origin of large spots observed as immense storms in great planets, e.g., the Great Red Spot in Jupiter and large spots in Saturn. It may be also useful for explanation of an origin of high-latitude spots in rapidly rotating late-type stars.