Nathan Kleeorin

Turbulence energetics in stably stratified geophysical flows: Strong and weak mixing regimes

Traditionally, turbulence energetics is characterised by turbulent kinetic energy (TKE) and modelled using solely the TKE budget equation. In stable stratification, TKE is generated by the velocity shear and expended through viscous dissipation and work against buoyancy forces. The effect of stratification is characterised by the ratio of the buoyancy gradient to squared shear, called the Richardson number, Ri. It is widely believed that at Ri exceeding a critical value, Ric, local shear cannot maintain turbulence, and the flow becomes laminar. We revise this concept by extending the energy analysis to turbulent potential and total energies (TPE, and TTE = TKE + TPE), consider their budget equations, and conclude that TTE is a conservative parameter maintained by shear in any stratification. Hence there is no ‘energetics Ric’, in contrast to the hydrodynamic-instability threshold, Ric−instability, whose typical values vary from 0.25 to 1. We demonstrate that this interval, 0.25 < Ri < 1, separates two different turbulent regimes: strong mixing and weak mixing rather than the turbulent and the laminar regimes, as the classical concept states. This explains persistent occurrence of turbulence in the free atmosphere and deep ocean at Ri ≫ 1, clarifies the principal difference between turbulent boundary layers and free flows, and provides the basis for improving operational turbulence closure models. Copyright

Generation of Magnetic Field by Combined Action of Turbulence and Shear

The feasibility of a mean-field dynamo in nonhelical turbulence with a superimposed linear shear is studied numerically in elongated shearing boxes. Exponential growth of the magnetic field at scales much larger than the outer scale of the turbulence is found. The characteristic scale of the field is lB proportional S(-1/2) and the growth rate is gamma proportional S, where S is the shearing rate. This newly discovered shear dynamo effect potentially represents a very generic mechanism for generating large-scale magnetic fields in a broad class of astrophysical systems with spatially coherent mean flows.

THE MEAN ELECTROMOTIVE FORCE FOR MHD TURBULENCE: THE CASE OF A WEAK MEAN MAGNETIC FIELD AND SLOW ROTATION

The mean electromotive force that occurs in the framework of mean-field magnetohydrodynamics is studied for cases in which magnetic field fluctuations are not only due to the action of velocity fluctuations on the mean magnetic field. The possibility of magnetic field fluctuations independent of a mean magnetic field, as they may occur as a consequence of a small-scale dynamo, is taken into account. Particular attention is paid to the effect of a mean rotation of the fluid on the mean electromotive force, although only small rotation rates are considered. Anisotropies of the turbulence due to gradients of its intensity or its helicity are admitted. The mean magnetic field is considered to be weak enough to exclude quenching effects. A � -approximation is used in the equation describing the deviation of the cross-helicity tensor from that for zero mean magnetic field, which applies in the limit of large hydrodynamic Reynolds numbers. For the effects described by the mean electromotive force like � -effect, turbulent diffusion of magnetic fields etc in addition to the contributions determined by the velocity fluctuations also those determined by the magnetic field fluctuations independent of the mean magnetic field are derived. Several old results are confirmed, partially under more general assumptions, and quite a few new ones are given. Provided the kinematic helicity and the current helicity of the fluctuations have the same signs the � -effect is always diminished by the magnetic fluctuations. In the absence of rotation these have, however, no influence on the turbulent diffusion. Besides the diamagnetic effect due to a gradient of the intensity of the velocity fluctuations there is a paramagnetic effect due to a gradient of the intensity of the magnetic fluctuations. In the absence of rotation these two effects compensate each other in the case of equipartition of the kinetic and magnetic energies of the fluctuations of the original turbulence, i.e. that with zero mean magnetic field, but the rotation makes the situation more complex. The:TJ-effect works in the same way with velocity fluctuations and magnetic field fluctuations. A contribution to the electromotive force connected with the symmetric parts of the gradient tensor of the mean magnetic field, which does not occur in the absence of rotation, was found in the case of rotation, resulting from velocity or magnetic fluctuations. The implications of the results for the mean electromotive force for mean-field dynamo models are discussed with special emphasis to dynamos working without � -effect. The results for the coefficients defining the mean electromotive force which are determined by the velocity fluctuations in the case of vanishing mean motion agree formally with the results obtained in the kinematic approach, specified by second-order approximation and high-conductivity limit. However, their range of validity is clearly larger.

Magnetic helicity evolution during the solar activity cycle: Observations and dynamo theory

We study a simple model for the solar dynamo in the framework of the Parker migratory dynamo, with a nonlinear dynamo saturation mechanism based on magnetic helicity conservation arguments. We find a parameter range in which the model demonstrates a cyclic behaviour with properties similar to that of Parker dynamo with the simplest form of algebraic α -quenching. We compare the nonlinear current helicity evolution in this model with data for the current helicity evolution obtained during 10 years of observations at the Huairou Solar Station of China. On one hand, our simulated data demonstrate behaviour comparable with the observed phenomenology, provided that a suitable set of governing dynamo parameters is chosen. On the other hand, the observational data are shown to be rich enough to reject some other sets of governing parameters. We conclude that, in spite of the very preliminary state of the observations and the crude nature of the model, the idea of using observational data to constrain our ideas concerning magnetic field generation in the framework of the solar dynamo appears promising.

Magnetic fluctuations and formation of large-scale inhomogeneous magnetic structures in a turbulent convection.

Magnetic fluctuations generated by a tangling of the mean magnetic field by velocity fluctuations are studied in a developed turbulent convection with large magnetic Reynolds numbers. We show that the energy of magnetic fluctuations depends on the magnetic Reynolds number only when the mean magnetic field is smaller than B(eq)/4Rm(1/4), where B(eq) is the equipartition mean magnetic field determined by the turbulent kinetic energy and Rm is the magnetic Reynolds number. Generation of magnetic fluctuations in a turbulent convection with a nonzero mean magnetic field results in a decrease of the total turbulent pressure and may cause the formation of large-scale inhomogeneous magnetic structures even in an originally uniform mean magnetic field. This effect is caused by a negative contribution of the turbulent convection to the effective mean Lorentz force. The inhomogeneous large-scale magnetic fields are formed due to excitation of the large-scale instability. The energy for this instability is supplied by small-scale turbulent convection. The discussed effects might be useful for understanding the origin of solar nonuniform magnetic fields: e.g., sunspots.

The role of magnetic helicity transport in nonlinear galactic dynamos

We consider the magnetic helicity balance for the galactic dynamo in the framework of the local dynamo problem, as well as in the no-z model (which includes explicitly the radial distribution of the magnetic elds). When calculating the magnetic helicity balance we take into account the redistribution of the small-scale and large-scale magnetic elds between the magnetic helicities, as well as magnetic helicity transport and diusion due to small- scale turbulence. We demonstrate that the magnetic helicity flux through the galactic disc boundaries leads to a steady-state magnetic eld with magnetic energy comparable to the equipartition energy of the turbulent motions of the interstellar medium. If such flux is ignored, the steady-state magnetic eld is found to be much smaller than the equipartition eld. The total magnetic helicity flux through the boundaries consists of both an advective flux and a diusive flux. The exact ratio of these contributions seems not to be crucial for determining the strength of the steady-state magnetic eld and its structure. However at least some diusive contribution is needed to smooth the magnetic helicity prole near to the disc boundaries. The roles of various transport coecients for magnetic helicity are investigated, and the values which lead to magnetic eld congurations comparable with those observed are determined.

Magnetic helicity tensor for an anisotropic turbulence

The evolution of the magnetic helicity tensor for a nonzero mean magnetic field and for large magnetic Reynolds numbers in an anisotropic turbulence is studied. It is shown that the isotropic and anisotropic parts of the magnetic helicity tensor have different characteristic times of evolution. The time of variation of the isotropic part of the magnetic helicity tensor is much larger than the correlation time of the turbulent velocity field. The anisotropic part of the magnetic helicity tensor changes for the correlation time of the turbulent velocity field. The mean turbulent flux of the magnetic helicity is calculated as well. It is shown that even a small anisotropy of turbulence strongly modifies the flux of the magnetic helicity. It is demonstrated that the tensor of the magnetic part of the alpha effect for weakly inhomogeneous turbulence is determined only by the isotropic part of the magnetic helicity tensor.

DETECTION OF NEGATIVE EFFECTIVE MAGNETIC PRESSURE INSTABILITY IN TURBULENCE SIMULATIONS

We present the first demonstration of the negative effective magnetic pressure instability in direct numerical simulations of stably stratified, externally forced, isothermal hydromagnetic turbulence in the regime of large plasma beta. By the action of this instability, an initially uniform horizontal magnetic field forms flux concentrations whose scale is large compared to the turbulent scale. We further show that the magnetic energy of these large-scale structures is only weakly dependent on the magnetic Reynolds number, provided its value is large enough for the instability to be excited. Our results support earlier mean-field calculations and analytic work which identified this instability. Applications to the formation of active regions in the Sun are discussed.

The radial distribution of magnetic helicity in the solar convective zone: observations and dynamo theory

We continue our attempt to connect observational data on current helicity in solar active regions with solar dynamo models. In addition to our previous results about temporal and latitudinal distributions of current helicity, we argue that some information concerning the radial profile of the current helicity averaged over time, and latitude can be extracted from the available observations. The main feature of this distribution can be presented as follows. Both shallow and deep active regions demonstrate a clear dominance of one sign of current helicity in a given hemisphere during the whole cycle. Broadly speaking, current helicity has opposite polarities in the Northern and Southern hemispheres, although there are some active regions that violate this polarity rule. The relative number of active regions violating the polarity rule is significantly higher for deeper active regions. A separation of active regions into ‘shallow’, ‘middle’ and ‘deep’ is made by comparing their rotation rate and the helioseismic rotation law. We use a version of Parker’s dynamo model in two spatial dimensions, which employs a nonlinearity based on magnetic helicity conservation arguments. The predictions of this model about the radial distribution of solar current helicity appear to be in remarkable agreement with the available observational data; in particular the relative volume occupied by the current helicity of ‘wrong’ sign grows significantly with the depth.

THE NEGATIVE EFFECTIVE MAGNETIC PRESSURE IN STRATIFIED FORCED TURBULENCE

To understand the basic mechanism of the formation of magnetic flux concentrations, we determine by direct numerical simulations the turbulence contributions to the mean magnetic pressure in a strongly stratified isothermal layer with large plasma beta, where a weak uniform horizontal mean magnetic field is applied. The negative contribution of turbulence to the effective mean magnetic pressure is determined for strongly stratified forced turbulence over a range of values of magnetic Reynolds and Prandtl numbers. Small-scale dynamo action is shown to reduce the negative effect of turbulence on the effective mean magnetic pressure. However, the turbulence coefficients describing the negative effective magnetic pressure phenomenon are found to converge for magnetic Reynolds numbers between 60 and 600, which is the largest value considered here. In all these models, the turbulent intensity is arranged to be nearly independent of height, so the kinetic energy density decreases with height due to the decrease in density. In a second series of numerical experiments, the turbulent intensity increases with height such that the turbulent kinetic energy density is nearly independent of height. Turbulent magnetic diffusivity and turbulent pumping velocity are determined with the test-field method for both cases. The vertical profile of the turbulent magnetic diffusivity is found to agree with what is expected based on simple mixing length expressions. Turbulent pumping is shown to be down the gradient of turbulent magnetic diffusivity, but it is twice as large as expected. Corresponding numerical mean-field models are used to show that a large-scale instability can occur in both cases, provided the degree of scale separation is large enough and hence the turbulent magnetic diffusivity small enough.