Helical Mode Absolute Statistical Equilibrium of Ideal Three-Dimensional Hall Magnetohydrodynamics

Using the Fourier helical decomposition,we obtain the absolute statistical equilibrium spectra of left-and righthanded helical modes in the incompressible ideal Hall magnetohydrodynamics(MHD). It is shown that the left-handed helical modes play a major role on the spectral transfer properties of turbulence when the generalized helicity and magnetic helicity are both positive. In contrast, the right-handed helical modes will play a major role when both are negative. Furthermore, we also find that if the generalized helicity and magnetic helicity have opposite signs, the tendency of equilibrium spectra to condense at the large or small wave numbers will be presented in different helical sectors. This indicates that the generalized helicity dominates the forward cascade and the magnetic helicity dominates the inverse cascade properties of the Hall MHD turbulence.     

Chandrasekhar–Kendall functions in astrophysical dynamos

Some of the contributions of Chandrasekhar to the field of magnetohydrodynamics are highlighted. Particular emphasis is placed on the Chandrasekhar–Kendall functions that allow a decomposition of a vector field into right- and left-handed contributions. Magnetic energy spectra of both contributions are shown for a new set of helically forced simulations at resolutions higher than what has been available so far. For a forcing function with positive helicity, these simulations show a forward cascade of the right-handed contributions to the magnetic field and nonlocal inverse transfer for the left-handed contributions. The speed of inverse transfer is shown to decrease with increasing value of the magnetic Reynolds number.     

Energy transfer and dual cascade in kinetic magnetized plasma turbulence

The question of how nonlinear interactions redistribute the energy of fluctuations across available degrees of freedom is of fundamental importance in the study of turbulence and transport in magnetized weakly collisional plasmas, ranging from space settings to fusion devices. In this Letter, we present a theory for the dual cascade found in such plasmas, which predicts a range of new behavior that distinguishes this cascade from that of neutral fluid turbulence. These phenomena are explained in terms of the constrained nature of spectral transfer in nonlinear gyrokinetics. Accompanying this theory are the first observations of these phenomena, obtained via direct numerical simulations using the gyrokinetic code AstroGK. The basic mechanisms that are found provide a framework for understanding the turbulent energy transfer that couples scales both locally and nonlocally.     

Local shell-to-shell energy transfer via nonlocal interactions in fluid turbulence

In this paper we analytically compute the strength of nonlinear interactions in a triad, and the energy exchanges between wave-number shells in incompressible fluid turbulence. The computation has been done using first-order perturbative field theory. In three dimensions, magnitude of triad interactions is large for nonlocal triads, and small for local triads. However, the shell-to-shell energy transfer rate is found to be local and forward. This result is due to the fact that the nonlocal triads occupy much less Fourier space volume than the local ones. The analytical results on three-dimensional shell-to-shell energy transfer match with their numerical counterparts. In two-dimensional turbulence, the energy transfer rates to the nearby shells are forward, but to the distant shells are backward; the cumulative effect is an inverse cascade of energy.     

The role of helicity in turbulent MHD flows

We have studied the behavior of a helical homogeneous small-scale MHD turbulent flow under the influence of a weak inhomogeneous large-scale disturbance. We have shown that turbulent energy redistribution in the presence of nonzero helicity occurs mainly over large scales. Helicity increases correlation time, leading to the weakening of a direct cascade and to the formation of steep spectra over small scales, with simultaneous turbulent energy growth over large scales. Furthermore, an expression for the effective viscosity of the mean flow is derived. It is shown that the magnetic field, in addition to the helicity, reduces the effective viscosity of the medium. This may be important in the study of MHD flow around obstacles in the presence of an external magnetic field. Zh. éksp. Teor. Fiz. 114, 171–181 (July 1998) Published in English in the original Russian journal. Reproduced here with stylistic changes by the Translation Editor.     

Kinetic helicity,magnetic helicity and fast dynamo action

The influence of the flow helicity on kinematic fast dynamo action is considered. Three different flows are studied, possessing identical chaotic properties but very different distributions of helicity (maximal helicity, zero net helicity and zero helicity density). All three flows provide strong evidence of fast dynamo action, indicating that helicity is not a crucial feature of fast dynamo flows. Comparisons are made between the magnetic fields generated by the three flows and it is established how certain key quantities scale with the magnetic Reynolds number. In particular, it is shown that the relative magnetic helicity tends to zero as the magnetic Reynolds number tends to infinity.     

Energy flow along the medium-induced parton cascade

We discuss the dynamics of parton cascades that develop in dense QCD matter, and contrast their properties with those of similar cascades of gluon radiation in vacuum. We argue that such cascades belong to two distinct classes that are characterized respectively by an increasing or a constant (or decreasing) branching rate along the cascade. In the former class, of which the BDMPS, medium-induced, cascade constitutes a typical example, it takes a finite time to transport a finite amount of energy to very soft quanta, while this time is essentially infinite in the latter case, to which the DGLAP cascade belongs. The medium induced cascade is accompanied by a constant flow of energy towards arbitrary soft modes, leading eventually to the accumulation of the initial energy of the leading particle at zero energy. It also exhibits scaling properties akin to wave turbulence. These properties do not show up in the cascade that develops in vacuum. There, the energy accumulates in the spectrum at smaller and smaller energy as the cascade develops, but the energy never flows all the way down to zero energy. Our analysis suggests that the way the energy is shared among the offsprings of a splitting gluon has little impact on the qualitative properties of the cascades, provided the kernel that governs the splittings is not too singular.     

Helicity redistribution during relaxation of astrophysical plasmas

We present the first 3D numerical MHD simulations that show that Taylor's relaxation conjecture is not satisfied in some MHD evolution of magnetic configurations encountered in solar physics. We show that magnetic helicity can be slowly injected through the boundary into a magnetic configuration which then evolves into a MHD disruption, with the formation in finite time of a current sheet through which reconnection occurs, leading to a release of magnetic energy. While helicity is well conserved during the process, it is shown that the relaxed state is far from the constant- alpha linear force-free field that would be predicted by Taylor's conjecture.     

Imprint of large-scale flows on turbulence

We investigate the locality of interactions in hydrodynamic turbulence using data from a direct numerical simulation on a grid of 1024(3) points; the flow is forced with the Taylor-Green vortex. An inertial range for the energy is obtained in which the flux is constant and the spectrum follows an approximate Kolmogorov law. Nonlinear triadic interactions are dominated by their nonlocal components, involving widely separated scales. The resulting nonlinear transfer itself is local at each scale but the step in the energy cascade is independent of that scale and directly related to the integral scale of the flow. Interactions with large scales represent 20% of the total energy flux. Possible explanations for the deviation from self-similar models, the link between these findings and intermittency, and their consequences for modeling of turbulent flows are briefly discussed.     

Inverse statistics of smooth signals: the case of two dimensional turbulence

The problem of inverse statistics (statistics of distances for which the signal fluctuations are larger than a certain threshold) in differentiable signals with power law spectrum, E(k) approximately k(-alpha), 3< or =alpha<5, is discussed. We show that for these signals, with random phases, exit-distance moments follow a bifractal distribution. We also investigate two dimensional turbulent flows in the direct cascade regime, which display a more complex behavior. We give numerical evidences that the inverse statistics of 2D turbulent flows is described by a multifractal probability distribution; i.e., the statistics of laminar events is not simply captured by the exponent alpha characterizing the spectrum.     

Experimental Simulation of the Generation of a Vortex Flow on a Water Surface by a Wave Cascade

The generation of a vortex flow by waves on a water surface, which simulate an energy cascade in a system of gravity waves at frequencies of 3, 4, 5, and 6 Hz, has been studied experimentally. It has been found that pumping is accompanied by the propagation of waves on the surface at different angles to the fundamental mode and by a nonlinear interaction between waves resulting in the generation of new harmonics. It has been shown that large-scale flows are formed by modes of the lowest frequency of 3 Hz intersecting at acute angles. The energy distribution of the vortex motion can be described by a power-law function of the wavenumber and is independent of the energy distribution in a system of surface waves. The energy coming to large-scale vortex flows directly from the wave system is transferred to small scales. A direct rather than inverse energy flux is established in the system of vortices.     

Transient vortex events in the initial value problem for turbulence

A vorticity surge event that could be a paradigm for a wide class of bursting events in turbulence is studied. The coherent mechanism is characterized by locally transverse vortex configurations that are intrinsically helical in both physical and Fourier space when there is a peak of the maximum vorticity parallel omega parallel(infinity)(t). At no time are nonhelical, antiparallel vorticity elements observed. This event precedes the appearance of the traditional signatures of an energy cascade such as strong growth of the dissipation, spectra approaching -5/3, and strongly Beltramized vortex tubes. Comparing how different large-eddy simulations reproduce these properties demonstrates the importance of properly modeling nonlinear transport of both energy and helicity.     

Statistical equilibrium solutions of the shallow water equations

A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2D fluid flow with a free surface, is described. The model contains a competing acoustic turbulent direct energy cascade, and a 2D turbulent inverse energy cascade. It is shown, nonetheless that, just as in the corresponding theory of the inviscid Euler equation, the infinite number of conserved quantities constrains the flow sufficiently to produce nontrivial large-scale vortex structures which are solutions to a set of explicitly derived coupled nonlinear partial differential equations.     

Direct numerical experiment on the measurement of the dispersion relation for gravity waves in the presence of the condensate

During previous numerical experiments on isotropic turbulence of surface gravity waves we observed formation of the long wave background (condensate). It was shown (Korotkevich, Phys. Rev. Lett. 101, 074504 (2008)) that the presence of the condensate changes a spectrum of direct cascade, corresponding to the flux of energy to the small scales from pumping region (large scales). Recent experiments show that the inverse cascade spectrum is also affected by the condensate. In this case mechanism proposed as a cause for the change of direct cascade spectrum cannot work. But inverse cascade is directly influenced by the linear dispersion relation for waves, as a result direct measurement of the dispersion relation in the presence of condensate is necessary. We performed the measurement of this dispersion relation from the direct numerical experiment. The results demonstrate that in the region of inverse cascade influence of the condensate cannot be neglected.     

On two-dimensionalization of three-dimensional turbulence in shell models

Applying a modified version of the Gledzer-Ohkitani-Yamada (GOY) shell model, the signatures of so-called two-dimensionalization effect of three-dimensional incompressible, homogeneous, isotropic fully developed unforced turbulence have been studied and reproduced. Within the framework of shell models we have obtained the following results: (i) progressive steepening of the energy spectrum with increased strength of the rotation, and, (ii) depletion in the energy flux of the forward forward cascade, sometimes leading to an inverse cascade. The presence of extended self-similarity and self-similar PDFs for longitudinal velocity differences are also presented for the rotating 3D turbulence case.     

Nonlinear instability of elementary stratified flows at large Richardson number

Elementary stably stratified flows with linear instability at all large Richardson numbers have been introduced recently by the authors [J. Fluid Mech. 376, 319-350 (1998)]. These elementary stratified flows have spatially constant but time varying gradients for velocity and density. Here the nonlinear stability of such flows in two space dimensions is studied through a combination of numerical simulations and theory. The elementary flows that are linearly unstable at large Richardson numbers are purely vortical flows; here it is established that from random initial data, linearized instability spontaneously generates local shears on buoyancy time scales near a specific angle of inclination that nonlinearly saturates into localized regions of strong mixing with density overturning resembling Kelvin-Helmholtz instability. It is also established here that the phase of these unstable waves does not satisfy the dispersion relation of linear gravity waves. The vortical flows are one family of stably stratified flows with uniform shear layers at the other extreme and elementary stably stratified flows with a mixture of vorticity and strain exhibiting behavior between these two extremes. The concept of effective shear is introduced for these general elementary flows; for each large Richardson number there is a critical effective shear with strong nonlinear instability, density overturning, and mixing for elementary flows with effective shear below this critical value. The analysis is facilitated by rewriting the equations for nonlinear perturbations in vorticity-stream form in a mean Lagrangian reference frame. (c) 2000 American Institute of Physics.     

Differential model for 2D turbulence

We present a phenomenological model for 2D turbulence in which the energy spectrum obeys a nonlinear fourth-order differential equation. This equation respects the scaling properties of the original Navier-Stokes equations, and it has both the −5/3 inverse-cascade and the −3 direct-cascade spectra. In addition, our model has Raleigh-Jeans thermodynamic distributions as exact steady state solutions. We use the model to derive a relation between the direct-cascade and the inverse-cascade Kolmogorov constants, which is in good qualitative agreement with the laboratory and numerical experiments. We discuss a steady state solution where both the enstrophy and the energy cascades are present simultaneously, and we discuss it in the context of the Nastrom-Gage spectrum observed in atmospheric turbulence. We also consider the effect of the bottom friction on the cascade solutions and show that it leads to an additional decrease and finite-wavenumber cutoffs of the respective cascade spectra, which agrees with the existing experimental and numerical results. The text was submitted by the authors in English.     

Phenomenology of two-dimensional stably stratified turbulence under large-scale forcing

In this paper, we characterise the scaling of energy spectra, and the interscale transfer of energy and enstrophy, for strongly, moderately and weakly stably stratified two-dimensional (2D) turbulence, restricted in a vertical plane, under large-scale random forcing. In the strongly stratified case, a large-scale vertically sheared horizontal flow (VSHF) coexists with small scale turbulence. The VSHF consists of internal gravity waves and the turbulent flow has a kinetic energy (KE) spectrum that follows an approximate k^?3 scaling with zero KE flux and a robust positive enstrophy flux. The spectrum of the turbulent potential energy (PE) also approximately follows a k^?3 power-law and its flux is directed to small scales. For moderate stratification, there is no VSHF and the KE of the turbulent flow exhibits Bolgiano–Obukhov scaling that transitions from a shallow k^?11/5 form at large scales, to a steeper approximate k^?3 scaling at small scales. The entire range of scales shows a strong forward enstrophy flux, and interestingly, large (small) scales show an inverse (forward) KE flux. The PE flux in this regime is directed to small scales, and the PE spectrum is characterised by an approximate k^?1.64 scaling. Finally, for weak stratification, KE is transferred upscale and its spectrum closely follows a k^?2.5 scaling, while PE exhibits a forward transfer and its spectrum shows an approximate k^?1.6 power-law. For all stratification strengths, the total energy always flows from large to small scales and almost all the spectral indicies are well explained by accounting for the scale-dependent nature of the corresponding flux.     

Stochastic density waves of granular flows: strong-intermittent dissipation fields with self-organization

The quantitative (scaling) results of a recent lattice-gas simulation of granular flows [1] are interpreted in terms of Kolmogorov-Obukhov approach revised for strong space-intermittent systems. Renormalised power spectrum with exponent '-4/3' seems to be an universal spectrum of scalar fluctuations convected by stochastic velocity fields in dissipative systems with inverse energy transfer (some other laboratory and geophysic turbulent flows with this power spectrum as well as an analogy between this phenomenon and turbulent percolation on elastic backbone are pointed out).