Kinetic energy repartition in MHD turbulence

Liquid-sodium-cooled breeder reactors may soon be operating at magnetic Reynolds numbers RM where magnetic fields can be self-excited by a dynamo mechanism (as first suggested by Bevir 1973). Such flows have kinetic Reynolds numbers RV of the order of 107 and are therefore highly turbulent.This leads us to investigate the behaviour of MHD turbulence with high RV and low magnetic Prandtl numbers. We use the eddy-damped quasi-normal Markovian closure applied to the MHD equations. For simplicity we restrict ourselves to homogeneous and isotropic turbulence, but we do include helicity.We obtain a critical magnetic Reynolds number RMc of the order of a few tens (non-helical case) above which magnetic energy is present. RMc is practically independent of RV (in the range 40 to 106). RMc can be considerably decreased by the presence of helicity: when the overall size of the flow L is much larger than the integral scale l0, RMc can drop below unity as suggested by an α-effect argument. When L ≈ l0 the drop can still be substantial (factor of 6) when helicity is a maximum. We examine how the turbulence is modified when RM crosses RMc: presence of magnetic energy, decreased kinetic energy, steepening of kinetic-energy spectrum, etc.We make no attempt to obtain quantitative estimates for a breeder reactor, but discuss some of the possible consequences of exceeding RMc, such as decreased turbulent heat transport. More precise information may be obtained from numerical simulations and experiments (including some in the subcritical regime).

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Inverse cascade of kinetic energy in two-dimensional β-Plane magnetohydrodynamic turbulence

10.5194/egusphere-egu2020-4497 ◽

2020 ◽

Author(s):

Timofey Zinyakov ◽

Arakel Petrosyan

Keyword(s):

Numerical Simulation ◽

Kinetic Energy ◽

Cascade Process ◽

Two Dimensional ◽

Mhd Turbulence ◽

Self Similarity ◽

Magnetohydrodynamic Turbulence ◽

Zonal Flows ◽

Inverse Cascade ◽

Kolmogorov Spectrum

Numerical studies of two-dimensional &#946;-plane homogeneous magnetohydrodynamic turbulence are presented. The study of the fundamental properties of such turbulence allows understanding the evolution of various astrophysical objects from the Sun and stars to planetary systems, galaxies, and galaxy clusters. Energy spectra and cascade process in two-dimensional &#946;-plane MHD are studied.In this work the equations of two-dimensional magnetohydrodynamics with the Coriolis force in the &#946;-plane approximation are used for the qualitative analysis and numerical simulation of processes in plasma astrophysics. The equations are solved on a square box of edge size 2&#960; with periodic boundary conditions applying a the pseudospectral method using the 2/3 rule for dealiasing. The results of numerical simulation of two-dimensional &#946;-plane MHD turbulence with a spatial resolution of 1024 &#215; 1024 and 4096 &#215; 4096 with different Rossby parameters &#946; and different Reynolds numbers are presented.It is found that only unsteady zonal flows with complex temporal dynamics are formed in two-dimensional &#946;-plane magnetohydrodynamic turbulence. It is shown that flow nonstationarity is due to the appearance of isotropic magnetic islands caused by the Lorentz force in the system. The formation of Iroshnikov&#8211;Kraichnan spectrum is shown in the early stages of evolution of two-dimensional &#946;-plane magnetohydrodynamic turbulence. The self-similarity of the decay of Iroshnikov&#8211;Kraichnan spectrum is studied. On long time scale violation of self-similarity of the decay and formation of Kolmogorov spectrum is discovered. The inverse cascade of kinetic energy, which is characteristic of the detected Kolmogorov spectrum, provides the formation of zonal flows.This work was supported by the Russian Foundation for Basic Research (project no. 19-02-00016).

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Multifractal scaling of the kinetic energy flux in solar wind turbulence

Annales Geophysicae ◽

10.1007/s00585-996-0259-4 ◽

1996 ◽

Vol 14 (3) ◽

pp. 259-269 ◽

Cited By ~ 30

Author(s):

E. Marsch ◽

C.-Y. Tu ◽

H. Rosenbauer

Keyword(s):

Solar Wind ◽

Kinetic Energy ◽

Energy Flux ◽

Dissipation Rate ◽

High Speed ◽

Velocity Structure ◽

Functional Dependence ◽

Scaling Exponent ◽

Mhd Turbulence ◽

Generalized Dimension

Abstract. The geometrical and scaling properties of the energy flux of the turbulent kinetic energy in the solar wind have been studied. Using present experimental technology in solar wind measurements we cannot directly measure the real volumetric dissipation rate, ε(t), but are constrained to represent it by its surrogate the energy flux near the dissipation range at the proton gyro scale. There is evidence for the multifractal nature of the so defined dissipation field ε(t), a result derived from the scaling exponents of its statistical moments. The generalized dimension Dq has been determined and reveals that the dissipation field has a multifractal structure, which is not compatible with a scale-invariant cascade. The related multifractal spectrum α has been estimated for the first time for MHD turbulence in the solar wind. Its features resemble those obtained for turbulent fluids and other nonlinear multifractal systems. The generalized dimension Dq can for turbulence in high-speed streams be fitted well by the functional dependence of the p-model with a comparatively large parameter p1=0.87, indicating a strongly intermittent multifractal energy cascade. The experimental value for Dp/3 used in the scaling exponent s(p) of the velocity structure function gives an exponent that can describe some of the observations. The scaling exponent μ of the autocorrelation function of ε (t) has also been directly evaluated, being 0.37. Finally, the mean dissipation rate was determined, which could be used in solar wind heating models.

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The decay of isotropic magnetohydrodynamics turbulence and the effects of cross-helicity

Journal of Plasma Physics ◽

10.1017/s0022377818000120 ◽

2018 ◽

Vol 84 (1) ◽

Cited By ~ 1

Author(s):

Antoine Briard ◽

Thomas Gomez

Keyword(s):

Magnetic Field ◽

Reynolds Number ◽

Energy Spectrum ◽

Kinetic Energy ◽

Total Energy ◽

Isotropic Turbulence ◽

Reynolds Numbers ◽

Inertial Range ◽

Mhd Turbulence ◽

Previous Prediction

Decaying homogeneous and isotropic magnetohydrodynamics (MHD) turbulence is investigated numerically at large Reynolds numbers thanks to the eddy-damped quasi-normal Markovian (EDQNM) approximation. Without any background mean magnetic field, the total energy spectrum $E$ scales as $k^{-3/2}$ in the inertial range as a consequence of the modelling. Moreover, the total energy is shown, both analytically and numerically, to decay at the same rate as kinetic energy in hydrodynamic isotropic turbulence: this differs from a previous prediction, and thus physical arguments are proposed to reconcile both results. Afterwards, the MHD turbulence is made imbalanced by an initial non-zero cross-helicity. A spectral modelling is developed for the velocity–magnetic correlation in a general homogeneous framework, which reveals that cross-helicity can contain subtle anisotropic effects. In the inertial range, as the Reynolds number increases, the slope of the cross-helical spectrum becomes closer to $k^{-5/3}$ than $k^{-2}$. Furthermore, the Elsässer spectra deviate from $k^{-3/2}$ with cross-helicity at large Reynolds numbers. Regarding the pressure spectrum $E_{P}$, its kinetic and magnetic parts are found to scale with $k^{-2}$ in the inertial range, whereas the part due to cross-helicity rather scales in $k^{-7/3}$. Finally, the two $4/3$rd laws for the total energy and cross-helicity are assessed numerically at large Reynolds numbers.

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On uniqueness of transfer rates in magnetohydrodynamic turbulence

Journal of Plasma Physics ◽

10.1017/s0022377819000710 ◽

2019 ◽

Vol 85 (5) ◽

Cited By ~ 1

Author(s):

Franck Plunian ◽

Rodion Stepanov ◽

Mahendra Kumar Verma

Keyword(s):

Kinetic Energy ◽

Transfer Rate ◽

Stokes Equations ◽

Magnetic Energy ◽

A Priori ◽

Magnetic Helicity ◽

Mhd Turbulence ◽

Magnetohydrodynamic Turbulence ◽

Transfer Rates ◽

Induction Equation

In hydrodynamic and MHD (magnetohydrodynamic) turbulence, formal expressions for the transfer rates rely on integrals over wavenumber triads $(\boldsymbol{k},\boldsymbol{p},\boldsymbol{q})$ satisfying $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0$ . As an example $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ denotes the kinetic energy transfer rate to the mode $\boldsymbol{k}$ , from the two other modes in the triad, $\boldsymbol{p}$ and $\boldsymbol{q}$ . However as noted by Kraichnan (Phys. Rev., vol. 111, 1958, pp. 1747–1747), in $S_{E}^{uu}(\boldsymbol{k}\mid \boldsymbol{p},\boldsymbol{q})$ , what fraction of the energy transferred to the mode $\boldsymbol{k}$ originated from $\boldsymbol{p}$ and which from $\boldsymbol{q}$ is unknown. Such an expression is thus incongruent with the customary description of turbulence in terms of two-scale energy exchange. Notwithstanding this issue, Dar et al. (Physica D, vol. 157 (3), 2001, pp. 207–225) further decomposed these transfers into separate contributions from $\boldsymbol{p}$ -to- $\boldsymbol{k}$ and $\boldsymbol{q}$ -to- $\boldsymbol{k}$ , thus introducing the concept of mode-to-mode transfers that they applied to MHD turbulence. Doing so, they had to set aside additional transfers circulating within each triad, but failed to calculate them. In the present paper we explain how to derive the complete expressions of the mode-to-mode transfers, including the circulating transfers. We do it for kinetic energy and kinetic helicity in hydrodynamic turbulence, for kinetic energy, magnetic energy and magnetic helicity in MHD turbulence. We find that the degree of non-uniqueness of the energy transfers derived from the induction equation is a priori higher than the one derived from the Navier–Stokes equations. However, separating the contribution of magnetic advection from magnetic stretching, the energy mode-to-mode transfer rates involving the magnetic field become uniquely defined, in striking contrast to the hydrodynamic case. The magnetic helicity mode-to-mode transfer rate is also found to be uniquely defined, contrary to kinetic helicity in hydrodynamics. We find that shell-to-shell transfer rates have the same properties as mode-to-mode transfer rates. Finally calculating the fluxes, we show how the circulating transfers cancel in accordance with conservation laws.

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Quasi-static magnetohydrodynamic turbulence at high Reynolds number

Journal of Fluid Mechanics ◽

10.1017/jfm.2011.207 ◽

2011 ◽

Vol 681 ◽

pp. 434-461 ◽

Cited By ~ 30

Author(s):

B. FAVIER ◽

F. S. GODEFERD ◽

C. CAMBON ◽

A. DELACHE ◽

W. J. T. BOS

Keyword(s):

Magnetic Field ◽

Reynolds Number ◽

Kinetic Energy ◽

High Reynolds Number ◽

Transverse Velocity ◽

Nonlinear Phenomenon ◽

Mhd Turbulence ◽

Linearized Theory ◽

High Reynolds ◽

The Magnetic Field

We analyse the anisotropy of homogeneous turbulence in an electrically conducting fluid submitted to a uniform magnetic field, for low magnetic Reynolds number, in the quasi-static approximation. We interpret contradictory earlier predictions between linearized theory and simulations: in the linear limit, the kinetic energy of transverse velocity components, normal to the magnetic field, decays faster than the kinetic energy of the axial component, along the magnetic field (Moffatt, J. Fluid Mech., vol. 28, 1967, p. 571); whereas many numerical studies predict a final state characterized by dominant energy of transverse velocity components. We investigate the corresponding nonlinear phenomenon using direct numerical simulation (DNS) of freely decaying turbulence, and a two-point statistical spectral closure based on the eddy-damped quasi-normal Markovian (EDQNM) model. The transition from the three-dimensional turbulent flow to a ‘two-and-a-half-dimensional’ flow (Montgomery & Turner, Phys. Fluids, vol. 25, 1982, p. 345) is a result of the combined effects of short-time linear Joule dissipation and longer time nonlinear creation of polarization anisotropy. It is this combination of linear and nonlinear effects which explains the disagreement between predictions from linearized theory and results from numerical simulations. The transition is characterized by the elongation of turbulent structures along the applied magnetic field, and by the strong anisotropy of directional two-point correlation spectra, in agreement with experimental evidence. Inertial equatorial transfers in both DNS and the model are presented to describe in detail the most important equilibrium dynamics. Spectral scalings are maintained in high-Reynolds-number turbulence attainable only with the EDQNM model, which also provides simplified modelling of the asymptotic state of quasi-static magnetohydrodynamic (MHD) turbulence.

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Magnetically dominated structures as an important component of the solar wind turbulence

Annales Geophysicae ◽

10.5194/angeo-25-1913-2007 ◽

2007 ◽

Vol 25 (8) ◽

pp. 1913-1927 ◽

Cited By ~ 40

Author(s):

R. Bruno ◽

R. D'Amicis ◽

B. Bavassano ◽

V. Carbone ◽

L. Sorriso-Valvo

Keyword(s):

Magnetic Field ◽

Solar Wind ◽

Kinetic Energy ◽

Magnetic Energy ◽

Radial Distance ◽

Radial Dependence ◽

Mhd Turbulence ◽

Heliographic Latitude ◽

Magnetic Structures ◽

Fast Wind

Abstract. This study focuses on the role that magnetically dominated fluctuations have within the solar wind MHD turbulence. It is well known that, as the wind expands, magnetic energy starts to dominate over kinetic energy but we lack of a statistical study apt to estimate the relevance of these fluctuations depending on wind speed, radial distance from the sun and heliographic latitude. Our results suggest that this kind of fluctuations can be interpreted as non-propagating structures, advected by the wind during its expansion. In particular, observations performed in the ecliptic revealed a clear radial dependence of these magnetic structures within fast wind, but not within slow wind. At short heliocentric distances (~0.3 AU) the turbulent population is largely dominated by Alfvénic fluctuations characterized by high values of normalized cross-helicity and a remarkable level of energy equipartition. However, as the wind expands, a new-born population, characterized by lower values of Alfvénicity and a clear imbalance in favor of magnetic energy becomes visible and clearly distinguishable from the Alfvénic population largely characterized by an outward sense of propagation. We estimate that more than 20% of all the analyzed intervals of hourly scale within fast wind are characterized by normalized cross-helicity close to zero and magnetic energy largely dominating over kinetic energy. Most of these advected magnetic structures result to be non-compressive and might represent the crossing of the border between adjacent flux tubes forming, as suggested in literature, the advected background structure of the interplanetary magnetic field. On the other hand, their features are also well fitted by the Magnetic Field Directional Turnings paradigm as proposed in literature.

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As a Matter of Tension: Kinetic Energy Spectra in MHD Turbulence

The Astrophysical Journal ◽

10.3847/1538-4357/abdd22 ◽

2021 ◽

Vol 909 (2) ◽

pp. 148

Author(s):

Philipp Grete ◽

Brian W. O’Shea ◽

Kris Beckwith

Keyword(s):

Kinetic Energy ◽

Energy Spectra ◽

Mhd Turbulence

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On nonlinear Alfvén-cyclotron waves in multi-species plasma

Journal of Plasma Physics ◽

10.1017/s0022377810000541 ◽

2010 ◽

Vol 77 (3) ◽

pp. 385-403 ◽

Cited By ~ 10

Author(s):

ECKART MARSCH ◽

DANIEL VERSCHAREN

Keyword(s):

Magnetic Field ◽

Kinetic Energy ◽

Electric Field ◽

Transverse Magnetic Field ◽

Acoustic Waves ◽

Mean Field ◽

Transverse Magnetic ◽

Longitudinal Electric Field ◽

Mhd Turbulence ◽

The Mean

AbstractLarge-amplitude Alfvén waves are ubiquitous in space plasmas and a main component of magnetohydrodynamic (MHD) turbulence in the heliosphere. As pump waves, they are prone to parametric instability by which they can generate cyclotron and acoustic daughter waves. Here, we revisit a related process within the framework of the multi-fluid equations for a plasma consisting of many species. The nonlinear coupling of the Alfvén wave to acoustic waves is studied, and a set of compressive and coupled-wave equations for the transverse magnetic field and longitudinal electric field is derived for waves propagating along the mean-field direction. It turns out that slightly compressive Alfvén waves exert, through induced gyro-radius and kinetic-energy modulations, an electromotive force on the particles in association with a longitudinal electric field, which has a potential that is given by the gradient of the transverse kinetic energy of the particles gyrating about the mean field. This in turn drives electric fluctuations (sound and ion-acoustic waves) along the mean magnetic field, which can nonlinearly react back on the transverse magnetic field. Mutually coupled Alfvén-cyclotron--acoustic waves are thus excited, a nonlinear process that can drive a cascade of wave energy in the plasma, and may generate compressive microturbulence. These driven electric fluctuations might have consequences for the dissipation of an MHD turbulence and, thus, for the heating and acceleration of particles in the solar wind.

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