JOINT INVERSE CASCADE OF MAGNETIC ENERGY AND MAGNETIC HELICITY IN MHD TURBULENCE

To understand the turbulent generation of large-scale magnetic fields and to advance beyond purely kinematic approaches to the dynamo effect like that introduced by Steenbeck, Krause & Radler (1966)’ a new nonlinear theory is developed for three-dimensional, homogeneous, isotropic, incompressible MHD turbulence with helicity, i.e. not statistically invariant under plane reflexions. For this, techniques introduced for ordinary turbulence in recent years by Kraichnan (1971 a)’ Orszag (1970, 1976) and others are generalized to MHD; in particular we make use of the eddy-damped quasi-normal Markovian approximation. The resulting closed equations for the evolution of the kinetic and magnetic energy and helicity spectra are studied both theoretically and numerically in situations with high Reynolds number and unit magnetic Prandtl number.Interactions between widely separated scales are much more important than for non-magnetic turbulence. Large-scale magnetic energy brings to equipartition small-scale kinetic and magnetic excitation (energy or helicity) by the ‘Alfvén effect’; the small-scale ‘residual’ helicity, which is the difference between a purely kinetic and a purely magnetic helical term, induces growth of large-scale magnetic energy and helicity by the ‘helicity effect’. In the absence of helicity an inertial range occurs with a cascade of energy to small scales; to lowest order it is a −3/2 power law with equipartition of kinetic and magnetic energy spectra as in Kraichnan (1965) but there are −2 corrections (and possibly higher ones) leading to a slight excess of magnetic energy. When kinetic energy is continuously injected, an initial seed of magnetic field will grow to approximate equipartition, at least in the small scales. If in addition kinetic helicity is injected, an inverse cascade of magnetic helicity is obtained leading to the appearance of magnetic energy and helicity in ever-increasing scales (in fact, limited by the size of the system). This inverse cascade, predicted by Frischet al.(1975), results from a competition between the helicity and Alféh effects and yields an inertial range with approximately — 1 and — 2 power laws for magnetic energy and helicity. When kinetic helicity is injected at the scale linjand the rate$\tilde{\epsilon}^V$(per unit mass), the time of build-up of magnetic energy with scaleL[Gt ] linjis$t \approx L(|\tilde{\epsilon}^V|l^2_{\rm inj})^{-1/3}.$

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Inverse cascade in simulated MHD turbulence driven by small scale source of magnetic helicity

Magnetohydrodynamics ◽

10.22364/mhd.52.1-2.29 ◽

2016 ◽

Vol 52 (1-2) ◽

pp. 261-268

Author(s):

R. Stepanov ◽

V. Titov

Keyword(s):

Magnetic Helicity ◽

Small Scale ◽

Mhd Turbulence ◽

Inverse Cascade

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Helical mode interactions and spectral transfer processes in magnetohydrodynamic turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.43 ◽

2016 ◽

Vol 791 ◽

pp. 61-96 ◽

Cited By ~ 14

Author(s):

Moritz Linkmann ◽

Arjun Berera ◽

Mairi McKay ◽

Julia Jäger

Keyword(s):

Energy Transfer ◽

Magnetic Fields ◽

Magnetic Helicity ◽

Asymmetric Effect ◽

Mhd Turbulence ◽

Dynamo Action ◽

Reverse Transfer ◽

Transfer Processes ◽

Inverse Cascade ◽

Mode Interactions

Spectral transfer processes in homogeneous magnetohydrodynamic (MHD) turbulence are investigated analytically by decomposition of the velocity and magnetic fields in Fourier space into helical modes. Steady solutions of the dynamical system which governs the evolution of the helical modes are determined, and a stability analysis of these solutions is carried out. The interpretation of the analysis is that unstable solutions lead to energy transfer between the interacting modes while stable solutions do not. From this, a dependence of possible interscale energy and helicity transfers on the helicities of the interacting modes is derived. As expected from the inverse cascade of magnetic helicity in 3-D MHD turbulence, mode interactions with like helicities lead to transfer of energy and magnetic helicity to smaller wavenumbers. However, some interactions of modes with unlike helicities also contribute to an inverse energy transfer. As such, an inverse energy cascade for non-helical magnetic fields is shown to be possible. Furthermore, it is found that high values of the cross-helicity may have an asymmetric effect on forward and reverse transfer of energy, where forward transfer is more quenched in regions of high cross-helicity than reverse transfer. This conforms with recent observations of solar wind turbulence. For specific helical interactions the relation to dynamo action is established. The present analysis provides new theoretical insights into physical processes where inverse cascade and dynamo action are involved, such as the evolution of cosmological and astrophysical magnetic fields and laboratory plasmas.

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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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Magnetic Helicity and the Geodynamo

Fluids ◽

10.3390/fluids6030099 ◽

2021 ◽

Vol 6 (3) ◽

pp. 99

Author(s):

John V. Shebalin

Keyword(s):

Magnetic Field ◽

Boundary Conditions ◽

Magnetic Energy ◽

Liquid Core ◽

Dipole Field ◽

Magnetic Helicity ◽

Quasi Equilibrium ◽

Mhd Turbulence ◽

Poloidal Magnetic Field ◽

Dipole Alignment

We present theoretical and computational results in magnetohydrodynamic turbulence that we feel are essential to understanding the geodynamo. These results are based on a mathematical model that focuses on magnetohydrodynamic (MHD) turbulence, but ignores compressibility and thermal effects, as well as imposing model-dependent boundary conditions. A principal finding is that when a turbulent magnetofluid is in quasi-equilibrium, the magnetic energy in the internal dipole component is equal to the magnetic helicity multiplied by the dipole wavenumber. In the case of the Earth, measurement of the exterior magnetic field gives us, through boundary conditions, the internal poloidal magnetic field. The connection between magnetic helicity and dipole field in the liquid core then gives us the toroidal part of the internal dipole field and a model value of 3 mT for the average core dipole magnetic field. Here, we present the theoretical analysis and numerical simulations that lead to these conclusions. We also test an earlier assertion that differential oblateness may be related to dipole alignment, and while there is an effect, rotation appears to be far more important. In addition, the relationship between dipole quasi-stationarity, broken ergodicity and broken symmetry is clarified. Lastly, we discuss how inertial waves in a rotating magnetofluid can affect dipole alignment.

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Inverse cascade of hybrid helicity in BΩ -MHD turbulence

Physical Review Fluids ◽

10.1103/physrevfluids.4.073701 ◽

2019 ◽

Vol 4 (7) ◽

Cited By ~ 3

Author(s):

Mélissa D. Menu ◽

Sébastien Galtier ◽

Ludovic Petitdemange

Keyword(s):

Mhd Turbulence ◽

Inverse Cascade

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Fully developed MHD turbulence near critical magnetic Reynolds number

Journal of Fluid Mechanics ◽

10.1017/s002211208100298x ◽

1981 ◽

Vol 104 ◽

pp. 419-443 ◽

Cited By ~ 63

Author(s):

J. Léorat ◽

A. Pouquet ◽

U. Frisch

Keyword(s):

Reynolds Number ◽

Kinetic Energy ◽

Isotropic Turbulence ◽

Magnetic Energy ◽

Reynolds Numbers ◽

Magnetic Reynolds Number ◽

Mhd Equations ◽

Liquid Sodium ◽

Turbulent Heat ◽

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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