Inverse cascade of hybrid helicity in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>B</mml:mi><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow></mml:math>-MHD turbulence

Menu, Mélissa; Galtier, Sébastien; Petitdemange, Ludovic

doi:10.1103/physrevfluids.4.073701

Cited by 5 publications

(8 citation statements)

References 58 publications

(73 reference statements)

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“…The denominators for the terms in G ± (φ) are positive because the arguments of the logarithms in (71), as well as α in (70), are all positive; also, from ( 37) and ( 40), we see that φ/H M = k M ≥ k min . For the Fourier case we are discussing here, k min = 1, and for the spherical shell model of the outer core developed by [34], k min ∼ = 1.8638.…”

Section: Entropymentioning

confidence: 90%

“…Above, the sum over k ′ means, again, that only independent modes k are included (if k, then not −k). The fact that there is only one unknown quantity φ in (70) means that the entropy functional (71) depends only on the one variable φ. As discussed by [57], finding the (single) minimum of S(φ) gives us the value φ = φ o = ⟨E M ⟩ that sets the values of α, β and γ, as well as the system entropy S o = S(φ o ).…”

Section: Entropymentioning

confidence: 99%

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The Statistical Mechanics of Ideal Magnetohydrodynamic Turbulence and a Solution of the Dynamo Problem

Shebalin

2024

Fluids

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We review and extend the theory of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence. The theory contains a solution to the ‘dynamo problem’, i.e., the problem of determining how a planetary or stellar body produces a global dipole magnetic field. We extend the theory to the case of ideal MHD turbulence with a mean magnetic field that is alignedwith a rotation axis. The existing theory is also extended by developing the thermodynamics of ideal MHD turbulence based on entropy. A mathematical model is created by Fourier transforming the MHD equations and dynamical variables, resulting in a dynamical system consisting of the independent Fourier coefficients of the velocity and magnetic fields. This dynamical system has a large but finite-dimensional phase space in which the phase flow is divergenceless in the ideal case. There may be several constants of the motion, in addition to energy, which depend on the presence, or lack thereof, of a mean magnetic field or system rotation or both imposed on the magnetofluid; this leads to five different cases of MHD turbulence that must be considered. The constants of the motion (ideal invariants)—the most important being energy and magnetic helicity—are used to construct canonical probability densities and partition functions that enable ensemble predictions to be made.These predictions are compared with time averages from numerical simulations to test whether or not the system is ergodic. In the cases most pertinent to planets and stars, nonergodicity is observed at the largest length-scales and occurs when the components of the dipole field become quasi-stationary and dipole energy is directly proportional to magnetic helicity. This nonergodicity is evident in the thermodynamics, while dipole alignment with a rotation axis may be seen as the result of dynamical symmetry breaking, i.e., ‘broken ergodicity’. The relevance of ideal theoretical results to real (forced, dissipative) MHD turbulence is shown through numerical simulation. Again, an important result is a statistical solution of the ‘dynamo problem’.

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Section: Entropymentioning

confidence: 90%

Section: Entropymentioning

confidence: 99%

The Statistical Mechanics of Ideal Magnetohydrodynamic Turbulence and a Solution of the Dynamo Problem

Shebalin

2024

Fluids

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“…Wave turbulence is a very common natural phenomenon found, for example, with/in gravity waves [6][7][8][9], capillary waves [see also the discussion in section IX, [10][11][12], quantum turbulence [13][14][15][16], nonlinear optics [17][18][19], inertial waves [20][21][22][23][24], magnetostrophic waves [25,26], elastic plates [27][28][29][30][31][32], plasma waves [33][34][35][36][37], or primordial gravitational waves [38][39][40]. These few examples demonstrate the vitality of the domain.…”

Section: Introductionmentioning

confidence: 99%

Wave turbulence: the case of capillary waves

Galtier

2020

Geophysical & Astrophysical Fluid Dynamics

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Capillary waves are perhaps the simplest example to consider for an introduction to wave turbulence. Since the first paper by Zakharov and Filonenko [1], capillary wave turbulence has been the subject of many studies but a didactic derivation of the kinetic equation is still lacking. It is the objective of this paper to present such a derivation in absence of gravity and in the approximation of deep water. We use the Eulerian method and a Taylor expansion around the equilibrium elevation for the velocity potential to derive the kinetic equation. The use of directional polarities for three-wave interactions leads to a compact form for this equation which is fully compatible with previous work. The exact solutions are derived with the so-called Zakharov transformation applied to wavenumbers and the nature of these solutions is discussed. Experimental and numerical works done in recent decades are also reviewed.

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“…Often, fluid motions in such rotating planetary and astrophysical objects are in a turbulent state in characterised by presence of propagating waves and vortices which affect the mean dynamical characteristics and transport properties of matter and energy, in particular the rate of accretion, the rate of conversion of the kinetic energy into heat and dynamo action of generation of large-scale mean magnetic field. An interesting new mechanism of preferential alignment of the large-scale planetary and stellar magnetic field and rotation based on the nonlinearly interacting waves in rotating MHD was recently put forward in [13]. In this paper, we will consider weakly and strongly nonlinear regimes of rotating MHD turbulence assuming that the motions are strongly subsonic and sub-Alfvenic.…”

Section: Introductionmentioning

confidence: 99%

Rotating magnetohydrodynamic turbulence

Bell¹,

Nazarenko²

2019

J. Phys. A: Math. Theor.

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Turbulence in rotating Magneto-hydrodynamic systems is studied theoretically and numerically. In the linear limit, when the velocity and magnetic perturbations are small, the system supports two types of waves. When the rotation effects are stronger than the ones of the external magnetic field, one of these waves contains most of the kinetic energy (inertial wave) and the other -most of the magnetic energy (magnetostrophic wave). The weak wave turbulence (WWT) theory for decoupled inertial and magnetospheric wave systems was previously derived by Galtier (2014). In the present paper, we derive theory of strong turbulence for such waves based on the critical balance (CB) approach conjecturing that the linear and nonlinear timescales are of similar magnitudes in a wide range of turbulent scales. Regimes of weak and strong wave turbulence are simulated numerically. The results appear to be in good agreement with the WWT and CB predictions, particularly for the exponents of the kinetic and magnetic energy spectra.

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Inverse cascade of hybrid helicity inBΩ-MHD turbulence

Cited by 5 publications

References 58 publications

The Statistical Mechanics of Ideal Magnetohydrodynamic Turbulence and a Solution of the Dynamo Problem

The Statistical Mechanics of Ideal Magnetohydrodynamic Turbulence and a Solution of the Dynamo Problem

Wave turbulence: the case of capillary waves

Rotating magnetohydrodynamic turbulence

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