Exact solution of the incompressible Hall magnetohydrodynamics

Mahajan, S. M.; Krishan, V.

doi:10.1111/j.1745-3933.2005.00028.x

Cited by 44 publications

(53 citation statements)

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“…Therefore the Hall effect disappears, and plasma behaves like ideal MHD in very strong magnetic field. Appendix A: Derivation of (13) In this section, we explicitly derive the relativistic HMHD dispersion relation (13). From (6) and (9), we get…”

Section: Discussionmentioning

confidence: 99%

“…Whereas the cyclotron frequency is proportional to the magnetic field strength, the Alfvén speed may not exceed the speed of light (see (14)); hence the modified skin depth is required to shrinks as the magnetic field strength increases. We note that the dispersion relation (13) and the modified inertial length δ i are valid as long as the relativistic two-fluid equations is correct for ion-electron plasma since (1)-(5) are rigorously derived by the relativistic two-fluid equations [19,20]. However, it is proven that there is limitations for nonrelativistic HMHD dispersion relation when it is derived from a kinetic theory [34,35].…”

Section: Introductionmentioning

confidence: 94%

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Modification of magnetohydrodynamic waves by the relativistic Hall effect

Kawazura

2017

Phys. Rev. E

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This study shows that a relativistic Hall effect significantly changes the properties of wave propagation by deriving a linear dispersion relation for relativistic Hall magnetohydrodynamics (HMHD). Whereas, in nonrelativistic HMHD, the phase and group velocities of fast magnetosonic wave become anisotropic with an increasing Hall effect, the relativistic Hall effect brings upper bounds to the anisotropies. The Alfvén wave group velocity with strong Hall effect also becomes less anisotropic than non-relativistic case. Moreover, the group velocity surfaces of Alfvén and fast waves coalesce into a single surface in the direction other than near perpendicular to the ambient magnetic field. It is also remarkable that a characteristic scale length of the relativistic HMHD depends on ion temperature, magnetic field strength, and density while the non-relativistic HMHD scale length, i.e., ion skin depth, depends only on density. The modified characteristic scale length increases as the ion temperature increases and decreases as the magnetic field strength increases.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

Modification of magnetohydrodynamic waves by the relativistic Hall effect

Kawazura

2017

Phys. Rev. E

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“…The calculation above is analogous to the calculation by Mahajan & Krishan (2005) for incompressible Hall MHD (i.e., essentially, the high-βe limit of the equations discussed in Appendix E), but the result is more general in the sense that it holds at arbitrary ion and electron betas. The Mahajan-Krishan solution in the EMHD limit amounts to noticing that Eq.…”

Section: Kinetic Alfvén Wavesmentioning

confidence: 99%

Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades In Magentized Weakly Collisional Plasmas

Schekochihin¹,

Cowley²,

W.³

et al. 2009

156

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This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulent motions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the "inertial range" above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvénic fluctuations and a passive cascade of density and magnetic-fieldstrength fluctuations. The former are governed by the Reduced Magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvénic component (in the collisional limit, these compressive fluctuations become the slow and entropy modes of the conventional MHD). In the "dissipation range" below ion gyroscale, there are again two cascades: the kinetic-Alfvén-wave (KAW) cascade governed by two fluid-like Electron Reduced Magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations both in space and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was Landau-damped at the ion gyroscale to collisional scales in the phase space and leads to ion heating. The KAW energy is similarly damped at the electron gyroscale and converted into electron heat. Kolmogorov-style scaling relations are derived for all of these cascades. The relationship between the theoretical models proposed in this paper and astrophysical applications and observations is discussed in detail.

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“…[16]]. These V and B have oscillatory amplitudes, thus the Bernoulli condition (22) demands a non-constant h (ρ).…”

Section: Arxiv:151101599v1 [Physicsplasm-ph] 5 Nov 2015mentioning

confidence: 99%

Nonlinear Alfvén waves in extended magnetohydrodynamics

Abdelhamid

Yoshida

2016

Physics of Plasmas

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Large-amplitude Alfvén waves are observed in various systems in space and laboratories, demonstrating an interesting property that the wave shapes are stable even in the nonlinear regime. The ideal magnetohydrodynamics (MHD) model predicts that an Alfvén wave keeps an arbitrary shape constant when it propagates on a homogeneous ambient magnetic field. However, such arbitrariness is an artifact of the idealized model that omits the dispersive effects. Only special wave forms, consisting of two component sinusoidal functions, can maintain the shape; we derive fully nonlinear Alfvén waves by an extended MHD model that includes both the Hall and electron inertia effects. Interestingly, these "small-scale effects" change the picture completely; the large-scale component of the wave cannot be independent of the small scale component, and the coexistence of them forbids the large scale component to have a free wave form. This is a manifestation of the nonlinearitydispersion interplay, which is somewhat different from that of solitons.PACS numbers: 52.35. Bj, 52.30.Cv, 52.35.Mw, 47.10.Df Alfvén waves are the most typical electromagnetic phenomena in magnetized plasmas. In particular, nonlinear Alfvén waves deeply influence various plasma regimes in laboratory as well as in space, which have a crucial role in plasma heating [1,2], turbulence [3][4][5], reconnection[6], etc. As an interesting property of the Alfvén waves, the amplitudes as well as wave forms are totally arbitrary when they propagate on a homogenous ambient magnetic field [7,8]. In fact, we often observe large-amplitude Alfvén waves in orderly propagation (for example [9]). To put it in theoretical language, the set A of Alfvén waves after an appropriate transformation (see [10]), is a closed linear subspace, i.e., every linear combination of the members of A gives solution to the fully nonlinear wave equation. Needless to say, the set of general solutions to a linear equation is, by definition, a linear subspace. However, it is remarkable that the nonlinear MHD equation has such a linear subspace A of solutions.Here we investigate the underlying mechanism producing such solutions in the context of a more accurate framework, generalized MHD. When we take into account dispersion effects (we consider both ion and electron inertial effects [11,12]), the wave forms are no longer arbitrary (remember that the ideal MHD model is dispersion free). Yet, we find that the generalized MHD system has a linear subspace of nonlinear solutions. The Casimir invariants of the system is the root cause of this interesting property [13].We start by reviewing how the nonlinear Alfvén waves are created in ideal MHD; we put the problem in the perspective of Hamiltonian mechanics. We then formulate the generalized MHD system in a Hamiltonian form. Via constructing equilibrium solutions (so-called Beltrami equilibriums) on Casimir leaves, we derive nonlinear wave solutions. The dispersion relation is exactly that of the linear theory, while the wave amplitude may be arbitrarily ...

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Exact solution of the incompressible Hall magnetohydrodynamics

Cited by 44 publications

References 12 publications

Modification of magnetohydrodynamic waves by the relativistic Hall effect

Modification of magnetohydrodynamic waves by the relativistic Hall effect

Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades In Magentized Weakly Collisional Plasmas

Nonlinear Alfvén waves in extended magnetohydrodynamics

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