A note on the stability of Beltrami fields for compressible fluid flows

Er‐Riani, Mustapha; Naji, Ahmed; Jarroudi, Mustapha El

doi:10.1016/j.ijnonlinmec.2014.09.009

Cited by 3 publications

(7 citation statements)

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“…The question of the ideal stability of the periodic shorter wavelength (α 0 > 1) Taylor states has a conflicted history (e.g. Moffatt 1986;Galloway & Frisch 1987;Er-Riani et al 2014), which has recently been resolved in East et al (2015). In that study, numerical simulations of both FFE and relativistic MHD revealed that α 0 > 1 Taylor states are linearly unstable to ideal perturbations.…”

Section: Linear Instability Of the Excited Taylor Statesmentioning

confidence: 99%

Freely Decaying Turbulence in Force-Free Electrodynamics

Zrake

East

2016

ApJ

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Freely decaying relativistic force-free turbulence is studied for the first time. We initiate the magnetic field at a short wavelength and simulate its relaxation toward equilibrium on two and three dimensional periodic domains, in both helical and non-helical settings. Force-free turbulent relaxation is found to exhibit an inverse cascade in all settings, and in 3D to have a magnetic energy spectrum consistent with the Kolmogorov 5/3 power law. 3D relaxations also obey the Taylor hypothesis; they settle promptly into the lowest energy configuration allowed by conservation of the total magnetic helicity. But in 2D, the relaxed state is a force-free equilibrium whose energy greatly exceeds the Taylor minimum, and which contains persistent force-free current layers and isolated flux tubes. We explain this behavior in terms of additional topological invariants that exist only in two dimensions, namely the helicity enclosed within each level surface of the magnetic potential function. The speed and completeness of turbulent magnetic free energy discharge could help account for rapidly variable gamma-ray emission from the Crab Nebula, gamma-ray bursts, blazars, and radio galaxies.

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Section: Linear Instability Of the Excited Taylor Statesmentioning

confidence: 99%

Freely Decaying Turbulence in Force-Free Electrodynamics

Zrake

East

2016

ApJ

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“…In this work we focus on space-periodic equilibria as a simple, computationally tractable setting free of the effect of confining boundaries (as in extended outflows). Though there is a rich literature studying such solutions [15,[22][23][24][25], important facts regarding their stability have not been appreciated. Focusing on a prototypical example, the "ABC" solutions [26] (defined below), in [24] it was claimed that such solutions are stable to incompressible perturbations (see also [25]).…”

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confidence: 99%

Spontaneous Decay of Periodic Magnetostatic Equilibria

et al. 2015

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In order to understand the conditions which lead a highly magnetized, relativistic plasma to become unstable, and in such cases how the plasma evolves, we study a prototypical class of magnetostatic equilibria where the magnetic field satisfies ∇ × B = αB, where α is spatially uniform, on a periodic domain. Using numerical solutions we show that generic examples of such equilibria are unstable to ideal modes (including incompressible ones) which are marked by exponential growth in the linear phase. We characterize the unstable mode, showing how it can be understood in terms of merging magnetic and current structures, and explicitly demonstrate its instability using the energy principle. Following the nonlinear evolution of these solutions, we find that they rapidly develop regions with relativistic velocities and electric fields of comparable magnitude to the magnetic field, liberating magnetic energy on dynamical timescales and eventually settling into a configuration with the largest allowable wavelength. These properties make such solutions a promising setting for exploring the mechanisms behind extreme cosmic sources of gamma rays.

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“…Indeed, the stability of LFF fields was the motivation for the consideration of the magnetoelastic wave problem by Moffatt (1986). Despite the apparent conflict between the result of Voslamber & Callebaut (1962) and Moffatt's stability theorem, it was not until much more recently that a counterexample was explicitly presented by Er-Riani et al (2014), who showed that there exist periodic force-free fields that are unstable to ideal periodic perturbations, as long as these perturbations are allowed to have wavevectors smaller than α. 11 Soon after, East et al (2015) were able to find energy-decreasing perturbations for a number of LFF fields by numerical minimisation of (A 5) under variation of the Fourier coefficients in the expansion of ξ .…”

Section: Resultsmentioning

confidence: 99%

“…The analysis in the preceding sections has been idealised because real tangled magnetic equilibria are generically unstable, even to ideal perturbations (Er-Riani, Naji & El Jarroudi 2014; East et al. 2015).…”

Section: Numerical Studymentioning

confidence: 99%

“…Despite the apparent conflict between the result of Voslamber & Callebaut (1962) and Moffatt's stability theorem, it was not until much more recently that a counterexample was explicitly presented by Er-Riani et al. (2014), who showed that there exist periodic force-free fields that are unstable to ideal periodic perturbations, as long as these perturbations are allowed to have wavevectors smaller than 11 . Soon after, East et al.…”

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confidence: 99%

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Elasticity of tangled magnetic fields

Hosking

Schekochihin

2020

J. Plasma Phys.

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The fundamental difference between incompressible ideal magnetohydrodynamics and the dynamics of a non-conducting fluid is that magnetic fields exert a tension force that opposes their bending; magnetic fields behave like elastic strings threading the fluid. It is natural, therefore, to expect that a magnetic field tangled at small length scales should resist a large-scale shear in an elastic way, much as a ball of tangled elastic strings responds elastically to an impulse. Furthermore, a tangled field should support the propagation of ‘magnetoelastic waves’, the isotropic analogue of Alfvén waves on a straight magnetic field. Here, we study magnetoelasticity in the idealised context of an equilibrium tangled field configuration. In contrast to previous treatments, we explicitly account for intermittency of the Maxwell stress, and show that this intermittency necessarily decreases the frequency of magnetoelastic waves in a stable field configuration. We develop a mean-field formalism to describe magnetoelastic behaviour, retaining leading-order corrections due to the coupling of large- and small-scale motions, and solve the initial-value problem for viscous fluids subjected to a large-scale shear, showing that the development of small-scale motions results in anomalous viscous damping of large-scale waves. Finally, we test these analytic predictions using numerical simulations of standing waves on tangled, linear force-free magnetic-field equilibria.

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A note on the stability of Beltrami fields for compressible fluid flows

Cited by 3 publications

References 35 publications

Freely Decaying Turbulence in Force-Free Electrodynamics

Freely Decaying Turbulence in Force-Free Electrodynamics

Spontaneous Decay of Periodic Magnetostatic Equilibria

Elasticity of tangled magnetic fields

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