Elasticity of tangled magnetic fields

Abstract

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.

MHD turbulence: a biased review

“…Hosking et al. (2020) have shown numerically that magnetoelastic waves do exist in certain tangled, force-free magnetic configurations, and are well described by (13.25) (modulo some further nuance that can mean that is somewhat reduced for tangled fields that are spatially intermittent). What is still unknown is whether they can propagate against the background of a saturated dynamo state or are quickly damped by small-scale motions and thus rendered irrelevant.…”

“…On a very crude level, it is perhaps obvious that this should be so, because ideal MHD equations have two types of exact solutions for which nonlinear interactions vanish: Elsasser states (, or ) and static force-free magnetic fields (, where ). If the system finds a way towards either of these solutions, globally or locally, concentrated on scales large enough to make dissipation small, it may, subject to this small dissipation, be able to linger in those states (‘may’ because the stability of the force-free states, e.g., is not guaranteed: see discussion and references in Appendix A of Hosking, Schekochihin & Balbus 2020). We shall see below that both magnetically dominated scenarios and convergence to Elsasser states are possible and that recent developments point to magnetic reconnection muscling its way into this topic as well.…”

On the construction of general large-amplitude spherically polarised Alfvén waves

“…The system is reminiscent of the tangled fields possible in magnetostatic force-free equilibria (those with u = 0 and B × (∇ × B) = 0, see e.g. Chandrasekhar & Woltjer 1958;Marsh 1996;Hosking, Schekochihin & Balbus 2020), but with a different class of equilibria that includes flows and has seen comparatively little study.…”

Tangled magnetic field model of QPOs

“…As a simplification, we ignore the dipole component, and treat the magnetic field inside the star as uniformly and isotropically tangled. The magnetic tangle gives an isotropic contribution to the shear stress with an effective shear modulus (for normal protons) 𝜇 𝐵 ≡ 𝐵 2 𝑡 /(4𝜋), where 𝐵 2 𝑡 1/2 is the spatially averaged tangled magnetic field (LvE; see also Hosking et al (2020) for a detailed treatment of tangled fields in MHD). 3 For SGR 1900, LvE estimate 𝜇 𝐵 6 × 10 29 ergs cm −3 , while the volume-averaged shear modulus of the relatively thin solid crust is about half this value.…”