Potential Vorticity Mixing in a Tangled Magnetic Field

Abstract

A theory of potential vorticity (PV) mixing in a disordered (tangled) magnetic field is presented. The analysis is in the context of β-plane MHD, with a special focus on the physics of momentum transport in the stably stratified, quasi-2D solar tachocline. A physical picture of mean PV evolution by vorticity advection and tilting of magnetic fields is proposed. In the case of weak-field perturbations, quasi-linear theory predicts that the Reynolds and magnetic stresses balance as turbulence Alfvénizes for a larger mean magnetic field. Jet formation is explored quantitatively in the mean field-resistivity parameter space. However, since even a modest mean magnetic field leads to large magnetic perturbations for large magnetic Reynolds number, the physically relevant case is that of a strong but disordered field. We show that numerical calculations indicate that the Reynolds stress is modified well before Alfvénization -i.e. before fluid and magnetic energies balance. To understand these trends, a double-average model of PV mixing in a stochastic magnetic field is developed. Calculations indicate that mean-square fields strongly modify Reynolds stress phase coherence and also induce a magnetic drag on zonal flows. The physics of transport reduction by tangled fields is elucidated and linked to the related quench of turbulent resistivity. We propose a physical picture of the system as a resisto-elastic medium threaded by a tangled magnetic network. Applications of the theory to momentum transport in the tachocline and other systems are discussed in detail.

Potential vorticity transport in weakly and strongly magnetized plasmas

“…This is accomplished by considering the scale ordering. In the two-scale average method proposed 7 , an average over an area is performed, with a scale (1/k avg ) larger than the scale of the stochastic fields (1/k st ) but smaller than the Magnetic Rhines scale 41 (k MR ), and Rossby wavelength (k Rossby ). This average is denoted as F ≡ dR 2 dB st • P (B st,x ,B st,y ) • F, where F is arbitrary function, dR 2 denotes integration over the region, and P (B st,x ,B st,y ) is probability distribution function for the random fields.…”

“…Stochastic fields are ubiquitous. One example is the tangled field of the solar tachocline 7,36 -a candidate site for the solar dynamo. The solar tachocline is a thin strongly stratified layer between the radiation and convection zones, located at ∼ 0.7 solar radius 36 , where magnetic fields are perturbed by 'pumping' from the convection zone.…”

“…1). Chen and Diamond 7 proposed that the effects of suppression by random-fields are already substantial (even for weak B 0 ) on account of Reynolds stress decoherence. They discussed a β -plane (quasi-2D) MHD model for the solar tachocline and studied how the zonal flow is suppressed by random fields.…”

“…They shows as mean magnetic field strong enough, zonal flow generation stops and the system is fully Alfvénized. Reproduced with permission fromChen and Diamond 7 , by permission of the AAS. Copyright 2020 The American Astronomical Society.…”

Elasticity of tangled magnetic fields

“…Williams (2004) has speculated on the importance of magnetoelasticity in astrophysical systems in which the presence of magnetic fields is traditionally modelled by magnetic viscosity. More recently, Chen & Diamond (2020) have developed a theory of potential vorticity mixing in the solar tachocline, accounting for the effect of a tangled magnetic-field elasticity using a similar model. Experimentally, elastic waves have been observed in viscoelastic flows of polymer solutions (Qin et al 2019), which obey a system of equations closely related to MHD (Ogilvie & Proctor 2003).…”