Transport and instability in driven two-dimensional magnetohydrodynamic flows

Durston, Sam; Gilbert, Andrew D.

doi:10.1017/jfm.2016.361

Cited by 6 publications

(7 citation statements)

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“…If the magnetic field is large enough, then the zonostrophic instability switches off, as shown numerically on a β −plane Durston & Gilbert 2016) and on a spherical surface (Tobias et al 2011). Theoretically, this suppression of the zonostrophic instability has been described via a straightforward application of QL theory (Tobias et al 2011;Constantinou & Parker 2018), though as we will show here, this approach does not capture the relevant physics.…”

Section: Comparison Of Theory With Numerical Calculationsmentioning

confidence: 83%

Potential Vorticity Mixing in a Tangled Magnetic Field

Chen

Diamond

2020

ApJ

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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.

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Section: Comparison Of Theory With Numerical Calculationsmentioning

confidence: 83%

Potential Vorticity Mixing in a Tangled Magnetic Field

Chen

Diamond

2020

ApJ

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“…For β = 0 this reduces to the PDE of Sivashinsky (1985) and simulations show that the inverse cascade of structures to large scales in y is arrested by the β-effect. These authors characterise the fundamental instability of the Kolmogorov flow as due to a negative eddy viscosity, in other words that the large-scale y-dependent modes have growth rate p = −ν E k 2 + • • • where the eddy viscosity ν E changes sign from positive below the threshold Re c = √ 2, to negative above, so destabilising the flow on large scales with the fastest growing modes determined by the next terms in this series (Dubrulle and Frisch 1991).…”

Section: Introductionmentioning

confidence: 99%

“…One family exists for magnetic Prandtl number P < 1, for arbitrarily strong magnetic fields, provided the Reynolds number is above a threshold depending on P . This is studied numerically and growth rates obtained through asymptotic approximations for k 1; these authors refer to these modes as Alfvén Dubrulle-Frisch modes, as the instability can be linked to a change of sign of the eddy viscosity (Dubrulle and Frisch 1991).…”

Section: Introductionmentioning

confidence: 99%

“…Constantinou and Parker (2018) analysed Kelvin-Orr shearing wave dynamics for Rossby/Alfvén waves and the interplay between Reynolds and Maxwell stresses, providing evidence for this B 2 0 ∼ η threshold for jet formation. Durston and Gilbert (2016) focused on the couplings between large-scale zonal flow and zonal field in the presence of waves, calculating an effective viscosity and effective magnetic diffusivity, plus an effective cross transport term in which current gradients can drive the zonal vorticity; this and other transport effects are discussed in Chechkin (1999), Kim (2007), and Leprovost and Kim (2009). Parker and Constantinou (2019) interpret the presence or otherwise of jets in terms of the competition between a positive magnetic eddy viscosity term and a negative, purely hydrodynamic eddy viscosity.…”

Section: Introductionmentioning

confidence: 99%

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Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow

Algatheem¹,

Gilbert²,

Hillier³

2023

Preprint

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This paper concerns the classic problem of stability of the Kolmogorov flow u = (0, sin x) in the infinite (x, y)-plane. A mean magnetic field of strength B 0 is introduced and the MHD linear stability problem is studied for modes with a wave-number k in the y-direction, and sometimes a Bloch wavenumber in the x-direction. The key parameters governing the problem are the Reynolds number ν −1 , magnetic Prandtl number P , and dimensionless magnetic field strength B 0 corresponding to an inverse magnetic Mach number. The mean magnetic field can also be taken to have an arbitrary direction in the (x, y)-plane and a mean x-directed flow U 0 can be incorporated, in the most general formulation.First the paper considers Kolmogorov flow with a y-directed mean magnetic field, which for convenience we refer to as 'vertical'. Taking = 0, the suppression of the classic hydrodynamic instability is observed with increasing field strength B 0 . A branch of strong-field instabilities occurs when the magnetic Prandtl number P is less than unity, as found recently by A.E. Fraser, I.G. Cresser and P. Garaud (J. Fluid Mech. 949, A43, 2022). Analytical results based on eigenvalue perturbation theory in the limit k → 0 support the numerics for both weak-field and strong-field instabilities, and show their origin in the coupling of large-scale weakly damped modes with x-wavenumber n = 0, to smaller-scale modes having n = ±1.The paper then considers the case of 'horizontal' or x-directed mean magnetic field. Here the unperturbed state consists of steady, wavey magnetic field lines distorted by the underlying Kolmogorov flow, and with the driving body force balancing both viscosity and the Lorentz force. As the magnetic field is increased from zero, the purely hydrodynamic instability is suppressed again, but for stronger fields a new branch of instabilities appears. Allowing a non-zero Bloch wavenumber allows further instability, and in some circumstances when the system is hydrodynamically stable, arbitrarily weak magnetic fields can give growing modes, via the instability taking place on large scales in x and y. Numerical results are presented together with eigenvalue perturbation theory in the limits k, → 0. The theory gives analytical approximations for growth rates and thresholds that are in good agreement with those computed.

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“…Idealized settings to study individual effects prove fruitful. For example, certain aspects of fluid physics under the influence of rotation and magnetization can be studied in the 2D β plane or 2D spherical surface [15][16][17][18][19]. A β plane is a Cartesian geometrical simplification of a rotating sphere that retains the physics associated with rotation and the latitudinal variation of the Coriolis effect [20].…”

Section: Introductionmentioning

confidence: 99%

Magnetic eddy viscosity of mean shear flows in two-dimensional magnetohydrodynamics

Parker

Constantinou

2019

Phys. Rev. Fluids

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Magnetic induction in magnetohydrodynamic fluids at magnetic Reynolds number (Rm) less than 1 has long been known to cause magnetic drag. Here, we show that when Rm 1 and the fluid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than the kinetic energy, induction due to a mean shear flow leads to a magnetic eddy viscosity. The magnetic viscosity is derived from simple physical arguments, where a coherent response due to shear flow builds up in the magnetic field until decorrelated by turbulent motion. The dynamic viscosity coefficient is approximately (B 2 p /2µ0)τcorr, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional case, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero viscosity whenever there is differential rotation. Hence, these results serve as a dynamical generalization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular momentum and may be of importance to zonal flows in astrophysical domains such as the interior of some gas giants. * parker68@llnl.gov

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Transport and instability in driven two-dimensional magnetohydrodynamic flows

Cited by 6 publications

References 54 publications

Potential Vorticity Mixing in a Tangled Magnetic Field

Potential Vorticity Mixing in a Tangled Magnetic Field

Zonostrophic instabilities in magnetohydrodynamic Kolmogorov flow

Magnetic eddy viscosity of mean shear flows in two-dimensional magnetohydrodynamics

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