High-speed shear-driven dynamos. Part 2. Numerical analysis

Abstract

This paper aims to numerically verify the large Reynolds number asymptotic theory of magneto-hydrodynamic (MHD) flows proposed in the companion paper Deguchi (2019). To avoid any complexity associated with the chaotic nature of turbulence and flow geometry, nonlinear steady solutions of the viscous-resistive magneto-hydrodynamic equations in plane Couette flow have been utilized. Two classes of nonlinear MHD states, which convert kinematic energy to magnetic energy effectively, have been determined. The first class of nonlinear states can be obtained when a small spanwise uniform magnetic field is applied to the known hydrodynamic solution branch of the plane Couette flow. The nonlinear states are characterised by the hydrodynamic/magnetic roll-streak and the resonant layer at which strong vorticity and current sheets are observed. These flow features, and the induced strong streamwise magnetic field, are fully consistent with the vortex/Alfvén wave interaction theory proposed in Deguchi (2019). When the spanwise uniform magnetic field is switched off, the solutions become purely hydrodynamic. However, the second class of 'self-sustained shear driven dynamos' at the zero-external magnetic field limit can be found by homotopy via the forced states subject to a spanwise uniform current field. The discovery of the dynamo states has motivated the corresponding large Reynolds number matched asymptotic analysis in Deguchi (2019). Here, the reduced equations derived by the asymptotic theory have been solved numerically. The asymptotic solution provides remarkably good predictions for the finite Reynolds number dynamo solutions. arXiv:1809.03855v2 [physics.flu-dyn]

High-speed shear-driven dynamos. Part 1. Asymptotic analysis

“…Despite the caveat, for some cases, the dynamos of slightly different structures can be driven even when the external magnetic field is switched off, as shown by the rather surprising numerical results in the companion paper Deguchi (2019). Section 4 successfully formulated the asymptotic theory for that purely shear driven dynamos (the S 3 dynamos), revealing that the magnetic field generation indeed survives at the large Reynolds number limit.…”

“…The flow remains always linearly stable. However, the nonlinear finite Reynolds number computation in Deguchi (2019) indeed found the non-trivial streamwise magnetic field amplification. The generation of that magnetic streak must be explained by the entire vortex/Alfvén wave interaction mechanism shown in the figure.…”

“…In fact, a dynamo theory is possible for the purely shear driven cases if we choose slightly smaller sized leading order magnetic fields. The asymptotic theory here is partially motivated by the numerical finding of such dynamos to be shown in Deguchi (2019).…”

“…However, this does not mean that the dynamo states cannot be realised in the zero external magnetic field limit. In fact, the numerical study of the shear driven dynamos in the companion paper Deguchi (2019) showed the production of the magnetic field at that limit. Motivated by that numerical result, section 4 is concerned with the mathematical description of such self-sustained shear driven dynamos, and show that the induced magnetic field is indeed slightly smaller than the vortex/Alfvén wave interaction states.…”

Subcritical magnetohydrodynamic instabilities: Chandrasekhar’s theorem revisited

“…(2013). The invariant solutions in the non-rotating MHD plane Couette flow have been studied only recently in Deguchi (2019 a , b ), together with their large-Reynolds-number asymptotic development. The present work will extend that result for broader magnetic Mach numbers.…”