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

TL;DR

This paper numerically verifies the asymptotic theory of shear-driven dynamos at high Reynolds numbers using steady MHD solutions in plane Couette flow, revealing two classes of nonlinear states and confirming theoretical predictions.

Contribution

It introduces two classes of nonlinear MHD states in plane Couette flow and validates asymptotic theory predictions through numerical solutions at finite Reynolds numbers.

Findings

Identification of shear-driven dynamo states without external magnetic fields.

Numerical solutions align closely with asymptotic theory predictions.

Discovery of self-sustained shear-driven dynamos at zero external magnetic field.

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 utilised. 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/Alfv\'en 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.