  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.
