Linear and nonlinear hydromagnetic stability in laminar and turbulent flows

We consider the evolution of arbitrarily large perturbations of a prescribed pure hydrodynamical flow of an electrically conducting fluid. We study whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process. For that purpose we derive a generalized Reynolds-Orr equation for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy. The flow is confined in a finite volume so the normal component of the velocity at the boundary is zero. The tangential component is left arbitrary in contrast with previous works. For the magnetic field we mostly employ the classical boundary conditions where the field extends in the whole space. We establish critical values of hydrodynamic and magnetic Reynolds numbers below which arbitrarily large initial perturbations of the hydrodynamic flow decay. This involves generalization of the Rayleigh-Faber-Krahn inequality for the smallest eigenvalue of an elliptic operator. For high Reynolds number turbulence we provide an estimate of critical magnetic Reynolds number below which arbitrarily large fluctuations of the magnetic field decay.

Figure 1

The radial structure of the slowest decaying mode of unsteady Stokes flow in a ball.

Figure 2

The radial structure of the components $B_Y$ and $B_P$ of the slowest decaying mode of the induction equation in the absence of flow in a ball ($B_Y$ and $B_P$ are the coefficients of ${\mathit{Y}}_{1m}$ and ${\mathit{\Psi }}_{1m}$, respectively).