A note on topology and magnetic energy in incompressible perfectly conducting fluids

Abstract

In an incompressible perfectly conducting fluid the Navier-Stokes equations become \[ \frac{\partial v}{\partial t} + v.\nabla v = -\nabla {\rm P} - \nu\Delta v + j\times {\boldmath H}, \] where j = curl H, div H = 0, div v = 0, and dH/dt = curl (v × H). The last equation follows from the Maxwell equation dH/dt = − curl E and the assumption of perfect conduction — that the electric field in the frame of the fluid vanishes, E′ = E + v × H = 0. In geometric terms, it says that the system evolves so that the time derivative of H is equal to minus its spatial Lie derivative: \[ \frac{{\rm d}{\boldmath H}}{{\rm d}t} = -L_{\rm v}{\boldmath H}. \]

Thus H is equivariant with respect to the evolution (or ‘frozen in the fluid’) as long as the evolution follows these equations. Since the first equation tends to dissipate magnetic energy E = ∫ ∥H∥² the question naturally rises whether the topology of H determines lower bounds on E. We treat this question in the general context of a divergence-free vector field H on a closed Riemannian 3-manifold M. We obtain a result bounding E from below but make no assertion on the existence of extremals. Arnol'd (1986) has defined a quadratic form for any ‘null-homologous’ H