The non-radial velocity theorem revisited

摘要

The article presents evidence that non-radial flows, i.e. flows v with v center dot r equivalent to 0, in a fluid of radially symmetric conductivity contained in a spherical volume B-R of radius R are not capable of magnetic Field generation. More precisely, it is proved that the L-1-norm integral(BR) vertical bar P(center dot, t)vertical bar dv, of the poloidal scalar P(r, t) of the magnetic field is bounded for all times by its initial value in the L-2-norm. Moreover, the L-1-norm of P over an arbitrary sphere S-r with radius r decays (in time) to zero uniformly with respect to r. In the case that the poloidal field has died out, we prove, furthermore, decay to zero of the toroidal scalar T in the norin (max(vertical bar r vertical bar=r) T(r, t) - min(vertical bar r vertical bar=r) T(r, t)) uniformly with respect to r. This implies, in particular, UrlifOrinly pointwise decay to zero of the toroidal scalar in B-R.