MAGNETIC DAMPING OF JETS AND VORTICES

It is well known that the imposition of a static magnetic field tends to suppress motion in an electrically conducting liquid. Here we look at the magnetic damping of liquid-metal flows where the Reynolds numbe r is large and the magnetic Reynolds number is small. The magnetic fie ld is taken as uniform and the fluid is either infinite in extent or e lse bounded by an electrically insulating surface S. Under these condi tions, we and that three general principles govern the flow. First, th e Lorentz force destroys kinetic energy but does not alter the net lin ear momentum of the fluid, nor does it change the component of angular momentum parallel to B. In certain flows, this implies that momentum, linear or angular, is conserved. Second, the Lorentz force guides the flow in such a way that the global Joule dissipation, D, decreases, a nd this decline in D is even more rapid than the corresponding fall in global kinetic energy, E. (Note that both D and E are quadratic in u. ) Third, this decline in relative dissipation, D/E, is essential to co nserving momentum, and is achieved by propagating linear or angular mo mentum out along the magnetic field lines. In fact, this spreading of momentum along the B-lines is a diffusive process, familiar in the con text of MHD turbulence. We illustrate these three principles with the aid of a number of specific examples. In increasing order of complexit y we look at a spatially uniform jet evolving in time, a three-dimensi onal jet evolving in space, and an axisymmetric vortex evolving in bot h space and time. We start with a spatially uniform jet which is dissi pated by the sudden application of a transverse magnetic field. This s imple (perhaps even trivial) example provides a clear illustration of our three general principles. It also provides a useful stepping-stone to our second example of a steady three-dimensional jet evolving in s pace. Unlike the two-dimensional jets studied by previous investigator s, a three-dimensional jet cannot be annihilated by magnetic braking. Rather, its cross-section deforms in such a way that the momentum flux of the jet is conserved, despite a continual decline in its energy fl ux. We conclude with a discussion of magnetic damping of axisymmetric vortices. As with the jet flows, the Lorentz force cannot destroy the motion, but rather rearranges the angular momentum of the flow so as t o reduce the global kinetic energy. This process ceases, and the flow reaches a steady state, only when the angular momentum is uniform in t he direction of the field lines. This is closely related to the tenden cy of magnetic fields to promote two-dimensional turbulence.