The geometry and symmetries of magnetohydrodynamic turbulence: Anomalies of spatial periodicity

It has become common to formulate theories and computations of magnetohydro dynamic turbulent effects in rectangular periodic boundary conditions, proc eeding by analogy with what is seen as a useful framework for Navier-Stokes fluid turbulence. It is shown here that because of certain features of Max well's equations for electrodynamics, it is inconsistent to invoke three-di mensional, rectangular, periodic boundary conditions and symmetry at the sa me time that the displacement current is neglected. The difficulty does not arise in the two-dimensional case. In three dimensions, the difficulty can be remedied by a reformulation in cylindrical geometry, imposing symmetry in the azimuthal and axial directions, but not in the radial one; a geometr y that is closer to laboratory possibilities than the wholly three-dimensio nal periodic assumption. The reformulation seems particularly necessary in cases with a net flux of magnetic field and/or electric currents through th e system. These cases no longer seem discontinuous from those without net m agnetic fluxes or currents. The price paid is a loss of some possibilities for dimensional analysis and identification of similarity variables. (C) 19 99 American Institute of Physics. [S1070-664X(99)02907-9].