Singularity formation in a class of stretched solutions of the equations for ideal magneto-hydrodynamics

A class of stretched solutions of the equations for three-dimensional, incompressible, ideal magneto-hydrodynamics (MHD) is studied. In Elsasser variables, V ± = U ± B, these solutions have the form V ± = (v±, v± 3 ) where v± = v±(x, y, t) and v± 3 (x, y, z, t) = zγ±(x, y, t) + β±(x, y, t). Twodimensional partial differential equations for γ±, v± and β± are obtained that are valid in a tubular domain which is infinite in the z-direction with periodic cross section. Pseudo-spectral computations of these equations provide evidence for a blow-up in finite time in the above variables. This apparent blow-up is an infinite energy process that gives rise to certain subtleties; while all the variables appear to blow-up simultaneously, the two-dimensional part of the magnetic field b = 1 2 ( v+ − v−) blows up at a very late stage. This singularity in b is hard to detect numerically but supporting analytical evidence of a Lagrangian nature is provided for its existence. In three dimensions these solutions correspond to magnetic vortices developing along the axis of the tube prior to breakdown. Mathematics Subject Classification: 35Q35, 76B03, 76W05