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 (MM) is studied. In Elsasser variables, V+/- = U +/- B, these solutions have the form V+/- = (upsilon (+/-), upsilon (+/-)(3)) where upsilon (+/-) = upsilon (+/-)(x, y, t) and upsilon (+/-)(3)(x, y, z, t) = zy(+/-)(x, y, t) + beta (+/-)(x, y, t). Two-dimensional partial differential equations for gamma (+/-), upsilon (+/-) and beta (+/-) 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(upsilon (+) - upsilon (-)) blows up at a very late stage. This 2 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.