The Growth of Cross Helicity in the Protogalactic Dynamo

The protogalactic dynamo theory of Kulsrud and coworkers raises the possibility that a strong magnetic field, which is coherent over galaxy-size length scales, emerges from a nonlinear turbulent dynamo during the protogalactic collapse. After galaxy formation, it is not clear that this galactic-scale field could be sustained by turbulence generated by supernovae because the turbulent stirring of supernovae is at the comparatively small length scale of 100 pc. However, it is known that if there is sufficient cross helicity <v \b.dot b> (the angle brackets denote an average over an ensemble of random motions) in the protogalactic turbulence, some portion of the protogalactic magnetic energy will be immune to the turbulent cascade of energy, and the field generated by the protogalactic dynamo will be long-lived. In this paper, the evolution of the cross helicity in the protogalactic dynamo is investigated under the assumptions that (1) the protogalactic turbulence is supported by external forcing due to shocks, (2) resistivity is negligible compared to viscosity, (3) the initial magnetic energy and initial cross helicity are minuscule, and (4) the magnetic helicity <a \b.dot b> and kinetic helicity <v \b.dot \b.nabla X v> are zero. A numerical calculation based upon the direct interaction approximation (DIA) shows that while the magnetic energy grows rapidly due to line stretching, the cross helicity either decays or at most grows very slowly. This suggests that cross helicity cannot be invoked as a mechanism for sustaining the protogalactic magnetic field. Other mechanisms, however, may exist.