Turbulence in superfluid helium: Steady homogeneous counterflow

The fully developed turbulent state of a superfluid is considered in the simple limit of homogeneous counterflow. The vortex tangle is described in statistical terms which focus on the distribution of line length with respect to the local self-induced velocity v₋ of the line. The equation for the motion of a line element is found to contain driving terms arising from the interaction of the normal fluid with the line element, and randomizing terms arising from its self-induced motion. A geometrical argument indicates that a random tangle has a characteristic distance over which all local derivatives randomize. A dynamical argument suggests that line-line crossings lead to a characteristic randomization distance equal to the typical interline spacing, and to a characteristic randomization time of order 〈{v₋〉}. Setting the geometric randomization distance equal to then allows one to model the effect of the self-induced motion as a kind of random walk. The resulting complicated differential equation predicts steady-state properties in good agreement with those observed experimentally. It is also found that the well-known Vinen equation follows as an accurate consequence of the theory. The physical interpretation of this equation, however, turns out to be different from that which was originally proposed.