Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L and in counterflow velocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from a second order expansion of the Vinen equation itself, and those which are not related to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.