  We review the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations for
superfluid Helium turbulence and discuss their implications for recent
measurements of superfluid turbulence decay.
  A new Hamiltonian formulation of these equations renormalizes the vortex line
velocity to incorporate finite temperature effects. These effects also
renormalize the coupling constant in the mutual friction force between the
superfluid and normal fluid components by a factor of rho_s / \rho (the
superfluid mass fraction) but they leave the vortex line tension unaffected.
Thus, the original HVBK form is recovered at zero temperature and its mutual
friction coefficients are renormalized at nonzero temperature. The HVBK
equations keep their form and no new parameters are added. However, a
temperature dependent trade-off does arise between the mutual friction coupling
and the vortex line tension.
  The renormalized HVBK equations obtained via this new Hamiltonian approach
imply a dynamical equation for the space-integrated vortex tangle length, which
is the quantity measured by second sound attenuation experiments in superfluid
turbulence. A Taylor-Proudman theorem also emerges for the superfluid vortices
that shows the steady vortex line velocity becomes columnar under rapid
rotation.
