Dissipation in quantum turbulence in superfluid $^4$He above 1K

There are two commonly discussed forms of quantum turbulence in superfluid $^4$He above 1K: in one there is a random tangle of quantizes vortex lines, existing in the presence of a non-turbulent normal fluid; in the second there is a coupled turbulent motion of the two fluids, often exhibiting quasi-classical characteristics on scales larger than the separation between the quantized vortex lines in the superfluid component. The decay of vortex line density, $L$, in the former case is often described by the equation $dL/dt=-chi_2 (kappa/2pi)L^2$, where $kappa$ is the quantum of circulation, and $chi_2$ is a dimensionless parameter of order unity. The decay of total turbulent energy, $E$, in the second case is often characterized by an effective kinematic viscosity, $ u$, such that $dE/dt=- u kappa^2 L^2$. We present new values of $chi_2$ derived from numerical simulations and from experiment, which we compare with those derived from a theory developed by Vinen and Niemela. We summarise what is presently known about the values of $ u$ from experiment, and we present a brief introductory discussion of the relationship between $chi_2$ and $ u$, leaving a more detailed discussion to a later paper.