Theory of quantum grid turbulence in superfluid ${}^3\mathrm{He}\text{−}B$

A theory is developed to describe grid turbulence in a superfluid in the case where the normal fluid is held stationary, as would be the case for superfluid ${}^3\mathrm{He}\text{−}B$ in which the normal fluid is very viscous. The theory is a straightforward development of earlier work, reviewed by Vinen and Niemela [J. Low Temp. Phys. 128, 167 (2002)], and it shows that on large length scales the turbulence is strongly damped by mutual friction. A comparison is made with recent work by Volovik and his colleagues [G. E. Volovik, JETP Lett. 78, 533 (2003); G. E. Volovik, J. Low Temp. Phys. (to be published); and L’vov, Nazarenko, and Volovik, JETP Lett. (to be published)], which was developed while our work was in progress.

Figure 1

(Color online) The solution of Eq. (25) with the boundary conditions explained in the text.

Figure 2

(Color online) $ϵ\left(k\right)∕ϵ\left(k_0\right)$ plotted against $k∕k_0$ for two values of $Q=R{\gamma }^{3∕2}{\mathbf{\left(}ϵ\left(\infty \right)\mathbf{\right)}}^{-1∕2}$. Upper graph: $Q=25$; lower graph: $Q=0.1$.

Figure 3

(Color online) Variation of vorticity with wave number.

Figure 4

Different turbulent regimes for different values of $r∕\mathcal{l}$ and $\alpha$.