Low-Lying Superfluid States in a Rotating Annulus

The low-lying states of rotating liquid He II in an annulus ($R_1<r<\mathrm{}{R}_2$) are studied with the model of a classical inviscid fluid. An exact hydrodynamic solution is obtained with the method of images for a system consisting of rectilinear vortices with circulation $\kappa$ combined with circulation ${\Gamma }_1$ about the inner cylinder. The energy and angular momentum are calculated, both for an arbitrary configuration of vortices and for the particular configuration of a symmetric ring of $l$ vortices. If ${\Gamma }_1$ is treated as a variational parameter, the critical angular velocity ${\Omega }_0$ for the appearance of vortices in a narrow annulus is $\left(\frac{\kappa }{\pi d^2}\right)\ln \left(\frac{2d}{\pi a}\right)$, where $d$ is the width of the annulus and $a$ is the radius of the vortex core. For $\Omega <{\Omega }_0$, the equilibrium state is an irrotational (vortex-free) flow with quantized circulation $n_{\kappa }(n=1,2,\cdots )$; these levels are equally spaced, and a given quantum state represents the lowest free energy only in a narrow angular-velocity interval of $\frac{\kappa }{2\pi R^2}$, where $R$ is the mean radius of the annulus. The maximum quantum number of irrotational circulation is $\frac{2\pi R^2{\Omega }_0}{\kappa }=2\mathrm{}{\left(\frac{R}{d}\right)}^2\ln \left(\frac{2d}{\pi a}\right)\gg 1$. For $\Omega >{\Omega }_0$, the vortices lie on the circumference of a ring midway between the walls, and the number of vortices increases rapidly with $\Omega$. If ${\Gamma }_1$ is constrained to vanish identically, the critical angular velocity ${\Omega }_c$ for the appearance of vortices in a narrow annulus is of order $\frac{\kappa }{2\pi \mathrm{Rd}}$; this is equivalent to Feynman's critical velocity $v_c=O\left(\frac{\hslash }{\mathrm{md}}\right)$ for singly quantized vortices with $\kappa =\frac{h}{m}$. In the opposite limit of a wide annulus ($R_1\ll R_2$), the equilibrium state is shown to agree with Vinen's earlier calculations.