Low-Lying Superfluid States in a Rotating Annulus

The low-lying states of rotating liquid He II in an annulus (R₁<r<R₂) 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 κ combined with circulation Γ₁ 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 Γ₁ is treated as a variational parameter, the critical angular velocity Ω₀ for the appearance of vortices in a narrow annulus is (κπd²)ln(2dπa), where d is the width of the annulus and a is the radius of the vortex core. For Ω<Ω₀, the equilibrium state is an irrotational (vortex-free) flow with quantized circulation n_κ(n=1,2,⋯); these levels are equally spaced, and a given quantum state represents the lowest free energy only in a narrow angular-velocity interval of κ2πR², where R is the mean radius of the annulus. The maximum quantum number of irrotational circulation is 2πR²Ω₀κ=2(R/d)²ln(2dπa)1. For Ω>Ω₀, the vortices lie on the circumference of a ring midway between the walls, and the number of vortices increases rapidly with Ω. If Γ₁ is constrained to vanish identically, the critical angular velocity Ωc for the appearance of vortices in a narrow annulus is of order κ2πRd; this is equivalent to Feynman's critical velocity vc=O(md) for singly quantized vortices with κ=h/m. In the opposite limit of a wide annulus (R₁R₂), the equilibrium state is shown to agree with Vinen's earlier calculations.