Vortices and Rotation

Abstract

As well as being free from viscosity, the Bose–Einstein condensate has another striking property—it is constrained to circulate only through the presence of whirlpools of fixed size and quantized circulation. In contrast, in conventional fluids, the eddies can have arbitrary size and circulation. Here we establish the form of these quantum vortices, their key properties, and how they are formed and modelled.

Problems

5.1

Consider the bucket of Sects. 5.5 and 5.6 to now feature a harmonic potential $V(r)=\frac{1}{2}m \omega _r^2 r^2$ perpendicular to the axis of the cylinder. Take the condensate to adopt the Thomas–Fermi profile.

• (a) Show that the energy of the vortex-free condensate is $E_0=\pi m n_0 \omega _r^2 H_0 R_r^4 / 6$, where $R_r$ is the radial Thomas–Fermi radius and $n_0$ is the density along the axis.

• (b) Now estimate the kinetic energy $E_\mathrm{kin}$ due to a vortex along the axis via Eq. (5.12). Use the fact that $a_0 \ll R_r$ to simplify your final result.

• (c) Estimate the angular momentum of the vortex state, and hence estimate the critical rotation frequency at which the presence of a vortex becomes energetically favourable.

5.2

Use the LIA (Eq. (5.49)) to determine the angular frequency of rotation of a Kelvin wave of wave length $\lambda =2\pi /k$ (where k is the wavenumber) on a vortex with circulation $\kappa$.

5.3

Using the vortex point method and the complex potential, determine the translational speed of a vortex-antivortex pair (each of circulation $\kappa$) separated by the distance 2D.

5.4

Using the vortex point method and the complex potential, determine the period of rotation of a vortex-vortex pair (each of circulation $\kappa$) separated by the distance 2D.

5.5

Consider a homogeneous, isotropic, random vortex tangle (ultra-quantum turbulence) of vortex line density L, contained in a cubic box of size D. Show that the kinetic energy is approximately

$$\begin{aligned} E \approx \frac{\rho \kappa ^2 L D^3}{4 \pi } \ln \left( \frac{\ell }{a_0} \right) , \end{aligned}$$

(5.60)

where $\rho$ is the density, $\kappa$ the quantum of circulation, $\ell \approx L^{-1/2}$ is the inter-vortex distance and $a_0$ is the vortex core radius.

5.6

In an ordinary fluid of kinematic viscosity $\nu$, the decay of the kinetic energy per unit mass, $E'$, obeys the equation

$$\begin{aligned} \frac{dE'}{dt}=-\nu \omega ^2, \end{aligned}$$

(5.61)

where $\omega$ is the rms vorticity. Consider ultra-quantum turbulence of vortex line density L. Define the rms superfluid vorticity as $\omega =\kappa L$, and show that the vortex line density obeys the equation,

$$\begin{aligned} \frac{dL}{dt}= -\frac{\nu }{c} L^2, \end{aligned}$$

(5.62)

where the constant c is,

$$\begin{aligned} c=\frac{1}{4 \pi } \ln \left( \frac{\ell }{a_0}\right) , \end{aligned}$$

(5.63)

hence show that, for large times, the turbulence decays as

$$\begin{aligned} L \sim \frac{c}{\nu } t^{-1}. \end{aligned}$$

(5.64)