1. Introduction
The linear response of a rapidly rotating Navier–Stokes fluid to an impulsive increase in the angular velocity of its container, or ‘spin-up’, has been extensively studied. In the seminal work of Greenspan & Howard (Reference Greenspan and Howard1963), three phases were identified: formation of a viscous boundary layer, Ekman pumping and the damping of residual inertial oscillations by viscosity. Co-rotation between the interior fluid and the container is established by Ekman pumping, during which a secondary flow recycles fluid from the boundary layers into the interior. The time scale for the spin-up, known as the Ekman time, is proportional to the viscosity to the minus half power, and achieves co-rotation much faster than viscous diffusion. Since the work of Greenspan & Howard (Reference Greenspan and Howard1963), spin-up has been studied in fluids with density stratification (Pedlosky Reference Pedlosky1967; Walin Reference Walin1969), stratified and compressible fluids (Abney & Epstein Reference Abney and Epstein1996; van Eysden & Melatos Reference van Eysden and Melatos2008), magnetized plasmas (Loper Reference Loper1971; Easson Reference Easson1979; van Eysden Reference van Eysden2014), multi-component fluids (Ungarish Reference Ungarish1990; Amberg & Ungarish Reference Amberg and Ungarish1993) and different geometries (Clark et al. Reference Clark, Clark, Thomas and Lee1971; van Eysden & Melatos Reference van Eysden and Melatos2013, Reference van Eysden and Melatos2014). For a review of spin-up in classical fluids, the reader is referred to Benton & Clark (Reference Benton and Clark1974).
The spin-up of superfluids like helium II has also received considerable attention both experimentally (Tsakadze & Tsakadze Reference Tsakadze and Tsakadze1980) and theoretically (Campbell & Krasnov Reference Campbell and Krasnov1982; Adams, Cieplak & Glaberson Reference Adams, Cieplak and Glaberson1985; Peradzynski, Filipkowski & Fiszdon Reference Peradzynski, Filipkowski and Fiszdon1990), with the aim of shedding light on the interaction between the quantized vortices and the container walls. A rigorous treatment was provided by Reisenegger (Reference Reisenegger1993), who solved the two-fluid equations of Chandler & Baym (Reference Chandler and Baym1986) between slowly accelerating parallel plates, and showed that the Ekman time is reduced by a factor depending on the superfluid mutual friction coefficients and the normal fluid density fraction. This was generalized to the impulsive acceleration of containers of arbitrary shape (van Eysden & Melatos Reference van Eysden and Melatos2013) and to include the self-consistent response of the container (van Eysden & Melatos Reference van Eysden and Melatos2014) to facilitate comparison with helium II experiments (van Eysden & Melatos Reference van Eysden and Melatos2011, Reference van Eysden and Melatos2012). However, theoretical work to date on the spin-up of two-component superfluids such as helium II assumes smooth-walled containers and neglects the effects of pinning.
The spin-up of helium II at zero temperature, where the normal fluid component vanishes and cannot facilitate Ekman pumping, was also investigated by Reisenegger (Reference Reisenegger1993). By introducing a frictional force to account for the sliding of vortex lines at the boundary, Reisenegger (Reference Reisenegger1993) showed that the superfluid spins up via an Ekman-like secondary flow with a time scale that depends on the strength of the frictional force. However, only the response of a single-component fluid to slowly accelerating parallel plates was studied.
There is good reason to expect that the impulsive acceleration problem produces different physics from that of slowly accelerating parallel plates. When an ideal magnetohydrodynamic plasma is slowly accelerated (i.e. the time scale for acceleration of the container is slower than the Alfvén crossing time), the plasma moves with the container as a rigid body. However, in response to an impulsive acceleration, Alfvén waves are excited which produce persistent oscillations of the container and the plasma (van Eysden Reference van Eysden2014). A final state of co-rotation between the container and the plasma is inconsistent with energy conservation if the fluid is dissipationless, hence the system oscillates persistently. Similar arguments apply for a superfluid at zero temperature rotating uniformly with a high density of vortex lines.
Two primary motivations exist for this study. The first is a series of experiments performed by Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1972, Reference Tsakadze and Tsakadze1973, Reference Tsakadze and Tsakadze1974, Reference Tsakadze and Tsakadze1980), in which the angular velocity of containers filled with uniformly rotating helium II was impulsively increased. The subsequent motion of the container, responding freely to the hydrodynamic torque of the contained fluid, was then recorded. In experiments where the container was coated with powder to facilitate vortex pinning, a sinusoidal oscillation of the container was observed. The experiments were intended to observe Tkachenko modes, arising from rigidity of the vortex lattice; however, the observed frequencies and their independence of vessel radius were inconsistent with theory. In contrast, the low-frequency columnar nature of the Ekman pumping mechanism presented here provides a more natural agreement with theory. Sonin (Reference Sonin1976, Reference Sonin1987) studied the eigenfrequencies of rigid vortex lattices including line bending and identified a low-frequency ‘inertial-wave resonance’, but did not solve the initial value problem.
The second application is the recovery of pulsar glitches, the original motivation for the experiments of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980). Glitches are tiny impulsive increases in the rotation frequency of the neutron star crust, typically followed by a quasi-exponential recovery that is believed to be associated with the response of the interior superfluid (Baym, Pethick & Pines Reference Baym, Pethick and Pines1969b ). Although the superfluid neutron vortices are believed to pin to lattice sites in the crust and to flux tubes arising from type II superconductivity of protons in the outer core, the Ekman pumping mechanism identified here may give rise to long-period oscillations if vortex creep is active.
In this paper, we revisit the problem of superfluid spin-up, by solving for the response of a uniformly rotating superfluid to an impulsive acceleration of the container. We focus on cylindrical geometry, where the vortices are strongly pinned to the top and bottom of the container. We apply the traditional Laplace transform techniques of Greenspan & Howard (Reference Greenspan and Howard1963), but solve self-consistently for the motion of the container as in van Eysden (Reference van Eysden2014) and van Eysden & Melatos (Reference van Eysden and Melatos2014). We demonstrate the presence of a low-frequency oscillation mode, with a period that scales as the vortex line parameter to the one-fourth power. This mode arises in the fast rotation limit, analogous to classical Ekman pumping. A secondary flow is present, which is periodically reversed by the tension in the vortex lines as they are sheared in the boundary layer. By solving the two-fluid equations of Chandler & Baym (Reference Chandler and Baym1986), we predict an oscillation period of 10 s in helium II, compared with that of 40 s observed in the experiments of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980). A direct comparison cannot be made because the experiment is not perfectly suited for the application of the linear theory of Chandler & Baym (Reference Chandler and Baym1986). Using a simplified pinning model, we predict that oscillations in neutron stars can have periods of days to years if the pinning of neutron vortices to flux tubes in the core is strong. These oscillations are weakly damped and are expected to be observable over time scales much longer than the period.
The paper is structured as follows. In § 2, we linearize the equations of Chandler & Baym (Reference Chandler and Baym1986) using a common Ekman pumping ansatz. These equations are solved in § 3 for a superfluid at zero temperature and perfect vortex pinning of vortices at the top and bottom of a cylindrical container, where it is shown that the solution is oscillatory and has an Ekman-like secondary circulation. In § 4, the full two-fluid equations are solved and compared with the experiments of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980) in helium II. Neutron star applications are considered in § 5. In § 6, we summarize our conclusions. Technical details of the solution are presented in the appendices along with some important limiting cases.
2. Governing equations
A convenient description of superfluids such as helium II, rotating with a high density of vortex lines, is given by the Hall–Vinen–Bekharevich–Khalatnikov (HVBK) equations. The fluid comprises two components: a ‘normal’ component, denoted by subscript
$n$
, with viscosity
${\it\eta}$
, and an inviscid ‘superfluid’ component, denoted by subscript
$s$
. Under rotation the inviscid component forms a dense array of quantized vortices, which are smooth-averaged in the hydrodynamic approximation, endowing the inviscid component with a macroscopic vorticity. The vortices mediate interactions between the normal and superfluid components, giving rise to a mutual friction force. An informative introduction to the HVBK equations is provided in Henderson & Barenghi (Reference Henderson and Barenghi2000). A more general set of equations, derived by Baym & Chandler (Reference Baym and Chandler1983) and Chandler & Baym (Reference Chandler and Baym1986), includes restoring forces experienced by the vortex lattice when vortices are displaced from their equilibrium configuration. This force produces Tkachenko oscillations, which are expected to be observed in superfluid experiments. Assuming that the fluid is incompressible, the equations of Chandler & Baym (Reference Chandler and Baym1986), written in HVBK form and in the laboratory frame, are
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial \boldsymbol{v}_{n}}{\partial t}+\boldsymbol{v}_{n}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{v}_{n}=-\boldsymbol{{\rm\nabla}}p_{n}+\frac{{\it\rho}_{s}}{{\it\rho}}\boldsymbol{F}+{\it\nu}_{n}{\rm\nabla}^{2}\boldsymbol{v}_{n}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial \boldsymbol{v}_{s}}{\partial t}+\boldsymbol{v}_{s}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{v}_{s}=-\boldsymbol{{\rm\nabla}}p_{s}-\boldsymbol{t}-{\bf\sigma}-\frac{{\it\rho}_{n}}{{\it\rho}}\boldsymbol{F}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}_{n}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}_{s}=0, & \displaystyle\end{eqnarray}$$
$\boldsymbol{v}_{n,s}$
,
${\it\rho}_{n,s}$
and
$p_{n,s}$
are the macroscopic velocities, densities and reduced pressures of the normal and superfluid components respectively. Throughout our analysis, both the normal and the superfluid components are considered to be incompressible; the validity of this assumption is assessed in §§ 4 and 5 for helium II and neutron stars respectively. The mutual friction force is 
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{F}={\textstyle \frac{1}{2}}B\hat{{\it\omega}}_{s}\times [{\bf\omega}_{s}\times (\boldsymbol{v}_{n}-\boldsymbol{v}_{s})-\boldsymbol{t}-{\bf\sigma}]+{\textstyle \frac{1}{2}}B^{\prime }[{\bf\omega}_{s}\times (\boldsymbol{v}_{n}-\boldsymbol{v}_{s})-\boldsymbol{t}-{\bf\sigma}], & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}{\bf\omega}_{s}=\boldsymbol{{\rm\nabla}}\times \boldsymbol{v}_{s}\end{eqnarray}$$
is the macroscopic vorticity of the superfluid,
$\hat{{\it\omega}}_{s}={\bf\omega}_{s}/|{\bf\omega}_{s}|$
is the vortex line direction vector, and
$B$
and
$B^{\prime }$
are dimensionless mutual friction coefficients. The vortex tension force per unit mass is
$$\begin{eqnarray}\boldsymbol{t}={\it\nu}_{s}{\bf\omega}_{s}\times (\boldsymbol{{\rm\nabla}}\times \hat{{\bf\omega}}_{s}),\end{eqnarray}$$
where the vortex line tension parameter is given by
$$\begin{eqnarray}{\it\nu}_{s}=\frac{{\it\Gamma}}{4{\rm\pi}}\log \left(\frac{b_{0}}{a_{0}}\right).\end{eqnarray}$$
Here,
${\it\Gamma}=\hbar{\rm\pi}/m$
is the quantum of circulation,
$m$
is the mass of one helium atom,
$b_{0}$
is the inter-vortex spacing and
$a_{0}$
is the size of the vortex core. The kinematic viscosity
${\it\nu}_{n}$
is defined as the shear viscosity divided by the normal fluid density,
${\it\eta}/{\it\rho}_{n}$
. The final parameter
${\bf\sigma}$
in (2.1)–(2.4) comes from the theory of Baym & Chandler (Reference Baym and Chandler1983) and describes the restoring force of the vortex lattice in response to shear deformations. It has the form
$$\begin{eqnarray}{\bf\sigma}=\frac{\hbar|{\bf\omega}_{s}|}{8m}[2\boldsymbol{{\rm\nabla}}_{\bot }(\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }{\bf\xi})-{\rm\nabla}_{\bot }^{2}{\bf\xi}],\end{eqnarray}$$
where
${\bf\xi}$
is the vortex line displacement vector and
$\boldsymbol{{\rm\nabla}}_{\bot }$
is the two-dimensional gradient operator. Both
${\bf\xi}$
and
$\boldsymbol{{\rm\nabla}}_{\bot }$
are two-dimensional in the plane orthogonal to the angular velocity of the background superfluid flow, i.e.
$\hat{{\it\omega}}_{s}\boldsymbol{\cdot }{\bf\xi}=\hat{{\it\omega}}_{s}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{\bot }=0$
. Equation (2.9) only applies to linear deformations of a rectilinear vortex array, hence (2.1)–(2.4) are only valid in the linear approximation when
${\bf\sigma}$
is included. When
${\bf\sigma}=0$
, (2.1)–(2.4) are the HVBK equations, which describe nonlinear flows including quasi-classical turbulence (Henderson, Barenghi & Jones Reference Henderson, Barenghi and Jones1995; Peralta et al.
Reference Peralta, Melatos, Giacobello and Ooi2008).
The vortex lines obey the vorticity conservation law
$$\begin{eqnarray}\frac{\partial {\bf\omega}_{s}}{\partial t}=\boldsymbol{{\rm\nabla}}\times (\boldsymbol{v}_{L}\times {\bf\omega}_{s}),\end{eqnarray}$$
where in the linear approximation the perturbation to the vortex line velocity is given by
$\partial {\it\xi}/\partial t$
. Taking the curl of (2.2) and comparing the result with (2.10) gives the equation of motion for the vortex lines,
$$\begin{eqnarray}{\bf\omega}_{s}\times (\boldsymbol{v}_{L}-\boldsymbol{v}_{s})=\boldsymbol{t}+{\bf\sigma}+\frac{{\it\rho}_{n}}{{\it\rho}}\boldsymbol{F}.\end{eqnarray}$$
Equations (2.1) and (2.2) can be combined into an equation for the total fluid,
$$\begin{eqnarray}\displaystyle \frac{\partial }{\partial t}({\it\rho}_{n}\boldsymbol{v}_{n}+{\it\rho}_{s}\boldsymbol{v}_{s})+\boldsymbol{{\rm\nabla}}_{j}({\it\rho}_{n}v_{ni}v_{nj}+{\it\rho}_{s}v_{si}v_{sj})=\boldsymbol{{\rm\nabla}}_{j}\unicode[STIX]{x1D61B}_{ij}, & & \displaystyle\end{eqnarray}$$
where
$$\begin{eqnarray}\unicode[STIX]{x1D61B}_{ij}=-p{\it\delta}_{ij}+\unicode[STIX]{x1D61B}_{ij}^{v}+\unicode[STIX]{x1D61B}_{ij}^{s}+\unicode[STIX]{x1D61B}_{ij}^{t}.\end{eqnarray}$$
The contributions to the stress are the total pressure,
$$\begin{eqnarray}p={\it\rho}_{n}p_{n}+{\it\rho}_{s}p_{s}-{\it\rho}_{s}{\it\nu}_{s}|{\bf\omega}_{s}|,\end{eqnarray}$$
the viscous stress,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}_{ij}^{v}={\it\rho}_{n}{\it\nu}_{n}(\boldsymbol{{\rm\nabla}}_{i}v_{j}+\boldsymbol{{\rm\nabla}}_{j}v_{i}), & & \displaystyle\end{eqnarray}$$
the vortex line tension,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}_{ij}^{s}={\it\rho}_{s}{\it\nu}_{s}|{\bf\omega}_{s}|(\hat{{\it\omega}}_{si}\hat{{\it\omega}}_{sj}-{\it\delta}_{ij}), & & \displaystyle\end{eqnarray}$$
and the stress arising from the displacement of vortices from the equilibrium configuration in the lattice,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D61B}_{ij}^{t}=\frac{{\it\rho}_{s}\hbar|{\bf\omega}_{s}|}{8m}[\boldsymbol{{\rm\nabla}}_{\bot i}{\it\xi}_{j}+\boldsymbol{{\rm\nabla}}_{\bot j}{\it\xi}_{i}-3{\it\delta}_{ij}(\partial _{k}{\it\xi}_{k})]. & & \displaystyle\end{eqnarray}$$
The term (2.17) is responsible for Tkachenko oscillations.
To study the coupled response of a superfluid and its container, we consider two infinite parallel plates with separation
$2L$
. This geometry has a long history in the study of the spin-up of rapidly rotating fluids in geophysics (Greenspan & Howard Reference Greenspan and Howard1963; Pedlosky Reference Pedlosky1967; Walin Reference Walin1969), magnetized plasmas (Loper Reference Loper1971; Easson & Pethick Reference Easson and Pethick1979; van Eysden Reference van Eysden2014), condensed matter (Reisenegger Reference Reisenegger1993) and astrophysics (Abney & Epstein Reference Abney and Epstein1996). At times
$t<0$
, the superfluid and its container rotate rigidly and uniformly about the cylindrical axis with angular velocity
${\it\Omega}$
. At time
$t=0$
, the magnitude of the angular velocity of the container is impulsively increased to
${\it\Omega}(1+{\it\epsilon})$
, where
${\it\epsilon}\ll 1$
is the Rossby number. For
$t>0$
the container and fluid are left to evolve freely, and we solve self-consistently for the coupled motion of the container and plasma as in previous studies (van Eysden & Melatos Reference van Eysden and Melatos2013, Reference van Eysden and Melatos2014; van Eysden Reference van Eysden2014). The sidewalls of the container are neglected; however, they typically play a secondary role in spin-up problems like that considered here. Sidewall effects in helium II experiments are discussed in § 4.3.
To solve for the motion of the container, we invoke angular momentum conservation between the container and the fluid,
$$\begin{eqnarray}I_{c}\frac{\text{d}{\it\bf\Omega}_{c}}{\text{d}t}=-\oint \boldsymbol{x}\times (\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D64F})\,\text{d}S+{\bf\tau}_{ext},\end{eqnarray}$$
where
$I_{c}$
is the moment of inertia of the container,
$\hat{\boldsymbol{n}}$
is the unit vector normal to the boundary and
$\text{d}S$
is an element of area on the boundary. The first term on the right-hand side of (2.18) is the hydrodynamic torque exerted on the crust by the fluid arising from viscous stresses and stress exerted by the vortex array, where
$\unicode[STIX]{x1D64F}$
is given by (2.12). The second term is an external torque which may arise from, e.g., friction in the apparatus for superfluid experiments or the magnetic dipole torque in pulsars.
The normal fluid co-rotates with the container, giving the boundary condition
$$\begin{eqnarray}\displaystyle \boldsymbol{v}_{n}={\it\bf\Omega}_{c}\times \boldsymbol{x}, & & \displaystyle\end{eqnarray}$$
where
$\boldsymbol{x}$
is the radial vector and
${\it\bf\Omega}_{c}(t)$
is the angular velocity of the container, which is a function of time. Equation (2.19) embodies the usual no-slip boundary conditions for viscous flows. Following Reisenegger (Reference Reisenegger1993), for the superfluid we choose the following boundary conditions:
$$\begin{eqnarray}\hat{\boldsymbol{n}}\times [{\it\rho}_{s}|{\bf\omega}_{s}|L{\it\gamma}(\boldsymbol{v}_{L}-{\it\bf\Omega}_{c}\times \boldsymbol{x})\pm (\hat{\boldsymbol{n}}\boldsymbol{\cdot }\unicode[STIX]{x1D64F})]=0,\quad \hat{\boldsymbol{n}}\boldsymbol{\cdot }\boldsymbol{v}_{s}=0,\end{eqnarray}$$
on surfaces that intersect vortex lines, where
$\hat{\boldsymbol{n}}$
is the unit vector normal to the surface. The dimensionless constant
${\it\gamma}$
governs the rate of vortex creep at the boundary. In the limit
${\it\gamma}\rightarrow \infty$
, the vortices are pinned to the boundary, and when
${\it\gamma}=0$
, the vortices exert no stress on the boundary, i.e. they are freely sliding. For the normal component of the superfluid we require no penetration.
In cylindrical coordinates
$(r,{\it\phi},z)$
, the initial conditions are
$$\begin{eqnarray}\displaystyle \boldsymbol{v}_{n,s}(0)=r{\it\Omega}\hat{{\it\phi}},\quad {\it\bf\Omega}_{c}(0)={\it\Omega}(1+{\it\epsilon})\hat{z}, & & \displaystyle\end{eqnarray}$$
i.e. we assume that the two fluids are initially co-rotating. Strictly speaking, the initial velocity for the fluid components obeys (2.21) everywhere except in an infinitely thin region adjacent to the boundary where it is spun up by the container.
For
${\it\epsilon}\ll 1$
, (2.1)–(2.21) can be linearized by perturbing around an equilibrium rotating with uniform angular velocity
${\it\Omega}$
about the
$z$
-axis. The external torque is also taken to be aligned with the rotation axis, i.e.
${\bf\tau}_{ext}={\it\tau}_{ext}\hat{z}$
. The geometry and initial conditions are axisymmetric, and the resulting flow is axisymmetric. The following substitutions are made for the velocity and pressure fields:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{v}_{n,s}(r,z,t)\rightarrow r{\it\Omega}\hat{{\it\phi}}+{\it\epsilon}{\it\Omega}L\left(r^{\ast }\frac{\partial {\it\chi}_{n,s}}{\partial z^{\ast }}\hat{r}+r^{\ast }V_{n,s}\hat{{\it\phi}}-2{\it\chi}_{n,s}\hat{z}\right),\\ \displaystyle \boldsymbol{v}_{L}(r,z,t)\rightarrow r{\it\Omega}\hat{{\it\phi}}+{\it\epsilon}{\it\Omega}Lr^{\ast }\left(\frac{\partial U_{{\it\xi}}}{\partial t^{\ast }}\hat{r}+\frac{\partial V_{{\it\xi}}}{\partial t^{\ast }}\hat{{\it\phi}}\right),\\ \displaystyle p_{n,s}(r,z,t)\rightarrow \frac{(r{\it\Omega})^{2}}{2}+{\it\epsilon}{\it\Omega}^{2}L^{2}\left(\frac{r^{\ast 2}P_{n,s}}{2}+2Q_{n,s}\right),\end{array}\!\!\right\} & & \displaystyle\end{eqnarray}$$
where
${\it\chi}_{n,s}(z^{\ast },t^{\ast })$
,
$V_{n,s}(z^{\ast },t^{\ast })$
,
$Q_{n,s}(z^{\ast },t^{\ast })$
and
$P_{n,s}(t^{\ast })$
are all dimensionless quantities. The functions
$r{\it\chi}_{n,s}$
are streamfunctions for the secondary flow. The asterisked quantities are defined as
$r^{\ast }=r/L$
,
$z^{\ast }=z/L$
,
$t^{\ast }={\it\Omega}t$
. Equation (2.22) is essentially the ‘von Kármán similarity’ form and is typically employed in studies of spin-up between parallel plates (Greenspan & Howard Reference Greenspan and Howard1963; Easson Reference Easson1979; Reisenegger Reference Reisenegger1993; van Eysden & Melatos Reference van Eysden and Melatos2013). The ansatz (2.22) automatically satisfies the continuity equations (2.3) and (2.4) and the conditions for rotational equilibrium for the background flow. The radial dependence of the azimuthal velocity is motivated by the boundary conditions (2.19), and chosen for the other variables such that
$r$
vanishes from the resulting equations in the most general way. Under these assumptions, the only non-vanishing component of the external torque is in the
$\hat{z}$
direction, hence
$$\begin{eqnarray}\displaystyle {\it\bf\Omega}_{c}(t^{\ast })\rightarrow {\it\Omega}\hat{z}+{\it\epsilon}{\it\Omega}f(t^{\ast })\hat{z}, & & \displaystyle\end{eqnarray}$$
where
$f$
is a function of
$t^{\ast }$
only. Henceforth, we drop the asterisk notation so that all variables are dimensionless.
Substituting (2.22) into the normal fluid momentum equation (2.1), we obtain
$$\begin{eqnarray}\displaystyle 0\! & = & \displaystyle \!\left(\frac{\partial }{\partial t}-E\frac{\partial ^{2}}{\partial z^{2}}+\frac{{\it\rho}_{s}B}{{\it\rho}}\right)\frac{\partial {\it\chi}_{n}}{\partial z}-\left(2-\frac{{\it\rho}_{s}B^{\prime }}{{\it\rho}}\right)V_{n}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\rho}_{s}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}}{\partial z}-\frac{{\it\rho}_{s}B^{\prime }}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}+P_{n},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \left(\frac{\partial }{\partial t}-E\frac{\partial ^{2}}{\partial z^{2}}+\frac{{\it\rho}_{s}B}{{\it\rho}}\right)V_{n}+\left(2-\frac{{\it\rho}_{s}B^{\prime }}{{\it\rho}}\right)\frac{\partial {\it\chi}_{n}}{\partial z}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\rho}_{s}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}+\frac{{\it\rho}_{s}B^{\prime }}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}}{\partial z},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0=\left(\frac{\partial }{\partial t}-E\frac{\partial ^{2}}{\partial z^{2}}\right){\it\chi}_{n}-\frac{\partial Q_{n}}{\partial z}, & & \displaystyle\end{eqnarray}$$
and from (2.2) we have for the superfluid
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \left[\frac{\partial }{\partial t}+\frac{{\it\rho}_{n}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\right]\frac{\partial {\it\chi}_{s}}{\partial z}-\left(1-\frac{{\it\rho}_{n}B^{\prime }}{2{\it\rho}}\right)\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\rho}_{n}B}{{\it\rho}}\frac{\partial {\it\chi}_{n}}{\partial z}-\frac{{\it\rho}_{n}B^{\prime }}{{\it\rho}}V_{n}+P_{s},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \left[\frac{\partial }{\partial t}+\frac{{\it\rho}_{n}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\right]V_{s}+\left(1-\frac{{\it\rho}_{n}B^{\prime }}{2{\it\rho}}\right)\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}}{\partial z}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\rho}_{n}B}{{\it\rho}}V_{n}+\frac{{\it\rho}_{n}B^{\prime }}{{\it\rho}}\frac{\partial {\it\chi}_{n}}{\partial z},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0=\frac{\partial {\it\chi}_{s}}{\partial t}-\frac{\partial Q_{s}}{\partial z}, & & \displaystyle\end{eqnarray}$$
where we define the dimensionless parameters
$$\begin{eqnarray}\displaystyle E=\frac{{\it\nu}_{n}}{L^{2}{\it\Omega}},\quad E_{s}=\frac{{\it\nu}_{s}}{L^{2}{\it\Omega}}. & & \displaystyle\end{eqnarray}$$
The dimensionless parameter
$E$
is the Ekman number and is a ratio of viscous forces and the rotational inertia in the flow. By analogy, we define the superfluid Ekman number
$E_{s}$
, which is a ratio of the vortex line tension and rotational inertia in the flow. Equation (2.18) becomes
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}f}{\text{d}t}=\mp \frac{K}{{\it\rho}}\left({\it\rho}_{s}E_{s}\frac{\partial ^{2}{\it\chi}_{s}}{\partial z^{2}}+{\it\rho}_{n}E\frac{\partial V_{n}}{\partial z}\right)+{\it\alpha}(1+K), & \displaystyle\end{eqnarray}$$
at
$z=\pm 1$
, where we define
$$\begin{eqnarray}K=\frac{{\rm\pi}{\it\rho}R^{4}L}{I_{c}},\quad {\it\alpha}=\frac{{\it\tau}_{ext}}{{\it\epsilon}{\it\Omega}^{2}I_{tot}}.\end{eqnarray}$$
In (2.31), the dimensionless parameter
$K$
denotes the ratio of the moments of inertia of the fluid and container, and
${\it\alpha}$
is the dimensionless external torque, where
$I_{tot}=I_{c}(1+K)$
is the total moment of inertia of the fluid and container.
The boundary conditions for the normal fluid (2.19) become
$$\begin{eqnarray}\displaystyle & \displaystyle V_{n}-rf=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\chi}_{n}}{\partial z}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{n}=0, & \displaystyle\end{eqnarray}$$
$z=\pm 1$
. For the superfluid, using the linearized forms of (2.11) and (2.2) to eliminate the vortex line velocity and mutual friction, (2.20a,b
) becomes 
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial V_{s}}{\partial t}\pm \frac{E_{s}}{{\it\gamma}}\frac{\partial V_{s}}{\partial z}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial }{\partial t}\frac{\partial {\it\chi}_{s}}{\partial z}-2f\pm \frac{E_{s}}{{\it\gamma}}\frac{\partial ^{2}{\it\chi}_{s}}{\partial z^{2}}+P_{s}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{s}=0, & \displaystyle\end{eqnarray}$$
$z=\pm 1$
. The initial conditions (2.21) become 
$$\begin{eqnarray}\displaystyle & \displaystyle V_{n}(z,0)=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle V_{s}(z,0)=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle f(0)=1. & \displaystyle\end{eqnarray}$$
${\bf\xi}$
, can be calculated by substituting (2.22) into the vortex line equation of motion (2.11) and using (2.2). We find 
$$\begin{eqnarray}\displaystyle & \displaystyle 2\frac{\partial V_{{\it\xi}}}{\partial t}=\frac{\partial }{\partial t}\frac{\partial {\it\chi}_{s}}{\partial z}+P_{s}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle 2\frac{\partial U_{{\it\xi}}}{\partial t}=-\frac{\partial V_{s}}{\partial t}. & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle U_{{\it\xi}}(z,0)=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle V_{{\it\xi}}(z,0)=0. & \displaystyle\end{eqnarray}$$
3. Pure superfluid
3.1. Exact solution
To illustrate the Ekman pumping mechanism, which is the principal result of this paper, we consider a superfluid at zero temperature, where
${\it\rho}_{n}=0$
. To solve the initial value problem, we take the Laplace transform,
$$\begin{eqnarray}\tilde{X}(z,s)=\int _{0}^{\infty }X(z,t)\text{e}^{-st}\,\text{d}t.\end{eqnarray}$$
Equations (2.27) and (2.28) are
$$\begin{eqnarray}\displaystyle & \displaystyle s\frac{\partial \tilde{{\it\chi}}_{s}}{\partial z}-\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\tilde{V}_{s}+\tilde{P}_{s}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle s\tilde{V}_{s}+\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial \tilde{{\it\chi}}_{s}}{\partial z}=0. & \displaystyle\end{eqnarray}$$
$Q_{s}$
for completeness. Taking the derivative of (3.2) we obtain 
$$\begin{eqnarray}\displaystyle s\frac{\partial ^{2}\tilde{{\it\chi}}_{s}}{\partial z^{2}}-\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial \tilde{V}_{s}}{\partial z}=0. & & \displaystyle\end{eqnarray}$$
Equations (3.3) and (3.4) can be solved for
$\tilde{{\it\chi}}_{s}$
and
$\tilde{V}_{s}$
;
$\tilde{P}_{s}$
is then obtained from (3.2). The equation for the container, (2.31), becomes
$$\begin{eqnarray}\displaystyle s\tilde{f}-1=\mp KE_{s}\left(\frac{\partial ^{2}\tilde{{\it\chi}}_{s}}{\partial z^{2}}\right), & & \displaystyle\end{eqnarray}$$
at
$z=\pm 1$
, and the boundary conditions are
$$\begin{eqnarray}\displaystyle & \displaystyle s\tilde{V}_{s}\pm \frac{E_{s}}{{\it\gamma}}\frac{\partial \tilde{V}_{s}}{\partial z}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle s\frac{\partial \tilde{{\it\chi}}_{s}}{\partial z}-2\tilde{f}\pm \frac{E_{s}}{{\it\gamma}}\frac{\partial ^{2}\tilde{{\it\chi}}_{s}}{\partial z^{2}}+\tilde{P}_{s}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \tilde{{\it\chi}}_{s}=0, & \displaystyle\end{eqnarray}$$
$z=\pm 1$
. The solution to (3.3)–(3.8) is 
$$\begin{eqnarray}\displaystyle \tilde{V}_{s} & = & \displaystyle \frac{2\text{i}\tilde{f}}{{\it\Delta}s^{2}}\left.\{\!C_{+}[{\it\kappa}_{-}s(\cosh {\it\kappa}_{-}-\cosh {\it\kappa}_{-}z)+(E_{s}/{\it\gamma}){\it\kappa}_{-}^{2}\sinh {\it\kappa}_{-}],\right.\nonumber\\ \displaystyle & & \displaystyle -\,\left.C_{-}[{\it\kappa}_{+}s(\cosh {\it\kappa}_{+}-\cosh {\it\kappa}_{+}z)+(E_{s}/{\it\gamma}){\it\kappa}_{+}^{2}\sinh {\it\kappa}_{+}]\!\},\right.\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \tilde{{\it\chi}}_{s}=\frac{2\tilde{f}}{{\it\Delta}s}[C_{+}(\sinh {\it\kappa}_{-}z-z\sinh {\it\kappa}_{-})+C_{-}(\sinh {\it\kappa}_{+}z-z\sinh {\it\kappa}_{+})], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \tilde{P}_{s}=\frac{2\tilde{f}}{{\it\Delta}s^{2}}(4+s^{2})(C_{+}\sinh {\it\kappa}_{-}+C_{-}\sinh {\it\kappa}_{+}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \tilde{f}=\frac{{\it\Delta}}{\bar{{\it\Delta}}s}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\kappa}_{\pm }=\sqrt{\frac{2\pm \text{i}s}{E_{s}}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle C_{\pm }={\it\kappa}_{\pm }\cosh {\it\kappa}_{\pm }+\frac{(E_{s}{\it\kappa}_{\pm }^{2}\mp 2\text{i}{\it\gamma})\sinh {\it\kappa}_{\pm }}{{\it\gamma}s}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\Delta}=C_{+}\left[C_{-}+\frac{2E_{s}{\it\kappa}_{-}^{2}\sinh {\it\kappa}_{-}}{s^{2}}\right]+C_{-}\left[C_{+}+\frac{2E_{s}{\it\kappa}_{+}^{2}\sinh {\it\kappa}_{+}}{s^{2}}\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle \bar{{\it\Delta}} & = & \displaystyle C_{+}\left[C_{-}+\frac{2E_{s}{\it\kappa}_{-}^{2}(1+K)\sinh {\it\kappa}_{-}}{s^{2}}\right]+C_{-}\left[C_{+}+\frac{2E_{s}{\it\kappa}_{+}^{2}(1+K)\sinh {\it\kappa}_{+}}{s^{2}}\right].\qquad\end{eqnarray}$$
To obtain the response of the container, we must find the inverse Laplace transform of
$\tilde{f}$
. This is readily done by realizing that there are simple poles at
$s=0$
and at the zeroes of
$\bar{{\it\Delta}}$
. The result is
$$\begin{eqnarray}f(t)=\left[\frac{{\it\Delta}}{\bar{{\it\Delta}}}\right]_{s=0}+\mathop{\sum }_{n}R(s_{n})\text{e}^{s_{n}t},\end{eqnarray}$$
where
$s_{n}$
are the roots of
$\bar{{\it\Delta}}$
and
$$\begin{eqnarray}R(s)=\frac{{\it\Delta}(s)}{s\displaystyle \frac{\text{d}\bar{{\it\Delta}}}{\text{d}s}}.\end{eqnarray}$$
The sum in (3.17) implies summing over all the zeroes of
$\bar{{\it\Delta}}$
. The solution to the initial value problem could also be obtained from a mode expansion and appropriate orthogonality relations; however, the Laplace transform more readily gives the result. In general, the eigenvalues occur in conjugate pairs. For
${\it\gamma}\rightarrow \infty$
we have
$\text{Re}(s_{n})=0$
and
$\text{Im}[R(s_{n})]=0$
, and the modes in (3.17) are purely oscillatory. When
${\it\gamma}$
is finite the friction force results in dissipation and
$s_{n}$
and
$R(s_{n})$
have real and imaginary components.
Before proceeding, let us consider the relevant range for
$E_{s}$
in the hydrodynamic approximation of the HVBK equations. The vorticity of the superfluid is given by the circulation per vortex multiplied by the number of vortices per unit area, hence
$$\begin{eqnarray}2{\it\Omega}={\it\Gamma}\times \frac{N}{{\rm\pi}R^{2}}.\end{eqnarray}$$
From (2.8), (2.30) and (3.19) we then obtain
$$\begin{eqnarray}E_{s}=\frac{R^{2}}{2L^{2}N}\ln \left(\frac{b_{0}}{a_{0}}\right).\end{eqnarray}$$
The
$\ln$
term is of order unity or an order of magnitude more, so that for a vessel of aspect ratio close to unity we have
$E_{s}\sim 1/N$
. We require
$N\gg 1$
for the hydrodynamic approximation to be valid, hence we must have
$E_{s}\ll 1$
.
Dimensionless angular velocity of the container,
$f(t)$
, for
$E_{s}=0.1$
(blue curve) and
$E_{s}=0.01$
(purple curve). We take
$K=1$
and
${\it\gamma}\rightarrow \infty$
.
Eigenfrequencies (a) and amplitudes (b) of the modes present in figure 1. Blue circles correspond to
$E_{s}=0.1$
and purple squares correspond to
$E_{s}=0.01$
.
In figure 1 we plot the response of the container for
$E_{s}=0.1$
(blue curve) and
$E_{s}=0.01$
(purple curve) for
${\it\gamma}\rightarrow \infty$
(perfect pinning) and
$K=1$
in both cases. The first 500 eigenmodes of (3.17) have been plotted. We find that the container oscillates about the centre of mass of the system in a smooth sinusoidal manner, with a period much greater than the rotation period. This smooth oscillation can be understood by looking at the modal decomposition of the solution, plotted in figure 2. On the left-hand side we present the eigenvalues, in increasing order of
$\text{Im}(s_{n})$
. On the right-hand side we plot the corresponding amplitudes of each mode. We see from figure 2(b) that the dominant contribution is from the fundamental mode, with very little contribution from the higher modes. As
$E_{s}$
is reduced, we find that all the power becomes concentrated in the fundamental mode. Therefore, for
$E_{s}\ll 1$
, the impulsive acceleration of the container only excites the fundamental mode of the system.
To obtain an analytic result for the fundamental mode, we investigate the limit
$E_{s}\ll 1$
. The frequency of oscillation is much smaller than the rotation frequency, hence we look for solutions in the limit
$s\ll 1$
. We then find
$$\begin{eqnarray}\bar{{\it\Delta}}\approx \frac{\text{e}^{2k}}{2s^{3}{\it\gamma}^{2}}(2+k{\it\gamma}s)[4(1+K){\it\gamma}+2s+k{\it\gamma}s^{2}],\end{eqnarray}$$
where
$k=\sqrt{2/E_{s}}\gg 1$
. Equation (3.21) has zeroes at
$$\begin{eqnarray}s_{\pm }=-\frac{1}{k{\it\gamma}}\left[1\pm \sqrt{1-4(1+K)k{\it\gamma}^{2}}\right].\end{eqnarray}$$
The zero at
$s=-2/(k{\it\gamma})$
is also a zero of
${\it\Delta}$
, and therefore not a pole of
$\tilde{f}$
. Evaluating
$R(s_{\pm })$
we find
$$\begin{eqnarray}f(t)=\frac{1}{1+K}\left[1+\left(\frac{K}{s_{+}-s_{-}}\right)(s_{+}\text{e}^{s_{-}t}-s_{-}\text{e}^{s_{+}t})\right].\end{eqnarray}$$
We now examine the behaviour of (3.23) by considering two limiting cases. To recover the oscillatory solution observed in figure 1, we consider the strong pinning limit
${\it\gamma}\gg [4(1+K)k]^{-1/2}$
. From (3.22) we have
$$\begin{eqnarray}s_{\pm }=\pm 2\text{i}\sqrt{\frac{1+K}{k}}\end{eqnarray}$$
and (3.23) becomes
$$\begin{eqnarray}f(t)=\frac{1}{1+K}\left[1+K\cos \left(2\sqrt{\frac{1+K}{k}}t\right)\right].\end{eqnarray}$$
We therefore obtain the period of oscillation
$$\begin{eqnarray}t_{P}=\frac{2{\rm\pi}}{{\it\Omega}|\text{Im}(s_{\pm })|}=\frac{{\rm\pi}}{{\it\Omega}}\sqrt{\frac{k}{1+K}}.\end{eqnarray}$$
This is the principal result of this paper. A single long-time-scale mode has emerged for
$E_{s}\ll 1$
, which has a period proportional to
$E_{s}^{-1/4}$
. This is analogous to classical Ekman pumping, where the Ekman time (proportional to
$E^{-1/2}$
) emerges for
$E\ll 1$
. However, whereas a viscous fluid is dissipative and results in relaxation, a superfluid is dissipationless and oscillates. For
$E_{s}=0.01$
we find
${\it\Omega}t_{P}=8.35$
, in agreement with figure 1. The result (3.36) agrees exactly with the inertial-wave resonance identified by Sonin (Reference Sonin1987); see (6.21) in that review.
In the weak pinning limit,
${\it\gamma}\ll [4(1+K)k]^{-1/2}$
, (3.22) gives
$$\begin{eqnarray}s_{+}=-\frac{2}{k{\it\gamma}},\quad s_{-}=-2(1+K){\it\gamma}.\end{eqnarray}$$
Because
$s_{-}\ll s_{+}$
in this limit, (3.23) becomes
$$\begin{eqnarray}f=\frac{1}{1+K}(1+K\text{e}^{s_{-}t}).\end{eqnarray}$$
This exponential decay is a result of the friction force of the vortex lines as they slide on the boundaries. When
$K=0$
the time scale for the relaxation is
$$\begin{eqnarray}t_{R}=\frac{1}{{\it\Omega}|s_{-}|}=\frac{1}{2{\it\Omega}{\it\gamma}},\end{eqnarray}$$
which is the result obtained in Reisenegger (Reference Reisenegger1993). Because the weak pinning limit was studied by Reisenegger (Reference Reisenegger1993), the principal focus of this paper will be on our new result in the perfect pinning limit.
3.2. Ekman pumping
To visualize the flow for
$E_{s}\ll 1$
we can invert the other variables in this limit. We obtain
$$\begin{eqnarray}\displaystyle & \displaystyle V_{s}=V_{s}^{I}+V_{s}^{B}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{s}={\it\chi}_{s}^{I}+{\it\chi}_{s}^{B}, & \displaystyle\end{eqnarray}$$
$I$
and
$B$
respectively), given by 
$$\begin{eqnarray}\displaystyle & \displaystyle V_{s}^{I}=\frac{1}{1+K}\left[1-\left(\frac{s_{+}\text{e}^{s_{-}t}-s_{-}\text{e}^{s_{+}t}}{s_{+}-s_{-}}\right)\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle V_{s}^{B}=\frac{1}{1+K}\left(\frac{s_{+}\text{e}^{s_{+}t}-s_{-}\text{e}^{s_{-}t}}{s_{+}-s_{-}}\right)[\text{e}^{k(z-1)}-\text{e}^{-k(z+1)}], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{s}^{I}=\frac{2z(\text{e}^{s_{-}t}-\text{e}^{s_{+}t})}{k(s_{+}-s_{-})}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{s}^{B}=-\frac{2(\text{e}^{s_{-}t}-\text{e}^{s_{+}t})}{k(s_{+}-s_{-})}[\text{e}^{k(z-1)}-\text{e}^{-k(z+1)}]. & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle P_{s}=\frac{2}{1+K}\left[1-\left(\frac{s_{+}\text{e}^{s_{-}t}-s_{-}\text{e}^{s_{+}t}}{s_{+}-s_{-}}\right)\right]. & & \displaystyle\end{eqnarray}$$
Equations (3.32)–(3.36) are akin to Ekman pumping in a viscous fluid (Greenspan & Howard Reference Greenspan and Howard1963; Benton & Clark Reference Benton and Clark1974; van Eysden & Melatos Reference van Eysden and Melatos2013), where the exponentially decaying term has been replaced with the
$\text{e}^{s_{\pm }t}$
terms. The variable
${\it\chi}_{s}$
is like a streamfunction for a secondary flow (see (2.22)), which draws fluid radially inwards in the interior and delivers it to the boundary layer where it is pumped radially outwards. However, when there is perfect pinning, the flow is oscillatory, with period given by (3.26). The Ekman circulation initially proceeds in the classical manner, clockwise in the region
$0<z<1$
and counterclockwise in the region
$-1<z<0$
, but is periodically reversed. This reversal is induced by vortex array. In the boundary layer, the vortex array is sheared, storing potential energy in the form of vortex line tension. The potential energy increases until the Ekman pumping is eventually slowed, halted and then reversed. As there is no dissipation in the system, this energy exchange continues indefinitely. Because the flow obeys the Taylor–Proudman theorem to leading order, the interior azimuthal flow is columnar, as in classical Ekman pumping. The vortices in the interior remain straight and move inwards and outwards with the secondary radial flow, increasing and decreasing the angular momentum of the fluid.
Dimensionless angular velocity interior fluid,
$f(t)$
(blue curve), angular velocity of the interior fluid,
$V_{s}^{I}(t)$
(purple curve), radial velocity of the interior fluid (divided by
$r$
),
$\partial {\it\chi}_{s}^{I}/\partial z$
(red curve), and interior radial vortex line displacement,
$U_{{\it\xi}}$
(brown curve). We take
$E_{s}=0.01$
,
$K=1$
and
${\it\gamma}\rightarrow \infty$
. Angular momentum is exchanged between the fluid and the container (blue and purple curves). The radial flow (red curve) is
${\rm\pi}/2$
radians out of phase with the radial vortex line displacement (brown curve).
In figure 3 we plot the dimensionless angular velocity of the container,
$f(t)$
(blue curve), the angular velocity of the interior fluid,
$V_{s}^{I}(t)$
(purple curve), the radial velocity of the interior fluid (divided by
$r$
),
$\partial {\it\chi}_{s}^{I}/\partial z$
(red curve), and the interior radial vortex line displacement,
$U_{{\it\xi}}$
(brown curve) for
$E_{s}=0.01$
,
$K=1$
and
${\it\gamma}\rightarrow \infty$
. The top two curves (blue and purple) show that the angular velocities of the container and the interior fluid are sinusoidal and
${\rm\pi}$
radians out of phase. Therefore, angular momentum is conserved between the fluid and the container, as required. The bottom curves show that the radial component of the secondary flow and the radial vortex line displacement are
${\rm\pi}/2$
radians out of phase. As the radial flow shears the vortices in the boundary layer, the restoring force from the line tension increases until the radial flow is halted, and then reversed. At this turning point in the radial flow the vortex displacement is at a maximum. The vortices are only displaced radially inwards, because the azimuthal flow velocity is always greater than or equal to its initial value.
Flow plots for
$E_{s}=0.01$
: (a)
$t=2$
(near secondary flow maximum), (b)
$t=4$
(near secondary flow minimum). The colour shading represents the magnitude of the azimuthal angular velocity of the interior fluid,
$V_{s}^{I}(t)$
, where blue is zero and white is unity (cf. figure 3). The dashed contours show streamlines for the secondary flow
${\it\chi}_{s}^{I}(t)$
with values
$\pm$
0.002, 0.01, 0.025, 0.05, 0.1. The solid lines show the radial vortex line displacement
$U_{{\it\xi}}^{I}$
, plotted with exaggerated amplitude for visibility.
Snapshots of the flow described by figure 3 are presented in figure 4 at the times
$t=2$
(a) and
$t=4$
(b). The magnitude of the fluid angular velocity
$V_{s}$
is shown in colour, where dark is zero and light is unity. The radial vortex line displacement
$U_{{\it\xi}}$
is overplotted as thick black lines, scaled appropriately for illustrative purposes. Contours for the streamfunction of the secondary flow,
${\it\chi}_{s}$
, are plotted as dashed lines for values of
$\pm$
0.002, 0.01, 0.025, 0.05, 0.1. In figure 4(a) (
$t=2$
), the secondary flow is near a maximum, represented by the large density of contours. The flow is clockwise in the upper-right quadrant and the secondary flow is moving the vortex lines inwards in the interior. The angular velocities of the container and the interior flow are matched, so there is no boundary layer correction. In figure 4(b) (
$t=4$
), the secondary flow is near a minimum. The radial component of the secondary flow is turning from inwards to outwards in the interior. The vortex lines are at their maximum extension, and will subsequently straighten to drive the outward radial flow in the interior. The difference between the angular velocities of the container and the fluid is greatest, and the boundary layer in the azimuthal velocity is prominent.
To make a final connection to Ekman pumping, we examine the governing equations in the limit
$E_{s}\ll 1$
. In the interior, we find from (3.32) that the azimuthal velocity evolves on the time scale
$E_{s}^{-1/4}{\it\Omega}^{-1}$
, and from (3.34) the streamfunction for the secondary flow,
${\it\chi}_{s}$
, scales as
$E_{s}^{1/4}$
. Applying these scalings, we find in exactly the same manner as for classical Ekman pumping (see Greenspan & Howard Reference Greenspan and Howard1963, § 5) that the solutions for the interior flow satisfy
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle 2V_{s}^{I}-P_{s},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle \frac{\partial V_{s}^{I}}{\partial t}+2\frac{\partial {\it\chi}_{s}^{I}}{\partial z}.\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle \left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}^{B},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle \left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}^{B}}{\partial z}.\end{eqnarray}$$
$$\begin{eqnarray}\frac{\partial }{\partial t}\frac{\partial {\it\chi}_{s}^{B}}{\partial z}-2f\pm \frac{E_{s}}{{\it\gamma}}\frac{\partial ^{2}{\it\chi}_{s}^{B}}{\partial z^{2}}+P_{s}=0,\end{eqnarray}$$
at
$z=\pm 1$
. Finally, the initial conditions are
$$\begin{eqnarray}\displaystyle & \displaystyle V_{s}^{I}(0)=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle f(0)=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{s}^{I}(0)=0. & \displaystyle\end{eqnarray}$$
The mechanism described above possesses many properties similar to Ekman pumping. In particular, boundary layers form on boundaries orthogonal to the rotation axis, driving a radial outflow. This flow is replenished by fluid from the interior, generating a secondary circulation. However, there are important differences. Classical Ekman pumping is dissipative, while the mechanism described here is dissipationless. Moreover, in classical Ekman pumping, the Coriolis force couples the azimuthal and radial velocities in the boundary layer, driving the secondary flow. Here, the velocity components in the boundary layer are uncoupled (see (3.39) and (3.40)), but it is the boundary condition itself (2.19) that couples the velocity components.
Finally, (3.37)–(3.41) can be solved to obtain a boundary condition for the interior flow analogous to that of Greenspan (Reference Greenspan1968) (see (2.6.13) in that paper), which relates the differential vorticity at the boundary with the secondary inflow into the boundary layer,
$$\begin{eqnarray}\left(\frac{1}{\sqrt{2{\it\Omega}{\it\nu}_{s}}}\frac{\partial }{\partial t}\mp \frac{1}{L{\it\gamma}}\right)\hat{k}\boldsymbol{\cdot }\boldsymbol{v}^{I}=\hat{k}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\times (\boldsymbol{v}^{I}-\boldsymbol{v}_{B}),\end{eqnarray}$$
at
$z=\pm 1$
, where
$\boldsymbol{v}^{I}$
denotes the velocity field in the interior. This relation applies at the wall in cylindrical geometry and applies only to a single fluid.
4. Helium II
4.1. Experiments
The response of a superfluid-filled container following an impulsive acceleration has been investigated in a series of experiments conducted by Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980). In § 5 of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980), plexiglass cylinders are coated with a powder to increase vortex adhesion, and oscillatory motion of the container is reported. The experiment is conducted at
$1.52~\text{K}$
with a rotational frequency of
$3~\text{rad}~\text{s}^{-1}$
. The vessel is
$64~\text{mm}$
in diameter,
$50~\text{mm}$
in height and
$0.2~\text{mm}$
thick. The response of the vessel is given in figure 27(c) of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980). The magnitude of the angular velocity after the spin-up is not given, but appears to be approximately
$3.6~\text{rad}~\text{s}^{-1}$
. The response comprises an exponential decay of
$0.3~\text{rad}~\text{s}^{-1}$
over approximately
$40~\text{s}$
, followed by steady oscillation with a period of
$40~\text{s}$
, decaying over a time scale of roughly
$480~\text{s}$
. The oscillation period also appears to decrease with time, probably as a result of nonlinear effects as the experiment has a Rossby number of
${\it\epsilon}\sim (3.6-3)/3=0.2$
.
Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980) also report the following effects. The oscillation period is unaffected by the radius of the cylinder. This was tested by inserting coaxial cylinders of both 22 and
$43~\text{mm}$
inside the main cylinder. However, the frequency was observed to double when the container height was halved, which was achieved by inserting disks into the cylinder. This behaviour is qualitatively consistent with the columnar flow of Ekman pumping and the solution presented in § 3.
To apply our theory to the Tsakadze experiments, we need to generalize the solution in § 3 to the two-fluid system in § 2. For the experiment described above, the dimensionless parameters at
$1.52~\text{K}$
are (Donnelly & Barenghi Reference Donnelly and Barenghi1998)
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle E=4.07\times 10^{-5}\left(\frac{{\it\nu}_{n}}{7.64\times 10^{-4}~\text{cm}^{2}~\text{s}^{-1}}\right)\left(\frac{{\it\Omega}}{3~\text{rad}~\text{s}^{-1}}\right)^{-1}\left(\frac{L}{2.5~\text{cm}}\right)^{-2},\\ \displaystyle E_{s}=5.95\times 10^{-5}\left(\frac{{\it\nu}_{s}}{1.12\times 10^{-3}~\text{cm}^{2}~\text{s}^{-1}}\right)\left(\frac{{\it\Omega}}{3~\text{rad}~\text{s}^{-1}}\right)^{-1}\left(\frac{L}{2.5~\text{cm}}\right)^{-2},\\ \displaystyle Ma=3.2\times 10^{-4}\left(\frac{u_{1}}{2.35\times 10^{4}~\text{cm}~\text{s}^{-1}}\right)^{-1}\left(\frac{{\it\Omega}}{3~\text{rad}~\text{s}^{-1}}\right)\left(\frac{L}{2.5~\text{cm}}\right),\\ \displaystyle B=1.2748,\\ \displaystyle B^{\prime }=0.2922,\\ \displaystyle {\it\rho}_{n}=0.1206{\it\rho}.\end{array}\!\!\right\} & & \displaystyle\end{eqnarray}$$
The third parameter,
$Ma$
, where
$u_{1}$
is the first sound velocity, is the Mach number and indicates the relative importance of compressibility effects. Clearly, for helium II the incompressibility approximation is valid. For the experiment described above, all the parameters of our theory are given by (4.1) except
$K$
. In experiments involving spheres and no pinning, Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980) present the steady-state spin-down of empty and filled vessels, from which one can determine
$K\sim 0.8$
(van Eysden & Melatos Reference van Eysden and Melatos2011). Unfortunately, Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980) do not provide enough data to determine
$K$
for this experiment, and we take
$K=1$
.
4.2. Two-fluid solution
Solution of the full system of equations in § 2 poses a formidable challenge. In helium II, we have
$E\ll 1$
and
$E_{s}\ll 1$
but we still have
$E_{s}\sim E$
, so that neither viscosity nor vortex tension is negligible. Moreover, because
$B,B^{\prime }\sim 1$
, the mutual friction coupling time between the two fluid components is of the order of the rotation period, and the normal fluid and superfluid are approximately locked together over the Ekman time (Reisenegger Reference Reisenegger1993; van Eysden & Melatos Reference van Eysden and Melatos2013, Reference van Eysden and Melatos2014).
To solve the two-fluid system we use a boundary layer approximation like that used in § 5 of Greenspan & Howard (Reference Greenspan and Howard1963) and presented in § 4.3. We assume that the azimuthal velocity is symmetric about the
$z=0$
plane, and hence
${\it\chi}_{n,s}$
are antisymmetric. In the interior we have
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle 2V_{n}^{I}-\frac{{\it\rho}_{s}B^{\prime }}{{\it\rho}}(V_{n}^{I}-V_{s}^{I})-P_{n},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle \frac{\partial V_{n}^{I}}{\partial t}+2\frac{\partial {\it\chi}_{n}^{I}}{\partial z}+\frac{{\it\rho}_{s}B}{{\it\rho}}(V_{n}^{I}-V_{s}^{I})-\frac{{\it\rho}_{s}B^{\prime }}{{\it\rho}}\left(\frac{\partial {\it\chi}_{n}^{I}}{\partial z}-\frac{\partial {\it\chi}_{s}^{I}}{\partial z}\right),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle 2V_{s}^{I}+\frac{{\it\rho}_{n}B^{\prime }}{{\it\rho}}(V_{n}^{I}-V_{s}^{I})-P_{s},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \displaystyle \frac{\partial V_{s}^{I}}{\partial t}+2\frac{\partial {\it\chi}_{s}^{I}}{\partial z}-\frac{{\it\rho}_{n}B}{{\it\rho}}(V_{n}^{I}-V_{s}^{I})+\frac{{\it\rho}_{n}B^{\prime }}{{\it\rho}}\left(\frac{\partial {\it\chi}_{n}^{I}}{\partial z}-\frac{\partial {\it\chi}_{s}^{I}}{\partial z}\right),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \left(-E\frac{\partial ^{2}}{\partial z^{2}}+\frac{{\it\rho}_{s}B}{{\it\rho}}\right)\frac{\partial {\it\chi}_{n}^{B}}{\partial z}-\left(2-\frac{{\it\rho}_{s}B^{\prime }}{{\it\rho}}\right)V_{n}^{B}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\rho}_{s}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}^{B}}{\partial z}-\frac{{\it\rho}_{s}B^{\prime }}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}^{B},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \left(-E\frac{\partial ^{2}}{\partial z^{2}}+\frac{{\it\rho}_{s}B}{{\it\rho}}\right)V_{n}^{B}+\left(2-\frac{{\it\rho}_{s}B^{\prime }}{{\it\rho}}\right)\frac{\partial {\it\chi}_{n}^{B}}{\partial z}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\rho}_{s}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}^{B}+\frac{{\it\rho}_{s}B^{\prime }}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}^{B}}{\partial z},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \frac{{\it\rho}_{n}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}^{B}}{\partial z}-\frac{{\it\rho}_{n}B}{{\it\rho}}\frac{\partial {\it\chi}_{n}^{B}}{\partial z}\nonumber\\ \displaystyle & & \displaystyle -\,\left(1-\frac{{\it\rho}_{n}B^{\prime }}{2{\it\rho}}\right)\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}^{B}-\frac{{\it\rho}_{n}B^{\prime }}{{\it\rho}}V_{n}^{B},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \frac{{\it\rho}_{n}B}{2{\it\rho}}\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)V_{s}^{B}-\frac{{\it\rho}_{n}B}{{\it\rho}}V_{n}^{B}\nonumber\\ \displaystyle & & \displaystyle +\,\left(1-\frac{{\it\rho}_{n}B^{\prime }}{2{\it\rho}}\right)\left(2-E_{s}\frac{\partial ^{2}}{\partial z^{2}}\right)\frac{\partial {\it\chi}_{s}^{B}}{\partial z}+\frac{{\it\rho}_{n}B^{\prime }}{{\it\rho}}\frac{\partial {\it\chi}_{n}^{B}}{\partial z}.\end{eqnarray}$$
These equations reduce to (5.3)–(5.10) in Greenspan & Howard (Reference Greenspan and Howard1963) and (3.37)–(3.40) in the present paper when
$B=B^{\prime }=0$
.
The governing equation for the motion of the container is (2.31). The boundary conditions for the superfluid are (2.36), (3.41) and (2.38). For the normal fluid, the required boundary conditions are (Greenspan & Howard Reference Greenspan and Howard1963, § 5)
$$\begin{eqnarray}\displaystyle & \displaystyle V_{n}-f=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\chi}_{n}^{B}}{\partial z}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{n}=0. & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}V_{n}^{I}(0)=0\end{eqnarray}$$
for the normal fluid. The general solution is presented in appendix A. Because of its complexity, we do not present the final result in algebraic form. For the numbers of (4.1) and taking
$K=1$
,
${\it\gamma}\rightarrow \infty$
and
${\it\Omega}=3~\text{rad}~\text{s}^{-1}$
, we obtain
$$\begin{eqnarray}\displaystyle f(t) & = & \displaystyle 0.500000+0.000023\text{e}^{-3.82908t}\nonumber\\ \displaystyle & & \displaystyle +\,\text{e}^{-0.004182t}[0.499977\cos (0.58827t)+0.001126\sin (0.58827t)],\end{eqnarray}$$
where
$t$
is in seconds. The first term is the steady-state term, which corresponds to the centre of mass of the system at
$1/(1+K)$
. The second term corresponds to a rapid damping and has low amplitude. This term represents the strong mutual friction between the two components. The third term is an oscillation, weakly damped by viscosity. It has a period of
$10.7~\text{s}$
and a damping time of
$239.1~\text{s}$
.
The result (4.14) suggests that viscosity and mutual friction have little effect on the final result. This is because the Ekman time and secondary flow scale as
$E^{1/2}$
in classical Ekman pumping, but scale as
$E_{s}^{1/4}$
for the oscillatory superfluid Ekman pumping mechanism identified in this paper. For the numbers in (4.1), we find
$$\begin{eqnarray}E_{s}^{1/4}=0.088,\quad E^{1/2}=0.006.\end{eqnarray}$$
Therefore, the superfluid mechanism is dominant. Neglecting viscosity (
$E=0$
) and assuming strong mutual friction coupling (
$B,B^{\prime }\sim 1$
), an approximate solution can be obtained. The details are presented in appendix B. The result has the form (3.23), with
$s_{\pm }$
now given by
$$\begin{eqnarray}s_{\pm }=-\frac{1}{k{\it\gamma}}\left[1\pm \sqrt{1-\frac{4(1+K)k{\it\gamma}^{2}{\it\rho}_{s}}{{\it\rho}}}\right].\end{eqnarray}$$
The oscillatory Ekman pumping mechanism identified in § 3 is operating in the superfluid, with the normal fluid locked to it as a result of the strong mutual friction coupling. This mass loading of the superfluid results in a factor of
${\it\rho}_{s}/{\it\rho}$
in the spin-up time. Only the leading-order normal fluid velocity is involved in the oscillation; there is no secondary flow in the normal fluid. The reader is referred to § B.2 for the solution for all variables in this limit.
Comparison of the general solution ((4.14), thick black curve) and the approximate solution ((3.23) and (B 12), thin grey curve), assuming zero viscosity (
$E=0$
) and strong mutual friction coupling (
$B,B^{\prime }\sim 1$
). Viscosity has a negligible effect on the period, which is the same for both solutions. The experimentally observed period decays from an initial maximum of
$40~\text{s}$
.
A comparison between the general solution (4.14) and the approximate solution ((3.23) and (B 12)) assuming
$E=0$
and
$B,B^{\prime }\sim 1$
is shown in figure 5. The period of the approximate solution is identical to the general result, suggesting that it depends weakly on viscosity. The approximate solution is dissipationless, whereas the general solution is damped on a time scale of a few hundred seconds. When
${\it\gamma}\rightarrow \infty$
, the period of oscillation at
$T=1.52~\text{K}$
is
$$\begin{eqnarray}t_{P}=10.7~\text{s}\left(\frac{3~\text{rad}~\text{s}^{-1}}{{\it\Omega}}\right)^{-1}\left(\frac{5.95\times 10^{-5}}{E_{s}}\right)^{-1/4}\left(\frac{0.8794}{{\it\rho}_{s}/{\it\rho}}\right)^{-1/2}\left(\frac{2}{1+K}\right)^{-1/2}.\end{eqnarray}$$
This is shorter than the maximum observed period of oscillation in the Tzakadze experiments, which decays from an initial maximum of approximately
$40~\text{s}$
. The period can be lengthened by decreasing the frictional pinning force (decreasing
${\it\gamma}$
); however, this quickly shortens the damping time to much less than that observed in the experiments. Because of the precision to which the parameters in (4.1) have been measured, the discrepancy must arise either because the linear hydrodynamic theory is invalid, or because one or more of the experimental parameters has been erroneously reported. The former is certainly likely; as discussed in § 4.1, the Rossby number is
$0.2$
, so nonlinear effects may be present. The observed decay of the oscillation period cannot be captured by the linear theory. The impulsive spin-up of the container cannot be performed instantaneously as required for a perfect comparison with theory, and this may explain the initial rapid decay of the container motion observed in the experiments. Errors in reporting the experimental parameters are also possible, because Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980) report some inconsistently (e.g. vessel radii), and others not at all (e.g. the vessel moment of inertia, which we have assumed equal to that of the fluid). A thorough comparison between theory and experiment will require new experiments. Such experiments could be combined with the well-known properties of helium II to provide a useful test of both the HVBK theory and the calculations in this paper, as in van Eysden & Melatos (Reference van Eysden and Melatos2011). Alternatively, the container response predicted here may be a useful check of known superfluid parameters in helium II, or used to constrain parameters in condensates where they remain unknown (van Eysden & Melatos Reference van Eysden and Melatos2012).
4.3. Tkachenko oscillations
In our calculations, we have solved the complete equations of Chandler & Baym (Reference Chandler and Baym1986), in order to apply the most sophisticated hydrodynamic theory available to the helium II experiments. Although this theory includes the extra term (2.9) over the HVBK theory, we find that the contribution from this force vanishes. This result was also obtained by Reisenegger (Reference Reisenegger1993). The reason for this lies in the ansatz (2.21), where the flow is assumed to be axisymmetric and the radial dependence of the azimuthal velocity matches the boundary conditions. The equation of motion for the vortex lines requires that
${\bf\xi}$
has the same radial dependence as the superfluid velocity, and upon substituting (2.21) into (2.9), we find that
${\bf\sigma}=0$
.
Physically, (2.9) describes the restoring force experienced by the vortex array as a result of displacements of vortices from their equilibrium configuration. Hence, the vortex array experiences a restoring force in response to shear deformations, giving rise to Tkachenko oscillations with speed
$c_{T}=(\hbar{\it\Omega}/4m)^{1/2}$
. However, in the spin-up problem considered here, the top and bottom boundaries are rigid bodies and the impulsive acceleration of the boundary does not displace any vortices from their equilibrium configuration. Therefore, Tkachenko modes are not excited by the top and bottom of the container.
Tkachenko modes may be excited at the sidewall of the container, which has been neglected in our analysis. However, solution for the eigenfrequencies for the full three-dimensional flow bounded at the top, bottom and sidewalls of the cylinder poses a formidable challenge. In the limit where the period of superfluid Ekman pumping is much shorter than the crossing time for a Tkachenko mode (
$t_{P}\ll c_{T}R$
), as in the Tsakadze experiments, the flow is columnar and the motion of the vortex array is determined by the superfluid Ekman pumping mechanism. In this limit one may expect a complicated boundary layer structure on the sidewalls, generalizing the nested Stewartson layers of thickness
$E^{1/4}L$
and
$E^{1/3}L$
that form in classical Ekman pumping. However, in the opposite limit (
$t_{P}\gg c_{t}R$
), which can be realized in a container of large aspect ratio (
$L\gg R$
), the Tkachenko mode is dominant (Sonin Reference Sonin1976, Reference Sonin1987). However, the viscous diffusion time
$E^{-1}{\it\Omega}^{-1}$
is comparable to or faster than the period of the fundamental Tkachenko mode,
$\text{ln}(b_{0}/a_{0})E_{s}^{-1}{\it\Omega}^{-1}$
(see (4.1)), so that excited Tkachenko modes will be rapidly damped.
Therefore, Tkachenko oscillations are unlikely to be relevant in the experiments of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980). Tkachenko oscillations have been imaged in rapidly rotating dilute-gas Bose–Einstein condensates, excited by blasting atoms or creating a dip in the trapping potential at the centre of the condensate (Coddington et al. Reference Coddington, Engels, Schweikhard and Cornell2003). Here, the experimental set-up and excitation mechanism are appropriate to excite Tkachenko modes, whereas in the experiments of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980) they are not.
5. Neutron stars
Neutron stars are compact stellar corpses formed in core collapse supernovae. The resulting compact object typically has a radius of 12–14 km and a mass of 1.4–2 times that of the Sun. The outer kilometre of the star is a rigid highly conducting crustal lattice interspersed with a neutron superfluid, while the core comprises approximately 95 % superfluid neutrons, 5 % electrons and superconducting protons, and a small fraction of muons. In the deep core, the composition remains uncertain. The fluid dynamics in the outer core is often modelled using the HVBK equations, where the ‘normal’ and ‘superfluid’ components refer to a proton–electron plasma and neutron superfluid respectively (Peralta et al. Reference Peralta, Melatos, Giacobello and Ooi2006a ,Reference Peralta, Melatos, Giacobello and Ooi b ; Peralta & Melatos Reference Peralta and Melatos2009).
Neutron stars are also endowed with strong magnetic fields, ranging from
$10^{8}$
to
$10^{15}$
G. In isolated radio pulsars, radio waves are beamed along the magnetic axis and create a lighthouse effect that is observed as pulsations on Earth. Occasionally, these objects are observed to ‘glitch’, where the rotational frequency of the star suddenly increases by up to one part in
$10^{5}$
(Espinoza et al.
Reference Espinoza, Lyne, Stappers and Kramer2011). The distribution of glitch sizes and waiting times is consistent with the hypothesis that they arise from a self-organized collective avalanche process (Anderson & Itoh Reference Anderson and Itoh1975; Melatos, Peralta & Wyithe Reference Melatos, Peralta and Wyithe2008; Warszawski & Melatos Reference Warszawski and Melatos2013). During a glitch, angular momentum stored in the decoupled superfluid components of the neutron star is transferred erratically to the solid crust (
${\sim}1\,\%$
of the total moment of inertia) (Link, Epstein & van Riper Reference Link, Epstein and van Riper1992), although the exact source of this angular momentum is uncertain (Andersson et al.
Reference Andersson, Glampedakis, Ho and Espinoza2012; Chamel Reference Chamel2013). The trigger is often ascribed to collective unpinning of quantized superfluid vortices, induced when the crust–superfluid differential rotation (and hence the Magnus force) exceeds a threshold (Baym, Epstein & Link Reference Baym, Epstein and Link1992; Warszawski, Melatos & Berloff Reference Warszawski, Melatos and Berloff2012).
The glitch event is followed by a recovery phase, during which the components inside the star adjust to a new equilibrium configuration (Baym et al.
Reference Baym, Pethick and Pines1969b
). This recovery is typically dominated by a step increase in the rotational frequency of the star, a step change in the frequency derivative and a quasi-exponential relaxation (van Eysden & Melatos Reference van Eysden and Melatos2010). However, quasi-exponential oscillations have also been reported following some glitches. A post-glitch oscillation with a period of approximately four months was reported in the Crab pulsar soon after its discovery (Ruderman Reference Ruderman1970a
,Reference Ruderman
b
). The 1975 and 1986 Crab glitches were observed to ‘overshoot’ their final rotation frequency during their recovery (Wong, Backer & Lyne Reference Wong, Backer and Lyne2001; van Eysden & Melatos Reference van Eysden and Melatos2010), which could potentially be explained by a damped oscillation with a period of approximately 200–300 days. Oscillations also appear to be present in timing residuals following the 1988 Christmas Vela glitch (see figure 2 of McCulloch et al.
Reference McCulloch, Hamilton, McConnell and King1990), which, if real, appear to have a period of
${\sim}20~\text{days}$
. These oscillations have received little attention observationally or theoretically, but may shed light on the interior of neutron stars. Here, we consider the possibility that the superfluid oscillations described in the previous sections may be operating in neutron stars, and ask what the observable consequences are for glitch recovery.
During glitch recovery, the proton–electron plasma in the core is coupled to the highly conducting crust via the magnetic field on a time scale of seconds (Easson Reference Easson1979; van Eysden Reference van Eysden2014), and hence these components are assumed to be rigidly locked together. The superfluid neutrons are usually assumed to respond to changes in the angular velocity of the crust via the interaction of the neutron vortices with the proton–electron plasma in the core (Alpar, Langer & Sauls Reference Alpar, Langer and Sauls1984; Mendell Reference Mendell1991a ,Reference Mendell b ; Glampedakis, Andersson & Samuelsson Reference Glampedakis, Andersson and Samuelsson2011). The electrons scatter from the neutron vortices that are magnetized by the entrainment of proton currents, giving the mutual friction force
$$\begin{eqnarray}\displaystyle \boldsymbol{F}=-\frac{{\it\rho}}{{\it\rho}_{n}}{\it\beta}_{0}\hat{{\it\omega}}_{s}\times [{\bf\omega}_{s}\times (\boldsymbol{v}_{L}-\boldsymbol{v}_{n})], & & \displaystyle\end{eqnarray}$$
where the subscript
$n$
refers the proton–electron plasma, which plays the role of the normal fluid component, and
$s$
refers to the neutron superfluid. The coefficient
${\it\beta}_{0}$
is given by (Mendell Reference Mendell1991b
)
$$\begin{eqnarray}{\it\beta}_{0}=1.1\times 10^{-2}\frac{(y-1)^{2}x^{7/6}}{y^{1/2}(1-x)},\end{eqnarray}$$
where
$x$
is the proton fraction (
${\it\rho}_{n}/{\it\rho}$
in the notation used here) and
$y$
is the normalized effective mass of the proton from Fermi liquid theory. The coefficient
${\it\beta}_{0}^{\prime }$
that appears in (5.1) in HVBK theory is typically taken as zero in neutron stars. The motion of magnetized neutron vortices is also inhibited by their pinning to flux tubes. When a vortex is pinned it moves with the normal fluid so that
$\boldsymbol{v}_{L}=\boldsymbol{v}_{n}$
. However, an unpinned vortex moves with velocity given by (2.11). Solving the vortex line equation of motion (2.11) gives
$$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{v}_{L}=\boldsymbol{v}_{n}-\frac{1}{1+{\it\beta}_{0}^{2}}\{\hat{{\it\omega}}_{s}\times [\hat{{\it\omega}}_{s}\times (\boldsymbol{v}_{s}-\boldsymbol{v}_{n})+{\bf\sigma}+\boldsymbol{t}]+{\it\beta}_{0}[\hat{{\it\omega}}_{s}\times (\boldsymbol{v}_{s}-\boldsymbol{v}_{n})+{\bf\sigma}+\boldsymbol{t}]\}.\qquad & \displaystyle\end{eqnarray}$$
At zero temperature, all vortices are pinned to flux tubes. At finite temperature a vortex can become thermally excited with activation energy
$A$
upon unpinning. The activation energy for unpinning depends on the magnetic energy between a vortex and a flux tube,
$E_{p}$
, the dimensionless vortex tension,
$\mathscr{T}$
, the angular velocity lag,
${\rm\Delta}{\it\omega}$
, and the critical angular velocity lag for unpinning,
${\rm\Delta}{\it\omega}_{c}$
, as
$$\begin{eqnarray}A=5.1E_{p}\mathscr{T}^{1/2}\left(1-\frac{{\rm\Delta}{\it\omega}}{{\rm\Delta}{\it\omega}_{c}}\right)^{5/4}.\end{eqnarray}$$
Therefore, the mutual friction force has a nonlinear dependence on the velocity through
${\rm\Delta}{\it\omega}$
. However, to simplify our analysis we take
$A$
as constant, noting that a rigorous calculation should include this nonlinear dependence.
The partition function for this two-state system is
$$\begin{eqnarray}Z=1+\text{e}^{-{\it\beta}A},\end{eqnarray}$$
where
${\it\beta}=(k_{B}T)^{-1}$
,
$k_{B}$
is Boltzmann’s constant and
$T$
is the temperature. When the thermal energy is much greater than the activation energy for unpinning a vortex,
${\it\beta}^{-1}\gg A$
, all vortices are thermally activated and move with average velocity given by (5.3). However, when the thermal energy is much less than the activation energy for unpinning,
${\it\beta}^{-1}\ll A$
, the vortex movement is exponentially suppressed, pinning the vortices to the normal fluid. For slow vortex slippage,
$\text{e}^{-{\it\beta}A}\ll 1$
and
$Z\approx 1$
and the ensemble average vortex line velocity is given by
$$\begin{eqnarray}\displaystyle \langle \boldsymbol{v}_{L}\rangle _{{\it\beta}}-\boldsymbol{v}_{n} & = & \displaystyle \langle \boldsymbol{v}_{L}-\boldsymbol{v}_{n}\rangle _{{\it\beta}}\approx Z^{-1}\text{e}^{-{\it\beta}A}(\boldsymbol{v}_{L}-\boldsymbol{v}_{n})\nonumber\\ \displaystyle & = & \displaystyle -\frac{1}{1+{\it\beta}_{0}^{2}}\left.\{\!\hat{{\it\omega}}_{s}\times [\hat{{\it\omega}}_{s}\times (\boldsymbol{v}_{s}-\boldsymbol{v}_{n})+{\bf\sigma}+\boldsymbol{t}]\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.{\it\beta}_{0}[\hat{{\it\omega}}_{s}\times (\boldsymbol{v}_{s}-\boldsymbol{v}_{n})+{\bf\sigma}+\boldsymbol{t}]\!\}\text{e}^{-{\it\beta}A}.\right.\end{eqnarray}$$
By substituting the vortex line velocity from (5.6) into the vortex line conservation equation (2.10) and integrating, we obtain an HVBK-like equation for the superfluid (2.2) with modified mutual friction coefficients,
$$\begin{eqnarray}\displaystyle & \displaystyle B=\frac{2{\it\rho}}{{\it\rho}_{n}}\frac{{\it\beta}_{0}}{1+{\it\beta}_{0}^{2}}\text{e}^{-{\it\beta}A}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle B^{\prime }=\frac{2{\it\rho}}{{\it\rho}_{n}}\left[1-\frac{1}{1+{\it\beta}_{0}^{2}}\text{e}^{-{\it\beta}A}\right]. & \displaystyle\end{eqnarray}$$
Before proceeding, we assess the validity of the incompressibility assumption for the present application. In neutron stars the speed of sound is approximately one-fifth of the speed of light (see, e.g., Reisenegger & Goldreich Reference Reisenegger and Goldreich1992), giving a Mach number of
$1.7\times 10^{-3}$
. Therefore, the incompressibility assumption in neutron stars is valid.
We can now apply the theory in § 2 to neutron stars. Because the normal fluid is rigidly locked to the crust via the magnetic field, we have
$\boldsymbol{v}_{n}={\it\bf\Omega}_{c}r$
. Therefore, it is a single-fluid problem involving only the superfluid, and we can repeat the analysis in § 3 with the addition of mutual friction terms. In this case, the crust plays the role of the container. We assume that the vortices are perfectly pinned to the crust with
${\it\gamma}\rightarrow \infty$
at
$z=\pm 1$
. In the core of the star (the region
$-1<z<1$
), we expect superfluid oscillations as illustrated in § 3, where vortex lines are straight in the core, except in a small boundary layer region adjacent to the crust. However, the motion of the neutron superfluid in the core is now impeded by the interaction with the flux tubes in the proton electron plasma, which is described by the mutual friction force.
For the superfluid oscillation mechanism to operate, vortices pinned in the crust must communicate with vortices in the core via vortex line tension. In the outer crust, neutrons pair in a
$^{1}S_{0}$
state, while in the core they are expected to pair in a
$^{3}P_{2}$
state. The
$^{1}S_{0}$
phase is expected to be lost near the base of the crust; however, current calculations of the pairing gaps for the
$^{3}P_{2}$
phase are not certain enough to know at what depth superfluidity will be restored. If the two phases are connected, then the relevant boundary conditions at the interface are also unknown. It may be that the vortices in the crust and the core are not connected, in which case the superfluid oscillations described here will not occur.
Repeating the analysis in § 3 with the mutual friction terms, the result (3.21) generalizes to
$$\begin{eqnarray}\displaystyle \displaystyle \bar{{\it\Delta}}\approx \frac{k\text{e}^{2k}}{2{\it\rho}s({\it\rho}s+{\it\rho}_{n}B)}\{(2{\it\rho}-{\it\rho}_{n}B^{\prime })[2K{\it\rho}_{s}+(2{\it\rho}-{\it\rho}_{n}B^{\prime })]+({\it\rho}s+{\it\rho}_{n}B)({\it\rho}ks+{\it\rho}_{n}B)\},\quad & & \displaystyle\end{eqnarray}$$
and the inverse Laplace transform is
$$\begin{eqnarray}f(t)=\frac{1}{(2{\it\rho}-{\it\rho}_{n}B^{\prime })+2K{\it\rho}_{s}}\left[(2{\it\rho}-{\it\rho}_{n}B^{\prime })+\left(\frac{2K{\it\rho}_{s}}{s_{+}-s_{-}}\right)(s_{+}\text{e}^{s_{-}t}-s_{-}\text{e}^{s_{+}t})\right],\end{eqnarray}$$
where
$$\begin{eqnarray}s_{\pm }=-\frac{{\it\rho}_{n}B}{2{\it\rho}}\pm \sqrt{\frac{k({\it\rho}_{n}B)^{2}-4(2{\it\rho}-{\it\rho}_{n}B^{\prime })[2K{\it\rho}_{s}+(2{\it\rho}-{\it\rho}_{n}B^{\prime })]}{4{\it\rho}^{2}k}}.\end{eqnarray}$$
In a typical neutron star, we have
${\it\rho}_{n}/{\it\rho}=x\sim 0.05$
,
${\it\Omega}=100~\text{rad}~\text{s}^{-1}$
,
$0.3\leqslant y=m_{p}^{\ast }/m_{p}\leqslant 0.7$
and
$K\approx 50$
(Mendell Reference Mendell1991a
,Reference Mendell
b
; van Eysden & Melatos Reference van Eysden and Melatos2010). The pinning fraction in the core is uncertain. Hence, we have
${\it\beta}_{0}\ll 1$
(see (5.2)), and also
$K{\it\rho}_{s}\gg {\it\rho}$
, because the neutron superfluid is the dominant contribution to the moment of inertia of the star. The superfluid Ekman number is approximately
$$\begin{eqnarray}\displaystyle E_{s}=3.06\times 10^{-17}\left(\frac{{\it\nu}_{s}}{3.06\times 10^{-3}~\text{cm}^{2}~\text{s}^{-1}}\right)\left(\frac{{\it\Omega}}{100~\text{rad}~\text{s}^{-1}}\right)^{-1}\left(\frac{L}{10^{6}~\text{cm}}\right)^{-2}.\qquad & & \displaystyle\end{eqnarray}$$
Under these assumptions (5.10) can be written as
$$\begin{eqnarray}f(t)=\text{e}^{-t/t_{d}}\left[\cos \left(\frac{2{\rm\pi}t}{t_{p}}\right)+\frac{t_{p}}{2{\rm\pi}t_{d}}\sin \left(\frac{2{\rm\pi}t}{t_{p}}\right)\right],\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle t_{d} & = & \displaystyle \frac{1}{{\it\Omega}|\text{Re}(s_{\pm })|}\approx \frac{\text{e}^{{\it\beta}A}}{{\it\Omega}{\it\beta}_{0}},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle t_{p} & = & \displaystyle \frac{2{\rm\pi}}{{\it\Omega}|\text{Im}(s_{\pm })|}\approx \frac{{\rm\pi}\text{e}^{{\it\beta}A/2}}{{\it\Omega}}\sqrt{\frac{{\it\rho}k}{K{\it\rho}_{s}}}.\end{eqnarray}$$
In figure 6 we plot the damping time and period for superfluid oscillations in a neutron star as a function of the pinning parameter
${\it\beta}A$
. We plot the full result for the damping time and period using (5.11). Results are plotted for
$y=m_{p}^{\ast }/m_{p}=0.3$
(blue) and
$0.7$
(purple). We find that the approximate results given in (5.14) and (5.15) give excellent agreement. The damping time becomes longer as pinning is increased, because the amount of vortices moving and dissipating energy through scattering is exponentially suppressed. The period also increases as pinning increases, because pinning is inhibiting the movement of vortices and hence the transport of angular momentum. In the limit of infinitely strong pinning, the vortices stop moving and the period becomes infinitely long.
Damping time (a) and period (b) for oscillations in a neutron star. Lines are plotted for
$y=m_{p}^{\ast }/m_{p}=0.3$
(blue) and
$0.7$
(purple). In (b) the period is independent of
$m_{p}^{\ast }$
and the lines overlap.
In principle, oscillations of almost any period are possible, depending on the pinning strength,
$A$
. The oscillation amplitude is unaffected by pinning and is always equal to the glitch amplitude. The damping time scale is always longer than the period, so the oscillations are only weakly damped and the second term in (5.13) is small. Post-glitch oscillations are observed in both the Crab and Vela pulsars that are candidates for the oscillatory Ekman pumping mechanism in this paper. In the Crab pulsar, oscillations with a period of roughly four months were reported following the earliest glitches (Ruderman Reference Ruderman1970a
,Reference Ruderman
b
). The ‘overshoot’ observed in the recovery from the 1975 and 1986 Crab glitches could also correspond to a damped oscillation with a period of 200–300 days (Wong et al.
Reference Wong, Backer and Lyne2001; van Eysden & Melatos Reference van Eysden and Melatos2010). The former would require
${\it\beta}A\approx 23.8$
, and for the latter
${\it\beta}A\approx 24.8$
. In the Vela pulsar, a damped periodic oscillation is clearly visible in the timing residuals following the 1988 glitch (see figure 2 of McCulloch et al.
Reference McCulloch, Hamilton, McConnell and King1990). These oscillations appear to have a period of
${\sim}20~\text{days}$
, requiring
${\it\beta}A\approx 20.2$
. Therefore, superfluid Ekman pumping can explain these oscillations if pinning between vortices and flux tubes is extremely strong, limiting vortices to a very slow creep.
These numbers can be compared with theoretical expectations using known parameters in the outer core of the pulsar. Assuming a typical pulsar temperature and using parameter estimates from Link (Reference Link2014), from (5.4) we find
$$\begin{eqnarray}{\it\beta}A=2.05\times 10^{5}\left(\frac{100~\text{MeV}}{E_{p}}\right)^{1/2}\left(\frac{0.12}{\mathscr{T}}\right)^{1/2}\left(\frac{10^{7}~\text{K}}{T}\right)^{-1},\end{eqnarray}$$
where we have taken
${\rm\Delta}{\it\omega}=0$
. Clearly, (5.16) leads to extremely long oscillation periods and damping times. However, this is a gross upper estimate because the lag
${\rm\Delta}{\it\omega}$
is not likely to be zero. If the lag is close to the critical lag for unpinning
${\rm\Delta}{\it\omega}_{c}$
, the estimate (5.16) can be greatly reduced, see (5.4). However, to correctly determine the lag the nonlinear calculation including velocity dependence in the mutual friction coefficients is required, which is beyond the scope of this paper.
Finally, we remark on limitations of the present model. Neutron stars are strongly stratified, typically with buoyancy frequencies
$N$
exceeding the rotation rate (Reisenegger & Goldreich Reference Reisenegger and Goldreich1992). Buoyancy inhibits Ekman pumping from penetrating the core, confining the circulation to a buoyancy layer with thickness
${\it\Omega}L/N$
(Walin Reference Walin1969; Abney & Epstein Reference Abney and Epstein1996; van Eysden & Melatos Reference van Eysden and Melatos2008). The spin-up time is reduced commensurately to
$E^{-1/2}N^{-1}$
because the effective size of the Ekman cell has been reduced by a fraction
${\it\Omega}/N$
. The final state is not co-rotation, but has persistent shear between the fluid in the buoyancy layer and that in the core, which in untouched by Ekman pumping (Melatos Reference Melatos2012). In a typical neutron star, we find
${\it\Omega}/N\sim 0.20$
(Reisenegger & Goldreich Reference Reisenegger and Goldreich1992), so that the spin-up time is reduced by a factor of 5.
A similar result is expected for the superfluid Ekman pumping mechanism discussed in this paper. As in the classical stratified Ekman pumping, we expect that buoyancy forces will confine the circulation to a depth
${\it\Omega}L/N$
. The Taylor–Proudman theorem no longer applies in the core, and the vortex lines will be sheared. Above
${\it\Omega}L/N$
the fluid undergoes superfluid oscillations, but at greater depths the fluid does not participate. To estimate the period of oscillations, we assume that the length scale for the Ekman cell is reduced from
$L$
to
${\it\Omega}L/N$
, and (3.26) becomes
$$\begin{eqnarray}t_{P}=\frac{{\rm\pi}}{{\it\Omega}}\sqrt{\frac{{\it\Omega}}{N}\left(\frac{k}{1+K}\right)}.\end{eqnarray}$$
In this case we have
$\sqrt{{\it\Omega}/N}\sim 0.45$
, shortening the period by a factor of approximately 2.2. Therefore, the superfluid oscillations are more weakly affected by stratification than classical Ekman pumping.
6. Conclusions
The response of a uniformly rotating superfluid threaded with a large density of vortex lines to an impulsive response of its container has been investigated. When vortices are strongly pinned to the container and there is no dissipation, the system oscillates persistently with a period of order
$E^{-1/4}{\it\Omega}^{-1}$
, where
$E_{s}$
is the dimensionless vortex line tension parameter. The low-frequency mode is generated by a secondary flow analogous to classical Ekman pumping, periodically reversed by the vortex tension. This secondary flow transports vortices radially inwards and outwards in the interior, changing the angular momentum of the fluid. The full set of equations of Chandler & Baym (Reference Chandler and Baym1986) has been solved and compared with experiments performed by Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980) in superfluid helium. We find qualitative agreement between theory and experiment; however, quantitative comparison is difficult because some experimental parameters are unknown, and the experiment may probe the nonlinear regime, which is beyond the scope of this theory. The columnar nature of superfluid Ekman pumping also reproduces the experimentally reported dependence of the frequency on vessel radius and height; however, the theory would benefit from comparison with new carefully conducted experiments. We argue that Tkachenko modes are unlikely to be excited by the boundary and initial conditions imposed in the spin-up experiments of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980). In neutron stars, the movement of neutron vortices is inhibited by pinning to type II superconducting protons in the core, slowing the superfluid Ekman pumping process. For very strong pinning, and hence slow vortex creep, the theory can explain post-glitch oscillations with periods of days to years, with much longer damping times.
Note added in proof
At the time of acceptance of this paper the author was unaware of the work by Sonin (Reference Sonin1976, Reference Sonin1987) on the experiments of Tsakadze & Tsakadze (Reference Tsakadze and Tsakadze1980). The frequency of the inertial resonance identified by Sonin (Reference Sonin1987) (see (6.55) in that paper) is identical to that of the superfluid Ekman pumping frequency (3.23) in this paper. However, comparisons for the two-fluid problem including pinning are difficult to make because of the complexity of the problem, differences in boundary conditions and differences in the notation used. The present paper studies the initial value problem, while Sonin (Reference Sonin1976, Reference Sonin1987) investigate resonances.
Acknowledgements
The author thanks Bennett Link for useful feedback on the manuscript.
Appendix A. General solution in the boundary layer approximation
Here, we present the solution to the governing equations (4.2)–(4.9). The solution is too large to present algebraically, so we outline the method and present key results. Because of the size of the equations, it is easier to perform the calculations in programs such as Mathematica or Maple and handle them numerically.
We assume that the container is rigid so that the upper and lower boundaries move identically. Therefore, the
$V_{n,s}$
are symmetric in
$z$
and the
${\it\chi}_{n,s}$
are antisymmetric. Applying these boundary conditions, we solve in the upper half-plane only.
The normal fluid has the solution to the interior flow equations (4.2)–(4.5),
$$\begin{eqnarray}\displaystyle V_{n}^{I}(t)=\mathop{\sum }_{i=i}^{3}V_{ni}^{I}\text{e}^{s_{i}t}+V_{n4}^{I}, & & \displaystyle\end{eqnarray}$$
and similarly for
$V_{s}$
. Three time scales exist in the problem and are obtained when applying the boundary conditions below. The streamfunction for the secondary flow is
$$\begin{eqnarray}\displaystyle {\it\chi}_{n}^{I}(z,t)=z\left[\mathop{\sum }_{i=i}^{3}{\it\chi}_{ni}^{I}\text{e}^{s_{i}t}+{\it\chi}_{n4}^{I}\right], & & \displaystyle\end{eqnarray}$$
and similarly for
${\it\chi}_{s}$
. The pressure is
$$\begin{eqnarray}\displaystyle P_{n}(t)=\mathop{\sum }_{i=i}^{3}P_{ni}\text{e}^{s_{i}t}+P_{n4}, & & \displaystyle\end{eqnarray}$$
and similarly for
$P_{s}$
. Substitution of (A 1)–(A 3) into (4.2)–(4.5) allows us to solve for the coefficients of
${\it\chi}_{n,s}$
and
$P_{n,s}$
in terms of those of
$V_{n,s}^{I}$
.
In the boundary layer, the solutions to (4.6)–(4.9) have the form
$$\begin{eqnarray}\displaystyle V_{n}^{B}(z,t) & = & \displaystyle \displaystyle \mathop{\sum }_{i=1}^{3}\text{e}^{s_{i}t}[V_{ni+}^{B}\text{e}^{-k_{+}(1-z)}+V_{ni-}^{B}\text{e}^{-k_{-}(1-z)}],\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle V_{s}^{B}(z,t) & = & \displaystyle \displaystyle \mathop{\sum }_{i=1}^{3}\text{e}^{s_{i}t}[V_{si+}^{B}\text{e}^{-k_{+}(1-z)}+V_{si-}^{B}\text{e}^{-k_{-}(1-z)}+V_{si}^{B}\text{e}^{-k(1-z)}],\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle k_{\pm }=\sqrt{\frac{\pm 2\text{i}{\it\rho}[B\pm \text{i}(2-B^{\prime })]}{E[{\it\rho}_{n}B\pm \text{i}(2{\it\rho}-{\it\rho}_{n}B^{\prime })]}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle k=\sqrt{\frac{2}{E_{s}}}. & \displaystyle\end{eqnarray}$$
${\it\chi}_{n,s}^{B}$
we have 
$$\begin{eqnarray}\displaystyle {\it\chi}_{n}^{B}(z,t) & = & \displaystyle \displaystyle \mathop{\sum }_{i=1}^{3}\text{e}^{s_{i}t}[{\it\chi}_{ni+}^{B}\text{e}^{-k_{+}(1-z)}+{\it\chi}_{ni-}^{B}\text{e}^{-k_{-}(1-z)}],\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\it\chi}_{s}^{B}(z,t) & = & \displaystyle \displaystyle \mathop{\sum }_{i=1}^{3}\text{e}^{s_{i}t}[{\it\chi}_{si+}^{B}\text{e}^{-k_{+}(1-z)}+{\it\chi}_{si-}^{B}\text{e}^{-k_{-}(1-z)}+{\it\chi}_{si}^{B}\text{e}^{-k(1-z)}].\end{eqnarray}$$
$k_{\pm }$
), the superfluid and normal fluid components are coupled, whereas for the superfluid Ekman pumping mechanism, the normal fluid is not involved. Substituting (A 4)–(A 9) into (4.6)–(4.9) and comparing like terms gives
$V_{si\pm }^{B}$
,
${\it\chi}_{si\pm }^{B}$
and
${\it\chi}_{ni\pm }^{B}$
in terms of
$V_{ni\pm }^{B}$
. The coefficients
$V_{si}^{B}$
and
${\it\chi}_{si}^{B}$
are unrelated to other boundary layer coefficients.
The motion of the container has the solution
$$\begin{eqnarray}\displaystyle f(t)=\mathop{\sum }_{i=i}^{3}f_{i}\text{e}^{s_{i}t}+f_{4}, & & \displaystyle\end{eqnarray}$$
where the
$f_{i}$
can be determined in terms of
$V_{ni\pm }^{B}$
,
$V_{si}^{B}$
and
${\it\chi}_{si}^{B}$
by substituting (A 4)–(A 9) into (2.31). It should be noted that only the boundary layer corrections to the velocity contribute to the torque on the container.
The coefficients
$V_{ni\pm }^{B}$
,
$V_{si}^{B}$
,
${\it\chi}_{si}^{B}$
,
$V_{si}^{I}$
and
$V_{s4}^{I}$
can be solved for in terms of
$V_{ni}^{I}$
through application of the boundary conditions (2.36), (3.41) and (2.38) for the superfluid and (4.10)–(4.12) for the normal fluid and comparing like terms. For each
$i$
, the six boundary conditions determine five unknowns in terms of
$V_{ni}^{I}$
; the final equation is an eigenvalue equation for
$s_{i}$
. The equation is cubic in
$s_{i}$
, giving solutions for the three
$s_{i}$
. The boundary conditions also give
$f_{4}=V_{s4}=V_{n4}$
.
The remaining four unknown coefficients
$V_{ni}^{I}$
and
$V_{n4}$
are determined by the initial conditions (3.42)–(3.44) and (4.13). The result is too cumbersome to present algebraically, but is presented for the experimentally measured parameters in helium II in (4.14).
Appendix B. Solutions in the boundary layer approximation for strong coupling
In helium II, we have
$B,B^{\prime }\sim 1$
, resulting in the strong coupling between the two fluid components. Any differential rotation is removed over the rotational time scale and the two fluid components are ‘locked together’ over the much longer Ekman time (Reisenegger Reference Reisenegger1993; van Eysden & Melatos Reference van Eysden and Melatos2013). This assumption can be used to find a more analytically tractable solution than that presented in appendix A.
When
$B=B^{\prime }=0$
, (4.3) and (4.5) describe the geostrophic balance in the interior, balancing the time rate of change of the azimuthal flow with secondary flow. In (4.3) all terms are of order
$E^{1/2}$
(Greenspan & Howard Reference Greenspan and Howard1963) and in (4.5) all terms are of order
$E_{s}^{1/4}$
(see § 2). However, when
$B,B^{\prime }\sim 1$
the third terms in (4.3) and (4.5) are of order unity, while the remainder are of the order of the secondary flow. Therefore, we perturb the leading-order velocity so that to leading order we have
$V_{n}^{I}=V_{s}^{I}$
and the velocity difference
${\it\delta}V^{I}$
is of the order of the secondary flow, i.e.
$$\begin{eqnarray}\displaystyle V_{s}^{I}=V_{n}^{I}-{\it\delta}V^{I}. & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle 2V_{n}^{I}-P_{n},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \frac{\partial V_{n}^{I}}{\partial t}+2\frac{\partial {\it\chi}_{n}^{I}}{\partial z}+\frac{{\it\rho}_{s}B}{{\it\rho}}{\it\delta}V^{I}-\frac{{\it\rho}_{s}B^{\prime }}{{\it\rho}}\left(\frac{\partial {\it\chi}_{n}^{I}}{\partial z}-\frac{\partial {\it\chi}_{s}^{I}}{\partial z}\right),\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle 2V_{s}^{I}-P_{s},\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle 0 & = & \displaystyle \frac{\partial V_{s}^{I}}{\partial t}+2\frac{\partial {\it\chi}_{s}^{I}}{\partial z}-\frac{{\it\rho}_{n}B}{{\it\rho}}{\it\delta}V^{I}+\frac{{\it\rho}_{n}}{{\it\rho}}B^{\prime }\left(\frac{\partial {\it\chi}_{n}^{I}}{\partial z}-\frac{\partial {\it\chi}_{s}^{I}}{\partial z}\right).\end{eqnarray}$$
$V_{n}^{I}$
has the same form as (A 1), and similarly for
${\it\delta}V^{I}$
we have 
$$\begin{eqnarray}\displaystyle {\it\delta}V^{I}(t)=\mathop{\sum }_{i=i}^{3}{\it\delta}V_{i}^{I}\text{e}^{s_{i}t}+{\it\delta}V_{4}^{I}. & & \displaystyle\end{eqnarray}$$
Using the same form for the streamfunctions (A 2) and pressures (A 3), their coefficients can be determined in terms of
$V_{I}^{ni}$
and
${\it\delta}V_{i}^{I}$
using (B 2)–(B 5).
The remaining procedure follows as in appendix A. However, we find that there are now only two time scales
$s_{1}$
and
$s_{2}$
; the third corresponded to mutual friction coupling which is now effectively instantaneous. Accordingly, we can no longer impose separate initial conditions on
$V_{s}^{I}$
and
$V_{n}^{I}$
, which must be the same. The resulting solutions are still algebraically cumbersome, so we present results in the limit of negligible viscosity and negligible vortex tension.
B.1. Solution for negligible viscosity
A useful result discussed in § 4.2 is the case when viscous effects are negligible. We argue in that section that this is relevant in helium II.
In the boundary layer we find that
$V_{n,si\pm }^{B}=0$
and
${\it\chi}_{n,si\pm }^{B}=0$
. We also do not need to satisfy the normal fluid boundary conditions for no slip (4.10) and (4.12). The result is
$$\begin{eqnarray}\displaystyle & \displaystyle f(t)=\frac{1}{1+K}\left[1+K\frac{s_{1}\text{e}^{s_{2}t}-s_{2}\text{e}^{s_{1}t}}{s_{1}-s_{2}}\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle V_{n}^{I}=\frac{1}{1+K}\left[1-\frac{s_{1}\text{e}^{s_{2}t}-s_{2}\text{e}^{s_{1}t}}{s_{1}-s_{2}}\right], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\delta}V^{I}=\frac{2(\text{e}^{s_{2}t}-\text{e}^{s_{1}t})}{{\it\beta}k(s_{1}-s_{2})}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{s}^{I}=\frac{2z(\text{e}^{s_{2}t}-\text{e}^{s_{1}t})}{k(s_{1}-s_{2})}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{n}^{I}=0, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}s_{1,2}=-\frac{1}{k{\it\gamma}}\left[1\pm \sqrt{1-\frac{4(1+K){\it\gamma}^{2}k{\it\rho}_{s}}{{\it\rho}}}\right]\end{eqnarray}$$
and
$$\begin{eqnarray}{\it\beta}=\frac{B}{2-B^{\prime }}.\end{eqnarray}$$
The implications of this result are discussed in § 4.2
B.2. Solution for negligible vortex pinning or vortex tension
The focus of this paper has been on pinned vortices; however, as a useful check we can recover the previous results of Reisenegger (Reference Reisenegger1993) and van Eysden & Melatos (Reference van Eysden and Melatos2013, Reference van Eysden and Melatos2014), where pinning is negligible.
When
${\it\gamma}\rightarrow 0$
, we obtain the result
$$\begin{eqnarray}\displaystyle & \displaystyle f(t)=\frac{1}{1+K}(1+K\text{e}^{s_{3}t}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle V_{n}^{I}=\frac{1}{1+K}(1-\text{e}^{s_{3}t}), & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\delta}V^{I}=\frac{{\it\rho}\text{e}^{s_{3}t}(k_{1}^{2}+k_{2}^{2})[(k_{1}+k_{2})+\text{i}{\it\beta}(k_{1}-k_{2})]}{4{\it\beta}^{2}k_{1}^{2}k_{2}^{2}{\it\rho}_{s}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{n}^{I}=\frac{\text{i}z\text{e}^{s_{3}t}(k_{1}-k_{2})}{2k_{1}k_{2}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\it\chi}_{s}^{I}={\it\chi}_{n}^{I}-\frac{{\it\rho}\text{e}^{s_{3}t}(k_{1}^{2}+k_{2}^{2})[(k_{1}+k_{2})+\text{i}{\it\beta}(k_{1}-k_{2})]}{4{\it\beta}^{2}k_{1}^{2}k_{2}^{2}{\it\rho}_{s}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}s_{3}=-\frac{E}{2}(k_{1}+k_{2})(1+K){\it\rho}_{n}.\end{eqnarray}$$
Interestingly, the same result is obtained by taking
$E_{s}=0$
. This is just a statement that when there is no vortex pinning, the vortex tension plays no active role in the spin-up, as discussed in van Eysden & Melatos (Reference van Eysden and Melatos2013). There is a slight difference in the boundary layer solution: for
$E_{s}=0$
,
$V_{s3}^{B}={\it\chi}_{s3}^{B}=0$
, while for
${\it\gamma}=0$
they are non-zero.
From (B 19) we obtain the spin-up time
$$\begin{eqnarray}\displaystyle t_{s} & = & \displaystyle \frac{1}{{\it\Omega}|s_{3}|}=\frac{{\it\rho}}{{\it\rho}_{n}E^{1/2}{\it\Omega}(1+K)}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{{\it\rho}\left[\frac{B^{2}+(2-B^{\prime })^{2}}{({\it\rho}_{n}B)^{2}+(2{\it\rho}-{\it\rho}_{n}B^{\prime })^{2}}\right]^{1/2}+\frac{2{\it\rho}_{s}{\it\rho}B}{({\it\rho}_{n}B)^{2}+(2{\it\rho}-{\it\rho}_{n}B^{\prime })^{2}}\right\}^{-1/2}.\end{eqnarray}$$
This result was obtained by Reisenegger (Reference Reisenegger1993) and explored in detail by van Eysden & Melatos (Reference van Eysden and Melatos2013). The term in the braces arises from the coupling between the two fluid components in the boundary layer and is of order unity. The pre-factor is the familiar Ekman time modified by the normal fluid density fraction. The two fluid components are strongly coupled (i.e. they achieve co-rotation within a rotation period as
$B\sim 1$
), but it is only the normal fluid providing the Ekman pumping, which is reflected in the spin-up time.
