Oscillatory superfluid Ekman pumping in Helium II and neutron stars
UUnder consideration for publication in J. Fluid Mech. Oscillatory superfluid Ekman pumping inHelium II and neutron stars
C. Anthony van Eysden , † Department of Physics, Montana State University, Bozeman, Montana, 59717, USA Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken23, SE-10691 Stockholm, Sweden(Received ?; revised ?; accepted ?. - To be entered by editorial office)
The linear response of a superfluid, rotating uniformly in a cylindrical container andthreaded with a large number of vortex lines, to an impulsive increase in the angularvelocity of the container is investigated. At zero temperature and with perfect pinning ofvortices to the top and bottom of the container, we demonstrate that the system oscillatespersistently with a frequency proportional to the vortex line tension parameter to thequarter power. This low-frequency mode is generated by a secondary flow analogousto classical Ekman pumping that is periodically reversed by the vortex tension in theboundary layers. We compare analytic solutions to the two-fluid equations of Chandler& Baym (1986) with the spin-up experiments of Tsakadze & Tsakadze (1980) in heliumII and find the frequency agrees within a factor of four, although the experiment is notperfectly suited to the application of the linear theory. We argue that this oscillatoryEkman pumping mode, and not Tkachenko modes provide a natural explanation forthe observed oscillation. In neutron stars, the oscillation period depends on the pinninginteraction between neutron vortices and flux tubes in the outer core. Using a simplifiedpinning model, we demonstrate that strong pinning can accommodate modes with periodsof days to years, which are only weakly damped by mutual friction over longer timescales.
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1. Introduction
The linear response of a rapidly rotating Navier-Stokes fluid to an impulsive increase inangular velocity of its container, or ‘spin-up’ has been extensively studied. In the seminalwork of Greenspan & Howard (1963), three phases were identified: formation of a viscousboundary layer, Ekman pumping, and the damping of residual inertial oscillations byviscosity. Co-rotation between the interior fluid and container is established by Ekmanpumping, during which a secondary flow recycles fluid from the boundary layers into theinterior. The time-scale for the spin-up, known as the Ekman time, is proportional to theviscosity to the minus half power, and achieves co-rotation much faster than viscous dif-fusion. Since the work of Greenspan & Howard (1963), spin-up has been studied in fluidswith density stratification (Pedlosky 1967; Walin 1969), stratified and compressible fluids(Abney & Epstein 1996; van Eysden & Melatos 2008) magnetized plasmas (Loper 1971; † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] J u l C. A. van Eysden
Easson 1979; van Eysden 2014), and multi-component fluids (Ungarish 1990; Amberg &Ungarish 1993) and different geometries (Clark et al. et al. et al. slowly accelerating parallel plates, and showed that the Ekmantime is reduced by a factor depending on the superfluid mutual friction coefficients andthe normal fluid density fraction. This was generalized to the impulsive accelerationof containers of arbitrary shape (van Eysden & Melatos 2013) and to include the self-consistent response of the container (van Eysden & Melatos 2014) to facilitate comparisonwith helium II experiments (van Eysden & Melatos 2011, 2012). However, theoreticalwork to date on the spin-up of two-component superfluids such as helium II assumessmooth-walled containers and neglects the effects of pinning.The spin up of helium II at zero temperature, where the normal fluid componentvanishes and cannot facilitate Ekman pumping was also investigated by Reisenegger(1993). By introducing a frictional force to account for the sliding of vortex lines at theboundary, Reisenegger (1993) showed that the superfluid spins up via an Ekman-likesecondary flow with a timescale that depends on the strength of the frictional force.However, only the response of a single-component fluid to slowly accelerating parallelplates was studied.There is good reason to expect that the impulsive acceleration problem produces dif-ferent physics than that of the slowly accelerating parallel plates. When an ideal magne-tohydrodynamic plasma is slowly accelerated (i.e., the time-scale for acceleration of thecontainer is the slower than the Alfv´en crossing time), the plasma moves with the con-tainer as a rigid body. However, in response to an impulsive acceleration, Alfv´en wavesare excited that produce persistent oscillations of the container and the plasma (van Eys-den 2014). A final state of co-rotation between the container and plasma is inconsistentwith energy conservation if the fluid is dissipation-less, hence the system oscillates persis-tently. Similar arguments apply for a superfluid at zero temperature rotating uniformlywith a high density of vortex lines.Two primary motivations exist for this study. The first is a series of experimentsperformed by Tsakadze & Tsakadze (1980), in which the angular velocity of containersfilled with uniformly rotating helium II was impulsively increased. The subsequent motionof the container, responding freely to the hydrodynamic torque of the contained fluid, wasthen recorded. In experiments where the container was coated with powder to facilitatevortex pinning, a sinusoidal oscillation of the container was observed. These oscillationsare have been interpreted as Tkachenko modes, however the dependence of the oscillationperiod on vessel radius predicted for these modes is inconsistent with measurements,whereas the columnar nature of the Ekman pumping mechanism presented here is not.The second application is the recovery of pulsar glitches; the original motivation forexperiments of Tsakadze & Tsakadze (1980). Glitches are tiny, impulsive increases in therotation frequency of the neutron star crust, typically followed by a quasi-exponentialrecovery that is believed to be associated with the response of the interior superfluidBaym et al. (1969 a ). Although the superfluid neutron vortices are believed to pin tolattice sites in the crust and to flux tubes arising from type II superconductivity of scillatory superfluid Ekman pumping impulsive acceleration of the container. We focuson cylindrical geometry, where the vortices are strongly pinned to the top and bottomof the container. We apply the traditional Laplace transform techniques of Greenspan& Howard (1963), but solve self-consistently for the motion of the container as in vanEysden & Melatos (2014) and van Eysden (2014). We demonstrate the presence of alow-frequency oscillation mode, with a period that scales as the vortex line parameter tothe one-fourth power. This mode arises in the fast rotation limit, analogous to classicalEkman pumping. A secondary flow is present, which is periodically reversed by thetension in the vortex lines as they are sheared in the boundary layer. By solving the two-fluid equations of Chandler & Baym (1986), we predict an oscillation period of 10 secondsin helium II, compared with that of 40 seconds observed in the experiments of Tsakadze& Tsakadze (1980). A direct comparison cannot be made because the experiment is notperfectly suited for the application of the linear theory of Chandler & Baym (1986). Usinga simplified pinning model, we predict that oscillations in neutron stars can have periodsof days to years if the pinning of neutron vortices to flux tubes in the core is strong.These oscillations are weakly damped and expected to be observable over timescalesmuch longer than the period.The paper is structured as follows. In § § T = 0 and perfect vortex pinning of vortices at the top and bottomof a cylindrical container, where it is shown that the solution is oscillatory and hasan Ekman-like secondary circulation. In §
4, the full two-fluid equations are solved andcompared with the experiments of Tsakadze & Tsakadze (1980) in helium II. Neutron starapplications are considered in §
5. In §
2. Governing equations
A convenient description of superfluids such as helium II, rotating with a high densityof 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 η , and an inviscid ‘superfluid’ component, denoted by subscript s . † Underrotation the inviscid component forms a dense array of quantized vortices, which aresmooth-averaged in the hydrodynamic approximation, endowing the inviscid componentwith a macroscopic vorticity. The vortices mediate interactions between the normal andsuperfluid components, giving rise to a mutual friction force. An informative introductionto the HVBK equations is provided in Henderson & Barenghi (2000). A more general setof equations, derived by Baym & Chandler (1983); Chandler & Baym (1986), includesrestoring forces experienced by the vortex lattice when vortices are displaced from theirequilibrium configuration. This force produces Tkachenko oscillations, which are expectedto be observed in superfluid experiments. Assuming that the fluid is incompressible, theequations of Chandler & Baym (1986), written in HVBK form and in the laboratory † In a neutron star, the ‘normal’ and ‘superfluid’ components refer to a proton-electron plasmaand neutron superfluid, respectively. This is discussed further in § C. A. van Eysden frame are ∂ v n ∂t + v n · ∇ v n = − ∇ p n + ρ s ρ F + ν n ∇ v n , (2.1) ∂ v s ∂t + v s · ∇ v s = − ∇ p s − t − σ − ρ n ρ F , (2.2) ∇ · v n = 0 , (2.3) ∇ · v s = 0 , (2.4)where v n,s , ρ n,s and p n,s are the macroscopic velocities, densities and reduced pressuresof the normal and superfluid components, respectively. Throughout our analysis, boththe normal and superfluid components are considered incompressible; the validity of thisassumption is assessed in § § F = 12 B ˆ ω s × [ ω s × ( v n − v s ) − t − σ ]+ 12 B (cid:48) [ ω s × ( v n − v s ) − t − σ ] , (2.5)and ω s = ∇ × v s , (2.6)is the macroscopic vorticity of the superfluid and ˆ ω s = ω s / | ω s | is the vortex line directionvector, and B and B (cid:48) are dimensionless mutual friction coefficients. The vortex tensionforce per unit mass is t = ν s ω s × ( ∇ × ˆ ω s ) . (2.7)where the vortex line tension parameter is given by ν s = Γ4 π log (cid:18) b a (cid:19) . (2.8)Here Γ = (cid:126) π/m is the quantum of circulation, m is the mass of one helium atom, b isthe inter-vortex spacing and a is the size of the vortex core. The kinematic viscosity ν n is defined as the shear viscosity divided by the normal fluid density, η/ρ n . The finalparameter σ in (2.1)–(2.4) comes from the theory of Baym & Chandler (1983) anddescribes the restoring force of the vortex lattice in response to shear deformations. Ithas the form σ = (cid:126) | ω s | m (cid:2) ∇ ⊥ ( ∇ · ξ ) − ∇ ⊥ ξ (cid:3) , (2.9)where ξ is the vortex line displacement vector and ∇ ⊥ is the two-dimensional gradientoperator. Both ξ and ∇ ⊥ are two-dimensional in the plane orthogonal to the angularvelocity of the background superfluid flow, i.e., ˆ ω s · ξ = ˆ ω s · ∇ ⊥ = 0. Equation (2.9)only applies to linear deformations of a rectilinear vortex array, hence (2.1)–(2.4) areonly valid in the linear approximation when σ is included. When σ = 0, (2.1)–(2.4) arethe HVBK equations, which describe non-linear flows including quasi-classical turbulence(Henderson et al. et al. ∂ ω s ∂t = ∇ × ( v L × ω s ) , (2.10)where in the linear approximation the perturbation to the vortex line velocity is given by ∂ξ/∂t . Taking the curl of (2.2) and comparing the result with (2.10) gives the equation scillatory superfluid Ekman pumping ω s × ( v L − v s ) = t + σ + ρ n ρ F . (2.11)Equations (2.1) and (2.2) can be combined into an equation for the total fluid, ∂∂t ( ρ n v n + ρ s v s ) + ∇ j ( ρ n v ni v nj + ρ s v si v sj ) = ∇ j T ij , (2.12)where T ij = − pδ ij + T vij + T sij + T tij , (2.13)The contributions to the stress are the total pressure p = ρ n p n + ρ s p s − ρ s ν s | ω s | . (2.14)the viscous stress T vij = ρ n ν n ( ∇ i v j + ∇ j v i ) , (2.15)the vortex line tension, T sij = ρ s ν s | ω s | (ˆ ω si ˆ ω sj − δ ij ) , (2.16)and the stress arising from the displacement of vortices from the equilibrium configurationin the lattice, T tij = ρ s (cid:126) | ω s | m [ ∇ ⊥ i ξ j + ∇ ⊥ j ξ i − δ ij ( ∂ k ξ k )] . (2.17)The term (2.17) is responsible for Tkachenko oscillations.To study the coupled response of a superfluid and its container, we consider two infiniteparallel plates with separation 2 L . This geometry has a long history in the study of thespin up of rapidly rotating fluids in geophysics (Greenspan & Howard 1963; Pedlosky1967; Walin 1969), magnetized plasmas (Loper 1971; Easson & Pethick 1979; van Eysden2014) , condensed matter (Reisenegger 1993) and astrophysics (Abney & Epstein 1996).At times t <
0, the superfluid and its container rotate rigidly and uniformly about thecylindrical axis with angular velocity Ω. At time t = 0, the magnitude of the angularvelocity of the container is impulsively increased to Ω(1 + (cid:15) ), where the (cid:15) (cid:28) t > § I c d Ω c dt = − (cid:73) x × ( ˆ n · T ) dS + τ ext , (2.18)where I c is the moment of inertia of the container, ˆ n is the unit vector normal to theboundary and dS is an element of area on the boundary. The first term on the right handside of (2.18) is the hydrodynamic torque exerted on the crust by the fluid arising fromviscous stresses and stress exerted by the vortex array, where T is given by (2.12). Thesecond term is an external torque which may arise from, e.g., friction in the apparatusfor superfluid experiments or the magnetic dipole torque in pulsars.The normal fluid co-rotates with the container, giving the boundary condition v n = Ω c × x , (2.19) C. A. van Eysden where x is the radial vector and Ω c ( t ) is the angular velocity of the container, which isa function of time. Equation (2.19) embodies the usual no-slip boundary conditions forviscous flows. Following Reisenegger (1993), for the superfluid we choose the followingboundary conditionsˆ n × [ ρ s | ω s | Lγ ( v L − Ω c × x ) ± ( ˆ n · T )] = 0 , ˆ n · v s = 0 , (2.20)on surfaces that intersect vortex lines, where ˆ n is the unit vector normal to the surface.The dimensionless constant γ governs the rate of vortex creep at the boundary. In thelimit γ → ∞ , the vortices are pinned to the boundary and when γ = 0, the vortices exertno stress on the boundary, i.e., they are freely sliding. For the normal component of thesuperfluid we require no-penetration.In cylindrical coordinates ( r, φ, z ), the initial conditions are v n,s (0) = r Ω ˆ φ , Ω c (0) = Ω(1 + (cid:15) )ˆ z , (2.21)i.e., we assume the two fluids are initially co-rotating. Strictly speaking, the initial ve-locity for the fluid components obey (2.21) everywhere except in an infinitely thin regionadjacent to the boundary where it is spun up by the container.For (cid:15) (cid:28)
1, equations (2.1)–(2.21) can be linearized by perturbing around an equilib-rium rotating with uniform angular velocity Ω about the z -axis. The external torque isalso taken to be aligned with the rotation axis, i.e., τ ext = τ ext ˆ z . The geometry andinitial conditions are axisymmetric, and the resulting flow axisymmetric. The followingsubstitutions are made for the velocity and pressure fields v n,s ( r, z, t ) → r Ω ˆ φ + (cid:15) Ω L (cid:20) r ∗ ∂χ n,s ∂z ∗ ˆ r + r ∗ V n,s ˆ φ − χ n,s ˆ z (cid:21) , v L ( r, z, t ) → r Ω ˆ φ + (cid:15) Ω Lr ∗ (cid:20) ∂U ξ ∂t ˆ r + ∂V ξ ∂t ˆ φ (cid:21) ,p n,s ( r, z, t ) → ( r Ω) (cid:15) Ω L (cid:18) r ∗ P n,s Q n,s (cid:19) , (2.22)where χ n,s ( z ∗ , t ∗ ), V n,s ( z ∗ , t ∗ ), Q n,s ( z ∗ , t ∗ ) and P n,s ( t ∗ ) are all dimensionless quantities.The functions rχ n,s are stream-functions for the secondary flow. The asterisked quantitiesare defined as r ∗ = r/L , z ∗ = z/L , t ∗ = Ω t . Equation (2.22) is essentially the “von-Karman similarity” form and is typically used in employed in studies of spin-up betweenparallel plates (Greenspan & Howard 1963; Easson 1979; Reisenegger 1993; van Eysden& Melatos 2013). 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 radialdependence 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 inthe most general way. Under these assumptions, the only non-vanishing component ofthe external torque is in the ˆ z direction, hence Ω c ( t ∗ ) → Ωˆ z + (cid:15) Ω f ( t ∗ )ˆ z , (2.23)where f is a function of t ∗ only. Henceforth, we drop the asterisk notation so that allvariables are dimensionless.Substituting (2.22) into the normal fluid momentum equation (2.1), we obtain0 = (cid:18) ∂∂t − E ∂ ∂z + ρ s Bρ (cid:19) ∂χ n ∂z − (cid:18) − ρ s B (cid:48) ρ (cid:19) V n scillatory superfluid Ekman pumping − ρ s B ρ (cid:18) − E s ∂ ∂z (cid:19) ∂χ s ∂z − ρ s B (cid:48) ρ (cid:18) − E s ∂ ∂z (cid:19) V s + P n , (2.24)0 = (cid:18) ∂∂t − E ∂ ∂z + ρ s Bρ (cid:19) V n + (cid:18) − ρ s B (cid:48) ρ (cid:19) ∂χ n ∂z − ρ s B ρ (cid:18) − E s ∂ ∂z (cid:19) V s + ρ s B (cid:48) ρ (cid:18) − E s ∂ ∂z (cid:19) ∂χ s ∂z , (2.25)0 = (cid:18) ∂∂t − E ∂ ∂z (cid:19) χ n − ∂Q n ∂z , (2.26)and from (2.2) we have for the superfluid0 = (cid:20) ∂∂t + ρ n B ρ (cid:18) − E s ∂ ∂z (cid:19)(cid:21) ∂χ s ∂z − (cid:18) − ρ n B (cid:48) ρ (cid:19) (cid:18) − E s ∂ ∂z (cid:19) V s − ρ n Bρ ∂χ n ∂z − ρ n B (cid:48) ρ V n + P s , (2.27)0 = (cid:20) ∂∂t + ρ n B ρ (cid:18) − E s ∂ ∂z (cid:19)(cid:21) V s + (cid:18) − ρ n B (cid:48) ρ (cid:19) (cid:18) − E s ∂ ∂z (cid:19) ∂χ s ∂z − ρ n Bρ V n + ρ n B (cid:48) ρ ∂χ n ∂z , (2.28)0 = ∂χ s ∂t − ∂Q s ∂z , (2.29)where we define the dimensionless parameters E = ν n L Ω , E s = ν s L Ω . (2.30)The dimensionless parameter E is the Ekman number and is a ratio of viscous forces andthe 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 dfdt = ∓ Kρ (cid:18) ρ s E s ∂ χ s ∂z + ρ n E ∂V n ∂z (cid:19) + α (1 + K ) , (2.31)at z = ±
1, where we define K = πρR LI c , α = τ ext (cid:15) Ω I tot . (2.32)In (2.31), the dimensionless parameter K denotes the ratio of the moments of inertia ofthe fluid and container, and α 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 V n − rf = 0 , (2.33) ∂χ n ∂z = 0 , (2.34) χ n = 0 , (2.35)at z = ±
1. For the superfluid, using the linearized forms of (2.11) and (2.2) to eliminate
C. A. van Eysden the vortex line velocity and mutual friction, (2.20) becomes ∂V s ∂t ± E s γ ∂V s ∂z = 0 , (2.36) ∂∂t ∂χ s ∂z − f ± E s γ ∂ χ s ∂z + P s = 0 , (2.37) χ s = 0 , (2.38)at z = ±
1. The initial conditions (2.21) become V n ( z,
0) = 0 , (2.39) V s ( z,
0) = 0 , (2.40) f (0) = 1 . (2.41)The vortex line displacements, ξ , can be calculated by substituting (2.22) into the vortexline equation of motion (2.11) and using (2.2). We find2 ∂V ξ ∂t = ∂∂t ∂χ s ∂z + P s , (2.42)2 ∂U ξ ∂t = − ∂V s ∂t . (2.43)For the initial conditions we require U ξ ( z,
0) = 0 , (2.44) V ξ ( z,
0) = 0 . (2.45)
3. Pure superfluid
Exact solution
To illustrate the Ekman pumping mechanism which is the principal result of this paper,we consider a superfluid at T = 0, where ρ n = 0. To solve the initial value problem, wetake the Laplace transform, ˜ X ( z, s ) = (cid:90) ∞ X ( z, t ) e − st d t . (3.1)Equations (2.27) and (2.28) are s ∂ ˜ χ s ∂z − (cid:18) − E s ∂ ∂z (cid:19) ˜ V s + ˜ P s = 0 , (3.2) s ˜ V s + (cid:18) − E s ∂ ∂z (cid:19) ∂ ˜ χ s ∂z = 0 . (3.3)Equation (2.29) is not required to solve the system, and is only necessary if one desiresto calculate Q s for completeness. Taking the derivative of (3.2) we obtain s ∂ ˜ χ s ∂z − (cid:18) − E s ∂ ∂z (cid:19) ∂ ˜ V s ∂z = 0 . (3.4)Equations (3.3) and (3.4) can be solved for ˜ χ s and ˜ V s ; ˜ P s is then obtained from (3.2).The equation for the container, (2.31) becomes s ˜ f − ∓ KE s (cid:18) ∂ ˜ χ s ∂z (cid:19) , (3.5) scillatory superfluid Ekman pumping z = ±
1, and the boundary conditions are s ˜ V s ± E s γ ∂ ˜ V s ∂z = 0 , (3.6) s ∂ ˜ χ s ∂z − f ± E s γ ∂ ˜ χ s ∂z + ˜ P s = 0 , (3.7)˜ χ s = 0 , (3.8)at z = ±
1. The solution to (3.3)–(3.8) is˜ V s = 2 i ˜ f ∆ s (cid:8) C + (cid:2) κ − s (cosh κ − − cosh κ − z ) + ( E s /γ ) κ − sinh κ − (cid:3) , − C − (cid:2) κ + s (cosh κ + − cosh κ + z ) + ( E s /γ ) κ sinh κ + (cid:3)(cid:9) , (3.9)˜ χ s = 2 ˜ f ∆ s [ C + (sinh κ − z − z sinh κ − ) + C − (sinh κ + z − z sinh κ + )] , (3.10)˜ P s = 2 ˜ f ∆ s (cid:0) s (cid:1) ( C + sinh κ − + C − sinh κ + ) , (3.11)˜ f = ∆¯∆ s , (3.12)where κ ± = (cid:114) ± isE s , (3.13) C ± = κ ± cosh κ ± + (cid:0) E s κ ± ∓ iγ (cid:1) sinh κ ± γs , (3.14)∆ = C + (cid:20) C − + 2 E s κ − sinh κ − s (cid:21) + C − (cid:20) C + + 2 E s κ sinh κ + s (cid:21) , (3.15)¯∆ = C + (cid:20) C − + 2 E s κ − (1 + K ) sinh κ − s (cid:21) + C − (cid:20) C + + 2 E s κ (1 + K ) sinh κ + s (cid:21) . (3.16)To obtain the response of the container, we must find the inverse Laplace transform of˜ f . This is readily done by realizing that there are simple poles at s = 0 and at the zeroesof ¯∆. The result is f ( t ) = (cid:20) ∆¯∆ (cid:21) s =0 + (cid:88) n R ( s n ) e s n t , (3.17)where s n are the roots of ¯∆ and R ( s ) = ∆( s ) s d ¯∆ ds . (3.18)The sum in (3.17) implies summing over all the zeroes of ¯∆. † In general, the eigenvaluesoccur in conjugate pairs. For γ → ∞ we have Re( s n ) = 0 and Im[ R ( s n )] = 0, andthe modes in (3.17) are purely oscillatory. When γ is finite the friction force results indissipation and s n and R ( s n ) have real and imaginary components. † The solution to the initial value problem and could also be obtained from a mode expansionand appropriate orthogonality relations, however, the Laplace transform more readily gives theresult. C. A. van Eysden E s (cid:61) s (cid:61) f (cid:72) t (cid:76) Figure 1.
Dimensionless angular velocity of the container, f ( t ), for E s = 0 . E s = 0 .
01 (purple curve). We take K = 1 and γ → ∞ . (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) I m (cid:72) s n (cid:76) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) R e (cid:64) R (cid:72) s n (cid:76) (cid:68) Figure 2.
Eigenfrequencies (left panel) and amplitudes (right panel) of the modes present inFigure 1. Blue circles correspond to E s = 0 .
1, purple squares correspond to E s = 0 . Before proceeding, let us consider the relevant range for E s in hydrodynamic approxi-mation of the HVBK equations. The vorticity of the superfluid is given by the circulationper vortex multiplied by the number of vortices per unit area, hence2Ω = Γ × NπR (3.19)From (2.8), (2.30) and (3.19) we then obtain E s = R L N ln (cid:18) b a (cid:19) . (3.20)The ln term is of order unity or an order of magnitude more, so that for a vessel ofaspect ratio close to unity we have E s ∼ /N . We require N (cid:29) E s (cid:28) E s = 0 . E s = 0 .
01 (purple curve) for γ → ∞ (perfect pinning) and K = 1 in both cases. The scillatory superfluid Ekman pumping s n ). On the right hand side weplot the corresponding amplitudes of each mode. We see from the right panel that thedominant contribution is from the fundamental mode, with very little contribution fromthe higher modes. As E s is reduces, we find that all the power becomes concentrated inthe fundamental mode. Therefore, for E s (cid:28)
1, the impulsive acceleration of the containeronly excites the fundamental mode of the system.To obtain an analytic result for the fundamental mode, we investigate the limit E s (cid:28) s (cid:28)
1. We then find¯∆ ≈ e k s γ (2 + kγs ) (cid:2) K ) γ + 2 s + kγs (cid:3) , (3.21)where k = (cid:112) /E s (cid:29)
1. Equation (3.21) has zeroes at s ± = − kγ (cid:104) ± (cid:112) − K ) kγ (cid:105) . (3.22)The zero at s = − / ( kγ ) is also a zero of ∆, and therefore not a pole of ˜ f . Evaluating R ( s ± ) we find f ( t ) = 11 + K (cid:20) (cid:18) Ks + − s − (cid:19) (cid:0) s + e s − t − s − e s + t (cid:1)(cid:21) . (3.23)We now examine the behavior of (3.23) by considering two limiting cases. To recoverthe oscillatory solution observed in Figure 1, we consider the strong pinning limit γ (cid:29) [4 (1 + K ) k ] − / . From (3.22) we have s ± = ± i (cid:114) Kk , (3.24)and (3.23) becomes f ( t ) = 11 + K (cid:34) K cos (cid:32) (cid:114) Kk t (cid:33)(cid:35) . (3.25)We therefore obtain the period of oscillation t P = 2 π Ω | Im( s ± ) | = π Ω (cid:114) k K . (3.26)This is the principal result of this paper. A single, long time-scale mode has emergedfor E s (cid:28)
1, which has a period proportional to E − / s . This is analogous to classicalEkman pumping, where the Ekman time (proportional to E − / ) emerges for E (cid:28) E s = 0 .
01 we find Ω t P = 8 .
35 in agreement withFigure 1.The weak pinning limit, γ (cid:28) [4 (1 + K ) k ] − / , (3.22) gives s + = − kγ , s − = − K ) γ . (3.27)2 C. A. van Eysden
Because s − (cid:28) s + in this limit, (3.23) becomes f = 11 + K (cid:0) Ke s − t (cid:1) . (3.28)This exponential decay is a result of the friction force of the vortex lines as they slide onthe boundaries. When K = 0 the timescale for the relaxation is t R = 1Ω | s − | = 12Ω γ . (3.29)which is the result obtained in Reisenegger (1993). Because the weak pinning limit wasstudied by Reisenegger (1993), the principal focus of this paper will be on our new resultin the perfect pinning limit. 3.2. Ekman pumping
To visualize the flow for E s (cid:28) V s = V Is + V Bs , (3.30) χ s = χ Is + χ Bs . (3.31)where we have separated the solution into the interior flow and boundary layer corrections(denoted with subscripts I and B respectively) given by, V Is = 11 + K (cid:20) − (cid:18) s + e s − t − s − e s + t s + − s − (cid:19)(cid:21) (3.32) V Bs = 11 + K (cid:18) s + e s + t − s − e s − t s + − s − (cid:19) (cid:104) e k ( z − − e − k ( z +1) (cid:105) , (3.33) χ Is = 2 z ( e s − t − e s + t ) k ( s + − s − ) , (3.34) χ Bs = − e s − t − e s + t ) k ( s + − s − ) (cid:104) e k ( z − − e − k ( z +1) (cid:105) , (3.35)There is no boundary layer component for the pressure, which is given by P s = 21 + K (cid:20) − (cid:18) s + e s − t − s − e s + t s + − s − (cid:19)(cid:21) . (3.36)Equations (3.32)–(3.36) are akin to Ekman pumping in a viscous fluid (Greenspan& Howard 1963; Benton & Clark 1974; van Eysden & Melatos 2013), where the expo-nentially decaying term has been replaced with the oscillatory e s ± t terms. The variable χ s is like a stream-function for a secondary flow [see (2.22)] which draws fluid radiallyinwards in the interior, delivers it to the boundary layer where it is pumped radiallyoutwards. However, when there is perfect pinning, the flow is oscillatory, with periodgiven by (3.26). The Ekman circulation initially proceeds in the classical manner, clock-wise in the region 0 < z < − < z <
0, but isperiodically 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 andthen reversed. As there is no dissipation in the system, this energy exchange continuesindefinitely. Because the flow obeys the Taylor-Proudman theorem to leading order, theinterior azimuthal flow is columnar, as in classical Ekman pumping. The vortices in theinterior remain straight and move inwards and outwards with the secondary radial flow,increasing and decreasing the angular momentum of the fluid. scillatory superfluid Ekman pumping f (cid:72) t (cid:76) V sI (cid:72) t (cid:76) (cid:182) Χ sI (cid:72) t (cid:76) (cid:182) z U Ξ I (cid:72) t (cid:76) (cid:45) (cid:45) Figure 3.
Dimensionless angular velocity f ( t ) (blue curve), azimuthal angular velocity, V Is ( t )(purple curve) and radial velocity (divided by r ) ∂χ Is /∂z (red curve) and the radial vortex linedisplacement U ξ (brown curve). We take E s = 0 . K = 1 and γ → ∞ . Angular momentum isexchanged between the azimuthal flow and the container (blue and purple curves). The radialflow (red curve) is π/ In Figure 3 we plot the dimensionless angular velocity of the container, f ( t ) (bluecurve), angular velocity of the fluid, V Is ( t ) (purple curve) and radial velocity of the fluid(divided by r ), ∂χ Is /∂z (red curve) and the radial vortex line displacement U ξ (browncurve) for E s = 0 . K = 1 and γ → ∞ . The top two curves (blue and purple) show thatthe angular velocity of the container and the interior fluid are sinusoidal and π radiansout of phase. Therefore angular momentum is conserved between fluid and container, asrequired. The bottom curves show that the radial component of the secondary flow andthe radial vortex line displacement are π/ t = 2(left panel) and t = 4 (right panel) The magnitude of the fluid angular velocity V s is shownin color, where dark is zero and light is unity. The radial vortex line displacement U ξ isover-plotted as thick-black lines, scaled appropriately for illustrative purposes. Contoursfor the stream-function of the secondary flow, χ s are plotted as dashed lines for valuesof ± t = 2) the secondary flow is neara maximum, represented by the large density of contours. The flow is clockwise in theupper-right quadrant and the secondary flow is moving the vortex lines inwards in theinterior. The angular velocity of the container and azimuthal flow are matched, so thereis no boundary layer correction. In the right panel ( t = 4), the secondary flow is neara minimum. The radial component of the secondary flow is turning from inwards tooutwards in the interior. The vortex lines are at their maximum extension, and willsubsequently straighten to drive the outward radial flow in the interior. The difference4 C. A. van Eysden
Figure 4.
Flow plots for E s = 0 .
01. Left: t = 2 (near secondary flow maximum), right: t = 4(near secondary flow maximum). The color shading represents the magntude of the azimuthalangular velocity, V Is ( t ), where blue zero and white is unity (c.f. Figure 3). The dashed con-tours show streamlines for the secondary flow χ Is ( t ) with values: ± U Iξ , plotted with exaggerated amplitude forvisibility. between the angular velocities of the container and fluid is greatest, and the boundarylayer in the azimuthal velocity is prominent.To make a final connection to Ekman pumping, we examine the governing equations inthe limit E s (cid:28)
1. In the interior, we find from (3.32) that the azimuthal velocity evolveson the timescale E − / s Ω − and from (3.34) the stream-function for the secondary flow, χ s , scales as E / s . Applying these scalings, we find in exactly the same manner as forclassical Ekman pumping [see Greenspan & Howard (1963), §
5] the solutions for theinterior flow satisfy 0 = 2 V Is − P s , (3.37)0 = ∂V Is ∂t + 2 ∂χ Is ∂z . (3.38)The second, (3.38) describes the geostrophic balance in the interior. In the boundarylayer we have 0 = (cid:18) − E s ∂ ∂z (cid:19) V Bs , (3.39)0 = (cid:18) − E s ∂ ∂z (cid:19) ∂χ Bs ∂z . (3.40)The boundary conditions are given by (2.36)–(2.38), where (2.37) is replaced by ∂∂t ∂χ Bs ∂z − f ± E s γ ∂ χ Bs ∂z + P s = 0 , (3.41)at z = ±
1. Finally, the initial conditions are V Is (0) = 0 , (3.42) scillatory superfluid Ekman pumping f (0) = 0 , (3.43) χ Is (0) = 0 , (3.44)Equations (3.37)–(3.44) are sufficient to recover the approximate solution (3.30)–(3.36).They are the equivalent of the boundary layer approximation derived for classic spin-upby Greenspan & Howard (1963). They will be used in § (cid:18) √ ν s ∂∂t ∓ Lγ (cid:19) ˆ k · v I = ˆ k · ∇ × (cid:0) v I − v B (cid:1) , (3.45)at z = ±
1, where v I denotes the velocity field in the interior. This relation applies atthe wall in cylindrical geometry and applies only to a single fluid.
4. Helium II
Experiments
The response of a superfluid-filled container following an impulsive acceleration has beeninvestigated in series of experiments conducted by Tsakadze & Tsakadze (1980). In § .
52 K with a rotational frequency of 3 rad s − . The vessel is 64 mmin diameter, 50 mm in height and 0 . . − . The response comprises anexponential decay of 0 . − over approximately 40 s, followed by steady oscillationwith a period of 40 s, decaying over a timescale of roughly 480 s. The oscillation period alsoappears to decrease with time, probably a result of non-linear effects as the experimenthas Rossby number (cid:15) ∼ (3 . − / . § § §
2. For the experiment described above, the dimensionless6
C. A. van Eysden parameters at 1 .
52 K are (Donnelly & Barenghi 1998) E = 4 . × − (cid:18) ν n . × − cm s − (cid:19) (cid:18) Ω3 rad s − (cid:19) − (cid:18) L . (cid:19) − ,E s = 5 . × − (cid:18) ν s . × − cm s − (cid:19) (cid:18) Ω3 rad s − (cid:19) − (cid:18) L . (cid:19) − , Ma = 3 . × − (cid:18) u . × cm s − (cid:19) − (cid:18) Ω3 rad s − (cid:19) (cid:18) L . (cid:19) ,B = 1 . ,B (cid:48) = 0 . ,ρ n = 0 . ρ . (4.1)The third parameter, Ma, where u is the first sound velocity, is the Mach number andindicates the relative importance of compressibility effects. Clearly for helium II theincompressibility approximation is valid. For the experiment described above, all theparameters of our theory are given by (4.1) except K . In experiments involving spheresand no pinning, Tsakadze & Tsakadze (1980) present the steady-state spin down of emptyand filled vessels from which one can determine K ∼ . K for this experiment, and we take K = 1.4.2. Two-fluid solution
Solving the full system of equations in § E (cid:28) E s (cid:28) E s ∼ E , so that neither viscosity or vortextension is negligible. Also, because B, B (cid:48) ∼
1, the mutual friction coupling time betweenthe two fluid components is of the order of the rotation period, and the normal fluid andsuperfluid are approximately locked together over the Ekman time (Reisenegger 1993;van Eysden & Melatos 2013, 2014).To solve the two-fluid system we use a boundary layer approximation like that used in § § z = 0 plane, and hence χ n,s are antisymmetric. In theinterior we have0 = 2 V In − ρ s B (cid:48) ρ (cid:0) V In − V Is (cid:1) − P n , (4.2)0 = ∂V In ∂t + 2 ∂χ In ∂z + ρ s Bρ (cid:0) V In − V Is (cid:1) − ρ s B (cid:48) ρ (cid:18) ∂χ In ∂z − ∂χ Is ∂z (cid:19) , (4.3)0 = 2 V Is + ρ n B (cid:48) ρ (cid:0) V In − V Is (cid:1) − P s , (4.4)0 = ∂V Is ∂t + 2 ∂χ Is ∂z − ρ n Bρ (cid:0) V In − V Is (cid:1) + ρ n B (cid:48) ρ (cid:18) ∂χ In ∂z − ∂χ Is ∂z (cid:19) , (4.5)while in the upper boundary layer we have0 = (cid:18) − E ∂ ∂z + ρ s Bρ (cid:19) ∂χ Bn ∂z − (cid:18) − ρ s B (cid:48) ρ (cid:19) V Bn − ρ s B ρ (cid:18) − E s ∂ ∂z (cid:19) ∂χ Bs ∂z − ρ s B (cid:48) ρ (cid:18) − E s ∂ ∂z (cid:19) V Bs , (4.6) scillatory superfluid Ekman pumping
170 = (cid:18) − E ∂ ∂z + ρ s Bρ (cid:19) V Bn + (cid:18) − ρ s B (cid:48) ρ (cid:19) ∂χ Bn ∂z − ρ s B ρ (cid:18) − E s ∂ ∂z (cid:19) V Bs + ρ s B (cid:48) ρ (cid:18) − E s ∂ ∂z (cid:19) ∂χ Bs ∂z , (4.7)0 = ρ n B ρ (cid:18) − E s ∂ ∂z (cid:19) ∂χ Bs ∂z − ρ n Bρ ∂χ Bn ∂z − (cid:18) − ρ n B (cid:48) ρ (cid:19) (cid:18) − E s ∂ ∂z (cid:19) V Bs − ρ n B (cid:48) ρ V Bn , (4.8)0 = ρ n B ρ (cid:18) − E s ∂ ∂z (cid:19) V Bs − ρ n Bρ V Bn + (cid:18) − ρ n B (cid:48) ρ (cid:19) (cid:18) − E s ∂ ∂z (cid:19) ∂χ Bs ∂z + ρ n B (cid:48) ρ ∂χ Bn ∂z . (4.9)These equations reduce to (5.3)–(5.10) in Greenspan & Howard (1963) and (3.37)–(3.40)in the present paper when B = B (cid:48) = 0.The governing equation for the motion of the container is (2.31). The boundary con-ditions for the superfluid are (2.36), (3.41) and (2.38). For the normal fluid, the requiredboundary conditions are [Greenspan & Howard (1963), § V n − f = 0 , (4.10) ∂χ Bn ∂z = 0 , (4.11) χ n = 0 . (4.12)The initial conditions are (3.42)–(3.44) for the superfluid and V In (0) = 0 , (4.13)for the normal fluid. The general solution is presented in § A. Because of its complexity,we do not present the final result in algebraic form. For the numbers (4.1) and taking K = 1, γ → ∞ and Ω = 3 rad s − , we obtain f ( t ) = 0 . . e − . t + e − . t [0 . . t ) + 0 . . t )] , (4.14)where t is in seconds. The first term is the steady-state term, which corresponds tothe centre-of-mass of the system at 1 / (1 + K ). The second term corresponds to a rapiddamping and has low amplitude. This term represents the strong mutual friction betweenthe two components. The third term is an oscillation, weakly damped by viscosity. It hasa period of 10 . . E / in clas-sical Ekman pumping, but scale as E / s for the oscillatory superfluid Ekman pumpingmechanism identified in this paper. For the numbers in (4.1), we find E / s = 0 . , E / = 0 . . (4.15)Therefore the superfluid mechanism is dominant. Neglecting viscosity ( E = 0) and as-suming strong mutual friction coupling ( B, B (cid:48) ∼ § B. The result has the form (3.23), with s ± now8 C. A. van Eysden (cid:72) s (cid:76) f (cid:72) t (cid:76) Figure 5.
Comparison of the general solution [(4.14), thick black curve] and the approximatesolution [(3.23) and (B 12), thin gray curve] assuming zero viscosity ( E = 0) and strong mutualfriction couping ( B, B (cid:48) ∼ given by s ± = − kγ (cid:34) ± (cid:115) − K ) kγ ρ s ρ (cid:35) . (4.16)The oscillatory Ekman pumping mechanism identified in § ρ s /ρ in the spin-up time. Only theleading order normal fluid velocity is involved in the oscillation; there is no secondaryflow in the normal fluid. The reader is referred to § B.2 for the solution for all variablesin this limit.A comparison between the general solution (4.14) and the approximate solution [(3.23)and (B 12)] assuming E = 0 and B, B (cid:48) ∼ γ → ∞ , the period of oscillationat T = 1 .
52 K is t P = 10 . (cid:18) − Ω (cid:19) − (cid:18) . × − E s (cid:19) − / (cid:18) . ρ s /ρ (cid:19) − / (cid:18)
21 + K (cid:19) − / . (4.17)This is shorter than the maximum observed period of oscillation in the Tzakadze exper-iments, which decays from an initial maximum of approximately 40 s. The period can belengthened by decreasing the frictional pinning force (decreasing γ ), however, this quicklyshortens the damping time to much less than that observed in the experiments. Becauseof the precision to which the parameters in (4.1) have been measured, the discrepancymust arise either because the linear hydrodynamic theory is invalid, or one or more of theexperimental parameters have been erroneously reported. The former is certainly likely;as discussed in § .
2, so non-linear effects may be present. Theobserved decay of the oscillation period cannot be captured by the linear theory. The im- scillatory superfluid Ekman pumping
Tkachenko oscillations
In our calculations, we have solved the complete equations of Chandler & Baym (1986),in order to apply the most sophisticated hydrodynamic theory available to the helium IIexperiments. 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 obtainedby Reisenegger (1993). The reason for this lies in the ansatz (2.21), where the flow isassumed to be axisymmetric and the radial dependence of azimuthal velocity matchesthe boundary conditions. The equation of motion for the vortex lines requires that ξ hasthe same radial dependence as the superfluid velocity, and upon substituting (2.21) into(2.9), we find that σ = 0.Physically, (2.9) describes the restoring force experienced by the vortex array as aresult of displacements of vortices from their equilibrium configuration. Hence, the vor-tex array experiences a restoring force in response to shear deformations, giving rise toTkachenko oscillations. However, in the spin-up problem considered here, the top andbottom boundaries are rigid bodies and the impulsive acceleration of the boundary doesnot displace any vortices from their equilibrium configuration. Therefore we do not neces-sarily expect Tkachenko modes to be excited in the experiments of Tsakadze & Tsakadze(1980).Tkachenko modes may be excited at the cylinder wall, which has been neglected in ouranalysis. However the required boundary conditions required at the cylinder wall are notdiscussed by Baym & Chandler (1983). It is known that it is energetically favorable forvortex-free region to form adjacent to the cylinder wall, making it difficult for changesin the angular velocity of the side-wall to be communicated to vortex array. Even if theywere excited, Tkachenko oscillations on timescales of order E − s Ω − , much less than theEkman pumping mechanism described above, which has timescale E − / s Ω − . Thereforethe motion of the vortex lines is determined by the vortex tension producing Ekmanpumping, and the restoring force from the lattice deformations is too slow to have anyeffect.Therefore, Tkachenko oscillations are unlikely to be excited in the experiments ofTsakadze & Tsakadze (1980). Tkachenko oscillations have been imaged in rapidly-rotatingdilute-gas Bose-Einstein condensates, excited by blasting atoms or creating a dip in thetrapping potential at the centre of the condensate (Coddington et al. C. A. van Eysden
5. Neutron stars
Neutron stars are compact stellar corpses formed by the core collapse of a massive starafter a supernova. The resulting compact object typically has a radius of 12-14 km anda mass of 1.4-2 times that of the sun. The outer kilometer of the star is a rigid, highlyconducting crustal lattice interspersed with a neutron superfluid, while the core comprisesapproximately 95% superfluid neutrons, 5% electrons and superconducting protons, anda small fraction of muons. In the deep core, the composition remains uncertain.Neutron stars are also endowed with strong magnetic fields, ranging from 10 –10 Gauss. In isolated radio pulsars, radio waves are beamed along the magnetic axis andcreate a lighthouse effect that is observed as pulsations on Earth. Occasionally, theseobjects are observed to ‘glitch’, where the rotational frequency of the star suddenlyincreases by up to one part in 10 (Espinoza et al. et al. ∼
1% of the total moment of inertia) (Link et al. et al. et al. et al. et al. b ). This recoveryis typically dominated by a step increase in the rotational frequency of the star, a stepchange in the frequency derivative, and a quasi-exponential relaxation (van Eysden &Melatos 2010). However, quasi-exponential oscillations have also been reported followingsome glitches. A post-glitch oscillation with a period of approximately four months wasreported in the Crab pulsar soon after its discovery (Ruderman 1970 a , b ). The 1975 and1986 Crab glitches were observed to “overshoot” their final rotation frequency duringtheir recovery (Wong et al. et al. (1990)], which, if real, appear to have aperiod of ∼
20 days. These oscillations have received little attention observationally ortheoretically, but may shed light on the interior of neutron stars. Here we consider thepossibility that the superfluid oscillations described in the previous sections may beoperating in neutron stars, and ask what the observable consequences are for glitchrecovery.During glitch recovery, the proton-electron plasma in the core is coupled to the highlyconducting crust via the magnetic field on a time-scale of seconds (Easson 1979; vanEysden 2014), and hence, these components are assumed to be rigidly locked together.The superfluid neutrons are usually assumed to respond changes in the angular velocityof the crust via the interaction of the neutron vortices with the proton-electron plasmain the core (Alpar et al. a , b ; Glampedakis et al. F = − ρρ n β ˆ ω s × [ ω s × ( v L − v n )] , (5.1)where the subscript n refers the the proton-electron plasma, which plays the role of the scillatory superfluid Ekman pumping s refers to the neutron superfluid. The coefficient β is givenby Mendell (1991 b ) β = 1 . × − ( y − x / y / (1 − x ) , (5.2)where x is the proton fraction ( ρ n /ρ in the notation used here) and y is the normalizedeffective mass of the proton from Fermi liquid theory. The coefficient β (cid:48) that appearsin (5.1) in HVBK theory is typically taken as zero in neutron stars. The motion ofmagnetized neutron vortices is also inhibited by their pinning to flux tubes. When avortex is pinned it moves with the normal fluid so that v L = v n . However, an unpinnedvortex moves with velocity given by (2.11). Solving the vortex line equation of motion(2.11) gives v L = v n −
11 + β { ˆ ω s × [ˆ ω s × ( v s − v n ) + σ + t ]+ β [ˆ ω s × ( v s − v n ) + σ + t ] } . (5.3)At zero temperature, all vortices are pinned to flux tubes. At finite temperature a vortexcan become thermally excited with activation energy A upon unpinning. The activationenergy for unpinning depends on the magnetic energy between a vortex and a flux tube E p , the dimensionless vortex tension T , the angular velocity lag ∆ ω and the criticalangular velocity lag for unpinning ∆ ω c as A = 5 . E p T / (cid:18) − ∆ ω ∆ ω c (cid:19) / . (5.4)Therefore the mutual friction force has a non-linear dependence on the velocity through∆ ω . However, to simplify our analysis we take A as constant, noting that a rigorouscalculation should include this non-linear dependence.The partition function for this two-state system is Z = 1 + e − βA , (5.5)where β = ( k B T ) − , k B is Boltzmann’s constant and T is the temperature. When thethermal energy is much greater than the activation energy for unpinning a vortex β − (cid:29) A , the 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 β − (cid:28) A , the vortex movement is exponentially suppressed pinning the vortices to thenormal fluid. For slow vortex slippage, e − βA (cid:28) Z ≈ (cid:104) v L (cid:105) β − v n = (cid:104) v L − v n (cid:105) β ≈ Z − e − βA ( v L − v n )= −
11 + β { ˆ ω s × [ˆ ω s × ( v s − v n ) + σ + t ]+ β [ˆ ω s × ( v s − v n ) + σ + t ] } e − βA . (5.6)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) withmodified mutual friction coefficients B = 2 ρρ n β β e − βA , (5.7) B (cid:48) = 2 ρρ n (cid:20) −
11 + β e − βA (cid:21) . (5.8)2 C. A. van Eysden
For a detailed derivation of the theory of vortex slippage, the reader is referred to Link(2014), where (5.6) and (5.8) are rigorously derived [see (48) and (51) in that reference].Before proceeding, we assess the validity of the incompressibility assumption for thepresent application. In neutron stars the speed of sound is approximately one fifth ofthe speed of light (see e.g., Reisenegger & Goldreich (1992)), giving a Mach number1 . × − . Therefore the incompressibility assumption in neutron stars is valid.We can now apply the theory in § v n = Ω c r . Therefore, it is a single-fluid problem involving only the superfluid and we can repeat the analysis in § γ → ∞ at z = ± − < z < § S state, while in the core they are expected to pair in a P state. The S phase is expected to be lost near the base of the crust, however current calculations of thepairing gaps for the P phase are not certain enough to know at what depth superfluiditywill be restored. If the two phases are connected, then the relevant boundary conditionat the interface are also unknown. It may be that the vortices in the crust and core arenot connected, in which case the superfluid oscillations described here will not occur.Repeating the analysis in § ≈ ke k ρs ( ρs + ρ n B ) { (2 ρ − ρ n B (cid:48) ) [2 Kρ s + (2 ρ − ρ n B (cid:48) )]+ ( ρs + ρ n B ) ( ρks + ρ n B ) } , (5.9)and the inverse Laplace transform is f ( t ) = 1(2 ρ − ρ n B (cid:48) ) + 2 Kρ s (cid:20) (2 ρ − ρ n B (cid:48) ) + (cid:18) Kρ s s + − s − (cid:19) (cid:0) s + e s − t − s − e s + t (cid:1)(cid:21) , (5.10)where s ± = − ρ n B ρ ± (cid:115) k ( ρ n B ) − ρ − ρ n B (cid:48) ) [2 Kρ s + (2 ρ − ρ n B (cid:48) )]4 ρ k . (5.11)In a typical neutron star, we have ρ n /ρ = x ∼ .
05, Ω = 100 rad s − , 0 . (cid:54) y = m ∗ p /m p (cid:54) . K ≈
50 (Mendell 1991 a , b ; van Eysden & Melatos 2010). The pinning fractionin the core is uncertain. Hence we have β (cid:28) Kρ s (cid:29) ρ becausethe neutron superfluid is the dominant contribution to the moment of inertia of the star.The superfluid Ekman number is approximately E s = 3 . × − (cid:18) ν s . × − cm s − (cid:19) (cid:18) Ω100 rad s − (cid:19) − (cid:18) L cm (cid:19) − , (5.12)Under these assumptions (5.10) can be written f ( t ) = e − t/t d (cid:20) cos (cid:18) πtt p (cid:19) + t p πt d sin (cid:18) πtt p (cid:19)(cid:21) , (5.13) scillatory superfluid Ekman pumping m p (cid:42) m p (cid:61) p (cid:42) m p (cid:61) yr10 yr10 yr Β A t d Β A t p Figure 6.
Damping time (left) and period (right) for oscillations in a neutron star. Lines areplotted for y = m ∗ p /m p = 0 . . m ∗ p and the lines overlap. where t d = 1Ω | Re( s ± ) | ≈ e βA Ω β , (5.14) t p = 2 π Ω | Im( s ± ) | ≈ πe βA/ Ω (cid:115) ρkKρ s . (5.15)In Figure 6 we plot the damping time and period for superfluid oscillations in a neutronstar as a function of the pinning parameter βA . We plot the full result for the dampingtime and period using (5.11). Results are plotted for y = m ∗ p /m p = 0 . . A . The oscillation amplitude is unaffected by pinning and always equal tothe glitch amplitude. The damping time-scale is always longer than the period, so theoscillations are only weakly damped and the second term in (5.13) is small. Post-glitchoscillations are observed in both the Crab and Vela pulsars that are candidates for theoscillatory Ekman pumping mechanism in this paper. In the Crab, oscillations with aperiod of roughly four months were reported following the earliest glitches (Ruderman1970 a , b ). The “overshoot” observed in the recovery the 1975 and 1986 Crab glitches couldalso correspond to a damped oscillation with period of 200-300 days (Wong et al. βA ≈ .
8, and for the latter βA ≈ .
8. In Vela, a damped periodic oscillation is clearly visible in the timing residualsfollowing the 1988 glitch [see Figure 2 of McCulloch et al. (1990)]. These oscillationsappear to have a period of ∼
20 days, requiring βA ≈ .
2. Therefore, superfluid Ekmanpumping can explain these oscillations if pinning between vortices and flux tubes isextremely strong, limiting vortices to a very slow creep.These numbers can be compared with theoretical expectations using known parame-ters in the outer core of the pulsar. Assuming a typical pulsar temperature and using4
C. A. van Eysden parameter estimates from Link (2014), from (5.4) we find βA = 2 . × (cid:18)
100 MeV E p (cid:19) / (cid:18) . T (cid:19) / (cid:18) K T (cid:19) − , (5.16)where we have taken ∆ ω = 0. Clearly (5.16) leads to extremely long oscillation periodsand damping times. However, this is a gross upper estimate because the lag ∆ ω is notlikely to be zero. If the lag is close to the critical lag for unpinning ∆ ω c , the estimate(5.16) can be greatly reduced, see (5.8). However, to correctly determine the lag thenon-linear calculation including velocity dependence in the mutual friction coefficients isrequired, which is beyond the scope of this paper.Finally, we remark on limitations of the present model. Neutron stars are stronglystratified, typically with buoyancy frequencies N exceeding the rotation rate (Reiseneg-ger & Goldreich 1992). Buoyancy inhibits Ekman pumping from penetrating the core,confining the circulation to a buoyancy layer with thickness Ω L/N (Walin 1969; Abney& Epstein 1996; van Eysden & Melatos 2008). The spin-up is time is reduced commensu-rately reduced to E − / N − because the effective size of the Ekman cell has been reducedby a fraction Ω /N . The final state is not co-rotation, but has persistent shear between thefluid in the buoyancy layer and that in the core which in untouched by Ekman pumping(Melatos 2012). In a typical neutron star, we find Ω /N ∼ .
20 (Reisenegger & 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 discussedin this paper. As in the classical stratified Ekman pumping, we expect that buoyancyforces will confine the circulation to a depth Ω
L/N . The Taylor-Proudman theorem nolonger applies in the core, and the vortex lines will be sheared. Above Ω
L/N the fluidundergoes in 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 Ekmancell is reduced from L to Ω L/N , and (3.26) becomes t P = π Ω (cid:115) Ω N (cid:18) k K (cid:19) . (5.17)In this case we have (cid:112) Ω /N ∼ .
45, shortening the period by a factor of approximately of2.2. Therefore the superfluid oscillations are more weakly affected by stratification thanclassical Ekman pumping.
6. Conclusions
The response of a uniformly rotating superfluid threaded with a large density of vor-tex lines to an impulsive response of its container has been investigated. When vorticesare strongly pinned to the container and there is no dissipation, the system oscillatespersistently with a period of order E − / Ω − , where E s is the dimensionless vortex linetension parameter. The low-frequency mode is generated by a secondary flow analogousto classical Ekman pumping, periodically reversed by the vortex tension. This secondaryflow transports vortices radially inwards and outwards in the interior, changing the an-gular momentum of the fluid. The full set of equations of Chandler & Baym (1986) havebeen solved and compared with experiments performed by Tsakadze & Tsakadze (1980)in superfluid helium. We find qualitative agreement between theory and experiment,however quantitative comparison is difficult because some experimental parameters areunknown, and the experiment may probe the non-linear regime, which is beyond thescope of this theory. The columnar nature of superfluid Ekman pumping also reproduces scillatory superfluid Ekman pumping 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 toolarge to present algebraically, so we outline the method and present key results. Becauseif the size of the equations, it is easier to perform the calculations in programs such asMathematica or Maple and handle them numerically.We assume the container is rigid so that the upper and lower boundaries are moveidentically. Therefore, the V n,s are symmetric in z and χ n,s are anti-symmetric. Applyingthese 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) V In ( t ) = (cid:88) i = i V Ini e s i t + V In , (A 1)and similarly for V s . Three time-scales exist in the problem and are obtained whenapplying the boundary conditions below. The stream-function for the secondary flow is χ In ( z, t ) = z (cid:34) (cid:88) i = i χ Ini e s i t + χ In (cid:35) , (A 2)and similarly for χ s . The pressure is P n ( t ) = (cid:88) i = i P ni e s i t + P n , (A 3)and similarly for P s . Substituting (A 1)–(A 3) into (4.2)–(4.5) allows to solve for thecoefficients of χ n,s and P n,s in terms of those of V In,s .In the boundary layer, the solutions to (4.6)–(4.9) have the form V Bn ( z, t ) = (cid:88) i =1 e s i t (cid:104) V Bni + e − k + (1 − z ) + V Bni − e − k − (1 − z ) (cid:105) , (A 4) V Bs ( z, t ) = (cid:88) i =1 e s i t (cid:104) V Bsi + e − k + (1 − z ) + V Bsi − e − k − (1 − z ) + V Bsi e − k (1 − z ) (cid:105) , (A 5)in the upper boundary layer, where k ± = (cid:115) ± iρ [ B ± i (2 − B (cid:48) )] E [ ρ n B ± i (2 ρ − ρ n B (cid:48) )] , (A 6) k = (cid:114) E s . (A 7)6 C. A. van Eysden
For χ Bn,s we have χ Bn ( z, t ) = (cid:88) i =1 e s i t (cid:104) χ Bni + e − k + (1 − z ) + χ Bni − e − k − (1 − z ) (cid:105) , (A 8) χ Bs ( z, t ) = (cid:88) i =1 e s i t (cid:104) χ Bsi + e − k + (1 − z ) + χ Bsi − e − k − (1 − z ) + χ Bsi e − k (1 − z ) (cid:105) . (A 9)Note that for the classical Ekman boundary layers (with exponents k ± ), the superfluidand normal fluid components are coupled, whereas, for the superfluid Ekman pumpingmechanism, the normal fluid is not involved. Substituting (A 4)–(A 9) into (4.6)–(4.9)and comparing like terms gives V Bsi ± , χ Bsi ± and χ Bni ± in terms of V Bni ± . The coefficients V Bsi and χ Bsi are unrelated to other boundary layer coefficients.The motion of the container has the solution f ( t ) = (cid:88) i = i f i e s i t + f , (A 10)where the f i can be determined in terms of V Bni ± , V Bsi and χsi B by substituting (A 4)–(A 9) into (2.31). Note that only the boundary layer corrections to the velocity contributeto the torque on the container.The coefficients V Bni ± , V Bsi and χ Bsi and V Isi and V Is can be solved for in terms of V Ini through application of the boundary conditions (2.36), (3.41) and (2.38) for thesuperfluid 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 Ini , the final equationis an eigenvalue equation for s i . The equation is cubic is s i giving solutions for the three s i . The boundary conditions also give f = V s = V n .The remaining four unknown coefficients V Ini and V n are determined by the initialconditions (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 equation(4.14). Appendix B. Solutions in the boundary layer approximation forstrong coupling
In helium II, we have
B, B (cid:48) ∼
1, resulting in the strong coupling between the twofluid components. Any differential rotation is removed over the rotational time-scaleand the two fluid-components are “locked together” over the much longer Ekman time(Reisenegger 1993; van Eysden & Melatos 2013). This assumption can be used to find amore analytically tractable solution to that presented in § A.When B = B (cid:48) = 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 / (Greenspan & Howard 1963) and in (4.5) all terms are order E / s (see § B, B (cid:48) ∼ V In = V Is and the velocitydifference δV I is of the order of the secondary flow, i.e., V Is = V In − δV I . (B 1) scillatory superfluid Ekman pumping V In − P n , (B 2)0 = ∂V In ∂t + 2 ∂χ In ∂z + ρ s Bρ δV I − ρ s B (cid:48) ρ (cid:18) ∂χ In ∂z − ∂χ Is ∂z (cid:19) , (B 3)0 = 2 V Is − P s , (B 4)0 = ∂V Is ∂t + 2 ∂χ Is ∂z − ρ n Bρ δV I + ρ n ρ B (cid:48) (cid:18) ∂χ In ∂z − ∂χ Is ∂z (cid:19) , (B 5)The solution for V In has the same form (A 1), and similarly for δV I we have δV I ( t ) = (cid:88) i = i δV Ii e s i t + δV I , (B 6)Using the same form for the stream-functions (A 2) and pressures (A 3), their coefficientscan be determined in terms of V niI and δV Ii using (B 2)–(B 5).The remaining procedure follows as in § A, however we find that there are now only twotime-scales s and s ; the third corresponded to mutual friction coupling which is noweffectively instantaneous. Accordingly, we can no longer impose separate initial conditionson V Is and V In , which must be the same. The resulting solutions are still algebraicallycumbersome, so we present results in the limit of negligible viscosity and negligible vortextension. B.1. Solution for negligible viscosity
A useful result discussed in § V Bn,si ± = 0 and χ Bn,si ± = 0. We also do not need tosatisfy the normal fluid boundary conditions for no slip (4.10) and (4.12). The result is f ( t ) = 11 + K (cid:20) K s e s t − s e s t s − s (cid:21) , (B 7) V In = 11 + K (cid:20) − s e s t − s e s t s − s (cid:21) , (B 8) δV I = 2 ( e s t − e s t ) βk ( s − s ) , (B 9) χ Is = 2 z ( e s t − e s t ) k ( s − s ) , (B 10) χ In = 0 , (B 11)where s , = − kγ (cid:34) ± (cid:115) − K ) γ kρ s ρ (cid:35) . (B 12)and β = B − B (cid:48) . (B 13)The implications of this result are discussed in § C. A. van Eysden
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 canrecover the previous results of Reisenegger (1993) and van Eysden & Melatos (2013,2014) where pinning is negligible.When γ →
0, we obtain the result f ( t ) = 11 + K (cid:0) Ke s t (cid:1) , (B 14) V In = 11 + K (cid:0) − e s t (cid:1) , (B 15) δV I = ρe s t (cid:0) k + k (cid:1) [( k + k ) + iβ ( k − k )]4 β k k ρ s , (B 16) χ In = ize s t ( k − k )2 k k , (B 17) χ Is = χ In − ρe s t (cid:0) k + k (cid:1) [( k + k ) + iβ ( k − k )]4 β k k ρ s , (B 18)where s = − E k + k ) (1 + K ) ρ n . (B 19)Interestingly the same result is obtained by taking E s = 0. This is just a statement thatwhen there is no vortex pinning, the vortex tension plays no active role in the spin-up, asdiscussed in van Eysden & Melatos (2013). There is a slight difference in the boundarylayer solution; for E s = 0, V Bs = χ Bs = 0, while for γ = 0 they are non-zero.From (B 19) we obtain the spin-up time t s = 1Ω | s | = ρρ n E / Ω (1 + K ) × ρ (cid:34) B + (2 − B (cid:48) ) ( ρ n B ) + (2 ρ − ρ n B (cid:48) ) (cid:35) / + 2 ρ s ρB ( ρ n B ) + (2 ρ − ρ n B (cid:48) ) − / . (B 20)This result was obtained by Reisenegger (1993) and explored in detail by van Eysden& Melatos (2013). The term in the curly braces arises from the coupling between thetwo fluid components is a boundary layer of order unity. The pre-factor is the familiarEkman time modified by the normal fluid density fraction. The two fluid components arestrongly coupled (i.e., they achieve co-rotation within a rotation period as B ∼ REFERENCESAbney, M. & Epstein, R. I.
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