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Oscillatory superfluid Ekman pumping in helium II and neutron stars

Published online by Cambridge University Press:  16 October 2015

C. Anthony van Eysden
C. Anthony van Eysden*
Affiliation:
Department of Physics, Montana State University, Bozeman, MT 59717, USA Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
*
†Email address for correspondence: anthonyvaneysden@montana.edu
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Abstract

The linear response of a superfluid, rotating uniformly in a cylindrical container and threaded with a large number of vortex lines, to an impulsive increase in the angular velocity of the container is investigated. At zero temperature and with perfect pinning of vortices to the top and bottom of the container, we demonstrate that the system oscillates persistently with a frequency proportional to the vortex line tension parameter to the quarter power. This low-frequency mode is generated by a secondary flow analogous to classical Ekman pumping that is periodically reversed by the vortex tension in the boundary layers. We compare analytic solutions to the two-fluid equations by Chandler & Baym (J. Low Temp. Phys., vol. 62, 1986, pp. 119–142) with the spin-up experiments by Tsakadze & Tsakadze (J. Low Temp. Phys., vol. 39, 1980, pp. 649–688) in helium II and find that the frequency agrees within a factor of four, although the experiment is not perfectly suited to the application of linear theory. We argue that this oscillatory Ekman pumping mode, and not Tkachenko modes, provides a natural explanation for the observed oscillation. In neutron stars, the oscillation period depends on the pinning interaction between neutron vortices and flux tubes in the outer core. Using a simplified pinning model, we demonstrate that strong pinning can accommodate modes with periods of days to years, which are only weakly damped by mutual friction over longer time scales.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Complex Fluids: Quantum fluids Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 783 , 25 November 2015 , pp. 251 - 282
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© 2015 Cambridge University Press 

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References

Abney, M. & Epstein, R. I. 1996 Ekman pumping in compact astrophysical bodies. J. Fluid Mech. 312, 327–340.Google Scholar
Adams, P. W., Cieplak, M. & Glaberson, W. I. 1985 Spin-up problem in superfluid $^{4}$ He. Phys. Rev. B 32, 171–177.Google Scholar
Alpar, M. A., Langer, S. A. & Sauls, J. A. 1984 Rapid postglitch spin-up of the superfluid core in pulsars. Astrophys. J. 282, 533–541.CrossRefGoogle Scholar
Amberg, G. & Ungarish, M. 1993 Spin-up from rest of a mixture: numerical simulation and asymptotic theory. J. Fluid Mech. 246, 443–464.Google Scholar
Anderson, P. W. & Itoh, N. 1975 Pulsar glitches and restlessness as a hard superfluidity phenomenon. Nature 256, 25–27.CrossRefGoogle Scholar
Andersson, N., Glampedakis, K., Ho, W. C. G. & Espinoza, C. M. 2012 Pulsar glitches: the crust is not enough. Phys. Rev. Lett. 109 (24), 241103.CrossRefGoogle ScholarPubMed
Baym, G. & Chandler, E. 1983 The hydrodynamics of rotating superfluids. I. Zero-temperature, nondissipative theory. J. Low Temp. Phys. 50, 57–87.CrossRefGoogle Scholar
Baym, G., Epstein, R. I. & Link, B. 1992 Dynamics of vortices in neutron stars. Physica B 178, 1–12.CrossRefGoogle Scholar
Baym, G., Pethick, C. & Pines, D. 1969a Spin up in neutron stars: the future of the Vela pulsar. Nature 224, 872–874.Google Scholar
Baym, G., Pethick, C. & Pines, D. 1969b Superfluidity in neutron stars. Nature 224, 673–674.Google Scholar
Benton, E. R. & Clark, A. 1974 Spin-up. Annu. Rev. Fluid Mech. 6, 257–280.Google Scholar
Campbell, L. J. & Krasnov, Y. K. 1982 Transient behavior of rotating superfluid helium. J. Low Temp. Phys. 49, 377–396.Google Scholar
Chamel, N. 2013 Crustal entrainment and pulsar glitches. Phys. Rev. Lett. 110 (1), 011101.Google Scholar
Chandler, E. & Baym, G. 1986 The hydrodynamics of rotating superfluids. II. Finite temperature, dissipative theory. J. Low Temp. Phys. 62, 119–142.Google Scholar
Clark, A., Clark, P. A., Thomas, J. H. & Lee, N.-H. 1971 Spin-up of a strongly stratified fluid in a sphere. J. Fluid Mech. 45, 131–149.Google Scholar
Coddington, I., Engels, P., Schweikhard, V. & Cornell, E. A. 2003 Observation of Tkachenko oscillations in rapidly rotating Bose–Einstein condensates. Phys. Rev. Lett. 91 (10), 100402.Google Scholar
Donnelly, R. J. & Barenghi, C. F. 1998 The observed properties of liquid helium at the saturated vapor pressure. J. Phys. Chem. Ref. Data 27, 1217–1274.CrossRefGoogle Scholar
Easson, I. 1979 Postglitch behavior of the plasma inside neutron stars. Astrophys. J. 228, 257–267.CrossRefGoogle Scholar
Easson, I. & Pethick, C. J. 1979 Magnetohydrodynamics of neutron star interiors. Astrophys. J. 227, 995–1012.Google Scholar
Espinoza, C. M., Lyne, A. G., Stappers, B. W. & Kramer, M. 2011 A study of 315 glitches in the rotation of 102 pulsars. Mon. Not. R. Astron. Soc. 414 (2), 1679–1704.CrossRefGoogle Scholar
van Eysden, C. A. 2014 Short-period pulsar oscillations following a glitch. Astrophys. J. 789, 142.Google Scholar
van Eysden, C. A. & Melatos, A. 2008 Gravitational radiation from pulsar glitches. Class. Quant. Grav. 25 (22), 225020.Google Scholar
van Eysden, C. A. & Melatos, A. 2010 Pulsar glitch recovery and the superfluidity coefficients of bulk nuclear matter. Mon. Not. R. Astron. Soc. 409, 1253–1268.CrossRefGoogle Scholar
van Eysden, C. A. & Melatos, A. 2011 Spin down of superfluid-filled vessels: theory versus experiment. J. Low Temp. Phys. 165, 1–14.Google Scholar
van Eysden, C. A. & Melatos, A. 2012 Interpreting superfluid spin up through the response of the container. J. Low Temp. Phys. 166, 151–170.Google Scholar
van Eysden, C. A. & Melatos, A. 2013 Spin-up of a two-component superfluid: analytic theory in arbitrary geometry. J. Fluid Mech. 729, 180–213.Google Scholar
van Eysden, C. A. & Melatos, A. 2014 Spin-up of a two-component superfluid: self-consistent container feedback. J. Fluid Mech. 744, 89–110.Google Scholar
Glampedakis, K., Andersson, N. & Samuelsson, L. 2011 Magnetohydrodynamics of superfluid and superconducting neutron star cores. Mon. Not. R. Astron. Soc. 410, 805–829.CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press.Google Scholar
Greenspan, H. P. & Howard, L. N. 1963 On a time-dependent motion of a rotating fluid. J. Fluid Mech. 17, 385–404.Google Scholar
Henderson, K. L. & Barenghi, C. F. 2000 The anomalous motion of superfluid helium in a rotating cavity. J. Fluid Mech. 406, 199–219.Google Scholar
Henderson, K. L., Barenghi, C. F. & Jones, C. A. 1995 Nonlinear Taylor–Couette flow of helium II. J. Fluid Mech. 283, 329–340.Google Scholar
Link, B. 2014 Thermally activated post-glitch response of the neutron star inner crust and core. I. Theory. Astrophys. J. 789 (2), 141 (18 pp).Google Scholar
Link, B., Epstein, R. I. & van Riper, K. A. 1992 Pulsar glitches as probes of neutron star interiors. Nature 359, 616–618.Google Scholar
Loper, D. E. 1971 Hydromagnetic spin-up of a fluid confined by two flat electrically conducting boundaries. J. Fluid Mech. 50, 609–623.Google Scholar
McCulloch, P. M., Hamilton, P. A., McConnell, D. & King, E. A. 1990 The VELA glitch of Christmas 1988. Nature 346, 822–824.Google Scholar
Melatos, A. 2012 Fast fossil rotation of neutron star cores. Astrophys. J. 761, 32.Google Scholar
Melatos, A., Peralta, C. & Wyithe, J. S. B. 2008 Avalanche dynamics of radio pulsar glitches. Astrophys. J. 672, 1103–1118.Google Scholar
Mendell, G. 1991a Superfluid hydrodynamics in rotating neutron stars. I. Nondissipative equations. Astrophys. J. 380, 515–529.Google Scholar
Mendell, G. 1991b Superfluid hydrodynamics in rotating neutron stars. II. Dissipative effects. Astrophys. J. 380, 530–540.Google Scholar
Pedlosky, J. 1967 The spin up of a stratified fluid. J. Fluid Mech. 28, 463–479.Google Scholar
Peradzynski, Z., Filipkowski, S. & Fiszdon, W. 1990 Spin-up of He II in a cylindrical vessel of finite height. Eur. J. Mech. (B/Fluids) 9, 259–272.Google Scholar
Peralta, C., Melatos, A., Giacobello, M. & Ooi, A. 2006a Gravitational radiation from nonaxisymmetric spherical Couette flow in a neutron star. Astrophys. J. 644, L53–L56.Google Scholar
Peralta, C., Melatos, A., Giacobello, M. & Ooi, A. 2006b Transitions between turbulent and laminar superfluid vorticity states in the outer core of a neutron star. Astrophys. J. 651, 1079–1091.Google Scholar
Peralta, C., Melatos, A., Giacobello, M. & Ooi, A. 2008 Superfluid spherical Couette flow. J. Fluid Mech. 609, 221–274.Google Scholar
Peralta, C. & Melatos, A. 2009 An unstable superfluid Stewartson layer in a differentially rotating neutron star. Astrophys. J. Lett. 701, L75–L78.Google Scholar
Reisenegger, A. 1993 The spin-up problem in helium II. J. Low Temp. Phys. 92, 77–106.CrossRefGoogle Scholar
Reisenegger, A. & Goldreich, P. 1992 A new class of g-modes in neutron stars. Astrophys. J. 395, 240–249.Google Scholar
Ruderman, M. 1970a Long period oscillations in rotating neutron stars. Nature 225, 619–620.Google Scholar
Ruderman, M. 1970b Pulsar wobble and neutron starquakes. Nature 225, 838–839.Google Scholar
Sonin, É. B. 1976 Vortex–lattice vibrations in a rotating helium II. Sov. J. Expl Theor. Phys. 43, 1027.Google Scholar
Sonin, E. B. 1987 Vortex oscillations and hydrodynamics of rotating superfluids. Rev. Mod. Phys. 59, 87–155.Google Scholar
Tsakadze, J. S. & Tsakadze, S. J. 1972 Relaxation phenomena at acceleration of rotation of a spherical vessel with helium II and relaxation in pulsars. Phys. Lett. A 41, 197–199.Google Scholar
Tsakadze, J. S. & Tsakadze, S. J. 1973 Measurement of the relaxation time on acceleration of vessels with helium II and superfluidity in pulsars. Sov. J. Expl Theor. Phys. 37, 918.Google Scholar
Tsakadze, J. S. & Tsakadze, S. J. 1974 On the problem of relaxation time determination in superfluids when their rotation is accelerated. Phys. Lett. A 47, 477–478.Google Scholar
Tsakadze, J. S. & Tsakadze, S. J. 1980 Properties of slowly rotating helium II and the superfluidity of pulsars. J. Low Temp. Phys. 39, 649.Google Scholar
Ungarish, M. 1990 Spin-up from rest of a mixture. Phys. Fluids 2, 160–166.Google Scholar
Walin, G. 1969 Some aspects of time-dependent motion of a stratified rotating fluid. J. Fluid Mech. 36, 289–307.Google Scholar
Warszawski, L. & Melatos, A. 2013 Knock-on processes in superfluid vortex avalanches and pulsar glitch statistics. Mon. Not. R. Astron. Soc. 428, 1911–1926.Google Scholar
Warszawski, L., Melatos, A. & Berloff, N. G. 2012 Unpinning triggers for superfluid vortex avalanches. Phys. Rev. B 85 (10), 104503.Google Scholar
Wong, T., Backer, D. C. & Lyne, A. G. 2001 Observations of a series of six recent glitches in the Crab pulsar. Astrophys. J. 548, 447–459.Google Scholar