Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T11:23:59.766Z Has data issue: false hasContentIssue false
  • English
  • Français

Experimental study of the stability and dynamics of a two-dimensional ideal vortex under external strain

Published online by Cambridge University Press:  01 June 2018

N. C. Hurst Open the ORCID record for N. C. Hurst [Opens in a new window] ,
J. R. Danielson ,
D. H. E. Dubin  and
C. M. Surko
N. C. Hurst*
Affiliation:
Department of Physics, University of California – San Diego, La Jolla, CA 92093, USA
J. R. Danielson
Affiliation:
Department of Physics, University of California – San Diego, La Jolla, CA 92093, USA
D. H. E. Dubin
Affiliation:
Department of Physics, University of California – San Diego, La Jolla, CA 92093, USA
C. M. Surko
Affiliation:
Department of Physics, University of California – San Diego, La Jolla, CA 92093, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The dynamics of two-dimensional (2-D) ideal fluid vortices is studied experimentally in the presence of an irrotational strain flow. Laboratory experiments are conducted using strongly magnetized pure electron plasmas, a technique which is made possible by the isomorphism between the drift–Poisson equations describing plasma dynamics transverse to the field and the 2-D Euler equations describing an ideal fluid. The electron plasma system provides an excellent opportunity to study the dynamics of a 2-D Euler fluid due to weak dissipation and weak 3-D effects, simple diagnosis and precise control. The plasma confinement apparatus used here was designed specifically to study vortex dynamics under the influence of external flow by applying boundary conditions in two dimensions. Additionally, vortex-in-cell simulations are carried out to complement the experimental results and to extend the parameter range of the studies. It is shown that the global dynamics of a quasi-flat vorticity profile is in good quantitative agreement with the theory of a piecewise-constant elliptical patch of vorticity, including the equilibria, dynamical orbits and stability properties. Deviations from the elliptical patch theory are observed for non-flat vorticity profiles; they include inviscid damping of the orbits and modified stability limits. The dependence of these phenomena on the flatness of the initial profile is discussed. The relationship of these results to other theoretical, numerical and experimental studies is also discussed.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models MHD and Electrohydrodynamics: Plasmas Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 848 , 10 August 2018 , pp. 256 - 287
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, A., Chesler, P. M. & Liu, H. 2014 Holographic turbulence. Phys. Rev. Lett. 112, 151602.CrossRefGoogle ScholarPubMed
Backhaus, E. Yu., Fajans, J. & Wurtele, J. S. 1999 Stability of highly asymmetric non-neutral plasmas. Phys. Plasmas 6 (1), 19–30.CrossRefGoogle Scholar
Balmforth, N. J., Smith, S. G. L. & Young, W. R. 2001 Disturbing vortices. J. Fluid Mech. 426, 95–133.CrossRefGoogle Scholar
Basdevant, C. & Philipovitch, T. 1994 On the validity of the ‘Weiss criterion’ in two-dimensional turbulence. Physica D 73, 17–30.Google Scholar
Chen, F. F. 1984 Introduction to Plasma Physics and Controlled Fusion, 2nd edn. Plenum Press.CrossRefGoogle Scholar
Chen, S., Maero, G. & Rome, M. 2017 Spectral analysis of forced turbulence in a non-neutral plasma. J. Plasma Phys. 83 (3), 705830303.CrossRefGoogle Scholar
Chu, R., Wurtele, J. S., Notte, J., Peurrung, A. J. & Fajans, J. 1993 Pure electron plasmas in asymmetric traps. Phys. Fluids B 5 (7), 2378–2386.CrossRefGoogle Scholar
Crosby, A., Johnson, E. R. & Morrison, P. J. 2013 Deformation of vortex patches by boundaries. Phys. Fluids 25, 023602.CrossRefGoogle Scholar
Danielson, J. R., Dubin, D. H. E., Greaves, R. G. & Surko, C. M. 2015 Plasma and trap-based techiques for science with positrons. Rev. Mod. Phys. 87 (1), 247–306.CrossRefGoogle Scholar
Driscoll, C. F. & Fine, K. S. 1990 Experiments on vortex dynamics in pure electron plasmas. Phys. Fluids B 2, 1359–1366.CrossRefGoogle Scholar
Dritschel, D. G. 1989 Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two dimensional, inviscid, incompressible flows. Comput. Phys. Rep. 10, 77–146.CrossRefGoogle Scholar
Dritschel, D. G. 1990 The stability of elliptical vortices in an external straining flow. J. Fluid Mech. 210, 223–261.CrossRefGoogle Scholar
Dritschel, D. G. & Legras, B. 1993 Modeling oceanic and atmospheric vortices. Phys. Today 46 (3), 44–51.CrossRefGoogle Scholar
Dubin, D. H. E. 1998 Collisional transport in non-neutral plasmas. Phys. Plasmas 5 (5), 1688–1694.CrossRefGoogle Scholar
Dubin, D. H. E. & O’Neil, T. M. 1999 Trapped nonneutral plasmas, liquids, and crystals (the thermal equilibrium states). Rev. Mod. Phys. 71 (1), 87–172.CrossRefGoogle Scholar
Durkin, D. & Fajans, J. 2000 Experiments on two-dimensional vortex patterns. Phys. Fluids 12 (2), 289–293.CrossRefGoogle Scholar
Eggleston, D. 1994 Experimental study of two-dimensional electron vortex dynamics in an applied irrotational shear flow. Phys. Plasmas. 1 (12), 3850–3856.CrossRefGoogle Scholar
Fajans, J., Backhaus, E. Yu. & Gilson, E. 2000 Bifurcations in elliptical, asymmetric non-neutral plasmas. Phys. Plasmas 7 (10), 3929–3933.CrossRefGoogle Scholar
Fine, K. S., Cass, A. C., Flynn, W. G. & Driscoll, C. F. 1995 Relaxation of 2d turbulence to vortex crystals. Phys. Rev. Lett. 75 (18), 3277–3280.CrossRefGoogle ScholarPubMed
Godon, P. & Livio, M. 1999 Vortices in protoplanetary disks. Astrophys. J. 523 (1), 350–356.CrossRefGoogle Scholar
Goodman, J., Hou, T. Y. & Lowengrub, J. 1990 Convergence of the point vortex method for the 2-D Euler equations. In Communications on Pure and Applied Mathematics, Vol. XLIII, pp. 415–430. John Wiley and Sons.Google Scholar
Haller, G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 1–26.CrossRefGoogle Scholar
Hasegawa, A. & Mima, K. 1978 Pseudo-three-dimensional turbulence in magnetized nonuniform plasma. Phys. Fluids 21 (87), 87–92.CrossRefGoogle Scholar
Hua, B. L. & Klein, P. 1998 An exact criterion for the stirring properties of nearly two-dimensional turbulence. Physica D 113, 98–110.Google Scholar
Hunt, J. C. R. & Carruthers, D. J. 1990 Rapid distortion theory and the ‘problems’ of turbulence. J. Fluid Mech. 212, 497–532.CrossRefGoogle Scholar
Hurst, N. C., Danielson, J. R., Dubin, D. H. E. & Surko, C. M. 2016 Evolution of a vortex in a strain flow. Phys. Rev. Lett. 117, 235001.CrossRefGoogle Scholar
Hurst, N. C., Danielson, J. R. & Surko, C. M. 2018 An electron plasma experiment to study vortex dynamics subject to externally imposed flows. AIP Conf. Proc. 1928, 020007.CrossRefGoogle Scholar
Kawai, Y., Kiwamoto, Y., Soga, Y. & Aoki, J. 2007 Turbulent cascade in vortex dynamics of magnetized pure electron plasmas. Phys. Rev. E 75, 066404.Google ScholarPubMed
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50 (10), 3517–3520.CrossRefGoogle Scholar
Kriesel, J. M. & Driscoll, C. F. 2000 Two regimes of asymmetry-induced transport in non-neutral plasmas. Phys. Rev. Lett. 85 (12), 2510–2513.CrossRefGoogle ScholarPubMed
Kriesel, J. M. & Driscoll, C. F. 2001 Measurements of viscosity in pure-electron plasmas. Phys. Rev. Lett. 87 (13), 135003.CrossRefGoogle ScholarPubMed
Legras, B., Dritschel, D. G. & Caillol, P. 2001 The erosion of a two-dimensional vortex in a background straining flow. J. Fluid Mech. 441, 369–398.CrossRefGoogle Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37 (3), 289–335.CrossRefGoogle Scholar
Lingevitch, J. F. & Bernoff, A. J. 1995 Distortion and evolution of a localized vortex in an irrotational flow. Phys. Fluids 7 (5), 1015–1026.CrossRefGoogle Scholar
Lithwick, Y. 2009 Formation, survival, and destruction of vortices in accretion disks. Astrophys. J. 693 (1), 85–96.CrossRefGoogle Scholar
Majda, A. J. & Bertozzi, A. L. 2002 Vorticity and Incompressible Flow. Cambridge University Press.Google Scholar
Mariotti, A., Legras, B. & Dritschel, D. G. 1994 Vortex stripping and the erosion of coherent structures in two-dimensional flows. Phys. Fluids 6 (12), 3954–3962.CrossRefGoogle Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 21–43.CrossRefGoogle Scholar
Meacham, S. P., Morrison, P. J. & Flierl, G. R. 1997 Hamiltonian moment reduction for describing vortices in shear. Phys. Fluids 9 (8), 2310–2328.CrossRefGoogle Scholar
Melander, M. V., Zabusky, N. J. & Styczek, A. S. 1986 A moment model for vortex interactions of the two-dimensional euler equations. Part 1. Computational validation of a Hamiltonian elliptical representation. J. Fluid Mech. 167, 95–115.CrossRefGoogle Scholar
Mitchell, T. B. & Driscoll, C. F. 1996 Electron vortex orbits and merger. Phys. Fluids 8 (7), 1828–1841.CrossRefGoogle Scholar
Mitchell, T. B. & Rossi, L. F. 2008 The evolution of Kirchoff elliptic vortices. Phys. Fluids 20, 054103.CrossRefGoogle Scholar
Moffatt, H. K. 2001 The topology of scalar fields in 2D and 3D turbulence. In IUTAM Symposium on Geometry and Statistics of Turbulence (ed. Kambe, T., Nakano, T. & Miyauchi, T.), pp. 13–22. Kluwer Academic Publishers.CrossRefGoogle Scholar
Montgomery, D. & Turner, L. 1980 Two-dimensional electrostatic turbulence with variable density and pressure. Phys. Fluids 23 (2), 264–268.CrossRefGoogle Scholar
Moore, D. W. & Saffman, P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence and its Detection (ed. Rogers, M., Olsen, J. H. & Goldburg, A.), pp. 339–354. Plenum Press.CrossRefGoogle Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467–521.CrossRefGoogle Scholar
Notte, J., Fajans, J., Chu, R. & Wurtele, J. S. 1993 Experimental breaking of an adiabatic invariant. Phys. Rev. Lett. 70 (25), 3900–3903.CrossRefGoogle ScholarPubMed
Okubo, A. 1970 Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. Oceanographic Abstracts 17 (3), 445–454.CrossRefGoogle Scholar
O’Neil, T. M. 1980 Cooling of a pure electron plasma by cyclotron radiation. Phys. Fluids 23 (4), 725–731.CrossRefGoogle Scholar
O’Neil, T. M. 1999 Trapped plasmas with a single sign of charge (from Coulomb crystals to 2d turbulence and vortex dynamics). Phys. Today 52 (24), 24–30.CrossRefGoogle Scholar
Peurrung, A. J. & Fajans, J. 1993 A limitation to the analogy between pure electron plasmas and two-dimensional inviscid fluids. Phys. Fluids B 5 (12), 4295–4298.CrossRefGoogle Scholar
Polvani, L. M. & Flierl, G. R. 1986 Generalized kirchoff vortices. Phys. Fluids 29, 2376–2379.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Schecter, D. A., Dubin, D. H. E., Cass, A. C., Driscoll, C. F., Lansky, I. M. & O’Neil, T. M. 2000 Inviscid damping of asymmetries on a two-dimensional vortex. Phys. Fluids 12 (10), 2397–2412.CrossRefGoogle Scholar
Soga, Y., Kiwamoto, Y., Sanpei, A. & Aoki, J. 2003 Merger and binary structure formation of two discrete vortices in a background vorticity distribution of a pure electron plasma. Phys. Plasmas 10, 3922–3926.CrossRefGoogle Scholar
Tabeling, P. 2002 Two-dimensional turbulence: a physicist approach. Phys. Rep. 362, 1–62.CrossRefGoogle Scholar
Terry, P. W. 2000 Suppression of turbulence and transport by sheared flow. Rev. Mod. Phys. 72 (1), 109–165.CrossRefGoogle Scholar
Trieling, R. R., Beckers, M. & Van Heijst, G. J. F. 1997 Dynamics of monopolar vortices in a strain flow. J. Fluid Mech. 345, 165–201.CrossRefGoogle Scholar
Turner, M. R. & Gilbert, A. D. 2008 Thresholds for the formation of satellites in two-dimensional vortices. J. Fluid Mech. 614, 381–405.CrossRefGoogle Scholar
Turner, M. R., Gilbert, A. D. & Bassom, A. P. 2008 Neutral modes of a two-dimensional vortex and their link to persistent cat’s eyes. Phys. Fluids 20, 021101.CrossRefGoogle Scholar
Vanneste, J. & Young, W. R. 2010 On the energy of elliptical vortices. Phys. Fluids 22, 081701.CrossRefGoogle Scholar
Voth, G. A., Haller, G. & Gollub, J. P. 2002 Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Lett. 88 (25), 254501.CrossRefGoogle ScholarPubMed
Weiss, J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294.Google Scholar
Zabusky, N. J. 1979 Contour dynamics for the euler equations in two dimensions. J. Comput. Phys. 30 (1), 96–106.CrossRefGoogle Scholar