Skip to Main Content
Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

Journals Home Search My Subscriptions Subscribe
Previous issue | This issue | Most recent issue | All issues | Next issue | Previous article | Recently published articles | Next article
   
 
 

 

Quasiperiodicity and chaos in the nonlinear evolution of the Kelvin-Helmholtz instability of supersonic anisotropic tangential velocity discontinuities


Authors: S. Roy Choudhury and Kevin G. Brown
Journal: Quart. Appl. Math. 61 (2003), 41-72
MSC: Primary 76X05; Secondary 37N10, 76E17, 76E25, 76E30, 85A15
DOI: https://doi.org/10.1090/qam/1955223
MathSciNet review: MR1955223
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, strongly anisotropic, magnetized, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations. For modes near the critical point of the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein-Gordon equation. The nonlinear coefficient turns out to be complex, which is, to the best of our knowledge, unlike previously considered cases and leads to completely different dynamics from that reported earlier. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability. In particular, the complex nonlinearity allows the existence of quasiperiodic and chaotic wave envelopes, unlike in earlier physical models governed by nonlinear Klein-Gordon equations. In addition, numerical diagnostics reveal the onset of chaos as a consequence of modulation of the external magnetic field.

References [Enhancements On Off] (What's this?)

  • Mark J. Ablowitz and Harvey Segur, Solitons and the inverse scattering transform, SIAM Studies in Applied Mathematics, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. MR 642018
    • Abraham-Shrauner, B. (1973). “Small Amplitude Hydromagnetic Waves for a Plasma with a Generalized Polytrope Law.” Plasma Phys. 15, 375–385. Abramowitz, M., and Stegun, I. A. (1964). “Handbook of Mathematical Functions.” Washington: National Bureau of Standards. Aref, H., and Siggia, E. D. (1981). “Evolution and Breakdown of a Vortex Street in Two Dimensions.” Journal of Fluid Mechanics 109, 435–463. Begelman, M. C., Blandford, R. D., and Rees, M. J. (1984). “Theory of Extragalactic Radio Sources". Review of Modern Physics 56, 255–351.
  • D. J. Benney and A. C. Newell, Sequential time closures for interacting random waves, J. Math. and Phys. 46 (1967), 363–393. MR 223133
    • Benney, D. J., and Roskes, G. J. (1969). Studies in Applied Math. 48, 377. Blandford, R. D., and Pringle, J. E. (1976). “Kelvin-Helmholtz Instability of Relativistic Beams.” Monthly Notices of the Royal Astronomical Society 176, 443–454. Blumen, W. (1970). “Shear Layer Instability of an Inviscid Compressible Fluid.” Journal of Fluid Mechanics 40, 769–781. Blumen, W., Drazin, P. G., and Billings, D. F. (1975). “Shear Layer Instability of an Inviscid Compressible Fluid, Part 2.” Journal of Fluid Mechanics 71, 305–316. Brandt, J. C., and Mendis, D. A. (1979). The Solar Wind, in “Solar System Plasma Physics.” (C. F. Kennel, L. Lanzerotti, and E. N. Parker, Eds.). Amsterdam: North Holland. Bridge, H. S., Belcher, J. W., Lazarus, A. J., Sullivan, J. D., Bagenal, F., McNutt, Jr., R. L., Oglivie, K. W., Scudder, J. D., and Sittler, E. C. (1979). “Plasma observations near Jupiter: Initial results". Science 206, 972–976. Brown, G. L., and Roshko, A. (1974). “On density effects and large structure in turbulent mixing layers.” Journal of Fluid Mechanics 64, 775–816.
  • Kevin G. Brown and S. Roy Choudhury, An analytical study of the Kelvin-Helmholtz instabilities of compressible, magnetized tangential velocity discontinuities with generalized polytrope laws, Quart. Appl. Math. 60 (2002), no. 4, 601–630. MR 1938343, DOI https://doi.org/10.1090/qam/1938343
  • Kevin G. Brown and S. Roy Choudhury, The initial-value problem for the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers with generalized polytrope laws, Quart. Appl. Math. 60 (2002), no. 4, 657–673. MR 1939005, DOI https://doi.org/10.1090/qam/1939005
  • S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961. MR 0128226
  • A. D. D. Craik, M. Nagata, and I. M. Moroz, Second-harmonic resonance in nonconservative systems, Wave Motion 15 (1992), no. 2, 173–183. MR 1150679, DOI https://doi.org/10.1016/0165-2125%2892%2990017-V
  • Alex D. D. Craik, Wave interactions and fluid flows, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge, 1985. MR 896268
  • A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London Ser. A 338 (1974), 101–110. MR 349126, DOI https://doi.org/10.1098/rspa.1974.0076
    • Dobrowolny, H. and D’Angelo, N. (1972). “Wave motion in type I comet tails.” Cosmic Plasma Physics (K. Schindler, ed.), New York: Plenum.
  • Roger K. Dodd, J. Chris Eilbeck, John D. Gibbon, and Hedley C. Morris, Solitons and nonlinear wave equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982. MR 696935
    • Drazin, P. G. (1970). Journal of Fluid Mechanics 42, 321.
  • P. G. Drazin and William Hill Reid, Hydrodynamic stability, Cambridge University Press, Cambridge-New York, 1981. Cambridge Monographs on Mechanics and Applied Mathematics. MR 604359
    • Ershkovich, A. I., Nusnov, A. A., and Chernikov, A. A. (1972). “Oscillations of type I comet tails.” Planetary and Space Science 20, 1235–1243. Ershkovich, A. I., and Chernikov, A. A. (1973). “Nonlinear waves in type I comet tails.” Planetary and Space Science 21, 663–673.
  • J. A. Fejer, Hydromagnetic stability at a fluid velocity discontinuity between compressible fluids, Phys. Fluids 7 (1964), 499–503. MR 163553, DOI https://doi.org/10.1063/1.1711229
    • Gerwin, R. A. (1968). “Stability of the Interface between two fluids in relative motion.” Rev. Modern Phys. 40, 652–658.
  • J. D. Gibbon, I. N. James, and I. M. Moroz, An example of soliton behaviour in a rotating baroclinic fluid, Proc. Roy. Soc. London Ser. A 367 (1979), no. 1729, 219–237. MR 547624, DOI https://doi.org/10.1098/rspa.1979.0084
  • J. D. Gibbon and M. J. McGuinness, Amplitude equations at the critical points of unstable dispersive physical systems, Proc. Roy. Soc. London Ser. A 377 (1981), no. 1769, 185–219. MR 626469, DOI https://doi.org/10.1098/rspa.1981.0121
    • Hasimoto, H., and Ono, H. (1972). Journal of the Physical Society of Japan 33, 805.
  • Robert C. Hilborn, Chaos and nonlinear dynamics, The Clarendon Press, Oxford University Press, New York, 1994. An introduction for scientists and engineers. MR 1263025
    • Infeld, E., Ziemkiewicz, J., and Rowlands, G. (1987). “Stability of Nonlinear Hydromagnetic Waves and Solitons.” Physics of Fluids 30, 2330.
  • E. Infeld and G. Rowlands, Stability of nonlinear ion sound waves and solitons in plasmas, Proc. Roy. Soc. London Ser. A 366 (1979), no. 1727, 537–554. MR 547762, DOI https://doi.org/10.1098/rspa.1979.0068
    • Jokipii, J. R., and Davis, L. (1969). “Long wavelength turbulence and the heating of the solar wind.” Astrophys. Journal 156, 1101–1106. Kaup, D. J., Roy Choudhury, S., and Thomas, G. E. (1988). “The full second-order theory of the diocotron and magnetron resonances.” Phys Rev. A 38, 1402. Landau, L. D. (1944). “The instability of moving superposed fluids.” Akad. Nauk SSSR C. R. Dokl. 44, 139–144. Lange, C. G., and Newell, A. C. (1971). “The post buckling problem for thin elastic shells.” SIAM Journal of Applied Math. 21, 605.
  • Charles G. Lange and Alan C. Newell, A stability criterion for envelope equations, SIAM J. Appl. Math. 27 (1974), 441–456. MR 381500, DOI https://doi.org/10.1137/0127034
    • Lerche, I. (1966). “Validity of the hydromagnetic approach in discussing instability of the Magnetospheric Boundary.” Journal of Geophys. Res. 71, 2365–2371.
  • A. Michalke, On the inviscid instability of the hyperbolic-tangent velocity profile, J. Fluid Mech. 19 (1964), 543–556. MR 184516, DOI https://doi.org/10.1017/S0022112064000908
    • Miles, J. W. (1957). “On wind over water.” Journal Acoust. Soc. Amer. 29, 226–230.
  • John W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid Mech. 4 (1958), 538–552. MR 97930, DOI https://doi.org/10.1017/S0022112058000653
    • Miura, A., and Pritchett, P. L. (1982). “Nonlocal stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma.” Journal of Geophy. Res. 87, 7431–7444. Miura, A. (1984). “Anomalous transport by magnetohydrodynamic Kelvin-Helmholtz instabilities in the solar wind-magnetosphere interaction.” Journal of Geophy. Res. 89, 801–818.
  • Youichi Murakami, A note on modulational instability of a nonlinear Klein-Gordon equation, J. Phys. Soc. Japan 55 (1986), no. 11, 3851–3856. MR 875754, DOI https://doi.org/10.1143/JPSJ.55.3851
  • Ali H. Nayfeh and Balakumar Balachandran, Applied nonlinear dynamics, Wiley Series in Nonlinear Science, John Wiley & Sons, Inc., New York, 1995. Analytical, computational, and experimental methods; A Wiley-Interscience Publication. MR 1310778
    • Nayfeh, A. H., and Saric, W. S. (1971). “The Kelvin-Helmholtz Instability I.” Journal of Fluid Mechanics 46, 209. Nayfeh, A. H., and Saric, W. S. (1972). “The Kelvin-Helmholtz Instability II.” Journal of Fluid Mechanics 55, 311.
  • Ali Hasan Nayfeh, Perturbation methods, John Wiley & Sons, New York-London-Sydney, 1973. Pure and Applied Mathematics. MR 0404788
    • Nepveu, M. (1980). “Cylindrical jets."Astronom. and Astrophys. 84, 14–21. Ness, N. F., Acuna, M. H., Lepping, R. P., Connerney, J. E. P., Behannon, K. W., and Burlaga, L. F. (1981). “Magnetic field studies by Voyager I.” Science 212, 211–217.
  • Alan C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (1969), no. 2, 279–303. MR 3363403, DOI https://doi.org/10.1017/S0022112069000176
  • Alan C. Newell, Envelope equations, Nonlinear wave motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972) Amer. Math. Soc., Providence, R.I., 1974, pp. 157–163. Lectures in Appl. Math., Vol. 15. MR 0380123
  • Paul K. Newton, Chaos in Rayleigh-Bénard convection with external driving, Phys. Rev. A (3) 37 (1988), no. 3, 932–934. MR 927227, DOI https://doi.org/10.1103/PhysRevA.37.932
    • Norman, M. L., Smarr, L., Winkler, K. H. A., and Smith, M. D. (1982). “Instabilities of cylindrical jets.” Astronom. and Astrophys. 113, 285–351.
  • Francisco Lara Ochoa and J. D. Murray, A nonlinear analysis for spatial structure in a reaction-diffusion model, Bull. Math. Biol. 45 (1983), no. 6, 917–930. MR 727354, DOI https://doi.org/10.1016/S0092-8240%2883%2980069-X
    • Parker, E. N. (1963). “Interplanetary Dynamical Processes.” New York: Interscience.
  • E. J. Parkes, The modulational instability of the nonlinear Klein-Gordon equation, Wave Motion 13 (1991), no. 3, 261–275. MR 1105952, DOI https://doi.org/10.1016/0165-2125%2891%2990063-T
    • Pearlstein, L. D., and Berk, H. L. (1969). “Bound states of a Schrödinger equation.” Phys. Rev. Lett. 23, 220. Pedlosky, J. (1970). “Finite amplitude baroclinic waves.” Journal Atmospheric Science 27, 15. Pedlosky, J. (1972). “Finite amplitude baroclinic wave packets.” Journal Atmospheric Science 29, 680. Pritchett, P. L., and Coroniti, F. V. (1984). “The collisionless macroscopic Kelvin-Helmholtz instability I. Transverse electrostatic mode.” Journal Geophys. Res. 89, 168–178. Pu, Z. Y., and Kivelson, M. G. (1983). “Kelvin-Helmholtz instability at the magnetopause: Solution for compressible plasmas.” Journal Geophys. Res. 88, 841–852 and “Energy flux into the magnetosphere,” 853–861. Ray, T. P. (1982). “The effects of a simple shear layer on the growth of Kelvin-Helmholtz instabilities.” Monthly Notices Roy. Astronom. Soc. 198, 617–625. Ray. T. P., and Erschkovich, A. I. (1983). “Kelvin-Helmholtz instabilities of magnetized shear layers.” Monthly Notices Roy. Astronom. Soc. 204, 821–826. Roy Choudhury, S., and Lovelace, R. V. E. (1986). “On the Kelvin-Helmholtz instabilities of high-velocity magnetized shear layers.” Astrophys. Journal 302, 188–199. Roy Choudhury, S. (1986). “Kelvin-Helmholtz instabilities of supersonic magnetized shear layers.” Journal Plasma Phys. 35, 375–392. Roy Choudhury, S., and Patel, V. L. (1985). “Kelvin-Helmholtz instabilities of high-velocity, magnetized anisotropic shear layers.” Physics of Fluids 28, 3292–3301. Roy Choudhury, S. (1990). “Global asymptotic analysis of the Kelvin-Helmholtz instability of supersonic shear layers.” Canad. Journal Phys. 68, 334–342.
  • S. Roy Choudhury, On bifurcations and chaos in predator-prey models with delay, Chaos Solitons Fractals 2 (1992), no. 4, 393–409. MR 1295920, DOI https://doi.org/10.1016/0960-0779%2892%2990015-F
  • Lokenath Debnath and S. Roy Choudhury (eds.), Nonlinear instability analysis, Advances in Fluid Mechanics, vol. 12, Computational Mechanics Publications, Southampton, 1997. MR 1481998
    • Sen, A. K. (1965). “Stability of the magnetosphere boundary.” Planet. Space Sci. 13, 131–141.
  • Amiya K. Sen, Effect of compressibiliy on Kelvin-Helmholtz instability in a plasma, Phys. Fluids 7 (1964), 1293–1298. MR 165823, DOI https://doi.org/10.1063/1.1711374
    • Southwood, D. J. (1968). “The hydromagnetic stability of the magnetospheric boundary.” Planet Space Sci. 16, 587–605. Southwood, D. J. (1974). “Some features of the field line resonances in the magnetosphere.” Planet Space Sci. 22, 483–491.
  • K. Stewartson and J. T. Stuart, A non-linear instability theory for a wave system in plane Poiseuille flow, J. Fluid Mech. 48 (1971), 529–545. MR 309420, DOI https://doi.org/10.1017/S0022112071001733
    • Stuart, J. T. (1960). “On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows.” Journal Fluid Mech. 8, 183. Sturrock, P. A., and Hartle, R. E. (1966). “Two-fluid mode of the solar wind.” Phys. Rev. Lett. 16, 628–631.
  • John W. Miles, On Kelvin-Helmholtz instability, Phys. Fluids 23 (1980), no. 9, 1915–1916. MR 585970, DOI https://doi.org/10.1063/1.863218
    • Talwar, S. P. (1965). “Kelvin-Helmholtz instability in an anisotropic plasma.” Physics of Fluids 8, 1295–1299. Turland, B. D. and Scheuer, P. A. G. (1976). “Instabilities of Kelvin-Helmholtz type for relativistic streaming.” Monthly Notices Roy. Astronom. Soc. 176, 421–441. Weissman, M. A. (1979). “Nonlinear Wave Packets in the Kelvin-Helmholtz instability.” Philos. Trans. Roy. Soc. London 290, 639.
  • G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
    • Winant, C. D., and Browand, F. K. (1974). “Vortex pairing: The mechanism of turbulent mixing-layer growth.” Journal Fluid Mech. 63, 237–255.
  • David J. Wollkind, Valipuram S. Manoranjan, and Limin Zhang, Weakly nonlinear stability analyses of prototype reaction-diffusion model equations, SIAM Rev. 36 (1994), no. 2, 176–214. MR 1278632, DOI https://doi.org/10.1137/1036052

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 76X05, 37N10, 76E17, 76E25, 76E30, 85A15

Retrieve articles in all journals with MSC: 76X05, 37N10, 76E17, 76E25, 76E30, 85A15

Additional Information

Article copyright: © Copyright 2003 American Mathematical Society

Previous issue | This issue | Most recent issue | All issues | Next issue | Previous article | Recently published articles | Next article