Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-29T15:02:26.069Z Has data issue: false hasContentIssue false

Dispersive shock waves in viscously deformable media

Published online by Cambridge University Press:  08 February 2013

Nicholas K. Lowman*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
M. A. Hoefer
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
*
Email address for correspondence: nklowman@ncsu.edu

Abstract

The viscously dominated, low-Reynolds-number dynamics of multi-phase, compacting media can lead to nonlinear, dissipationless/dispersive behaviour when viewed appropriately. In these systems, nonlinear self-steepening competes with wave dispersion, giving rise to dispersive shock waves (DSWs). Example systems considered here include magma migration through the mantle as well as the buoyant ascent of a low-density fluid through a viscously deformable conduit. These flows are modelled by a third-order, degenerate, dispersive, nonlinear wave equation for the porosity (magma volume fraction) or cross-sectional area, respectively. Whitham averaging theory for step initial conditions is used to compute analytical, closed-form predictions for the DSW speeds and the leading edge amplitude in terms of the constitutive parameters and initial jump height. Novel physical behaviours are identified including backflow and DSW implosion for initial jumps sufficient to cause gradient catastrophe in the Whitham modulation equations. Theoretical predictions are shown to be in excellent agreement with long-time numerical simulations for the case of small- to moderate-amplitude DSWs. Verifiable criteria identifying the breakdown of this modulation theory in the large jump regime, applicable to a wide class of DSW problems, are presented.

Type
Papers
Copyright
©2013 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

Ablowitz, M. J., Baldwin, D. E. & Hoefer, M. A. 2009 Soliton generation and multiple phases in dispersive shock and rarefaction wave interaction. Phys. Rev. E 80, 016603.CrossRefGoogle ScholarPubMed
Barcilon, V. & Lovera, O. M. 1989 Solitary waves in magma dynamics. J. Fluid Mech. 204, 121133.Google Scholar
Barcilon, V. & Richter, F. M. 1986 Nonlinear waves in compacting media. J. Fluid Mech. 164, 429448.Google Scholar
Chanson, H. 2010 Tidal Bores, Aegir and Pororoca: The Geophysical Wonders. World Scientific.Google Scholar
Conti, C., Fratalocchi, A., Peccianti, M., Ruocco, G. & Trillo, S. 2009 Observation of a gradient catastrophe generating solitons. Phys. Rev. Lett. 102, 083902.Google Scholar
Dutton, Z., Budde, M., Slowe, C. & Hau, L. V. 2001 Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose–Einstein condensate. Science 293, 663.Google Scholar
El, G. A. 2005 Resolution of a shock in hyperbolic systems modified by weak dispersion. Chaos 15, 037103.CrossRefGoogle ScholarPubMed
El, G. A., Grimshaw, R. H. J. & Smyth, N. F. 2006 Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027104.Google Scholar
El, G. A., Grimshaw, R. H. J. & Smyth, N. F. 2009 Transcritical shallow-water flow past topography: finite-amplitude theory. J. Fluid Mech. 640, 187.CrossRefGoogle Scholar
Elperin, T., Kleeorin, N. & Krylov, A. 1994 Nondissipative shock waves in two-phase flows. Physica D 74, 372385.Google Scholar
Esler, J. G. & Pearce, J. D. 2011 Dispersive dam-break and lock-exchange flows in a two-layer fluid. J. Fluid Mech. 667, 555585.Google Scholar
Flashka, H., Forest, M. G. & McLaughlin, D. W. 1980 Multiphase averaging and the inverse spectral transform of the Korteweg–de Vries equation. Commun. Pure Appl. Maths 33, 739784.CrossRefGoogle Scholar
Fowler, A. C. 1985 A mathematical model of magma transport in the asthenosphere. Geophys. Astrophys. Fluid Dyn. 6396.CrossRefGoogle Scholar
Grava, T. & Tian, F.-R. 2002 The generation, propagation, and extinction of multiphases in the KdV zero-dispersion limit. Commun. Pure Appl. Maths 55, 15691639.Google Scholar
Gurevich, A. V. & Pitaevskii, L. P. 1974 Nonstationary structure of a collissionless shock wave. Sov. Phys. JETP 33, 291297.Google Scholar
Harris, S. E. 1996 Conservation laws for a nonlinear wave equation. Nonlinearity 9, 187208.Google Scholar
Harris, S. E. & Clarkson, P. A. 2006 Painleve analysis and similarity reductions for the magma equation. SIGMA 2, 68.Google Scholar
Helfrich, K. R. & Whitehead, J. A. 1990 Solitary waves on conduits of buoyant fluid in a more viscous fluid. Geophys. Astrophys. Fluid Dyn. 51, 3552.Google Scholar
Hoefer, M. A. & Ablowitz, M. J. 2007 Interactions of dispersive shock waves. Physica D 236, 4464.CrossRefGoogle Scholar
Hoefer, M. A., Ablowitz, M. J., Coddington, I., Cornell, E. A., Engels, P. & Schweikhard, V. 2006 Dispersive and classical shock waves in Bose–Einstein condensates and gas dynamics. Phys. Rev. A 74, 023623.Google Scholar
Jorge, M. C., Minzoni, A. A. & Smyth, N. F. 1999 Modulation solutions for the Benjamin–Ono equation. Physica D 132, 118.Google Scholar
Kamchatnov, A. M., Kuo, Y. H., Lin, T. C., Horng, T. L., Gou, S. C., Clift, R., El, G. A. & Grimshaw, R. H. J. 2012 Undular bore theory for the Gardner equation. Phys. Rev. E 86, 036605.Google Scholar
Katz, R. F., Knepley, M., Smith, B., Spiegelman, M. & Coon, E. 2007 Numerical simulation of geodynamic processes with the portable extensible toolkit for scientific computation. Phys. Earth Planet. Inter. 163, 5268.Google Scholar
Kodama, Y., Pierce, V. U. & Tian, F.-R. 2008 On the Whitham equations for the defocusing complex modified KdV equation. SIAM J. Math. Anal. 41, 2658.Google Scholar
Marchant, T. R. & Smyth, N. F. 2005 Approximate solutions for magmon propagation from a reservoir. IMA J. Appl. Maths 70, 793813.Google Scholar
McKenzie, D. 1984 The generation and compaction of partially molten rock. J. Petrol. 25, 713765.Google Scholar
Nakayama, M. & Mason, D. P. 1992 Rarefactive solitary waves in two-phase fluid flow of compacting media. Wave Motion 15, 357392.Google Scholar
Olson, P. & Christensen, U. 1986 Solitary wave propagation in a fluid conduit within a viscous matrix. J. Geophys. Res. 91, 63676374.CrossRefGoogle Scholar
Ostrovsky, L. A. & Potapov, A. I. 2002 Modulated Waves: Theory and Applications. Johns Hopkins University.Google Scholar
Pierce, V. U. & Tian, F.-R. 2007 Self-similar solutions of the non-strictly hyperbolic Whitham equations for the KdV hierarchy. Dyn. PDE 4, 263282.Google Scholar
Porter, V. A. & Smyth, N. F. 2002 Modelling the morning glory of the Gulf of Carpentia. J. Fluid Mech. 454, 120.CrossRefGoogle Scholar
Richter, F. M. & McKenzie, D. 1984 Dynamical models for melt segregation from a deformable matrix. J. Geol. 92, 729740.Google Scholar
Scheidegger, A. E. 1974 The Physics of Flow Through Porous Media. University of Toronto.Google Scholar
Scott, D. R. & Stevenson, D. J. 1984 Magma solitons. Geophys. Res. Lett. 11, 11611164.Google Scholar
Scott, D. R. & Stevenson, D. J. 1986 Magma ascent by porous flow. Geophys. Res. Lett. 91, 92839296.Google Scholar
Scott, D. R., Stevenson, D. J. & Whitehead, J. A. 1986 Observations of solitary waves in a viscously deformable pipe. Nature 319, 759761.Google Scholar
Simpson, G. & Spiegelman, M. 2011 Solitary wave benchmarks in magma dynamics. J. Sci. Comput. 49, 268290.CrossRefGoogle Scholar
Simpson, G., Spiegelman, M. & Weinstein, M. I. 2007 Degenerate dispersive equations arising in the study of magma dynamics. Nonlinearity 20.CrossRefGoogle Scholar
Simpson, G., Spiegelman, M. & Weinstein, M. I. 2010 A multiscale model of partial melts I: effective equations. J. Geophys. Res.-Sol. Ea. 115, B04410.Google Scholar
Simpson, G & Weinstein, M. I. 2008 Asymptotic stability of ascending solitary magma waves. SIAM J. Math. Anal. 40, 13371391.Google Scholar
Spiegelman, M 1993a Flow in deformable porous media I. Simple analysis. J. Fluid Mech. 247, 1738.Google Scholar
Spiegelman, M. 1993b Flow in deformable porous media II. Numerical analysis—the relationship between shock waves and solitary waves. J. Fluid Mech. 247, 3963.CrossRefGoogle Scholar
Spiegelman, M., Kelemen, P. B. & Aharonov, E. 2001 Causes and consequences of flow organization during melt transport: the reaction infiltration instability in compactible media. J. Geophys. Res. 106, 20612077.Google Scholar
Takahashi, D., Sachs, J. R. & Satsuma, J. 1990 Properties of the magma and modified magma equations. J. Phys. Soc. Japan 59, 19411953.Google Scholar
Taylor, R. J., Baker, D. R. & Ikezi, H. 1970 Observation of collisionless electrostatic shocks. Phys. Rev. Lett. 24, 206209.Google Scholar
Tran, M. Q., Appert, K., Hollenstein, C., Means, R. W. & Vaclavik, J. 1977 Shocklike solutions of the Korteweg–de Vries equation. Plasma Phys. 19, 381.Google Scholar
Wan, W., Jia, S. & Fleischer, J. W. 2007 Dispersive superfluid-like shock waves in nonlinear optics. Nat. Phys. 3, 4651.Google Scholar
Whitehead, J. A. & Helfrich, K. R. 1986 The Korteweg–deVries equation from laboratory conduit and magma migration equations. Geophys. Res. Lett. 13, 545546.Google Scholar
Whitham, G. B. 1965 Nonlinear dispersive waves. Proc. R. Soc. Lond., Ser. A 283, 238261.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley and Sons.Google Scholar