Skip to main content

Coherent structures and the saturation of a nonlinear dynamo

  • Erico L. Rempel (a1), Abraham C.-L. Chian (a1) (a2) (a3), Axel Brandenburg (a4) (a5), Pablo R. Muñoz (a1) and Shawn C. Shadden (a6)...

Eulerian and Lagrangian tools are used to detect coherent structures in the velocity and magnetic fields of a mean-field dynamo, produced by direct numerical simulations of the three-dimensional compressible magnetohydrodynamic equations with an isotropic helical forcing and moderate Reynolds number. Two distinct stages of the dynamo are studied: the kinematic stage, where a seed magnetic field undergoes exponential growth; and the saturated regime. It is shown that the Lagrangian analysis detects structures with greater detail, in addition to providing information on the chaotic mixing properties of the flow and the magnetic fields. The traditional way of detecting Lagrangian coherent structures using finite-time Lyapunov exponents is compared with a recently developed method called function $M$ . The latter is shown to produce clearer pictures which readily permit the identification of hyperbolic regions in the magnetic field, where chaotic transport/dispersion of magnetic field lines is highly enhanced.

Corresponding author
Email address for correspondence:
Hide All
Archontis V., Dorch S. B. F. & Nordlund A. 2003 Numerical simulations of kinematic dynamo action. Astron. Astrophys. 397, 393399.
Baggaley A. W., Barenghi C. F., Shukurov A. & Subramanian K. 2009 Reconnecting flux-rope dynamo. Phys. Rev. E 80, 055301.
Beron-Vera F. J., Olascoaga M. J. & Goni G. J. 2010 Surface ocean mixing inferred from different multisatellite altimetry measurements. J. Phys. Oceanogr. 40, 24662480.
Blackman E. G. 1996 Overcoming the backreaction on turbulent motions in the presence of magnetic fields. Phys. Rev. Lett. 77, 26942697.
Borgogno D., Grasso D., Pegoraro F. & Schep T. J. 2011 Barriers in the transition to global chaos in collisionless magnetic reconnection. I. Ridges on the finite time Lyapunov exponent field. Phys. Plasmas 18, 102307.
Brandenburg A. 2001 The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence. Astrophys. J. 550, 824840.
Brandenburg A., Jennings R. L., Nordlund A., Rieutord M., Stein R. F. & Tuominen I. 1996 Magnetic structures in a dynamo simulation. J. Fluid Mech. 306, 325352.
Brandenburg A., Klapper I. & Kurths J. 1995 Generalized entropies in a turbulent dynamo simulation. Phys. Rev. E 52, R4602R4605.
Brandenburg A. & Subramanian K. 2005 Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417, 1209.
Branicki M., Mancho A. M. & Wiggins S. 2011 A Lagrangian description of transport associated with a fronteddy interaction: application to data from the North-Western Mediterranean Sea. Physica D 240, 282304.
Cattaneo F., Hughes D. W. & Kim E.-J. 1996 Suppression of chaos in a simplified nonlinear dynamo model. Phys. Rev. Lett. 76, 20572060.
de la Cámara A., Mancho A. M., Ide K., Serrano E. & Mechoso C. R. 2012 Routes of transport across the Antarctic polar vortex in the southern spring. J. Atmos. Sci. 69, 741752.
Chakraborty P., Balachandar S. & Adrian R. J. 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189214.
Chertkov M., Falkovich G., Kolokolov I. & Vergassola M. 1999 Small-scale turbulent dynamo. Phys. Rev. Lett. 83, 40654068.
Chong M. S., Perry A. E. & Cantwell B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids A 2, 765777.
Chuychai P., Ruffolo D., Matthaeus W. H. & Rowlands G. 2005 Suppressed diffusive escape of topologically trapped magnetic field lines. Astrophys. J. 633, L49L52.
Démoulin P. 2006 Extending the concept of separatrices to QSLs for magnetic reconnection. Adv. Space Res. 37, 12691282.
Donzis D. A., Yeung P. K. & Sreenivasan K. R. 2008 Dissipation and enstrophy in isotropic turbulence: resolution effects and scaling in direct numerical simulations. Phys. Fluids 20, 045108.
Evans T. E., Roeder R. K. W., Carter J. A. & Rapoport B. I. 2004 Homoclinic tangles, bifurcations and edge stochasticity in diverted tokamaks. Contrib. Plasma Phys. 44, 235240.
Farazmand M. & Haller G. 2012 Computing Lagrangian coherent structures from their variational theory. Chaos 22, 013128.
Grasso D., Borgogno D., Pegoraro F. & Schep T. J. 2010 Barriers to eld line transport in 3D magnetic congurations. J. Phys.: Conf. Ser. 260, 012012.
Green M. A., Rowley C. W. & Haller G. 2007 Detection of Lagrangian coherent structures in three-dimensional turbulence. J. Fluid Mech. 572, 111120.
Haller G. 2001 Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149, 248277.
Haller G. 2005 An objective definition of a vortex. J. Fluid Mech. 525, 126.
Haller G. 2011 A variational theory of hyperbolic Lagrangian coherent structures. Physica D 240, 574598.
Haller G. & Beron-Vera F. J. 2012 Geodesic theory of transport barriers in two-dimensional flows. Physica D 241, 16801702.
Haller G. & Yuan G. 2000 Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D 147, 352370.
Hunt J. C. R., Wray A. A. & Moin P. 1988 Eddies, streams, and convergence zones in turbulent flows. Center for Turbulence Research Report CTR-S88,
Jeong J. & Hussain F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.
Lawson S. J. & Barakos G. N. 2010 Computational fluid dynamics analyses of flow over weapons-bay geometries. J. Aircraft 47, 16051623.
Leoncini X., Agullo O., Muraglia M. & Chandre C. 2006 From chaos of lines to Lagrangian structures in flux conservative fields. Eur. Phys. J. B 53, 351360.
Madrid J. A. J. & Mancho A. M. 2009 Distinguished trajectories in time dependent vector fields. Chaos 19, 013111.
Mendoza C. & Mancho A. M. 2010 Hidden geometry of ocean flows. Phys. Rev. Lett. 105, 038501.
Mendoza C., Mancho A. M. & Rio M.-H. 2010 The turnstile mechanism across the Kuroshio current: analysis of dynamics in altimeter velocity fields. Nonlinear Process. Geophys. 17, 103111.
Moffatt H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Peacock T. & Dabiri J. 2010 Introduction to focus issue: Lagrangian coherent structures. Chaos 20, 017501.
Rempel E. L., Chian A. C.-L. & Brandenburg A. 2011 Lagrangian coherent structures in nonlinear dynamos. Astrophys. J. Lett. 735, L9 (7pp).
Rempel E. L., Chian A.C.-L. & Brandenburg A. 2012 Lagrangian chaos in an ABC-forced nonlinear dynamo. Phys. Scr. 86, 018405.
Ruffolo D., Matthaeus W. H. & Chuychai P. 2003 Trapping of solar energetic particles by the small-scale topology of solar wind turbulence. Astrophys. J. 597, L169L172.
Santos J. C., Büchner J., Madjarska M. S. & Alves M. V. 2008 On the relation between DC current locations and an EUV bright point: a case study. Astron. Astrophys. 490, 345352.
Seripienlert A., Ruffolo D., Matthaeus W. H. & Chuychai P. 2010 Dropouts in solar energetic particles: associated with local trapping boundaries or current sheets? Astrophys. J. 711, 980989.
Servidio S., Matthaeus W. H., Shay M. A., Dmitruk P., Cassak P. A. & Wan M. 2010 Statistics of magnetic reconnection in two-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 17, 032315.
Shadden S. C. 2011 Lagrangian coherent structures. In Transport and Mixing in Laminar Flows, pp. 5989. Wiley-VCH Verlag GmbH & Co. KGaA.
Shadden S. C., Lekien F. & Marsden J. E. 2005 Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows. Physica D 212, 271304.
Varun A. V., Balasubramanian K. & Sujith R. I. 2008 An automated vortex detection scheme using the wavelet transform of the d2 field. Exp. Fluids 45, 857868.
Voth G. A., Haller G. & Gollub J. P. 2002 Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Lett. 88, 254501.
Weiss J. 1991 The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273294.
Yeates A. R. & Hornig G. 2011 A generalized flux function for three-dimensional magnetic reconnection. Phys. Plasmas 18, 102118.
Yeates A. R., Hornig G. & Welsch B. T. 2012 Lagrangian coherent structures in photospheric flows and their implications for coronal magnetic structures. Astron. Astrophys. 539, A1 (9pp).
Zel’dovich Ya. B., Ruzmaikin A. A., Molchanov S. A. & Sokoloff D. D. 1984 Kinematic dynamo problem in a linear velocity field. J. Fluid Mech. 144, 111.
Zhong J., Huang T. S. & Adrian R. J. 1998 Extracting 3D vortices in turbulent fluid flow. IEEE Trans. Pattern Anal. Mach. Intell. 20, 193199.
Zhou J., Adrian R. J., Balachandar S. & Kendall T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353396.
Zienicke E., Politano H. & Pouquet A. 1998 Variable intensity of Lagrangian chaos in the nonlinear dynamo problem. Phys. Rev. Lett. 81, 46404643.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 35 *
Loading metrics...

Abstract views

Total abstract views: 125 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd January 2018. This data will be updated every 24 hours.