Skip to main content Accessibility help

Dynamo action in complex flows: the quick and the fast



We consider the kinematic dynamo problem for a velocity field consisting of a mixture of turbulence and coherent structures. For these flows the dynamo growth rate is determined by a competition between the large flow structures that have large magnetic Reynolds number but long turnover times and the small ones that have low magnetic Reynolds number but short turnover times. We introduce the concept of a quick dynamo as one that reaches its maximum growth rate in some (small) neighbourhood of its critical magnetic Reynolds number. We argue that if the coherent structures are quick dynamos, the overall dynamo growth rate can be predicted by looking at those flow structures that have spatial and temporal scales such that their magnetic Reynolds number is just above critical. We test this idea numerically by studying 2.5-dimensional dynamo action which allows extreme parameter values to be considered. The required velocities, consisting of a mixture of turbulence with a given spectrum and long-lived vortices (coherent structures), are obtained by solving the active scalar equations. By using spectral filtering we demonstrate that the scales responsible for dynamo action are consistent with those predicted by the theory.



Hide All
Batchelor, G. K. 1950 On the spontaneous magnetic field in a conducting liquid in a turbulent motion. Proc. R. Soc. Lond. A 201, 405416.
Boldyrev, S. & Cattaneo, F. 2004 Magnetic-field generation in Kolmogorov turbulence. Phys. Rev. Lett. 92 (14), 144501.
Brummell, N. H., Cattaneo, F. & Tobias, S. M. 2001 Linear and nonlinear dynamo properties of time-dependent ABC flows. Fluid Dyn. Res. 28, 237265.
Cattaneo, F. & Tobias, S. M. 2005 Interaction between dynamos at different scales. Phys. Fluids 17, 127105 16.
Constantin, P., Nie, Q. & Schörghofer, N. 1998 Nonsingular surface quasi-geostrophic flow. Phys. Lett. A 241, 168172.
Galloway, D. J. & Proctor, M. R. E. 1992 Numerical calculations of fast dynamos in smooth velocity fields with realistic diffusion. Nature 356, 691693.
Held, I. M., Pierrehumbert, R. T., Garner, S. T. & Swanson, K. L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 120.
Kazantsev, A. P. 1968 Enhancement of a magnetic field by a conducting fluid. Sov. Phys. JETP 26, 10311031.
Kraichnan, R. H. & Nagarajan, S. 1967 Growth of turbulent magnetic fields. Phys. Fluids 10, 859870.
Llewellyn-Smith, S. G. & Tobias, S. M. 2004 Vortex dynamos. J. Fluid Mech. 498, 121.
McWilliams, J. C. 1984 The emergence of coherent isolated vortices in turbulent flow. J. Fluid Mech. 146, 2143.
Otani, N. F. 1993 A fast kinematic dynamo in two-dimensional time-dependent flows. J. Fluid Mech. 253, 327340.
Parker, E. N. 1979 Cosmical Magnetic Fields: Their Origin and their Activity. Clarendon; Oxford University Press.
Pierrehumbert, R. T., Held, I. M. & Swanson, K. L. 1995 Spectra of local and nonlocal two dimensional turbulence. Chaos, Solitons Fractals 4, 11111116.
Ponty, Y., Mininni, P. D., Montgomery, D. C., Pinton, J.-F., Politano, H. & Pouquet, A. 2005 Numerical study of dynamo action at low magnetic Prandtl number. Phys. Rev. Lett. 94, 164502-14
Ponty, Y., Politano, H. & Pinton, J.-F. 2004 Simulation of induction at low magnetic Prandtl number. Phys. Rev. Lett. 92, 144503-14
Roberts, G. O. 1972 Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc. Lond. A 271, 411454.
Saffman, P. G. 1963 On the fine-scale structure of vector fields convected by a turbulent fluid. J. Fluid Mech. 16, 545572.
Schekochihin, A. A., Cowley, S. C., Maron, J. L. & McWilliams, J. C. 2004 Critical Magnetic Prandtl Number for Small-Scale Dynamo. Phys. Rev. Lett. 92 (5), 054502–+.
Schekochihin, A. A., Haugen, N. E. L., Brandenburg, A., Cowley, S. C., Maron, J. L. & McWilliams, J. C. 2005 The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers. Astrophys. J. 625, L115L118.
Thomson, D. J. & Devenish, B. J. 2005 Particle pair separation in kinematic simulations. J. Fluid Mech. 526, 277302.
Vainshtein, S. I. & Kichatinov, L. L. 1986 The dynamics of magnetic fields in a highly conducting turbulent medium and the generalized Kolmogorov-Fokker-Planck equations. J. Fluid Mech. 168, 7387.
Zel'dovich, Y. B. 1957 The magnetic field in the two dimensional motion of a conducting turbulent liquid. Sov. Phys. JETP 4, 460462.
MathJax is a JavaScript display engine for mathematics. For more information see

Dynamo action in complex flows: the quick and the fast



Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.