Skip to main content Accessibility help
×
Home
Hostname: page-component-564cf476b6-dr4jh Total loading time: 0.3 Render date: 2021-06-21T14:43:59.454Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit

Published online by Cambridge University Press:  14 June 2012

David G. Dritschel
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
Steven M. Tobias
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Corresponding
E-mail address:

Abstract

In this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number . When , the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour–spectral method, the ‘combined Lagrangian advection method’, formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

Type
Papers
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below.

References

1. Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, 233239.CrossRefGoogle Scholar
2. Biskamp, D. 2003 Magnetohydrodynamic Turbulence. Cambridge University Press.CrossRefGoogle ScholarPubMed
3. Biskamp, D. & Welter, H. 1989 Dynamics of decaying two-dimensional magnetohydrodynamic turbulence. Phys. Fluids B 1, 19641979.CrossRefGoogle Scholar
4. Brandenburg, A. & Nordlund, Å 2011 Astrophysical turbulence modelling. Rep. Prog. Phys. 74, 046901.CrossRefGoogle Scholar
5. Cattaneo, F. & Tobias, S. M. 2005 Interaction between dynamos at different scales. Phys. Fluids 17, 127105.CrossRefGoogle Scholar
6. Cattaneo, F. & Vainshtein, S. I. 1991 Suppression of turbulent transport by a weak magnetic field. Astrophys. J. 376, L21.CrossRefGoogle Scholar
7. Diamond, P. H., Itoh, S.-I., Itoh, K. & Silvers, L. J. 2007 -plane MHD turbulence and dissipation in the solar tachocline. In The Solar Tachocline (ed. Hughes, D. W., Rosner, R. & Weiss, N. O. ), p. 213. Cambridge University Press.CrossRefGoogle Scholar
8. Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields. Q. J. R. Meteorol. Soc. 123, 10971130.CrossRefGoogle Scholar
9. Dritschel, D. G. & Fontane, J. 2010 The combined Lagrangian advection method. J. Comput. Phys. 229, 54085417.CrossRefGoogle Scholar
10. Dritschel, D. G. & McIntyre, M. E. 2008 Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci. 65, 855874.CrossRefGoogle Scholar
11. Dritschel, D. G. & Scott, R. K. 2009 On the simulation of nearly inviscid two-dimensional turbulence. J. Comput. Phys. 228, 27072711.CrossRefGoogle Scholar
12. Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V. 2008a Unifying scaling theory for vortex dynamics in two-dimensional turbulence. Phys. Rev. Lett. 101 (9), 094501.CrossRefGoogle ScholarPubMed
13. Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V. 2008b Late time evolution of unforced inviscid two-dimensional turbulence. J. Fluid Mech. 640, 215233.CrossRefGoogle Scholar
14. Dritschel, D. G. & Viúdez, Á. 2003 A balanced approach to modelling rotating stably stratified geophysical flows. J. Fluid Mech. 488, 123150.CrossRefGoogle Scholar
15. Fontane, J. & Dritschel, D. G. 2009 The HyperCASL algorithm: a new approach to the numerical simulation of geophysical flows. J. Comput. Phys. 228, 64116425.CrossRefGoogle Scholar
16. Gill, A. E. 1982 Atmosphere-Ocean Dynamics. Academic.Google Scholar
17. Gilman, P. A. 2000 Magnetohydrodynamic ‘Shallow Water’ equations for the solar tachocline. Astrophys. J. 544, L7982.CrossRefGoogle Scholar
18. Gough, D. O. 2007 An introduction to the solar tachocline. In The Solar Tachocline (ed. Hughes, D. W., Rosner, R. & Weiss, N. O. ), p. 3 Cambridge University Press.CrossRefGoogle Scholar
19. Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential-vorticity maps. Q. J. R. Meteorol. Soc. 111, 877946.CrossRefGoogle Scholar
20. Iroshnikov, P. S. 1967 Turbulence of a conducting fluid in a strong magnetic field. Sov. Astron. 7, 566571.Google Scholar
21. Kraichnan, R. H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 13851387.CrossRefGoogle Scholar
22. Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 14171423.CrossRefGoogle Scholar
23. McKiver, W. & Dritschel, D. G. 2008 Balance in non-hydrostatic rotating stratified turbulence. J. Fluid Mech. 596, 201219.CrossRefGoogle Scholar
24. Miesch, M. S. 2005 Large-scale dynamics of the convection zone and tachocline. Living Rev. Solar Phys. 2, 1.CrossRefGoogle Scholar
25. Ossendrijver, M. 2003 The solar dynamo. Astron. Astrophys. Rev. 11, 287367.CrossRefGoogle Scholar
26. 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.CrossRefGoogle Scholar
27. Tobias, S. M. 2005 The solar tachocline: formation, stability and its role in the solar dynamo. In Fluid Dynamics and Dynamos in Astrophysics and Geophysics (ed. Soward, A. M., Jones, C. A., Hughes, D. W. & Weiss, N. O. ), p. 193. CRC.Google Scholar
28. Tobias, S. M. 2010 The solar tachocline: a strudy in stably stratified MHD turbulence. In IUTAM Symposium on Turbulence in the Atmosphere and Oceans (ed. Dritschel, D. G. ), pp. 169179. Springer.CrossRefGoogle Scholar
29. Tobias, S. M. & Cattaneo, F. 2008 Dynamo action in complex flows: the quick and the fast. J. Fluid Mech. 601, 101122.CrossRefGoogle Scholar
30. Tobias, S. M., Cattaneo, F. & Boldyrev, S. 2012 MHD dynamos & turbulence. In The Nature of Turbulence. Cambridge University Press.Google Scholar
31. Tobias, S. M., Dagon, K. & Marston, J. B. 2011 Astrophysical fluid dynamics via direct statistical simulation. Astrophys. J. 727, 127.CrossRefGoogle Scholar
32. Tobias, S. M., Diamond, P. H. & Hughes, D. W. 2007 -plane magnetohydrodynamic turbulence in the solar tachocline. Astrophys. J. 667, L113L116.CrossRefGoogle Scholar
33. Tobias, S. & Weiss, N. 2007 The solar dynamo and the tachocline. In The Solar Tachocline (ed. Hughes, D. W., Rosner, R. & Weiss, N. O. ), pp. 319350. Cambridge University Press.CrossRefGoogle Scholar
34. Weiss, N. O. 1966 The expulsion of magnetic flux by eddies. Proc. R. Soc. Lond. A 293, 310328.CrossRefGoogle Scholar
35. Wood, T. S. & McIntyre, M. E. 2011 Polar confinement of the Sun’s interior magnetic field by laminar magnetostrophic flow. J. Fluid Mech. 677, 445482.CrossRefGoogle Scholar
11
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *