Hostname: page-component-6bf8c574d5-pdxrj Total loading time: 0 Render date: 2025-03-07T02:25:31.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean-field dynamo due to spatio-temporal fluctuations of the turbulent kinetic energy

Published online by Cambridge University Press:  19 October 2023

Kishore Gopalakrishnan Open the ORCID record for Kishore Gopalakrishnan [Opens in a new window]  and
Nishant K. Singh Open the ORCID record for Nishant K. Singh [Opens in a new window]
Kishore Gopalakrishnan*
Affiliation:
Inter-University Centre for Astronomy & Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
Nishant K. Singh*
Affiliation:
Inter-University Centre for Astronomy & Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India
*
†Email addresses for correspondence: kishoreg@iucaa.in, nishant@iucaa.in
†Email addresses for correspondence: kishoreg@iucaa.in, nishant@iucaa.in
Get access Rights & Permissions [Opens in a new window]

Abstract

In systems where the standard $\alpha$ effect is inoperative, one often explains the existence of mean magnetic fields by invoking the ‘incoherent $\alpha$ effect’, which appeals to fluctuations of the mean kinetic helicity at a mesoscale. Most previous studies, while considering fluctuations in the mean kinetic helicity, treated the mean turbulent kinetic energy at the mesoscale as a constant, despite the fact that both these quantities involve second-order velocity correlations. The mean turbulent kinetic energy affects the mean magnetic field through both turbulent diffusion and turbulent diamagnetism. In this work, we use a double-averaging procedure to analytically show that fluctuations of the mean turbulent kinetic energy at the mesoscale (giving rise to $\eta$-fluctuations at the mesoscale, where the scalar $\eta$ is the turbulent diffusivity) can lead to the growth of a large-scale magnetic field even when the kinetic helicity is zero pointwise. Constraints on the operation of such a dynamo are expressed in terms of dynamo numbers that depend on the correlation length, correlation time and strength of these fluctuations. In the white-noise limit, we find that these fluctuations reduce the overall turbulent diffusion, while also contributing a drift term which does not affect the growth of the field. We also study the effects of non-zero correlation time and anisotropy. Turbulent diamagnetism, which arises due to inhomogeneities in the turbulent kinetic energy, leads to growing mean-field solutions even when the $\eta$-fluctuations are statistically isotropic.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 973 , 25 October 2023 , A29
Check for updates
Copyright
© The Author(s), 2023. Published by 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

Abramenko, V.I., Carbone, V., Yurchyshyn, V., Goode, P.R., Stein, R.F., Lepreti, F., Capparelli, V. & Vecchio, A. 2011 Turbulent diffusion in the photosphere as derived from photospheric bright point motion. Astrophys. J. 743 (2), 133.CrossRefGoogle Scholar
Bahng, J. & Schwarzschild, M. 1961 Lifetime of solar granules. Astrophys. J. 134, 312–323.CrossRefGoogle Scholar
Brandenburg, A., Rädler, K.-H., Rheinhardt, M. & Käpylä, P.J. 2008 Magnetic diffusivity tensor and dynamo effects in rotating and shearing turbulence. Astrophys. J. 676 (1), 740–751.CrossRefGoogle Scholar
Brandenburg, A. & Subramanian, K. 2005 Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417 (1–4), 1–209.CrossRefGoogle Scholar
Busse, F.H. & Wicht, J. 1992 A simple dynamo caused by conductivity variations. Geophys. Astrophys. Fluid Dyn. 64 (1–4), 135–144.CrossRefGoogle Scholar
Giesecke, A., Nore, C., Stefani, F., Gerbeth, G., Léorat, J., Luddens, F. & Guermond, J.-L. 2010 Electromagnetic induction in non-uniform domains. Geophys. Astrophys. Fluid Dyn. 104 (5–6), 505–529.CrossRefGoogle Scholar
Gressel, O., Rüdiger, G. & Elstner, D. 2023 Alpha tensor and dynamo excitation in turbulent fluids with anisotropic conductivity fluctuations. Astron. Nachr. 344 (3), e210039. https://onlinelibrary.wiley.com/doi/pdf/10.1002/asna.20210039.CrossRefGoogle Scholar
Harris, C.R., et al. 2020 Array programming with NumPy. Nature 585 (7825), 357–362.CrossRefGoogle ScholarPubMed
Hunter, J.D. 2007 Matplotlib: a 2d graphics environment. Comput. Sci. Engng 9 (3), 90–95.CrossRefGoogle Scholar
Jones, C.A. 2011 Planetary magnetic fields and fluid dynamos. Annu. Rev. Fluid Mech. 43 (1), 583–614.CrossRefGoogle Scholar
Knobloch, E. 1977 The diffusion of scalar and vector fields by homogeneous stationary turbulence. J. Fluid Mech. 83 (1), 129–140.CrossRefGoogle Scholar
Kraichnan, R.H. 1976 Diffusion of weak magnetic fields by isotropic turbulence. J. Fluid Mech. 75 (part 4), 657–676.CrossRefGoogle Scholar
Kraichnan, R.H. 1977 Eulerian and Lagrangian renormalization in turbulence theory. J. Fluid Mech. 83 (2), 349–374.CrossRefGoogle Scholar
Krause, F. & Rädler, K.-H. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory, 1st edn. Pergamon.Google Scholar
Moffatt, H.K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Moffatt, H.K. 1983 Transport effects associated with turbulence with particular attention to the influence of helicity. Rep. Prog. Phys. 46 (5), 621–664.CrossRefGoogle Scholar
Monin, A.S. & Yaglom, A.M. 1971 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. MIT.Google Scholar
Monin, A.S. & Yaglom, A.M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2. MIT.Google Scholar
Nicklaus, B. & Stix, M. 1988 Corrections to first order smoothing in mean-field electrodynamics. Geophys. Astrophys. Fluid Dyn. 43 (2), 149–166.CrossRefGoogle Scholar
Pouquet, A., Frisch, U. & Léorat, J. 1976 Strong MHD helical turbulence and the nonlinear dynamo effect. J. Fluid Mech. 77 (2), 321–354.CrossRefGoogle Scholar
Pétrélis, F., Alexakis, A. & Gissinger, C. 2016 Fluctuations of electrical conductivity: a new source for astrophysical magnetic fields. Phys. Rev. Lett. 116, 161102.CrossRefGoogle ScholarPubMed
Rincon, F. 2019 Dynamo theories. J. Plasma Phys. 85 (4), 205850401.CrossRefGoogle Scholar
Rincon, F. & Rieutord, M. 2018 The Sun's supergranulation. Living Rev. Solar Phys. 15 (1), 6.CrossRefGoogle Scholar
Roberts, P.H. & Soward, A.M. 1975 A unified approach to mean field electrodynamics. Astron. Nachr. 296 (2), 49–64.CrossRefGoogle Scholar
Rogers, T.M. & McElwaine, J.N. 2017 The hottest hot jupiters may host atmospheric dynamos. Astrophys. J. Lett. 841 (2), L26.CrossRefGoogle Scholar
Roudier, Th. & Muller, R. 1986 Structure of the solar granulation. Sol. Phys. 107 (1), 11–26.CrossRefGoogle Scholar
Ruzmaikin, A.A., Shukurov, A.M. & Sokoloff, D.D. 1988 Magnetic Fields of Galaxies, Astrophysics and Space Science Library, vol. 113. Kluwer Academic Publishers.CrossRefGoogle Scholar
Rädler, K.-H., Kleeorin, N. & Rogachevskii, I. 2003 The mean electromotive force for MHD turbulence: the case of a weak mean magnetic field and slow rotation. Geophys. Astrophys. Fluid Dyn. 97 (3), 249–274.CrossRefGoogle Scholar
Silant'ev, N.A. 1999 Large-scale amplification of a magnetic field by turbulent helicity fluctuations. J. Expl Theor. Phys. 89, 45–55.CrossRefGoogle Scholar
Silant'ev, N.A. 2000 Magnetic dynamo due to turbulent helicity fluctuations. Astron. Astrophys. 364, 339–347.Google Scholar
Singh, N.K. 2016 Moffatt-drift-driven large-scale dynamo due to $\alpha$ fluctuations with non-zero correlation times. J. Fluid Mech. 798, 696–716.CrossRefGoogle Scholar
Singh, N.K. & Jingade, N. 2015 Numerical studies of dynamo action in a turbulent shear flow. I. Astrophys. J. 806 (1), 118.CrossRefGoogle Scholar
Sokolov, D.D. 1997 The disk dynamo with fluctuating spirality. Astron. Rep. 41 (1), 68–72.Google Scholar
Sridhar, S. & Singh, N.K. 2014 Large-scale dynamo action due to ${\alpha }$ fluctuations in a linear shear flow. Mon. Not. R. Astron. Soc. 445 (4), 3770–3787.CrossRefGoogle Scholar
Vainshtein, S.I. & Kichatinov, L.L. 1983 The macroscopic magnetohydrodynamics of inhomogeneously turbulent cosmic plasmas. Geophys. Astrophys. Fluid Dyn. 24 (4), 273–298.CrossRefGoogle Scholar
Vishniac, E.T. & Brandenburg, A. 1997 An incoherent ${\alpha }-{\varOmega }$ dynamo in accretion disks. Astrophys. J. 475 (1), 263–274.CrossRefGoogle Scholar
Yousef, T.A., Heinemann, T., Schekochihin, A.A., Kleeorin, N., Rogachevskii, I., Iskakov, A.B., Cowley, S.C. & McWilliams, J.C. 2008 Generation of magnetic field by combined action of turbulence and shear. Phys. Rev. Lett. 100, 184501.CrossRefGoogle ScholarPubMed