The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers

A. A. Schekochihin; N. E.L. Haugen; A. Brandenburg; S. C. Cowley; J. L. Maron; J. C. McWllliams

doi:10.1086/431214

The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers

A. A. Schekochihin, N. E.L. Haugen, A. Brandenburg, S. C. Cowley, J. L. Maron, J. C. McWllliams

Research output: Contribution to journal › Review article › peer-review

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Abstract

We study numerically the dependence of the critical magnetic Reynolds number Rm_c for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rm_c → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rm_c/Re = Pm_c → const as Re → ∞ (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rm_c grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rm_c ∼ 500 for Re ∼ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rm_c, is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve Rm_c(Re).

Original language	English (US)
Pages (from-to)	L115-L118
Journal	Astrophysical Journal
Volume	625
Issue number	2 II
DOIs	https://doi.org/10.1086/431214
State	Published - Jun 1 2005
Externally published	Yes

All Science Journal Classification (ASJC) codes

Astronomy and Astrophysics
Space and Planetary Science

Keywords

MHD
Magnetic fields
Methods: numerical
Turbulence

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10.1086/431214

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@article{397dd76a87d24a5db269098b41fcc909,

title = "The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers",

abstract = "We study numerically the dependence of the critical magnetic Reynolds number Rmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rmc → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rmc/Re = Pmc → const as Re → ∞ (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rmc grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rmc ∼ 500 for Re ∼ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rmc, is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve Rmc(Re).",

keywords = "MHD, Magnetic fields, Methods: numerical, Turbulence",

author = "Schekochihin, {A. A.} and Haugen, {N. E.L.} and A. Brandenburg and Cowley, {S. C.} and Maron, {J. L.} and McWllliams, {J. C.}",

note = "Funding Information: It is a pleasure to acknowledge discussions with participants of the program “Magnetohydrodynamics of Stellar Interiors” at the Isaac Newton Institute, Cambridge (UK), especially S. Fauve, N. Kleeorin, Y. Ponty, M. Proctor, and I. Rogachevskii. We also thank C. Forest and J.-F. Pinton for discussions of both laboratory and numerical dynamos. Simulations were done at the UKAFF (Leicester), NCSA (Illinois), Norwegian High Performance Computing Consortium (Trondheim and Bergen), and the Danish Center for Scientific Computing. This work was supported in part by NSF grant AST 00-98670 and by the US DOE Center for Mutiscale Plasma Dynamics. A. A. S. was supported by the UKAFF Fellowship. N. E. L. H. was supported in part by the David Crighton Visiting Fellowship (DAMTP, Cambridge).",

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doi = "10.1086/431214",

language = "English (US)",

volume = "625",

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T1 - The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers

AU - Schekochihin, A. A.

AU - Haugen, N. E.L.

AU - Brandenburg, A.

AU - Cowley, S. C.

AU - Maron, J. L.

AU - McWllliams, J. C.

N1 - Funding Information: It is a pleasure to acknowledge discussions with participants of the program “Magnetohydrodynamics of Stellar Interiors” at the Isaac Newton Institute, Cambridge (UK), especially S. Fauve, N. Kleeorin, Y. Ponty, M. Proctor, and I. Rogachevskii. We also thank C. Forest and J.-F. Pinton for discussions of both laboratory and numerical dynamos. Simulations were done at the UKAFF (Leicester), NCSA (Illinois), Norwegian High Performance Computing Consortium (Trondheim and Bergen), and the Danish Center for Scientific Computing. This work was supported in part by NSF grant AST 00-98670 and by the US DOE Center for Mutiscale Plasma Dynamics. A. A. S. was supported by the UKAFF Fellowship. N. E. L. H. was supported in part by the David Crighton Visiting Fellowship (DAMTP, Cambridge).

PY - 2005/6/1

Y1 - 2005/6/1

N2 - We study numerically the dependence of the critical magnetic Reynolds number Rmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rmc → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rmc/Re = Pmc → const as Re → ∞ (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rmc grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rmc ∼ 500 for Re ∼ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rmc, is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve Rmc(Re).

AB - We study numerically the dependence of the critical magnetic Reynolds number Rmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rmc → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rmc/Re = Pmc → const as Re → ∞ (no small-scale dynamo exists at low Pm). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, Rmc grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rmc ∼ 500 for Re ∼ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rmc, is not sensitive to the particular form of the viscous cutoff. This work represents a significant extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve Rmc(Re).

KW - MHD

KW - Magnetic fields

KW - Methods: numerical

KW - Turbulence

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U2 - 10.1086/431214

DO - 10.1086/431214

M3 - Review article

AN - SCOPUS:22544441344

SN - 0004-637X

VL - 625

SP - L115-L118

JO - Astrophysical Journal

JF - Astrophysical Journal

IS - 2 II

ER -

The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers

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