Magnetic Eddy Viscosity of Mean Sheared Flows in Two-Dimensional Magnetohydrodynamics

Parker, J B; Constantinou, N C

doi:10.1103/PhysRevFluids.4.083701

Title: Magnetic Eddy Viscosity of Mean Sheared Flows in Two-Dimensional Magnetohydrodynamics

Journal Article · Tue Aug 27 00:00:00 EDT 2019 · Physical Review Fluids

DOI:https://doi.org/10.1103/PhysRevFluids.4.083701· OSTI ID:1568002

Parker, J B ^[1]; Constantinou, N C ^[2]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Australian National Univ., Canberra, ACT (Australia)

Magnetic induction in magnetohydrodynamic fluids at magnetic Reynolds number (Rm) less than 1 has long been known to cause magnetic drag. Here we show that when Rm $$\gg 1$$ and the fluid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than the kinetic energy, induction due to a mean shear flow leads to a magnetic eddy viscosity. The magnetic viscosity is derived from simple physical arguments, where a coherent response due to shear flow builds up in the magnetic field until decorrelated by turbulent motion. The dynamic viscosity coefficient is approximately $$(B^2_p/2_{μ_0}) τ_{corr}$$, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional case, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero viscosity whenever there is differential rotation. Hence, these results serve as a dynamical generalization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular momentum and may be of importance to zonal flows in astrophysical domains such as the interior of some gas giants.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Cite

Export

Save

Research Organization:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1568002

Alternate ID(s):: OSTI ID: 1559072

Report Number(s):: LLNL-JRNL-767360; 957784; TRN: US2100227

Journal Information:: Physical Review Fluids, Vol. 4, Issue 8; ISSN 2469-990X

Publisher:: American Physical Society (APS)Copyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 4 works

Citation information provided by
Web of Science

References (55)

Eddy Viscosity in Two and Three Dimensions Kraichnan, Robert H. Journal of the Atmospheric Sciences, Vol. 33, Issue 8 https://doi.org/10.1175/1520-0469(1976)033%3C1521:EVITAT%3E2.0.CO;2	journal	August 1976
β-Plane Magnetohydrodynamic Turbulence in the Solar Tachocline Tobias, Steven M.; Diamond, Patrick H.; Hughes, David W. The Astrophysical Journal, Vol. 667, Issue 1 https://doi.org/10.1086/521978	journal	September 2007
Jupiter’s atmospheric jet streams extend thousands of kilometres deep Kaspi, Y.; Galanti, E.; Hubbard, W. B. Nature, Vol. 555, Issue 7695 https://doi.org/10.1038/nature25793	journal	March 2018
Measurement and implications of Saturn’s gravity field and ring mass Iess, L.; Militzer, B.; Kaspi, Y. Science, Vol. 364, Issue 6445 https://doi.org/10.1126/science.aat2965	journal	January 2019
Emergence and Equilibration of Jets in Beta-Plane Turbulence: Applications of Stochastic Structural Stability Theory Constantinou, Navid C.; Farrell, Brian F.; Ioannou, Petros J. Journal of the Atmospheric Sciences, Vol. 71, Issue 5 https://doi.org/10.1175/JAS-D-13-076.1	journal	May 2014
Interaction between eddies and mean flow in Jupiter's atmosphere: Analysis of Cassini imaging data Salyk, Colette; Ingersoll, Andrew P.; Lorre, Jean Icarus, Vol. 185, Issue 2 https://doi.org/10.1016/j.icarus.2006.08.007	journal	December 2006
Observation of turbulent-driven shear flow in a cylindrical laboratory plasma device Tynan, G. R.; Holland, C.; Yu, J. H. Plasma Physics and Controlled Fusion, Vol. 48, Issue 4 https://doi.org/10.1088/0741-3335/48/4/S05	journal	March 2006
Zonal flow magnetic field interaction in the semi-conducting region of giant planets Cao, Hao; Stevenson, David J. Icarus, Vol. 296 https://doi.org/10.1016/j.icarus.2017.05.015	journal	November 2017
Observation of Turbulent-Driven Shear Flow in a Cylindrical Laboratory Plasma Device Holland, C.; Yu, J. H.; James, A. Physical Review Letters, Vol. 96, Issue 19 https://doi.org/10.1103/PhysRevLett.96.195002	journal	May 2006
Magnetic Suppression of Zonal Flows on a Beta Plane Constantinou, Navid C.; Parker, Jeffrey B. The Astrophysical Journal, Vol. 863, Issue 1 https://doi.org/10.3847/1538-4357/aace53	journal	August 2018
Generation of Mean Flows and Jets on a Beta Plane and over Topography Vallis, Geoffrey K.; Maltrud, Matthew E. Journal of Physical Oceanography, Vol. 23, Issue 7 https://doi.org/10.1175/1520-0485(1993)023%3C1346:GOMFAJ%3E2.0.CO;2	journal	July 1993
Zonal flow as pattern formation Parker, Jeffrey B.; Krommes, John A. Physics of Plasmas, Vol. 20, Issue 10 https://doi.org/10.1063/1.4828717	journal	October 2013
Ab Initio Simulations for Material Properties Along the Jupiter Adiabat French, Martin; Becker, Andreas; Lorenzen, Winfried The Astrophysical Journal Supplement Series, Vol. 202, Issue 1 https://doi.org/10.1088/0067-0049/202/1/5	journal	August 2012
Saturn's Gravitational Field, Internal Rotation, and Interior Structure Anderson, J. D.; Schubert, G. Science, Vol. 317, Issue 5843 https://doi.org/10.1126/science.1144835	journal	September 2007
Anelastic spherical dynamos with radially variable electrical conductivity Dietrich, W.; Jones, C. A. Icarus, Vol. 305 https://doi.org/10.1016/j.icarus.2018.01.003	journal	May 2018
Interactions of disparate scales in drift-wave turbulence Krommes, John A.; Kim, Chang-Bae Physical Review E, Vol. 62, Issue 6 https://doi.org/10.1103/PhysRevE.62.8508	journal	December 2000
Saturn's Deep Atmospheric Flows Revealed by the Cassini Grand Finale Gravity Measurements Galanti, E.; Kaspi, Y.; Miguel, Y. Geophysical Research Letters, Vol. 46, Issue 2 https://doi.org/10.1029/2018GL078087	journal	January 2019
Structure and Spacing of Jets in Barotropic Turbulence Farrell, Brian F.; Ioannou, Petros J. Journal of the Atmospheric Sciences, Vol. 64, Issue 10 https://doi.org/10.1175/JAS4016.1	journal	October 2007
Dynamics of zonal flows: failure of wave-kinetic theory, and new geometrical optics approximations Parker, Jeffrey B. Journal of Plasma Physics, Vol. 82, Issue 6 https://doi.org/10.1017/S0022377816001021	journal	November 2016
Generation of zonal flows through symmetry breaking of statistical homogeneity Parker, Jeffrey B.; Krommes, John A. New Journal of Physics, Vol. 16, Issue 3 https://doi.org/10.1088/1367-2630/16/3/035006	journal	March 2014
The Non-uniform Rotation of the Sun and its Magnetic Field Ferraro, V. C. A. Monthly Notices of the Royal Astronomical Society, Vol. 97, Issue 6 https://doi.org/10.1093/mnras/97.6.458	journal	April 1937
Cassini Ring Seismology as a Probe of Saturn’s Interior. I. Rigid Rotation Mankovich, Christopher; Marley, Mark S.; Fortney, Jonathan J. The Astrophysical Journal, Vol. 871, Issue 1 https://doi.org/10.3847/1538-4357/aaf798	journal	January 2019
S3T Stability of the Homogeneous State of Barotropic Beta-Plane Turbulence Bakas, Nikolaos A.; Constantinou, Navid C.; Ioannou, Petros J. Journal of the Atmospheric Sciences, Vol. 72, Issue 5 https://doi.org/10.1175/JAS-D-14-0213.1	journal	May 2015
Constraints on deep-seated zonal winds inside Jupiter and Saturn Liu, J.; Goldreich, P.; Stevenson, D. Icarus, Vol. 196, Issue 2 https://doi.org/10.1016/j.icarus.2007.11.036	journal	August 2008
Jovian atmospheric dynamics: an update after Galileo and Cassini Vasavada, Ashwin R.; Showman, Adam P. Reports on Progress in Physics, Vol. 68, Issue 8 https://doi.org/10.1088/0034-4885/68/8/R06	journal	July 2005
On the Changes in the Spectral Distribution of Kinetic Energy for Twodimensional, Nondivergent Flow Fjørtoft, Ragnar Tellus, Vol. 5, Issue 3 https://doi.org/10.3402/tellusa.v5i3.8647	journal	January 1953
Note Concerning the Nature of the Large-Scale Eddies in the Atmosphere Starr, Victor P. Tellus, Vol. 5, Issue 4 https://doi.org/10.3402/tellusa.v5i4.8694	journal	January 1953
Anelastic dynamo models with variable electrical conductivity: An application to gas giants Duarte, Lúcia D. V.; Gastine, Thomas; Wicht, Johannes Physics of the Earth and Planetary Interiors, Vol. 222 https://doi.org/10.1016/j.pepi.2013.06.010	journal	September 2013
Zonal Flows in Two-Dimensional Decaying Magnetohydrodynamic Turbulence on a β-Plane Zinyakov, T. A.; Petrosyan, A. S. JETP Letters, Vol. 108, Issue 2 https://doi.org/10.1134/S002136401814014X	journal	July 2018
Saturn eddy momentum fluxes and convection: First estimates from Cassini images Delgenio, A.; Barbara, J.; Ferrier, J. Icarus, Vol. 189, Issue 2 https://doi.org/10.1016/j.icarus.2007.02.013	journal	August 2007
Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics Gama, S.; Vergassola, M.; Frisch, U. Journal of Fluid Mechanics, Vol. 260 https://doi.org/10.1017/S0022112094003459	journal	February 1994
Transport and instability in driven two-dimensional magnetohydrodynamic flows Durston, Sam; Gilbert, Andrew D. Journal of Fluid Mechanics, Vol. 799 https://doi.org/10.1017/jfm.2016.361	journal	June 2016
Constraints on Saturn’s tropospheric general circulation from Cassini ISS images Del Genio, Anthony D.; Barbara, John M. Icarus, Vol. 219, Issue 2 https://doi.org/10.1016/j.icarus.2012.03.035	journal	June 2012
On the Mechanism Underlying the Spontaneous Emergence of Barotropic Zonal Jets Bakas, Nikolaos A.; Ioannou, Petros J. Journal of the Atmospheric Sciences, Vol. 70, Issue 7 https://doi.org/10.1175/JAS-D-12-0102.1	journal	July 2013
Statistical state dynamics of vertically sheared horizontal flows in two-dimensional stratified turbulence Fitzgerald, Joseph G.; Farrell, Brian F. Journal of Fluid Mechanics, Vol. 854 https://doi.org/10.1017/jfm.2018.560	journal	September 2018
A complex dynamo inferred from the hemispheric dichotomy of Jupiter’s magnetic field Moore, Kimberly M.; Yadav, Rakesh K.; Kulowski, Laura Nature, Vol. 561, Issue 7721 https://doi.org/10.1038/s41586-018-0468-5	journal	September 2018
Magnetic damping of jets and vortices Davidson, P. A. Journal of Fluid Mechanics, Vol. 299 https://doi.org/10.1017/S0022112095003466	journal	September 1995
Astrophysical Fluid Dynamics via Direct Statistical Simulation Tobias, S. M.; Dagon, K.; Marston, J. B. The Astrophysical Journal, Vol. 727, Issue 2 https://doi.org/10.1088/0004-637X/727/2/127	journal	January 2011
A dynamo model of Jupiter’s magnetic field Jones, C. A. Icarus, Vol. 241 https://doi.org/10.1016/j.icarus.2014.06.020	journal	October 2014
Zonostrophic Instability Srinivasan, Kaushik; Young, W. R. Journal of the Atmospheric Sciences, Vol. 69, Issue 5 https://doi.org/10.1175/JAS-D-11-0200.1	journal	May 2012
Magnetic drag on hot Jupiter Atmospheric Winds Perna, Rosalba; Menou, Kristen; Rauscher, Emily The Astrophysical Journal, Vol. 719, Issue 2 https://doi.org/10.1088/0004-637X/719/2/1421	journal	July 2010
On the relationship between zonal jets and dynamo action in giant planets: ZONAL JETS AND DYNAMO ACTION Heimpel, Moritz; Gómez Pérez, Natalia Geophysical Research Letters, Vol. 38, Issue 14 https://doi.org/10.1029/2011GL047562	journal	July 2011
A suppression of differential rotation in Jupiter’s deep interior Guillot, T.; Miguel, Y.; Militzer, B. Nature, Vol. 555, Issue 7695 https://doi.org/10.1038/nature25775	journal	March 2018
Waves and turbulence on a beta-plane Rhines, Peter B. Journal of Fluid Mechanics, Vol. 69, Issue 3 https://doi.org/10.1017/S0022112075001504	journal	June 1975
On the Changes in the Spectral Distribution of Kinetic Energy for Twodimensional, Nondivergent Flow FjØRtoft, Ragnar Tellus, Vol. 5, Issue 3 https://doi.org/10.1111/j.2153-3490.1953.tb01051.x	journal	August 1953
Note Concerning the Nature of the Large-Scale Eddies in the Atmosphere Starr, Victor P. Tellus, Vol. 5, Issue 4 https://doi.org/10.1111/j.2153-3490.1953.tb01079.x	journal	November 1953
Astrophysical Fluid Dynamics via Direct Statistical Simulation Tobias, S. M.; Dagon, K.; Marston, J. B. arXiv https://doi.org/10.48550/arxiv.1009.2684	text	January 2010
Emergence and equilibration of jets in beta-plane turbulence: applications of Stochastic Structural Stability Theory Constantinou, Navid C.; Farrell, Brian F.; Ioannou, Petros J. arXiv https://doi.org/10.48550/arxiv.1208.5665	text	January 2012
Zonal Flow as Pattern Formation Parker, Jeffrey B.; Krommes, John A. arXiv https://doi.org/10.48550/arxiv.1301.5059	text	January 2013
S3T stability of the homogeneous state of barotropic beta-plane turbulence Bakas, Nikolaos A.; Constantinou, Navid C.; Ioannou, Petros J. arXiv https://doi.org/10.48550/arxiv.1407.3354	text	January 2014
Dynamics of Zonal Flows: Failure of Wave-Kinetic Theory, and New Geometrical Optics Approximations Parker, Jeffrey B. arXiv https://doi.org/10.48550/arxiv.1604.06904	text	January 2016
Statistical State Dynamics of Vertically Sheared Horizontal Flows in Two-Dimensional Stratified Turbulence Fitzgerald, Joseph G.; Farrell, Brian F. arXiv https://doi.org/10.48550/arxiv.1612.03243	preprint	January 2016
Zonal Flow Magnetic Field Interaction in the Semi-Conducting Region of Giant Planets Cao, Hao; Stevenson, David J. arXiv https://doi.org/10.48550/arxiv.1703.10273	text	January 2017
Cassini Ring Seismology as a Probe of Saturn's Interior I: Rigid Rotation Mankovich, Christopher; Marley, Mark S.; Fortney, Jonathan J. arXiv https://doi.org/10.48550/arxiv.1805.10286	text	January 2018
Saturn's deep atmospheric flows revealed by the Cassini Grand Finale gravity measurements Galanti, Eli; Kaspi, Yohai; Miguel, Yamila arXiv https://doi.org/10.48550/arxiv.1902.04268	text	January 2019

Similar Records

Eddy viscosity and laminarization of sheared flow in three dimensional reduced magnetohydrodynamic turbulence

Journal Article · Wed Aug 01 00:00:00 EDT 2001 · Physics of Plasmas · OSTI ID:1568002

Kim, Eun-jin; Hahm, T S; Diamond, P H

Three-fluid, three-dimensional magnetohydrodynamic solar wind model with eddy viscosity and turbulent resistivity

Journal Article · Tue Jun 10 00:00:00 EDT 2014 · Astrophysical Journal · OSTI ID:1568002

Usmanov, Arcadi V.; Matthaeus, William H.

Eddy viscosity and flow properties of the solar wind: Co-rotating interaction regions, coronal-mass-ejection sheaths, and solar-wind/magnetosphere coupling

Journal Article · Mon May 15 00:00:00 EDT 2006 · Physics of Plasmas · OSTI ID:1568002

Borovsky, Joseph E

Related Subjects

79 ASTRONOMY AND ASTROPHYSICS

Dedalus output from 2D incompressible MHD fluid on a magnetized beta plane Constantinou, Navid C.; Parker, Jeffrey B. https://doi.org/10.5281/zenodo.3892088 The current record is supplemented by this dataset	dataset	January 2020
Dedalus output from 2D incompressible MHD fluid on a magnetized beta plane Constantinou, Navid C.; Parker, Jeffrey B. https://doi.org/10.5281/zenodo.3892089 The current record is supplemented by this dataset	dataset	January 2020

Title: Magnetic Eddy Viscosity of Mean Sheared Flows in Two-Dimensional Magnetohydrodynamics

Citation Formats

References (55)

Linked Research (2)

Similar Records

Related Subjects