We examine the complex nonlinear flow-magnetic field dynamics in magneto-hydrodynamic (MHD) turbulence. Using direct numerical simulations (DNS), we investigate the dynamical interactions subject to the influence of a uniform applied background magnetic field. The initial magnetic and kinetic Reynolds numbers (based on Taylor microscale) are 45 and there are no initial magnetic field fluctuations. The sum total of turbulent magnetic and kinetic energies decays monotonically. With time, the turbulent magnetic fluctuations grow by extracting energy from velocity fluctuations. Expectedly, the distribution of energy between kinetic and magnetic fluctuations exhibits large periodic oscillations from the equipartition state due to Alfvén waves. We perform a detailed analysis of the flow-magnetic field coupling and posit a simple model for the energy interchange. Such dynamical analysis can provide the insight required for turbulence control and closure modeling strategies.

References

References
1.
Alboussière
,
T.
,
Cardin
,
P.
,
Debray
,
F.
,
La Rizza
,
P.
,
Masson
,
J.
,
Plunian
,
F.
,
Ribeiro
,
A.
, and
Schmitt
,
D.
,
2011
, “
Experimental Evidence of Alfvén Wave Propagation in a Gallium Alloy
,”
Phys. Fluids
,
23
, p.
096601
.10.1063/1.3633090
2.
Brandenburg
,
A
.,
2003
, “
Computational Aspects of Astrophysical MHD and Turbulence
,”
Advances in Nonlinear Dynamos, The Fluid Mechanics of Astrophysics and Geophysics
, Vol.
9
,
A. M.
Soward
,
C. A.
Jones
, and
D. W.
Hughes
, eds.,
CRC Press
,
Boca Raton, FL
, pp.
269
344
.
3.
Brandenburg
,
A.
, and
Käpylä
,
P.
,
2007
, “
Magnetic Helicity Effects in Astrophysical and Laboratory Dynamos
,”
New J. Phys.
,
9
, pp.
305
330
.10.1088/1367-2630/9/8/305
4.
Mininni
,
P.
,
Ponty
,
Y.
,
Montgomery
,
D.
,
Pinton
,
J.
,
Politano
,
H.
, and
Pouquet
,
A.
,
2005
, “
Dynamo Regimes With a Nonhelical Forcing
,”
Astrophys. J.
,
626
, pp.
853
863
.10.1086/429911
5.
Pouquet
,
A.
, and
Patterson
,
G.
,
1978
, “
Numerical Simulation of Helical Magnetohydrodynamic Turbulence
,”
J. Fluid Mech.
,
85
(
2
), pp.
305
323
.10.1017/S0022112078000658
6.
Davidson
,
P. A.
,
2001
,
An Introduction to Magnetohydrodynamics
,
Cambridge University Press
,
Cambridge, UK
.
7.
Pope
,
S.
,
2000
,
Turbulent Flows
,
Cambridge University Press
,
Cambridge, UK
.
8.
Davidson
,
P.
,
2004
,
Turbulence: An Introduction for Scientists and Engineers
,
Oxford University Press
,
New York
.
9.
Roy
,
R. I. S.
,
Hastings
,
D. E.
, and
Taylor
,
S.
,
1996
, “
Three-Dimensional Plasma Particle-in-Cell Calculations of Ion Thruster Backflow Contamination
,”
J. Comput. Phys.
,
128
, pp.
6
18
.10.1006/jcph.1996.0192
10.
Haas
,
J. M.
, and
Gallimore
,
A. D.
,
2001
, “
Internal Plasma Potential Profiles in a Laboratory-Model Hall Thruster
,”
Phys. Plasma
,
38
(
2
), pp.
652
660
.10.1063/1.1338535
11.
Tarditi
,
A. G.
, and
Shebalin
,
J. V.
,
2003
, “
Magnetic Nozzle Plasma Exhaust Simulation for the VASIMR Advanced Propulsion Concept
,”
Proceedings of the 28th International Electric Propulsion Conferenc
e
.
12.
Riley
,
B. M.
,
Girimaji
,
S. S.
, and
Richard
,
J. C.
,
2009
, “
Magnetic Field Effects on Axis-Switching and Instabilities in Rectangular Plasma Jets
,”
Flow, Turbul. Combust.
,
82
(
3
), pp.
375
390
.10.1007/s10494-008-9182-y
13.
Macheret
,
S. O.
,
Shneider
,
M. N.
, and
Miles
,
R. B.
,
2002
, “
Magnetohydrodynamic Control of Hypersonic Flows and Scramjets Using Electron Beam Ionization
,”
AIAA J.
,
40
(
1
), pp.
74
81
.10.2514/2.1616
14.
Biskamp
,
D.
,
2003
,
Magnetohydrodynamic Turbulence
,
Cambridge University Press
,
Cambridge, UK
.
15.
Balsara
,
D.
, and
Pouquet
,
A.
,
1999
, “
The Formation of Large-Scale Structures in Supersonic Magnetohydrodynamic Flows
,”
Phys. Plasmas
,
6
, pp.
89
100
.10.1063/1.873263
16.
Chen
,
F. F.
,
1984
,
Introduction to Plasma Physics and Controlled Fusion
,
2nd ed.
,
Springer
,
New York
.
17.
Christensson
,
M.
,
Hindmarsh
,
M.
, and
Brandenburg
,
A.
,
2001
, “
Inverse Cascade in Decaying Three-Dimensional-Magnetohydrodynamic Turbulence
,”
Phys. Rev. E
,
64
(
5
), p.
056405
.10.1103/PhysRevE.64.056405
18.
Frisch
,
U.
,
Pouquet
,
A.
,
Léorat
,
J.
, and
Mazure
,
A.
,
1975
, “
Possibility of an Inverse Cascade of Magnetic Helicity in Magnetohydrodynamic Turbulence
,”
J. Fluid Mech.
,
68
(
4
), pp.
769
778
.10.1017/S002211207500122X
19.
Knaepen
,
B.
,
Kassinos
,
S. C.
, and
Carati
,
D.
,
2004
, “
Magnetohydrodynamic Turbulence at Moderate Magnetic Reynolds Numbers
,”
J. Fluid Mech.
,
513
, pp.
199
220
.10.1017/S0022112004000023
20.
Knaepen
,
B.
, and
Moreau
,
R.
,
2008
, “
Magnetohydrodynamic Turbulence at Low Magnetic Reynolds Number
,”
Ann. Rev. Fluid Mech.
,
40
, pp.
25
45
.10.1146/annurev.fluid.39.050905.110231
21.
Matthaeus
,
W. H.
,
Ghosh
,
S.
,
Oughton
,
S.
, and
Roberts
,
D. A.
,
1996
, “
Anisotropic Three-Dimensional MHD Turbulence
,”
J. Geophys. Res. Space Phys.
,
101
(
A4
), pp.
7619
7629
.10.1029/95JA03830
22.
Miller
,
R. S.
,
Mashayek
,
F.
,
Adumitoraie
,
V.
, and
Givi
,
P.
,
1996
, “
Structure of Homogeneous Nonhelical Magnetohydrodynamic Turbulence
,”
Phys. Plasmas
,
3
(
9
), pp.
3304
3317
.10.1063/1.871599
23.
Ponty
,
Y.
,
Mininni
,
P. D.
,
Montgomery
,
D. C.
,
Pinton
,
J.-F.
,
Politano
,
H.
, and
Pouquet
,
A.
,
2005
. “
Numerical Study of Dynamo Action at Low Magnetic Prandtl Numbers,”
Phys. Rev. Lett.
,
94
(
164502
), p.
164502
.10.1103/PhysRevLett.94.164502
24.
Müller
,
W. C.
, and
Grappin
,
R.
,
2005
, “
Spectral Energy Dynamics in Magnetohydrodynamic Turbulence
,”
Phys. Rev. Lett.
,
95
, p.
114502
.10.1103/PhysRevLett.95.114502
25.
Richard
,
J.
,
Riley
,
B.
, and
Girimaji
,
S.
,
2011
, “
Magnetohydrodynamic Turbulence Decay Under the Influence of Uniform or Random Magnetic Fields
,”
J. Fluids Eng.
,
133
, p.
081205
.10.1115/1.4003985
26.
Shebalin
,
J. V.
,
Matthaeus
,
W. H.
, and
Montgomery
,
D. C.
,
1983
, “
Anisotropy in MHD Turbulence Due to a Mean Magnetic Field
,”
J. Plasma Phys.
,
29
, pp.
525
547
.10.1017/S0022377800000933
27.
Shebalin
,
J. V.
,
2005
, “
Theory and Simulation of Real and Ideal Magnetohydrodynamic Turbulence
,”
Discrete Contin. Dyn. Syst., Ser. B
,
5
(
1
), p.
153174
.10.3934/dcdsb.2005.5.153
28.
Shebalin
,
J
.,
2009
, “
Plasma Relaxation and the Turbulent Dynamo
,”
Phys. Plasmas
,
16
, p.
072301
.10.1063/1.3159866
29.
Yoshizawa
,
A.
,
Itoh
,
S.-I.
, and
Itoh
,
K.
,
2003
,
Plasma and Fluid Turbulence: Theory and Modeling
,
Institute of Physics
,
London
.
30.
Alfvén
,
H.
,
1942
, “
Existence of Electromagnetic-Hydrodynamic Waves
,”
Nature
,
150
(
3805
), pp.
405
406
.10.1038/150405d0
31.
d'Humiéres
,
D.
,
Ginzburg
,
I.
,
Krafczyk
,
M.
,
Lallemand
,
P.
, and
Luo
,
L.-S.
,
2002
, “
Multiple-Relaxation-Time Lattice Boltzmann Models in Three Dimensions
,”
Philos. Trans. R. Soc. Lond. A
,
220
, pp.
437
451
.10.1098/rsta.2001.0955
32.
Eggels
,
J. G. M.
,
1996
, “
Direct and Large-Eddy Simulation of Turbulent Fluid Flow Using the Lattice-Boltzmann Scheme
,”
Int. J. Heat Fluid Flow
,
17
, pp.
307
323
.10.1016/0142-727X(96)00044-6
33.
Girimaji
,
S. S.
,
2007
, “
Boltzmann Kinetic Equation for Filtered Fluid Turbulence
,”
Phys. Rev. Lett.
,
99
, p.
034501
.10.1103/PhysRevLett.99.034501
34.
He
,
X.
, and
Luo
,
L.-S.
,
1997
, “
A Priori Derivation of the Lattice Boltzmann Equation
,”
Phys. Rev. E.
,
55
, pp.
R6333
R6337
.10.1103/PhysRevE.55.R6333
35.
He
,
X.
, and
Luo
,
L.-S.
,
1997
, “
Theory of the Lattice Boltzmann Method: From the Boltzmann Equation to the Lattice Boltzmann Equation a Priori Derivation of the Lattice Boltzmann Equation
,”
Phys. Rev. E.
,
56
,
p
.
6811
–6817.
36.
Lee
,
K.
,
Yu
,
D.
, and
Girimaji
,
S. S.
,
2006
, “
Lattice Boltzmann DNS of Decaying Compressible Isotropic Turbulence With Temperature Fluctuations
,”
Int. J. Comput. Fluid Dyn.
,
20
(
6
), pp.
401
413
.10.1080/10618560601001122
37.
Luo
,
L.-S.
,
1998
, “
Unified Theory of Lattice Boltzmann Models for Nonideal Gases
,”
Phys. Rev. Lett.
,
81
, pp.
1618
1621
.10.1103/PhysRevLett.81.1618
38.
Luo
,
L.-S.
, and
Girimaji
,
S. S.
,
2003
, “
Theory of the Lattice Boltzmann Method: Two-Fluid Model for Binary Mixtures
,”
Phys. Rev. E.
,
67
, p.
036302
.10.1103/PhysRevE.67.036302
39.
Shan
,
X.
, and
Doolen
,
G. D.
,
1993
, “
Lattice Boltzmann Model for Simulating Flows With Multiple Phases and Components
,”
Phys. Rev. E.
,
47
(
3
), pp.
1815
1819
.10.1103/PhysRevE.47.1815
40.
Martys
,
N. S.
,
Shan
,
X.
, and
Chen
,
H.
,
1998
, “
Evaluation of the External Force Term in the Discrete Boltzmann Equation
,”
Phys. Rev. E.
,
58
(
5
), pp.
6855
6857
.10.1103/PhysRevE.58.6855
41.
Yu
,
H.
,
Girimaji
,
S.
, and
Luo
,
L.
,
2005
, “
Lattice Boltzmann Simulations of Decaying Homogeneous Isotropic Turbulence
,”
Phys. Rev. E
,
71
(
1
), p.
016708
.10.1103/PhysRevE.71.016708
42.
Yu
,
D.
, and
Girimaji
,
S.
,
2005
, “
DNS of Homogenous Shear Turbulence Revisited with the Lattice Boltzmann Method
,”
J. Turbul.
6
(
6
), pp.
1
17
.10.1080/14685240500103200
43.
Yu
,
H.
,
Girimaji
,
S. S.
, and
Luo
,
L.-S.
,
2005
, “
DNS and LES of Decaying Isotropic Turbulence With and Without Frame Rotation using Lattice Boltzmann Method
,”
J. Comput. Phys.
,
209
(
2
), pp.
599
616
.10.1016/j.jcp.2005.03.022
44.
Yu
,
D.
, and
Girimaji
,
S. S.
,
2006
, “
Direct Numerical Simulations of Homogeneous Turbulence Subject to Periodic Shear
,”
J. Fluid Mech.
,
566
, pp.
117
151
.10.1017/S0022112006001832
45.
Yu
,
H.
, and
Girimaji
,
S. S.
,
2005
, “
Near-Field Turbulent Simulations of Rectangular Jets Using Lattice Boltzmann Method
,”
Phys. Fluids
,
17
(
12
), p.
125106
.10.1063/1.2140021
46.
Dellar
,
P
.,
2002
, “
Lattice Kinetic Schemes for MHD
,”
J. Comput. Phys.
,
179
, pp.
95
126
.10.1006/jcph.2002.7044
47.
Riley
,
B. M.
,
Richard
,
J. C.
, and
Girimaji
,
S. S.
,
2008
, “
Assessment of Magnetohydrodynamic Lattice Boltzmann Schemes in Turbulence and Rectangular Jets
,”
Int. J. Modern Phys. C, Comput. Phys. Phys. Comput.
,
18
(
8
), pp.
1211
1220
.10.1142/S0129183108012881
48.
Riley
,
B. M.
,
2007
, “
Magnetohydrodynamic Lattice Boltzmann Simulations of Turbulence and Rectangular Jet Flow
,” M.S. thesis, Texas A&M University, College Station, TX.
49.
Krafczyk
,
M.
,
Tölke
,
J.
, and
Luo
,
L.-S.
,
2003
, “
Large-Eddy Simulations With a Multiple-Relaxation-Time LBE Model
,”
Int. J. Mod. Phys. B
,
17
, pp.
33
39
.10.1142/S0217979203017059
50.
Chen
,
S.
, and
Doolen
,
G.
,
2003
, “
Lattice Boltzmann Method for Fluid Flows
,”
Ann. Rev. Fluid Mech.
,
30
(
1
), pp.
329
364
.10.1146/annurev.fluid.30.1.329
51.
Dong
,
Y.
, and
Sagaut
,
P.
,
2008
, “
A Study of Time Correlations in Lattice Boltzmann-Based Large-Eddy Simulation of Isotropic Turbulence
,”
Phys. Fluids
,
20
, p.
035105
.10.1063/1.2842381
52.
Premnath
,
K.
,
Pattison
,
M.
, and
Banerjee
,
S.
,
2009
, “
Dynamic Subgrid Scale Modeling of Turbulent Flows Using Lattice-Boltzmann Method
,”
Phys. A: Stat. Mech. Appl.
,
388
(
13
), pp.
2640
2658
.10.1016/j.physa.2009.02.041
53.
Kerimo
,
J.
, and
Girimaji
,
S.
,
2007
, “
Boltzmann–BGK Approach to Simulating Weakly Compressible 3D Turbulence: Comparison Between Lattice Boltzmann and Gas Kinetic Methods
,”
J. Turbul.
,
8
(
1
), pp.
1
16
.10.1080/14685240701528551
54.
Gekelman
,
W.
,
Vincena
,
S.
, and
Collette
,
A.
,
2008
, “
Visualizing Three-Dimensional Reconnection in a Colliding Laser Plasma Experiment
,”
IEEE Trans. Plasma Sci.
,
36
(
4
), pp.
1122
1123
.10.1109/TPS.2008.922928
55.
Mininni
,
P.
,
Lee
,
E.
,
Norton
,
A.
, and
Clyne
,
J.
,
2008
, “
Flow Visualization and Field Line Advection in Computational Fluid Dynamics: Application to Magnetic Fields and Turbulent Flows
,”
New J. Phys.
,
10
, p.
125007
.10.1088/1367-2630/10/12/125007
56.
Zhang
,
Y.
,
Boehmer
,
H.
,
Heidbrink
,
W.
,
McWilliams
,
R.
,
Leneman
,
D.
, and
Vincena
,
S.
,
2007
, “
Lithium Ion Sources for Investigations of Fast Ion Transport in Magnetized Plasmas
,”
Rev. Sci. Instrum.
,
78
, p.
013302
.10.1063/1.2431086
57.
Drozdenko
,
T.
, and
Morales
,
G.
,
2001
, “
Nonlinear Effects Resulting From the Interaction of a Large-Scale Alfvén Wave With a Density Filament
,”
Phys. Plasmas
,
8
, pp.
3265
3277
.10.1063/1.1376652
You do not currently have access to this content.