Visco-resistive magnetohydrodynamic turbulence, driven by a two-dimensional unstable shear layer that is maintained by an imposed body force, is examined by decomposing it into dissipationless linear eigenmodes of the initial profiles. The down-gradient momentum flux, as expected, originates from the large-scale instability. However, continual up-gradient momentum transport by large-scale linearly stable but nonlinearly excited eigenmodes is identified and found to nearly cancel the down-gradient transport by unstable modes. The stable modes effectuate this by depleting the large-scale turbulent fluctuations via energy transfer to the mean flow. This establishes a physical mechanism underlying the long-known observation that coherent vortices formed from nonlinear saturation of the instability reduce turbulent transport and fluctuations, as such vortices are composed of both the stable and unstable modes, which are nearly equal in their amplitudes. The impact of magnetic fields on the nonlinearly excited stable modes is then quantified. Even when imposing a strong magnetic field that almost completely suppresses the instability, the up-gradient transport by the stable modes is at least two-thirds of the down-gradient transport by the unstable modes, whereas for weaker fields, this fraction reaches up to 98%. These effects are persistent with variations in magnetic Prandtl number and forcing strength. Finally, continuum modes are shown to be energetically less important, but essential for capturing the magnetic fluctuations and Maxwell stress. A simple analytical scaling law is derived for their saturated turbulent amplitudes. It predicts the falloff rate as the inverse of the Fourier wavenumber, a property which is confirmed in numerical simulations.
References
1.
E. C.
Harding
, J. F.
Hansen
, O. A.
Hurricane
, R. P.
Drake
, H. F.
Robey
, C. C.
Kuranz
, B. A.
Remington
, M. J.
Bono
, M. J.
Grosskopf
, and R. S.
Gillespie
, “Observation of a Kelvin-Helmholtz instability in a high-energy-density plasma on the omega laser
,” Phys. Rev. Lett.
103
, 045005
(2009
).2.
H.
Hasegawa
, M.
Fujimoto
, T.-D.
Phan
, H.
Rème
, A.
Balogh
, M. W.
Dunlop
, C.
Hashimoto
, and R.
TanDokoro
, “Transport of solar wind into Earth's magnetosphere through rolled-up Kelvin-Helmholtz vortices
,” Nature
430
, 755
(2004
).3.
D. W.
Waugh
, A. H.
Sobel
, and L. M.
Polvani
, “What is the polar vortex and how does it influence weather?
,” Bull. Am. Meteorol. Soc.
98
, 37
(2017
).4.
P. L.
Read
, R. M. B.
Young
, and D.
Kennedy
, “The turbulent dynamics of Jupiter's and Saturn's weather layers: Order out of chaos?
,” Geosci. Lett.
7
, 10
(2020
).5.
J.
Fuller
, A. L.
Piro
, and A. S.
Jermyn
, “Slowing the spins of stellar cores
,” Mon. Not. R. Astron. Soc.
485
, 3661
(2019
).6.
M. E.
Pessah
, C.-K.
Chan
, and D.
Psaltis
, “The signature of the magnetorotational instability in the Reynolds and Maxwell stress tensors in accretion discs
,” Mon. Not. R. Astron. Soc.
372
, 183
(2006
).7.
J.
Goodman
and G.
Xu
, “Parasitic instabilities in magnetized, differentially rotating disks
,” Astrophys. J.
432
, 213
(1994
).8.
J.
Alves
, C.
Zucker
, A. A.
Goodman
, J. S.
Speagle
, S.
Meingast
, T.
Robitaille
, D. P.
Finkbeiner
, E. F.
Schlafly
, and G. M.
Green
, “A Galactic-scale gas wave in the solar neighbourhood
,” Nature
578
, 237
(2020
).9.
R.
Fleck
, “The ‘Radcliffe Wave’ as a Kelvin–Helmholtz instability
,” Nature
583
, E24
(2020
).10.
A.
Miura
, “Self-organization in the two-dimensional Kelvin-Helmholtz instability
,” Phys. Rev. Lett.
83
, 1586
(1999
).11.
D.
Lecoanet
, M.
McCourt
, E.
Quataert
, K. J.
Burns
, G. M.
Vasil
, J. S.
Oishi
, B. P.
Brown
, J. M.
Stone
, and R. M.
O'Leary
, “A validated non-linear Kelvin-Helmholtz benchmark for numerical hydrodynamics
,” Mon. Not. R. Astron. Soc.
455
, 4274
(2016
).12.
C.
Ho
and P.
Huerre
, “Perturbed free shear layers
,” Annu. Rev. Fluid Mech.
16
, 365
(1984
).13.
F. K.
Browand
and C. M.
Ho
, “The mixing layer, an example of quasi two-dimensional turbulence
,” J. Mec. Theor. Appl.
2
, 99
(1983
); available at https://ui.adsabs.harvard.edu/abs/1983JMTAS.......99B/abstract.14.
V. P.
Starr
and N. E.
Gaut
, “Negative viscosity
,” Sci. Am.
223
, 72
(1970
).15.
A.
Miura
and T.
Sato
, “Theory of vortex nutation and amplitude oscillation in an inviscid shear instability
,” J. Fluid Mech.
86
, 33
(1978
).16.
W.
Horton
, T.
Tajima
, and T.
Kamimura
, “Kelvin–Helmholtz instability and vortices in magnetized plasma
,” Phys. Fluids
30
, 3485
(1987
).17.
P. W.
Terry
, P.-Y.
Li
, M. J.
Pueschel
, and G. G.
Whelan
, “Threshold heat-flux reduction by near-resonant energy transfer
,” Phys. Rev. Lett.
126
, 025004
(2021
).18.
G. G.
Whelan
, M. J.
Pueschel
, and P. W.
Terry
, “Nonlinear electromagnetic stabilization of plasma microturbulence
,” Phys. Rev. Lett.
120
, 175002
(2018
).19.
M. J.
Pueschel
, B. J.
Faber
, J.
Citrin
, C. C.
Hegna
, P. W.
Terry
, and D. R.
Hatch
, “Stellarator turbulence: Subdominant eigenmodes and quasilinear modeling
,” Phys. Rev. Lett.
116
, 085001
(2016
).20.
K. D.
Makwana
, P. W.
Terry
, M. J.
Pueschel
, and D. R.
Hatch
, “Subdominant modes in zonal-flow-regulated turbulence
,” Phys. Rev. Lett.
112
, 095002
(2014
).21.
D. R.
Hatch
, F.
Jenko
, A. B.
Navarro
, and V.
Bratanov
, “Transition between saturation regimes of gyrokinetic turbulence
,” Phys. Rev. Lett.
111
, 175001
(2013
).22.
D. R.
Hatch
, P. W.
Terry
, F.
Jenko
, F.
Merz
, and W. M.
Nevins
, “Saturation of gyrokinetic turbulence through damped eigenmodes
,” Phys. Rev. Lett.
106
, 115003
(2011
).23.
M. J.
Pueschel
, P.-Y.
Li
, and P. W.
Terry
, “Predicting the critical gradient of ITG turbulence in fusion plasmas
,” Nucl. Fusion
61
, 054003
(2021
).24.
P.-Y.
Li
and P. W.
Terry
, “Assessing physics of ion temperature gradient turbulence via hierarchical reduced-model representations
,” Phys. Plasmas
29
, 042301
(2022
).25.
G. G.
Whelan
, M. J.
Pueschel
, P. W.
Terry
, J.
Citrin
, I. J.
McKinney
, W.
Guttenfelder
, and H.
Doerk
, “Saturation and nonlinear electromagnetic stabilization of ITG turbulence
,” Phys. Plasmas
26
, 082302
(2019
).26.
P. W.
Terry
, B. J.
Faber
, C. C.
Hegna
, V. V.
Mirnov
, M. J.
Pueschel
, and G. G.
Whelan
, “Saturation scalings of toroidal ion temperature gradient turbulence
,” Phys. Plasmas
25
, 012308
(2018
).27.
A. E.
Fraser
, M. J.
Pueschel
, P. W.
Terry
, and E. G.
Zweibel
, “Role of stable modes in driven shear-flow turbulence
,” Phys. Plasmas
25
, 122303
(2018
).28.
K. D.
Makwana
, P. W.
Terry
, J.-H.
Kim
, and D. R.
Hatch
, “Damped eigenmode saturation in plasma fluid turbulence
,” Phys. Plasmas
18
, 012302
(2011
).29.
K. D.
Makwana
, P. W.
Terry
, and J.-H.
Kim
, “Role of stable modes in zonal flow regulated turbulence
,” Phys. Plasmas
19
, 062310
(2012
).30.
P. W.
Terry
, D. A.
Baver
, and S.
Gupta
, “Role of stable eigenmodes in saturated local plasma turbulence
,” Phys. Plasmas
13
, 022307
(2006
).31.
R. H.
Levy
and R. W.
Hockney
, “Computer experiments on low-density crossed-field electron beams
,” Phys. Fluids
11
, 766
(1968
).32.
N. J.
Zabusky
and G. S.
Deem
, “Dynamical evolution of two-dimensional unstable shear flows
,” J. Fluid Mech.
47
, 353
(1971
).33.
L.-S.
Huang
and C.-M.
Ho
, “Small-scale transition in a plane mixing layer
,” J. Fluid Mech.
210
, 475
(1990
).34.
R. D.
Moser
and M. M.
Rogers
, “The three-dimensional evolution of a plane mixing layer: Pairing and transition to turbulence
,” J. Fluid Mech.
247
, 275
(1993
).35.
J. J.
Riley
and R. W.
Metcalfe
, “Direct numerical simulation of a perturbed turbulent mixing layer
,” AIAA Paper No. 1980-0274, 1980
.36.
D.
Oster
and I.
Wygnanski
, “The forced mixing layer between parallel streams
,” J. Fluid Mech.
123
, 91
(1982
).37.
Y.
Ito
, K.
Nagata
, Y.
Sakai
, and O.
Terashima
, “Momentum and mass transfer in developing liquid shear mixing layers
,” Exp. Therm. Fluid Sci.
51
, 28
(2013
).38.
A.
López Zazueta
and L.
Zavala Sansón
, “Self-oscillations of a two-dimensional shear flow with forcing and dissipation
,” Phys. Fluids
30
, 044101
(2018
).39.
A.
VanDine
, H. T.
Pham
, and S.
Sarkar
, “Turbulent shear layers in a uniformly stratified background: DNS at high Reynolds number
,” J. Fluid Mech.
916
, A42
(2021
).40.
A. K. M. F.
Hussain
and K. B. M. Q.
Zaman
, “An experimental study of organized motions in the turbulent plane mixing layer
,” J. Fluid Mech.
159
, 85
(1985
).41.
A. K. M. F.
Hussain
, “Coherent structures and turbulence
,” J. Fluid Mech.
173
, 303
(1986
).42.
L.
Landau
, “On the problem of turbulence
,” C. R. Acad. Sci. U. R. S. S.
44
, 311
(1944
).43.
A. E.
Fraser
, P. W.
Terry
, E. G.
Zweibel
, and M. J.
Pueschel
, “Coupling of damped and growing modes in unstable shear flow
,” Phys. Plasmas
24
, 062304
(2017
).44.
A. E.
Fraser
, P. W.
Terry
, E. G.
Zweibel
, M. J.
Pueschel
, and J. M.
Schroeder
, “The impact of magnetic fields on momentum transport and saturation of shear-flow instability by stable modes
,” Phys. Plasmas
28
, 022309
(2021
).45.
B.
Tripathi
, A. E.
Fraser
, P. W.
Terry
, E. G.
Zweibel
, and M. J.
Pueschel
, “Mechanism for sequestering magnetic energy at large scales in shear-flow turbulence
,” arXiv:2205.01298 (2022
).46.
S.
Chandrasekhar
, Hydrodynamic and Hydromagnetic Stability
(Clarendon Press
, Oxford
, 1961
).47.
J.
Mak
, S. D.
Griffiths
, and D. W.
Hughes
, “Vortex disruption by magnetohydrodynamic feedback
,” Phys. Rev. Fluids
2
, 113701
(2017
).48.
A. A.
Schekochihin
, J. L.
Maron
, S. C.
Cowley
, and J. C.
McWilliams
, “The small-scale structure of magnetohydrodynamic turbulence with large magnetic Prandtl numbers
,” Astrophys. J.
576
, 806
(2002
).49.
D.
Biskamp
, Magnetohydrodynamic Turbulence
(Cambridge University Press
, Cambridge
, 2003
).50.
F.
Ebrahimi
, S. C.
Prager
, and D. D.
Schnack
, “Saturation of magnetorotational instability through magnetic field generation
,” Astrophys. J.
698
, 233
(2009
).51.
M. J.
Pueschel
, D.
Told
, P. W.
Terry
, F.
Jenko
, E. G.
Zweibel
, V.
Zhdankin
, and H.
Lesch
, “Magnetic reconnection turbulence in strong guide fields: Basic properties and application to coronal heating
,” Astrophys. J., Suppl. Ser.
213
, 30
(2014
).52.
J. B.
Marston
, E.
Conover
, and T.
Schneider
, “Statistics of an unstable barotropic jet from a cumulant expansion
,” J. Atmos. Sci.
65
, 1955
(2008
).53.
K. M.
Smith
, C. P.
Caulfield
, and J. R.
Taylor
, “Turbulence in forced stratified shear flows
,” J. Fluid Mech.
910
, A42
(2021
).54.
A.
Allawala
, S. M.
Tobias
, and J. B.
Marston
, “Dimensional reduction of direct statistical simulation
,” J. Fluid Mech.
898
, A21
(2020
).55.
K. J.
Burns
, G. M.
Vasil
, J. S.
Oishi
, D.
Lecoanet
, and B. P.
Brown
, “Dedalus: A flexible framework for numerical simulations with spectral methods
,” Phys. Rev. Res.
2
, 023068
(2020
).56.
C. M.
Bender
, PT Symmetry: In Quantum and Classical Physics
(World Scientific Publishing
, 2019
).57.
D. R.
Hatch
, F.
Jenko
, A. B.
Navarro
, V.
Bratanov
, P. W.
Terry
, and M. J.
Pueschel
, “Linear signatures in nonlinear gyrokinetics: Interpreting turbulence with pseudospectra
,” New J. Phys.
18
, 075018
(2016
).58.
Y.
Fu
and H.
Qin
, “The physics of spontaneous parity-time symmetry breaking in the Kelvin-Helmholtz instability
,” New J. Phys.
22
, 083040
(2020
).59.
K. M.
Case
, “Stability of inviscid plane Couette flow
,” Phys. Fluids
3
, 143
(1960
).60.
P. W.
Terry
, D. A.
Baver
, and D. R.
Hatch
, “Reduction of inward momentum flux by damped eigenmodes
,” Phys. Plasmas
16
, 122305
(2009
).61.
J. B.
Marston
, G. P.
Chini
, and S. M.
Tobias
, “Generalized quasilinear approximation: Application to zonal jets
,” Phys. Rev. Lett.
116
, 214501
(2016
).62.
G. K.
Batchelor
, “On the spontaneous magnetic field in a conducting liquid in turbulent motion
,” Proc. Roy. Soc. London, Ser. A
201
, 405
(1950
).63.
G. K.
Batchelor
and I.
Proudman
, “The effects of rapid distortion of a fluid in turbulent motion
,” Q. J. Mech. Appl. Math
7
, 83
(1954
).64.
A. A.
Townsend
, The Structure of Turbulent Shear Flow
, 2nd ed. (Cambridge University Press
, Cambridge
, 1976
).65.
A.
Alexakis
, P. D.
Mininni
, and A.
Pouquet
, Phys. Rev. E
72
, 046301
(2005
).66.
K. J.
Burns
, “Flexible spectral algorithms for simulating astrophysical and geophysical flows
,” Ph.D. thesis (Massachusetts Institute of Technology
, 2018
).67.
A. E.
Fraser
, “Role of stable eigenmodes in shear-flow instability saturation and turbulence
,” Ph.D. thesis (University of Wisconsin-Madison
, 2020
).68.
S. A.
Orszag
, “Analytical theories of turbulence
,” J. Fluid Mech.
41
, 363
(1970
).69.
K.
Taira
, S. L.
Brunton
, S. T. M.
Dawson
, C. W.
Rowley
, T.
Colonius
, B. J.
McKeon
, O. T.
Schmidt
, S.
Gordeyev
, V.
Theofilis
, and L. S.
Ukeiley
, “Modal analysis of fluid flows: An overview
,” AIAA J.
55
, 4013
(2017
).70.
P.
Garaud
, “Double-diffusive convection at low Prandtl number
,” Annu. Rev. Fluid Mech.
50
, 275
(2018
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.
