Straining of magnetic fields by large-scale shear flow, which is generally assumed to lead to intensification and generation of small scales, is reexamined in light of the persistent observation of large-scale magnetic fields in astrophysics. It is shown that, in magnetohydrodynamic turbulence, unstable shear flows have the unexpected effect of sequestering magnetic energy at large scales due to counteracting straining motion of nonlinearly excited large-scale stable eigenmodes. This effect is quantified via dissipation rates, energy transfer rates, and visualizations of magnetic field evolution by artificially removing the stable modes. These analyses show that predictions based upon physics of the linear instability alone miss substantial dynamics, including those of magnetic fluctuations.

1.
P. L.
Johnson
, “
Energy transfer from large to small scales in turbulence by multiscale nonlinear strain and vorticity interactions
,”
Phys. Rev. Lett.
124
,
104501
(
2020
).
2.
G. K.
Batchelor
, “
On the spontaneous magnetic field in a conducting liquid in turbulent motion
,”
Proc. R. Soc. London, Ser. A
201
,
405
(
1950
).
3.
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
).
4.
A. A.
Townsend
,
The Structure of Turbulent Shear Flow
, 2nd ed. (
Cambridge University Press
,
Cambridge
,
1976
).
5.
J.
Maron
,
S.
Cowley
, and
J.
McWilliams
, “
The nonlinear magnetic cascade
,”
Astrophys. J.
603
,
569
(
2004
).
6.
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
).
7.
A.
Neronov
and
I.
Vovk
, “
Evidence for strong extragalactic magnetic fields from Fermi observations of TeV blazars
,”
Science
328
,
73
(
2010
).
8.
R. M.
Kulsrud
, “
A critical review of galactic dynamos
,”
Annu. Rev. Astron. Astrophys.
37
,
37–64
(
1999
).
9.
A.
Brandenburg
and
K.
Subramanian
, “
Astrophysical magnetic fields and nonlinear dynamo theory
,”
Phys. Rep.
417
,
1–209
(
2005
).
10.
S. M.
Tobias
and
F.
Cattaneo
, “
Shear-driven dynamo waves at high magnetic Reynolds number
,”
Nature
497
,
463
(
2013
).
11.
J.
Squire
and
A.
Bhattacharjee
, “
Generation of large-scale magnetic fields by small-scale dynamo in shear flows
,”
Phys. Rev. Lett.
115
,
175003
(
2015
).
12.
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
).
13.
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
).
14.
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
).
15.
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
).
16.
G. G.
Whelan
,
M. J.
Pueschel
, and
P. W.
Terry
, “
Nonlinear electromagnetic stabilization of plasma microturbulence
,”
Phys. Rev. Lett.
120
,
175002
(
2018
).
17.
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
).
18.
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
).
19.
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
).
20.
D. R.
Hatch
,
F.
Jenko
,
A. B.
Navarro
, and
V.
Bratanov
, “
Transition between saturation regimes of gyrokinetic turbulence
,”
Phys. Rev. Lett.
111
,
175001
(
2013
).
21.
K. D.
Makwana
,
P. W.
Terry
, and
J.-H.
Kim
, “
Role of stable modes in zonal flow regulated turbulence
,”
Phys. Plasmas
19
,
062310
(
2012
).
22.
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
).
23.
P. W.
Terry
,
D. A.
Baver
, and
S.
Gupta
, “
Role of stable eigenmodes in saturated local plasma turbulence
,”
Phys. Plasmas
13
,
022307
(
2006
).
24.
D.
Biskamp
,
Magnetohydrodynamic Turbulence
(
Cambridge University Press
,
2003
).
25.
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
).
26.
J. B.
Marston
,
E.
Conover
, and
T.
Schneider
, “
Statistics of an unstable barotropic jet from a cumulant expansion
,”
J. Atmos. Sci.
65
,
1955
(
2008
).
27.
K. M.
Smith
,
C. P.
Caulfield
, and
J. R.
Taylor
, “
Turbulence in forced stratified shear flows
,”
J. Fluid Mech.
910
,
A42
(
2021
).
28.
A.
Allawala
,
S. M.
Tobias
, and
J. B.
Marston
, “
Dimensional reduction of direct statistical simulation
,”
J. Fluid Mech.
898
,
A21
(
2020
).
29.
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
).
30.
This is because the dissipationless linear operator respects the parity-time-reversal symmetry operation, and hence, its eigenmodes can be proven to form a complete basis, see Ref. 31.
31.
C. M.
Bender
,
PT Symmetry: In Quantum and Classical Physics
(
World Scientific
,
2019
).
32.
B.
Tripathi
,
A. E.
Fraser
,
P. W.
Terry
,
E. G.
Zweibel
, and
M. J.
Pueschel
, “
Near-cancellation of up- and down-gradient momentum transport in forced magnetized shear-flow turbulence
,”
Phys. Plasmas
(submitted) (
2022
).
33.
Energy norm is used to normalize the eigenmodes.
34.
Some similarities can be identified between the state-space reconstruction via this method and other approaches of modeling turbulence, e.g., exact coherent structures35 and proper orthogonal decomposition.36 
35.
B.
Suri
,
J.
Tithof
,
R. O.
Grigoriev
, and
M. F.
Schatz
, “
Forecasting fluid flows using the geometry of turbulence
,”
Phys. Rev. Lett.
118
,
114501
(
2017
).
36.
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
).
37.
A.
Ishizawa
,
Y.
Kishimoto
, and
Y.
Nakamura
, “
Multi-scale interactions between turbulence and magnetic islands and parity mixture—A review
,”
Plasma Phys. Controlled Fusion
61
,
054006
(
2019
).
38.
M.
Sato
and
A.
Ishizawa
, “
Nonlinear parity mixtures controlling the propagation of interchange modes
,”
Phys. Plasmas
24
,
082501
(
2017
).
39.
S.
Chandrasekhar
,
Hydrodynamic and Hydromagnetic Stability
(
Clarendon Press
,
Oxford
,
1961
).
40.
K. M.
Case
, “
Stability of inviscid plane Couette flow
,”
Phys. Fluids
3
,
143
(
1960
).
41.
Thus, reconstructed flow includes components that strain the magnetic field efficiently. The turbulent magnetic field, in contrast, requires many eigenmodes for each wavenumber. This is consistent with the fluid-straining-induced generation of small scales of the magnetic field which, when projected onto the eigenmode space, covers a large number of marginally stable continuum eigenmodes.
42.
Y.
Fu
and
H.
Qin
, “
The physics of spontaneous parity-time symmetry breaking in the Kelvin-Helmholtz instability
,”
New J. Phys.
22
,
083040
(
2020
).
43.
Essentially, the same conclusion has been found by recomputing Q1 and Q2 where they refer to the transfer rates between the fluctuations and the instantaneous mean profiles.
44.
J.
Fuller
,
A. L.
Piro
, and
A. S.
Jermyn
, “
Slowing the spins of stellar cores
,”
Mon. Not. R. Astron. Soc.
485
,
3661
(
2019
).
45.
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
).
46.
J.
Goodman
and
G.
Xu
, “
Parasitic instabilities in magnetized, differentially rotating disks
,”
Astrophys. J.
432
,
213
(
1994
).
47.
P.
Garaud
, “
Double-diffusive convection at low Prandtl number
,”
Annu. Rev. Fluid Mech.
50
,
275
(
2018
).
48.
J.
Mak
,
S. D.
Griffiths
, and
D. W.
Hughes
, “
Vortex disruption by magnetohydrodynamic feedback
,”
Phys. Rev. Fluids
2
,
113701
(
2017
).
49.
B.
Tripathi
,
A. E.
Fraser
,
P. W.
Terry
,
E. G.
Zweibel
,
M. J.
Pueschel
, and
E. H.
Anders
, “
Testing quasilinear theories in shear-flow turbulence
,” (unpublished).
50.
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
).

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