To comprehensively understand the saturation of two-dimensional (2D) magnetized Kelvin–Helmholtz-instability-driven turbulence, energy transfer analysis is extended from the traditional interaction between scales to include eigenmode interactions, by using the nonlinear couplings of linear eigenmodes of the ideal instability. While both kinetic and magnetic energies cascade to small scales, a significant fraction of turbulent energy deposited by unstable modes in the fluctuation spectrum is shown to be re-routed to the conjugate-stable modes at the instability scale. They remove energy from the forward cascade at its inception. The remaining cascading energy flux is shown to attenuate exponentially at a small scale, dictated by the large-scale stable modes. Guided by a widely used instability-saturation assumption, a general quasi-linear model of instability is tested by retaining all nonlinear interactions except those that couple to the large-scale stable modes. These complex interactions are analytically removed from the magnetohydrodynamic equations using a novel technique. Observations are an explosive large-scale vortex separation instead of the well-known merger of 2D, a dramatic enhancement in turbulence level and spectral energy fluxes, and a reduced small-scale dissipation length scale. These show the critical role of the stable modes in instability saturation. Possible reduced-order turbulence models are proposed for fusion and astrophysical plasmas, based on eigenmode-expanded energy transfer analyses.

1.
J.
Fuller
,
A. L.
Piro
, and
A. S.
Jermyn
, “
Slowing the spins of stellar cores
,”
Mon. Not. R. Astron. Soc.
485
,
3661
(
2019
).
2.
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
).
3.
J.
Goodman
and
G.
Xu
, “
Parasitic instabilities in magnetized, differentially rotating disks
,”
Astrophys. J.
432
,
213
(
1994
).
4.
P.
Garaud
, “
Double-diffusive convection at low Prandtl number
,”
Annu. Rev. Fluid Mech.
50
,
275
(
2018
).
5.
A. J.
Barker
,
C. A.
Jones
, and
S. M.
Tobias
, “
Angular momentum transport by the GSF instability: Non-linear simulations at the equator
,”
Mon. Not. R. Astron. Soc.
487
,
1777
(
2019
).
6.
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
).
7.
P. W.
Terry
,
D. A.
Baver
, and
S.
Gupta
, “
Role of stable eigenmodes in saturated local plasma turbulence
,”
Phys. Plasmas
13
,
022307
(
2006
).
8.
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
).
9.
G. G.
Whelan
,
M. J.
Pueschel
, and
P. W.
Terry
, “
Nonlinear electromagnetic stabilization of plasma microturbulence
,”
Phys. Rev. Lett.
120
,
175002
(
2018
).
10.
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
).
11.
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
).
12.
D. R.
Hatch
,
P. W.
Terry
,
F.
Jenko
,
F.
Merz
,
M. J.
Pueschel
,
W. M.
Nevins
, and
E.
Wang
, “
Role of subdominant stable modes in plasma microturbulence
,”
Phys. Plasmas
18
,
055706
(
2011
).
13.
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
).
14.
P.-Y.
Li
,
P. W.
Terry
,
G. G.
Whelan
, and
M. J.
Pueschel
, “
Saturation physics of threshold heat-flux reduction
,”
Phys. Plasmas
28
,
102507
(
2021
).
15.
P.-Y.
Li
and
P. W.
Terry
, “
Assessing physics of ion temperature gradient turbulence via hierarchical reduced-model representations
,”
Phys. Plasmas
29
,
042301
(
2022
).
16.
G.
Salvesen
,
K.
Beckwith
,
J. B.
Simon
,
S. M.
OŃeill
, and
M. C.
Begelman
, “
Quantifying energetics and dissipation in magnetohydrodynamic turbulence
,”
Mon. Not. R. Astron. Soc.
438
,
1355
(
2014
).
17.
K. M.
Smith
,
C. P.
Caulfield
, and
J. R.
Taylor
, “
Turbulence in forced stratified shear flows
,”
J. Fluid Mech.
910
,
A42
(
2021
).
18.
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
).
19.
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
).
20.
K. M.
Case
, “
Stability of inviscid plane Couette flow
,”
Phys. Fluids
3
,
143
(
1960
).
21.
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
).
22.
J.
Mak
,
S. D.
Griffiths
, and
D. W.
Hughes
, “
Vortex disruption by magnetohydrodynamic feedback
,”
Phys. Rev. Fluids
2
,
113701
(
2017
).
23.
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
,”
Phys. Plasmas
29
,
070701
(
2022
).
24.
B.
Tripathi
,
A. E.
Fraser
,
P. W.
Terry
,
E. G.
Zweibel
, and
M. J.
Pueschel
, “
Near-cancellation of up- and down-gradient momentum transports in magnetized shear flow turbulence due to stable modes
,”
Phys. Plasmas
29
,
092301
(
2022
).
25.
M. K.
Verma
,
Energy Transfers in Fluid Flows: Multiscale and Spectral Perspectives
(
Cambridge University Press
,
Cambridge
,
2019
).
26.
C. C.
Hegna
,
P. W.
Terry
, and
B. J.
Faber
, “
Theory of ITG turbulent saturation in stellarators: Identifying mechanisms to reduce turbulent transport
,”
Phys. Plasmas
25
,
022511
(
2018
).
27.
D.
Biskamp
,
Magnetohydrodynamic Turbulence
(
Cambridge University Press
,
Cambridge
,
2003
).
28.
R. H.
Kraichnan
, “
The structure of isotropic turbulence at very high Reynolds numbers
,”
J. Fluid Mech.
5
,
497
(
1959
).
29.
P.
Grete
,
B. W.
O'Shea
,
K.
Beckwith
,
W.
Schmidt
, and
A.
Christlieb
, “
Energy transfer in compressible magnetohydrodynamic turbulence
,”
Phys. Plasmas
24
,
092311
(
2017
).
30.
B.
Teaca
,
M. K.
Verma
,
B.
Knaepen
, and
D.
Carati
, “
Energy transfer in anisotropic magnetohydrodynamic turbulence
,”
Phys. Rev. E
79
,
046312
(
2008
).
31.
B.
Teaca
,
A. B.
Navarro
, and
F.
Jenko
, “
The energetic coupling of scales in gyrokinetic plasma turbulence
,”
Phys. Plasmas
21
,
072308
(
2014
).
32.
B.
Teaca
,
F.
Jenko
, and
D.
Todd
, “
Gyrokinetic turbulence: Between idealized estimates and a detailed analysis of nonlinear energy transfers
,”
New J. Phys.
19
,
045001
(
2017
).
33.
M. K.
Verma
, “
Variable energy flux in turbulence
,”
J. Phys. A
55
,
013002
(
2021
).
34.
C.
Dong
,
L.
Wang
,
Y. M.
Huang
,
L.
Comisso
,
T. A.
Sandstrom
, and
A.
Bhattacharjee
, “
Reconnection-driven energy cascade in magnetohydrodynamic turbulence
,”
Sci. Adv.
8
(
49
),
eabn7627
(
2022
).
35.
A.
Alexakis
,
P. D.
Mininni
, and
A.
Pouquet
, “
Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence
,”
Phys. Rev. E
72
,
046301
(
2005
).
36.
O.
Debliquy
,
M. K.
Verma
, and
D.
Carati
, “
Energy fluxes and shell-to-shell transfers in three-dimensional decaying magnetohydrodynamic turbulence
,”
Phys. Plasmas
12
,
042309
(
2005
).
37.
A.
Allawala
,
S. M.
Tobias
, and
J. B.
Marston
, “
Dimensional reduction of direct statistical simulation
,”
J. Fluid Mech.
898
,
A21
(
2020
).
38.
J. B.
Marston
,
E.
Conover
, and
T.
Schneider
, “
Statistics of an unstable barotropic jet from a cumulant expansion
,”
J. Atmos. Sci.
65
,
1955
(
2008
).
39.
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
).
40.
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
).
41.
S. A.
Orszag
, “
Analytical theories of turbulence
,”
J. Fluid Mech.
41
,
363
(
1970
).
42.
K. D.
Makwana
,
P. W.
Terry
, and
J.-H.
Kim
, “
Role of stable modes in zonal flow regulated turbulence
,”
Phys. Plasmas
19
,
062310
(
2012
).
43.
L.
Biferale
,
S.
Musacchio
, and
F.
Toschi
, “
Inverse energy cascade in three-dimensional isotropic turbulence
,”
Phys. Rev. Lett.
108
,
164501
(
2012
).
44.
A.
Alexakis
,
P. D.
Mininni
, and
A.
Pouquet
, “
Imprint of large-scale flows on turbulence
,”
Phys. Rev. Lett.
95
,
264503
(
2005
).
45.
Y.-H.
Pao
, “
Structure of turbulent velocity and scalar fields at large wavenumbers
,”
Phys. Fluids
8
,
1063
(
1965
).
46.
P. W.
Terry
,
A. F.
Almagri
,
G.
Fiksel
,
C. B.
Forest
,
D. R.
Hatch
,
F.
Jenko
,
M. D.
Nornberg
,
S. C.
Prager
,
K.
Rahbarnia
,
Y.
Ren
, and
J. S.
Sarff
, “
Dissipation range turbulent cascades in plasmas
,”
Phys. Plasmas
19
,
055906
(
2012
).
47.
P. W.
Terry
and
V.
Tangri
, “
Magnetohydrodynamic dissipation range spectra for isotropic viscosity and resistivity
,”
Phys. Plasmas
16
,
082305
(
2009
).
48.
D.
Hu
,
A.
Bhattacharjee
, and
Y.-M.
Huang
, “
Energy spectrum of tearing mode turbulence in sheared background field
,”
Phys. Plasmas
25
,
062305
(
2018
).
49.
J. B.
Marston
,
G. P.
Chini
, and
S. M.
Tobias
, “
Generalized quasilinear approximation: Application to zonal jets
,”
Phys. Rev. Lett.
116
,
214501
(
2016
).
50.
S.
Chandrasekhar
,
Hydrodynamic and Hydromagnetic Stability
(
Clarendon Press
,
Oxford
,
1961
).
51.
A. A.
Schekochihin
, “
MHD turbulence: A biased review
,”
J. Plasma Phys.
88
,
155880501
(
2022
).
You do not currently have access to this content.