Tangled magnetic fields, often coexisting with an ordered mean field, have a major impact on turbulence and momentum transport in many plasmas, including those found in the solar tachocline and magnetic confinement devices. We present a novel mean field theory of potential vorticity mixing in β-plane magnetohydrodynamic (MHD) and drift wave turbulence. Our results show that mean square stochastic fields strongly reduce Reynolds stress coherence. This decoherence of potential vorticity flux due to stochastic field scattering leads to suppression of momentum transport and zonal flow formation. A simple calculation suggests that the breaking of the shear-eddy tilting feedback loop by stochastic fields is the key underlying physics mechanism. A dimensionless parameter that quantifies the increment in power threshold is identified and used to assess the impact of stochastic field on the L-H transition. We discuss a model of stochastic fields as a resisto-elastic network.

1.
J.
Pedlosky
,
Geophysical Fluid Dynamics
, Springer study edition (
Springer Verlag
,
1979
).
2.
A.
Bracco
,
A.
Provenzale
,
E.
Spiegel
, and
P.
Yecko
, “
Spotted disks
,” arXiv preprint astro-ph/9802298 (
1998
).
3.
M. E.
McIntyre
, “
Solar tachocline dynamics: Eddy viscosity, anti-friction, or something in between
,” in
Stellar Astrophysical Fluid Dynamics
, edited by
M. J.
Thompson
and
J.
Christensen-Dalsgaard
(
Cambridge University Press
,
Cambridge
,
2003
), pp.
111
130
.
4.
P. H.
Diamond
,
S.-I.
Itoh
,
K.
Itoh
, and
T. S.
Hahm
, “
Topical review: Zonal flows in plasma a review
,”
Plasma Phys. Controlled Fusion
47
,
R35
+ (
2005
).
5.
S. R.
Keating
and
P. H.
Diamond
, “
Turbulent diffusion of magnetic fields in two-dimensional magnetohydrodynamic turbulence with stable stratification
,”
Phys. Rev. Lett.
99
,
224502
(
2007
).
6.
S.
Durston
and
A. D.
Gilbert
, “
Transport and instability in driven two-dimensional magnetohydrodynamic flows
,”
J. Fluid Mech.
799
,
541
578
(
2016
).
7.
C.-C.
Chen
and
P. H.
Diamond
, “
Potential vorticity mixing in a tangled magnetic field
,”
Astrophys. J.
892
,
24
(
2020
).
8.
G. I.
Taylor
, “
I. eddy motion in the atmosphere
,”
Philosoph. Trans. R. Soc. London. Ser. A
215
,
1
26
(
1915
).
9.
H.
Poincare
, “
Chapitre premier: Théorème de helmholtz
,” in
Théorie Des Tourbillons, Cours de Physique Mathematique
(
G. Carre
,
Paris
,
1893
), pp.
3
29
.
10.
A.
Hasegawa
and
K.
Mima
, “
Pseudo-three-dimensional turbulence in magnetized nonuniform plasma
,”
Phys. Fluids
21
,
87
92
(
1978
).
11.
N.
Leprovost
and
E-j
Kim
, “
Effect of rossby and alfvén waves on the dynamics of the tachocline
,”
Astrophys. J.
654
,
1166
(
2007
).
12.
R. B.
Wood
and
M. E.
McIntyre
, “
A general theorem on angular-momentum changes due to potential vorticity mixing and on potential-energy changes due to buoyancy mixing
,”
J. Atmos. Sci.
67
,
1261
1274
(
2010
).
13.
L. D.
Landau
, “
61–On the vibrations of the electronic plasma
,” in
The Collected Papers of L. D. Landau
, edited by
D.
ter Haar
(
Pergamon
,
1965
), pp.
445
460
.
14.
E. N.
Parker
, “
A Solar Dynamo Surface Wave at the Interface between Convection and Nonuniform Rotation
,”
Astrophys. J.
408
,
707
(
1993
).
15.
A. V.
Gruzinov
and
P. H.
Diamond
, “
Nonlinear mean field electrodynamics of turbulent dynamos
,”
Phys. Plasmas
3
,
1853
1857
(
1996
).
16.
S.
Tobias
, “
The solar tachocline: A study in stably stratified MHD turbulence
,” in
IUTAM Symposium on Turbulence in the Atmosphere and Oceans
, edited by
D.
Dritschel
(
Springer
,
Dordrecht
,
2005
), p.
193
.
17.
D.
Fyfe
and
D.
Montgomery
, “
High-beta turbulence in two-dimensional magnetohydrodynamics
,”
J. Plasma Phys.
16
,
181
191
(
1976
).
18.
N. H.
Brummell
,
S. M.
Tobias
,
J. H.
Thomas
, and
N. O.
Weiss
, “
Flux pumping and magnetic fields in the outer penumbra of a sunspot
,”
Astrophys. J.
686
,
1454
1465
(
2008
).
19.
T. E.
Evans
, “
Resonant magnetic perturbations of edge-plasmas in toroidal confinement devices
,”
Plasma Phys. Controlled Fusion
57
,
123001
(
2015
).
20.
T.
Evans
,
R.
Moyer
,
J.
Watkins
,
T.
Osborne
,
P.
Thomas
,
M.
Becoulet
,
J.
Boedo
,
E.
Doyle
,
M.
Fenstermacher
,
K.
Finken
,
R.
Groebner
,
M.
Groth
,
J.
Harris
,
G.
Jackson
,
R. L.
Haye
,
C.
Lasnier
,
S.
Masuzaki
,
N.
Ohyabu
,
D.
Pretty
,
H.
Reimerdes
,
T.
Rhodes
,
D.
Rudakov
,
M.
Schaffer
,
M.
Wade
,
G.
Wang
,
W.
West
, and
L.
Zeng
, “
Suppression of large edge localized modes with edge resonant magnetic fields in high confinement DIII-d plasmas
,”
Nucl. Fusion
45
,
595
607
(
2005
).
21.
T.
Evans
,
M.
Fenstermacher
,
R.
Moyer
,
T.
Osborne
,
J.
Watkins
,
P.
Gohil
,
I.
Joseph
,
M.
Schaffer
,
L.
Baylor
,
M.
Bécoulet
,
J.
Boedo
,
K.
Burrell
,
J.
deGrassie
,
K.
Finken
,
T.
Jernigan
,
M.
Jakubowski
,
C.
Lasnier
,
M.
Lehnen
,
A.
Leonard
,
J.
Lonnroth
,
E.
Nardon
,
V.
Parail
,
O.
Schmitz
,
B.
Unterberg
, and
W.
West
, “
RMP ELM suppression in DIII-d plasmas with ITER similar shapes and collisionalities
,”
Nucl. Fusion
48
,
024002
(
2008
).
22.
A. W.
Leonard
,
A. M.
Howald
,
A. W.
Hyatt
,
T.
Shoji
,
T.
Fujita
,
M.
Miura
,
N.
Suzuki
, and
S.
Tsuji
, “
Effects of applied error fields on the H-mode power threshold of JFT-2M
,”
Nucl. Fusion
31
,
1511
1518
(
1991
).
23.
P.
Gohil
,
T.
Evans
,
M.
Fenstermacher
,
J.
Ferron
,
T.
Osborne
,
J.
Park
,
O.
Schmitz
,
J.
Scoville
, and
E.
Unterberg
, “
L–h transition studies on DIII-d to determine h-mode access for operational scenarios in ITER
,”
Nucl. Fusion
51
,
103020
(
2011
).
24.
S.
Kaye
,
R.
Maingi
,
D.
Battaglia
,
R.
Bell
,
C.
Chang
,
J.
Hosea
,
H.
Kugel
,
B.
LeBlanc
,
H.
Meyer
,
G.
Park
, and
J.
Wilson
, “
L–h threshold studies in NSTX
,”
Nucl. Fusion
51
,
113019
(
2011
).
25.
F.
Ryter
,
S. K.
Rathgeber
,
L. B.
Orte
,
M.
Bernert
,
G. D.
Conway
,
R.
Fischer
,
T.
Happel
,
B.
Kurzan
,
R. M.
McDermott
,
A.
Scarabosio
,
W.
Suttrop
,
E.
Viezzer
,
M.
Willensdorfer
, and
E.
Wolfrum
, “
Survey of the h-mode power threshold and transition physics studies in ASDEX upgrade
,”
Nucl. Fusion
53
,
113003
(
2013
).
26.
S.
Mordijck
,
T. L.
Rhodes
,
L.
Zeng
,
E. J.
Doyle
,
L.
Schmitz
,
C.
Chrystal
,
T. J.
Strait
, and
R. A.
Moyer
, “
Effects of resonant magnetic perturbations on turbulence and transport in DIII-d l-mode plasmas
,”
Plasma Phys. Controlled Fusion
58
,
014003
(
2016
).
27.
R.
Scannell
,
A.
Kirk
,
M.
Carr
,
J.
Hawke
,
S. S.
Henderson
,
T.
O'Gorman
,
A.
Patel
,
A.
Shaw
, and
A.
Thornton
, “
Impact of resonant magnetic perturbations on the l-h transition on MAST
,”
Plasma Phys. Controlled Fusion
57
,
075013
(
2015
).
28.
Y.
In
,
J.-K.
Park
,
Y.
Jeon
,
J.
Kim
,
G.
Park
,
J.-W.
Ahn
,
A.
Loarte
,
W.
Ko
,
H.
Lee
,
J.
Yoo
 et al., “
Enhanced understanding of non-axisymmetric intrinsic and controlled field impacts in tokamaks
,”
Nucl. Fusion
57
,
116054
(
2017
).
29.
L.
Schmitz
,
D.
Kriete
,
R.
Wilcox
,
T.
Rhodes
,
L.
Zeng
,
Z.
Yan
,
G.
McKee
,
T.
Evans
,
C.
Paz-Soldan
,
P.
Gohil
,
B.
Lyons
,
C.
Petty
,
D.
Orlov
, and
A.
Marinoni
, “
L–h transition trigger physics in ITER-similar plasmas with applied n = 3 magnetic perturbations
,”
Nucl. Fusion
59
,
126010
(
2019
).
30.
P.
Diamond
,
Y.-M.
Liang
,
B.
Carreras
, and
P.
Terry
, “
Self-regulating shear flow turbulence: A paradigm for the l to h transition
,”
Phys. Rev. Lett.
72
,
2565
(
1994
).
31.
E-j
Kim
and
P.
Diamond
, “
Mean shear flows, zonal flows, and generalized kelvin–helmholtz modes in drift wave turbulence: A minimal model for l h transition
,”
Phys. Plasmas
10
,
1698
1704
(
2003
).
32.
M.
Malkov
and
P.
Diamond
, “
Weak hysteresis in a simplified model of the lh transition
,”
Phys. Plasmas
16
,
012504
(
2009
).
33.
T.
Estrada
,
C.
Hidalgo
,
T.
Happel
, and
P.
Diamond
, “
Spatiotemporal structure of the interaction between turbulence and flows at the lh transition in a toroidal plasma
,”
Phys. Rev. Lett.
107
,
245004
(
2011
).
34.
H. K.
Moffatt
,
Magnetic Field Generation in Electrically Conducting Fluids
(
Cambridge University Press
,
Cambridge
1978
).
35.
P. A.
Gilman
, “
Magnetohydrodynamic “shallow water” equations for the solar tachocline
,”
Astrophys. J.
544
,
L79
L82
(
2000
).
36.
J.
Christensen-Dalsgaard
and
M. J.
Thompson
, “
Observational results and issues concerning the tachocline
,” in
The Solar Tachocline
, edited by
D. W.
Hughes
,
R.
Rosner
, and
N. O.
Weiss
(
Cambridge University Press
,
2007
), pp.
53
86
.
37.
S. M.
Tobias
,
P. H.
Diamond
, and
D. W.
Hughes
, “
β-plane magnetohydrodynamic turbulence in the solar tachocline
,”
Astrophys. J.
667
,
L113
L116
(
2007
).
38.
A. B.
Rechester
and
M. N.
Rosenbluth
, “
Electron heat transport in a tokamak with destroyed magnetic surfaces
,”
Phys. Rev. Lett.
40
,
38
41
(
1978
).
39.
Y. B.
Zel'dovich
, “
Percolation properties of a random two-dimensional stationary magnetic field
,”
ZhETF Pisma Redaktsiiu
38
,
51
(
1983
).
40.
R.
Kubo
, “
Stochastic liouville equations
,”
J. Math. Phys.
4
,
174
183
(
1963
).
41.
P. B.
Rhines
, “
Waves and turbulence on a beta-plane
,”
J. Fluid Mech.
69
,
417
443
(
1975
).
42.
S. M.
Tobias
and
F.
Cattaneo
, “
Dynamo action in complex flows: The quick and the fast
,”
J. Fluid Mech.
601
,
101
122
(
2008
).
43.
A.
Skal
and
B.
Shklovskii
, “
Influence of the impurity concentration on the hopping conduction in semiconductors
,”
Sov. Phys. Semicond
7
,
1058
1061
(
1974
).
44.
P.-G.
De Gennes
, “
On a relation between percolation theory and the elasticity of gels
,”
J. Phys. Lett.
37
,
1
2
(
1976
).
45.
T.
Nakayama
,
K.
Yakubo
, and
R. L.
Orbach
, “
Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations
,”
Rev. Mod. Phys.
66
,
381
(
1994
).
46.
L.
Mestel
,
Stellar Magnetism
(
Cambridge University Press
,
1999
), Vol.
410
, pp.
374
378
.
47.
E. A.
Spiegel
and
J.-P.
Zahn
, “
The solar tachocline
,”
Astron. Astrophys.
265
,
106
114
(
1992
).
48.
D. O.
Gough
and
M. E.
McIntyre
, “
Inevitability of a magnetic field in the Sun's radiative interior
,”
Nature
394
,
755
757
(
1998
).
49.
D. M.
Kriete
,
G. R.
McKee
,
L.
Schmitz
,
D.
Smith
,
Z.
Yan
,
L.
Morton
, and
R.
Fonck
, “
Effect of magnetic perturbations on turbulence-flow dynamics at the Lh transition on DIII-D
,”
Phys. Plasmas
27
,
062507
(
2020
).
50.
P. H.
Diamond
and
M. N.
Rosenbluth
, “
Theory of the renormalized dielectric for electrostatic drift wave turbulence in tokamaks
,”
Phys. Fluids
24
,
1641
1649
(
1981
).
51.
Y. B.
Zel'dovich
, “
7: A magnetic field in the two-dimensional motion of a conducting turbulent fluid
,” in
Selected Words of Yakov Borisovich Zeldovich
, Volume I, edited by
R. A.
Sunyaev
(
Princeton University Press
,
2014
), pp.
93
96
.
52.
M.
Rosenbluth
,
R.
Sagdeev
,
J.
Taylor
, and
G.
Zaslavski
, “
Destruction of magnetic surfaces by magnetic field irregularities
,”
Nucl. Fusion
6
,
297
300
(
1966
).
53.
E-j
Kim
and
P. H.
Diamond
, “
Zonal flows and transient dynamics of the lh transition
,”
Phys. Rev. Lett.
90
,
185006
(
2003
).
54.
Ö.
Gürcan
,
P.
Diamond
,
T.
Hahm
, and
R.
Singh
, “
Intrinsic rotation and electric field shear
,”
Phys. Plasmas
14
,
042306
(
2007
).
55.
P.
Diamond
,
C.
McDevitt
,
Ö.
Gürcan
,
T.
Hahm
, and
V.
Naulin
, “
Transport of parallel momentum by collisionless drift wave turbulence
,”
Phys. Plasmas
15
,
012303
(
2008
).
56.
Y.
Kosuga
,
P.
Diamond
, and
Ö. D.
Gürcan
, “
On the efficiency of intrinsic rotation generation in tokamaks
,”
Phys. Plasmas
17
,
102313
(
2010
).
57.
L.
Schmitz
,
L.
Zeng
,
T. L.
Rhodes
,
J. C.
Hillesheim
,
E. J.
Doyle
,
R. J.
Groebner
,
W. A.
Peebles
,
K. H.
Burrell
, and
G.
Wang
, “
Role of zonal flow predator-prey oscillations in triggering the transition to h-mode confinement
,”
Phys. Rev. Lett.
108
,
155002
(
2012
).
58.
G. D.
Conway
,
C.
Angioni
,
F.
Ryter
,
P.
Sauter
, and
J.
Vicente
(
ASDEX Upgrade Team
), “
Mean and oscillating plasma flows and turbulence interactions across the lh confinement transition
,”
Phys. Rev. Lett.
106
,
065001
(
2011
).
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