Inverse cascades, zonal jets and turbulence/transport ... · Sergey Nazarenko, Warwick, UK Balk,...
Transcript of Inverse cascades, zonal jets and turbulence/transport ... · Sergey Nazarenko, Warwick, UK Balk,...
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Inverse cascades, zonal jets and
turbulence/transport suppression in
CHM model
Sergey Nazarenko, Warwick, UK
Balk, Connaughton, Dyachenko, Manin, Nadiga, Quinn, Zakharov,
1988-2010
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Drift waves in fusion
devices
Rossby waves in atmospheres
of rotating planets
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Charney-Hasegawa-Mima
equation
• Ψ – streamfunction (electrostatic potential).
• ρ – Deformation radius (ion Larmor radius).
• β – PV gradient (diamagnetic drift).
• x – east-west (poloidal arc-length)
• y – south-north (radial length).
2 2
2 2 0t x x y y x
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Turbulence-ZF system. LH-transition
• Small-scale turbulence generates zonal flows
• ZF’s suppress waves
• Hence transport barriers, LH transition
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Zonal Jets in Earth’s Atmosphere
Average oceanic winds on Earth (QSCAT)
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Zonal Jets in Earth’s Oceans
Eddy-resolving simulation of Earths oceans (Earth Simulator Center/JAMSTEC)
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Barotropic governor in GFD• James and Gray’ 1986
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Mechanisms of zonal flow generation:
• (part 1) Anizotropic inverse cascade
• (part 2) Modulational instability
(part 3) Feedback of ZF onto turbulence:
turbulence suppression, LH transition
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Part 1
ZF generation by anisotropic inverse cascades
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• Ψ – streamfunction (electrostatic potential).
• ρ – Deformation radius (ion Larmor radius).
• β – PV gradient (diamagnetic drift).
• x – east-west (poloidal arc-length)
• y – south-north (radial length).
2 2
2 2 0t x x y y x
2D Euler equation
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Conservation laws for Euler
- energy spectrum( ) ( ) ( ) ik xE k u x r u x e dr
2
2 2
( )
( ) ( )
u E k dk
u k E k dk
- energy
-enstrophy
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FjØrtoft’53 argument for 2D turbulence.
• Produce turbulence at kf and have two dissipation regions at k- and k+separated by large inertial ranges.
• Production rates for energy and enstrophy are related as kf2 .
• If energy is dissipated at k+ at rate ~ then enstrophy is dissipated at a rate k+
2 >> kf2 which is a contradiction. Therefore, energy must be
dissipated at k- – inverse energy cascade.
• Similar ad absurdum argument is used to show that enstrophy cannot be dissipated at k-, and therefore cascades forward in k.
2
2 2
( )
( ) ( )
u E k dk
u k E k dk
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Rhines scale crossover
• Nonlinear=linear → Rhinesscale.
• “Lazy 8” separates vortex-dominated and wave-dominated scales (Rhines’75, Holloway’84)
• Outside of lazy-8: Kraichnan’s isotropicinverse cascade.
• Inside lazy-8 the cascade is anisotropic and dominated by triad wave resonances.
2 2
2 2 0t x x y y x
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Extra quadratic invariant on β-plane
• Balk, Nazarenko & Zakharov (1990)
• Adiabatic for the original β-plane equation: requires small nonlinearity and possibly random phases.
• For case kρ >>1:
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Anisotropic cascades in β-plane turbulence
• 3 cascades cannot be isotropic.
• Let us produce turbulence near some k0, surround it by a large non-dissipative area and dissipate at large k and at small kx and small ky.
•Fjortoft’s argument separates the k-space into three non-intersecting sectors to which the energy, enstrophy and zonostrophy can cascade.
•Zonostrophy Φ forces energy E to the scales corresponding to zonal flows.
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Numerics• Pseudo-spectral, no dissipation.
• Initial condition:
Quantify cascades via trajectory
Of centroids for E, Z and Φ:
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Weak nonlinearity run, NL/L =k02/kβ
2=0.07
• Initial turbulence is well
within the dumbbell.
• Because of slow weakly
nonlinear evolution, we
compare with a non-
conserved quantity (red).
• Energy and zonostrophy
are well conserved,
enstrophy less well.
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Weak nonlinearity run, NL/L =k02/kβ
2=0.07
• All three invariants
cascade as predicted.
• Energy cascades to
zonal scales along
the boundary of its
sector.
• Zonostrophy cascade
is slightly anisotropic.
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Strong nonlinearity run, NL/L =k02/kβ
2=0.7
• Initial turbulence is at the
border of the dumbbell.
• Zonostropy is not
conserved initially, but is
conserved later.
• This is because the
nonlinearity weakens as
the inverse cascade enters
into the dumbbell.
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Strong nonlinearity run, NL/L =k02/kβ
2=0.7
• E, Z and Φ cascade
similar to the weakly
nonlinear case.
• Faster and less
chaotic trajectories.
• Enstrophy and
Zonostrophy
cascades are almost
isotropic.
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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Vorticity field
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2D spectrum
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2D spectrum
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2D spectrum
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2D spectrum
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2D spectrum
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2D spectrum
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2D spectrum
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Part 2: Modulational Instability
• These waves are solutions of CHM equation for any amplitude. Are
they stable? (Lorentz 1972, Gill 1973).
· ·
0 0 0( , ) i i t i i tt e e k x k xx
2frequency of linear v( a es) w .xk
k F
k
0 1
1
( ,0) ( ) ( ),
( ) ( ) ( ) perturbati( o .n)Z
x x x
x x x x
· ·
· ·
· ·
( ) (0,zonal part
satellite
satellite
)
( )
( ) .
,
,
i i
Z
i i
i i
ae ae q
b e b e
b e b e
q x q x
p x p x
p x p x
x q
x p k q
x p k q
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Instability dispersion relation
2 2 2 22 22 2 2
0 2 2( ) ( ) 0
( )( ) ( )( )x
x x
k kq F q k q
F
p
F
p
p p p p
k q
3
0 nonlinearity param t .e erk
M
Euler limit (Rayleigh instabilty);
weak monlinearity: resonant wave inetr0 ac .tion
M
M
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Structure of instability as a function of M
Unstable region collapses onto the resonant curve. For small M
the most unstable disturbance is not zonal.
M=10 M=1
M=0.5
M=0.1
1 2
1 2( ) ( (
+ = ,
+ ) = )
k k k
k k k
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Nonlinear stage of modulational instability
Pinching of jets predicted by Manin & SN, 1994. Transport barriers.
Growth of the q-mode compared to
predictions of linear stability. Zonal velocity profile (averaged over x).
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Strong wave case (M = 10)
• Jet pinch, roll-up into a double vortex street. After long time, the
street breaks via a vortex pairing instability, leading to turbulence with
a PV staircase structure.
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Weak wave case (M = 0.1)
Original drift wave experiences self-focusing, but jets do not roll into
vortices. Energy oscillates between 4-modes as predicted by the 4-mode
truncation. At long time: transition to turbulence with inclined jets.
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Ocean jets
• From Maximenko et al 2008.
• Slightly off-zonal jets.
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Summary for part 2.
• MI of a travelling drift wave exists for any nonlinearity M. Two limits :
Euler limit for M>>1 vs weak resonance interaction for M<<1.
• Most unstable disturbance is zonal for large M’s and an inclined wave
for small M. Inclined jets are seen of small M for long-time nonlinear
stage.
• ZF’s are mostly eastward due to the beta-effect.
• Nonlinear pinching of ZF’s (for any M). Simplest model for the
transport barriers.
• Role of MI for broad initial wave spectra?
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Part 3: LH transition
• Small-scale turbulence causes anomalous transport, hence L-mode.
• Negative feedback loop.
• Suppressed turbulence →no transport →improved confinement & H-mode.
Balk, SN and Zakharov 1990
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Cartoon of nonlocal interaction
• Eddy scale L decreases via shearing by ZF
• Potential enstrophy Z is conserved.
• => Eddy energy E =Z L2 is decreasing
• Total E is conserved, => E is transferred from the eddy to ZF
• Wrong! Both smaller and larger L’s are produced. The energy of the eddy is unchanged. (Kraichnan 1976).
Victor P. Starr,Physics of Negative
Viscosity Phenomena (McGraw Hill Book
Co., New York 1968).
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Small-scale energy conservation
• Energy in SS eddies is conserved if they
are initially isotropic (Kraichnan 1976)
• 1. Dissipation: ellipse cannot get too thin.
• 2. Anisotropic initial eddies
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Wave instabilities
• Maximum on the kx-axis at kρ ~ 1.
• γ=0 line crosses k=0 point.
Access to stored free energy:
GFD: Baroclinic instability.
In plasmas: ITG, ETG instablities.
2 2
2 2 ˆt x x y y x
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Forced CHM simulation• Generation of ZF and
suppression of small-scale
turbulence
• Diffusion on k-space curves
(as predicted in Balk et al
1991)
• ZF saturation:
* max
ZF
f
v LU
k
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Summary for all parts.
• ZF’s can be generated via modulational
instability (for narrowband initial data) and by
anisotropic inverse cascade (for broadband
initial data)
• ZF’s suppress turbulence thereby causing
transport barriers.
• All effects present in forced/dissipated CHM
model.
• Examine 2-potential models which includes
instabilities.