Stable organized motion and coherent structures in fusion ... 2006... · Publications September...

Stable organized motion and coherent structures in fusion

plasmas

• Introduction• Calculation of the spectrum in the problem

of perturbed zonal flow (similar to Orr-Sommerfeld) ; numerical description

• Continuous model for the stationary states of the Hasegawa-Mima equation

Plasma Theory Group

Members: F. SpineanuM. VladL. Anton (long term visit in UK)

Main subjects of interest :

Studies on instabilities, turbulence and transport in confined plasmasStatistical physics of test particle motionFirst principle models of plasma self-organised flows, coherent structures

Major recent results• Determination of the effects of intrinsic trajectory trapping or eddying

produced by the ExB drift in non-linear regimes.

Analytical methods for determining the statistical properties of test particles evolving in turbulent fields:

- DCT (Decorrelation Trajectory Method);- NSM (Nested Subensemble Method)

– Applications

• Field theoretical methodsA fundamental theoretical structure, largely extending the classical concepts in fluid/plasma description– Applications

PublicationsSince the beginning of Association (2000):

35 articles in international refereed journals40 International Conferences (Proceedings)

• Physical Review Letters : 6 articles• Physical Review : 8 articles• Physics of Plasmas : 2 articles• PPCF : 5 articles• Nuclear Fusion : 1 article

Average impact factor/person/year = 5.23

NOVA Science Publishers

(M.V. and F. S)

Publications September 2005 – September 20061. M. Vlad, F. Spineanu, S. Benkadda, “Impurity pinch from a ratchet process”, Physical Review Letters 96

(2006) 085001.2. F. Spineanu, M. Vlad, ´Statistical properties of an ensemble of vortices interacting with a turbulent field´,

Physics of Plasmas 12 (2005) 112303.3. M. Vlad, F. Spineanu, S. Benkadda, “Mass charge effects for impurity ratchet pinch”, in preparation (2006). 4. F. Spineanu, M. Vlad, “Density pinch from self-organizing vorticity in cilindrical plasma”, in preparation (2006).5. M. Vlad, F. Spineanu, “Turbulent pinch in non-homogeneous confining magnetic field”, ”, in preparation (2006).6. F. Spineanu, M. Vlad, K. Itoh, S.-I. Itoh, ‘Review of analytical treatments of barrier-type problems in plasma

theory’, Progress in chemical Research, Editor A. N. Linke, Nova Publishers (2006), ISBN: 1-59454-483-2.7. F. Spineanu, M. Vlad, ‘Soliton self-modulation of the turbulence amplitude and plasma rotation’, Progress in

Soliton Research, Editor L. V. Chen, Nova Publisher (2006), ISBN 1-59454-769-6.8. M. Vlad, F. Spineanu, “From Brownian diffusion to the non-linear transport by continuous movements”, in

‘Albert Einstein Century’, Editor J. Alimi, American Institute of Physics (2006) (in print).9. F. Spineanu, M. Vlad, “A background trend to ordered states in confined plasmas”, 11th European Fusion

Theory Conference, Aix en Provence, 26 September 2005.10. M. Vlad, F. Spineanu, “Test particles, test modes and self-consistent turbulence”, 11th European Fusion

Theory Conference, Aix en Provence, 26 September 2005.11. M. Vlad, F. Spineanu, P. Mantica and EFDA JET contributors, „Ratchet pinch and density peaking in H-mode

JET plasmas”, 11th EU-US Transport Task Force Workshop, Marseille 2006.12. F. Spineanu, M. Vlad, „The common ground of the density pinch and the transition to high confinement

regimes in tokamak”, 11th EU-US Transport Task Force Workshop, Marseille 2006.13. F. Spineanu, M. Vlad, „Field theoretical methods in theory of turbulence relaxation”, BG-URSI School and

Workshop on Waves and Turbulence Phenomena in Space Plasmas, 2006, Kiten, Bulgaria, invited paper.14. M. Vlad, F. Spineanu, „Nonlinear effects in charged particle transport in turbulent magnetic fields”, BG-URSI

School and Workshop on Waves and Turbulence Phenomena in Space Plasmas, 2006, Kiten, Bulgaria, invited paper.

15. F. Spineanu, M. Vlad, „Helicity fluctuation, generation of link number and effect on resistivity”, 21st IAEAFusion Energy Conference, Chengdu, China, 2006, accepted.

16. M. Vlad, F. Spineanu, „Test particle statistics and turbulence in magnetically confined plasmas”, 21st IAEAFusion Energy Conference, Chengdu, China, 2006, accepted.

Collaborations

CEA – Cadarache + Universite de Provence (Marseille)

ENEA - Frascati

ContactsK. Itoh and S.-I. Itoh (Japan)D. Montgomery (Dartmouth, USA)P. H. Diamond (San Diego, USA)K. Spatschek (Juelich, Germany)F. Jenko (Garching, Germany)

JET, Transport Task Force

Plasma rotation, density pinch, impurity inflow

Stable organised motion and coherent structures in fusion plasmas

Generation and stability of coherent flows in tokamak plasmasSimulation of alpha particle transport by collective modes

Stable organized motion and coherent structures in fusion plasmas

Most of the work in the period January – September 2006 was concentrated on the following milestones of our Work Plan:

• Assessment of the effect of the magnetic field inhomogeneity on test particle transport and calculation of the impurity pinch and diffusion coefficients in turbulent plasmas using the decorrelation trajectory method.

• Calculation of the spectrum in the problem of perturbed zonal flows (similar to the Orr-Sommerfeld); numerical description.

• Continuous model for the stationary states of the Hasegawa-Mima equation (small scale decay of the ITG turbulence).

We have also continued several studies concerning the transport coefficients in turbulent plasmas and we have initiated a new subject

1

Milestone : Calculation of the spectrum

in the problem of perturbed zonal flow

(similar to Orr-Sommerfeld); numerical description

Previous work : derivation of the exact analytic solution of the Flierl-

Petviashvili equation.

Present objectives:

• explain the streamers and intermittency of the ITB

• understand the role of the tilting instability

Target:

• JET experiments on ITB

• gyrokinetic simulations of sheared flow

• ITER-relevant : threshold for the ITB formation

– Cluj 2006 PTG Milestones I

2

Pattern of the zonal flow on

Jupiter (data from Voyager 1 and

2, cf. Busse, Chaos, 1994).

Suggested form (Hasegawa,

Kodama) and numerical

simulation

(compare L-H transition flows)


3

The first major problem: generation of ZF

Class A

1. Rayleigh-Benard analogy : Reynolds stress drives cell-flow transfor-

mation (Howard-Krishnamurti, Finn et al. for RB, Pogutse et al.

for interchange);

2. interaction of a set of essential spectral components (four-wave

paradigm) (Chen, White, Zonca);

3. thermodynamic view: wave-kinetic Fisher equation for k-space

pseudoparticles: predator-pray model (Diamond et al.);

Class B

Correlated growth of secondary instabilities of elongated ITG cells

Reynolds stress + convective cell transversal instability

(see numerical simulations of Kishimoto, Idomura et al.)


4

Possible scenario: reconnection of strongly tilted eddiesof the ITG mode

GyroFluid simulations

(Kishimoto)

The main driving mechanism

is Reynolds stress:

1. The tilting instability;

2. Gradual transformation:

ITG to electron drift;

3. The periodic layered

flow is an attractor;

(still to explain the super-fast

transition (∼ exp(exp(t)))


5

ITG versus Drift wave dynamics

∂∂t

(1 − ∇2

⊥

)φ −

[(−∇⊥φ × ez

)· ∇⊥

]∇2

⊥φ

+∇‖v‖i + vD

(1 +

1+ηiτ

∇2⊥

)∂φ∂y

= 0

∂v‖i∂t

+[(

−∇⊥φ × ez)· ∇⊥

]v‖i

= −∇‖p − ∇‖φ + µ∇2‖

v‖i

∂p∂t

+[(

−∇⊥φ × ez)· ∇⊥

]p

+vD1+ηi

τ∂p∂y

= −Υ∇‖v‖i

∂∂t

∇2⊥φ +

[(−∇⊥φ × ez

)· ∇⊥

]∇2

⊥φ

+κnτ

∂∂y

∇2⊥φ − µ∇2

⊥φ + ∇‖V‖e = 0

V‖e =mi

meη∇‖ (φ − p)

∂p∂t

+(−∇⊥φ × ez

)· ∇⊥p

−κn∂φ∂y

+ ∇‖V‖e = 0


6

The Flierl-Petviashvili equation

∆φ = αφ − βφ2 (1)

The exact solution

φ (x, y) =α

2β+ s℘

(iay + ibx + ω|g2 =

3α2

(sβ)2

)(2)

with the condition

a2 + b

2 =sβ

6(3)


7

Figure 1: Periodic solution of the Flierl-Petviashvili eq.


8

Comparison with experimental observations

(Doublet III-D, Coda et al., PRL2001)

Experiment Analytic solution φs

ϕrms > 10 V ϕrms > 17 V

ωE×B ∼ 2 × 105 s−1 ωE×B ∼ 2.2 × 105 s−1

λr ∈ (15...30) ρs λr ≃ 17.4ρs


9

−10 −8 −6 −4 −2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

kx (cm−1)

S(k

x)

The spectrum of the stationary potential perturbation

Figure 2: Experimental and theoretical spectrum.


10

Comparison with numerical simulations

(Gyrokinetic, Lausanne group, Alfey et al., Varenna, 2002)

Simulation Analytic solution φs

Er ∼ (−24 · · · + 24) × 103 (V/m) Er ∼ (−23 · · · + 23) × 103 (V/m)

ωE×B ∼ 2 × 10−3 Ω−1 ωE×B ∼ 10 × 10−3 Ω−1

δx ∼ (8.1 · · · 13.5) ρs δx ≃ 13.6ρs


11

The second major problem: destruction of ZF

Stability and structural stability

Figure 3: A weak and large scale perturbation evolves to a system of

vortices.


12

The stability of sheared flows

Typical instabilities

1. Kelvin-Helmholtz instability; cat’s eye Kelvin-Stuart (Reynolds con-

dition: existance of an inflection point in the velocity profile)

2. dissipative laminar sheared flow: Orr-Sommerfeld (our first order

from the barotropic eq.)

3. Kolmogorov flow: perturbation on a spatially periodic flow : chaos


13

The linear stability of the equation with scalar

nonlinearity

The equation of drift waves with the presence of a temperature gradient

∂

∂t

(1 −∇2

⊥

)φ =

∂

∂y

[(−∇2

⊥ + 4η2)φ]− φ

∂φ

∂y

The stationary solution is perturbed by a small function ε (x, y, t)

φ → φs (x, y) + ε (x, y, t)

where φs (x, y) is the periodic flow solution (expressed in terms of the

Weierstrass function). We obtain

∂

∂t(1 − ∆) ε =

∂

∂y

(−∆ε + 4η

2ε − φsε −

1

2ε2

)

We look for the linear dispersion relation.


14

The dispersion relation

∂2ε

∂x2+

iky

iω + iky

(− ω

ky− ωky − k

2y − 4η

2 + φs

)ε = 0

Notations q1 ≡ 1

1+ ωky

and a1 ≡ − ωky

− ωky − k2y − 4η2

∂2ε

∂x2+ (q1a1 + q1φs) ε = 0

φs (x, y) = φ0 + s℘ (x)

where φ0 = 3α2β

. The equation becomes

∂2ε

∂x2+ [q1a1 + q1φ0 + q1s℘ (x)] ε = 0 (4)


15

The Mathieu equation is an approximation

1

ky=

ω ± Q1/2

P + 1

1 ∓

[1 +

2 (P + 1)

(−ω + Q1/2)2

]1/2 (5)

1

ky≃ ω ∓√

Q

P + 1

(1 ∓

√2√Q

)

We obtain λ⊥ = 3ρs and from numerical simulations, λ⊥ = 6ρs

A necessary alternative: transversal stability of the flow in the

elongated ITG cells


16

Conclusions to this milestone The stationary sheared flow discovered

by us previsously as a solution of the Flierl-Petviashvili equation is very

close to experiment and simulation. The formation of such a structure is

connected with the tilting instability and reconnection of eddies aligned

along the poloidal direction.

The stability shows first destruction and generation of cells of the order

few ρs.


17

Future workAnalyticFind the description of the late phase of the tilting instability and

of the reconnection of eddies (Eckhaus instability).

NumericTime dependent simulations of of tilting and reconnection of eddies.


18

The collaborationsJETIn Transport Task Force. Pedestal physics (Andreev, etc.)

FrascatiLimit cycle of low order system for the quasi-periodic destruction

of the flow (Zonca, Annibaldi, Briguglio, Gregorio Vlad)


1

Milestone : Continuous model for the stationary states

of the Hasegawa-Mima equation

Previous work : derivation of the sinh-Poisson equation for the Euler

fluid.

Present objectives:

• explain the density pinch in tokamak and generation of background

gradients

• explain the low- to high- confinement transition

Target:

• JET experiments on density pinch

• DIII-D experiments on poloidal velocity profile at LH transition

• ITPA scaling for ITER

– PTG Milestones II

2

First part : What is behind the

evolution of vorticity to coherent

structures

Euler equation. D. Montgomery et

al., Phys. Fluids A4 (1992) 3.


3

There is a fundamental dynamics of the ideal plasma/neutral fluid leading

to evolution toward regular structures and coherent flows. It exists in the

background of the instabilities and turbulence.

It is justified to say that the vorticity self-organises

Two systems related with plasma/fluid vorticity elements exhibit

self-organization

• The model of point-like vortices interacting via a potential allows the

representation of the vorticity as a self-organising fluid, which acts as

a drive for passive fields like density. It has a negative temperature

(Taylor, Edwards);

• the continuum limit of the system of point-like vortices can be

described as the equations of continuity and momentum conservation

for an ideal compressible fluid; this fluid has negative pressure


4

There are two fundamental models :

• the line evolving under Nambu-Goto action leads to a fluid with

negative pressure and strange polytropic (Chaplygin gas).

• a fluid/plasma with vorticity is equivalent with a system of point-

like vortices interacting in plane via a potential

– long-range (Coulombian, logarithmic) : Euler fluid. No density

(ρ = const) The stationary states sinh-Poisson equation. Exactly

integrable.

– short-range (screened, K0) : Charney-Hasegawa-Mima

plasma/atmosphere. The third direction is implicitely present.

The density is not constant.

How are they connected: both reduce to the sinh-Poisson equation, along

two ways: at self-duality and as condition of emdedding in R3 according

to Gauss-Codazzi eqs.


5

The fact that the density follows the vorticity is due to the existence of an

intrinsic finite length in 2D plasma: the Larmor radius

The Charney-Hasegawa-Mima equationThe equation (CHM) derived for the two-dimensional plasma drift

waves and for Rossby waves in meteorology is:

(1 −∇

2

⊥

) ∂φ∂t

− κ∂φ

∂y− [(−∇⊥φ× n) · ∇⊥]∇2

⊥φ = 0 (1)

This is the equation governing the stationary states of the CHM eq.

∆ψ +1

2p2sinhψ (coshψ − p) = 0


6

Second part. How the vorticity evolves to a co-

herent structure and how the density responds

The conservation laws are less useful here. We need dynamical equations

• the aggregation-coagulation statistical process (Phase I, at the

beginning of the discharge when a random distribution of vortices is

generated. The merging takes place at a rate t−3/4 and the result is a

central vortex and possibly a ring of opposite vorticity).

• the equations of motion of the field theoretical model (Phase II)

• the equations of the ideal fluid with negative pressure

• the Manton’s method : dynamics on the parameter manifold, close to

self-duality


7

L = −κεµνρtr

(∂µAνAρ +

2

3AµAνAρ

)(2)

−tr[(Dµ

φ)† (Dµφ)]

−V(φ, φ

†

)

Sixth order potential

V(φ, φ

†

)=

1

4κ2tr

[([[φ, φ

†

], φ

]− v

2φ)†([[

φ, φ†

], φ

]− v

2φ)]

. (3)

The Euler Lagrange equations are

DµDµφ =

∂V

∂φ†(4)

−κενµρ

Fµρ = iJν (5)


8

The density reaction is via the effective Larmor radius

• The vorticity builds up a central cluster surrounded by a ring of

vorticity of opposite sign. The density must follow the vorticity

(Ertel’s theorem). This is only possible due to the polarisation drift,

compressible, which implies (but does not explicitely uses) the third

dimension.

• The pinch of vorticity is at the origin of the pinch of density; the

density develops gradients and in consequence, diamagnetic flow

• The combination of diamagnetic flow vd and of rotation speed u

induces a change in a basic parameter: ρs is replaced by an effective

Larmor radius1

(ρ

effs

)2=

1

ρ2s

(1 − vd

u

)

• When the density dragged by the vorticity develops higher gradients,


9

the effective Larmor radius becomes very large ρeff → ∞. The model

of point-like vortices with short-range interaction (Hasegawa-Mima)

evolves to the point-like vortices with long-range interaction (Euler

fluid). The density decouples from the vorticity;


10

Confirmation that our equation is the correct description :

Numerical solution for L = 307 : mono- and multipolar vortex


11

Third part. The space of solutions andhow the pinch arises.

The numerical study of the equation reveals the existance of three type

of states

• smooth vortex, stable

• strongly localised (narrow) vortex, a physical approximation of the

singular vortex

• a class of intermediate quasi-solutions, organized along a line of

minimum departure from the action extremum.

The smooth vortices are accessible from a wide range of initial conditions.

The narrow vortices are only accessible from a subset of initial functions.

There are differences: for narrow vortices the maximal vorticity is much

higher compared to the smooth vortex. There is a difference in both

energy and vorticity between them.


12


13

The H (high) confinement state as coherent

structureCoherent structure (Lagrangian) vs Driven-dissipative (spectrum).

0 20 40 60 80 100 120 140 160−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

radius

ψ s

trea

mfu

nctio

n

The solution streamfunction ψ(r)

Figure 1: Natural edge sheared flow in TJ-II Stellarator (Alonso et al.)

and our solution for the potential.


14

Comparison with the self-organisation of the vorticity obtained in a

statistical approach (HW)

Figure 2: A theory of self-organization of drift turbulence (Hasegawa-

Wakatani) leads to this radial profile for the potential. Actually their

model does not explicitely need excitation of drift waves.


15

Edge electric field in DIII and JET


16

−400 −300 −200 −100 0 100 200 300 400−8

−6

−4

−2

0

2

4

6

8

10

12x 10

−3

r

vψ

The magnitude of the velocity vector tangent to the streamlines

Results for the poloidal velocity


17


18

Transition from smooth to narrow requires an input of kinetic en-

ergy and localised vorticity

−20

−10

0

10

20

−10−5

05

1015

0

100

200

300

400

ln(Efin

)

ln(|Ωfin

|)

L

Figure 3: The squares are smooth and the dots are narrow vortices. The

plot presents the difference in both kinetic energy and vorticity. At right is

a qs. for L = 401 with slow intermediate variation of vθ.


19

Scatterplot (Energy, Vorticity) of final results for pinch and for

sheared velocity vθ

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01−1

−0.5

0

0.5

Final energy

Fin

al v

ortic

ity Ω

Small dots: all data; green squares: evolution; red circle: narrow; black square: solution

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−20

−10

0

10

20

30

40

50

60

70

Final energy

Fin

al v

ortic

ity Ω

Small dots: all data; green squares: evolution; red circle: narrow; black square: solution

Figure 4: Black squares: smooth solutions; green: intermediate quasi-

solutions; red: singular solutions. In addition, for barriers: yellow: stable

qs. with ring; blue: intermediate forms evolving to stable ring qs.


20

The role of the effective Larmor radius

vd/u ρeff

s/ρs L = a/ρeff

s|vθbottom|

∣∣∣vphys

θbottom

∣∣∣ (m/s) |Er| (kV/m)

0.2 1.118 273 1 × 10−3 979 2.45

0.4 1.29 236 1 × 10−3 979 2.45

0.85 2.58 95 2.2 × 10−3 2152 5.38

0.95 4.47 63 0.02 19580 48.95


21

Conclusions to this milestone As part of the phenomenology of struc-

ture generation in fluids and plasma it appears convenient to take the

vorticity as the active factor of self-organization. This is supported by

the model of point-like vortices interacting in plane, model that has pro-

vided us with the description of the coherent stationary states. Examin-

ing the solutions of the coherent vortical type and their neighborhood in

the function space it is found that there are states with properties similar

to the physical states of 2D plasma. The pinch of density is the evolution

of the system along a string of quasi-solutions, between a smooth vortex

and a singular vortex. The increase of the effective Larmor radius due

to the increase of the density gradient slows down this process without

suppressing it. The transition between the L to H states appears as the

transition between the smooth vortex and a state with high vθ shear (a

quasi-solution which is close to the extremum of action), of which it is

separated by a gap in energy and vorticity.


22

Future workAnalyticProve that intermediate profiles are softly unstable. Find the equa-

tion of evolution of vorticity close to the absolute extremum (singu-

lar vortex), via Manton’s method. Derive a unique chain of equa-

tions for density pinch, from Ertel’s theorem and effective ρs.

NumericTime dependent simulations of CHM.


23

The collaborationsJETIn Transport Task Force (Mantica, Naulin, Tala, Rassmussen,

Parail, etc.). Contacts with experiments (Andrew) for poloidal

rotation, quite recent.

CEA-Marseille and ENS-ParisMuch better time dependent simulations using Adaptive Mesh Re-

finement and comparison with our stationary solutions (Kai, Farge,

Benkadda, Agullo, etc.)

2006 : Two invited researchers in the last year (from Frascati and

from CEA-Marseille) One month mobility for a member of our group,

in France.


24

Interesting theoretical development (spin-off)

We are able to provide an accurate description of the

large-scale atmospheric vortex : tropical cyclone (ty-

phoon, hurricane)This is confirmed by quantitative comparisons with ob-served profiles of azimuthal velocity, radius of the eye-wall, decay of velocity.


Stable organized motion and coherent structures in fusion ... 2006... · Publications September...

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