Order and chaos in some Hamiltonian systems of interest

in plasma physics Boris Weyssow

Universite Libre de Bruxelles

Dana ConstantinescuUniversity of Craiova, Romania

Emilia Petrisor, University of Timisoara, Romania

Jacques Misguich, CEA Cadarache, France

A class of Hamiltonian systems is studied in order to describe, from a mathematical point of view, the structure of the magnetic field in tokamaks with reversed shear

configuration. The magnetic transport barriers are analytically located and described for various safety factors and perturbations.  General explanations for some experimental observations concerning the transport barriers are issued from the analytical properties of the models:

-   the transport barriers are obtained in the presence of a reversed magnetic shear in the negative or low shear region (Litaudon X (1998), Maget P (2003), Neudatchin S. V. (2004))

-     zones with reduced transport appear when the minimum value of the safety factor closed to a low rational number (Lopez Cardozo N.J. (1997), Gormezano C. (1999), Garbet X. (2001), Jofrin E. (2002), Neudatchin S. V.(2004)

Tokamaks are toroidal devices used in thermocontrolled nuclear fusion

JET (EUR) Magnetic field

Toroidal Poloidal

component + component

helical magnetic field lines on nested tori sorrunding

the magnetic axis (the ideal case)

Toroidal coordinates

 = toroidal angle

= polar coordinates in a poloidal cross-section

,rcst

The magnetic field equations Hamiltonian system

0

0

B

B

d

d

Clebsch representation

is the magnetic field

B

,,r ,,

B

is the poloidal flux is the toroidal flux

Unperturbed case: 0 regular (helical) magnetic lines

Perturbed case ,,0 PK chaotic+regular magnetic lines

The discrete system

),(,: 11 KK TRSRST

( Poincare map associated with the poloidal cross-section )

hgK

hgKWTK

'

)1(mod':

is an area-preserving map compatible with the toroidal geometry

)(()( ATareaAarea K

00 0,0, KT

)

(the magnetic axis is invariant)

02

2

r(because

d

dW 0

Wq

1

ln

ln

d

qds

is the winding function

is the safety factor (the q-profile)

is the magnetic shear

hgP , is the perturbation

K is the stochasticity parameter

2cos2

1,,

2

2

hgPW

2cos2

1,,

2

2

PW

2cos2

1,,1)( 2 PbaW

Chirikov-Taylor (1979)

Wobig (1987)

the tokamap, R. Balescu (1998)

the rev-tokamap, R. Balescu (1998)

D. Del Castillo Negrete (1996)

2cos14

1,,222

4 22

P

wW

2cos14

1,1,0

,1,,11

210

0

102

PWwWw

ww

wwC

w

wwACAwW

Area-preserving maps

Twist : Non-twist:

(Monotonous winding function)

(positive or negative shear)

 

Poincare H. (1893) foundation of dynamical systems theory

Birkhoff G. D, (1920-1930) fundamental theorems

KAM (Moser 1962) (persistence of invariant circles)

Greene J. M,

Aubry M & Mather J. N.

MacKay R. S. Percival I. C.(1976-...)

(break-up of invariant circles,converse KAM theory etc)

D. del Castillo Negrete, Greene J.M.,Morrison P.J. (1996,1997)

(routes to chaos in standard map systems)

Delshams A., R. de la Llave (2000)

(KAM theory for non-twist maps)

Simo C. (1998) (invariant curves in perturbed n-t maps)

Petrisor E. (2001, 2002) (n-t maps with symmetry group, reconnection)

(Non monotonous winding function)

(reversed shear)

RS 1,,0, 1,,0, Cfor

1mod2cos1

1

411

42sin2

12sin2

12

1

:

2

2

2

KCAw

KK

TK

The rev-tokamap

),(,: 11 KK TRSRST

25.0,3333.0,9626.0,35.3 10 wwwK

1667.0,3333.0,67.0 10 www

K=3.5 K=4.5 K=5.5 K=6.21

Robust invariant circles (ITB) separating two invariant chaotic zones

Rev-tokamap is a non-twist map

1,,0, Cfor

The nontwist annulus (NTA) is the closure of all orbits starting from the critical twist circle.

0''':1 hgkWC

(the critical twist circle)

The revtokamap is closed to an almost integrable map in an annulus surrounding the curve C/1

NTA contains the most robust invariant circles

The magnetic transport barrier surrounds the shearless curveA magnetic transport barrier appears near 0 shear curve, even in systems involving monotonous q-profile

ITB (the physical transport barrier)

For K<3.923916 twist invariant circles

exist in the upper part of ITB

For K>3.923916 all invariant circles in the upper part of ITB are nontwist

1667.0,3333.0,67.0 10 www

K=1.6 K=1.7

The destruction of invariant circles

0,:

f

0,,*|, N

Unbounded component in the negative twist region

Bounded component in the positive twist region

No invariant circle pass through the points of

No invariant circle pass through the points of

as long as A belongs to the negative twist region.

K=0.5 K=4.1375

K=5 K=6

f

f

f

*,5.0 A

is the intersection of with the line

f

5.0

A A

A A

Theorem

1667.0,3333.0,67.0,3 10 wwwK

f

f

1667.0,3333.0,67.0,5.5 10 wwwK

1C

Reconnection phenomena(global bifurcation of the invariant manifolds of regular hyperbolic points of two Poincare-Birkhoff chains with the same rotation number)

Before reconnection: heteroclinic connections between the hyperbolic points in each chain

Reconnection: -connections between the hyperbolic points of distinct chains-heteroclinic connections in the same chain

After reconnection: homoclinic+heteroclinic connections in each chainThe chains are separated by meanders

The reconnection of twin Poincare-Birkhoff chainsoccurs in the NTA

Theorem

)(9.2)(267.2)(2.2,2.0,35.0,5/481.0 10 dkckbkwww

Scenario for reconnection(the same perturbation, modified W)

the P-B chains enter NTA but they arestill separated by rotational circles

the hyperbolic points reconnect

the meanders separate the P-B chainshaving homoclinic connection

the first collision-annihilation occurs

the second collision-annihilation occurs

there are no more periodic orbitsof type (n,m)

for w>n/m the two P-B chains of type (n,m)are outside NTA

w decreases

w decreases

w decreases

w decreases

w decreases

w decreases

w=0.53 w=0.51

w=0.50 w=0.49

Conclusions

 The rev-tokamap model was used for the theoretical study of magnetic transport barriers observed in reversed shear tokamaks.Analytical explanations were proposed for-the existence of transport barriers in the low shear regions-the enlargement of the transport barriers when the minimum value of the q-profile is closed to a low order rational.Results:

-A magnetic transport barrier appears near 0 shear curve, even in systems involving monotonous q-profile.

-In the rev-tokamap model the shape of the winding function has only quantitative importance in the size of NTA.

The enlargement of NTA is directly related to the maximum value of the winding function (corresponding to the minimum value of the safety factor).