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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).