Order and chaos in some Hamiltonian systems of interest in plasma physics

1. Order and chaos in some Hamiltonian systems of interest in plasma physics Boris Weyssow Universite Libre de Bruxelles Dana Constantinescu University of Craiova, Romania Emilia Petrisor, University of Timisoara, Romania Jacques Misguich, CEA Cadarache, France

2. 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 describedfor 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)

3. JET (EUR) Magnetic field Toroidal Poloidal component + component helical magnetic field lines on nested tori sorrunding the magnetic axis (the ideal case) Tokamaks are toroidal devices used in thermocontrolled nuclear fusion

4. Toroidal coordinates  = toroidal angle = polar coordinates in a poloidal cross-section The magnetic field equations Hamiltonian system Clebsch representation is the magnetic field is the poloidal flux is the toroidal flux Unperturbed case: regular (helical) magnetic lines chaotic+regularmagnetic lines Perturbed case

5. The discrete system ( Poincare map associated with the poloidal cross-section ) is an area-preserving map compatible with the toroidal geometry (the magnetic axis is invariant) (because ) is the winding function is the safety factor (the q-profile) is the magnetic shear is the perturbation K is thestochasticity parameter

6. Chirikov-Taylor (1979) Wobig (1987) D. Del Castillo Negrete (1996) the tokamap, R. Balescu (1998) the rev-tokamap, R. Balescu (1998)

7. Area-preserving maps Non-twist: Twist : (Monotonous winding function) (positive or negative shear) (Non monotonous winding function) (reversed shear) PoincareH. (1893) foundation of dynamical systems theory Birkhoff G. D, (1920-1930) fundamentaltheorems KAM (Moser 1962)(persistence of invariant circles) Greene J. M, Aubry M & Mather J. N. MacKayR. S. Percival I. C.(1976-...) (break-up of invariant circles,converse KAM theory etc) D. delCastillo 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)

8. The rev-tokamap K=6.21 K=3.5 K=4.5 K=5.5 Robust invariant circles (ITB) separating two invariant chaotic zones

9. Rev-tokamap is a non-twist map (the critical twist circle) The nontwist annulus (NTA) is the closure of all orbits starting from the critical twist circle.

10. The revtokamap is closed to an almost integrable map in an annulus surrounding the curve NTA contains the most robust invariant circles The magnetic transport barrier surrounds the shearless curve A magnetic transport barrier appears near 0 shear curve, even in systems involving monotonous q-profile

11. 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 K=1.6 K=1.7

12. The destruction of invariant circles Unbounded component in the negative twist region Bounded component in the positive twist region Theorem A A No invariant circle pass through the points of No invariant circle pass through the points of K=0.5 K=4.1375 A A as long as A belongs to the negative twist region. K=5 K=6 is the intersection of with the line

13. 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 chain The chains are separated by meanders Theorem The reconnection of twin Poincare-Birkhoff chains occurs in the NTA

14. Scenario for reconnection (the same perturbation, modified W) for w>n/m the two P-B chains of type (n,m) are outside NTA w decreases the P-B chains enter NTA but they are still separated by rotational circles w decreases the hyperbolic points reconnect w decreases the meanders separate the P-B chains having homoclinic connection w decreases the first collision-annihilation occurs w=0.53 w=0.51 w decreases the second collision-annihilation occurs w decreases there are no more periodic orbits of type (n,m) w=0.50 w=0.49

15. 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 minimumvalue 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).