Presentation on theme: "Boris Weyssow Universite Libre de Bruxelles Dana Constantinescu"— Presentation transcript:
1 Order and chaos in some Hamiltonian systems of interest in plasma physics Boris WeyssowUniversite Libre de BruxellesDana ConstantinescuUniversity of Craiova, RomaniaEmilia Petrisor, University of Timisoara, RomaniaJacques 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 described for various safety factors and perturbations.General explanations for some experimental observations concerning the transportbarriers are issued from the analytical properties of the models:- the transport barriers are obtained in the presence of a reversed magnetic shearin 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 Tokamaks are toroidal devices used in thermocontrolled nuclear fusion JET (EUR)Magnetic fieldToroidal Poloidalcomponent componenthelical magnetic field lines on nested tori sorrunding the magnetic axis (the ideal case)
4 The magnetic field equations Hamiltonian system Toroidal coordinates = toroidal angle= polar coordinates in a poloidal cross-sectionThe magnetic field equations Hamiltonian systemClebsch representationis the magnetic fieldis the poloidal fluxis the toroidal fluxUnperturbed case:regular (helical) magnetic lineschaotic+regular magnetic linesPerturbed case
5 is an area-preserving map compatible with the toroidal geometry 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 functionis the safety factor (the q-profile)is the magnetic shearis the perturbationK is the stochasticity parameter
6 the tokamap, R. Balescu (1998) Chirikov-Taylor (1979)Wobig (1987)D. Del Castillo Negrete (1996)the tokamap, R. Balescu (1998)the rev-tokamap, R. Balescu (1998)
7 D. del Castillo Negrete, Greene J.M.,Morrison P.J. (1996,1997) Area-preserving mapsNon-twist:Twist :(Monotonous winding function)(positive or negative shear)(Non monotonous winding function)(reversed shear)Poincare H. (1893) foundation of dynamical systems theoryBirkhoff G. D, ( ) fundamental theoremsKAM (Moser 1962) (persistence of invariant circles)Greene J. M,Aubry M & Mather J. N.MacKay R. S. Percival I. C.( )(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)
8 Robust invariant circles (ITB) separating two invariant chaotic zones The rev-tokamapK=6.21K=3.5K=4.5K=5.5Robust 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 circlesThe magnetic transport barrier surrounds the shearless curveA magnetic transport barrier appears near 0 shear curve,even in systems involving monotonous q-profile
11 ITB (the physical transport barrier) For K< twist invariant circlesexist in the upper part of ITBFor K> all invariant circles in the upper part of ITB are nontwistK=1.6K=1.7
12 The destruction of invariant circles Unbounded component in the negative twist regionBounded component in the positive twist regionTheoremAANo invariant circle pass through the points ofNo invariant circle pass through the points ofK=0.5K=4.1375AAas long as A belongs to the negative twist region.K=5K=6is the intersection ofwith the line
13 Reconnection phenomena (global bifurcation of the invariant manifolds of regular hyperbolic pointsof two Poincare-Birkhoff chains with the same rotation number)Before reconnection: heteroclinic connectionsbetween the hyperbolic points in each chainReconnection:-connections between the hyperbolic points of distinct chains-heteroclinic connections in the same chainAfter reconnection: homoclinic+heteroclinicconnections in each chainThe chains are separated by meandersTheoremThe reconnection of twin Poincare-Birkhoff chainsoccurs 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 NTAw decreasesthe P-B chains enter NTA but they arestill separated by rotational circlesw decreasesthe hyperbolic points reconnectw decreasesthe meanders separate the P-B chainshaving homoclinic connectionw decreasesthe first collision-annihilation occursw=0.53w=0.51w decreasesthe second collision-annihilation occursw decreasesthere are no more periodic orbitsof type (n,m)w=0.50w=0.49
15 ConclusionsThe rev-tokamap model was used for the theoretical study of magnetic transport barriersobserved 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-profileis closed to a low order rational.Results:A magnetic transport barrier appears near 0 shear curve, even in systems involvingmonotonous q-profile.In the rev-tokamap model the shape of the winding function has only quantitativeimportance in the size of NTA.The enlargement of NTA is directly related to the maximum value of the windingfunction (corresponding to the minimum value of the safety factor).