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Noble Cantor sets acting as partial internal transport barriers in fusion plasmas Article in Plasma Physics and Controlled Fusion · June 2002 DOI: 10.1088/0741-3335/44/7/101
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INSTITUTE OF PHYSICS PUBLISHING
PLASMA PHYSICS AND CONTROLLED FUSION
Plasma Phys. Control. Fusion 44 (2002) L29–L35
PII: S0741-3335(02)34783-3
LETTER TO THE EDITOR
Noble Cantor sets acting as partial internal transport barriers in fusion plasmas J H Misguich1 , J-D Reuss1 , D Constantinescu2 , G Steinbrecher2 , M Vlad3 , F Spineanu3 , B Weyssow4 and R Balescu4 1 Association Euratom-C.E.A. sur la Fusion, CEA/DSM/DRFC, C.E.A.-Cadarache, F-13108 Saint-Paul-lez-Durance, France 2 Association Euratom-N.A.S.T.I., Department of Physics, University of Craiova, Str. A.I. Cuza No 13, Craiova-1100, Romania 3 Association Euratom-N.A.S.T.I., National Institute of Laser, Plasma and Radiation Physics, PO Box MG-36, Magurele, Bucharest, Romania 4 Association Euratom-Etat Belge sur la Fusion, Universit´ e Libre de Bruxelles, CP 231, Campus Plaine, Boulevard du Triomphe, B-1050 Bruxelles, Belgium
E-mail:
[email protected]
Received 12 March 2002 Published 14 June 2002 Online at stacks.iop.org/PPCF/44/L29 Abstract In hot laboratory plasmas, internal transport barriers (ITBs) have recently been observed, localized in the radial profile ‘around’ rational values of the winding number ω(r) = 1/q(r). Such barriers are obviously related to the perturbed magnetic structure, described by a 1 + 1/2 Hamiltonian in presence of a perturbation. From the point of view of non-linear Hamiltonian dynamical systems [1], this experimental result appears highly paradoxical since rational q-values generally correspond to the less robust tori. We have studied the appearance of chaos of toroidal magnetic lines by a discrete area-preserving map named ‘tokamap’ [2]. By increasing the perturbation, we have observed in a wide chaotic sea the destruction of the last confining Kolmogorov–Arnold–Moser surfaces, broken and transformed into permeable Cantor sets (Cantori). The flux across a Cantorus has been computed by using refined mathematical techniques due to MacKay, Mather and Aubry. We have proved that the ITB observed in the tokamap is actually composed [3] of two permeable Cantori with ‘noble’ values of ω (in the definition of Percival), each Cantorus forming a partial (permeable) barrier inhibiting the magnetic line motion. More generally, between the dominant chains of rational islands q = m/(m − 1), the most resistant barriers between q = (m + 2)/(m + 1) and (m + 1)/m have been checked (Greene, MacKay and Stark) to be localized on the ‘most irrational’ numbers in these Farey intervals, i.e. on the noble numbers N (1, m) ≡ 1 + [1/(m + 1/G)] (where G is the Golden number), defined by their continuous fraction expansion N (i, m) = [i, m, (1)∞ ].
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Letter to the Editor
In conclusion, the study of the tokamap mapping allowed us to predict on mathematical basis that ITB can occur in tokamak plasmas not only ‘around’ rational magnetic surfaces but more precisely on noble q−values of irrational surfaces, and to localize them by the Fibonacci series of their convergents.
1. Chaotic magnetic lines In the way towards thermonuclear fusion energy, tokamak plasmas are the most studied today. Long time confinement of a hot plasma is achieved by a strong toroidal and poloidal magnetic field, with magnetic lines remaining a priori on perfect toroidal magnetic surfaces, described by one degree of freedom Hamilton equations of ‘motion’ along the torus. The helical motion of the line is characterized by a surface quantity, the smallradius dependent winding number ω(r), inverse of the safety factor q(r) used in tokamak physics. However, coil imperfections as well as internal instabilities (tearing) are responsible for magnetic perturbations, with a 1 + 1/2 Hamiltonian structure, allowing for surface breaking and appearance of (measured) magnetic island and chaotic layers. This remains a situation of incomplete magnetic chaos because several undestroyed magnetic surfaces—Kolmogorov– Arnold–Moser (KAM) tori—remain robust between chaotic layers and island remnants. These KAM surfaces may act as transport barriers inhibiting radial transport of heat and particles which predominantly follow magnetic lines (in the absence of particle collisions and magnetic drifts). Strong gradients of density or temperature have been measured either on the plasma edge in the case of H-modes, or inside the plasma ‘near’ rational values of ω(r) in the case of internal transport barriers (ITB). We do not intend to claim that the ITB observed in experiments are of purely magnetic origin, as in the model studied here. We simply develop a model for magnetic lines, in which the situation is simple and analogous to that occurring in toroidal plasmas, at least in the ideal case (no magnetic drifts, no electric field, no collisions). Perfect magnetic surfaces are in this case true KAM tori and form barriers that magnetic lines (and particles) cannot cross: KAM tori are robust barriers. In our magnetic model, the imposed magnetic perturbation deeply modifies this ideal picture and destroys part of the KAM surfaces: (a) some magnetic surfaces remain robust KAM barriers which cannot be crossed, or are changed into island chains (always partly chaotic around hyperbolic points), (b) some other magnetic surfaces are completely destroyed and form a chaotic zone, but (c) some other magnetic surfaces are broken into permeable Cantor sets (Cantori) forming ‘partial’ barriers which do not suppress but simply inhibit the radial motion of the magnetic lines, hence the transport. We have focused our attention on a perturbation amplitude L slightly above the threshold where the last KAM has been broken in the tokamap. In this case, what remains of the destroyed most robust last KAM surfaces are the two Cantori across which a slow intermittent transport has been observed: these are not robust barriers, but ‘partial’ barriers inside the plasma. In the present purely magnetic model, these Cantori are true internal barriers for transport. Of course, in more complex situations, many other effects may influence the nature and position of a transport barrier, but it is generally recognized that the position of the barrier in a toroidal plasma is strongly related to the q-value, which lead us to think that magnetic effects are indeed the more important ones.
Letter to the Editor
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2. Transport barriers in the tokamap From the point of view of non-linear dynamical systems theory (see e.g. [1]), this experimental result appears highly paradoxical since rational values of ω are known to correspond to the less robust circles (i.e. the first KAM tori which are broken when the perturbation is increased)1 . We have used the technique of discrete area-preserving Hamiltonian mappings in order to study chaotic magnetic lines in a toroidal geometry, namely in the ‘tokamap’ [2] with a realistic radial ω profile. The motion of a magnetic line in the poloidal section is described in the phase space (ψ, θ ) where the toroidal flux ψ = x 2 is simply related to the radial variable x, and θ is the poloidal angle. For increasing values of a perturbation or stochasticity parameter L, this mapping describes successive breaking of existing magnetic surfaces—KAM barriers— and the appearance of a slow radial motion of magnetic lines across the remnants of broken magnetic surfaces which appear as permeable Cantori. As expected from non-linear dynamical systems theories, the most resistant KAM surfaces have most irrational ω(r)2 . For a value of the perturbation parameter L very near the one necessary for the breaking of the KAM on the edge, a typical wide chaotic sea is found encircling a protected plasma core. Such a chaotic sea is described by a single very long magnetic line, and an intermittent motion is observed across an ITB [3]. By using refined mathematical techniques due to MacKay, Matter and Aubry, we have proved [4] that this ITB is actually composed of two permeable Cantori with irrational, ‘noble’ values of ω (in the definition of Percival [5]) given by N (1, 8) and N(1, 7) defined below in (2), as seen in the phase portrait of figure 1. 3. Barrier localization in q -profiles 3.1. The q-comb model From the experimental point-of-view, the localization of such ITB appears to be important in plasma simulations models, like the ‘q-comb’ model [6, 7] which considers a series of plasma shells with ‘good’ and ‘bad’ confinement properties (low and high values of thermal diffusivity χe ) alternating on precise radial positions which have been chosen, in agreement with experiments, ‘near’ rational values of q = 1/ω. An example is given in figure 2. In figure 2, the positions of the barriers are obtained from experimental evidences, but the precise location of the low χe regions (barriers) are not particularly related to any most resistant KAM or Cantorus, nor to noble q-values, but rather on a main rational (1, 4/3, 3/2 and 2) and a neighbouring surface. 3.2. Cantori on noble q-values The above results [4, 8] allow us to determine the values of ω where such transitions occur in the vicinity of a given rational chain, for instance the transition between a good confinement (low χe ) region with regular magnetic surfaces or island chains, and a bad confinement (high χe ) region with stochastic zones (see figure 2). Let us consider, for instance, the main series of rational islands m q = Q(m) ≡ (1) m−1 1
Let us remind that KAM surfaces are known as ‘tori’ or ‘circles’ in the poloidal plane, according to the number of dimension involved. 2 The Golden number G however was shown to play no particular role in the tokamap.
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radius 1.2
N ( 4 , 2)
1
0.8
0.6
0.4
N( 1 , 11) 0.2
0
N( 1 , 7) N( 1 , 8) 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
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0.9
1
angle Figure 1. The main √ noble transport barriers are represented in bold lines in normalized coordinates ‘radius’ = x = ψ and angle = θ. The robust KAM separating the central protected plasma core (in white) from the chaotic shell is characterized by q = N (1, 11) around the magnetic axis. The two semi-permeable Cantori N (1, 8) and N (1, 7) form an internal barrier resulting in a very slow and intermittent motion towards the chaotic sea. The robust KAM torus on the plasma edge N (4, 2) has been identified to have a vanishing flux.
Figure 2. Schematic drawing of a q-comb as presented in figure 5.4, p 60 in [6] (by courtesy of A Schilham). The position of the barriers are obtained from experimental evidences, but the precise location of the low χe regions (barriers) are not particularly related to any most resistent KAM or Cantorus, nor to noble q-values, but rather on a main rational (1, 4/3, 3/2 and 2) and a neighbouring surface.
Letter to the Editor
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separated by chaotic zones. The most resistant barriers between q = (m + 2)/(m + 1) and (m + 1)/m are expected, according to Greene et al [9], to be localized on the ‘most irrational’ numbers in these Farey intervals, i.e. on the noble numbers 1 (2) N (1, m) ≡ 1 + m + 1/G where √ 5+1 = N (1, 1) (3) G= 2 is the Golden number. The numbers introduced in (2) are thus defined by their continuous fraction expansion N (i, m) = [i, m, (1)∞ ] ≡ i + 1/(m + 1/ (1 + 1/(1 + · · ·)))
(4)
the rational approximants of which (also named ‘convergents’) have numerators and denominators following two Fibonacci series [10]. It is simple to prove that those noble numbers N (1, m) (2) are alternating with the main rationals Q(m) as illustrated in figure 3 Q(m + 1) > N (1, m) > Q(m + 2)
(5)
In the tokamap with L = 4.875/2π for instance, one main ITB has been observed in the chaotic sea. It is formed by the two noble Cantori q = N (1, 7) and N (1, 8), surrounding the chain of rational island remnants q = 9/8. For instance, we have proved [4] that in the tokamap with an monotonously increasing q-profile, the surface q = N (1, 7) located between q = 9/8 and q = 8/7 is indeed a Cantorus for a stochasticity parameter L = 4.875/2π . This value is slightly obove the threshold value where the most robust KAM surfaces have been destroyed inside the tokamap, giving rise to Cantori. We have proved indeed that the sequence of its convergent chains (6) qυ = 9/8, 17/15, 26/23, 43/38, 69/61, 112/99, 81/160, 293/259, 474/419, 767/678, 1241/1097, . . . (7) 1 ⇒ N (1, 7) ≡ 1 + = 1.311 267 464 . . . 7 + 1/G (where numerators and denominators separately follow Fibonacci series [10]) satisfies indeed the Greene supercritical condition [11] for Cantori. This condition is proved to be satisfied by calculating the residues Rυ+ and Rυ− for orbits qυ = nυ /mυ : the residue Rυ+ of a ‘maxmin’ orbit Cantorus
m=4
N(1,2)
Separatrix
Separatrix
C antorus
N(1,3)
C antorus
N(1,4)
m=5 Separatrix
m=6 Cantorus
N(1,5)
Figure 3. Schematical situation of several island chains in a dominant series q = Q(m) = (m + 1)/m with here m = 4, 5 and 6. We exhibit here the most resistant surfaces in these Farey intervals between q = Q(m + 2) and q = Q(m + 1), which are the noble q-values N (1, m) where the Cantori are localized in this simple case.
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Letter to the Editor
passing through elliptic or inverse hyperbolic points is going indeed to +∞, and the residue Rυ− of an ‘extremizing’ orbit passing through direct hyperbolic points is going to −∞, proving the existence of a Cantorus on q = N (1, 7). Moreover, the flux across this surface has been computed and is compatible with a permeable Cantorus: along the series of convergents (6) the flux indeed converges towards a finite non-zero value of the order of 10−9 [4], in contrast with the flux across a KAM which is checked to converge to zero. The general structure of the noble Cantori alternating with main rational chains q = Q(m) = m/(m − 1) is given in figure 3. This schema represents a typical situation of several island chains in a dominant series q = Q(m + 1) = (m + 1)/m where the chains m = 4, 5 and 6 are presented. It appears from the tokamap example [8, 4] that the most resistant surface in one of these Farey intervals, between q = Q(m + 2) and q = Q(m + 1), actually corresponds to the ‘most noble’ q-value which is exactly q = N (1, m) (see equation (5)). The surfaces with q = N (1, m) are the positions where we find the most robust KAM’s, which after breaking for increased values of L, give rise to the most resistant barriers, the Cantori. This can be seen in the JAVA animation movie [3] in which long sojourn times can be observed in most chaotic layers surrounding the main rational chains, each layer being limited by two Cantori. As Greene wrote 20 years ago, the breakup of magnetic surfaces ‘is a problem of number theory’ [12], as is verified here again. 4. Conclusion In conclusion, the study of the tokamap mapping has allowed us to predict on mathematical basis that ITBs can occur in the magnetic field of tokamak plasmas not only ‘around’ rational magnetic surfaces but more precisely on the edge of chaotic layers, i.e. on irrational surfaces, with noble q-values. These surfaces can be localized by the Fibonacci series of their convergents (see e.g. [6]). We have shown that the experimental measurements [7] which identify the presence of ITBs ‘around’ rational surfaces are actually in qualitative agreement with the above theoretical approach of a magnetic description of ITBs [4, 8], since the noble irrational surfaces are located in the near vicinity of rational surfaces, but alternating with the latter. Such precise localization can be used, for instance, to build more refined q-comb models for transport in stratified plasmas [7]. We want to thank Robert MacKay, Yves Elskens and Emilia Petrisor for many fruitful discussions about mathematical aspects. Four of the authors (DC, GS, MV and FS) have benefitted in 2001 from grants from the french Minist`ere des Affaires Etrang`eres through C.E.A. Partial support is acknowledged from NATO, Linkage Grants PST.CLG.971784 and 977397. References [1] MacKay R S 1993 Renormalisation in Area-preserving Maps (Singapore: World Scientific) [2] Balescu R, Vlad M and Spineanu F 1998 A Hamiltonian twist map for magnetic field lines in a toroidal geometry Phys. Rev. E 58 951 [3] Reuss J-D 2000 Tokamap intermittency gif-movie (Cadarache) (See EPAPS document No E-PHPAEN-8972105 for GIF and JAVA animation movies showing the time behaviour of the intermittent motion of a magnetic line across the two noble Cantori of the transport barrier in the poloidal plane. This document may be retrieved via the EPAPS homepage . See the EPAPS homepage for more information. An actualized version can be found on .) [4] Misguich J H, Reuss J-D, Constantinescu D, Steinbrecher G, Vlad M, Spineanu F, Weyssow B and Balescu R 2002 Noble internal transport barriers and radial subdiffusion of toroidal magnetic lines Report EUR-CEA-FC 1724 (Cadarache, France) [5] Percival I C 1982 Physica 6 D 67 [6] Hogeweij G M D, Lopes Cardozo N J, de Baar M R and Schilham A M R 1998 Nucl. Fusion 38 1881 [7] Schilham A 2001 Stratified thermonuclear plasmas PhD Thesis Eindhoven University of Technology, Eindhoven Schilham A M R, Hogeweij G M D and Lopes Cardozo N J 2001 Plasma Phys. Control. Fusion 43 1699 [8] Misguich J H 2001 Dynamics of chaotic magnetic lines: intermittency and noble internal transport barriers in the tokamap Phys. Plasmas 8 2132 [9] Greene J, MacKay R and Stark J 1986 Physica 21D 267 [10] Fibonacci (Leonardo Pisano): Liber abaci (Pisa, 1202) (see also .) [11] Greene J M 1979 J. Math. Phys. 20 1183–201 Greene J M 1980 Annals of New York Acad. Sci. 357 80–89 [12] Greene J M 1984 Renormalization and the breakup of magnetic surfaces Statistical Physics and chaos in Fusion Plasmas ed C W Horton Jr and L E Reichl (Wiley series in nonequilibrium problems in the physical sciences and biology, vol 3) I Prigogine and G Nicolis ser. ed (New York: Wiley) p 3
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