Kei. Hirose

T. Kamimura

Yoshi H. Ichikawa

National Institute for Fusion Science (NIFS-Repository)

§18. Anomalous Diffusion periodic Accelerator Standard Map and Modes Multiple-in the Kei Hirose, T.Kamimura, Yoshi H. lchikawa(Chubu Univ.) In order to achieve confinement of high tempera-ture plasma in toroidal magnetic devices, it is crucial to have good knowledge of magnetic surface. It is the central issues of controlled fusion research to un-derstand the long time behavior of plasma particles confined in the magnetic vessels!). Since the configu-ration of magnetic field lines is described in terms of the Hamiltonian canonical equations, many studies of the magnetic surface have been carried out by adopt-ing the methods of nonlinear dynamic theory 2) and 3). In particular, the Poincare's mapping approach is useful to deal with the conserved Hamiltonian system of a few degrees of freedom. The magnetic surface of toroidal configuration such as those of tokamaks, sterallators and other helical . devices has been analyzed by full use of the two di-mensional area preserving maps. Almost all of two dimensional area preserving maps are reduced to the standard map by local linear approximation, qn+l qn + Pn+l Pn+l Pn+F(qn)=Pn+Asin(27rqn) (1) When the nonlinear parameter A is sufficiently small, orbits described by (1) encircles the fixed point (qo = 0, p0 = 0) with closed curves representing regular mo-tion. Greene 4) has determined the critical value of A at which the last invariant surface called KAM sur-face is destroyed, and the system undergoes transition to the state of global chaos. The stochastic properties of the system in the global chaos is best characterized by the stochastic diffusion of the orbits in the phase space. N umeri-cal observation of the mean square fluctuation of the momentum variables can define the diffusion coeffi-cient. As for the standard map (1), the characteristic function gives rise to asymptotic expression of the diffusion coefficient 5), D = A 2 1- 2Jf(27r A)- Ji(27r A)+ 2Jj(27r A) (2) 4 (1 + J2(21rA)) 2 where J(x) is the n-th order Bessel function. Al-though 2) has been derived under the asymptotic condition of 27r A ~ 1, the numerically observed dif-fusion processes are consistent with this theoretical prediction, except in the region of 1 Ill< A< (1+ GY)' Ill (3) where the fundamental accelerator modes 1 . -1 ( l) qo = 27r sm A Po= 0 (4) are stable. Here, l is an integer measuring the size of acceleration in momentum direction. Surround-ing this fundamental accelerator mode, there occur the Poincare-Birkhoff bifurcation to generate higher order periodic accelerator modes. In numerical observation of the stochastic diffu-sion 6), we have found that even in small values of the nonlinear parameter A, there appear many sharp peaks of anomalous enhancement of diffusion, where the fundamental accelerator mode can not exist. These peaks of anomalous diffusion are at-tributed to multiple-periodic accelerator mode which can exist even if the nonlinear parameter A is smaller than 1. The purpose of the present paper is to de-velop quantitative analysis of the structure, and the stability of the period-3 and the period-5 accelera-tor modes, and to give quantitative account of the numerically observed diffusion coefficient. In the present studies of the multi-periodic acceler-ator mode of the standard map, we have determined quant.itatively the regions of the anomalous enhance-men of diffusion with the contribution of the · acceler-ator modes. It is very interesting to notice that the accelerator modes of large periodicity manifest their contribution at smaller nonlinear parameter A. In the present analysis, we have some indication of ob-serving the contribution of the period-7 accelerator mode at the value of A = 0.5. It is reserved for fu-ture studies to account quantitatively the stickiness around the regular orbits, and to explore the scaling law of the transport property in the system described by the standard map. References 1) Hamiltonian Dynamical Systems, compiled by R.S.MacKay and J .D.Meiss Adam Hilger, 1987 2) M.N .Rosenbluth, R.Z.Sagdeev, J .B.Taylor and G. M. Zaslafsky Nuclear Fusion 6 (1966) 297 3) N .N .Filonenko, R.Z.Sagdeev and G.M.Zasalavsky Nuclear Fusion7 (1967) 253 4) J .M.Greene J.Math.Phys 9 (1968) 760 5) J .D.Meis, J .R.Cary, C.Grebogi, J .D.Crawford and H.D.I.Abarbanel Physica 6D (1983) 6) Y.H.Ichikawa,T.Kamimura and T.Hatori Physica 29D (1987) 247 241 

Anomalous Diffusion and Multiple-periodic Accelerator Modes in the Standard Map

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Anomalous Diffusion and Multiple-periodic Accelerator Modes in the Standard Map

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