Bifurcations and onset of chaos on the ergodic magnetic limiter mapping

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Abstract

We propose a Hamiltonian formulation to study the magnetic field line structure in a tokamak with ergodic magnetic limiter. An analytical stroboscopic mapping, derived from this formulation, is used to investigate the onset of global field line chaos at the plasma edge and the Hamiltonian bifurcations of magnetic axes.

Introduction

Among the large number of fusion-oriented plasma devices, the tokamak seems to be one of the most promising candidates for a future thermonuclear power plant. Many factors conspire against the obtention of long lasting plasma confinement in tokamaks, however [1], [2]. One of them is the ubiquitous presence of plasma instabilities, that may destroy plasma confinement due to a variety of causes [3], [4], [5]. Another major problem in tokamak physics is the control of plasma contamination by impurities released from the inner wall by surface processes [6].

Tokamaks are toroidal pinches in which plasmas are generated by ohmic heating of a filling gas and confined by externally applied magnetic fields: a toroidal field produced by external coils, and a poloidal field generated by the plasma column itself [1], [2]. This combination will be called the equilibrium field. The corresponding magnetic field lines have helical shape so that, at least in a first approximation, particles are confined by them. We may think of these field lines as lying on magnetic surfaces with topology of nested tori.

From the point of view of a magnetohydrodynamical (MHD) theory, these surfaces are also isobaric ones, on which the plasma expansion caused by a pressure gradient is counterbalanced by the Lorentz force resulting from the interaction between the equilibrium magnetic field and the plasma electric current density. The existence of closed toroidal magnetic surfaces is a necessary, albeit not sufficient, condition for plasma confinement in tokamaks [3], [7].

In order to control the abovementioned plasma–wall interactions, that may lead to loss of confinement, it has been proposed to create a cold boundary layer of chaotic field lines in the periphery of the tokamak vessel [8], [9], [10]. This region comprises the outer plasma column and the vacuum region that surrounds it from the inner tokamak wall. This can be accomplished by destroying some, but not all, magnetic surfaces located in this region. A way to do this, without spoiling the plasma column itself, is to generate external magnetic fields that interact with the equilibrium field and cause a selective destruction of magnetic surfaces, which is the basic principle of the ergodic magnetic limiter concept.

Chaotic magnetic field lines are taken here from a magneto-static perspective, i.e., there is sensibility to initial conditions in the sense that two field line points, very close from each other, evolve through a large number of revolutions around the tokamak so that the distance between the resulting field lines deviates with a positive exponential rate [3], [7]. By identifying field line equations with canonical equations, we can build a Hamiltonian theory for the magnetic field line structure in symmetrical systems [11], [12]. The equilibrium field is a one degree-of-freedom, hence integrable, Hamiltonian system. In this framework we identify magnetic surfaces with KAM tori and chaos appears from the mechanism widely understood for this type of dynamical systems [13].

The application of magnetostatic perturbations due to currents external to the tokamak vessel, causes the destruction of some, but not all, magnetic surfaces [8], [9], [10]. In the Hamiltonian theory of near-integrable systems, which is applicable if the perturbing currents are not too large, we know that some surfaces are destroyed, producing island-shaped structures that resemble the orbit structure of a nonlinear pendulum [14], [15], [16]. In the plasma physics literature these structures are named magnetic islands, and they have a complicated structure near their hyperbolic (unstable) fixed points, in the sense that the invariant manifolds that stem from these points intercept each other in a rather complicated way, forming the so-called homoclinic figure. The field line dynamics on these homoclinic points is chaotic [14].

According to the KAM theorem however, there remains a large number of surviving, albeit distorted, magnetic surfaces. They act as barriers, preventing large scale field line diffusion [12]. On the other hand, the use of an ergodic magnetic limiter requires a wide region of chaotic field lines in the tokamak periphery. The transition to such a situation occurs in an abrupt way, since one requires that no undestructed magnetic surfaces should exist between neighbor magnetic islands. If this is true, the locally chaotic regions related to each islands' separatrices may coalesce and yield large scale chaotic motion [15].

As we further increase the perturbation strength, other phenomena take place. Even after a widespread chaotic region is created, the islands' centers are still stable fixed points and surrounded by an increasingly small number of KAM tori. At some another critical perturbation intensity, however, even those centers can lose stability and become unstable. Moreover, after we reach this critical parameter value, two new stable fixed points appear. This configures a bifurcation, that has important consequences, since it implies both in the disappearance of a magnetic axis (degenerate surface with zero radius) as well as in the formation of two new magnetic axes, altering in a dramatic way the topology of magnetic confinement.

Besides the ergodic limiter, the ergodic divertor has received great attention in modern tokamak research. In the ergodic divertor, the separatrix between the last confining magnetic surface and the open surfaces is replaced by a layer of chaotic field lines that divert plasma particles to divertor plates, where they can be recycled and pumped to reduce impurity levels in the plasma [17], [18]. Ergodic divertors have received treatments based on simple analytical maps for field lines [19] and more recently there were used canonical mappings from a Hamiltonian treatment [20], [21]. It turns out that some simple twist maps like the Chirikov–Taylor standard map may not be appropriate to model field line behavior [22], [23].

The ergodic limiter has been studied by means of a simplified mapping [15] that has been later improved with toroidal corrections and parameters describing the equilibrium and perturbed magnetic fields [24], [25]. The ergodic limiter is a symmetry breaking form of perturbation, in the sense that it spoils axisymmetry of the equilibrium tokamak field. The influence of the type of symmetry-breaking perturbation was studied from the point of view of analytical and numerical field line maps [26]. A Hamiltonian treatment of ergodic limiters has been proposed in a rectangular geometry [27]. We have recently proposed a Hamiltonian map in a realistic toroidal geometry and using magnetic fields consistent with a general MHD equilibrium theory [28]. This map may be derived from the canonical equations with a field line Hamiltonian where the ergodic limiter action is supposed to be a sequence of delta-functions in the toroidal direction.

The purpose of this paper is twofold: we will study the onset of magnetic field line chaos produced by an ergodic limiter, by analyzing the interaction between adjacent magnetic islands. Secondly, we investigate bifurcations that occur before the mainly chaotic region is generated in the tokamak periphery. We will use an analytically obtained field line mapping [28]. The advantage of this procedure, in comparison with a direct numerical integration of field line equations, is the higher computation speed of map iterations compared with usual integration schemes, like predictor–corrector methods for differential equation. This difference may be crucial if long-term behavior of field lines is being considered, as in numerical studies of anomalous diffusion [29]. However, the use of oversimplified physical models for both the equilibrium and the ergodic limiter magnetic field may lead to misleading results, so that we use in this work an appropriate geometry to fully incorporate toroidicity effects, and a MHD equilibrium model from which the equilibrium field is obtained.

This paper is organized as follows: in Section 2 we present the model fields for the equilibrium and the ergodic limiter perturbation, an analytically obtained field line mapping, and an explicit form of a field line Hamiltonian. Section 3 is devoted to an application of standard perturbation techniques to describe the magnetic island structure, which gives the location and width of each island of interest. Section 4 focuses on the onset of chaos, describing the application of a modified Chirikov criterion, and discusses the conditions under which we get chaotic behavior in the tokamak periphery. In Section 5 we study bifurcations that occur after global field line chaos. Our conclusions are left to the last section.

Section snippets

Model fields

In Fig. 1 we depict the basic tokamak geometry to be used throughout this paper. The tokamak vessel has a minor radius b and a major radius R0, so that an aspect ratio A=R0/b can be defined. Polar coordinates (r,θ) may be defined from the minor axis, with Φ as a toroidal angle. This choice of coordinates may give inaccurate results, since the resulting coordinate surfaces may not match, even in an approximate way, actual equilibrium magnetic surfaces. This has led us to the use of a

Derivation of the symplectic mapping

The magnetic field line equations B×dℓ=0, corresponding to the model fields described in the previous section, are written, using , , , and (25), asdrtdϕt=−1rtBT1−2rtR0cosθtθtAL3(rttt),dθtdϕt=1rtBT1−2rtR0cosθtrtΨp0(rt)+AL3(rttt),where BT≡−μ0I/R0 is the toroidal magnetic field at the magnetic axis.

Since the equilibrium field is axisymmetric, we may set the angle ϕt=t as a time-like variable, and put field line , in a Hamiltonian formdJdt=−Hϑ,dϑdt=HJ,where (J,ϑ) are the

Pendular islands

In this section we will study the effect of resonances caused by the magnetic field produced by an ergodic limiter. In the phase space the exact resonance will be the center of a magnetic island with pendular shape. In order to use standard results of perturbation theory we have to start from an expansion of the perturbing Hamiltonian in modes related to the angles ϑ and t. The Hamiltonian for the tokamak with ergodic limiters, Eq. (42), may be rewritten by Fourier-expanding the periodic delta

Onset of chaos

The fate of the equilibrium magnetic surfaces, after a perturbation breaks the system integrability, is basically determined by their safety factors. KAM theory predicts that, for those irrational surfaces with safety factors sufficiently far from a rational m:n, the topology is preserved, and the surfaces are only slightly deformed from the unperturbed tori (KAM surfaces) [14]. On a rational surface and in a neighborhood about it the KAM theorem fails, and we have to resort to the

Bifurcation phenomena

As the current limiter builds up, new phenomena are expected to appear, besides the enlargement of the locally chaotic regions in the neighborhood of the islands' separatrices [14], [36]. Let us fix our attention on the center of a primary island chain, where there exists a stable fixed point of the field line mapping. For example, the centers in a 4:1 chain are periodic points of a stable period-4 orbit. As the limiter current is further increased, it may happen that this periodic orbit looses

Conclusions

In this paper we derived a symplectic mapping to follow magnetic field lines in a tokamak with an ergodic magnetic limiter. The advantages of our procedure are: (a) we adopted a coordinate system which naturally embodies the plasma toroidal configurations [1], [2], [12]; (b) model fields were derived from sound physical assumptions: the equilibrium fields were not introduced in an ad hoc fashion [25], [26], but came from the solution of an MHD equilibrium set of equations; (c) the limiter field

Acknowledgements

This work was made possible with partial financial support of the following agencies: CNPq (Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior), FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) , and Fundação Araucária (State of Paraná, Brazil).

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