The fractal structure of magnetic island is studied qualitatively and quantitatively. The fractal dimension of magnetic island is obtained, and the formation of fractal structure of magnetic island is explained.

#### Keywords

- magnetic island
- fractal
- tokamak plasma

Fractal geometry was developed by Mandelbrot [1]. He provided a new perspective on nature. Fractal geometry exists not only in our imagination but also in the physical world. Star clusters in the universe, clouds in the sky, coastlines on earth and so on, all assume the shapes of fractal structures. In tokamak plasma, various MHD instabilities deteriorate plasma confinement and usually change the magnetic topology. In ideal MHD, plasma is perfect in conducting and magnetic field lines are frozen to plasma. In this case, reconnection of the magnetic field lines is forbidden. Taking into account plasma resistance, it can be stated that magnetic field lines are no longer attached to plasma. On a rational magnetic surface, magnetic field lines break and reconnect, forming magnetic islands. The formation of magnetic islands is generally associated with resistive instabilities and particularly tearing modes. The perturbed radial magnetic field is the key factor for the formation of magnetic islands. The ideal magnetic surface structure is nested and there is no radial magnetic field [2,3]. In fact, a radial magnetic field can be generated by plasma perturbation or an external magnetic field. As a result, the magnetic surface breaks into magnetic island chains and the toroidal symmetry is destroyed [4,5]. Soft X-ray is a powerful experimental tool for studying magnetic island structure [6,7,8,9].

In this paper, the fractal structure of magnetic island in tokamak plasma is studied. The fractal dimension of magnetic island is obtained. Then, an example of fractal structure of magnetic island is given, and finally, the formation of fractal structure of magnetic island is discussed.

A magnetic island can be thought of as a closed helical tube with its own magnetic axis. Helical flux is used to describe magnetic islands. Under the first-order approximation, a pendulum Hamilton equation can be obtained [10]. The projection of a magnetic island on the poloidal cross section can be easily expressed by using this equation.

The pendulum Hamiltonian equation can be written as [11]:

In a perturbed equilibrium, Ω_{1} is the label of the perturbed flux surface. The point at which the magnetic axis of the magnetic island intersects the poloidal cross section is called O-point. The boundary of the magnetic island is called the separatrix. Ω_{1} ranges from −1 at the O-point to +1 on the separatrix. The magnetic island can be described by a local magnetic surface coordinate system (Ω_{1}, _{1}, _{1}) similar to the flux coordinate describing the equilibrium field. The magnetic axis of the magnetic island is a helical curve. Its helicity is _{s}_{1} times around the O-point after circuiting _{1} times around the magnetic axis of the magnetic island. The magnetic islands formed on the equilibrium magnetic surface are called the first-order magnetic islands. The magnetic axis of the first-order magnetic island is a closed magnetic field line, the safety factor of which is equal to _{s}

As the order of magnetic island increases, the volume of magnetic island becomes smaller and smaller. On the tokamak poloidal cross section, the projection of magnetic islands with different orders presents a fractal structure. Perturbations are assumed to occur on rational magnetic surfaces with a safety factor equal to 2/1. Figure 1 shows the projection of the first-order magnetic island on a poloidal cross section. It consists of two parts, left and right. The basic pattern is similar to the figure. Its size decreases with increases in magnetic island order. It also has left and right parts. Figure 2 shows the second-order magnetic island, which consists of two basic patterns. Figure 3 shows the third-order magnetic island, which consists of four basic patterns. Figure 4 shows three magnetic islands of different orders: first, second and third. We can continue drawing higher order magnetic islands by inserting basic patterns in the left and right parts.

Several types of dimension are used to describe fractal structure. Among them, box dimension is suitable for calculating the magnetic island dimension. R is the radius of the minimum circle covering the basic pattern of the first-order magnetic island. Assuming that the basic pattern's size is r times smaller than that of the lower order magnetic island, and that r is a constant greater than 2, the box dimension of the magnetic island can be written as the following:
_{B}_{B}_{B}_{B}

The projections of magnetic islands of different orders can be obtained quantitatively on a given poloidal cross section. The first-order magnetic islands are described by magnetic surface coordinate system (Ω_{1}, _{1}, _{1}); so, the k-th order magnetic island's coordinate system is expressed by (Ω_{k}, _{k}_{k}_{k}, _{k}_{k}

However, this coordinate transformation is complicated since the parameters of perturbed plasma are needed. For simplification, the _{i}_{i}) does not change. The perturbed flux of the i-th order magnetic island _{i}_{is} − Ω_{iaxis}), where _{is} is the flux of rational surface and Ω_{iaxis} is the flux of the magnetic axis of the i-th order magnetic island. The map of (_{k}, ζ_{k}_{k−1}, _{k−1}) is supposed to be linear. Using Eq. (1) repeatedly under these assumptions, (Ω_{k}, θ_{k}, ζ_{k}) can be transformed to (Ω, _{0} = 1.7 m, the minor radius _{p}

In this case, flux perturbation that occurs on the rational surfaces with _{1} can be expressed by Eq. (1), and Ω_{2} can be written as

With fixed toroidal angle _{2}, _{2}, _{2}) is converted into the coordinate system (Ω_{1}, _{1}, _{1}) by using Eq. (4). Next, the coordinate system (Ω_{1}, _{1}, _{1}) is transformed into the coordinate system (_{2}, _{2}, _{2}) is transformed into the coordinate system (

In this paper, it is assumed that the radial magnetic field perturbation is in its simplest form, a single harmonic term. The perturbation forming the first-order magnetic island is written as
_{1}, _{1}, _{1}) can be written as F_{1} (Ω_{1}, _{1}, _{1}). Further, the function F_{1} can be decomposed into harmonic superposition. The harmonic term is cos(_{1}_{1} − _{1}_{1}), where _{1} and _{1} are integers. For the simplest case, the perturbation forming the second-order magnetic island is written as

It can be concluded that the first-order magnetic island has been observed experimentally. Further, the fractal structure of magnetic islands is expected to be seen in tokamak plasma.

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