Fractal structure of magnetic island in 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]: (1) Ω=qs′2qs(ψ−ψs)2/ψ˜−cosα \Omega = {{q_s^\prime} \over {2{q_s}}}{\left( {\psi - {\psi _s}} \right)^2}/\tilde \psi - \cos \alpha (2) α=mθ−nζ \alpha = m\theta - n\zeta where ψ is the unperturbed poloidal flux, ψ˜ \tilde \psi is the strength of flux perturbation and Ω is the helical flux. m and n are the poloidal and toroidal numbers of the resonant surface, θ and ζ are the poloidal and toroidal angles and q is the safety factor. The prime denotes the derivative with respect to ψ, and the subscripted S indicates that a quantity on a resonant rational surface is being evaluated. (ψ, θ, ζ) represent the magnetic flux coordinates of the equilibrium tokamak plasma. (R, Z, φ) represent the cylindrical coordinates of tokamak. These two coordinate systems can be transformed into each other.

In a perturbed equilibrium, Ω₁ 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. Ω₁ 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 (Ω₁, θ₁, ζ₁) similar to the flux coordinate describing the equilibrium field. The magnetic axis of the magnetic island is a helical curve. Its helicity is q_s with respect to the equilibrium magnetic axis. The magnetic island in the local coordinate system is helical. Magnetic field lines in the magnetic island form the nested magnetic surfaces and the structure is similar to the equilibrium magnetic surfaces. The helicity of the magnetic field lines in the magnetic island with respect to the magnetic axis of the island can be described by the equation q1=m1n1 {q_1} = {{{m_1}} \over {{n_1}}} . The magnetic field lines in the magnetic island rotate n₁ times around the O-point after circuiting m₁ 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 q_s with respect to the equilibrium magnetic axis. It is a straight line in the (θ, ζ) plane. The perturbation on the rational magnetic surface of the first-order magnetic island will lead to the formation of a magnetic island. We call these magnetic islands as the second-order magnetic islands. By analogy we can get the third-order magnetic island, the fourth-order magnetic island and so on. These magnetic islands have a structure similar to those of nested magnetic surfaces.

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.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

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: (3) DB=limk→∞ln2k−ln[R(1/r)k/2]=2ln2lnr {D_B} = \mathop {\lim }\limits_{k \to \infty } {{\ln {2^k}} \over { - \ln [R{{(1/r)}^{k/2}}]}} = {{2\ln 2} \over {\ln r}} where k is the order of magnetic island and D_B is less than 2. Extending to the general case, DB=2lnmlnr {D_B} = {{2\ln m} \over {\ln r}} , where r > m, D_B < 2 and m and n are poloidal number and toroidal number, respectively. It is independent of toroidal number n. The basic pattern consists of m parts. Fractal dimensions involve more complex structures. Given m, if r → ∞ then D_B → 0. Given r, D_B increases with the value of m. The magnetic island structure evolves from low dimensional to high dimensional. The magnetic island first occurs on the rational magnetic surface with poloidal number m = 1, then for the poloidal number m = 2, 3… and so on. It may be hypothesised that the more complex the structure of the magnetic island is, the more difficult it is to make excited and the more the energy needed. Therefore, the simpler the structure of the magnetic island is, the easier it is to make excited. One idea is that the more complex the physical system is, the more unstable it is. A system can be stabilised by simplifying the structure. So, the more complex the magnetic island is, the more unstable it is.

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 (Ω₁, θ₁, ζ₁); so, the k-th order magnetic island's coordinate system is expressed by (Ω_k, θ_k, ζ_k). By the transformation of (Ω_k, θ_k, ζ_k) → (ψ, θ, ζ), the projection of the k-th order magnetic island on the poloidal cross section (the toroidal angle ζ = const) can be obtained.

However, this coordinate transformation is complicated since the parameters of perturbed plasma are needed. For simplification, the q profile is assumed to be the same in magnetic islands of different orders. It means that the function q_i(Ω_i) does not change. The perturbed flux of the i-th order magnetic island ψ_i is assumed to be ɛ(Ω_is − Ω_iaxis), where ɛ is a small constant, Ω_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 (θ_k, ζ_k) → (θ_k−1, ζ_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, θ, ζ). By this method, the second-order magnetic island can be calculated. The plasma parameters are set with the plasma major radius R₀ = 1.7 m, the minor radius a = 0.4 m, the elongation k = 1.88, the triangularity δ = 0.75, the polodial beta β = 1.6, the toroidal magnetic field on magnetic axis B = 3.5 T and the plasma current I_p = 1.0 MA. Figure 5 shows the plasma boundary and the rational magnetic surface at q = 2. The first-order magnetic island occurs on the rational surface. The first-order magnetic island is shown in Figure 6. Figure 7 shows the rational magnetic surface with q = 2 in the first-order magnetic island. Figure 8 shows the rational surface of the first-order magnetic island in the coordinate system (ψ, α). The second-order magnetic island occurs on the rational surface. Figure 9 shows the second-order magnetic island. The horizontal lines in Figures 8 and 9 represent the rational magnetic surface with q = 2.

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

In this case, flux perturbation that occurs on the rational surfaces with q = 2. Ω₁ can be expressed by Eq. (1), and Ω₂ can be written as (4) Ω2=q1s′2q1s(Ω1−Ω1s)2/ψ˜1−cosα1 {\Omega _2} = {{q_{1s}^\prime} \over {2{q_{1s}}}}{\left( {{\Omega _1} - {\Omega _{1s}}} \right)^2}/{\tilde \psi _1} - \cos {\alpha _1} (5) α1=mθ1−nζ1 {\alpha _1} = m{\theta _1} - n{\zeta _1}

With fixed toroidal angle ζ, the projection of the first-order magnetic island on the poloidal cross section can be obtained by Eq. (1). The first-order magnetic island can be obtained by changing ζ continuously. The coordinate system (Ω₂, θ₂, ζ₂) is converted into the coordinate system (Ω₁, θ₁, ζ₁) by using Eq. (4). Next, the coordinate system (Ω₁, θ₁, ζ₁) is transformed into the coordinate system (ψ, θ, ζ) by Eq. (1). Then, the coordinate system (Ω₂, θ₂, ζ₂) is transformed into the coordinate system (ψ, θ, ζ); thus, the second-order magnetic island is obtained.

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 ψ˜cos(mθ−nς) \widetilde \psi \cos (m\theta - n\varsigma ) , whereas the perturbation in the surface coordinate system (Ω₁, θ₁, ζ₁) can be written as F₁ (Ω₁, θ₁, ζ₁). Further, the function F₁ can be decomposed into harmonic superposition. The harmonic term is cos(m₁θ₁ − n₁ς₁), where m₁ and n₁ are integers. For the simplest case, the perturbation forming the second-order magnetic island is written as ψ1˜cos(mθ1−nς1) \widetilde {{\psi _1}}\cos (m{\theta _1} - n{\varsigma _1}) . The perturbation resonates with the rational magnetic surface (q = m/n) in the first-order magnetic island. By analogy, the perturbation forming the k-th order magnetic island can be written as ψ˜k−1cos(mθk−1−nςk−1) {\widetilde \psi _{k - 1}}\cos (m{\theta _{k - 1}} - n{\varsigma _{k - 1}}) . Fourier transformation can be performed repeatedly on an external perturbation under various magnetic surface coordinate systems. It contains harmonic terms forming magnetic islands of various orders. This can be considered as the reason for the formation of fractal structure in a magnetic island.

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.