Homoclinic tangle of the ideal separatrix in the DIII-D tokamak from (30, 10) + (40, 10) perturbation

Large modern tokamaks such as ITER (meaning “The Way” in Latin),¹ Joint European Torus (JET),² and DIII-D³ have divertors.⁴ In divertor tokamaks, the separatrix defines the plasma edge and separates the closed magnetic surfaces from the open surfaces. The separatrix is designed to ensure that the plasma exhaust from the tokamak goes into the divertor chamber. An ideal tokamak would be axisymmetric and the magnetic field lines just outside the separatrix will enter the divertor chamber through a circular annulus. Magnetic asymmetries always exist, and even small asymmetries turn the separatrix into a complicated structure and radically change the behavior of magnetic field lines near the separatrix. Both the confinement of the plasma and the feasibility of divertors are sensitive to the behavior of the magnetic field lines near the separatrix in the presence of non-axisymmetric magnetic perturbations. In particular, non-axisymmetric perturbations cause the magnetic field lines to form a homoclinic tangle. Non-ideal effects of plasma diffusion fill these lobes and spread the heat load. This is generally a good effect. For sufficiently rapid diffusion, the effects of tangle on plasma footprint on collector plate are washed out.⁵ 

Poincaré discovered homoclinic tangle in his study of three body problem.^6,7 At a hyperbolic singular point, four manifolds join together; the two incoming stable separatrix manifolds, and the two outgoing unstable separatrix manifolds. The stable manifold leading to and the unstable manifold emanating from the hyperbolic point have extremely irregular behavior. This is because the stable and unstable manifolds cannot intersect themselves but the unstable manifold can intersect the stable manifold at points that are called homoclinic points. Between each homoclinic point and the hyperbolic fixed point, there are an infinite number of homoclinic points. Thus, the stable and the unstable manifold form an extremely complex network called the homoclinic tangles.⁸ For manifolds connected to neighboring hyperbolic points, these structures are called heteroclinic tangles. Without perturbation, the axisymmetric separatrix manifold is degenerate, and the stable and unstable manifolds coincide.

Trajectories of magnetic field lines in space are a 1½ degree of freedom Hamiltonian system.^9–11 The manifold on which magnetic field lines evolve is a topological space with a symplectic structure. A symplectic structure is the essence of physics of any Hamiltonian system. The new mathematical model for mechanics is a symplectic manifold with a Hamiltonian vector field that preserves the symplectic structure.¹² Arnold has stated that Hamiltonian mechanics is geometry in phase space and phase space has the structure of symplectic manifold.¹³ The symplectic property requires that the sum of the signed areas formed by the projections of volume on conjugate planes in phase space must be preserved at all times.^14,15 This condition is called the symplectic invariance of Hamiltonian systems.

The most efficient way to calculate homoclinic tangle of separatrix is to use a symplectic map.¹⁵ This is done, for example, in Refs. 16 and 17, in cases where the unperturbed tokamak plasma is near circular. However, in large modern tokamaks such as ITER, DIII-D, JET, and others the magnetic surfaces in the edge and the separatrix have a shape that is fundamentally different from that of near-circular plasma. Recently, we have developed a new method⁵ to study this fundamentally more complicated case. In this method, we calculate the successive homoclinic tangles formed as the separatrix manifold moves forward and backward in canonical time and the two intersect after each toroidal turn. This is fundamentally different approaches than calculating the homoclinic tangle in asymptotic limits as φ → ±∞ or by integrating the trajectories for very long times used by several researchers, as, for example, Ref. 18 or 19.

Evans et al. have calculated the homoclinic tangles in the DIII-D^1–4 from the time-asymptotic limits of field line trajectories, looked for experimental signatures of the tangles in the particle and heat deposition patterns on the plasma facing components, and studied a homoclinic tangle model for the evolution of ELMs.^18,20–22 Kroetz et al. have investigated tangles in the tokamaks with reversed magnetic shear and ergodic limiter to calculate the escape pattern and footprints of field lines.²³ Portela et al. have calculated homoclinic tangles using simple maps for integration of field line trajectories in divertor tokamaks.²⁴ Wingen et al. have calculated homoclinic tangle in circular plasmas.¹⁶ Silva et al. use the asymptotic limits to calculate homoclinic tangle.¹⁷ In these works, the approach is to calculate the homoclinic tangle in asymptotic limits as φ → ±∞ or by integrating the trajectories for very long times. We do not use this approach. We calculate the successive homoclinic tangles formed as the separatrix manifold moves forward and backward in canonical time and the two intersect after each toroidal turn.

Our new method calculates the homoclinic tangles of the perturbed separatrix as it moves forward and backward in toroidal angle, and every time the two intersect in a poloidal plane for the unique magnetic geometry of a specific device subject to perturbation and constrained by two topological invariants. Our method is based on the preservation of the two invariants for a Hamiltonian system—the symplectic invariant and the topological neighborhood.⁵ These two invariants are universal for Hamiltonian systems. Such mathematical constraints of high universality are of great importance in physics.²⁵ These two invariants impose severe constraints on the path of motion of system in phase space. As the separatrix of the Hamiltonian system moves in phase space, it changes in time and space and forms complicated structure—the homoclinic tangle—to preserve the two invariants. The homoclinic tangle generates lobes. The formation of lobes when the separatrix is perturbed is a venerable subject. We see a fascinating and complex phenomenon of dynamical formation of the tangle and its lobes.⁵ We combine our method with our new recent symplectic approach.^26,27 At the heart of the new approach is a set of new canonical coordinates called the natural canonical coordinates (NCC).^26–28 

Here, our motivation is to calculate the tangle of ideal separatrix. So we have chosen the unperturbed axisymmetric separatrix as our initial conditions. The tangle calculated here is not for the perturbed separatrix but for the ideal separatrix. The tangle of ideal separatrix here approximates the tangle of the perturbed separatrix. Our new method can readily calculate the tangle of perturbed separatrix as well. The actual perturbed tangle is not calculated directly because the proper initial conditions are unknown.

There are three sets of canonical coordinates for magnetic field lines: magnetic, physical, and natural.^26–28 Magnetic coordinates are averages over surfaces and have a singularity on the separatrix. For this reason, we cannot use the magnetic coordinates to integrate across the separatrix. The magnetic coordinates generally cannot be mapped to physical space. Maps in physical coordinates can be integrated across the separatrix. Also, the physical coordinates can be mapped to physical space. But the expressions for the Fourier components of perturbations in toroidal geometry become too complicated and cumbersome in the physical coordinates. We have developed a new set of canonical coordinates called the natural canonical coordinates.^26–28 Natural coordinates can be mapped to the standard coordinates such as the rectangular or cylindrical coordinates and give the position of field line in physical space. All single-null divertor tokamaks have the same magnetic topology but can have vastly different magnetic geometries. For example, the DIII-D and the simple map^29–31 have the same topology—a separatrix with a single-null—but the shape of the separatrix and the magnetic surfaces in the edge of the DIII-D and the simple map are fundamentally different.⁵ The DIII-D has a very complicated shape, while the simple map has the simplest possible shape for a single-null divertor tokamak. In these new coordinates, called the natural canonical coordinates, we can analytically express shapes of the unperturbed axisymmetric separatrix and the magnetic surfaces in the edge very accurately and very realistically using the Grad-Shafranov equilibrium solver data.^26,27 So in the natural coordinates, we can calculate the homoclinic tangle and the lobes for the unique geometry of a specific device such as the DIII-D, ITER, JET, etc.

In our new method, we combine the preservation of two invariants and the map in the natural coordinates. This combination has shown to be valuable and effective in studying the tangles and the lobes and their associated physical parameters in magnetic confinement devices. Our new method can be used to study the homoclinic and heteroclinic tangles and lobes in tokamaks with the same topology and vastly different shapes or geometries, such as the simple map, the DIII-D, ITER, JET, etc. Primary separatrix exists even when there is no perturbation. Internal separatrices arise because of magnetic perturbation. Our method can calculate homoclinic tangles of internal separatrices of very complicated shapes. Single-null divertors, double-null divertors, and snow-flake divertors have distinct topologies. For example, the primary separatrix in single-null divertors have a single hyperbolic point and perturbation produces homoclinic tangle; but double-null divertors have two hyperbolic points, so they cannot produce homoclinic tangle; they produce heteroclinic tangle. Single-null divertors with open lines going to infinity is topologically distinct from single-null divertors with shape of the number eight, where open lines close on themselves. Our method can calculate, compare, and contrast the homoclinic and heteroclinic tangles for different magnetic topologies. Our method can be extended to other Hamiltonian, near-Hamiltonian, and higher degrees of freedom Hamiltonian systems²⁹ with a single separatrix or with multiple separatrices.

We have applied this method to the DIII-D tokamak and the simple map for different types of magnetic perturbations and published the preliminary results.⁵ Here, we use our method to study the basic features of homoclinic tangle of the separatrix in the DIII-D tokamak in the presence of two Fourier components to represent peeling-ballooning modes of equal amplitude and no radial dependence.

This paper is organized as follows: In Sec. II, we describe Poincaré's discovery of homoclinic tangles in 1892, and give the reasons why and how homoclinic tangle is formed. We look at the formation of the homoclinic tangle from the topological as well as divertor tokamak angles. In Sec. III, we give the analytic expression for the equilibrium poloidal flux as a function of the natural canonical coordinates in the DIII-D, and show that we can very accurately and realistically represent the very complicated axisymmetric flux surfaces of the DIII-D in the natural coordinates. In Sec. IV, we give the DIII-D symplectic map equations in the natural coordinates, and describe the two Fourier component magnetic perturbation representing the peeling-ballooning modes. In Sec. V, we show in detail the homoclinic tangle of the separatrix in the DIII-D from the perturbation. In Sec. VI, we give an analysis of the data on the homoclinic tangle in the DIII-D. Analysis includes the lengths of field lines, differential poloidal rotation, normal displacement, diffusion, and the flux loss on the separatrix manifold. In Sec. VII, we summaries the results, give our conclusions, and discuss the implications of the perturbed separatrix for the edge physics in divertor tokamaks.

Poincaré discovered the homoclinic tangle in his study of the three body problem. He studied the three body problem in very simple terms. He set up the problem as two gravitating bodies perturbed by a third body that has so small a mass that the perturbation from it has negligible effect on the motion of the original two bodies. His approach was geometrical in nature. He visualized the behavior of system around the homoclinic fixed point, which is a saddle point where stable and unstable trajectories intersect. He discovered that the trajectories cross an infinite number of times and each of the two curves can never cross itself but it must fold back on itself an infinite number of times. Poincaré was struck by the extreme complexity of motion in phase space.^6,7

At a hyperbolic singular point, four manifolds join together; the two incoming stable separatrix manifolds, and the two outgoing unstable separatrix manifolds. For a point x on the stable manifold, the repeated forward transformation in canonical time Tⁿx brings x closer to the hyperbolic point as n approaches infinity; and for a point x on the unstable manifold, the repeated backward transformation in canonical time T^-nx brings x closer to the hyperbolic point as n approaches infinity. On the separatrix manifold, the period is infinite. So the motion of point towards the singular point becomes increasingly slow as the hyperbolic point is approached. The stable manifold $MS$ leading to and the unstable manifold $MU$ emanating from the hyperbolic point have extremely irregular behavior. This is because these two manifolds cannot intersect themselves but the unstable manifold $MU$ can intersect the stable manifold $MS$ at points that are called homoclinic points. The transformation T is continuous and the homoclinic point is not a fixed point, the repeated transformations Tⁿ produce new homoclinic points. An infinite number of transformations T must be applied to approach the hyperbolic point along the stable manifold $MS$⁠. Therefore, between each homoclinic point and the hyperbolic fixed point, there are an infinite number of homoclinic points. Thus, the stable manifold $MS$ and the unstable manifold $MU$ form an extremely complex network called the homoclinic tangles.⁸ For manifolds connected to neighboring hyperbolic points, these structures are called heteroclinic tangles. In equilibrium, the axisymmetric separatrix manifold $S$ is degenerate, and the stable and the unstable manifolds coincide, $S=MS∪MU$⁠. Non-axisymmetric perturbation breaks the symmetry and removes the degeneracy, forming the homoclinic and heteroclinic tangles.

In divertor tokamaks, we can expect lobes whenever the separatrix is perturbed to form homoclinic tangle by non-axisymmetric magnetic fields. Another way we can best understand this is to use the mapping of field lines from one toroidal transit to the next in tokamak. During each toroidal transit of the tokamak, a magnetic field line goes from a point x to a point x', which actually means from R to R' in major radius and form Z to Z' in vertical position. We can write this change using a mapping function x' = T(x). After two circuits of the torus, a field line goes from a point x to x', where x' = T²(x) ≡ T(T(x)). After n transits in tokamak, we can write the map as x' = Tⁿ(x). The X-point on separatrix is a fixed point x_f of the map T(x). This means x_f = T(x_f). The separatrix manifold in an axisymmetric tokamak as a whole is a set of points x_s in phase space. What forms this set of points x_s is that when we iterate it forward using the map T it approaches the X-point arbitrarily closely, lim_n→∞ T(x_s) → x_f. Again the same separatrix manifold is formed by the set of points x_u such that when iterated backwards the points approach the X-point arbitrarily closely, lim_n→−∞ T(x_u) → x_f. When we perturb the axisymmetric equilibrium magnetic field by a non-axisymmetric magnetic perturbation in tokamak, the mapping function x' = T(x) retains a fixed point but the curve formed by field lines that under a forward mapping approach the X-point and the curve formed by field lines that under a backward mapping approach the X-point are not the same curve. The first curve is the stable manifold $MS$ and the second curve is the unstable manifold $MU$ of the map. For small perturbations, the curves that correspond to the stable and the unstable manifolds cross. The point of crossing, or equivalently a point that lies on both manifolds is a homoclinic point x_h. The definition of the curves that form the stable and unstable manifolds implies that the mapping of any homoclinic point is to another homoclinic point, so Tⁿ(x_h) is a homoclinic point for arbitrary positive or negative n. The stable and unstable manifolds that pass through two successive homoclinic points bound a region of space, and that boundary can be mapped forward using T(x). The continuity of T(x) implies that the points enclosed by an arbitrary closed curve cannot escape being enclosed by the mapping of the curve. If the map were area preserving, which is equivalent to ∇·T = 0, then the area enclosed by a closed boundary would be unchanged by the mapping. The magnetic field is divergence free. So we can use magnetic flux coordinates and make ∇·T = 0. Using ordinary spatial coordinates, the area enclosed by the boundary times the average toroidal magnetic field is a constant. The term tangle comes from considering what happens as the closed curve defined by the stable and unstable manifolds are mapped using Tⁿ(x). The homoclinic points associated with successive iterations of the map come ever closer to the X-point but never reach the X-point. The implication is that the successive homoclinic points become ever closer together. Nevertheless, the area enclosed by the stable and manifolds between the two successive homoclinic points must remain essentially constant. The implication is that as n →∞, the curve that represents the unstable manifold must make wilder and wilder excursions. Similarly as n →−∞, the curve that represents the stable manifold must make wild excursions.

The magnetic field in toroidal plasmas can be expressed as $B=∇ψ×∇θ+∇φ×∇χ(θ,ψ,φ)$⁠,^9–11 where χ is a poloidal magnetic flux inside a magnetic surface and it is the Hamiltonian for the trajectories of field lines, θ is a poloidal angle and it is the generalized coordinate, ψ is a toroidal magnetic flux inside a surface and it is the generalized momentum, and φ is a toroidal angle and it is the generalized time. There are three sets of canonical coordinates for toroidal plasmas—the magnetic, natural, and physical. We use the natural canonical coordinates (NCC) (ψ,θ,φ). Here, ψ = B₀r²/2, θ = tan⁻¹(y/x), θ is the poloidal angle, φ is the toroidal angle, x = R-R₀, y = Z-Z₀, r = √[x²+ y²] is the radial distance from the magnetic axis, B₀ is the magnetic field on magnetic axis, and (R₀, Z₀) is the position of magnetic axis.^26,27 The natural coordinates can be readily mapped to the standard rectangular (X, Y, Z) or cylindrical (R, Z, φ) coordinates.

The total Hamiltonian in natural coordinates is $χ(ψ,θ,φ)=χ0(ψ,θ)+χ1(ψ,θ,φ)$⁠, where χ₀(ψ,θ) is the equilibrium poloidal flux and $χ1(ψ,θ,φ)$ is the magnetic perturbation. χ₀(ψ,θ) is called equilibrium generating function (EGF) for the symplectic map. The EGF can be very simple, as for the simple map,^29–31 or it can be a very complicated function, as for the DIII-D, depending on the shape (or the geometry) of the equilibrium magnetic surfaces in a specific device. The EGF for a specific device can be calculated from the equilibrium Grad-Shafranov data for the device.^26,27 The EGF so calculated can very accurately represent the magnetic geometry unique to the device. The EGF for the DIII-D in NCC is calculated from the Grad-Shafranov equilibrium data for the shot 115467 at 3000 ms. The EGF for the DIII-D is a very complicated function. It is a bivariate polynomial in u and v, given by χ₀(ψ,θ) = $∑i=15aiui+∑j=16bjvj+∑i=1,j=1i=5,j=6cijuivj$⁠, where u = √(ψ)cos(θ) and v = √(ψ)sin(θ).^26,27 It has a total of forty one terms. It very accurately represents the complicated shapes of the unperturbed magnetic surfaces in the DIII-D in the edge for the shot 11 5467 at 3000 ms.^26,27 See Figs. 1 and 2 in Ref. 26 and Fig. 1 in Ref. 27. In Fig. 1, we show the Grad-Shafranov equilibrium data for the DIII-D separatrix and the unperturbed separatrix calculated from the DIII-D EGF. In comparison, the EGF in NCC for the simple map is χ₀(ψ,θ) = $u2+v2+b̃3v3$⁠, where $b̃3$ = a constant. The simple map is the simplest map that has the magnetic topology of a single-null divertor tokamak and the magnetic surfaces of the simple map have the simplest possible shape. The EGF for the simple map has only three terms compared to forty one terms for the DIII-D. The EGF for simple map has no cross terms in u and v, while the EGF for the DIII-D has thirty cross terms. In Fig. 1, we show the unperturbed separatrices for the DIII-D and the simple map for comparison.

For the DIII-D shot 115467 at 3000 ms, B₀ = 1.589 T, the O-point is at (R₀,Z₀) = (1.758 m, −0.023 m), poloidal flux inside the separatrix is χ_SEP = 0.283 Weber, the X-point is at θ_X = 4.2832 radians, the inboard plate is at R_plate = R₀–0.65 m, and the outboard plate is at Z_plate = Z₀–1.367 m. The unperturbed separatrix manifold is made of N points or field lines with θ_i = 2πi/N, i = 1,2,…,N. N is large, N = 360 K, so the lines on the manifold are distributed uniformly in the poloidal angle, and ψ_i is the solution of χ₀(ψ_i,θ_i) = χ_SEP. This manifold is compact and closed.

We advance the DIII-D equilibrium separatrix manifold with the DIII-D map in NCC both forward and backward in the toroidal angle φ. When the forward and backward advanced manifolds intersect in φ = 0 plane, the preservation of the symplectic invariant and the topological neighborhoods generate the successive homoclinic tangles. The DIII-D map equations in NCC consist of two equations.^26,27 The equations are

where k is the map parameter, and it is the step size of the symplectic integration. We choose k = 360/2π. $χ(ψ,θ,φ)$ is the total Hamiltonian for the field lines, $χ(ψ,θ,φ)=χ0(ψ,θ)+χ1(ψ,θ,φ)$⁠. $χ0(ψ,θ)$ is the unperturbed Hamiltonian; it is the EGF for the equilibrium separatrix manifold. $χ1(ψ,θ,φ)$ is the perturbation; it is generally given by $χ1(ψ,θ,φ)=∑(m,n)fmn(ψ) cos(mθ−nφ+δmn)$⁠. m and n are the poloidal and toroidal mode numbers of the Fourier components of the perturbation.

Each of the 360 K points on the unperturbed separatrix manifold in the DIII-D is taken as the initial condition for the map. For the forward map, the first equation is solved for $ψn+1$ and then the second equation is solved for $θn+1 θn+1$⁠. For the backward map, the second equation is solved for $θn$ and then the first equation is solved for $ψn$⁠. Thus, the entire unperturbed manifold is advanced forward and backward in the toroidal angle. When the two advanced manifolds, the forward and backward, meet and intersect in the φ = 0 plane after every toroidal circuit, we see the successive homoclinic tangles.

The model of the perturbation is simplified in two ways. First, only a cosine series is used in a Fourier decomposition of the perturbation in the poloidal and toroidal angles; and second, the radial dependence of the perturbation is ignored. For perturbation, we choose the peeling-ballooning modes represented by Fourier components (m, n) = (30, 10) + (40, 10) with amplitude δ = 10⁻⁴ and 10⁻³. The perturbation is $χ1=δ[cos (30θ−10φ)+cos (40θ−10φ)]$⁠. The peeling-ballooning modes represent the type I Edge Localized Modes (ELMs).²⁶ Field lines that reach (R-R₀) < −0.8 m or (Z-Z₀) < −1.5 m are abandoned. We consider the surfaces (R−R₀) = −0.8 m as the inboard collector plate and (Z-Z₀) = −1.5 m as the outboard collector plate of the DIII-D. So when a field line reaches (R-R₀) < −0.8 m or (Z-Z₀) < −1.5 m, it has struck the plate and it is considered as lost. Then, the integration of that line is terminated.

In Fig. 2, we show the homoclinic tangles of the separatrix after a single toroidal turn for δ = 10⁻³ and 10⁻⁴ for comparison. In Fig. 3, we show the homoclinic tangle after 1, 5, and 10 turns and their enlarged views near the X-point when δ = 10⁻³. Homoclinic tangle of the unperturbed separatrix when δ = 10⁻⁴ is shown in Ref. 5. In Fig. 4, we show the normal displacement Δr from the equilibrium separatrix versus the poloidal angle θ after a single turn for the forward and backward lines.

From these detailed figures, we see that the homoclinic tangle is very complicated and becomes extremely complicated as more turns are taken, especially near the X-point. Near the X-point, the lobes are very pronounced. Lobes on the inboard side are larger than on the outboard side. It is very difficult to count the number of the lobes. Fig. 4 gives us an idea of the structure of the lobes. In the first turn for δ = 10⁻⁴, no line is terminated. So it is possible to picture both forward and backward manifolds (Figure 4); but it is very difficult to do so the same for the second turn onwards. For larger amplitude δ = 10⁻³, the extremely complicated tangle becomes even more complicated, and the large number of lobes becomes even larger. Heteroclinic tangle for larger surfaces is more pronounced than that for smaller surfaces, and more pronounced on inboard side than on the outboard side.⁵ 

We now analyze the data on the homoclinic tangle of the ideal separatrix manifold in the DIII-D.

After each iteration of the DIII-D map, we map the natural canonical coordinates to the rectangular coordinates, $(ψ,θ,φ)→(X,Y,Z)$⁠. We use the rectangular coordinates to calculate the distance between the positions of a line at the end of two successive iterations of the map. We approximate the length traversed by the line between the two successive iterations by this distance. Sum of these line segments gives us an approximate measure of the length l of a field line.

In Figure 5(a), we show the lengths l of the 360 K field lines as they move a single toroidal circuit forward and backward as functions of their starting poloidal angle position θ₀ on the equilibrium separatrix. Above and slightly below the midplane (6.24 < θ₀ and 0 ≤ θ₀ < 3.47), the backward moving lines move faster than the forward moving lines, and for the most part below the midplane (3.47 < θ₀ < 6.24), forward moving lines move faster than the backward moving lines. The average length of forward moving lines in a single toroidal turn is slightly larger than for the backward lines, 9.37 m versus 9.26 m, respectively. During the first circuit not a single line either from the forward or the backward lines is lost. We show the lengths after two toroidal turns in Figure 5(b). During the second turn roughly twice as many forward lines are lost as compared to the backward lines, 16% for the forward lines versus 9% for the backward lines. This trend of the asymmetric loss continues up to seven turns and then it reverses (see Figures 5(c) and 5(d)). During the second turn, the loss of forward lines occurs close to the X-point on the inboard side below the midplane and the loss of backward lines occurs above the midplane on the outboard side. By the end of seven circuits, an equal fraction of both the forward and backward lines are lost and the fractional loss is 56% (see below). All the field lines on the inboard side close to the X-point are lost. Most of the forward lines on the outboard side above the midplane are also lost. Below the midplane on the inboard side, from θ₀ = 2.5 to θ₀ = θ_X, we see large gaps for forward lines (see Fig. 5(c)). From the X-point to the outboard midplane, there is minimal loss of the forward lines (see Fig. 5(c)). Very similar scenario entails for the backward lines with one major difference. This difference is that the loss of backward lines from the inboard side close to the X-point is very small. The reason for these features is that in the close neighborhood of the X-point on the separatrix, the safety factor is extremely large and so the lines there move away or towards the X-point extremely slowly. This is also the reason for the difficulties in calculation of homoclinic tangle by tracing a single field line. The features seen at the end of seven turns continue up to ten turns except the difference that now the backward lines are lost faster than the forward lines after the equalization of loss rate at the end of seventh turn. By end of tenth turn, 62% of 360 K forward lines are lost and 66% of the backward lines are lost.

In Figure 6, we show the average length of field lines that start on the equilibrium separatrix as a function of the number of toroidal circuits for both the forward and the backward lines. A linear least square fit gives an excellent fit to the data. The fit tells us that on average the forward and backward moving lines traverse a distance of 9.37 and 9.26 m per toroidal circuit, respectively.

We calculate the differential poloidal rotation of the separatrix manifold from the perturbation. We calculate the poloidal rotation of each line on the equilibrium separatrix during a given toroidal turn as a function of the starting poloidal position θ₀. We denote the differential poloidal rotation by Δθ. We show the results in Figure 7. Poloidal position of the X-point in the DIII-D is θ_X = 4.2832 radians. Poloidal rotation close to the X-point is very slow. Near the X-point, it is as much as six orders of magnitude slower. Differential poloidal rotation of the manifold appears to show a self-similar structure with smallest scale of self-similarity near the X-point and largest scale in the region from the X-point up to the outboard midplane.

We calculate the displacement of a line normal to the equilibrium separatrix surface. We denote it by Δr. We consider Δr to be positive if the line is outside the equilibrium separatrix surface and negative if the line is inside. In Figure 8, we show Δr as a function of the starting poloidal angle θ₀ for the forward and backward lines after 1, 2, 5, and 10 toroidal turns. For the first circuit, the largest displacement is outwards and is about 6 cm for the forward manifold. For the forward lines, the large displacements are on the inboard side. For the backward lines, the large displacements are on the inboard side. After two circuits, the displacement jumps from 6 cm to as much as 40 cm; it is large on the inboard side for the forward lines and on the outboard side above the midplane for the backward lines. Such large displacements are overestimates because the collector plates are located at x_plate = R_plate–R₀ = −0.65 m on inboard side and y_plate= Z_plate–Z₀ = −1.367 m. We terminate a line when |R-R₀|> 0.8 m and |Z-Z₀| > 1.5 m, which is beyond the location of the plates.

In Figure 9, we show the average radial displacement of the lines normal to the ideal separatrix in the φ = 0 plane of the DIII-D as a function of toroidal turns. For both the forward and backward lines, the outward displacement is larger than the inward displacement. In Figure 10, we show the ratios of the number of lines with Δr > 0 to those with Δr < 0 as a function of the toroidal turns N_{tor cir} for the forward and the backward lines. This ratio is smaller than unity for all toroidal turns except for the first turn for the forward lines. So generally, more lines move into the ideal separatrix compared to those that move out of it. The data in last two figures may play a role in the nature of the radial electric field that may be generated by homoclinic tangle near the separatrix.

We calculate the diffusion coefficient of field lines normal to the equilibrium separatrix as D_F = (Δr)²/2Δφ after each turn. We separate the diffusion coefficient into two groups, one for the lines that diffuse inside the ideal separatrix, and the other for lines that diffuse out of the separatrix. We do this for both the forward lines and the backward lines. We show D_F versus the starting position θ₀ for the forward lines and the backward lines for 1, 2, 5, and 10 turns in Figures 11 and 12, respectively. D_F can vary over as much as 10 to 12 orders of magnitude; it can have extremely large poloidal gradients.

The equilibrium separatrix manifold of the DIII-D is made of 360 K field lines. These lines are uniformly distributed in the poloidal angle. If the position of a line after an iteration has $| R−R0 |>0.8 m$ or $| Z−Z0 |>1.5 m$⁠, we abandon that line and do not integrate it any further. We denote the fraction of 360 K starting lines that are abandoned by f_lost. We show f_lost as a function of toroidal circuits N_{tor cir} in Figure 13.

For the first seven circuits, the forward lines are lost faster than the backward lines. For the seventh through tenth circuit, the situation reverses. Within first 5 circuits, half of the forward lines are lost, and within 6 circuits, half of the backward lines are lost. The loss rate is fastest for the first 4 circuits. From the fifth to the eighth circuits, the loss rate slows down; and the loss rate slows down further for the remaining two circuits. Within first 5 circuits, half of all the forward and backward lines will be definitely lost. By the end of tenth circuit, 62% of forward lines are lost and 65% of backward lines are lost. From the data in Figure 13, it appears that the loss rate becomes very slow and may continue to be very slow. In the DIII-D, the inboard collector plate is located at $xPLATE=| R−R0 |=−0.655 m$ and the outboard plate is at $yPLATE=| Z−Z0 |=−1.3671 m$⁠. Since $xPLATE>−0.8 m$ and $yPLATE>−1.5 m$⁠, the field lines will strike the plates faster than the scenario that we have described for f_lost. Half of the lines in DIII-D will definitely strike the plates in less than 6 circuits. Consequently, half of the plasma particles that arrive on the separatrix from inside of it will strike the plates in less than six circuits as plasma particles diffuse outward normal to the separatrix.

Plasma confinement and feasibility of divertor are sensitive to the structure of the separatrix. The perturbed separatrix is vastly different from the ideal separatrix. This because the magnetic asymmetries create extremely complicated structure called the homoclinic tangle and its lobes. Very complicated shape of the separatrix such as in the DIII-D is radically different from a near-circular one.

In our new method, we use the natural canonical coordinates to represent the unperturbed surfaces and the map to advance the unperturbed separatrix forward and backward in toroidal angle. We can map the natural coordinates to the physical space. So we get an explicit and detailed picture of the evolution of the tangle and lobes in physical space after each turn for vastly complicated magnetic geometries in the real devices. Our approach is not limited to divertor tokamaks; it can be used for different kinds of magnetic topologies, for example, the limiter, single-null, double-null, doublet, snow-flake, tokamaks, stellarators, etc. It can be used for Hamiltonian systems, near-Hamiltonian systems, and higher degrees of freedom Hamiltonian systems.

Here, we have applied this new method to the DIII-D tokamak. We have considered the peeling-ballooning modes represented by two Fourier components with equal amplitude. We have captured the very complicated shape of the separatrix and the flux surfaces in the edge of the DIII-D. Every time the forward and backward manifolds meet in the φ = 0 plane of the DIII-D, they intersect and form the homoclinic tangle and its lobes to preserve the two topological invariants of symplecticity and neighborhood.

The homoclinic tangle of the ideal separatrix in the DIII-D is a very complicated structure and it becomes extremely complicated as more toroidal turns are taken, especially near the X-point. The tangle is the most complicated near the X-point and it forms the largest lobes there. On average the field lines cover a distance of about 9 m per toroidal turn. The poloidal rotation has large gradients in the poloidal direction. The average normal displacement of the lines on the ideal separatrix varies from 5 mm to 7 cm. The average outward displacement of the lines normal to the ideal separatrix is considerably larger than the average inward displacement. However, on the average, more lines are displaced inside the separatrix than outside. The field lines diffusion normal to the separatrix has extremely wide variation and very large poloidal gradients. Half of all the field lines are lost from the ideal separatrix in less than 6 toroidal turns.

The average distance 〈l〉 covered by the field lines in a single toroidal turn in the DIII-D is roughly about 9 m. For simplicity, we set 〈l〉 ∼ 9 m. We let T = Te = Ti = 50 eV in the DIII-D edge. T, T_e, and T_i are the plasma, electron, and ion temperature, respectively. So the thermal velocity of electrons is $vTe$ ≈ 3 × 10⁶m/s. Then, the time scale for a single toroidal turn for electrons is 〈l〉/$vTe$ ∼ 3 μs. 〈Δr〉 is the average normal displacement of the lines from the tangles from the magnetic perturbation per toroidal circuit of the DIII-D. The average normal displacement of the field lines on the ideal separatrix surface is roughly about 5 mm to 5 cm. So the general picture that emerges is that in a single turn on the time scale of 1 μs, field lines on the surface travel about 9 m, and are displaced from surface about 5 mm to 7 cm. The entire discussion here gives rise for a reasonable hypothesis at this point to the possibility that such displacement of electrons on this time scale of 1 μs may generate complicated electric fields in the edge, especially near the X-point. The magnetic field will not respond on this fast time scale; but the structure of the flux surfaces may become sufficiently spatially complicated on an iron gyroradius length scale that complicated electric potentials may be required to maintain the neutrality of the plasma. The E × B drifts from these complicated electric fields may modify plasma confinement and influence the edge physics. This is our outlook on the future studies.

We thank Allen Boozer of Columbia University and Todd Evans of General Atomics for valuable discussions, suggestions, and comments. This work was supported by Grant Nos. DE-FG02-01ER54624 and DE-FG02-04ER54793. This research used resources of the NERSC, supported by the Office of Science, US DOE, under Contract No. DE-AC02-05CH11231.