In the H-mode regime of diverted tokamaks, the presence of strong pressure gradients in the pedestal gives rise to a sizable bootstrap current, together with the Ohmic and Pfirsch–Schlueter currents, close to the separatrix. For such equilibria, the presence of finite current density close to the separatrix requires the reexamination of equilibrium properties. It is almost universally assumed that the two branches of the separatrix (the stable and unstable manifolds) are straight as they cross at the X-point. However, the opposite angles of the plasma-filled segment and vacuum one cannot be equal if the current density does not vanish at the separatrix on the plasma side. We solve this difficulty by chipping off a thin layer of plasma edge so that the sharp corner of the plasma-filled segment becomes a hyperbola. Using the conformal transformation, we found that in the assumption of a hyperbolic boundary, the X point moves beyond the plasma boundary to fall in the vacuum region. An acute angle of the plasma-filled segment leads to an obtuse opposite angle of vacuum segment and vice versa. In the case of an acute angle of the plasma-filled segment, the new X point shifts inside the X point formed by the asymptotes of a hyperbolic boundary; in the case of an obtuse angle of the plasma-filled segment, the new X point shifts outside the X point formed by the asymptotes of a hyperbolic plasma boundary. The results are important for understanding the X point features, which affect the tokamak edge stability and transport.

## I. INTRODUCTION

Understanding the X point structure at tokamak plasma edge is important to tokamak physics. It affects the studies of plasma edge transport and stability, as well as the assessment of the divertor heat load distribution.

It is almost universally assumed that the two branches of the separatrix (the stable and unstable manifolds) are straight as they cross at the X-point. One consequence of this assumption is that the angle subtended by the private flux region (the region where magnetic field lines are anchored to the vacuum vessel at both ends) equals the angle subtended by the plasma on the other side of the X-point. Actually, according to the magnetohydrodynamic (MHD) equilibrium equation and the Grad–Shafranov equation,^{1,2} the opposite angles of the plasma-filled segment and vacuum one cannot be equal if the current density does not vanish at the separatrix on the plasma side. The primitive numerical trace of this phenomenon appears to have been seen in early free boundary equilibrium calculations, such as in Fig. 3(a) of Ref. 3. However, their results are only reported for regions far away from X points, and features near X points have not been exploited and detailed. This was only clarified recently in the computation of the vacuum solution of Solovev's equilibrium in Ref. 4, which was confirmed later in Ref. 5.

MHD equilibria near a separatrix were analyzed for a simplified model in Ref. 6 and for narrow islands in the low beta limit in Refs. 7 and 8. Asymptotic equilibrium solution at X-point tip was later developed systematically in Ref. 9 with the same toroidal current density in each segment bordering the X point. Since then, further investigations have been made on the effects of finite beta, transport, symmetry, and etc.^{10–12} To adapt to reality, one has to deal with the case with one side having a finite toroidal current density and the other side being vacuum across the plasma boundary. In the conventional picture, at X point, the plasma-filled segment borders three vacuum segments. Furthermore, the angle of the plasma-filled segment is not a right angle, while each of three vacuum segments has to be right angle when both the normal and tangential components of the magnetic field are required to be continuous across the plasma–vacuum interface, as will be reviewed later in Sec. II. This makes that the sum of the angles of four segments is not $ 2 \pi $. This is certainly unacceptable.

In this work, we prove that there is asymptotic vacuum solution near the X-point tip without assuming the presence of plasma boundary surface current. This is based on the matching of the plasma and vacuum solutions across the hyperbola type of plasma boundary. As is well-known, the asymptotes of hyperbola are two crossing straight lines forming an angle, with both legs tangent to the hyperbola itself. On one hand, hyperbola is an equilibrium solution of the plasma-filled segment and on the other hand a hyperbola can well approximate the sharp corner of the plasma-filled segment. Note that there is a minimum length scale across the magnetic surface for MHD, e.g., the Larmor radius in the classical transport picture. Therefore, chipping off a thin layer to reduce a sharp corner to a hyperbola is reasonable. We also point out that the control surface approach with a thin edge layer chipped off was used in the numerical computation of the free boundary equilibrium with X points. It successfully produces the X point structure with the true vacuum region in Solovev's equilibrium case.^{4,5}

We show that with the sharp corner of the X point chipped off, the hyperbola boundary allows a conformal transformation to be performed so that an asymptotic vacuum solution at the X-point tip can be obtained. The solution is obtained through matching the poloidal flux and its derivate without assuming the presence of surface current. Therefore, it extends the asymptotic solution of the X point equilibrium in Ref. 9 to the case with a true vacuum surrounding the plasma torus. We find that the X point does reappear but falls in the vacuum region, instead of on the plasma–vacuum interface. Since the new X point lies in vacuum, the asymptotic solution with four vacuum segments with right angles becomes acceptable.

This paper is arranged as follows: in Sec. II, the existing asymptotic theory is reviewed and the total angle issue of the X point solution is pointed out; in Sec. III, the asymptotic vacuum solution is presented; in Sec. IV, the numerical results are presented; in Sec. V, conclusions and discussion are given.

## II. REVIEW OF EXISTING THEORY AND THE TOTAL ANGLE ISSUE OF X POINT SOLUTION

In this section, we review the existing theory about the asymptotic solution at the X-point tip with the same toroidal current density in each segment bordering the X point^{9} and explain the total angle issue of the X point solution.

^{1,2}

*ψ*is the poloidal magnetic flux,

*R*is the major radius,

*p*denotes plasma pressure,

*f*represents the poloidal current flux, and prime denotes the derivative with respect to

*ψ*. The right hand side of Eq. (1) specifies the toroidal current density. In this representation, the magnetic field is expressed as $ B = \u2207 \varphi \xd7 \u2207 \psi + f \u2207 \varphi $, with $\varphi $ being the axisymmetric toroidal angle.

^{9}A local coordinate system (

*x*,

*y*) as shown in Fig. 1, where the dashed lines represent the plasma boundary in the conventional approach. In this expansion, the source term on the right hand side of Eq. (1) is expanded as follows:

^{9}

*c*being an arbitrary constant, which determines the plasma boundary shape, i.e., the included angle of the plasma boundary given by

Here, more explanation is given as follows. The constant *c* in Eq. (5) actually introduces a homogeneous solution of the polynomial type to Eq. (3). In principle, more general homogeneous solution [Eq. (8) to be derived later] can be added to the solution. However, we note that the solution in Eq. (4) satisfies both the equilibrium equation, Eq. (3), and the boundary condition in Eq. (6), which represents the wedge angle. Based on the solution uniqueness, one can conclude that the homogeneous solution is completely determined by the constant c in the plasma segment.

The solution in the plasma-filled segment affects the vacuum solution as well. In addition to the boundary condition for *ψ* in Eq. (6), the normal derivative of *ψ* needs to be continuous as well across the plasma–vacuum interface to assure the tangential magnetic field to be continuous in the case without a finite sheet current on the plasma–vacuum interface. The continuity of the normal derivative of *ψ* makes the vacuum solution become the same type of polynomial as in the plasma segment solution. This is more clearly proved in the polar coordinate description to be presented later.

In the plasma-filled segment with a finite toroidal current density, i.e., $ s 0 = 0$, the angle of plasma segment is not a right angle due to $ a = b$. As discussed above (and later in the polar coordinates), the vacuum solution has to assume the same type of polynomial as the solution in the plasma segment. We can therefore use Eq. (4) to discuss the vacuum solution as well. In the vacuum regions, instead, one has $ s 0 = 0$ so that *a* = *b* and three vacuum segments must have right angles. Obviously, the sum of angles for four segments are not $ 2 \pi $ as soon as $ s 0 = 0$ in the plasma region. This indicates that the asymptotic solution cannot be found at the X point tip if a true vacuum facing plasma torus is considered and both the normal and tangential components of the magnetic field are required to be continuous across the plasma–vacuum interface. This can be alternatively proved as follows.

*c*and

_{v}*ν*are constants to be determined by the boundary conditions and the symmetric condition for $ \xb1 \theta $ has been taken into account. In passing, it is pointed out here that the vacuum solution in Eq. (8) can be also regarded as the general homogeneous solution of Eq. (3). Because the special solution is proportional to

*r*

^{2}in the plasma segment, to satisfy the boundary condition in Eq. (6) uniformly in

*r,*the index

*ν*in the homogeneous solution in Eq. (8) has to be 2, i.e., the polynomial of second degree. Therefore, the homogeneous solution is fully described by constant c in Eq. (5) in the rectangle coordinates.

*c*is a constant and the boundary condition in Eq. (6) has been taken into account with

_{p}*θ*

_{0}being the plasma–vacuum interface. The boundary conditions on the plasma–vacuum interface are that the poloidal flux and its radial derivative are continuous for the case without surface current. Therefore, the solution of plasma segment gives rise to two boundary conditions on the interface $ \theta = \theta 0$

*r*, the index

*ν*in Eq. (8) has to be 2. Therefore, the vacuum solution in Eq. (8) is reduced to

One has to find a physical solution in this case. There is a possibility to assume that there is a plasma surface current so that the tangential component of magnetic field can be discontinuous. Noting that the surface current may cause the tearing mode instabilities, we therefore proceed to solve this problem with both normal and tangential components of magnetic field being continuous across the plasma–vacuum interface in Sec. III.

## III. ASYMPTOTIC VACUUM SOLUTION

In this section, we are going to solve the angle sum issue at the X point as pointed out in Sec. II. We exclude the possibility to have a finite sheet current on the plasma–vacuum interface. This leads to the tangential component of magnetic field be continuous across the plasma–vacuum interface as well as the normal component. On one hand, there is an indeterminacy issue for the introduction of infinitesimally thin sheet current. Different amounts of thin sheet current introduced on the plasma–vacuum interface can lead to a different vacuum solution in extending the fixed boundary equilibrium solution to the free boundary one. On the other hand, the existence of the thin sheet current on the plasma–vacuum interface can be potentially unstable.

We solve this issue by proving that, although the plasma–vacuum boundary formed by the intersection of straight lines indeed has no mathematical solution, the hyperbolic plasma boundary does have a regular solution. Note that the intersection of two straight lines can actually be the asymptotes of a hyperbola. This shows that there are asymptotic solutions to the X point problem.

### A. The boundary value problem in vacuum

In this subsection, we describe the mathematic boundary value problem in vacuum region with the plasma–vacuum boundary given by a hyperbola.

*ψ*= 0, the plasma is confined locally in a segment with the intersection of two straight lines. Instead, we assume that the plasma–vacuum boundary is specified by $ \psi = \psi 0 > 0$ with

*ψ*

_{0}small but finite. This is equivalent to chip off a thin layer at the plasma edge. The local plasma equilibrium is still described by Eq. (13). With a thin layer chipped off, the plasma–vacuum boundary becomes

*ψ*itself and its normal derivative $ \u2202 \psi / \u2202 n$ are continuous across the plasma–vacuum boundary constitutes a complete boundary value problem (actually an initial value problem). The continuations of

*ψ*itself and its derivative just assure the normal and tangential components of the magnetic field

**B**are continuous across the plasma–vacuum interface.

### B. Conformal transformation

*u*. The solution is

*u*has been taken to be the same as

*x*.

*v*. The solution is

*v*has been taken to be the same as

*y*.

*x*,

*y*) space to that in the (

*u*,

*v*) space. The hyperbola boundary is changed to the straight line boundary with Laplace's equation remaining unchanged in this transformation. This can be seen as follows. Comparing the boundary equation in Eq. (14) with Eq. (19), one obtains

*u*,

*v*) space, the boundary shape in Eq. (14) becomes simply a straight line

*u*=

*u*

_{0}. This is shown in Fig. 1.

Here, we point out that the transformation contains a singular point in the vacuum region: $ u = \u2212 \pi / 2$ as shown in Fig. 1. Since we are only interested in the asymptotic solution at the vertex of the hyperbola boundary, this singularity does not limit us much for this purpose.

### C. Boundary conditions in the transformed space

In Sec. III B, we showed that the transformation in Eq. (16) simplifies the cutoff plasma–vacuum boundary in hyperbola to a constant line *u* = *u*_{0} in (*u*, *v*) space. In this subsection, we are going to work out the changes of the boundary conditions.

*u*=

*u*

_{0}in the (

*u*,

*v*) space, it is required that

*ψ*on the boundary

*u*=

*u*

_{0}to be continuous, we note from Eqs. (17) and (18) that

*u*,

*v*) space. Therefore, one has

*ψ*in the (

*u*,

*v*) space

### D. Solution in the transformed space

*u*,

*v*) space

*u*,

*v*) space according to Eq. (29)

*x*,

*y*) space, one just needs to transform the solution in Eq. (32) from the (

*u*,

*v*) space back to the (

*x*,

*y*) space using the coordinate transform formula in Eqs. (21) and (23). This concludes this section.

## IV. NUMERICAL RESULTS

With the analytical asymptotic solution obtained in Sec. III, we are going to display the solution numerically in this section to exploit its features.

The solution in Eq. (32) can be plotted in the (*x*, *y*) space with (*u*, *v*) being given Eqs. (21) and (23). From Eqs. (20) and (22), one can see that *u* and *v* depend only on $ x \xaf \u2261 x / A$ and $ y \xaf \u2261 y / A$. This leads us to use $ x \xaf$ and $ y \xaf$ to plot the results. As discussed in Sec. II, the parameters *a* and *b* in Eq. (24) are determined by the toroidal current density and boundary shape and *ψ*_{0} determines how close the vertex of the hyperbolic plasma–vacuum boundary to the X point formed by the asymptotes. Therefore, *A* is actually also related to the distance between the vertex of the plasma hyperbolic boundary and the X point forming by the asymptotes of hyperbola. The relation between *A* and *ψ*_{0} is given in Eq. (24). To show the poloidal magnetic flux *ψ* in the vacuum with Eq. (32), three cases are displayed numerically with the acute, obtuse, and right angles of the included angle formed by the asymptotes of the hyperbolic plasma boundary.

*ψ*in the $ ( x / A , y / A )$ space for the case with

*a*= 1 and $ \theta p = 0.3 \pi $, and the arbitrary applicable

*A*(or

*ψ*

_{0}), where

*θ*is the included angle formed by the two asymptotes of plasma hyperbolic boundary as shown in Fig. 4. It can be expressed as follows:

_{p}The poloidal flux contour for the same case is given in Fig. 3. In Fig. 4, the *X* point structure is extracted from the contour plot in Fig. 3. The red solid curve represents the hyperbolic plasma–vacuum boundary, the dashed red lines are the asymptotes to the hyperbolic plasma boundary, and the black curves show the new X point structure in the vacuum as given by Eq. (32). From these figures, one can see that the new X-type point appears in the vacuum region, instead of on the plasma–vacuum interface. On a small scale, one can see that the angle of each four vacuum segments is a right angle. This is consistent with the local theory reviewed in Sec. II. On a larger scale, however, an acute angle *θ _{p}* leads to an obtuse opposite angle of the vacuum segment. The new X point shifts toward the vertex of hyperbola and sits between the X-point due to the asymptotes and the vertex of the hyperbolic plasma–vacuum boundary.

Next, we describe the case with an obtuse angle formed by the two asymptotes of the plasma hyperbolic boundary. Figure 5 gives the poloidal flux contour for the case with *a* = 1, $ \theta p = 0.7 \pi $ and the arbitrary applicable *A* (or *ψ*_{0}). In Fig. 6, the *X* point structure is extracted from the contour plot in Fig. 5. The red solid curve represents the plasma–vacuum boundary, the dashed red lines are the asymptotes to the hyperbolic plasma boundary, and the black curves show the new X point structure in the vacuum. From these figures, one can see that the new X point appears in the vacuum region, instead of on the plasma–vacuum interface. On a small scale, one can see that the angle of each four vacuum segments is a right angle. On a larger scale, however, an obtuse angle *θ _{p}* leads to an acute opposite angle of the vacuum segment. The new X point shifts outward from the vertex of hyperbola and sits beyond the X-point formed by the asymptotes.

The case with a right angle of the plasma segment is also studied. In general, as reviewed in Sec. II, the solution with an X point lies on the plasma–vacuum boundary can be found in this case. One does not need to start with the hyperbola type of boundary to obtain the vacuum solution in this case. Nevertheless, the case with a hyperbola plasma–vacuum boundary and a right included angle formed by the two asymptotes is also investigated. The result is given in Fig. 7, which shows that the X point in the vacuum region is still in a right angle, overlapping the X point formed by the asymptotes. This is consistent with the case with the X point on the plasma boundary and the angle of the plasma-filled segment is $ 90 \xb0$.

Here, let us summarize the computation results about the corresponding angle of the segment opposite that with plasma like in Figs. 4, 6, and 7. We find that the angle of each segment bordering the new X point is a right angle on a small scale since the new X point resides in vacuum.^{16} However, from Figs. 4 and 6, one can see that the segment borders are not strictly straight lines. On a larger scale, the included angle of the vacuum segment opposing to the segment containing plasma becomes no longer a right angle depending on the angle formed by the asymptotes of the hyperbolic plasma boundary, *θ _{p}*. The angle of the vacuum segment opposing to the segment containing plasma can expand to be larger than $ 90 \xb0$ if $ \theta p < 90 \xb0$ or contract to become less than $ 90 \xb0$ if $ \theta p > 90 \xb0$. It remains to be $ 90 \xb0$ if $ \theta p = 90 \xb0$. The smaller

*θ*the larger the expansion of the corresponding angle of the segment opposite to that with plasma and vice versa. The current solution is just an asymptotic solution; one can anticipate that the angle can change further on an even larger scale. In this case, the numerical vacuum solver becomes necessary.

_{p}^{4,5}

*θ*. The equation for the new X point position can be obtained by

_{p}*u*is related to

*x*in Eq. (17) with

*v*= 0 and

*u*

_{0}to

*θ*in Eq. (33). This is confirmed by the X point structure plots, e.g., in Figs. 4, 6, and 7. The position of the X point by the asymptotes of the plasma hyperbolic boundary is at $ x \xaf = 0$. Note that the shift is small when the hyperbolic plasma boundary is close to the X point formed by the hyperbola asymptotes since

_{p}*A*is small in this case. This explains that the X point shift phenomenon may not be seen in the global numerical calculation with the control surface method since its effect can be negligible numerically.

^{4,5}

## V. CONCLUSIONS AND DISCUSSION

In this paper, the existing theory about the asymptotic MHD equilibrium solution near the X point is first reviewed. This introduces an outstanding mathematical problem that the asymptotic solution cannot be found at the X point tip if a true vacuum facing plasma torus is considered. We then propose a solution for it by chipping off a thin layer of plasma edge so that the sharp corner of the plasma-filled segment becomes a hyperbola. This is reasonable since the hyperbola remains to be the asymptotic solution of the Grad–Shafranov equation in the plasma-filled region and the hyperbolic boundary can approach closely to the asymptotes forming the usual X point picture on the plasma boundary. We also note that the control surface approach with the thin edge layer chipped off was used in the numerical computation of the free boundary equilibrium with X points and successfully produces the X point structure with a true vacuum region in Solovev's equilibrium case.^{4} It is also noted that the applicability condition for the MHD description contains the lower scale length limit, e.g., the Larmor radius.

We proved that with a hyperbolic plasma–vacuum boundary, the asymptotic analytical vacuum solution at the vertex of a hyperbola can be obtained by the conformal transformation. The analytical solution makes it possible to exploit the subtle features of the X point structure in the case of a true vacuum outside the plasma torus. We found that with the assumption of the hyperbolic plasma boundary, the X point moves beyond the plasma boundary to fall in the vacuum region. The angle of each vacuum segment bordering the new X point is a right angle on a small scale. This is consistent with the conventional theory as reviewed in Sec. II. However, on a larger scale, we find that an acute angle of the asymptotes of the hyperbolic plasma boundary leads to an obtuse opposite angle of vacuum segment and vice versa. Noting that our analytical theory is an asymptotic solution, in even larger scale, the global numerical computation is required.^{3–5} We also found that in the case of an acute angle of the asymptotes of the hyperbolic plasma boundary, the new X point shifts inwardly inside the X point formed by the asymptotes of the hyperbola boundary but remains outside the plasma boundary. In the case of an obtuse angle of the asymptotes of the hyperbolic plasma boundary, the new X point shifts outwardly beyond the X point formed by the asymptotes of a hyperbola boundary. Nevertheless, the shift is small when the hyperbolic plasma boundary is close to the X point by the hyperbola asymptotes. Its effect could be negligible numerically. The results indicate that one may need to adjust the tokamak edge picture. The boundary picture with the presence of the X point on the plasma boundary and finite toroidal current simultaneously needs to be changed to a quasi X point picture, with some edge portion being chipped off. The present calculation offers a mathematical interpretation for the control surface method.^{4,5}

The analytic asymptotic solution at the X point tip obtained in this paper provides a realistic picture for the X point without requiring the presence of plasma surface current. Note that even slightly chipping off a thin edge layer in the configuration with X point, the safety factor value, *q*, can change dramatically. There will be no concern about the infinite *q* issue at the tokamak plasma edge. The results can affect the stability and transport analyses, as well as the divertor design^{13} and the pedestal formation analyses.^{14,15} Therefore, the results are important.

The authors would like to acknowledge the helpful discussion with Dr. Richard Fitzpatrick, especially regarding the control surface treatment. This research was supported by the U. S. Department of Energy, Office of Fusion Energy Science under Grant No. DE-FG02-04ER54742.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Linjin Zheng:** conceptualization (equal). **Michael Kotschenreuther:** conceptualization (equal). **Francois L. Waelbroeck:** conceptualization (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.