Magnetic surface topology in decaying plasma knots

Torus-knot solitons have recently been formulated as solutions to the ideal incompressible magnetohydrodynamics (MHD) equations. We investigate numerically how these fields evolve in resistive, compressible, and viscous MHD. We find that certain decaying plasma torus knots exhibit magnetic surfaces that are topologically distinct from a torus. The evolution is predominantly determined by a persistent zero line in the field present when the poloidal winding number n p ≠ 1 . Dependence on the toroidal winding number nt is less pronounced as the zero line induced is contractible and disappears. The persistent zero line intersects the new magnetic surfaces such that, through the Hopf–Poincaré index theorem, the sum of zeroes on the new surfaces equals their (in general non-zero) Euler characteristic. Furthermore we observe the formation of magnetic islands between the surfaces. These novel persistent magnetic structures are of interest for plasma confinement, soliton dynamics and the study of dynamical systems in general.

Due to the generally high electrical conductivity of plasma described by magnetohydrodynamics (MHD), large electrical currents can flow and plasmas are heavily influenced by the resulting magnetic forces. The zerodivergence magnetic fields can lead to closed magnetic field lines, field lines that ergodically fill a magnetic surface, and field lines that chaotically fill a region of space. In ideal (zero-resistance) MHD the magnetic flux through a perfect conducting fluid element cannot change, leading to frozen in magnetic fields in the plasma [14]. This implies that in ideal MHD magnetic topology and magnetic helicity is conserved [15][16][17].
In 1982 Kamchatnov described an intrinsically stable plasma configuration [13] with a magnetic topology based on fibers of the Hopf map [18]. This type of MHD equilibrium, where the fluid velocity is parallel to the field and equal to the local Alfvén speed, was shown by Chandrasekhar to be stable [19], even in specific cases in the presence of dissipative forces [20]. Quasi stable self-organizing magnetic fields with similar magnetic topology to Kamchatnov's field (but different flow) have recently been demonstrated to occur in full-MHD simulations [12]. Here the final configuration is not a Taylor state, which is consistent with recent findings in [21].
Recently the class of topologically non-trivial solutions to Maxwell's equations has been extended by including torus knotted fields [22]. Another way of obtaining such solutions, for massless fields of various spins, is to use twistor theory [23]. The magnetic fields of the t=0 solutions in [22], have been used to construct novel plasma torus knots [24], solutions to the ideal incompressible MHD equations.
To investigate the potential importance of plasma torus knots for realistic plasma the influence of dissipation has to be investigated. Dissipation can lead to breaking and reconnection of field lines and thereby change the magnetic topology. In this article we show numerically that novel persistent magnetic structures emerge that are characterized by a non-zero Euler characteristic. Through the Poincareé Hopf index theorem this leads to

Plasma torus knots
An ideal MHD soliton, as defined in [13,24], is a static configuration of magnetic field B, fluid velocity u, and pressure p that satisfies the ideal, incompressible MHD equations. The fluid field and pressure that solve this can be inferred from the momentum equation, which can be written as: and the induction equation: In the ideal and incompressible case the fluid field and pressure corresponding to the soliton are given by respectively. Already in 1956 in a paper titled 'On the stability of the simplest solution of the equations of hydromagnetics', Chandrasekhar noted that this represents an exact stationary solution and proved it to be stable against linear perturbations [19].
Kamchatnov analyzed this solution for the specific case where the magnetic field and velocity field are given by the Hopf map, every single field line is linked with every other. The static magnetic field of the Kamchatnov-Hopf soliton, whose magnetic field is identical to the t=0 magnetic field of an electromagnetic solution in free space [9] can be obtained in various ways. Bateman's construction [25] of describing a (null) electromagnetic field via two Euler potentials α and β such that , is well suited to generate new solutions to Maxwells equations with torus knotted field lines [22]. The class of torus knot solitons, of which the Kamchatnov-Hopf soliton is an element, are the solutions to the ideal MHD equations where the velocity and fluid field are identical to the magnetic field of these EM solutions [24].
The torus knot solitons are constructed using the following complex-valued Euler potentials 2 ) to rescale the configuration. Taking gives the magnetic field of the Kamchatnov-Hopf soliton. Generalizing this construction by raising α to a positive integer power n t which we call the toroidal integer, and β to a positive integer power n p which we call the poloidal integer, the magnetic field of a ( ) n n , t p plasma torus knot is obtained via: where c is a scaling constant with correct dimensions and , integrated over all space, is a normalization factor that provides all magnetic field configurations with the same magnetic energy.
As noted, the plasma (1, 1) torus knot is the Kamchatnov-Hopf soliton: every magnetic field line is a circle that is linked with every other field line. Otherwise the field configuration is characterized by a finite set of core field lines that each form an ( ) n n , t p torus knot and around which other field lines span nested magnetic surfaces [22,24]. Figures 1(a) and (b) show an example of a plasma torus knot magnetic field configuration with toroidal integer n t =3 and a poloidal integer n p =2. Figure 1(a) shows that there are two core field lines that each form a ( ) 3, 2 torus knot, and figure 1(b) shows two magnetic surfaces spanned by field lines close to the two core field lines.
The magnetic helicity, defined as is an invariant of ideal MHD. The total magnetic helicity of the plasma torus knots is found by integrating equation (6) over all space, and is given by . The magnetic field of the torus knots goes to zero at infinity, such that this helicity is gauge invariant. The magnetic helicity is reduced in the plasma torus knots with higher values of n p and n t . This reduction is qualitatively understood by noting that the core field lines trace a path around the torus with a left-handed writhe while the magnetic surfaces have a right-handed twist around the core field lines. Since both writhe and twist contribute to the helicity the opposite signs cancel each other [26,27] and higher values of toroidal and poloidal integers reduce the total magnetic helicity.
The conservation of energy, angular momentum, and magnetic helicity, can provide stability to certain plasma configurations. As shown in [24] only the plasma torus knots with n p =1 have non-zero angular momentum and are able to form stable configurations in ideal MHD. We now investigate the solutions represented by equations (3) and (5) in the case of a non-ideal plasma by adding compressibility, magnetic diffusivity, and viscosity.

New emerging surfaces
We numerically study the dynamics of plasma torus knots using the PENCIL code, a high-order finite difference code for compressible MHD [28,29]. In resistive MHD the l.h.s. of equation (1)  We observe a dramatic change in the magnetic topology during a rapid first phase in the dynamics characterized by field lines pinching off and reconnecting. During this reconfiguration new magnetic surfaces are formed which have in general a different topology. After this first phase the new configurations persist in time and have an unchanging magnetic topology and a decreasing field strength due to the finite resistivity. For example figures 1(c)-(e) give an angled view, a side view, and a top view of a new magnetic surface in the (3, 2) plasma torus knot.
The magnetic fields close to the center of the initial plasma configuration have strong components in opposite directions along the z axis. The dynamical principle of the formation of the new surfaces is that regions of largest counter propagating fields (here the center of the configuration) will have highest dissipation which pinches off field lines. The remaining part of the field lines will find nearby pinched-off fields lines with whom they connect resulting in the new magnetic surfaces.
An interesting interpretation of the emergence of the observed structures is given by the mechanism of the flow of helicity across scales as demonstrated by Scheeler et al in [30]. They show that the component of fluid helicity caused by the linking of vorticity filaments in fluids will, through localized reconnections translate to larger scale writhe, and eventually to twist, present on the largest scale, approximately conserving total helicity. The parallels between fluid and magnetic helicity have been established since their conception [15]. In this interpretation the effect of the localized reconnection at the center of the configuration can be seen as removing the writhing and linking of the initial magnetic surfaces, resulting in a configuration where helicity is present as twist in the antiparallel legs of the new magnetic structure.
The topology of a surface is characterized by its genus g, which is equal to the number of holes in that surface. A torus has only one hole, and therefore genus 1, but the new (gray) emerging surface is a surface with three holes, a triple torus. The genus of a surface is related to the Euler characteristic by c =g 2 2 , which is an important topological invariant. The Poincaré-Hopf index theorem relates the Euler characteristic to zero points of a vector field on the surface as we shall show in section 4 and is therefore an important quantity in analyzing the new plasma structures.
The new surface drawn in gray is part of a set of nested surfaces with the same topology. This can be seen from the Poincaré plot in figure 2(f). Note that there also appear, as expected from references [12,31], magnetic islands (see the section on islands).
Simulations have been performed for the nine combinations with n t =1, 2, 3 and n p =1, 2, 3. The created surface depends on n p directly. When n p =1 the new surface is a torus, when n p =2 a triple torus (g = 3), and when n p =3 a surface which has genus g=5. These magnetic surfaces consist of 2n p twisted 'legs' that run parallel to the z-axis. These legs meet on the z-axis above and below the z=0 plane. In half the legs the magnetic field is oriented in the positive z-direction, in the other half in the negative z-direction. Where the legs meet the field exhibits a magnetic cusp geometry and the field lines switch over from one leg to the other.

Magnetic decay
If we look at the time evolution of the mean magnetic energy á ñ á ñ B B 2 0 2 , shown in figure 3, we see that the configurations with n p =1 (red curves) show the lowest energy loss, and thus the highest stability, followed by the fields with n p =2, and finally the fields with n p =3. The lowest loss of energy for fields where n p =1 is consistent with the analytical stability analysis in [24] that predicts plasma torus knots with n p =1 to be stable. Figure 3(b) shows the evolution of normalized magnetic helicity (magnetic helicity divided by magnetic energy). Because the fields start out normalized by energy, their initial value corresponds with the result of equation (7). The increase in normalized magnetic helicity can be seen as the result of the magnetic energy decaying faster than the helicity.
We observe that during the entire simulation the fluid velocity u remains to a high degree parallel and equal in (dimensionless) magnitude to the magnetic field B, even as the magnetic topology changes. The stability of the configuration is predicated on this condition being maintained. Balancing the momentum equation in this way is fundamentally different from the stability exhibited by a solution to the Grad Shafranov equation [32] or a force-free field [14].

Zero lines
The initial fields of the plasma torus knots contain nonsingular points where the magnetic field strength vanishes if either n p or n t is not unity. This can easily be seen by expanding equation (5) into: the field vanishes along the entire z-axis. The field only approaches zero linearly around the circular null if n t is two, and similarly for the straight null if n p =2, otherwise the null is higher-order. Null points have been described extensively in the literature [34,35] because of their role in reconnection [36]. We analyze the null lines in the field using the method presented in [35], by constructing the linearization matrix M whose indices are given by = ¶ M B ij j i , (see appendix B). The field around a null can be characterized by the three eigenvectors of this matrix [35] that represent the directions from which field lines approach and leave the null. Around the null lines, these three vectors are orthogonal, with the zero-eigenvalue eigenvector in the direction of the null line. The other two vectors determine two orthogonal directions, one from which the magnetic field lines approach the null, and the other determines the direction in which field lines leave the null. These two vectors rotate around the zero line as one moves along the zero line. The number of rotations of these vectors around the straight null is n t , and the number of rotations around the circular null is n p (see appendix B).
The zero circle disappears by contracting to a point on the origin. This process is shown in figure 4, and it explains why the final configuration depends predominantly on the poloidal index n p , which determines the topology of the new magnetic surfaces.
The Poincaré-Hopf index theorem states that the sum of the indices of the zeroes of a vector field on a compact, oriented manifold is equal to the Euler characteristic χ of that manifold. A magnetic surface is such a manifold, and the magnetic field necessarily is a vector field in that surface. A toroidal surface (c = 0) allows a smooth vector field without any zero points. A magnetic surface can only form the observed triple torus (c = -4), or quintuple torus c = - ( 8) if there are points where the magnetic field is zero, and the indices of these zeroes then must sum to −4 or −8, respectively. These zeroes are provided by the zero line on the z-axis, that intersects the surfaces exactly four times and carries index −1 or −2 (where the index is defined by the field restricted to the two-dimensional surface perpendicular to the zero line) if n p =2 or n p =3, respectively. Therefore these surfaces can exist by grace of the zero line, and they can only persist if the zero line persists. It is generally assumed that extended zeroes areunstable [17,35,37]. In our simulations however we observe that if the magnetic field starts at zero on the straight null, this zero line persists during entire simulation. This is in contrast to the zero line on thecircle in the z=0 plane, which disappears quickly, and is not present in the final configuration, as is shown in figure 4.
The Poincaré-Hopf index of a nonsingular zero point can be defined, as for example in [38]. Here it is shown that the vector field on the boundary of a neighborhood of the nonsingular null can be extended to a field in the interior of that neighborhood with finitely many singular nulls, and the sum of these indices is defined as the total index of the null. The total sum of the indices of nonsingular nulls is equal to the Euler characteristic of the manifold in which these nulls reside. The Euler characteristic of  3 is 0, and therefore the indices of the null lines sum to zero. This explains how the zero on thecircle can contract to a point, and disappear, leaving no zero point behind, or merge with the null line on the z-axis.
The null on the z-axis does not move, and persists in time. This does not necessarily imply that the zero line is stable, as any external perturbation of the form B z will make the null disappear, but the internal plasma dynamics do not make the null disappear. This persistence can be understood as a consequence of the symmetry of the field around the z-axis. The symmetry of the matrix M entails that the eigenvectors of M are orthogonal around the zero line, such that the field approaches the null from two orthogonal directions, in a plane perpendicular to the null line. The restriction of the vector field to this two-dimensional plane has a topological singularity (an X-point), and we observe that no field in the z-direction is created, allowing the zero line to persist, and the non-toroidal surfaces to exist.

Magnetic islands
The new magnetic surfaces are part of a set of nested surfaces with the same topology. There are, similar as what is observed from breaking up of toroidal magnetic surfaces [12,31], also magnetic islands between these surfaces. Between the non-toroidal magnetic surfaces these islands can take very different, and complicated forms, tracing out intricate knots and links in the space between the intact non-toroidal surfaces (see figure 5).
Magnetic islands are generally described by casting magnetic field line flow in terms of a Hamiltonian dynamical system, and the magnetic surfaces are invariant tori in this system. In the unperturbed system, the field will consist only of nested toroidal surfaces characterized by a rotational transform. A perturbation will cause some tori to break up into island chains. Close to rational surfaces, (surfaces where the rotational transform is rational, i.e. field lines form closed curves) field lines will form surfaces that lie around one of the closed field lines of the original rational surface.
The islands in these non-toroidal surfaces also originate from an intact surface, as can be seen in figure 5(a). The field in the cross section of one upwards oriented 'leg' of the branched surface rotates, and then is split over the two downwards oriented legs, to be split and combined again. Because of this a rotational transform of the surface cannot be defined. Even though every point of the cross section will eventually map back to the cross section, this mapping is not necessarily continuous, and not in general a rotation. We do observe field structures that strongly resemble magnetic islands, but these island chains need not be symmetrically distributed across the surface, as is the case for the structure in figure 5(b). Because these islands form around a field line lying on a non-toroidal surface, their path need not be a torus knot, but can follow a differently knotted path.

Conclusions and discussion
We have studied a class of plasma torus knots in resistive plasma, which consist of linked and knotted core field lines surrounded by nested toroidal surfaces. n p has strongest influence on the time evolution of these structures. Large values for n t slow down the decrease of á ñ B 2 only at high t. These dynamics can be understood by the dynamics of the zero lines, where the circular null, caused by > n 1 t is able to contract to a point and disappear, but the straight null caused by > n 1 p persists, and allows for new magnetic ordering with low energy loss. In the resistive simulations the magnetic topology is not conserved, and new and interesting magnetic topologies are created. Structures with genus 3 and 5 are observed, and persist in the simulations. These structures also exhibit magnetic island formation, which has a different character to islands that emerge from toroidal magnetic surfaces.
Given the crucial role played by the extended zero in the plasma configuration resulting from initial symmetry, it seems unlikely that magnetic surfaces with non-zero Euler characteristic occur in the turbulent plasmas encountered in an astrophysical context. It is however possible to engineer this symmetry. A zero line can occur where sets of opposing fields meet, and can be created by sets of opposing coils, as is done in the cusp geometry [39]. These non-toroidal surfaces suggest new topologies for magnetic confinement fusion devices, essentially a marriage of the cusp fusion concept [39] or polywell fusion concept [40] and the stellarator concept [41].
The expressions for the initial conditions in the simulations were generated from equation (5) using Mathematica. The source for the PENCIL-CODE is freely available on http://pencil-code.nordita.org/.
To simulate the plasma dynamics we assume an isothermal plasma, we take the background pressure to be the pressure for an isothermal gas r = p c s 2 , where ρ is the density and c 2 s is the speed of sound squared. We solve the coupled equations in terms of u, ρ, and A, from which the magnetic field is calculated by = B A . The vector potential of the initial condition is calculated by = j B and transforming that to the vector potential using the inverse Laplace transform.
The equation of motion for an isothermal plasma is where ν is the kinematic viscosity and S is the traceless rate of strain tensor The continuity equation in terms of the logarithmic density has the form The induction equationcan be written terms of the vector potential as where η is the magnetic diffusivity.  The matrix M is symmetric, so the field around the null is characterized by three orthogonal eigenvectors, given by: Here we have filled in the values of γ and δ.
The field around the z-axis is characterized by the null eigenvector in the z-direction indicating the direction of the null. The other two eigenvectors lie in the x y , -plane, with v 2 determining the direction from which field lines approach the null line and v 3 the direction in which field lines leave. v 1 and v 2 are orthogonal in the x y , plane, and their direction depends on z. In order to determine the angle of these vectors in this plane as a function of z, we write the vector v 2 as the complex number x = + v v i x y 2 2 , where the superscripts denote the x-respectively the y-component of the vector. After some manipulation we get: