# Topological Constraints on Magnetic Relaxation磁弛豫的拓扑约束

arXiv:1007.4925v1 [astro-ph.SR] 28 Jul 2010 Topological Constraints on Magnetic Relaxation A. R. Yeates,?G. Hornig, and A. L. Wilmot-Smith Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK (Dated: August 6, 2018) Abstract The fi nal state of turbulent magnetic relaxation in a reversed fi eld pinch is well explained by Taylor’s hypothesis. However, recent resistive-magnetohydrodynamic simulations of the relaxation of braided solar coronal loops have led to relaxed fi elds far from the Taylor state, despite the conservation of helicity. We point out the existence of an additional topological invariant in any fl ux tube with non-zero fi eld: the topological degree of the fi eld line mapping. We conjecture that this constrains the relaxation, explaining why only one of three example simulations reaches the Taylor state. PACS numbers: 52.30.Cv, 52.65.Kj, 52.35.Vd, 96.60.Hv 1 The landmark paper of J. B. Taylor [1] showed how the fi nal, relaxed, magnetic fi eld in a turbulent plasma experiment could be predicted theoretically by assuming conservation of the total magnetic helicity H = R V A · Bd3x, where B = ? × A. Taylor hypothesized that, in a turbulent plasma with small but non-vanishing resistivity, H is the only relevant constraint on the relaxation. The resulting minimum energy state is then a linear force-free fi eld [2], ? × B = αB, where α is a constant depending on the value of H. The success of this theory in predicting the fi nal state of relaxation in a reversed fi eld pinch has led to speculation that Taylor’s hypothesis might apply more generally, for example, to magnetic structures in the Sun’s atmosphere [3–6]. Here, the nature of the fi nal state is of great interest because it limits the magnetic energy available for conversion to heat during the relaxation. This is key to understanding how magnetic fi elds produce the extreme coronal temperatures in the Sun and other stars. To apply Taylor’s hypothesis in such magnetically open domains, H is replaced by the relative helicity HR with respect to a reference fi eld [4, 7], typically a potential fi eld. A local domain, such as an isolated loop, is taken (the corona cannot be globally a linear force-free fi eld) and the hypothesis has met with some notable success [6]. However, counter-examples are known [8, 9] and it now seems likely that constraints beyond the conservation of global magnetic helicity may be important for general relaxation events. For example, while HRmeasures only the second-order linkage of fi eld lines, higher-order invariants may play a role [10–13] (although this remains under debate [14]). This Letter presents a conjecture on one possible topological constraint and was motivated by recent resistive-magnetohydrodynamic simulations of the relaxation of a magnetic fl ux tube, intended to model a solar coronal loop. In all of our experiments the initial fl ux tube contains a braided magnetic fi eld of non-trivial topology but with no net current or helicity [9, 15]. Despite undergoing a turbulent relaxation with myriad small-scale current structures and magnetic reconnections, on a timescale short compared to the resistive timescale, the magnetic fi eld relaxes to a fi nal state which directly contradicts that predicted by Taylor’s hypothesis. Although force-free, the fi nal state is not linear force-free, because the coeffi cient α varies strongly across fi eld lines. For the fi rst experiment, where we started from a fi eld modeled on the pigtail braid, the current is concentrated into two fl ux tubes of opposite twist (sign of the parallel current). The initial and fi nal states of this braiding experiment are shown in Fig. 1. If HR were the only relevant constraint on relaxation then the fi nal state would be a near-uniform, vertical, potential fi eld (B = ?ψ), because the initial confi guration is chosen such that HR= 0. Note that HRis still well conserved during the turbulent relaxation, so the departure of the fi nal state from the Taylor prediction is not due to changes in HR. Rather, the result seems to indicate the presence of additional constraint(s). In this Letter, we propose that such a constraint is given by a property of the fi eld line mapping called its topological degree. This is a topological property that can be defi ned for any continuous mapping from one compact manifold to another. In the braiding experiment, the mapping of magnetic fi eld lines from the lower boundary S0(where they enter the domain) to the upper boundary S1(where they exit) forms such a mapping. This is because (a) all fi eld lines connect, in the same direction, these two boundaries (the vertical fl ux through any horizontal cross-section is constant through the domain), and (b) the magnetic fi eld is everywhere non-zero (B 6= 0) within the domain. Although the fi elds considered in this paper are defi ned in rectangular, Cartesian domains such that the fl ux tube is aligned along the z-axis and the boundaries are rectangles in the (x,y) plane, the ideas are more generally applicable because every fl ux tube of any shape with uni-directional fi eld B 6= 0 2 0 -2 2 x -2 0 2 y -20 -15 -10 -5 0 5 10 15 20 z 0 -2 2 x -2 0 2 y -20 -15 -10 -5 0 5 10 15 20 z -0.27 -0.18-0.09 0.00 0.09 0.18 0.27 (a)(b) FIG. 1. (Color online) Magnetic fi eld lines in the original braiding experiment [15] for (a) the initial state at t = 0, and (b) the relaxed state at t = 290 (in units of the Alfv′ en time). Field lines are traced from the same starting points in each case. In (b), color contours show α = j·B/B2on the S0plane. is topologically equivalent to a straight cylinder [16]. To defi ne the topological degree of the fi eld line mapping, let F(x0) ∈ S1be the end-point of the fi eld line starting at the point x0∈ S0 . The fi eld line mapping will have one or more periodic orbits xp 0∈ S0where F(x p 0) = x p 0 [17]. As a fi xed point of the mapping F?I (where I denotes the identity map), each xp 0 is characterized by a fi xed point index, defi ned as the local Brouwer degree of the mapping (see e.g. [18]). This index takes integer values, ±1 for generic, structurally stable, isolated periodic orbits. The case +1 corresponds to an elliptic null point of the local linearisation of F?I, or an elliptic periodic orbit, while ?1 indicates a hyperbolic periodic orbit. The sum T = X xp 0 index(xp 0) (1) over all isolated periodic orbits is called the Lefschetz number or topological degree of the mapping F. By the Lefschetz-Hopf theorem [19] it is a conserved quantity, providing that no periodic orbits cross the side boundary of the domain. Periodic orbits can therefore be created or annihilated only in pairs of opposite index. The topological degree T may be computed by evaluating the Kronecker integral around the boundary of S0[20], or by other numerical methods [21]. Here, we adopt the graphical 3 t = 0. -3-2-10123 x -3 -2 -1 0 1 2 3 y -3-2-10123 x -3 -2 -1 0 1 2 3 y t = 35. -3-2-10123 x -3 -2 -1 0 1 2 3 y -3-2-10123 x -3 -2 -1 0 1 2 3 y t = 50. -3-2-10123 x -3 -2 -1 0 1 2 3 y -3-2-10123 x -3 -2 -1 0 1 2 3 y t = 80. -3-2-10123 x -3 -2 -1 0 1 2 3 y -3-2-10123 x -3 -2 -1 0 1 2 3 y t = 110. -3-2-10123 x -3 -2 -1 0 1 2 3 y -3-2-10123 x -3 -2 -1 0 1 2 3 y t = 290. -3-2-10123 x -3 -2 -1 0 1 2 3 y -3-2-10123 x -3 -2 -1 0 1 2 3 y FIG. 2. (Color online) Sequence of color maps for the original braiding experiment [15], where total degree T = 2. color map technique of [20]. For example, Fig. 2 shows the colour maps at various times in the braiding simulation. Every point x0= (x0,y0) on the lower boundary of the domain (S0 ) is assigned one of four colours, according to its fi eld line mapping F(x0) = (Fx,Fy) to the upper boundary S1. We use red if Fx x0and Fy y0; yellow if Fx y0; green if Fx x0and Fy y0. On the resulting color map, periodic orbits correspond either to red-green or yellow-blue boundaries; isolated, generic periodic orbits are points where all four colours meet. The index of an isolated periodic orbit may be read from the sequence of colors passed through in an anti-clockwise direction around a small circle around the point. An elliptic orbit (index 1) has red-yellow-green-blue (r-y-g-b), while a hyperbolic orbit (index ?1) has r-b-g-y. To determine T, either sum the individual indices or simply record the sequence of colors around the boundary of S0. For example, in Fig. 2(a), the initial state for the braiding experiment, we fi nd 12 periodic orbits with index 1, and 10 with index -1, giving a net topological degree of 2. This corresponds with the (anti-clockwise) sequence of colors on the boundary of r-y-g-b-r-y-g-b . The later color maps in Fig. 2 for the braiding experiment show that, during the turbu- lent relaxation, changes in the magnetic topology by reconnection lead to the annihilation 4 -1.0-0.50.00.51.0 x -1.0 -0.5 0.0 0.5 1.0 y -1.0-0.50.00.51.0 x -1.0 -0.5 0.0 0.5 1.0 y FIG. 3. (Color online) Color map for the initial magnetic fi eld of the Browning et al. [6] simulation (case 1). of periodic orbits in pairs of index +1 and ?1. This is consistent with (1) as the reconnec- tion processes occur strictly in the interior of the domain such that T remains unchanged. Eventually only two elliptic periodic orbits remain in the fi nal relaxed state, lying within the two fl ux tubes of opposite twist shown in Fig. 1(b). This leads us to suggest an expla- nation for the discrepancy between this relaxed state and that predicted by Taylor theory: for a turbulent relaxation in a magnetic fl ux tube that leaves the side boundary fi xed, the topological degree T of the initial fi eld line mapping is an additional topological constraint that can prevent the system from reaching the Taylor state. We now consider another two resistive-MHD simulations of turbulent relaxation in mag- netic fl ux tubes which further support our degree conjecture. The fi rst is a simulation by Browning et al. [6] where the Taylor hypothesis succeeds in predicting the relaxed state. Using their expressions for B, we have computed the color map (shown in Fig. 3) of the initial force-free magnetic fi eld in their case 1. The fi eld consists of an axisymmetric twisted magnetic fl ux tube of unit radius, with a piecewise-constant profi le of α with radius. Fig. 3 shows that this initial confi guration has a single isolated periodic orbit at the origin. In addition, there are four rational surfaces on which fi eld lines have a winding number that exactly divides 2π, forming four continuous circles of periodic orbits in the S0plane. How- ever, it is easy to show that such rational surfaces do not contribute to the total degree, so T = 1, in accordance with the pattern of colors around the tube boundary, r = 1. The initial fl ux tube in the Browning et al. simulation is kink unstable, and a turbulent relaxation is triggered by an initial velocity perturbation. The system relaxes to a force-free equilibrium on a comparable timescale to our original braiding experiment. However, the diff erence here is that this fi nal equilibrium is a constant-α “Lundquist” magnetic fi eld, in accordance with expectations from Taylor theory [22]. The simulation still supports our degree conjecture because the fi nal state retains a total degree T = 1, with a single twisted fl ux tube. The diff erence between this and our braiding experiment is that here the Taylor state is compatible with the degree of the initial fi eld. Our fi nal example is a new simulation of an initially braided magnetic fi eld with degree T = 3, solving the same resistive-MHD equations as in our earlier braiding simulation [15]. The original initial condition with T = 2 was modeled on the pigtail braid (see Fig. 2 of Ref. [23]), and was formed from a uniform vertical fi eld super-imposed with six localized “twist” regions (Eq. 2 of [23]) arranged in two columns. The T = 2 property arose from 5 the arrangement of the positive twist regions in one column and the negative regions in the other. We obtain a topologically stable T = 3 confi guration by using four columns of twist regions arranged on a circle, where each contribute an elliptic periodic orbit. The total degree is 3 because the remainder of the fi eld contributes a net degree of ?1. The initial fi eld we used is shown in Fig. 4(a). In the notation of Ref. [23], the twist regions have param- eters c1= (r0,0,?18,1, √2,2), c 2 = (?r0,0,?18,1, √2,2), c 3 = (0,r0,?6,?1, √2,2), c4= (0,?r0,?6,?1, √2,2), c 5 = (r0,0,6,1, √2,2), c 6 = (?r0,0,6,1, √2,2) , c 7 = (0,r0,18,?1, √2,2), and c 8 = (0,?r0,18,?1, √2,2), where r 0 = 1.27.Note that the total helicity of this fi eld vanishes, as in the T = 2 experiment. The color map for this initial fi eld (Fig. 4c) confi rms that T = 3. The T = 3 initial condition leads again to a turbulent relaxation, reaching a relaxed state on a similar timescale to the T = 2 experiment. The fi nal color map is shown in Fig. 4(d). As in the T = 2 case, the Taylor theory fails, but the degree conjecture is upheld, with T = 3 maintained in the fi nal state. In fact, there are four elliptic periodic orbits and one hyperbolic periodic orbit, so the four initial columns have led to four twisted fl ux tubes in the fi nal state. The degree constraint itself would not preclude the annihilation of the hyperbolic orbit with one of the elliptic orbits. There must be some further topological constraint on the evolution preventing this. This remains to be fully explored. In conclusion, our examples show that preservation of the topological degree T leads to non-trivial constraints on braided magnetic fl ux tubes. We therefore claim that the topolog- ical degree imposes a constraint on the relaxation beyond that of helicity conservation. In particular, the Taylor state is not reached if its degree diff ers from that of the initial state, at least in cases like our examples where the boundary is unaff ected by the dynamics. Note that, although these simulations use line-tied boundary conditions (in z), the idea extends to the case of a z-periodic boundary and hence to a toroidal domain. Finally, we point out that a series of further invariants may be constructed by taking the topological degree of multiple iterations of the fi eld line mapping, that is, by considering the degree of periodic orbits with periods greater than one. This work was supported by the UK Science Phys. Plasmas, 7, 1623 (2000). [2] L. Woltjer, Proc. Nat. Acad. Sci., 44, 489 (1958). [3] J. Heyvaerts and E. R. Priest, Astron. Astrophys., 137, 63 (1984). [4] A. M. Dixon, M. A. Berger, P. K. Browning, and E. R. Priest, Astron. Astrophys., 225, 156 (1989). [5] G. E. Vekstein, E. R. Priest, and C. D. C. Steele, Astrophys. J., 417, 781 (1993). [6] P. K. Browning, C. Gerrard, A. W. Hood, R. Kevis, and R. A. M. van der Linden, Astron. Astrophys., 485, 837 (2008). [7] M. A. Berger and G. B. Field, J. Fluid Mech., 147, 133 (1984). [8] T. Amari and J. F. Luciani, Phys. Rev. Lett., 84, 1196 (2000). [9] D.