Magnetic field decay of three interlocked flux rings with zero linking number
aa r X i v : . [ a s t r o - ph . S R ] M a r Magnetic field decay of three interlocked flux rings with zero linking number
Fabio Del Sordo,
Simon Candelaresi,
1, 2 and Axel Brandenburg NORDITA, AlbaNova University Center, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden Department of Astronomy, Stockholm University, SE 10691 Stockholm, Sweden (Dated: Received 22 October 2009; revised 27 January 2010; published 3 March 2010)The resistive decay of chains of three interlocked magnetic flux rings is considered. Depending on the relativeorientation of the magnetic field in the three rings, the late-time decay can be either fast or slow. Thus, thequalitative degree of tangledness is less important than the actual value of the linking number or, equivalently,the net magnetic helicity. Our results do not suggest that invariants of higher order than that of the magnetichelicity need to be considered to characterize the decay of the field.
PACS numbers: PACS Numbers : 52.65.Kj, 52.30.Cv, 52.35.Vd
I. INTRODUCTION
Magnetic helicity plays an important role in plasma physics[1–3], solar physics [4–6], cosmology [7–9], and dynamo the-ory [10, 11]. This is connected with the fact that magnetichelicity is a conserved quantity in ideal magnetohydrodynam-ics [12]. The conservation law of magnetic helicity is ulti-mately responsible for inverse cascade behavior that can berelevant for spreading primordial magnetic field over largelength scales. It is also likely the reason why the magneticfields of many astrophysical bodies have length scales that arelarger than those of the turbulent motions responsible for driv-ing these fields. In the presence of finite magnetic diffusivity,the magnetic helicity can only change on a resistive time scale.Of course, astrophysical bodies are open, so magnetic helicitycan change by magnetic helicity fluxes out of or into the do-main of interest. However, such cases will not be consideredin the present paper.In a closed or periodic domain without external energy sup-ply, the decay of a magnetic field depends critically on thevalue of the magnetic helicity. This is best seen by consid-ering spectra of magnetic energy and magnetic helicity. Themagnetic energy spectrum M ( k ) is normalized such that Z M ( k ) dk = h B i / µ , (1)where B is the magnetic field, µ is the magnetic permeabil-ity, and k is the wave number (ranging from 0 to ∞ ). Themagnetic helicity spectrum H ( k ) is normalized such that Z H ( k ) dk = h A · B i , (2)where A is the magnetic vector potential with B = ∇ × A .In a closed or periodic domain, H ( k ) is gauge-invariant, i.e.it does not change after adding a gradient term to A . For fi-nite magnetic helicity, the magnetic energy spectrum is boundfrom below [12] such that M ( k ) ≥ k | H ( k ) | / µ . (3)This relation is also known as the realizability condition [13].Thus, the decay of a magnetic field is subject to a correspond-ing decay of its associated magnetic helicity. Given that in a closed or periodic domain the magnetic helicity changes onlyon resistive time scales [14], the decay of magnetic energyis slowed down correspondingly. More detailed statementscan be made about the decay of turbulent magnetic fields,where the energy decays in a power-law fashion proportionalto t − σ . In the absence of magnetic helicity, h A · B i = 0 , wehave a relatively rapid decay with σ ≈ . [15], while with h A · B i 6 = 0 , the decay is slower with σ between 1/2 [9] and2/3 [16].The fact that the decay is slowed down in the helical case iseasily explained in terms of the topological interpretation ofmagnetic helicity. It is well known that the magnetic helicitycan be expressed in terms of the linking number n of discretemagnetic flux ropes via [13] Z A · B d V = 2 n Φ Φ , (4)where Φ i = Z S i B · d S (for i = 1 and 2) (5)are the magnetic fluxes of the two ropes with cross-sectionalareas S and S . The slowing down of the decay is then plau-sibly explained by the fact that a decay of magnetic energy isconnected with a decay of magnetic helicity via the realizabil-ity condition (3). Thus, a decay of magnetic helicity can beachieved either by a decay of the magnetic flux or by mag-netic reconnection. Magnetic flux can decay through anni-hilation with oppositely oriented flux. Reconnection on theother hand reflects a change in the topological connectivity, asdemonstrated in detail in Ref. [17, p.28].The situation becomes more interesting when we considera flux configuration that is interlocked, but with zero linkingnumber. This can be realized quite easily by considering aconfiguration of two interlocked flux rings where a third fluxring is connected with one of the other two rings such that thetotal linking number becomes either 0 or 2, depending on therelative orientation of the additional ring, as is illustrated inFig. 1. Topologically, the configuration with linking numbersof 0 and 2 are the same except that the orientation of the fieldlines in the upper ring is reversed. Nevertheless, the simpletopological interpretation becomes problematic in the case ofzero linking number, because then also the magnetic helicity FIG. 1: Visualization of the triple ring configuration at the initialtime. Arrows indicate the direction of the field lines in the rings, cor-responding to a configuration with n = 0 (left) and n = 2 (center).On the right the non-interlocked configuration with n = 0 is shown. is zero, so the bound of M from below disappears, and M cannow in principle freely decay to zero. One might expect thatthe topology should then still be preserved, and that the link-ing number as defined above, which is a quadratic invariant,should be replaced with a higher order invariant [18–20]. Itis also possible that in a topologically interlocked configura-tion with zero linking number the magnetic helicity spectrum H ( k ) is still finite and that bound (3) may still be meaning-ful. In order to address these questions we perform numericalsimulations of the resistive magnetohydrodynamic equationsusing simple interlocked flux configurations as initial condi-tions. We also perform a control run with a non-interlockedconfiguration and zero helicity in order to compare the mag-netic energy decay with the interlocked case.Magnetic helicity evolution is independent of the equationof state and applies hence to both compressible and incom-pressible cases. In agreement with earlier work [21] we as-sume an isothermal gas, where pressure is proportional to den-sity and the sound speed is constant. However, in all cases thebulk motions stay subsonic, so for all practical purposes ourcalculations can be considered nearly incompressible, whichwould be an alternative assumption that is commonly made[22]. II. MODEL
We perform simulations of the resistive magnetohydrody-namic equations for a compressible isothermal gas where thepressure is given by p = ρc , with ρ being the density and c s being the isothermal sound speed. We solve the equationsfor A , the velocity U , and the logarithmic density ln ρ in the form ∂ A ∂t = U × B + η ∇ A , (6) D U D t = − c ∇ ln ρ + J × B /ρ + F visc , (7) D ln ρ D t = − ∇ · U , (8)where F visc = ρ − ∇ · νρ S is the viscous force, S isthe traceless rate of strain tensor, with components S ij = ( U i,j + U j,i ) − δ ij ∇ · U , J = ∇ × B /µ is the currentdensity, ν is the kinematic viscosity, and η is the magneticdiffusivity.The initial magnetic field is given by a suitable ar-rangement of magnetic flux ropes, as already illustratedin Fig. 1. These ropes have a smooth Gaussian cross-sectional profile that can easily be implemented in termsof the magnetic vector potential. We use the Pencil Code( http://pencil-code.googlecode.com ), wherethis initial condition for A is already prepared, except thatnow we adopt a configuration consisting of three interlockedflux rings (Fig. 1) where the linking number can be chosento be either 0 or 2, depending only on the field orientation inthe last (or the first) of the three rings. Here, the two outerrings have radii R o , while the inner ring is slightly bigger andhas the radius R i = 1 . R o , but with the same flux. We use R o as our unit of length. The sound travel time is given by T s = R o /c s .In the initial state we have U = and ρ = ρ = 1 . Ourinitial flux, Φ = R B · d S , is the same for all tubes with Φ = 0 . c s R √ µ ρ . This is small enough for compressibil-ity effects to be unimportant, so the subsequent time evolu-tion is not strongly affected by this choice. For this reason,the Alfv´en time, T A = √ µ ρ R / Φ , will be used as our timeunit. In all our cases we have T A = 10 T s and denote the di-mensionless time as τ = t/T A . In all cases we assume thatthe magnetic Prandtl number, ν/η , is unity, and we choose ν = η = 10 − R o c s = 10 − R /T A . We use meshpoints.We have chosen a fully compressible code, because it isreadily available to us. Alternatively, as discussed at the end of § I, one could have chosen an incompressible code by ignoringthe continuity equation and computing the pressure such that ∇ · U = 0 at all times. Such an operation breaks the localityof the physics and is computationally more intensive, becauseit requires global communication. III. RESULTS
Let us first discuss the visual appearance of the three inter-locked flux rings at different times. In Fig. 2 we compare thethree rings for the zero and finite magnetic helicity cases atthe initial time and at τ = 0 . . Note that each ring shrinks asa result of the tension force. This effect is strongest in the core FIG. 2: Visualization of the triple ring configuration at τ = 0 (left),as well as at τ = 0 . with zero linking number (center) and finitelinking number (right). The three images are in the same scale. Thechange in the direction of the field in the upper ring gives rise to acorresponding change in the value of the magnetic helicity. In thecenter we can see the emergence of a new flux ring encompassingthe two outer rings. Such a ring is not seen on the right. of each ring, causing the rings to show a characteristic inden-tation that was also seen in earlier inviscid and non-resistivesimulations of two interlocked flux rings [21].At early times, visualizations of the field show little differ-ence, but at time τ = 0 . some differences emerge in that theconfiguration with zero linking number develops an outer ringencompassing the two rings that are connected via the innerring; see Fig. 2. This outer ring is absent in the configurationwith finite linking number.The change in topology becomes somewhat clearer if weplot the magnetic field lines (see Fig. 3). For the n = 2 con-figuration, at time τ = 4 one can still see a structure of threeinterlocked rings, while for the n = 0 case no clear structurecan be recognized. Note that the magnitude of the magneticfield has diminished more strongly for n = 0 than for n = 2 .This is in accordance with our initial expectations.The differences between the two configurations becomeharder to interpret at later times. Therefore we compare inFig. 4 cross-sections of the magnetic field for the two cases.The xy cross-sections show clearly the development of thenew outer ring in the zero linking number configuration. Fromthis figure it is also evident that the zero linking number casesuffers more rapid decay because of the now anti-aligned magnetic fields (in the upper panel B x is of opposite signabout the plane y = 0 while it is negative in the lower panel).The evolution of magnetic energy is shown in Fig. 5 for thecases with zero and finite linking numbers. Even at the time τ ≈ . , when the rings have just come into mutual contact,there is no clear difference in the decay for the two cases.Indeed, until the time τ ≈ the magnetic energy evolvesstill similarly in the two cases, but then there is a pronounceddifference where the energy in the zero linking number caseshows a rapid decline (approximately like t − / ), while in thecase with finite linking number it declines much more slowly(approximately like t − / ). However, power law behavior isonly expected under turbulent conditions and not for the rel-atively structured field configurations considered here. The FIG. 3: (Color online) Magnetic flux tubes at time τ = 4 for thecase of zero linking number (upper picture) and finite linking num-ber (lower picture). The colors represent the magnitude of the mag-netic field, where the scale goes from red (lowest) over green to blue(highest). energy decay in the zero linking number case is roughly thesame as in a case of three flux rings that are not interlocked.The result of a corresponding control run is shown as a dot-ted line in Fig. 5. At intermediate times, . < τ < , themagnetic energy of the control run has diminished somewhatfaster than in the interlocked case with n = 0 . It is possiblethat this is connected with the interlocked nature of the fluxrings in one of the cases. Alternatively, this might reflect thepresence of rather different dynamics in the non-interlockedcase, which seems to be strongly controlled by oscillations onthe Alfv´en time scale. Nevertheless, at later times the decaylaws are roughly the same for non-interlocked and interlockednon-helical cases. FIG. 4: (Color online) Cross-sections in the xy plane of the magnetic field with zero linking number (upper row) and finite linking number(lower row). The z component (pointing out of the plane) is shown together with vectors of the field in the plane. Light (yellow) shadesindicate positive values and dark (blue) shades indicate negative values. Intermediate (red) shades indicate zero value. The time when the rings come into mutual contact ismarked by a maximum in the kinetic energy at τ ≈ . . Thiscan be seen from Fig. 6, where we compare kinetic and mag-netic energies separately for the cases with finite and zero link-ing numbers. Note also that in the zero-linking number casemagnetic and kinetic energies are nearly equal and decay inthe same fashion.Next we consider the evolution of magnetic helicity inFig. 7. Until the time τ ≈ . the value of the magnetic helic-ity has hardly changed at all. After that time there is a gradualdecline, but it is slower than the decline of magnetic energy.Indeed, the ratio h A · B i / h B i , which corresponds to a lengthscale, shows a gradual increase from . R o to nearly . R o atthe end of the simulation. This reflects the fact that the field has become smoother and more space-filling with time.Given that the magnetic helicity decays only rather slowly,one must expect that the fluxes Φ i of the three rings also onlychange very little. Except for simple configurations whereflux tubes are embedded in field-free regions, it is in generaldifficult to measure the actual fluxes, as defined in Eq. (5).On the other hand, especially in observational solar physics,one often uses the so-called unsigned flux [23, 24], which isdefined as P = Z S | B | dS. (9)For a ring of flux Φ that intersects the surface in the middle atright angles the net flux cancels to zero, but the unsigned fluxgets contributions from both intersection, so P = 2 | Φ | . In FIG. 5: Decay of magnetic energy (normalized to the initial value)for linking numbers 2 (solid line) and 0 (dashed line). The dottedline gives the decay for a control run with non-interlocked rings. Thedash-dotted lines indicate t / and t / scalings for comparison. Theinset shows the evolution of the maximum field strength in units ofthe thermal equipartition value, B th = c s ( ρ µ ) / .FIG. 6: Comparison of the evolution of kinetic and magnetic en-ergies in the cases with finite and with vanishing linking numbers.Note that in both cases the maximum kinetic energy is reached at thetime τ ≈ . . The two cases begin to depart from each other after τ ≈ . In the non-helical case the magnetic energy shows a sharpdrop and reaches equipartition with the kinetic energy, while in thehelical case the magnetic energy stays always above the equipartitionvalue. three-dimensional simulations it is convenient to determine P = Z V | B | dV. (10) FIG. 7: Evolution of magnetic helicity in the case with finite linkingnumber. In the upper panel, h A · B i is normalized to its initial value(indicated by subscript 0) while in the lower panel it is normalized tothe magnetic energy divided by R o .FIG. 8: Decay of the unsigned magnetic flux P (normalized to theinitial value P ) for the cases with n = 0 and n = 2 . The dotted linegives the decay for a control run with non-interlocked rings. For several rings, all with radius R , we have P = 2 πR N X i =1 | Φ i | = πN RP , (11)where N is the number of rings. In Fig. 8 we compare the evo-lution of P (normalized to the initial value P ) for the caseswith n = 0 and n = 2 . It turns out that after τ = 1 the valueof p is nearly constant for n = 2 , but not for n = 0 .Let us now return to the earlier question of whether a fluxconfiguration with zero linking number can have finite spec-tral magnetic helicity, i.e. whether H ( k ) is finite but of oppo-site sign at different values of k . The spectra M ( k ) and H ( k ) are shown in Fig. 9 for the two cases at time τ = 5 . This fig-ure shows that in the configuration with zero linking number H ( k ) is essentially zero for all values of k . This is not the FIG. 9: Comparison of spectra of magnetic energy and magnetichelicity in the case with zero linking number (upper panel) and fi-nite linking number (lower panel) at τ = 5 . Stretches with negativevalues of H ( k ) are shown as dotted lines. case and, in hindsight, is hardly expected; see Fig. 9 for thespectra of M ( k ) and k | H ( k ) | / µ in the two cases at τ = 5 .What might have been expected is a segregation of helicity notin the wave-number space, but in the physical space for posi-tive and negative values of y . It is then possible that magnetichelicity has been destroyed by locally generated magnetic he-licity fluxes between the two domains in y > and y < .However, this is not pursued further in this paper.In order to understand in more detail the way the energy isdissipated, we plot in Fig. 10 the evolution of the time deriva-tive of the magnetic energy E M = µ R B dV (upper panel)and the kinetic energy E K = R ρ U dV (lower panel). Inthe lower panel we also show the rate of work done by theLorentz force, W L = R U · ( J × B ) dV , and in the upperpanel we show the rate of work done against the Lorentz force, − W L . All values are normalized by E M0 /T s , where E M0 isthe value of E M at τ = 0 .The rates of magnetic and kinetic energy dissipation, ǫ M and ǫ K , respectively, can be read off as the difference betweenthe two curves in each of the two panels in Fig. 10. Indeed,we have − W L − dE M /dt = ǫ M , (12) W L + W C − dE K /dt = ǫ K , (13)where the compressional work term W C = R p ∇ · U d V isfound to be negligible in all cases. Looking at Fig. 10 wecan say that at early times ( < τ < . ) the magnetic fieldcontributes to driving fluid motions ( W L > ) while at latertimes some of the magnetic energy is replenished by kineticenergy ( W L < ), but since magnetic energy dissipation still FIG. 10: Evolution of the rate of work done against the Lorentzforce, − W L , together with d E M / d t (upper panel), as well as the rateof work done by the Lorentz force, + W L , together with d E K / d t (lower panel), all normalized in units of E M /T s , for the case withfinite linking number. The inset shows − W L at late times for thecase with n = 0 (solid line) and n = 2 (dashed line). dominates, the magnetic energy is still decaying ( dE M /dt < . The maximum dissipation occurs around the time τ = 0 . .The magnetic energy dissipation is then about twice as large asthe kinetic energy dissipation. We note that the ratio betweenmagnetic and kinetic energy dissipations should also dependon the value of the magnetic Prandtl number, Pr M = ν/η ,which we have chosen here to be unity. In this connection itmay be interesting to recall that one finds similar ratios of ǫ K and ǫ M both for helical and non-helical turbulence [25]. Atsmaller values of Pr M the ratio of ǫ K to ǫ K + ǫ M diminisheslike Pr − / for helical turbulence [26]. In the present case thedifference between n = 0 and 2 is, again, small. Only at latertimes there is a small difference in W L , as is shown in the insetof Fig. 10. It turns out that, for n = 2 , W L is positive while for n = 0 its value fluctuates around zero. This suggests that the n = 2 configuration is able to sustain fluid motions for longertimes than the n = 0 configuration. This is perhaps somewhatunexpected, because the helical configuration ( n = 2 ) shouldbe more nearly force free than the non-helical configuration.However, this apparent puzzle is simply explained by the factthat the n = 2 configuration has not yet decayed as much asthe n = 0 configuration has. IV. CONCLUSIONS
The present work has shown that the rate of magnetic en-ergy dissipation is strongly constrained by the presence ofmagnetic helicity and not by the qualitative degree of knotted-ness. In our example of three interlocked flux rings we con-sidered two flux chains, where the topology is the same exceptthat the relative orientation of the magnetic field is reversed inone case. This means that the linking number switches from 2to 0, just depending on the sign of the field in one of the rings.The resulting decay rates are dramatically different in the twocases, and the decay is strongly constrained in the case withfinite magnetic helicity.The present investigations reinforce the importance of con-sidering magnetic helicity in studies of reconnection. Recon-nection is a subject that was originally considered in two-dimensional studies of X-point reconnection [27, 28]. Three-dimensional reconnection was mainly considered in the last20 years. An important aspect is the production of currentsheets in the course of field line braiding [29]. Such currentsheets are an important contributor to coronal heating [30]. The crucial role of magnetic helicity has also been recog-nized in several papers [31, 32]. However, it remained unclearwhether the decay of interlocked flux configurations with zerohelicity might be affected by the degree of tangledness. Ourpresent work suggests that a significant amount of dissipationshould only be expected from tangled magnetic fields thathave zero or small magnetic helicity, while tangled regionswith finite magnetic helicity should survive longer and are ex-pected to dissipate less efficiently.
Acknowledgments
We acknowledge the allocation of computing resourcesprovided by the Swedish National Allocations Committee atthe Center for Parallel Computers at the Royal Institute ofTechnology in Stockholm and the National SupercomputerCenters in Link¨oping. This work was supported in part bythe European Research Council under the AstroDyn ResearchProject No. 227952 and the Swedish Research Council GrantNo. 621-2007-4064. [1] J. B. Taylor, Phys. Rev. Lett. , 1139 (1974).[2] T. H. Jensen and M. S. Chu, Phys. Fluids , 2881 (1984).[3] M. Berger and G. B. Field, J. Fluid Mech. , 133 (1984).[4] D. M. Rust and A. Kumar, Sol. Phys. , 69 (1994).[5] D. M. Rust and A. Kumar, Astrophys. J. , L199 (1996).[6] B. C. Low, Sol. Phys. , 217 (1996).[7] A. Brandenburg, K. Enqvist, and P. Olesen, Phys. Rev. D ,1291 (1996).[8] G. B. Field and S. M. Carroll, Phys. Rev. D , 103008 (2000).[9] M. Christensson, M. Hindmarsh, and A. Brandenburg, Astron.Nachr. , 393 (2005).[10] A. Pouquet, U. Frisch, and J. L´eorat, J. Fluid Mech. , 321(1976).[11] A. Brandenburg and K. Subramanian, Phys. Rep. , 1 (2005).[12] L. Woltjer, Proc. Natl. Acad. Sci. U.S.A. , 489 (1958).[13] H. K. Moffatt, J. Fluid Mech. , 117 (1969).[14] M. Berger, Geophys. Astrophys. Fluid Dyn. , 79 (1984).[15] M.-M. Mac Low, R. S. Klessen, and A. Burkert, Phys. Rev. Lett. , 2754 (1998).[16] D. Biskamp and W.-C. M¨uller, Phys. Rev. Lett. , 2195 (1999).[17] E. Priest and T. Forbes, Magnetic Reconnection, CambridgeUniversity Press, 2000 [18] A. Ruzmaikin and P. Akhmetiev, Phys. Plasmas , 331 (1994).[19] G. Hornig and C. Mayer, J. Phys. A , 3945 (2002).[20] R. Komendarczyk, Commun. Math. Phys. , 431 (2009).[21] R. M. Kerr and A. Brandenburg, Phys. Rev. Lett. , 1155(1999).[22] R. Grauer and C. Marliani, Phys. Rev. Lett. , 4850 (2000).[23] Zwaan, C., Sol. Phys. , 397 (1985).[24] C. J. Schrijver and K. L. Harvey, Sol. Phys. , 1 (1994).[25] N. E. L. Haugen, A. Brandenburg, and W. Dobler, Astrophys.J. , L141 (2003).[26] A. Brandenburg, Astrophys. J. , 1206 (2009).[27] E. N. Parker, J. Geophys. Res. , 509 (1957).[28] D. Biskamp, Phys. Fluids , 1520 (1986).[29] M. A. Berger, Phys. Rev. Lett. , 705 (1993).[30] K. Galsgaard and ˚A. Nordlund, J. Geophys. Res. , 13445(1996).[31] Y. Q. Hu, L. D. Xia, X. Li, J. X. Wang, and G. X. Ai, Sol. Phys. , 283 (1997).[32] Y. Liu, H. Kurokawa, C. Liu, D. H. Brooks, J. Dun, T. T. Ishii,and H. Zhang, Sol. Phys.240