Magnetic field generated by current filaments

Transcription

1 Journal of Phsics: Conference Series OPEN ACCESS Magnetic fiel generate b current filaments To cite this article: Y Kimura 2014 J. Phs.: Conf. Ser View the article online for upates an enhancements. Relate content - Multiple breathers on a vorte filament H Salman - Variational analsis of topological stationar barotropic MHD in the case of single-value magnetic surfaces A Yahalom - Lower an upper bouns on the energ of braie magnetic fiels J-J Al This content was ownloae from IP aress on 11/09/2018 at 16:35

2 Flu in QCS Magnetic fiel generate b current filaments Y. Kimura Grauate School of Mathematics, Nagoa Universit Furo-cho, Chikusa-ku, Nagoa , Japan kimura@math.nagoa-u.ac.jp Abstract. We investigate the magnetic fiel generate b two straight current filaments using the analog between stea MHD an Euler flows. Using the Biot-Savart law, we present a namical sstem escribing the etension of magnetic lines aroun the current filaments. It is emonstrate that, if two current filaments are non-parallel, a magnetic line starting near one current goes to infinit b the rifting effect of the other. 1. Introuction It is well-known that a practicall important analog resies between stea Euler an MHD flows. If we enote incompressible velocit an vorticit as u an ω for stea Euler flows, an a magnetic fiel (or the magnetic flu ensit) an an electric current istribution as B an j for stea MHD flows, the following analog for the two sets hols eactl : 1 u = ω u = 0 (1) B = j B = 0 (2) From the Biot-Savart law, which is equivalent to take the inverse of rotation, we have u() = 1 ω( ) ( ) 4π 3 V, (3) for u(), an B() = 1 j( ) ( ) 4π 3 V, (4) for B(). We shoul note that this analog breaks if we eal with a time-epenent problem, because corresponing namics, the Euler equation for u an the magnetic inuction equations for B, involve u an B in a ifferent manner. As long as we sta in stea problems, however, we can etract useful information from one sie an appl to the other. The content of this article make full use of this analog. Using the Biot-Savart law (3), we have investigate the particle motion aroun a vorte soliton (Kimura & Koikari (2004)). It is foun that particles near the loop part of a vorte soliton are trappe an move insie a knotte torus aroun the loop of the vorte soliton. 1 We have omitte the permeabilit ine µ 0 which woul appear when S. I. units are use. Content from this work ma be use uner the terms of the Creative Commons Attribution 3.0 licence. An further istribution of this work must maintain attribution to the author(s) an the title of the work, journal citation an DOI. Publishe uner licence b Lt 1

3 Flu in QCS To unerstan this torus formation, a simple to moel, which replace a vorte soliton with a couple of vorte filaments tangent to the soliton filament was introuce an name the chopsticks moel. A variet of particle motions incluing chaotic ones are foun insie the torus, an the iscover of such convolute particle motions even for a simple setting of vorte filaments is the irect motivation to stu possible complication of the magnetic fiel structure aroun a simple configuration of electric current filaments. For similar motivations, reaer is referre to, for eample, Gascon & Peralta-Salas (2005) an Aguirre et al. (2008) an the references therein. 2. Formulation Let us first calculate the magnetic fiel generate b a single electric current filament of a (stea) current J which passes 0 with the unit tangent vector t (Figure 1). A irect integration of (4) results J t ( 0 ) B() =. (5) 2π t ( 0 ) 2 Now a couple of current filaments, J1 an J2, are place as in Figure 2. We ma assume that J1 is on the -ais, an J2 is on a plane parallel to the -plane with a istance of, an is tilte b the angle θ from the irection. For the formula (5), we have t = (0, 0, 1), 0 = (0, 0, 0) for J1 an t = (0, sin θ, cos θ), 0 = (, 0, 0) for J2. The total magnetic fiel is a superposition of the contributions from each current filament. To see how a given magnetic line etens in the 3D space, we start at the initial point an connect segments of the magnetic vector fiel nicel. For the present case, it is equivalent to integrate the following namical sstem sin θ cos θ j2 j1 ( ) cos θ, + (6) = ( )2 + ( sin θ cos θ) 0 ( ) sin θ where we have replace J1 /2π j1 an J2 /2π j2. J1 θ J r t J2 0 B 0 Figure 1. A Single electric current filament J an the magnetic fiel B generate b J. Figure 2. The configuration of a couple of electric current filaments J1 an J2. We shoul note that the right han sie of (6) can be written with a vector potential as 2

4 Flu in QCS follows, = j 1 2 log(2 + 2 ) j log { ( sin θ cos θ) 2 + ( ) 2} 0 sin θ. cos θ (7) 3. Comparison with the vorte chopsticks moel With the vorte chopsticks moel, we obtaine the following epression corresponing to (7), = [ Γ 1 2π log t 1 ( 1 ) t 1 Γ ] 2 2π log t 2 ( 2 ) t 2 [ ] (ν2 + τ 2 )G( )ẑ + [ τg( )ẑ ], (8) where Γ 1, Γ 2 are the circulation of vorte filaments. In the above equation, the first two terms on the right han sie correspon irectl to the right han sie of (7), but there are aitional terms. These terms attribute to the stea rotational an translational motion of a vorte soliton, which can be parametrie b a half of the maimum curvature ν, a constant torsion τ an the local inuction constant G of a vorte soliton(hasimoto (1972), Kia (1981)). In Kimura & Koikari (2004), a wie variet of combinations of these parameters are eamine in integrating (8) to calculate the volume of the torus as a flui mass transporte b the vorte chopsticks. While eamining the parameter epenence on the volume of the torus, we notice a tenenc that the volume ecrease rasticall with ecreasing value of the torsion parameter τ, an in the limit when τ 0, we preict that the volume of torus also goes to 0. On the other han, as a finite volume remains even for ero ν (with nonero τ), we ma assume that the last term in (8), which correspons to a uniform translation of the sstem, is essential for a finite volume torus to eist. So if we use the analog between a vorte filament an a current filament, we ma epect that the translation of the sstem, which is unerstoo as an antirifting motion, is also essential for the eistence of close magnetic lines or their stabilit, if an. To corroborate this, we will investigate (6) for some special cases. For simplicit, if we assume that the initial point (,, ) is much closer to J 1 than J 2 in Figure 2, i.e / 1, then (6) is approimate as = j j 2 / ((/) sin θ) (/) sin θ cos θ sin θ. (9) From the above equation, we see that a trajector is near a circle aroun J 1 which is perturbe b the effect of J 2. We are concerne with how the effect of perturbation remains if becomes larger. To see this we check the evelopment of the -component of the above equation. The -component is = (j 2 /) sin θ ( 2 / 2 ) sin 2 θ + 1, an if we enote a = sin 2 θ/ 2 an b = (j 2 /) sin θ, an assume the initial conition (0) = 0, we have an initial value problem of / = b/(a 2 + 1), (0) = 0, (10) 3

5 Flu in QCS which can be integrate easil to give a 3 + = bt 3 1/3 3b 1/3 3 j2 t t =. a sin θ (11) The spee epens on the parameters, but (t) goes to plus or minus infinit as t for an values of θ ecept 0 an π. From (9) the equation for the square istance of trajector from J1, r2 (t) = (t)2 + (t)2, is given b 2j2 r2 = 2 2 ((/) sin θ cos θ). ( / ) sin2 θ + 1 (12) Using (11), the right han sie of this equation is evaluate as t 1/3 for t which provies the asmptotic behavior r2 t2/3. a) b) c) ) e) f) Figure 3. Simulation results of (6). Parameters are set j1 = j2 = 1 an = 1, an θ = π/6 for a), b), c) an θ = π/3 for ), e), f ). The initial conitions are (0.1, 0.333, 0) for a), ), (0.3, 0.333, 0) for b), e), (0.5, 0.333, 0) for c), f ). Figure 3 shows some plots of the trajectories (magnetic lines) obtaine b integrating (6). The parameters are set j1 = j2 = 1 an = 1, an θ = π/6 for a), b), c) an θ = π/3 for ), e), f ). All the initial conitions are set on the = 0 plane with = , an the component is 0.1 for a), ), 0.3 for b), e) an 0.5 for c), f ). If the initial positions are close to J1, the trajectories spiral out along J1 ( a), b), ), e)). On the other han, if the are close to 4

6 Flu in QCS Figure 4. The time evelopment of (t) for Figure 3a). Asmptoticall, it approach the line of t 1/3. Figure 5. The time evelopment of r 2 = (t) 2 +(t) 2 for Figure 3a). With oscillations, it approach the line of t 2/3 asmptoticall. the (approimate) hperbolic point (0.5, 0, 0), the trajectories meaner between J 1 an J 2, but still the component tens to go to infinit. In Figure 4 an Figure 5, the time evelopment of (t) an of r 2 (t) are plotte respectivel for the case of Figure 3a). As estimate, the show the asmptotic epenence of t 1/3 an t 2/3, respectivel. We have emonstrate that a magnetic line which passes a point close to one of the two straight current filaments in a skewe configuration goes to infinit b the rifting effect of the other. Although this observation implies that such a magnetic line can not be close, it still remains an open problem whether an magnetic lines not close to either current filament coul make a close loop or not. To answer to this question, a thorough investigation of (6) is necessar. Acknowlegments Comments an references from Professor Peralta-Salas(ICMAT) are gratefull acknowlege. References Aguirre, J., Giné, J. & Peralta-Salas, D Integrabilit of magnetic fiel create b current istributions. Nonlinearit 21, Gascon, F. & Peralta-Salas, D Some properties of magnetic fiels generate b smmetric configurations of wires. Phsica D 206, Hasimoto, H A soliton on a vorte filament. Journal of Flui Mechanics pp Kia, K A vorte filament moving without change of form. Journal of Flui Mechanics pp Kimura, Y. & Koikari, S Particle transport b a vorte soliton. Journal of Flui Mechanics 510,