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Magnetic Fronts in Galaxies A Astronomy Reports, Vol. 45, No. 7, 2001, pp. 497–501. Translated from AstronomicheskiÏ Zhurnal, Vol. 78, No. 7, 2001, pp. 579–584. Original Russian Text Copyright © 2001 by Petrov, Sokoloff, Moss. Magnetic Fronts in Galaxies A. P. Petrov1, D. D. Sokoloff 2, and D. L. Moss3 1Russian University of Peoples’ Friendship, Moscow, Russia 2Moscow State University, Moscow, Russia 3University of Manchester, Manchester, England Received October 18, 2000 Abstract—Magnetic-field reversals observed in the Milky Way at one or more Galactocentric radii are inter- preted as long-lived transient structures containing memory of the seed magnetic fields. These magnetic fronts can be described using the so-called no-z approximation for the nonlinear, mean-field dynamo equations, which are the topic of the paper. We obtain asymptotic estimates for the speed of propagation of discontinuities in the solution (internal fronts). These estimates agree well with numerical solutions of the no-z equations, and sup- port our interpretation of the magnetic-field reversals as long-lived transients. © 2001 MAIK “Nauka/Interpe- riodica”. 1. INTRODUCTION Such a discussion is the aim of this paper. Based on our analysis, we demonstrate that estimates for the life- The large-scale magnetic fields of spiral galaxies times of a magnetic front obtained in [3] are robust and appear to result from the action of the galactic dynamo, do not depend on the detailed vertical structure of the driven by the differential rotation of the galactic disk and galaxy. the helicity of the interstellar turbulence. The azimuthal component of the galactic magnetic field is dominant. In the Milky Way, however, the azimuthal component 2. EQUATIONS, GENERATION MECHANISM, reverses between the Orion and Sagittarius arms, and AND EQUILIBRIUM SOLUTION there may be additional reversals ([1], see [2] for a dis- cussion of the current state of this problem and the pos- We consider a galactic disk of radius R and uniform sibility of observing reversals in external galaxies). semi-thickness h. The equations of the no-z model for the azimuthal (Bφ) and radial (B ) components of an One possible explanation of the field reversals is the r following. Let us suppose that, by chance, the seed mag- axisymmetric magnetic field averaged over the vertical netic field has a reversal somewhere between Galactocen- cross section of the galactic disk are tric radii r and r . In the nonlinear stage of magnetic-field ∂ 1 2 Br 2 ∂ 1 ∂ evolution, a magnetic front with sharp field reversals -------- = – Rαα()B Bφ – B + λ ----- -------- ()rB , ∂t r ∂r r ∂r r appears somewhere between r1 and r˙2 . We identify the (1) ∂ front with this reversal. The front moves slowly in the Bφ 2 ∂ 1 ∂ --------- = RωGB – Bφ + λ ----- -------- ()rBφ . Galactic disk, and the estimates of [3] show that its life- ∂t r ∂r r ∂r time can be on the order of the age of the Galaxy. ∂Ω ∂ Ω However, some of the estimates in [3] require Here, G = r / r, where (r) is the rotation law, α η Ω 2 η detailed information about the vertical structure of the Rα = 0h/ and Rω = 0h / are dimensionless num- Galactic disk, while observations mainly provide infor- bers specifying the amplitudes of the helicity and dif- α Ω mation about the properties of the magnetic field and ferential rotation, respectively, where 0 and 0 are turbulence averaged over a vertical cross section of the representative values of α and Ω, and λ = h/R is the disk disk. Fortunately, the galactic dynamo equations can be aspect ratio. The ÐBr and ÐBφ terms in the first and sec- presented in a form known as the no-z model, which ond equations of (1) approximate the (dominant) diffu- focuses attention on the field structure in the galactic sion term perpendicular to the galactic disk. We allow plane and ignores structure in the vertical direction ([4], the dynamo to be saturated via α quenching: see also [5]). α() α() α()2 The no-z model appears to be very robust and quite B ==Bφ 0 fgBφ , (2) adequate for making comparisons with most observa- α 2 tions (see [6]). However, until now, it has been consid- where 0 is chosen such that f(0) = 1, g = 1/B0 , and B0 ered in the context of the particular physical problem of is the characteristic magnetic-field strength at which the galactic dynamo (see e.g. [3, 5]), and its mathemat- nonlinear effects in the dynamo become important. ical properties have not been discussed systematically. More generally, as in Eqs. (1), α will depend on the 1063-7729/01/4507-0497 $21.00 © 2001 MAIK “Nauka/Interperiodica” 498 PETROV et al. Br here, the ÐBr and ÐBφ terms in (3) are each multiplied π2 ≈ by /4, and so Dcrit 6. L2 A one-component asymptotic analysis ([7], see also S [3, 9]) supposes that the field has the form Ð (), Γt () () (), Γt () M Br tr ==e Qrbr r , Bφ rt e Qrb, (5) L where Q(r) can be obtained by solving the equation 1 M0 Bφ 2 1 ' ΓQ = γ ()r Q + λ ---()rQ ' , (6) r which coincides with the asymptotic expansion in [7]. T1 + M In the nonlinear generalization of (6) [10], γ(r) is replaced by the nonlinear term γ(r, Q). In contrast, deal- ing with the no-z model, we introduce nonlinearity into T2 Eqs. (1) directly. We now turn to the nonlinear properties of the The phase plane of the dynamical system (3): trajectories no-z model. Let us consider the steady-state problem are attracted to the states M+ and MÐ (for notation, see the for Eqs. (3): text). α ()2 Rα 0 fgBφ Bφ +0,Br ==RωGBr –0.Bφ (7) strength of the total field B, but for analytical conve- Equations (7) has the trivial solution Br = 0, Bφ = 0 αω 0 nience we take the simpler form (2). In the approx- (denoted below as M in the (Bφ, B ) plane) and, if the | | ≈ r imation (used here), B Bφ, and the differences magnetic-field generation is strong enough, there are between these forms for α quenching are small. The two nontrivial solutions: 2 function f is usually thought to decay with Bφ , to pro- () () ± ± kr ± ± 1 kr vide negative feedback on the magnetic-field genera- Bφ ==----------() , Br ----------- ----------() , tion. gr RωG gr If the seed magnetic field is sufficiently smooth and denoted below as M±. Here, k(r) means the nonvanish- remains smooth during the subsequent field evolution, ing positive root of the equation the radial diffusion terms in Eqs. (1) can be neglected, α ()2 yielding 1 +0DG 0 fgBφ = (8) (if the root does not exist, then neither do the nontrivial dBr α ()2 --------- = – Rα 0 fgBφ Bφ – Br, solutions; if several roots exist, we mean the smallest dt (3) one). dBφ ± ± --------- = RωGB – Bφ. If the solutions Bφ , B exist, then they are stable dt r r 0 0 and the solution Bφ , B is unstable. If, as usual, G < 0, Note that r can be considered a parameter in Eqs. (3). r Bφ and Br for the nontrivial equilibrium solution have We begin with the kinematic problem, neglecting opposite signs. nonlinear terms and looking for a separable solution of γt γt For the supercritical dynamo, direct calculation the form B (r, t) = b (r)e , Bφ = be . This eigenvalue r r shows that M0 is a saddle point, and M± are stable focal problem has the solution |α | |α | ≤ points if 0DG > 2 or stable nodal points if 0DG 2. γ ()r ==– 1 + –,Dα G br() γ ()r , Thus, the solution for the field tends to a stable point 0 (4) through oscillatory states if the field generation is () α br r = Rα 0. strong enough and monotonically if the generation is weaker. The excitation conditions are G < 0 and D > D = cr We present in the figure the phase portrait for Eqs. (3) α Ð1 min( 0G) , which is in qualitative agreement with the for the oscillatory case. The curves L , L represent the corresponding result for the complete mean-field equa- 1 2 ≈ − α 2 1 tions, namely Dcr 8 (e.g., [7]). Phillips [8] established functions Br = Rα 0f(gBφ )Bφ, Br = ----------- Bφ, respec- a close correspondence between no-z and asymptotic RωG solutions, and suggested a correction factor to be tively. The trajectories of system (3) are vertical when inserted in the no-z formulation that largely removes crossing the curve L2 and horizontal when crossing the the quantitative disagreement. If we insert this factor curve L1. Some typical trajectories are denoted T1 and T2. ASTRONOMY REPORTS Vol. 45 No. 7 2001 MAGNETIC FRONTS IN GALAXIES 499 τ λ The trajectories in the “southeast” (“northwest”) half- Here, = (r Ð r0)/ is a stretched variable and r0 is a sta- plane are attracted to M1 (M2). The separatrix S is given by ble reversal point (terms up to order λ are presented). 2 The smooth part of expansion (14) approximates the dB Rαα gBφ Bφ + B | | λ0 ---------r = ----------------------------------------0 r , (9) solution where r Ð r0 ~ , while the boundary func- dBφ – RωGBr + Bφ tions are introduced to make the expansion valid when ≈ r r0. The boundary functions must decrease rapidly with the initial conditions B = 0, Bφ = 0 (more accu- r (usually exponentially). rately, B 0 when Bφ 0). r Substituting (14) into (12) and equating terms of The solution of (9) for small B and Bφ—i.e., for a r order λ0, we obtain u = v = ±1, or u = v = 0. For weak seed field—is given by 0 0 0 0 a front solution, we set u = v = –1 for r < r and α 0 0 0 Rα 0 u = v = 1 for r > r .
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