c2 Sc 2 A c 2 T = c2 A

Transcription

1 Physics of Plasmas, in press Solitary perturbations of magnetic flux tubes M. Stix 1, Y. D. Zhugzhda 1,2, and R. Schlichenmaier 1 1 Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, Freiburg, Germany 2 IZMIRAN, Troitsk, Moscow Region, , Russia Abstract The thin-tube expansion, in the two-mode approximation [Y.D. Zhugzhda, Phys. Plasmas 3, 10 (1996) and 9, 971 (2002)], is applied to magnetic flux tubes. The propagation of finite-amplitude perturbations is studied numerically. A behavior is found that closely resembles solitary waves. In the two-mode approximation the slow body wave possesses the dispersion that is needed to prevent shock formation. There is qualitative agreement with an analytic treatment of weakly non-linear slow body waves [Y.D. Zhugzhda, Phys. Plasmas 11, 2256 (2004)], where the problem was reduced to a new model equation. I. Introduction Magnetic flux concentrations, or flux tubes, which occur in stellar atmospheres (see, for example, Ref [1]), have been described by an expansion in terms of the distance from the tube axis [2]. In the leading-order this expansion supports so-called tube waves [3] that have no dispersion and propagate with the tube speed, c T, defined through c2 Sc 2 A c 2 T = c 2 S +, (1) c2 A where ( ) dp c 2 S =, c 2 A = B2 (2) dρ µρ S are the squared sound and Alfvén speeds, respectively. As these tube waves are the only wave mode that is supported in the leading order, we shall also call this scheme the one-mode approximation. The tube waves appear as a thin-tube limit of slow body or surface waves, depending on the parameters of the external plasma [2, 4, 5]. The one-mode approximation does not take into account the dispersion, which is a crucial shortcoming since all wave modes that occur in real in flux tubes are dispersive. For weakly non-linear waves this drawback was removed by adding the dispersive term derived from the exact dispersion relation to the Leibovich Roberts equation [5, 6, 7]. However, so far all numerical treatments of the non-linear dynamics of flux tubes based on the one-mode approximation have ignored the wave dispersion [8, 9, 10, 11]. If, on the other hand, second-order terms of the expansion are retained, such as in the study of Ferriz Mas and Schüssler [12], then the wave propagation becomes dispersive. Zhugzhda [13, 14] has called this second-order truncation the two-mode approximation, because of the occurrence of a second time derivative and, therefore, of a second (fast) body wave. In Ref. [15] a scheme has been described for the small-amplitude body waves that occur at arbitrary levels of truncation. The two-mode approximation takes into account the geometrical dispersion which is due to the finite tube diameter. This enables a direct derivation of equations for weakly non-linear waves. For the special case of long wavelength (compared to the tube radius) Zhugzhda and Nakariakov [16, 17] were able to reduce 1

2 the non-linear wave equation to a Korteweg de Vries equation, and to explore their solitary and periodic solutions. In Ref. [18] an approximate form of the dispersion has been consideredthat allows to abandon the long-wavelength approximation and to obtain an evolutionary equation for non-linear body waves at arbitrary wavelength. This is a new kind of evolutionary equation; it will be called the Zhugzhda equation in the following. In distinction to well-known evolutionary equations, for example the Korteweg de Vries equation, both dark and bright solitary waves, i. e. perturbations of either sign, obey that equation. On the other hand, the Zhugzhda equation is not completely integrable, and its solitary solutions are therefore not solitons [19]. In the present investigation a numerical scheme is developed for the exact non-linear two-mode equations. The goal is to compare the theory of weakly non-linear body waves with the numerical solutions of the exact two-mode equations. Just as the one-mode approximation, the two-mode approximation ignores the dispersion due to the reaction of the external medium. But the latter is essential only in the long-wavelength regime; for short wavelengths the geometrical dispersion dominates [18]. II. Equations The equations which govern the two-mode approximation are well-known; they are repeated here for completeness. We consider a straight tube with no gravitational stratification. Let z be the coordinate along the tube, r the distance from the tube axis, and R the tube radius; let a prime denote a derivative with respect to z and a dot a time derivative. Axisymmetric variations of the tube then obey (e. g., [13, 14]) ρ 0 = 2ρ 0v r1 (ρ 0v z0), (3) ρ 0 v r1 + ρ 0v z0v r1 + ρ 0v 2 r1 = 2Π µ Bz0B r1, (4) ρ 0 v z0 + ρ 0v z0v z0 = P 0, (5) Ḃ z0 = 2(v z0b r1 v r1b z0), (6) ρ 0 P 0 γp 0 ρ 0 + (ρ 0P 0 γρ 0P 0)v z0 = 0, (7) B z0 + 2B r1 = 0. (8) P ( 2µ B2 z0 + R 2 Π ) 2µ B2 r1 = Π b, (9) πr 2 B z0 = Φ. (10) Equations (3) (10) are 8 equations for the 8 unknown variables P 0, ρ 0, v z0, v r1, B z0, B r1, R, and Π 2. The last of these variables is defined as Π 2 = P Bz0Bz2. (11) µ The subscripts denote the vector components and the order of the diverse coefficients within the thin-tube expansion, e. g., v = [v z0(z, t) + v z2(z, t) r 2, v r1(z, t) r]. In the present study the total external pressure Π b is considered as a known constant. The coupling of the external medium to the perturbation of the tube is neglected. Equation (9) is the condition of total pressure continuity at r = R, and (10) is the condition of constant magnetic flux. Equation (7) expresses energy conservation for adiabatic variations. The two-mode approximation is distinguished from the leading-order approximation in that equations (4) and (10) are added, and that (9) includes an additional term of order R 2. The additional time derivative in (4) allows for an additional mode of oscillation, which is a fast body wave as shown in Refs. [4] and [14]. In equations (3) (10) the non-linearity of the original magnetohydrodynamic equations has been retained. These equations have been used to study the propagation of tube waves in the non-linear regime: In the leading-order, and for the long-wavelength limit, Roberts [6] derived an equation that is related to the Benjamin Ono equation (which describes the case of a magnetic slab), while Zhugzhda & Nakariakov [16, 17, 20], using the two-mode approximation, considered cases which lead to a Korteweg de Vries equation. In Ref. [18] the study of Roberts has been extended to the case of an arbitrary wavelength. 2

3 III. Solitary perturbations A. Analytical approximation From the above equations an evolutionary equation for the (finite-amplitude) perturbation of the zero-order tube field B z0 (hereafter B 0) has been derived in Ref. [18]. In a frame that moves with velocity c T along the tube, this equation has two solutions of the form where and B 0(z, t) = B asech 2( (z V t)/l ), (12) V = ɛba 3, L = V c S + c T L, (13) V ɛ = ct B 0 3β + γ + 1 2β(β + 1), L = β = c2 S c 2 A (14) R0 β β + 1 2( β + 1 1). (15) The parameter β is γ/2 times the plasma beta. The half-width L of the solitary wave tends to L as the absolute amplitude B a increases. The radius R 0 is the equilibrium radius of the tube. The amplitude B a can have positive or negative values. The former case has been called the dark, the latter case the bright solitary wave. In the present context these names are appropriate because, as we shall see, the dark solitary wave has a negative pressure perturbation, and for an adiabatic variation of state this entails a negative temperature perturbation; the bright solitary wave has pressure and temperature perturbations of the opposite sign. Since the dark wave goes with a local contraction of the magnetic tube, we shall call this type of perturbation the traveling tube contraction; a perturbation that, like a bright solitary wave, has a local tube widening, will be called the traveling tube expansion. The solutions of the Zhugzhda equation are similar to the solutions of the Korteweg de Vries equation. In fact, the Korteweg de Vries equation has only one family of solutions; the family consists either of dark or of bright solitons, depending on the relative signs of the non-linear and dispersive terms. In the laboratory frame the solitary waves travel with velocity c T + V. Thus, the dark perturbation is slower than the tube speed, the bright perturbation is faster than the tube speed. The requirement of a real half-width L poses no formal restriction to the dark solitary wave, if the effect of pressure collapse is ignored, because V < 0 and, as always, c S > c T. On the other hand, there is a minimum amplitude B a for which a bright perturbation of form (12) can exist. The necessary condition is V > c S c T, or B a > 3(cS ct) ɛ. (16) B. Numerical scheme For the numerical examples presented in this contribution we use the program described in Ref. [11]. With this code it is possible to treat moving magnetic flux tubes. We do not utilize this possibility here; we also switch off the gravitational effects by setting g = 0. On the other hand, we have extended the code to the two-mode approximation; in the original form only the leading terms of the thin-tube expansion had been retained. Thus, the equations solved numerically are those described in Sect. II. The full set of the moving-tube equations in the two-mode approximation, including stratification and non-adiabatic effects, will be presented elsewhere. For the actual calculations we use reflecting boundary conditions at both ends of the tube. This is however of no concern for the results reported here because we consider the traveling perturbations only for the period before the first reflection. The numerical method used is a two-step predictor-corrector method as described in Ref. [21]. For our example tube we choose parameter values that, by order of magnitude, resemble the situation in the atmosphere of a star like the Sun. As initial condition we use a magnetic field perturbation of form (12), with t = 0. The initial perturbation of the longitudinal velocity then follows from the above equations as c 2 A v z(z, 0) = B 0(z, 0). (17) (c T + V )B 0 Since our code is structured such that it starts with a velocity, we use (17) for the actual calculation. From this initial perturbation a compressional signal, with a local tube expansion, travels into the z direction, and a rarefaction signal, with a tube contraction, travels into the +z direction. Each of these signals have one-half of the original amplitude of (17). 3

4 FIG. 1. A traveling tube contraction: Longitudinal velocity and magnetic field, pressure, and tube radius. Scale along the tube is in 10 3 km. The signal starts at the left edge, and the ensuing perturbations are shown after 8 (grey) and 16 (black) minutes. Each + marks a node of the computational grid. FIG. 2. Above: Wave packets of small amplitude: One mode, no dispersion (left), and two modes, with dispersion (right). Below: Large-amplitude wave packets: One mode, with shock formation (left), and two modes, with behaviour like a solitary wave (right). In all cases the magnetic signal starts at the left edge and is shown after 8 min (grey) and 16 min (black). 4

5 FIG. 3. Half-width L and velocity V, in a frame moving with c T, of a dark solitary wave (solid) and of the corresponding numerically calculated perturbation (dashed, after 16 min), as functions of the magnetic field amplitude. C. Traveling tube contraction For the traveling tube contraction we find a behaviour that indeed closely resembles a solitary wave, and for small (but still finite) amplitude even quantitatively agrees with the analytical result as given in Ref. [18]. Figure 1 shows a typical example. The input parameters of this example are: Total flux Φ = Mx, equilibrium density and temperature ρ 0 = gcm 3 and T 0 = 10 4 K, external pressure Π b = dyn. Other parameters are derived from this input, such as the equilibrium tube pressure, field, and radius, viz. P 0 = dyn, B 0 = G, R 0 = km, the tube speed c T = km/s and the dimensionless parameter β = Consistently with (17), the velocity perturbation is negative and, as already mentioned, the pressure and radius perturbations are also negative. In order to demonstrate the soliton-like behaviour of this example we show in Fig. 2 the same perturbation at small amplitude, and in the leading-order approximation of the thin-tube expansion, where only one mode exists. For a better comparison these small-amplitude perturbations are initiated not with the width according to (13), but with the same width as in the case of the large amplitude. In the one-mode approximation, at small amplitude, there is no dispersion, and the perturbation travels exactly with the tube speed, with no change of shape. At large amplitude (the same as in Fig. 1) a shock forms as a consequence of the non-linearity. The shock occurs at the trailing edge of the perturbation as a rarefaction shock, because v z is opposite to the direction of travel. In the two-mode approximation, on the other hand, dispersion occurs, visible already at small amplitude; this dispersion tends to advance the peak of the perturbation, as opposed to the backwards inclined peak of the one-mode, large-amplitude case. Thus, as in a real soliton, dispersion and non-linearity act against each other. Close inspection of Fig. 1 shows that the shape of our traveling perturbation is not strictly invariant. Nevertheless, at moderate amplitude, we find that the half-width L and the velocity V in the co-moving frame agree reasonably well with the predictions (13) above. Figure 3 shows these two parameters in comparison to those of the numerical calculation. For the latter, the velocity is the distance of the peak from the initial position, divided by the elapsed time; the half-width is measured at 42 % of the peak amplitude. In both cases, analytical and numerical, the half-width decreases with increasing amplitude. The velocity V becomes increasingly negative; in the laboratory frame, therefore, the perturbation lags the more behind the tube speed the larger its amplitude. D. Traveling tube expansion The analog to the bright solitary wave is a perturbation with negative magnetic field amplitude and, according to (17), positive longitudinal velocity. The perturbations of pressure and radius are also positive. The traveling tube perturbations behave in a similar way for a wide range of tube parameters. Figures 4 and 5 show the velocity and magnetic field of an example with Φ = Mx, ρ 0 = gcm 3, and otherwise as in the example above. These values were chosen in order to make β small, for a better agreement with the dispersion relation derived in Ref. [18]; here we have β = 0.247, and the tube speed is c T = km/s. This example has been calculated with L = 200 km in the initial perturbation. Although it is evident that the shape of the perturbation is not conserved and that, therefore, we should not speak of a solitary wave, there are several common properties: The perturbation travels faster than the tube speed; the travel velocity increases with increasing amplitude; and again the dispersion introduced by the twomode approximation counteracts the tendency of the one-mode approximation to form a shock. This time the shock is compressional because the longitudinal velocity is in the direction of the traveling perturbation. As the 5

6 FIG. 4. Velocity and magnetic field of a traveling tube expansion, calculated in the one-mode approximation. Horizontal scale in 10 3 km. The signal starts at the left edge; the ensuing perturbations are shown after 6 (grey) and 12 (black) minutes. FIG. 5. Velocity and magnetic field of a traveling tube expansion, calculated in the two-mode approximation. Otherwise as Fig. 4. figures illustrate, the amplitude of the traveling expansion gradually decreases for the two-mode example, while it increases towards the shock in the one-mode example. Quantitatively, there is less agreement with the bright solitary wave than we found for the traveling contraction and the dark solitary wave. In particular, we found no evidence for a critical amplitude like (16) beyond which soliton-like behaviour would occur. IV. Discussion There are several reasons why the numerical results presented in this contribution should deviate from the solitary waves found in Ref. [18]. The first is that the linear part of the Korteweg de Vries equation derived by Zhugzhda rests on a simplified form of the dispersion relation. In the two-mode approximation the phase speed V of smallamplitude body waves can be calculated in dimensionless form, Ω = V ph /c T, through Ω 2 = ε ± ε 2 2ε. (18) The dimensionless phase speed depends on a single parameter (see Ref. [15]), ε = c2 S + c 2 A 2c 2 T (1 + 4 α 2 ) = (β + 1)2 2β (1 + 4 ), (19) α 2 where α = kr 0 is a dimensionless wave number. For each k there are two modes, a slow and a fast body wave. Only the slow mode is present in the approximate dispersion relation used in Ref. [18]. Figure 6 compares this dispersion with the exact dispersion of the slow mode derived form (18). For small β there is good agreement, but less so for larger values of β. The second reason of disagreement is the different non-linearity. While the Zhugzhda equation has a non-linear term derived from an expansion in terms of an amplitude parameter (see Ref. [16]), we use here the non-linearity of the original magneto-hydrodynamic equations. Probably this difference is more severe than the difference in the linear terms, in particular at small β. In the above example of a traveling tube expansion we had β = and good agreement of the dispersion, so the deviating behaviour of the numerical solution should rather be caused by the non-linear term. 6

7 A third reason of disagreement between the analytical solitary solutions and the numerical results is the absence of the fast body mode in the Zhugzhda equation. This equation is of first order in the time derivative. The essence of the two-mode approximation is not only that it renders the slow tube mode dispersive, but also that it introduces a second wave mode since it is of second order in the time derivative. This is the reason for its name. The presence of the fast mode is manifest in the wiggles that accompany the traveling perturbations shown in this paper. Thus, it comes as no surprise that the half-width and velocity of the dark perturbations obtained from the numerical treatment and from the solution of the Zhugzhda equation differ from each other; the latter does not take into account the dependence of the non-linearity on wave number. The reason for the absence of a critical amplitude for bright perturbations is that the critical amplitude of overturning of a bright wave packet is smaller than the critical amplitude for the occurrence of a bright solitary wave. The theory of weakly non-linear waves fails to take into account the effect of overturning it assumes a balance between dispersion and non-linearity and, consequently, permits solitary waves of any amplitude. In reality, the amplitude of solitary waves can not exceed some critical value beyond which the dispersion is no longer able to compensate the non-linearity. It seems, that this critical amplitude is smaller than the critical amplitude for the occurrence of bright solitary waves. This is a consequence of rather weak dispersion for β 1, in which case the difference between tube and sound speed is small. FIG. 6. Dimensionless phase velocity Ω = V ph /c T of the slow body mode, as a function of the dimensionless wave number α = kr 0 and the parameter β = c 2 S/c 2 A, according to the exact dispersion relation (solid) and the approximation (dashed) of Ref. [18]. One might argue that the numerical solutions could asymptotically develop into real solitary waves, and this should be seen if only we would run them for a sufficient period of time. However, in view of the arguments discussed, this hope appears not to be justified. The traveling tube expansion that we have calculated in analogy to the bright solitary wave might be of interest for the flow in magnetic flux tubes in the penumbra of a sunspot. Indeed, there is observational evidence [22] that shocks might exist in penumbral filaments. Schlichenmaier et al. [11] observe the formation of shocks in their numerical models, and ascribe this to the shortcomings of the leading-order truncation. Indeed, during the phase of maximum tube expansion the additional pressure provided by the two-mode approximation the term R 2 in Eq. (9) is negative and so opposes the tendency of shock formation. The examples shown in the present contribution demonstrate that, in addition, the dispersive effect is essential for the mechanism. ACKNOWLEDGMENTS Y. Z. has been supported by DFG Grant No. 436 RUS 17/18/03 and by RFFI Grant No

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