Magnetoacoustic solitary waves in a pressure anisotropic high-β plasma

Physics Letters A 366 (2007) 105–109 www.elsevier.com/locate/pla

Magnetoacoustic solitary waves in a pressure anisotropic high-β plasma Bryan Simon a,∗ , George Rowlands b a Centre for Fusion, Space and Astrophysics, Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom b Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom

Received 10 November 2006; received in revised form 10 January 2007; accepted 5 February 2007 Available online 14 February 2007 Communicated by F. Porcelli

Abstract Oblique stationary wave solutions of one-dimensional Hall-MHD with explicit pressure anisotropy are investigated using asymptotic analysis, and the results are compared to numerics. An analytic condition is found the existence of solitary waves. Pressure anisotropy is found to be a crucial for their existence. © 2007 Elsevier B.V. All rights reserved. PACS: 52.35.Bj; 52.35.Mw Keywords: Nonlinear waves; Hall magnetohydrodynamics; Magnetohydrodynamic waves

1. Introduction Magnetoacoustic solitary waves have recently been identified as a possible candidate to explain magnetic holes (MHs), which are localised depressions of the magnetic field. MHs are found in a variety of space plasma environments such as the interplanetary magnetic field [1,2]; the planetary magnetosheaths of Earth [3,4], Jupiter [5] and Saturn [6]; in the region around comets [7]; and the heliosheath [8]. Although MHs are often associated with the mirror-mode instability, a large number of holes are also found to have mirror stable physical conditions. Recent work by Baumgärtel [9] suggests that the MHs may be associated with solitary solutions of the Derivative Nonlinear Schrödinger (DNLS) equation. The DNLS equation is derived from Hall-MHD [10–12]; however, the applicability to the plasma conditions in the vicinity of MHs is at least contentious [13]. Another model, based on fully-nonlinear stationary wave solutions of Hall-MHD was proposed by McKenzie and Doyle [14], and expanded on by Stasiewicz [15]. The model proposed is one-dimensional and considers an oblique background * Corresponding author.

E-mail address: [email protected] (B. Simon). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.02.026

magnetic field. It also includes an explicit pressure anisotropy dependence, and can be used to model a variety of wave structures, including MHs. In this Letter, we investigate the effect of pressure anisotropy on the possible existence and structure of solitary waves which are possible solutions to the model proposed by Stasiewicz [15]. These solitary waves could possibly be candidates for magnetic holes. 2. Governing equations The study is based on Hall-MHD, in the form given by Krall and Trivelpiece [16]. We cast the governing momentum and Ohm’s law equations as N mi

d v = J × B − ∇ · P, dt

1    J × B = E + v × B. eN

(1) (2)

 E Here, N is the number density, mi the ion mass, v and B,  and J are, respectively, the magnetic field and the electric field and current density, whilst P = p⊥ δii + (p − p⊥ )bˆi bˆj is the pressure tensor, where bˆi are the components of the unit vector  of the magnetic field B/B [15].

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We assume a general relation for the perpendicular pressure given by [15] p⊥ = p⊥0 (N/N0 )γ (B/B0 )κ

(3)

and introduce a pressure anisotropy parameter given by ap = p /p⊥ − 1.

(4)

We consider oblique perturbations of the MHD parameters in the form of finite amplitude plane waves. We move into a stationary wave frame moving at an angle α to the ambient magnetic field B0 = B0 (cos α, 0, sin α) with a constant velocity (Vx0 , 0, 0). Here, β = 2μ0 p⊥0 /B02 is the plasma beta, M  A = Vx0 /VA is the Alfvén Mach number, where VA =  0 B 2 /μ0 N0 mi is the Alfvén speed. Furthermore, b = B/B 0

is the dimensionless absolute value of the field. In this frame of reference, the continuity equation ∂ρ + ∇ · (ρ v) = 0 ∂t can be expressed as Vx N = Vx0 N0 .

2     2M¯ A2 bx0 2 +ε (1 − n)b2 + b2 b2 − λ = 0. (10) βnγ b2 + bx0 n 2 ) and b2 = b2 + b2 + b2 . Eq. (10) Here, λ = 1 + β(1 + ap bx0 y z x0 can be cast as a quadratic equation in b2 . One writes   2  2 2 2M¯ A2 bx0 2 b − b2 λ − βnγ − ε = 0. (1 − n) + βnγ ap bx0 n (11) Now, to O(1), we can ignore terms O(ε), and thus

βnγ = (5)

(6)

The density can be normalised as n = N/N0 . Furthermore, one can eliminate the velocity using Eq. (6): Vx /Vx0 = n−1 . One can write the normalised x component of the momentum equation as [15]     1 2MA2 − 1 + β nγ b κ − 1 + b 2 − 1 n   2 + bx0 ap β nγ bκ−2 − 1 = 0. (7) The transverse components of the generalised Ohm’s law (2) [15] can also be rewritten as    λi ∂by β n = bz0 n − 2 1 − ap Mx ∂x 2 Mx    β n − bz 1 − 2 1 − ap nγ bκ−2 (8) 2 Mx and

   λi ∂bz β γ κ−2 n = by 1 − 2 1 − ap n b . Mx ∂x 2 Mx

and ε is a small parameter. All other physical quantities have been normalised O(1). We now treat ε as an expansion parameter. Furthermore, we consider κ = 0; that is, we only consider polytropic plasmas. This restriction is made to simplify mathematical analysis. Then Eq. (7) can be expressed as

(9)

Here, we have introduced Mx = MA / cos α, and the ion inertial length, λi = VA /ωci , where ωci is the ion cyclotron frequency. Note that x is the spatial coordinate in the stationary wave frame. These three Eqs. (7), (8) and (9), as derived by Stasiewicz [15], form a complete set of equations for the field variables n, by and bz , containing the dimensionless parameters MA , ap , β, α, γ and κ. 3. An approximate solution for the density In the following, we look for a particular class of analytical solution of Eqs. (7)–(9). We consider plasma structures moving slowly with respect to the plasma wave frame; that is, MA is small and write MA2 = ε M¯ A2 , where M¯ A2 is a constant O(1)

(λ − b2 )b2 . 2 a + b2 bx0 p

(12)

We are searching for solitary wave solutions; and as part of this process we consider stationary solutions of the system of equations. From Eq. (9), it is clear that by = 0 is a condition for the equilibrium solution. Since we are considering a onedimensional problem in the x-direction, ∇ · B = 0 manifests 2 + b2 . itself as bx = constant. Hence, b2 = bx0 z Thus, one can rewrite Eq. (12) as βnγ =

2 ) (λ¯ − b¯ 2 )(b¯ 2 + bx0 , b2 (1 + ap ) + b¯ 2

(13)

x0

2 , and b¯ = b . Thus, n → 0 as b¯ 2 → λ ¯. where λ¯ = λ − bx0 z When making this approximation, we need to consider in detail what happens as n → 0, i.e. as b2 → λ, as the denominator of Eq. (12) is positive in almost all physical situations. Since 2 a + b2 = b2 (ap + 1) + b2 + b2 , condition ap  −1 and bx0 p y z x0 would break down in the absence of p and any transverse magnetic field perturbations, a rather exotic case. This has to be done, since it is not clear a priori that the ε term stays small compared to the other terms in this case. We write Eq. (10) in the form b2 = λ − δ; it can be expressed as  2 2  2M¯ A2 bx0 2 γ (λ + ap bx0 ) b =λ− ε (14) + βn . n λ 2 ) − [(λ ¯ − b2 ) − δ] = b¯ 2 − (λ¯ − δ). Now b2 − (λ − δ) = (b¯ 2 + bx0 x0 Thus, from Eq. (14), 2 2  2M¯ A2 bx0 d  ¯2 γ −1 (λ + ap bx0 ) − βγ n b − λ¯ = ε . dn λ n2

(15)

¯ deFrom Eq. (14), it is clear that |b2 − λ|, and therefore |b¯ 2 − λ| creases as n decreases to a minimum density. For any n smaller than this minimum, the difference starts increasing again and tends to −∞. Thus this expression also has a maximum, i.e. the right-hand side of Eq. (15) is zero, where nγ +1 = ε

2 λ 2M¯ A2 bx0 2 )β γ (λ + ap bx0

≈ε

2 2M¯ A2 bx0

1 γβ

(16)

and this gives a lower limit on the density below which we cannot exclude the ε term. Hence, apart from solutions where

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2 /(γβ)]1/1+γ , we can, to O(1), ignore the ε term n  [ε2M¯ A2 bx0 in Eq. (10). Then Eq. (12) gives a good O(1) approximation of the relationship between the density and magnetic field. In the next section we will show that this approximation gives good quantitative results when compared to numerical approximations of the full set of equations with small MA .

4. Location of fixed points Note that Eqs. (8) and (9) can be written exactly as ∂by = f (b) − bz g(b), ∂s ∂bz = by g(b), ∂s

(17) (18)

2 + b2 + b2 , where b2 = bx0 y z

 2  bx0 ap β 1− n, f (b) = bz0 1 − ε 2   b2 n ap β nγ g(b) = 1 − x0 1 − , ε 2 b2 

(19) (20)

and n is a known function of b2 only and given by Eq. (10). We have introduced a dimensionless variable s = Mλix x. The above constitutes a set of two coupled non-linear first order differential equations whose solutions can be studied qualitatively using phase space methods. For solitary wave solutions to exist it is necessary that fixed points of the above equations exist and further one must have at least a centre fixed point and a saddle point. Thus we now study the candidates for the existence of fixed points and whether they are centre or saddle points. ∂ = 0 at the fixed points, Eq. (17) leads to anNow, since ∂s other condition, namely ¯ f (b) = bg(b). One can simplify Eq. (19) by writing    ap β bz0 n 2 ¯ A2 − bx0 f (b) = ε M 1 − 2 ε M¯ A2   2 n bz0 bx0 ap β −1 , ≈ 2 ε M¯ A2

(21)

(22)

where we have used the fact that ε bx0 (1 − ap β/2). Since bx0 = cos α, this breaks down as α → 90◦ . In practise, this means that our approach is valid for α  85◦ . To simplify Eq. (20), we keep away from n → 0, then  2 n bx0 ap β n γ −1 . g(b) ≈ (23) 2 b2 ε M¯ A2 Thus, Eq. (21) can be expressed as     ap β n γ ap β ¯ bz0 −1 . −1 =b 2 + b¯ 2 2 2 bx0 Using Eq. (13), this can be rewritten as

(24)

Fig. 1. Comparison between the approximate theoretical fixed points of bz against ap and the numerical results. 3 denotes a numerical solution of the full Hall-MHD system described by Eqs. (7)–(9), and the straight black lines indicate the solution of the approximation given by Eq. (26), which uses the O(1) approximation for the density discussed in Section 3. There is good quantitative agreement between the theory and the numerics in the range of the validity of the theory where n ∼ O(1). Here β = 2, γ = 1.6 and MA = 0.1 in the numerical solution.

 

a β 2 2 ¯b − bz0 1 − p (1 + ap ) b¯ + bx0 2 ¯ 

 ap b 2 2 = (25) − bx0 − b¯ 2 . 1 + β 1 + ap bx0 2 This is a quadratic in ap , i.e. Eq. (25) can be expressed as



A0 ap2 + A1 ap + A2 = 0,

(26)

with A0 =

β 2 b (bz0 − bz ), 2 x0

 bz0 β  2 2 bz + bx0 2   2 bz  2 , 1 + β − bz + bx0 − 2   2 2 A2 = bz + bx0 (bz − bz0 ).

(27)

2 A1 = bx0 (bz − bz0 ) +

(28) (29)

The system has a trivial solution bz = bz0 . Solving Eq. (26) allows us to determine the positions and nature of the fixed points as function of the parameters bz and ap . In particular, we can calculate the possible bz at fixed points for different pressure anisotropies. We can also calculate the density by using Eq. (12). Thus, we are able to infer plasma conditions at fixed points using this simple approximation. Thus, Eq. (26) is an approximation to the general system defined by Eqs. (7)–(9). The advantage of Eq. (26) is that it is a quadratic equation, readily solvable analytically. The equation links the value of the magnetic field perturbation in the stationary wave frame with the local anisotropy of the medium. Fig. 1 compares the locations of fixed points obtained by solving Eq. (26) with those obtained from a direct numerical calculation using a two-dimensional Newton–Raphson method to solve Eqs. (7) and (21) when by = 0 and κ = 0. We restrict our analysis to values of −1/3 < ap < 1/2 (β = 2) as for values of ap obtained outside this range the plasma is prone to mirror or fire-hose instabilities. There is very good quantitative agreement between the theoretical approach and the numerics in the

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limit of applicability of the theory, i.e. where the density is of order unity. This justifies neglecting terms of order ε and less from the original equations. We have chosen a fairly oblique propagation angle. We note that the behaviour is different for negative and positive anisotropy, suggesting that the local pressure conditions affect the possibility of the existence of solitary waves. 5. Classification of fixed points We consider the eigenvalues around the stationary points to calculate the nature of the fixed point. To do this,  we note that at a fixed point, bx = bx0 and by = 0. If bT = b2 + b¯ 2 is the x0

total magnetic field at a fixed point, then for a small perturbation around the fixed point, one has 2 2 ¯ z + δby2 + (b¯ + δbz )2 ≈ bx0 + b¯ 2 + 2bδb b2 = bx0   ¯ z 2bδb ≈ bT2 1 + 2 bT

(30)

for linear perturbations. Thus, b ≈ bT +

b¯ δbz bT

near the fixed point. Now, using Taylor’s theorem,   b¯ b¯ f (b) ≈ f bT + δbz ≈ f (bT ) + f (bT ) δbz bT bT

(31)

(32)

and

  b¯ b¯ g(b) ≈ g bT + δbz ≈ g(bT ) + g (bT ) δbz . bT bT

(33)

¯ T ), we can consider Noting that at a fixed point, f (bT ) = bg(b the evolution of small perturbations using the approximate expansions of f and g given in Eqs. (32) and (33). We see that   b¯ 2 d

− g(bT ) δbz δby = f (bT ) − g (bT ) (34) ds bT and d (35) δby = g(bT )δbz , ds where we have discarded nonlinear terms in the perturbations. Hence, the evolution of perturbations in magnetic field around the fixed point can be written as      ¯2 d δby δby 0 f (bT ) − g (bT ) bbT − g(bT ) = . δbz 0 g(bT ) ds δbz (36) The eigenvalue equation yields two types of stationary points: centres, where both eigenvalues are imaginary and periodic oscillatory solutions exist in the phase space of these points, and saddle points, whose separatricies allow solitary wave solutions. Thus the condition

2 b¯ b¯ 2 − g(bT )g (bT ) ≶0 − g(bT ) + f (bT )g(bT ) bT bT

(37)

Fig. 2. The computational solutions as indicated by 3 for the fixed points as given by Eqs. (7)–(9) as shown in Fig. 1 and their respective topology classified according to Eq. (37). Here, α = 75◦ , β = 2, and γ = 1.6. The blue shaded areas correspond to the location of saddle points, which admit solitary wave solutions, and the unshaded regions correspond to the location of centres, which allow periodic wave solutions around them. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this Letter.)

 saddle points, respectively. corresponds to centre Fig. 2 illustrates the classification of the theoretical fixed points as saddle and centre points, respectively. The 3 correspond to the fixed points according to Eqs. (7)–(9) overlaid onto a coloured background. The white shading corresponds to areas which would correspond to centre points, and the dark areas correspond to possible locations of saddle points, admitting solitary wave solutions. The results indicate that in the parameter range considered, the equilibrium solution bz0 is always a centre, admitting periodic solutions. If we increase the anisotropy beyond the range considered here, then the bz0 solution does change from a centre to a saddle and allow solitary solutions. However, in these regions, the plasma is not stable to the microinstabilities caused by the anisotropy (the mirror or fire-hose instability), so it is not clear that these solutions themselves would remain stable. Fig. 2 suggests the existence of solitary waves at large bz solutions which are independent of anisotropy. Whilst these solutions certainly exist, they necessarily have a low density. This means that our theory is not necessarily valid for these solutions. Furthermore, we are not interested in these low density solutions, as they do not fit the phenomenology of magnetic holes. We are primarily interested in solutions where n ∼ O(1) and hence restrict attention to the region |bz |  1, and note that anisotropy is crucial for the existence of saddle points and hence for the existence of solitary wave solutions. We also note that there is a qualitative difference between the phase space portraits for positive and negative anisotropy, as illustrated in Fig. 3. For positive anisotropy, we have solitary solutions with a negative bz configuration, where the local bz has undergone a reversal of direction with respect to the equilibrium field, whilst for negative anisotropy a solitary solution fixed-point is sandwiched between two centre solutions allowing periodic waves.

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Fig. 3. Sketch solutions showing typical phase space portraits for negative (left) and positive (right) anisotropy. There is a qualitative difference in the phase space: for negative anisotropy a solitary wave fixed point is sandwiched between two centre solutions allowing periodic waves, whereas for positive anisotropy has a solitary wave fixed point below two centre solutions. Here, α = 75◦ , β = 2, γ = 1.6 and the anisotropy parameter is ap = −0.3 (left) ad ap = 0.5 (right).

6. Conclusions In this Letter, we present simple analytic approximations to the one-dimensional Hall-MHD equations to find stationary propagating solutions of density and magnetic field in a high-β plasma. We perform an ε expansion in the Alfvén Mach. Assuming slow propagation is physically relevant in space plasma physics, there is Cluster data with MA ∼ 0.1–0.3 [15]. We have obtained simple analytic formulae which give the regions of parameter space (e.g., pressure anisotropy, propagation direction) where solitary pulse solutions can exist. The analysis is appropriate for plasma structures which are moving with a low Alfvén Mach. A quantitative condition is obtained, Eq. (16), which gives the range of validity of the present work. This work extends earlier work by Stasiewicz [15], who searched for solitons around the bz ∼ 1 equilibrium. In the parameter range considered by this study, Stasiewicz [15] finds no solitary waves. Our work corroborates this: Figs. 2 and 3 show that the equilibrium at bz ∼ 1 is indeed a centre for all ranges of anisotropy parameter, unable to support solitary waves. However, investigation of the full magnetic field topology in the low Alfvén Mach propagation limit shows that additional equilibrium solutions exist for pressure anisotropic plasmas, which have not been considered before. We find that one of these additional solutions is a centre; wave solutions around this point are periodic in nature. However, the other additional solution is a saddle. This equilibrium allows solitary wave solutions in parameter space previously considered to only allow periodic solutions. We find that pressure anisotropy is crucial for the existence of solitary waves propagating at a oblique angles  85◦ in the

density domain where our approximation is valid. Furthermore, positive and negative anisotropy, corresponding to increased parallel and perpendicular pressure respectively, have different fixed point topologies, and hence distinct physical solutions. At this stage of the study, we can say nothing about the stability of these different solutions; further numerical and analytical stability analysis is required to assess the physical applicability to these results. Acknowledgements Bryan Simon would like to acknowledge a PPARC CASE studentship in partnership with UKAEA Culham and useful discussions with V.M. Nakariakov. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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