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On the possibility of Alfvén wave resonance in collisionless magnetic reconnection
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ScienceDirect Advances in Space Research 55 (2015) 1278–1284 www.elsevier.com/locate/asr
On the possibility of Alfve´n wave resonance in collisionless magnetic reconnection M. Hosseinpour ⇑ Department of Plasma Physics, Faculty of Physics, University of Tabriz, Tabriz, Iran Received 21 June 2014; received in revised form 17 November 2014; accepted 21 November 2014 Available online 28 November 2014
Abstract Alfve´n wave resonance and magnetic reconnection are among the potential candidates for efficient dissipation of magnetic energy in space and astrophysical plasmas. In this paper, the correspondence between Alfve´n resonance and the electron-inertia driven reconnection in a sheared force-free magnetic field is discussed. By analytical scaling the linear regimes of compressible tearing instability in the two-fluid magnetohydrodynamic (MHD) model, we present parametric conditions for the possibility of Alfve´n resonance existence. Meanwhile, it is argued that the slow MHD Alfve´nic resonance can take place only in the “intermediate” – called Hall-MHD regime when b > l. b is the ratio of plasma thermal pressure to the pressure in equilibrium magnetic field and l is the electron to ion mass ratio. There is no room for such a resonant dissipation phenomenon either in the single-fluid MHD or the electron-MHD regimes. Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Magnetohydrodynamics; Tearing instability; Magnetic reconnection; Alfve´n wave resonance
1. Introduction Among various schools of thoughts concerning the efficient dissipation of magnetic energy in the highly conductive environments of space plasmas there are two main ones: Alfve´n mode resonance which can explain the enhanced dissipation in dynamical processes such as magnetohydrodynamic (MHD) waves (Hasegawa and Chen, 1974; Wang et al., 1998; Erde´lyi and Goossens, 1995; Ruderman et al., 1997), and magnetic reconnection which is usually associated with longer time scales compared with the Alfve´n time (Birn and Priest, 2007; Priest and Forbes, 2000; Yamada et al., 2010; Zweibel and Yamada, 2009). In a collisionless plasma with negligible plasma resistivity, the electron-inertia is a potential means of breaking the frozen-in condition to convert magnetic energy into the kinetic and thermal energy of plasma. The inclusion of electron ⇑ Tel.: +98 413 339 3356.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.asr.2014.11.022 0273-1177/Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved.
inertia, inevitably, requires that the two-fluid MHD description of reconnection is to be considered, in which Hall effect plays an important role by facilitating the pace of magnetic reconnection (Drake et al., 2008; Fitzpatrick and Porcelli, 2004; Mirnov et al., 2004; Hosseinpour et al., 2009; Shay et al., 2001; Terasawa, 1983). In the Hall-MHD reconnection via tearing instability the current sheet acquires a double-layer structure: The inertial region surrounded by a much wider layer, where the electron-inertia plays no role, but the poloidal magnetic field is still advected towards the reconnection region by the Hall effect rather than by the bulk plasma motion. Relative importance of Hall effect and the electron inertia is characterized by the following non-dimensional parameters: d e c=ðxpe lÞ, the scaled electron inertial skin-depth (l is the magnetic shear length scale), for the electron inertia; d i c=ðxpi lÞ ¼ l1=2 d e d e ; l me =mi 1, the scaled ion skin depth, for the Hall effect. c and xp i=e are the speed of light and the ion/electron plasma frequency, respectively.
M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
At sufficiently small values of d i (see below), the Hall effect turns out to be less important, so that, the single-fluid MHD can adequately describe the dynamics of reconnection (“standard”- MHD regime). In contrast, at much larger values of d i (see below), ion component of plasma does not play any role in reconnection dynamics and therefore, the electron-MHD (EMHD) regime dominates. In the intermediate values of d i both electron and ion components determine the dynamics of reconnection, we call it an “intermediate” Hall-MHD regime. Each of these linear regimes of tearing instability are characterized by their respective current sheet width and the instability growth rate. In such a tearing instability, most of the magnetic energy is stored in the low-frequency MHD waves such as Alfve´n waves that can be generated by magnetic reconnection and propagated along the reconnected field lines (Kigure, 2010; Wang, 2002; Ma et al., 1995). One of the proposed mechanisms for dissipation of Alfve´n waves is resonant absorption (Poedts et al., 1989; Ionson, 1978; Hasegawa and Chen, 1974). Wang et al. (2011) has reported that the MHD perturbations due to tearing mode reconnection on the outer rational surface mediate in generating two Alfve´n resonance layers on the both sides of the inner rational surface and then prevent the formation of island growth (tearing mode suppression). However, in the Hall-MHD reconnection with the large guide field different types of waves can be excited such as kinetic or inertial Alfve´n waves. Kinetic Alfve´n waves can propagate at b > l, while inertial Alfve´n waves are excited at b < l (Huang, 2010; Rogers, 2001; Lyask and Lotko, 1996; Hasegawa and Chen, 1975; Hasegawa, 1976; Goertz and Boswell, 1979). These waves determine the electron dynamics in the ion diffusion region. Even, Whistler waves can propagate inside the electron inertial length scale (Wei, 2013; Wang and Luan, 2013). In order to concentrate our discussions on the Alfve´n resonance, we assume that the wave vector of fluctuations has no component along the guide field, thus it leaves no room for the kinetic/inertial Alfve´n waves and only the slow Alfve´nic waves are permitted to propagate regardless of the excitation mechanism for Alfve´n waves. In magnetic reconnection with nonuniform equilibrium magnetic field, the location of resonance layer varies with the Alfve´n frequency (see below). If the location of resonance layer lies within the current sheet formed by magnetic reconnection, then Alfve´n resonance, in fact, cannot occur and therefore, the dynamics of tearing mode is not influenced by the resonance phenomenon. It is of interest to investigate the possibility of occurring the Alfve´n resonance in each of the two-fluid MHD regimes of collisionless reconnection. It is worthy of note that, previously, the relation between Alfve´n resonance and magnetic reconnection has been argued by Uberoi (1994) and Vekstein (2000), but in the case of forced magnetic reconnection. In this type of reconnection the externally sinusoidal perturbation of
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plasma boundaries with a certain frequency leads to the magnetic reconnection at the surface where poloidal field changes its sign. Furthermore, the sinusoidal boundary perturbation can excite Alfve´nic waves. Uberoi (1994) in the absence of Hall effect has interpreted forced magnetic reconnection as the Alfve´n resonance with zero frequency, from which it was deduced in Uberoi and Zweibel (1999) that the theory of forced magnetic reconnection is actually embedded in the Alfve´n resonance theory. On the other hand, Vekstein (2000) discussed that these phenomena are operating separately, and transition from resonant absorption to forced reconnection occurs when the frequency of the external driver becomes sufficiently small. However, in the tearing instability, Alfve´n waves can be generated following the onset of magnetic reconnection and then propagated along the magnetic fields (Kigure, 2010; Wang, 2002; Ma et al., 1995; Sakai et al., 2000; Lazarian and Vishniak, 1998). Here, unlike the forced reconnection, there is no externally driven mechanism. In this study, we first analytically scale both the width of current sheet and the growth rate of collisionless tearing instability at different linear regimes within the two-fluid MHD framework and then investigate the possibility of occurring Alfve´n resonance at each of these regimes. To do so, we compare the location of Alfve´n resonance layer with the width of current sheet. If the resonance layer is located inside the current sheet, then the appearance of Alfve´n resonance is not expected. It should be note that, in this study, the mechanisms which can generate Alfve´n resonance will not be discussed, but only the possibility of Alfve´n resonance existence in the presence of linear tearing instability will be investigated. The structure of paper is as follows: Section 2 gives the description of the model and basic equations. Analytical analysis of collisionless reconnection regimes are included in Section 3 and discussions regarding the possibility of Alfve´n resonance occurrence are presented in Section 4, which is followed by a brief summary and conclusion in Section 5. 2. The model and basic equations A planar slab of a highly conductive uniform plasma [density n0 ] embedded in a sheared force-free magnetic field ð0Þ Bð0Þ ðxÞ ¼ zBð0Þ z ðxÞ þ yBy ðxÞ;
ð1Þ
with Byð0Þ ¼ B0 f ðxÞ;
Bzð0Þ ¼ B0 ½1 f 2 ðxÞ
1=2
ð2Þ
is surrounded by two perfectly conducting walls at x ¼ l. For the tearing perturbation of the form expðikyÞ reconnection occurs at the x ¼ 0, where the poloidal field component Bð0Þ y changes its sign: f ðxÞ ax for jxj l. Therefore, the wave number has only one component which is along the ‘y’ direction. The governing equations defining temporal evolution of perturbations about the initial equilibrium are as follows:
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The equation of bulk plasma motion dV 1 q ¼ ðj BÞ rP; dt c where j ¼
c ðr 4p
Vz ¼ ð3Þ /00 ¼
BÞ; the Faraday equation
@B ¼ cðr EÞ; @t
ð4Þ
and the generalized Ohm’s law that follows from the equation of motion for electrons: 1 me dVe 1 E þ ðVe BÞ ¼ rP e ; c ne e dt
ð5Þ
where the electron component velocity Ve ¼ V j=ne. The inertial term on the right-hand-side (RHS) of (5) is significant only inside the current sheet at x ¼ 0. Thus, the inerme @j e ne tial term can be simplified there as mee dV 2 @t. The dt perturbed magnetic field can be written as Bðx; y; tÞ ¼ rwðx; y; tÞ z þ Bz ðx; y; tÞz;
ð6Þ
where the poloidal flux function takes the form wðx; y; tÞ ¼ w0 ðxÞ þ w1 ðxÞexpðikyÞexpðctÞ
ð7Þ
ð0Þ
(w0 corresponds R to the initial field B given by Eqs. (1) and (2) as w0 ¼ B0 f ðxÞdx), and the toroidal field component Bz ðx; y; tÞ ¼ Bzð0Þ ðxÞ þ Bð1Þ z expðikyÞexpðctÞ:
ð8Þ
The plasma velocity V is represented as V ¼ ðr/ zÞ þ rv þ V z z;
ð9Þ
where the stream-function /ðx; y; tÞ ¼ /ðxÞexpðiky þ ctÞ and the velocity potential vðx; y; tÞ ¼ vðxÞexpðiky þ ctÞ correspond, respectively, to the vortical and compressional components of the poloidal plasma flow. By introducing non-dimensional variables with length normalized by l; V z by cl; / and v by cl2 ; w1 by B0 l and Bzð1Þ by B0 , Eqs. (3)–(5) yield the following set of equations linearized with respect to perturbations valid inside the current sheet around the field reversal plane x ¼ 0. Thus, w1 ¼ xðik/Þ þ ld i 2 w001
ikd i x b; ðcsA Þ
ikd i x 00 b ¼ ikxV z v þ ld i b þ w ; ðcsA Þ 1 00
2 00
ð10Þ ð11Þ
where b Bzð1Þ w1 , and al. The latter parameter indicates strength of the guide field, so that 1 corresponds to the strong guide field approximation. Introduction of the auxiliary perturbation function b allows a straightforward generalization for the case of an arbitrary guide field strength. Thus, in all estimates that follow it is assumed, for simplicity, that 1. A similar assumption is put for the perturbation wave vector, k 1. Note that this makes R 00 the tearing stability factor, D0 ¼ w1 w1 dx, of the order 1 ð0Þ of unity too, which ensures applicability of the constant-w approximation. The Hall effect is represented by the last terms on the RHS of Eqs. (10) and (11). Further, the linearized equation of motion (3) yields:
v00 ¼
ikx ðcsA Þ
2
b;
ð12Þ
2
w001 ;
ð13Þ
ikx ðcsA Þ
d2 ðbv00 bÞ: 2 ðcsA Þ d x2 1
ð14Þ
Here b C 2s =V 2A , where C s and V A are, respectively, the pffiffiffiffiffiffiffiffi sound and Alfve´n velocity (V A ¼ B0 = 4pq). Also, sA ¼ l=V A . Eqs. (10) and (11) are the ‘x’ and ‘z’ components of Eq. (4) respectively, where the curl of the gradient of scalar electron pressure is zero, r rP e ¼ 0. Therefore, there is no term including scalar electron pressure in Eqs. (10) and (11). Also, the linear forms of the adiabatic equation of state, @=@tðP =qC Þ ¼ 0, and the continuity equation, @q=@t þ r ðqVÞ ¼ 0, are @P ð1Þ =@t ½Cpð0Þ =qð0Þ @qð1Þ =@t ¼ 0 and @qð1Þ =@t þ qð0Þ r Vð1Þ ¼ 0 respectively. Note that, Vð0Þ rqð1Þ ¼ 0 and Vð0Þ rqð0Þ ¼ 0. Combination of these linearized equations yield that P ¼ ðqð0Þ C 2s Þ= cðr VÞ ¼ ðqð0Þ C 2s Þ=cr2 v, where the P and V are the perturbed quantities. Here C s and C are the sound speed and the ratio of specific heats. Thus, the gradient of the plasma pressure, rP , in Eq. (3) is included through the first term on the right hand side of Eq. (14). Note that due to cancelation, there is no term including plasma pressure in Eq. (13). Finally, it is worth to note about the symmetry of the functions involved in Eqs. (10)–(14). The poloidal flux function wðxÞ has even parity, while the streamfunction /ðxÞ is an odd function of x. The other functions under discussion namely bðxÞ and vðxÞ are, generally speaking, of mixed parity. However, only their odd components are of interest for the Hall affected magnetic reconnection (see Bian and Vekstein (2007) for more details), therefore, this particular parity is assumed in Eqs. (10)–(14), with the terms of “wrong” parity being omitted. 3. Reconnection regimes For R the constant-w case one can put w1 ¼ 1, so that D0 w001 dx w001 ðDxÞ 1, where ðDxÞ is width of the current sheet (the reconnection layer). On the other hand, it follows now from Eq. (10) that l d i 2 w001 w1 1, hence, 1
w001 ðDxÞ ;
ðDxÞ l d i 2 1:
ð15Þ
Consider first the collisionless “standard”- MHD regime, where Hall effect does not play a role. Then Eqs. (10) and (13) combine into a single equation for /: 2
ld i 2 ðcsA Þ 00 / kx/ iw1 ð0Þ ¼ 0; kx which under the transformations 1=2 k n¼ x; csA l1=2 d i
ð16Þ
ð17Þ
M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
v¼
1=2 i csA l1=2 d i k /; w1 ð0Þ
ð18Þ
1
iw1 ð0Þ ¼ l2 d i 3 ðkxÞ 1 ðcsA Þb00 d i ðkxÞðcsA Þ b:
ð1Þ
v00 n2 v þ n ¼ 0:
ð19Þ
Therefore, one gets for the instability factor Z 0 1 D ¼ w1 ð0Þ w001 dx Z 1 00 v 1=2 3=4 3=2 1=2 ¼ ðcsA Þ l di ðkÞ dn 1 n 2p Cð3=4Þ ðcsA Þ1=2 l3=4 d i 3=2 ðkÞ1=2 ; ¼ Cð1=4Þ which results in the instability growth rate: 2 2p Cð3=4Þ 2 ðcsA Þ ¼ kl3=2 d i 3 ðD0 Þ : Cð1=4Þ
ðcsA Þ
2
v00 ¼
b
n ¼ ðkÞ
d i ðDxÞ 00 v
w 2 2 2 ðcsA Þ 1 ðDxÞ ðcsA Þ ðDxÞ di
; i:e: b l7=2 d i 8 : ðcsA Þ v
ð20Þ
1=2 1=2
l
d i 3=2 ðcsA Þ
1=2
ð28Þ
b;
l
1=2
d i 1=2 ðcsA Þ
ð29Þ
x;
ð21Þ
Under this magnitude of b the Hall term in Eq. (10) is equal to
Z
w00 dx ¼ ðcsA Þ1=2 ðkÞ1=2 l1=2 d i 3=2
Z
1
v00 dn; 1 n
ð31Þ
which yields the following expression for EMHD regime growth rate: 2 Cð3=4Þ 2 ðcsA Þ ¼ 2 p k d i 3 lðD0 Þ : ð32Þ Cð1=4Þ Again, for k 1 and 1 we obtain the following scaling: ðcsA Þ l d i 3 : Thus, a direct transition from the “standard” MHD to the EMHD regime occurs at d i d i l3=8 (Fig. 1). If plasma b exceeds b , its compressibility becomes determined not by the inertial force but by the thermal pressure force, which balances the magnetic force in Eq. (22) yielding v00 ¼ b=b:
ð33Þ
Then, Eq. (11) results in the following magnitude of b: v00 ¼
ð23Þ
ð30Þ
The parameter D0 can then be expressed as D0 ¼ w1 1 ð0Þ
b
d i ðDxÞ b l3 d i 8 ; ðcsA Þ
1=2 1=2
v00 n2 v n ¼ 0:
By using Eqs. (15) and (21) for ðDxÞ and ðcsA Þ, one finds 5 that for b < b ðld i 2 Þ 1, the inertial term [the first one on the LHS of Eq. (22)] exceeds the thermal pressure 2 contribution, so that v b=ðcsA Þ . Thus, Eq. (11) yields then the following estimate for b: 00
ð0ÞðkÞ
Eq. (27) reduces to the “standard” equation
ð22Þ
ðcsA Þ2
v ¼ iw1 and
Thus, ðcsA Þ l3=2 d 3i for k 1 and 1. In order to decide when Hall effect becomes important, one needs to estimate first the magnitude of the guide field component perturbation, b, generated by the Hall term in Eq. (11). This involves the compressional component of the plasma flow, v, which is governed by Eq. (14): b
ð27Þ
By introducing the following new variables,
takes the form
v
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b d i ðDxÞ 00 di
w
; b ðcsA Þ 1 ðcsA Þ
i:e: b
b di : ðcsA Þ
This makes the Hall term in Eq. (10) equal to d i ðDxÞ b d i 2 ðDxÞ b
l2 d 2 i b; csA ðcsA Þ2
ð34Þ
ð35Þ
ð24Þ
hence, it becomes significant (reaching the order of unity) for d i > d i l3=8 1. If d i > d i , the electron magnetohydrodynamic (EMHD) regime takes place, for which the magnitude of b generated by Hall effect is determined not by the plasma compressibility, as it occurs above, but by the magnetic field diffusion. Therefore, this regime can be fully described by two equations for w1 and b that follow from Eqs. (10) and (11): 1
w1 ð0Þ ¼ l d i 2 w001 ikd i xðcsA Þ b; 00
ld i 2 b ¼ ikd i xðcsA Þ which combine into
1
w001 ;
ð25Þ ð26Þ
Fig. 1. ðb d i Þ diagram for the two-fluid collisionless tearing instability. Region (1): “standard” MHD regime; Region (2): “intermediate” HallMHD regime, and Region (3): electron-MHD regime. Dashed region shows the parametric space where the Alfve´n resonance can take place.
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M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
where expressions (15) and (21) for ðDxÞ and ðcsA Þ are used. As seen from (35), Hall effect remains negligible when b < b1 l2 d i 2 (note, that b1 b for d i l3=8 ). For b > b1 transition to the “intermediate” Hall-MHD regime occurs (Fig. 1). In this case, the appropriate forms of (10) and (11) which take into account (33): w1 ð0Þ ¼ l d i 2 w001
ik d i x b; csA
ð36Þ
b ik d i x 00 þ w1 ¼ 0; b csA
ð37Þ
yield " w001 ¼ w1 ð0Þ l d i 2 þ
b ðkd i xÞ ðcsA Þ
# 2 1
2
ð38Þ
;
hence, the matching requirement 0
D ¼ ¼
w1 1 ð0Þ
Z
1
1
w001 dx
pl1=2 d 2 i ðcsA Þ ð kÞb1=2
¼
1 ðl d 2i Þ
Z
1
1þ 1
bð k xÞ
2
l ðcsA Þ2
;
!1 dx ð39Þ
results in the following instability growth rate 1 ðcsA Þ ¼ k l 1=2 b 1=2 d i 2 D0 : p
ð40Þ
Therefore, in this regime, under assumptions of k 1 and 1, respective instability growth rate scales as ðcsA Þ l1=2 b1=2 d 2i . Further increase in plasma b eventually brings about the EMHD regime of collisionless reconnection described by Eq. (32). This is because the magnetic diffusion term in Eq. (11), l d i 2 b00 , becomes important in this case. Indeed, the latter can be estimated as l d i 2 b00 l d i 2 b= ðDxÞ2 b=ld i 2 , which exceeds the plasma compression effect v00 b=b when b > b2 ld i 2 (note that b2 b1 ). On the other hand, the EMHD regime takes place irrespective of the plasma b if the scaled ion inertial skin depth is very large: d i > d i l3=8 (Fig. 1). Altogether, the occurrence of various collisionless tearing mode regimes can be summarized as follows. The standard MHD regime takes place for b < b1 l2 d i 2 and d i < d i l3=8 . The intermediate Hall-MHD regime holds for b1 < b < b2 and d i < d i . Elsewhere in the (b d i ) plane the EMHD regime takes place. Fig. 1 shows different regimes of reconnection in ðb d i Þ space: Region (1): “standard” MHD regime; Region (2): “intermediate” Hall-MHD regime, and Region (3): electron-MHD regime. Bold lines separate different regimes. 4. Alfve´n resonance Outside the reconnection layer, where the non-ideal MHD effects are being ignored, the ideal MHD approximation holds. By assuming expðixtÞ form for the
time evolution of fluctuations with frequency x, the linearized equations of plasma motion (3) and magnetic induction yield (4) the following equation for the x-component, nx , of the plasma displacement (Uberoi, 1972): d 2 nx 2a2 k 2 V 2A x dnx x2 2 2 k 2 nx ¼ 0: ð41Þ VA dx2 x a2 k 2 V 2A x2 dx Therefore, the resonance condition is x2 ¼ a2 k 2 V 2A x2 ¼ 2 with two Alfve´n resonances located at s2 A ðaxÞ xA ðxsA Þl. This resonance condition can be satisfied for a broad interval of the fluctuation frequency, x, ranging from xmax ¼ kV A s1 A down to x ¼ 0. For low fre, two Alfve´n resonances located, quencies, x s1 A according to Uberoi (1994), at xA ðxÞ ðxsA Þl, are very close to each other. Therefore, when their separation becomes less than the finite width of each resonance they overlap, creating a single current sheet at x ¼ 0, which is a signature of magnetic reconnection. Now, the question is that under what parametric condition the Alfve´n resonance layer is located outside the current sheet. To answer this question, one should compare the width of current sheet, Dx, with the location of Alfve´n resonance, xA , at each regimes of reconnection discussed above. First consider the single-fluid MHD regime with ´ n resonance can Dx l1=2 d 1 i ðcsA Þl. Therefore, Alfve 1=2 1 appear if Dx < xA , i.e., l d i ðcsA Þl < ðxsA Þl. Since for this regime ðcsA Þ l3=2 d 3i , then it turns out that if x > x ¼ ld 2i s1 A , then there would be a room for appear1 ance of Alfve´n resonance. Note that, x ðsA Þ since it has been assumed that l; d i < 1 (this is the case for some applications of interest: by assuming a completely collisionless hydrogen plasma with l 5:4 104 , for tokamak plasmas with typical parameters of current sheet width l 30 100 ; cm, magnetic field, B 103 104 G, the normalized ion inertial skin depth is d 0:1 0:3 and b 0:01 0:1, and for the Solar flare plasmas with Interestl 104 km;B 102 G;d i 105 ; b 0:1 1). ingly, for very slow MHD waves generated by magnetic reconnection, with x c, the resonance cannot be expected because the above inequality l1=2 d 1 i ðcsA Þl < ðxsA Þl ðcsA Þl cannot be satisfied as l1=2 d 1 i > 1. In other words, in this case, Dx > xA . Therefore, in the (b d i ) parametric space of the “standard” MHD regime in Fig. 1, region (1), there is no room for the formation of Alfve´n resonance. A similar kind of scaling for the “intermediate”-Hall MHD regime with ðcsA Þ l1=2 b1=2 d 2i , and Dx l1=2 b1=2 ðcsA Þl shows that again if x > x ¼ ld 2i s1 A , then Alfve´n resonance can take place but, unlike the “standard” MHD regime, here Alfve´n resonance can occur even at small frequencies on the order of x c since at this regime the condition of Dx < xA yields that l1=2 b1=2 ðcsA Þl < ðxsA Þl ðcsA Þl, i.e., b > l. In the “intermediate” Hall-MHD regime the plasma-b might be greater than l (see the Fig. 1). Therefore, the dashed region in Fig. 1
M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
shows the parametric space in (b d i ) diagram where the possibility of Alfve´n resonance occurrence can be expected. Finally, consider the electron-MHD regime with csA ld 3i and Dx csA d 1 i l. Similar to previous regimes, if x > x ¼ ld 2i s1 , then Alfve´n resonance can appear. A However, for slow MHD Alfve´n waves with x c, the inequality Dx < xA results in csA d 1 i l < ðxsA Þl ðcsA Þl i.e., d 1 < 1 which is not satisfied according to the adopted i assumption that d i < 1. This means that in the EMHD regime similar to the “standard”- MHD one, there is no room for Alfve´n resonance at such small frequencies. 5. Summary To summarize, the relation between collisionless magnetic reconnection driven by the electron inertia and the Alfve´n resonance was investigated. There are three main linear regimes of reconnection in compressible tearing instability of a sheared force-free magnetic field: first, the “standard”- MHD regime where Hall effect becomes no important and the single-fluid MHD regime adequately describes the dynamics of reconnection; second, an ‘intermediate’- called Hall-MHD regime, in which Hall effect is playing an important role in pushing field lines to the reconnection site, and finally, the electron-MHD regime, where the dynamics of reconnection is determined by the electron component only, and ions are assumed to be at rest. Each of these regimes are characterized by their respective current sheet width and the growth rate of instability. On the other hand, the Alfve´n resonance can occur in such dynamic systems when another kind of wave mode converts to the Alfve´n mode or Alfve´n waves are directly generated for example by the magnetic reconnection. Thus, the frequency of existing fluctuations becomes the same as the slow Alfve´nic wave frequency. As a result, two layers of Alfve´n resonance can be formed provided that the location of resonance layers are to be outside of the current sheet. Our scaling analysis presented parametric condition determining when the Alfve´n resonance layers are outside of the current sheet. Moreover, it has been argued that, in the case of small frequencies on the order of the tearing instability growth rate (or reconnection rate), in the “standard”- MHD and electron-MHD regimes the location of resonance layer is inside the current sheet and hence, there would be no room for occurring the Alfve´n resonance. However, in the intermediate Hall-MHD regime, when b > l, the respective location of resonance layer is outside of the reconnection current sheet, and so the Alfve´n resonance can take place, otherwise, when b < l, the Alfve´n resonance would not be existed. The author thinks that the discussions presented in this study can be of interest in more understanding of the relation between Alfve´n resonance phenomenon and the magnetic reconnection via collisionless tearing instability in space plasma environments.
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References Bian, N., Vekstein, G., 2007. On the two-fluid modification of the resistive tearing instability. Phys. Plasmas 14, 072107. Birn, J., Priest, E., 2007. Reconnection of Magnetic Fields: Magnetohydrodynamic and Collisionless Theory and Observations. Cambridge University Press. Drake, J.F., Shay, M.A., Swisdak, M., 2008. The Hall fields and fast magnetic reconnection. Phys. Plasmas 15, 042306. Erde´lyi, R., Goossens, M., 1995. Resonant absorption of Alfve´n waves in coronal loops in visco-resistive MHD. A&A 294, 575. Fitzpatrick, R., Porcelli, F., 2004. Collisionless magnetic reconnection with arbitrary guide- field. Phys. Plasmas 11, 4713. Goertz, C.K., Boswell, R.W., 1979. Magnetosphere–ionosphere coupling. J. Geophys. Res. 84, 7239. Hasegawa, A., 1976. Particle acceleration by MHD surface wave and formation of aurora. J. Geophys. Res. 81, 5083. Hasegawa, A., Chen, L., 1974. Plasma heating by Alfve´n-wave phase mixing. Phys. Rev. Lett. 32, 454. Hasegawa, A., Chen, L., 1975. Kinetic process of plasma heating due to Alfve´n wave excitation. Phys. Rev. Lett. 35, 370. Hosseinpour, M., Bian, N., Vekstein, G., 2009. Two-fluid regimes of the resistive and collisionless tearing instability. Phys. Plasmas 16, 012104. Huang, S.Y. et al., 2010. Wave properties in the magnetic reconnection diffusion region with high b: application of the k-filtering method to cluster multi-spacecraft data. J. Geophys. Res. 115, A12211. Ionson, J.A., 1978. Resonant absorption of Alfve´nic surface waves and the heating of solar coronal loops. ApJ 226, 650. Kigure, H. et al., 2010. Generation of Alfve´n waves by magnetic reconnection. Astron. Soc. Jpn. 62, 993. Lazarian, A., Vishniak E., 1998. Reconnection via the tearing instability, arXiv:astro-ph/9804166. Lyask, L.R., Lotko, W., 1996. On the kinetic dispersion relation for shear Alfve´n waves. J. Geophys. Res. 101, 5085. Ma, Z.W., Lee, L.C., Otto, A., 1995. Generation of field-aligned currents and Alfve´n waves by 3D magnetic reconnection. Geophys. Res. Lett. 22, 1737. Mirnov, V.V., Hegna, C.C., Prager, S., 2004. Two-fluid tearing instability in force-free magnetic configuration. Phys. Plasmas 11, 4468. Poedts, S., Goossens, M., Kerner, W., 1989. Numerical simulation of coronal heating by resonant absorption of Alfve´n waves. Sol. Phys. 123, 83. Priest, E., Forbes, T., 2000. Magnetic Reconnection: MHD Theory and Applications. Cambridge University Press. Rogers, B.N. et al., 2001. Role of dispersive waves in collisionless magnetic reconnection. Phys. Rev. Lett. 87, 195004. Ruderman, M.S., Berghmans, D., Goossens, M., Poedts, S., 1997. Direct excitation of resonant torsional Alfve´n waves by footpoint motions. A&A 320, 305. Sakai, J.I., Haruki, T., Kazimura, Y., Neubert, T., 2000. Generation of electromagnetic waves and Alfve´n waves during coalescence of magnetic islands in pair plasmas. Phys. Scr. T. 84, 42. Shay, M.A., Drake, J.F., Rogers, B.N., Denton, R.E., 2001. Alfve´nic collisionless magnetic reconnection and the Hall term 2001. J. Geophys. Res. 106, 3759. Terasawa, T., 1983. Hall current effect on tearing mode instability. Geophys. Res. Lett. 10, 475. Uberoi, C., 1972. Alfve´n waves in inhomogeneous magnetic fields. Phys. Fluids 15, 1673. Uberoi, C., 1994. Resonant absorption of Alfve´n waves near a neutral point. J. Plasma Phys. 52, 215. Uberoi, C., Zweibel, E.G., 1999. Alfve´n resonances and forced reconnection. J. Plasma Phys. 62, 345. Vekstein, G., 2000. On the correspondence between forced magnetic reconnection and Alfve´n resonances. Phys. Plasmas 7, 3808. Wang, X. et al., 2002. Alfve´n waves in regions of magnetic reconnection and the acceleration of newborn ions. Chin. Astron. Astrophys. 26, 29.
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Wang, X., Luan, Q., 2013. Low frequency whistler waves excited in fast magnetic reconnection processes. Front. Phys. 8, 585. Wang, X., Bhattacharjee, A., Ma, Z.W., Ren, C., Hegna, C.C., Callen, J.D., 1998. Structure and dynamics of current sheets at Alfve´n resonances in a differentially rotating plasma. Phys. Plasmas 5, 2291. Wang, Z., Wei, L., Wang, X., Liu, Y., 2011. Self-suppression of double tearing modes via Alfve´n resonance in rotating tokamak plasmas. Phys. Plasmas 18, 050701.
Wei, X.H. et al., 2013. Generation mechanism of the whistler-mode waves in the plasma sheet prior to magnetic reconnection. Adv. Space Res. 52, 205. Yamada, M., Kulsrud, R., Ji, H., 2010. Magnetic reconnection. Rev. Mod. Phys. 82, 1. Zweibel, E.G., Yamada, M., 2009. Magnetic reconnection in astrophysical and laboratory plasmas. Ann. Rev. Astron. Astrophys. 47, 291–332.
ScienceDirect Advances in Space Research 55 (2015) 1278–1284 www.elsevier.com/locate/asr
On the possibility of Alfve´n wave resonance in collisionless magnetic reconnection M. Hosseinpour ⇑ Department of Plasma Physics, Faculty of Physics, University of Tabriz, Tabriz, Iran Received 21 June 2014; received in revised form 17 November 2014; accepted 21 November 2014 Available online 28 November 2014
Abstract Alfve´n wave resonance and magnetic reconnection are among the potential candidates for efficient dissipation of magnetic energy in space and astrophysical plasmas. In this paper, the correspondence between Alfve´n resonance and the electron-inertia driven reconnection in a sheared force-free magnetic field is discussed. By analytical scaling the linear regimes of compressible tearing instability in the two-fluid magnetohydrodynamic (MHD) model, we present parametric conditions for the possibility of Alfve´n resonance existence. Meanwhile, it is argued that the slow MHD Alfve´nic resonance can take place only in the “intermediate” – called Hall-MHD regime when b > l. b is the ratio of plasma thermal pressure to the pressure in equilibrium magnetic field and l is the electron to ion mass ratio. There is no room for such a resonant dissipation phenomenon either in the single-fluid MHD or the electron-MHD regimes. Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Magnetohydrodynamics; Tearing instability; Magnetic reconnection; Alfve´n wave resonance
1. Introduction Among various schools of thoughts concerning the efficient dissipation of magnetic energy in the highly conductive environments of space plasmas there are two main ones: Alfve´n mode resonance which can explain the enhanced dissipation in dynamical processes such as magnetohydrodynamic (MHD) waves (Hasegawa and Chen, 1974; Wang et al., 1998; Erde´lyi and Goossens, 1995; Ruderman et al., 1997), and magnetic reconnection which is usually associated with longer time scales compared with the Alfve´n time (Birn and Priest, 2007; Priest and Forbes, 2000; Yamada et al., 2010; Zweibel and Yamada, 2009). In a collisionless plasma with negligible plasma resistivity, the electron-inertia is a potential means of breaking the frozen-in condition to convert magnetic energy into the kinetic and thermal energy of plasma. The inclusion of electron ⇑ Tel.: +98 413 339 3356.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.asr.2014.11.022 0273-1177/Ó 2014 COSPAR. Published by Elsevier Ltd. All rights reserved.
inertia, inevitably, requires that the two-fluid MHD description of reconnection is to be considered, in which Hall effect plays an important role by facilitating the pace of magnetic reconnection (Drake et al., 2008; Fitzpatrick and Porcelli, 2004; Mirnov et al., 2004; Hosseinpour et al., 2009; Shay et al., 2001; Terasawa, 1983). In the Hall-MHD reconnection via tearing instability the current sheet acquires a double-layer structure: The inertial region surrounded by a much wider layer, where the electron-inertia plays no role, but the poloidal magnetic field is still advected towards the reconnection region by the Hall effect rather than by the bulk plasma motion. Relative importance of Hall effect and the electron inertia is characterized by the following non-dimensional parameters: d e c=ðxpe lÞ, the scaled electron inertial skin-depth (l is the magnetic shear length scale), for the electron inertia; d i c=ðxpi lÞ ¼ l1=2 d e d e ; l me =mi 1, the scaled ion skin depth, for the Hall effect. c and xp i=e are the speed of light and the ion/electron plasma frequency, respectively.
M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
At sufficiently small values of d i (see below), the Hall effect turns out to be less important, so that, the single-fluid MHD can adequately describe the dynamics of reconnection (“standard”- MHD regime). In contrast, at much larger values of d i (see below), ion component of plasma does not play any role in reconnection dynamics and therefore, the electron-MHD (EMHD) regime dominates. In the intermediate values of d i both electron and ion components determine the dynamics of reconnection, we call it an “intermediate” Hall-MHD regime. Each of these linear regimes of tearing instability are characterized by their respective current sheet width and the instability growth rate. In such a tearing instability, most of the magnetic energy is stored in the low-frequency MHD waves such as Alfve´n waves that can be generated by magnetic reconnection and propagated along the reconnected field lines (Kigure, 2010; Wang, 2002; Ma et al., 1995). One of the proposed mechanisms for dissipation of Alfve´n waves is resonant absorption (Poedts et al., 1989; Ionson, 1978; Hasegawa and Chen, 1974). Wang et al. (2011) has reported that the MHD perturbations due to tearing mode reconnection on the outer rational surface mediate in generating two Alfve´n resonance layers on the both sides of the inner rational surface and then prevent the formation of island growth (tearing mode suppression). However, in the Hall-MHD reconnection with the large guide field different types of waves can be excited such as kinetic or inertial Alfve´n waves. Kinetic Alfve´n waves can propagate at b > l, while inertial Alfve´n waves are excited at b < l (Huang, 2010; Rogers, 2001; Lyask and Lotko, 1996; Hasegawa and Chen, 1975; Hasegawa, 1976; Goertz and Boswell, 1979). These waves determine the electron dynamics in the ion diffusion region. Even, Whistler waves can propagate inside the electron inertial length scale (Wei, 2013; Wang and Luan, 2013). In order to concentrate our discussions on the Alfve´n resonance, we assume that the wave vector of fluctuations has no component along the guide field, thus it leaves no room for the kinetic/inertial Alfve´n waves and only the slow Alfve´nic waves are permitted to propagate regardless of the excitation mechanism for Alfve´n waves. In magnetic reconnection with nonuniform equilibrium magnetic field, the location of resonance layer varies with the Alfve´n frequency (see below). If the location of resonance layer lies within the current sheet formed by magnetic reconnection, then Alfve´n resonance, in fact, cannot occur and therefore, the dynamics of tearing mode is not influenced by the resonance phenomenon. It is of interest to investigate the possibility of occurring the Alfve´n resonance in each of the two-fluid MHD regimes of collisionless reconnection. It is worthy of note that, previously, the relation between Alfve´n resonance and magnetic reconnection has been argued by Uberoi (1994) and Vekstein (2000), but in the case of forced magnetic reconnection. In this type of reconnection the externally sinusoidal perturbation of
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plasma boundaries with a certain frequency leads to the magnetic reconnection at the surface where poloidal field changes its sign. Furthermore, the sinusoidal boundary perturbation can excite Alfve´nic waves. Uberoi (1994) in the absence of Hall effect has interpreted forced magnetic reconnection as the Alfve´n resonance with zero frequency, from which it was deduced in Uberoi and Zweibel (1999) that the theory of forced magnetic reconnection is actually embedded in the Alfve´n resonance theory. On the other hand, Vekstein (2000) discussed that these phenomena are operating separately, and transition from resonant absorption to forced reconnection occurs when the frequency of the external driver becomes sufficiently small. However, in the tearing instability, Alfve´n waves can be generated following the onset of magnetic reconnection and then propagated along the magnetic fields (Kigure, 2010; Wang, 2002; Ma et al., 1995; Sakai et al., 2000; Lazarian and Vishniak, 1998). Here, unlike the forced reconnection, there is no externally driven mechanism. In this study, we first analytically scale both the width of current sheet and the growth rate of collisionless tearing instability at different linear regimes within the two-fluid MHD framework and then investigate the possibility of occurring Alfve´n resonance at each of these regimes. To do so, we compare the location of Alfve´n resonance layer with the width of current sheet. If the resonance layer is located inside the current sheet, then the appearance of Alfve´n resonance is not expected. It should be note that, in this study, the mechanisms which can generate Alfve´n resonance will not be discussed, but only the possibility of Alfve´n resonance existence in the presence of linear tearing instability will be investigated. The structure of paper is as follows: Section 2 gives the description of the model and basic equations. Analytical analysis of collisionless reconnection regimes are included in Section 3 and discussions regarding the possibility of Alfve´n resonance occurrence are presented in Section 4, which is followed by a brief summary and conclusion in Section 5. 2. The model and basic equations A planar slab of a highly conductive uniform plasma [density n0 ] embedded in a sheared force-free magnetic field ð0Þ Bð0Þ ðxÞ ¼ zBð0Þ z ðxÞ þ yBy ðxÞ;
ð1Þ
with Byð0Þ ¼ B0 f ðxÞ;
Bzð0Þ ¼ B0 ½1 f 2 ðxÞ
1=2
ð2Þ
is surrounded by two perfectly conducting walls at x ¼ l. For the tearing perturbation of the form expðikyÞ reconnection occurs at the x ¼ 0, where the poloidal field component Bð0Þ y changes its sign: f ðxÞ ax for jxj l. Therefore, the wave number has only one component which is along the ‘y’ direction. The governing equations defining temporal evolution of perturbations about the initial equilibrium are as follows:
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The equation of bulk plasma motion dV 1 q ¼ ðj BÞ rP; dt c where j ¼
c ðr 4p
Vz ¼ ð3Þ /00 ¼
BÞ; the Faraday equation
@B ¼ cðr EÞ; @t
ð4Þ
and the generalized Ohm’s law that follows from the equation of motion for electrons: 1 me dVe 1 E þ ðVe BÞ ¼ rP e ; c ne e dt
ð5Þ
where the electron component velocity Ve ¼ V j=ne. The inertial term on the right-hand-side (RHS) of (5) is significant only inside the current sheet at x ¼ 0. Thus, the inerme @j e ne tial term can be simplified there as mee dV 2 @t. The dt perturbed magnetic field can be written as Bðx; y; tÞ ¼ rwðx; y; tÞ z þ Bz ðx; y; tÞz;
ð6Þ
where the poloidal flux function takes the form wðx; y; tÞ ¼ w0 ðxÞ þ w1 ðxÞexpðikyÞexpðctÞ
ð7Þ
ð0Þ
(w0 corresponds R to the initial field B given by Eqs. (1) and (2) as w0 ¼ B0 f ðxÞdx), and the toroidal field component Bz ðx; y; tÞ ¼ Bzð0Þ ðxÞ þ Bð1Þ z expðikyÞexpðctÞ:
ð8Þ
The plasma velocity V is represented as V ¼ ðr/ zÞ þ rv þ V z z;
ð9Þ
where the stream-function /ðx; y; tÞ ¼ /ðxÞexpðiky þ ctÞ and the velocity potential vðx; y; tÞ ¼ vðxÞexpðiky þ ctÞ correspond, respectively, to the vortical and compressional components of the poloidal plasma flow. By introducing non-dimensional variables with length normalized by l; V z by cl; / and v by cl2 ; w1 by B0 l and Bzð1Þ by B0 , Eqs. (3)–(5) yield the following set of equations linearized with respect to perturbations valid inside the current sheet around the field reversal plane x ¼ 0. Thus, w1 ¼ xðik/Þ þ ld i 2 w001
ikd i x b; ðcsA Þ
ikd i x 00 b ¼ ikxV z v þ ld i b þ w ; ðcsA Þ 1 00
2 00
ð10Þ ð11Þ
where b Bzð1Þ w1 , and al. The latter parameter indicates strength of the guide field, so that 1 corresponds to the strong guide field approximation. Introduction of the auxiliary perturbation function b allows a straightforward generalization for the case of an arbitrary guide field strength. Thus, in all estimates that follow it is assumed, for simplicity, that 1. A similar assumption is put for the perturbation wave vector, k 1. Note that this makes R 00 the tearing stability factor, D0 ¼ w1 w1 dx, of the order 1 ð0Þ of unity too, which ensures applicability of the constant-w approximation. The Hall effect is represented by the last terms on the RHS of Eqs. (10) and (11). Further, the linearized equation of motion (3) yields:
v00 ¼
ikx ðcsA Þ
2
b;
ð12Þ
2
w001 ;
ð13Þ
ikx ðcsA Þ
d2 ðbv00 bÞ: 2 ðcsA Þ d x2 1
ð14Þ
Here b C 2s =V 2A , where C s and V A are, respectively, the pffiffiffiffiffiffiffiffi sound and Alfve´n velocity (V A ¼ B0 = 4pq). Also, sA ¼ l=V A . Eqs. (10) and (11) are the ‘x’ and ‘z’ components of Eq. (4) respectively, where the curl of the gradient of scalar electron pressure is zero, r rP e ¼ 0. Therefore, there is no term including scalar electron pressure in Eqs. (10) and (11). Also, the linear forms of the adiabatic equation of state, @=@tðP =qC Þ ¼ 0, and the continuity equation, @q=@t þ r ðqVÞ ¼ 0, are @P ð1Þ =@t ½Cpð0Þ =qð0Þ @qð1Þ =@t ¼ 0 and @qð1Þ =@t þ qð0Þ r Vð1Þ ¼ 0 respectively. Note that, Vð0Þ rqð1Þ ¼ 0 and Vð0Þ rqð0Þ ¼ 0. Combination of these linearized equations yield that P ¼ ðqð0Þ C 2s Þ= cðr VÞ ¼ ðqð0Þ C 2s Þ=cr2 v, where the P and V are the perturbed quantities. Here C s and C are the sound speed and the ratio of specific heats. Thus, the gradient of the plasma pressure, rP , in Eq. (3) is included through the first term on the right hand side of Eq. (14). Note that due to cancelation, there is no term including plasma pressure in Eq. (13). Finally, it is worth to note about the symmetry of the functions involved in Eqs. (10)–(14). The poloidal flux function wðxÞ has even parity, while the streamfunction /ðxÞ is an odd function of x. The other functions under discussion namely bðxÞ and vðxÞ are, generally speaking, of mixed parity. However, only their odd components are of interest for the Hall affected magnetic reconnection (see Bian and Vekstein (2007) for more details), therefore, this particular parity is assumed in Eqs. (10)–(14), with the terms of “wrong” parity being omitted. 3. Reconnection regimes For R the constant-w case one can put w1 ¼ 1, so that D0 w001 dx w001 ðDxÞ 1, where ðDxÞ is width of the current sheet (the reconnection layer). On the other hand, it follows now from Eq. (10) that l d i 2 w001 w1 1, hence, 1
w001 ðDxÞ ;
ðDxÞ l d i 2 1:
ð15Þ
Consider first the collisionless “standard”- MHD regime, where Hall effect does not play a role. Then Eqs. (10) and (13) combine into a single equation for /: 2
ld i 2 ðcsA Þ 00 / kx/ iw1 ð0Þ ¼ 0; kx which under the transformations 1=2 k n¼ x; csA l1=2 d i
ð16Þ
ð17Þ
M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
v¼
1=2 i csA l1=2 d i k /; w1 ð0Þ
ð18Þ
1
iw1 ð0Þ ¼ l2 d i 3 ðkxÞ 1 ðcsA Þb00 d i ðkxÞðcsA Þ b:
ð1Þ
v00 n2 v þ n ¼ 0:
ð19Þ
Therefore, one gets for the instability factor Z 0 1 D ¼ w1 ð0Þ w001 dx Z 1 00 v 1=2 3=4 3=2 1=2 ¼ ðcsA Þ l di ðkÞ dn 1 n 2p Cð3=4Þ ðcsA Þ1=2 l3=4 d i 3=2 ðkÞ1=2 ; ¼ Cð1=4Þ which results in the instability growth rate: 2 2p Cð3=4Þ 2 ðcsA Þ ¼ kl3=2 d i 3 ðD0 Þ : Cð1=4Þ
ðcsA Þ
2
v00 ¼
b
n ¼ ðkÞ
d i ðDxÞ 00 v
w 2 2 2 ðcsA Þ 1 ðDxÞ ðcsA Þ ðDxÞ di
; i:e: b l7=2 d i 8 : ðcsA Þ v
ð20Þ
1=2 1=2
l
d i 3=2 ðcsA Þ
1=2
ð28Þ
b;
l
1=2
d i 1=2 ðcsA Þ
ð29Þ
x;
ð21Þ
Under this magnitude of b the Hall term in Eq. (10) is equal to
Z
w00 dx ¼ ðcsA Þ1=2 ðkÞ1=2 l1=2 d i 3=2
Z
1
v00 dn; 1 n
ð31Þ
which yields the following expression for EMHD regime growth rate: 2 Cð3=4Þ 2 ðcsA Þ ¼ 2 p k d i 3 lðD0 Þ : ð32Þ Cð1=4Þ Again, for k 1 and 1 we obtain the following scaling: ðcsA Þ l d i 3 : Thus, a direct transition from the “standard” MHD to the EMHD regime occurs at d i d i l3=8 (Fig. 1). If plasma b exceeds b , its compressibility becomes determined not by the inertial force but by the thermal pressure force, which balances the magnetic force in Eq. (22) yielding v00 ¼ b=b:
ð33Þ
Then, Eq. (11) results in the following magnitude of b: v00 ¼
ð23Þ
ð30Þ
The parameter D0 can then be expressed as D0 ¼ w1 1 ð0Þ
b
d i ðDxÞ b l3 d i 8 ; ðcsA Þ
1=2 1=2
v00 n2 v n ¼ 0:
By using Eqs. (15) and (21) for ðDxÞ and ðcsA Þ, one finds 5 that for b < b ðld i 2 Þ 1, the inertial term [the first one on the LHS of Eq. (22)] exceeds the thermal pressure 2 contribution, so that v b=ðcsA Þ . Thus, Eq. (11) yields then the following estimate for b: 00
ð0ÞðkÞ
Eq. (27) reduces to the “standard” equation
ð22Þ
ðcsA Þ2
v ¼ iw1 and
Thus, ðcsA Þ l3=2 d 3i for k 1 and 1. In order to decide when Hall effect becomes important, one needs to estimate first the magnitude of the guide field component perturbation, b, generated by the Hall term in Eq. (11). This involves the compressional component of the plasma flow, v, which is governed by Eq. (14): b
ð27Þ
By introducing the following new variables,
takes the form
v
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b d i ðDxÞ 00 di
w
; b ðcsA Þ 1 ðcsA Þ
i:e: b
b di : ðcsA Þ
This makes the Hall term in Eq. (10) equal to d i ðDxÞ b d i 2 ðDxÞ b
l2 d 2 i b; csA ðcsA Þ2
ð34Þ
ð35Þ
ð24Þ
hence, it becomes significant (reaching the order of unity) for d i > d i l3=8 1. If d i > d i , the electron magnetohydrodynamic (EMHD) regime takes place, for which the magnitude of b generated by Hall effect is determined not by the plasma compressibility, as it occurs above, but by the magnetic field diffusion. Therefore, this regime can be fully described by two equations for w1 and b that follow from Eqs. (10) and (11): 1
w1 ð0Þ ¼ l d i 2 w001 ikd i xðcsA Þ b; 00
ld i 2 b ¼ ikd i xðcsA Þ which combine into
1
w001 ;
ð25Þ ð26Þ
Fig. 1. ðb d i Þ diagram for the two-fluid collisionless tearing instability. Region (1): “standard” MHD regime; Region (2): “intermediate” HallMHD regime, and Region (3): electron-MHD regime. Dashed region shows the parametric space where the Alfve´n resonance can take place.
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M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
where expressions (15) and (21) for ðDxÞ and ðcsA Þ are used. As seen from (35), Hall effect remains negligible when b < b1 l2 d i 2 (note, that b1 b for d i l3=8 ). For b > b1 transition to the “intermediate” Hall-MHD regime occurs (Fig. 1). In this case, the appropriate forms of (10) and (11) which take into account (33): w1 ð0Þ ¼ l d i 2 w001
ik d i x b; csA
ð36Þ
b ik d i x 00 þ w1 ¼ 0; b csA
ð37Þ
yield " w001 ¼ w1 ð0Þ l d i 2 þ
b ðkd i xÞ ðcsA Þ
# 2 1
2
ð38Þ
;
hence, the matching requirement 0
D ¼ ¼
w1 1 ð0Þ
Z
1
1
w001 dx
pl1=2 d 2 i ðcsA Þ ð kÞb1=2
¼
1 ðl d 2i Þ
Z
1
1þ 1
bð k xÞ
2
l ðcsA Þ2
;
!1 dx ð39Þ
results in the following instability growth rate 1 ðcsA Þ ¼ k l 1=2 b 1=2 d i 2 D0 : p
ð40Þ
Therefore, in this regime, under assumptions of k 1 and 1, respective instability growth rate scales as ðcsA Þ l1=2 b1=2 d 2i . Further increase in plasma b eventually brings about the EMHD regime of collisionless reconnection described by Eq. (32). This is because the magnetic diffusion term in Eq. (11), l d i 2 b00 , becomes important in this case. Indeed, the latter can be estimated as l d i 2 b00 l d i 2 b= ðDxÞ2 b=ld i 2 , which exceeds the plasma compression effect v00 b=b when b > b2 ld i 2 (note that b2 b1 ). On the other hand, the EMHD regime takes place irrespective of the plasma b if the scaled ion inertial skin depth is very large: d i > d i l3=8 (Fig. 1). Altogether, the occurrence of various collisionless tearing mode regimes can be summarized as follows. The standard MHD regime takes place for b < b1 l2 d i 2 and d i < d i l3=8 . The intermediate Hall-MHD regime holds for b1 < b < b2 and d i < d i . Elsewhere in the (b d i ) plane the EMHD regime takes place. Fig. 1 shows different regimes of reconnection in ðb d i Þ space: Region (1): “standard” MHD regime; Region (2): “intermediate” Hall-MHD regime, and Region (3): electron-MHD regime. Bold lines separate different regimes. 4. Alfve´n resonance Outside the reconnection layer, where the non-ideal MHD effects are being ignored, the ideal MHD approximation holds. By assuming expðixtÞ form for the
time evolution of fluctuations with frequency x, the linearized equations of plasma motion (3) and magnetic induction yield (4) the following equation for the x-component, nx , of the plasma displacement (Uberoi, 1972): d 2 nx 2a2 k 2 V 2A x dnx x2 2 2 k 2 nx ¼ 0: ð41Þ VA dx2 x a2 k 2 V 2A x2 dx Therefore, the resonance condition is x2 ¼ a2 k 2 V 2A x2 ¼ 2 with two Alfve´n resonances located at s2 A ðaxÞ xA ðxsA Þl. This resonance condition can be satisfied for a broad interval of the fluctuation frequency, x, ranging from xmax ¼ kV A s1 A down to x ¼ 0. For low fre, two Alfve´n resonances located, quencies, x s1 A according to Uberoi (1994), at xA ðxÞ ðxsA Þl, are very close to each other. Therefore, when their separation becomes less than the finite width of each resonance they overlap, creating a single current sheet at x ¼ 0, which is a signature of magnetic reconnection. Now, the question is that under what parametric condition the Alfve´n resonance layer is located outside the current sheet. To answer this question, one should compare the width of current sheet, Dx, with the location of Alfve´n resonance, xA , at each regimes of reconnection discussed above. First consider the single-fluid MHD regime with ´ n resonance can Dx l1=2 d 1 i ðcsA Þl. Therefore, Alfve 1=2 1 appear if Dx < xA , i.e., l d i ðcsA Þl < ðxsA Þl. Since for this regime ðcsA Þ l3=2 d 3i , then it turns out that if x > x ¼ ld 2i s1 A , then there would be a room for appear1 ance of Alfve´n resonance. Note that, x ðsA Þ since it has been assumed that l; d i < 1 (this is the case for some applications of interest: by assuming a completely collisionless hydrogen plasma with l 5:4 104 , for tokamak plasmas with typical parameters of current sheet width l 30 100 ; cm, magnetic field, B 103 104 G, the normalized ion inertial skin depth is d 0:1 0:3 and b 0:01 0:1, and for the Solar flare plasmas with Interestl 104 km;B 102 G;d i 105 ; b 0:1 1). ingly, for very slow MHD waves generated by magnetic reconnection, with x c, the resonance cannot be expected because the above inequality l1=2 d 1 i ðcsA Þl < ðxsA Þl ðcsA Þl cannot be satisfied as l1=2 d 1 i > 1. In other words, in this case, Dx > xA . Therefore, in the (b d i ) parametric space of the “standard” MHD regime in Fig. 1, region (1), there is no room for the formation of Alfve´n resonance. A similar kind of scaling for the “intermediate”-Hall MHD regime with ðcsA Þ l1=2 b1=2 d 2i , and Dx l1=2 b1=2 ðcsA Þl shows that again if x > x ¼ ld 2i s1 A , then Alfve´n resonance can take place but, unlike the “standard” MHD regime, here Alfve´n resonance can occur even at small frequencies on the order of x c since at this regime the condition of Dx < xA yields that l1=2 b1=2 ðcsA Þl < ðxsA Þl ðcsA Þl, i.e., b > l. In the “intermediate” Hall-MHD regime the plasma-b might be greater than l (see the Fig. 1). Therefore, the dashed region in Fig. 1
M. Hosseinpour / Advances in Space Research 55 (2015) 1278–1284
shows the parametric space in (b d i ) diagram where the possibility of Alfve´n resonance occurrence can be expected. Finally, consider the electron-MHD regime with csA ld 3i and Dx csA d 1 i l. Similar to previous regimes, if x > x ¼ ld 2i s1 , then Alfve´n resonance can appear. A However, for slow MHD Alfve´n waves with x c, the inequality Dx < xA results in csA d 1 i l < ðxsA Þl ðcsA Þl i.e., d 1 < 1 which is not satisfied according to the adopted i assumption that d i < 1. This means that in the EMHD regime similar to the “standard”- MHD one, there is no room for Alfve´n resonance at such small frequencies. 5. Summary To summarize, the relation between collisionless magnetic reconnection driven by the electron inertia and the Alfve´n resonance was investigated. There are three main linear regimes of reconnection in compressible tearing instability of a sheared force-free magnetic field: first, the “standard”- MHD regime where Hall effect becomes no important and the single-fluid MHD regime adequately describes the dynamics of reconnection; second, an ‘intermediate’- called Hall-MHD regime, in which Hall effect is playing an important role in pushing field lines to the reconnection site, and finally, the electron-MHD regime, where the dynamics of reconnection is determined by the electron component only, and ions are assumed to be at rest. Each of these regimes are characterized by their respective current sheet width and the growth rate of instability. On the other hand, the Alfve´n resonance can occur in such dynamic systems when another kind of wave mode converts to the Alfve´n mode or Alfve´n waves are directly generated for example by the magnetic reconnection. Thus, the frequency of existing fluctuations becomes the same as the slow Alfve´nic wave frequency. As a result, two layers of Alfve´n resonance can be formed provided that the location of resonance layers are to be outside of the current sheet. Our scaling analysis presented parametric condition determining when the Alfve´n resonance layers are outside of the current sheet. Moreover, it has been argued that, in the case of small frequencies on the order of the tearing instability growth rate (or reconnection rate), in the “standard”- MHD and electron-MHD regimes the location of resonance layer is inside the current sheet and hence, there would be no room for occurring the Alfve´n resonance. However, in the intermediate Hall-MHD regime, when b > l, the respective location of resonance layer is outside of the reconnection current sheet, and so the Alfve´n resonance can take place, otherwise, when b < l, the Alfve´n resonance would not be existed. The author thinks that the discussions presented in this study can be of interest in more understanding of the relation between Alfve´n resonance phenomenon and the magnetic reconnection via collisionless tearing instability in space plasma environments.
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