Twisted shear Alfvén waves with orbital angular momentum

Twisted shear Alfvén waves with orbital angular momentum

Physics Letters A 376 (2012) 2792–2794 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla Twisted she...

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Physics Letters A 376 (2012) 2792–2794

Contents lists available at SciVerse ScienceDirect

Physics Letters A www.elsevier.com/locate/pla

Twisted shear Alfvén waves with orbital angular momentum P.K. Shukla a,b,∗ a b

International Centre for Advanced Studies in Physical Sciences & Institute for Theoretical Physics, Faculty of Physics and Astronomy, Ruhr University Bochum, D-44780 Bochum, Germany Department of Mechanical and Aerospace Engineering & Centre for Energy Research, University of California San Diego, La Jolla, CA 92093, USA

a r t i c l e

i n f o

Article history: Received 26 July 2012 Accepted 16 August 2012 Available online 20 August 2012 Communicated by V.M. Agranovich Keywords: Twisted dispersive Alfvén waves Orbital angular momentum Space and laboratory plasmas

a b s t r a c t It is shown that a dispersive shear Alfvén wave (DSAW) in a magnetized plasma can propagate as a twisted Alfvén vortex beam carrying orbital angular momentum (OAM). We obtain a wave equation from the generalized ion vorticity equation and the magnetic field-aligned electron momentum equation that couple the scalar and vector potentials of the DSAW. A twisted shear Alfvén vortex beam can trap and transport plasma particles and energy in magnetoplasmas, such as those in the Earth’s auroral zone, in the solar atmosphere, and in Large Plasma Device (LAPD) at University of California, Los Angeles. © 2012 Elsevier B.V. All rights reserved.

The Alfvén wave [1] is one of the fundamental eigen-modes in magnetized plasmas. In the Alfvén wave, the restoring force comes from the pressure of the magnetic fields, and the ion mass provides the inertia [2]. The propagation of the low-frequency (in comparison with the ion gyrofrequency) Alfvén wave is governed by the magnetohydrodynamic (MHD) equations [1,3] composed of the ion continuity, the ion momentum (with the J × B force and the plasma pressure gradient, where J is the plasma current and B the magnetic field), and Faraday’s law in which the wave electric field is eliminated by using Ohm’s law. The dispersion [4–10] to the Alfvén wave comes from the finite frequency (ω/ωci ) and the parallel electron inertial force effects in cold plasmas, as well as from the finite ion gyroradius and the parallel electron pressure gradient effects in warm plasmas, Due to non-ideal effects [5–7, 10], Alfvén waves thus couple to fast and slow magnetoacoustic waves, electromagnetic ion-cyclotron waves, the inertial and kinetic Alfvén waves [4,9,10] in a uniform magnetoplasma. It is well known that multifaceted dispersive Alfvén waves [11,13] play a significant role in the Earth’s auroral and magnetospheric plasmas, as well as in solar and laboratory plasmas with regard to acceleration and heating of plasma particles [15], reconnection of magnetic field lines, wave–wave and wave– particle interactions [16,17], and the formation of braided magnetic fields/magnetic field ropes [12–14], solitary [18] and vortical structures [19]. Specifically, it is widely thought that dispersive Alfvén waves power the auroral activities [20] in the Earth’s iono-

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Correspondence to: International Centre for Advanced Studies in Physical Sciences & Institute for Theoretical Physics, Faculty of Physics and Astronomy, Ruhr University Bochum, D-44780 Bochum, Germany. Tel.: +49 234 3223759; fax: +49 234 3214733. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physleta.2012.08.025

sphere, the heating of solar coronal plasmas [21], as well as play a decisive role for the formation of magnetic flux ropes in the solar atmosphere and in laboratory plasmas [13]. In this Letter, it is shown that a three-dimensional DSAW can propagate as a twisted vortex beam or in the form of a magnetic flux rope. The latter, which is also associated with a magnetic field whirl or a magnetic tornado, can trap and transport plasma particles and energy from one region to another in magnetized plasmas. Let us consider a magnetized electron–ion plasma in the presence of low-frequency (in comparison with the electron gyrofrequency ωce = e B 0 /me c, where e is the magnitude of the electron charge, B 0 the strength of the external magnetic field zˆ B 0 , me the electron mass, and c the speed of light in vacuum, zˆ the unit vector along the z-axis in a Cartesian coordinate system) dispersive shear Alfvén waves (DSAWs) with the electric and magnetic fields E = −∇φ − (1/c )ˆz∂ A z /∂ t and B⊥ = ∇ A z × zˆ , respectively, where φ and A z are the scalar and magnetic field-aligned vector potentials, respectively. In the electromagnetic fields, the electron fluid velocity is given by

ve ≈

c B0

zˆ × ∇φ −

ck B T e B 0 n0

zˆ × ∇ ne1 + zˆ v ez ,

(1)

where ne1 ( n0 ) is a small electron density perturbation in the equilibrium density n0 , k B the Boltzmann constant, T e the electron temperature, and the magnetic field-aligned electron fluid velocity v ez is determined from the z-component of Ampère’s law

v ez ≈

c ∇2 Az, 4π en0 ⊥

(2)

which neglects the displacement current, since the DSAW phase speed is much smaller than c. The parallel ion current has been

P.K. Shukla / Physics Letters A 376 (2012) 2792–2794

neglected, since it is much smaller than the parallel electron current. The perpendicular (to zˆ ) component of the ion fluid velocity perturbation vi ⊥ is determined from



 2 c ωci ∂2 c ωci ∂∇⊥ φ 2 + ω zˆ × ∇φ − , ci vi ⊥ = 2 B0 B0 ∂t ∂t

(3)

which is obtained by manipulating the perpendicular component of the ion momentum equation, where ωci = e B 0 /mi c is the ion gyrofrequency and mi the ion mass. We have assumed that the DAW phase speed is much larger than the ion thermal speed, and therefore neglected in Eq. (3) the contribution of the ion pressure gradient. The ions are confined in a two-dimensional (x– y) plane perpendicular to zˆ . Substituting (1) and (2) into the linearized electron continuity equation, we have 2 Az ∂ ne1 c ∂∇⊥ + = 0. ∂t 4π e ∂ z

(4)

Furthermore, from the linearized ion continuity equation and Eq. (3) we obtain



 n0 c ωci 2 ∂2 2 + ω ∇⊥ φ = 0, ci ni1 − 2 B0 ∂t

(5)

where ni1 ( n0 ) is a small ion number density perturbation. Invoking the quasi-neutrality condition ne1 = ni1 , we can eliminate ni1 from Eq. (5) by using Eq. (4), obtaining

∂φ c + 2 ∂t ω pi



 ∂2 2 ∂ Az + ω = 0, ci ∂z ∂t2



In obtaining (7), we have used the parallel electric field E z = −(∂φ/∂ z) − (1/c )∂ A z /∂ t. The parallel phase speed of the DSAW

is assumed to be much larger than the electron thermal speed, and therefore the parallel electron pressure gradient in Eq. (7) has been neglected. We now eliminate φ from (7) by using Eq. (6), obtaining the wave equation for the DSAWs

 ∂2 Az ∂t2

+

c2



ω2pi

 2 ∂2 2 ∂ Az + ωci = 0. ∂t2 ∂ z2

(8)

Within the framework of a plane-wave approximation, assuming that A z is proportional to exp(−i ωt + ik · r), where ω and k (= k⊥ + zˆ k z ) are the angular frequency and the wave vector, respectively, we Fourier analyze (8) to obtain the dispersion relation for the modified (by the ω/ωci effect) DSAWs [7]

ω2 =

2 k2z V 2A (1 − ω2 /ωci )

(1 + k2⊥ λe2 )

,

(9)

where k⊥ and√ k z are the components of k across and along zˆ , and V A = B 0 / 4π n0 mi the Alfvén speed. Eq. (9) gives the angular frequency of the low-frequency (in comparison with ωci ) inertial

(10)

ω  ωci and  1 yields ϕc = arctg(ω/ω L ), where √ ωL = ωce ωci is the lower-hybrid resonance frequency. 2

k⊥ λe2

which for

In order to study the property of a twisted DSAW, we now seek a solution of Eq. (8) in the form

A z = Ψl (r ) exp(ik z z − i ωk t ),

(11)

where Ψl (r ) is a slowly varying function of z, and r = (x + y ) . By using Eq. (11) we can write Eq. (8) in a paraxial approximation (viz. ∂ 2 Ψl /∂ z2  k2z Ψl ) as 2



2i

 ∂ 2 Ψl = 0, + ∇⊥ ∂z

2 1/ 2

(12)

where z and ∇⊥ are in units of λz = 1/k z and 1/λe , respectively. 2 Furthermore, we have denoted the operator ∇⊥ Ψl = (1/r )(∂/∂ r )(r ∂Ψl /∂ r ) + (1/r 2 )∂ 2 Ψl /∂θ 2 , and introduced the cylindrical coordinates with r = (r , θ, z). The solution of Eq. (12) can be written as a superposition of Laguerre–Gaussian (LG) modes [22–24], each of them representing a state of orbital angular momentum, characterized by the quantum number l, such that



Ψ pl F pl (r , z) exp(ilθ),

(13)

pl

1/ 2

 2 ∂ Az

2 1 − λe2 ∇⊥

ϕc = arctg(k z /k⊥ ),

Ψl =

∂φ 1 − λe2 ∇⊥ +c = 0, (7) ∂t ∂z where λe = c /ω pe is the electron skin depth or electron inertial length, and ω pe = (4π n0 e 2 /me )1/2 the electron plasma frequency.



Alfvén wave, ωk = k z V A /(1 + k2⊥ λe2 )1/2 , while for k2⊥ λe2  1, we have the angular frequency, ω = k z V A /(1 + k2z λ2i )1/2 , of the magnetic field-aligned dispersive Alfvén wave, where λi = c /ω pi is the ion inertial length. Furthermore, Eq. (9) predicts that the energy of the DSAWs can propagate along a resonance cone spreading at an angle ϕc , given by

(6)

where ω pi = (4π n0 e /mi ) is the ion plasma frequency. Eq. (6) represents the generalized ion vorticity equation that relates the 2 vorticity and electron current, which are proportional to ∇⊥ φ 2 and ∇⊥ A z . By using Eq. (2) into the parallel component of the electron momentum equation, we obtain 2

2793

where the mode structure function is |l|

F pl (r , z) = H pl X |l| L p ( X ) exp(− X /2),

(14)

with X = r / w ( z), and w ( z) is the DSA beam width. The normal2

2

|l|

ization factor H pl and the associated Laguerre polynomial L p (x) are, respectively,

1 H pl = √ 2 π



(l + p )! p!

1/2 (15)

,

and |l|

Lp (X) =

exp( X ) d p Xp !

dXp



X l+ p exp(− X ) ,

(16)

where p and l are the radial and angular mode numbers of the DSAW orbital angular momentum state. In a special case with l = 0 and p = 0, we have a Gaussian beam. The LG solutions, given by Eq. (13), describe the feature of a twisted DSAV beam carrying OAM. In a twisted DSAV beam, the wavefront would rotate around the beam’s propagation direction in a spiral that looks like fusilli pasta (or a bit like a DNA double helix), creating a vortex and leading to the DSAV beam with zero intensity at its center. A twisted DSAV beam (or an Alfvénic tornado or a braided magnetic field structure) can be created with the help of two oppositely propagating three-dimensional DSAWs that are colliding in a magnetoplasma. Twisting of the DSAWs would occur because different sections of the wavefront would bounce off different steps, introducing a delay between the reflection of neighboring sections and, therefore, causing the wavefront to be twisted due to entanglement of the wavefronts, and take on the shape of the reflector. Thus, due to angular symmetry, Noether theorem guards OAM conservation for a DSAV beam that is associated

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P.K. Shukla / Physics Letters A 376 (2012) 2792–2794

with the parallel electric field, the sheared wave magnetic field, and finite density perturbations. To summarize, we have shown that a DSAW in a uniform magnetoplasma can propagate as a twisted vortex beam or a magnetic flux rope. The latter can trap plasma particles and energy and transport them from one region to another in space, solar wind and astrophysical, and laboratory plasmas [13,25]. The twisted Alfvénic swirls can be identified as observational signatures of rapidly rotating magnetic flux ropes (or magnetic tornadoes/braided magnetic fields), which can provide an alternative mechanism for particle and energy transport in the Earth’s auroral zone, in the solar atmosphere [26], and in laboratory experiments [11]. Furthermore, the present investigation of a twisted DSAV beam can also be exploited for diagnostic purposes, when the DSAW frequencies are near the ion gyrofrequency and the magnetic field-aligned wavelengths are of the order of the ion inertial length [5]. In closing, we mention that the present investigation is complimentary to previous studies on twisted high-frequency electromagnetic waves in astrophysical environments [27], and is also in focus [28,29] in connection with twisted ultrasound pulses. Acknowledgements

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20] [21] [22]

This research was partially supported by the Deutsche Forschungsgemeinschaft (DFG), Bonn, through the project SH21/3-2 of the Research Unit 1048. References [1] H. Alfvén, Nature (London) 150 (1942) 405. [2] P.K. Shukla, J.M. Dawson, Astrophys. J. Lett. 276 (1984) L49. [3] N.F. Cramer, The Physics of Alfvén Wave, Wiley, Berlin, 2001.

[23] [24] [25] [26] [27] [28] [29]

R.J. Stefant, Phys. Fluids 13 (1971) 440. P.K. Shukla, L. Stenflo, Phys. Fluids 28 (1985) 1576. G. Brodin, L. Stenflo, Contrib. Plasma Phys. 30 (1990) 413. P.K. Shukla, L. Stenflo, in: T. Passot, P.L. Sulem (Eds.), Nonlinear MHD Waves and Turbulence, Springer, Berlin, 1999, pp. 1–30. B.B. Kadomtsev, in: V.D. Shafranov (Ed.), Reviews of Plasma Physics, Kluwer Academic Publishers, New York, 2001, pp. 189–192. D. Dastgeer, P.K. Shukla, Phys. Rev. Lett. 102 (2009) 045004. P.A. Damiano, A.N. Wright, J.F. McKenzie, Phys. Plasmas 16 (2009) 062901. C.A. Kletzing, D.J. Thuecks, F. Skiffs, S.R. Bounds, S. Vincena, Phys. Rev. Lett. 104 (2010) 095001. S.K.P. Tripathi, W. Gekelman, Phys. Rev. Lett. 105 (2010) 075005. W. Gekelman, S. Vincena, B. Van Compernolle, G.J. Morales, J.E. Maggs, P. Pribyl, T.A. Carter, Phys. Plasmas 18 (2011) 055501. W. Gekelman, E. Lawrence, B. Van Compernolle, Astrophys. J. 753 (2012) 131. P.K. Shukla, R. Bingham, J.F. McKenzie, I. Axford, Solar Phys. 186 (1998) 161. A. Hasegawa, L. Chen, Phys. Rev. Lett. 36 (1976) 1362. P.K. Shukla, B. Eliasson, L. Stenflo, R. Bingham, in: J. Weiland (Ed.), Recent Research Developments in Plasma Physics, Transworld Research Network, Trivendrum, Kerala, India, 2007, pp. 51–74. K. Stasiewicz, P.K. Shukla, G. Gustafsson, et al., Phys. Rev. Lett. 90 (2003) 085002. D. Sundkvist, V. Krasnoselskikh, P.K. Shukla, et al., Nature (London) 436 (2005) 825. C.C. Chaston, C. Salem, J.W. Bonnell, et al., Phys. Rev. Lett. 100 (2008) 175003; C. Chaston, J. Bonnel, J.P. McFadden, et al., Geophys. Res. Lett. 35 (2008) L17508. S.W. McIntosh, B.D. Pontieu, M. Carlsson, V. Hansteen, P. Boerner, M. Goosens, Nature (London) 475 (2011) 477. L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, J.P. Woerdman, Phys. Rev. A 45 (1992) 8185. L. Allen, S.M. Barnett, M.J. Padgett, Orbital Angular Momentum, Institute of Physics, Bristol, 2003. J.T. Mendonça, B. Thidé, H. Then, Phys. Rev. Lett. 102 (2009) 185005. K. Asamura, C.C. Chaston, Y. Hoh, et al., Astrophys. J. Lett. 745 (2012) L9. S. Widemeyer-Böhm, E. Scullion, O. Steiner, et al., Nature (London) 486 (2012) 505. M. Harwit, Astrophys. J. 597 (2003) 1266. J. Leckner, J. Phys.: Condens. Matter 18 (2006) 6149. A. Anhäuser, R. Wunenburger, E. Brasselet, Phys. Rev. Lett. 109 (2012) 034301.