Addendum on “Collisionless magnetohydrodynamic turbulence in two dimensions”, Ann. Phys. 317, 1 (2005)

Annals of Physics 322 (2007) 1247–1248 www.elsevier.com/locate/aop

Addendum

Addendum on ‘‘Collisionless magnetohydrodynamic turbulence in two dimensions’’, Ann. Phys. 317, 1 (2005) Bhimsen K. Shivamoggi University of Central Florida, Orlando, FL 32816-1364, USA Received 22 May 2006; accepted 11 October 2006 Available online 29 December 2006

Recent in situ multi-satellite observations (Sundkvist et al. [1]) reported detection of short-scale (of the order of ion-sound gyroradius qs) vortices in the earth’s magnetospheric cusp region and interpreted them as a manifestation of kinetic Alfve´n turbulence. We wish to point out that these vortices may be theoretically viewed as consequences of spectral energy pile-up in kinetic Alfve´n turbulence (Shivamoggi [2]). Shivamoggi [2] in fact predicted the existence of vortices with length scales of the order of qs in this system as found in the multi-satellite observations [1]. Shivamoggi [2] gave a formulation of two-dimensional (2D) collisionless magnetohydrodynamic (MHD) turbulence that included the effects of both electron inertia and electron pressure (or parallel electron compressibility) and is applicable to a strongly magnetized collisionless plasma like that in the magnetosphere. Assuming the existence of an inertial range of the Kolmogorov type which is in a state of statistical equilibrium and the ‘‘energy’’ (associated with the Elsa¨sser variables Z  V  B, V and B being the flow velocity and magnetic field, respectively) cascades in this range smoothly through nonlinear processes toward small scales, the ‘‘energy’’ spectrum was deduced to be ( EðkÞ 

q

1=2

e1=2 V A k 3=2 ; kqs  1 1=2

e1=2 V A k 1=2 ; kqs  1

DOI of original article: 10.1016/j.aop.2004.12.006. E-mail address: [email protected]

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ð1Þ

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Addendum / Annals of Physics 322 (2007) 1247–1248

Fig. 1. The ‘‘energy’’ spectra in 2D collisionless MHD turbulence.

k being the wavenumber, VA being the speed of the shear Alfve´n waves in the magnetic field of the large-scale eddies and e being the mean ‘‘energy’’ dissipation (or transfer) rate.1 Eq. (1) is schematically shown in Fig. 1. The ‘‘energy’’ spectrum for the classical MHD regime (kqs  1) corresponds to the Iroshnikov–Kraichnan [3,4] spectrum in 2D classical MHD turbulence. The shallower ‘‘energy’’ spectrum corresponds to the collisionless MHD regime (kqs  1) and implies the possibility of an ‘‘energy’’ pile-up at scales k1  qs, and hence a bump in the ‘‘energy’’ spectrum in the spectral interval around k  q1 s , which may then be expected to lead to the creation of vortex/current-filaments of length scales of the order of qs that appear to be those found in the multi-satellite observations [1]. References [1] D. Sundkvist, V. Krasnoselskikh, P.K. Shukla, A. Vaivads, M. Andre, S. Buchert, H. Reme, Nature 436 (2005) 825. [2] B.K. Shivamoggi, Ann. Phys. 317 (2005) 1. [3] I.S. Iroshnikov, Sov. Astron. 7 (1964) 568. [4] R.H. Kraichnan, Phys. Fluids 8 (1965) 1385. 1

The collisionless regime result ðE  k 2 Þ in (1) can be traced to the fact that the eddy turn over time ^s given by (4.7) in [2] is modified for this regime as follows:  s ^s  s 1 þ k 2 d 2e : ð2Þ sA 1

So, collisionless effects (kde „ 0) cause further inhibition of the energy cascade!