Anisotropic scaling of magnetohydrodynamic turbulence

arXiv:0807.3713v1 [physics.plasm-ph] 23 Jul 2008 APS/123-QED Anisotropic scaling of magnetohydrodynamic turbulence Timothy S. Horbury∗ The Blackett Laboratory Imperial College London SW7 2AZ U.K. Miriam Forman Stony Brook University, Stony Brook, N. Y. 11794, U.S.A. Sean Oughton Department of Mathematics, University of Waikato, Hamilton, New Zealand (Dated: November 2, 2018) We present a quantitative estimate of the anisotropic power and scaling of magnetic field fluctu- ations in inertial range magnetohydrodynamic turbulence, using a novel wavelet technique applied to spacecraft measurements in the solar wind. We show for the first time that, when the local magnetic field direction is parallel to the flow, the spacecraft-frame spectrum has a spectral index near 2. This can be interpreted as the signature of a population of fluctuations in field-parallel wavenumbers with a k−2 ∥ spectrum but is also consistent with the presence of a “critical balance” style turbulent cascade. We also find, in common with previous studies, that most of the power is contained in wavevectors at large angles to the local magnetic field and that this component of the turbulence has a spectral index of 5/3. PACS numbers: 52.35.Ra,96.50.Bh,52.30.Cv,95.75.Wx Magnetised plasmas fill most of the Universe and in many regions, turbulence plays important roles in the transport of energy and momentum and the accelera- tion and scattering of charged particles. Many aspects of plasma turbulence remain poorly understood, however. Here we present results on one of these, the anisotropy of the energy spectrum of magnetohydrodynamic (MHD) turbulence with respect to the magnetic field. In classical hydrodynamics, velocity fluctuations δuk with a wavenumber k decay and transfer energy to smaller scales on the shear timescale, τS ≈1/(kδuk). Within the steady inertial range, far from the energy in- put (“outer”) and dissipation scales, this leads to the dimensional result (δuk)3 ∝ǫ/k, where ǫ is the energy dissipation rate per unit mass. This gives the familiar Kolmogorov energy spectrum P(k) ∝k−5/3, widely ob- served in hydrodynamic turbulence. In a plasma, fluc- tuations can also propagate, as Alfv´en waves parallel to the magnetic field, and this leads to the Alfv´en timescale, τA ≈1/(k∥VA), being dynamically important. Here k∥is the component of the wavevector of the fluctuation par- allel to the local magnetic field and VA the Alfv´en speed. If τA ≪τS and assuming isotropy with respect to the local field, this leads to Iroshnikov-Kraichnan turbulence [1, 2] where (δuk)4 ∝ǫVA/k and P(k) ∝k−3/2 [e.g. 3, 4]. However, measurements in both space plasmas and ter- restrial plasma devices have shown that turbulent fluc- tuations are not isotropic. They typically have much longer correlation lengths along the field than across it ∗Electronic address: t.horbury@imperial.ac.uk; URL: http://www.imperial.ac.uk/people/t.horbury [5, 6, 7, 8, 9, 10] and the spectral index for the mag- netic energy is nearer 5/3 than 3/2 [11, 12]. When there is an energetically significant large-scale magnetic field, anisotropic models of MHD turbulence are required [13, 14, 15, 16, 17, 18]. For example, in the “critical bal- ance” framework [14], turbulent energy evolves towards wavevectors where the shear and Alfv´en timescales are balanced and most power resides in wavevectors where τS ≤τA, i.e. k∥≤k2/3 ⊥ǫ1/3V −1 A . The solar wind is a unique environment in which to study space plasma turbulence: it is relatively accessible and can be directly measured in exquisite detail using spacecraft instruments [e.g. 12, 19, 20, 21]. The solar wind flows radially away from the Sun at a velocity V of several hundred km s−1, much faster than spacecraft motions (a few km s−1) or the plasma wave speeds (tens of km s−1). As a result, in the plasma frame spacecraft measure along a radial line. Using Taylor’s hypothesis [22], one can relate the spacecraft frame energy spectrum P(f) to the wavevector spectrum P(k) [23]: P(f) = Z d3k P(k) δ(2πf −k · V ). (1) Anisotropies in P(k) with respect to the magnetic field can be analysed by measuring how P(f) varies with the angle of the magnetic field to the flow, θB. The exact form of this anisotropy is unknown, but one can make approximations motivated by theory and com- pare predictions with observations. One simple approx- imation is to assume that P(k) = 0 except for wavevec- tors exactly parallel (so-called “slab”) or perpendicular (“2D”) to the local magnetic field [24]. The correspond- 2 ing frequency spectrum can be deduced from Eq. 1: P(f; θB) = Cslabf −γslab |cos θB|γslab−1 +C2Df −γ2D |sin θB|γ2D−1 , (2) where Cslab and C2D are constants and γslab and γ2D are the spectral indexes of these components. P is thus in- sensitive to the slab component when the field is perpen- dicular to the flow and insensitive to 2D when the field is parallel [24]. Crucially, one can determine the scaling of both components by measuring the spectral index α of P ≈f −α separately for θB

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