Scale dependent alignment between velocity and magnetic field fluctuations in the solar wind and comparisons to Boldyrevs phenomenological theory

Phenomenological turbulence theories developed independently by Iroshnikov [1] and Kraichnan [2] extended and adapted the ideas of Kolmogorv's well known theory of hydrodynamic turbulence to incompressible MHD turbulence. Both Iroshnikov and Kraichnan predicted an equipartition of energy between kinetic and magnetic field fluctuations in the inertial range and an energy spectrum proportional to k -3/2 . But these early studies neglected the anisotropy of the turbulence which was subsequently found to be ubiquitous in both laboratory plasma experiments and in theoretical studies based on analysis and simulations of the equations of resistive incompressible MHD [3]. Turbulence in magnetized plasmas is spatially anisotropic with the local mean magnetic field providing a natural axis of symmetry.

More recent theories of incompressible MHD turbulence incorporate the anisotropy of the turbulence into the theory in a fundamental way. An influential theory of this kind is the theory of Goldreich and Sridhar (1995) [4], hereafter GS95, with important corrections by Goldreich and Sridhar (1997) [5]. GS95 introduced the idea of ‘critical balance’ in which there is a balance between the eddy turnover time, or energy cascade time, and the Alfvén crossing time of two wavepackets propagating in opposite directions along the local mean magnetic field. As a consequence of critical balance, the GS95 theory predicts that for a typical wavepacket the wavelengths parallel and perpendicular to the local field satisfy the anisotropic relation λ ⊥ ∝ λ 2/3 and the perpendicular energy spectrum is proportional to k -5/3 ⊥ . The decade following the publication of GS95 saw improved simulations of incompressible MHD turbulence in two and three dimensions, many of which showed that for plasmas with a strong mean magnetic field |B B B 0 | comparable to or greater than the r.m.s. magnetic field fluctuations the perpendicular energy spectrum scales like k

in contradiction to the GS95 theory [6,7,8,9]. To resolve this discrepancy Boldyrev [10,11] developed a new phenomenological theory with a critical balance condition different from that of GS95. A somewhat different theory was derived by Beresnyak and Lazarian [12]. The key new idea introduced by Boldyrev is that as energy cascades from large to small scales through the inertial range the velocity and magnetic field fluctuations undergo an alignment process whereby the average angle θ between δ v v v ⊥ and δ b b b ⊥ is a monotonically decreasing function of scale. In his theory, Boldyrev predicts that the angle obeys the scaling law θ ∝ λ 1/4 ⊥ and that the perpendicular energy spectrum is proportional to k

Evidence for Boldyrev’s alignment process and for the scaling law θ ∝ λ 1/4 ⊥ have been obtained through direct numerical simulations of forced, steady state incompressible MHD turbulence in three dimensions by Mason et al. [13,14]. These simulations demonstrate that for a number of different types of forcing functions the system develops an inertial range spanning approximately one decade in wavenumber where the perpendicular energy spectrum is proportional to k prompted us to consider whether this alignment process may also take place in the solar wind, a naturally occuring turbulent plasma that is directly accessible to in-situ spacecraft measurements [15,16,17,18]. Therefore a study was undertaken to investigate the possible existence of a scale dependent alignment between velocity and magnetic field fluctuations in the solar wind. The results of this study shall now be briefly summarized.

Boldyrev and his colleagues measure the angle θ using the formula [13,14]

where

) onto the plane perpendicular to the local mean magnetic field B B B 0 (x x x), respectively, and the displacement r r r is perpendicular to B B B 0 (x x x). It is desirable but not possible to employ precisely the same formula (1) when analyzing solar wind data. A single spacecraft essentially performs measurements along the solar wind flow direction. Because the flow is super-Alfvénic, Taylor’s “frozen turbulence” hypothesis may be used to relate the time τ between measurements to a spatial separation r = V sw τ along the average flow direction, approximately the radial direction in heliocentric coordinates. The angle between the mean magnetic field and the average flow direction varies significantly due to variations in the direction of the mean magnetic field, but it is often near 45 degrees, the inclination of the Parker spiral at 1 AU. Even though the theory is formulated for fluctuations between two points x x x and x x x + r r r inclined at 90 degrees to the local mean magnetic field, we expect that any alignment (if it exists) will still be measureable at inclinations of 45 degrees [19].

The solar wind data employed in this study consists of simultaneous measurements of the average velocity vector and magnetic field vector from instruments on the Wind spacecraft. The data was acquired near the orbi

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