Evidence for wave scattering effects on protons

6.2 Evidence for wave scattering effects on protons

The diffusion in velocity space of plasma ions being in resonance with waves is an old subject (Kennel and Engelmann, 1966 ) of plasma physics and has been studied in the literature in very much detail. Yet only recently have Galinsky and Shevchenko (2000

), Isenberg et al. (2000 ), Isenberg et al. (2001

), Cranmer (2001

), Vocks and Marsch (2001

), Vocks and Marsch (2002

) and Vocks and Mann (2003

) applied QLT also to the solar corona and solar wind (see the subsequent Section 7). QLT predicts that ions in resonance with transverse ion-cyclotron waves, propagating parallel to the magnetic field, undergo merely pitch-angle diffusion, while conserving their total kinetic energy in a frame moving with the wave phase speed $vM (k)$ . If the wave growth rate remains small, the slowly varying part of the ion VDF is controlled by diffusion. Its time evolution will, if the wave power is large enough, lead to a time-asymptotic state given by

where $v∥0$ and $v⊥ 0$ are the initial values, and whereby $v∥$ has to satisfy the resonance condition, $ω (k ) − k v − Ω ∥ ∥ ∥ j$ . Here $k ∥$ is the parallel wave-vector component, and $ω (k ) ∥$ the frequency in the inertial frame where the waves are assumed to propagate. In the case of plateau formation, the particles conserve their energy in this wave frame.

Observational evidence from Helios plasma data has been obtained for the occurrence of proton pitch-angle diffusion (Marsch and Tu, 2001b ). A comparison of the cyclotron-wave diffusion plateau, as it is predicted by using the cold plasma dispersion relation in the plateau condition of Equation (56 ), with the Helios observations is shown in Figure 14 . The VDFs in the left and right frames show the plateaus defined by vanishing pitch-angle gradients (also implying marginal plasma stability). Parts of the isodensity contours in velocity space shown in Figure 14 are outlined well by a sequence of segments of circles centered at the respective phase speeds (bold dots indicate its location), which are assumed to vary slightly, and due to dispersion are smaller than the local Alfvén speed. For the contours between $0.2$ and $0.4$ of the maximum density, the plateau can be as wide as 70 degrees in pitch angle as calculated in the wave frame. The horizontal axis refers to the parallel proton velocity component, whereby an outward velocity has a positive value. The dotted lines show the density contours observed by Helios at $0.3 AU$ in a high-speed wind. In the right frame, $R = 0.3 AU$ , $Vsw = 678 km s−1$ , and $VA = 184 km s−1$ . The diffusion plateaus of protons in resonance with left hand polarised cyclotron waves are shown by the solid lines. For $v∥ < 0$ , a proton is in resonance with outward waves, and for $v∥ > 0$ with inward propagating waves. The solid lines are the numerical solutions of Equation (56 ).

Figure 14:

Left: Comparison of measured proton velocity contours with the quasilinear plateau. The horizontal axis gives $v∥$ and the vertical $v⊥$ (in units of $km s− 1$ ). The measured contours correspond to fractions of $0.8$ , $0.6$ , $0.4$ , $0.2$ of the maximum. The dotted lines (circular arcs) delineate the theoretical contours shaped by diffusion Marsch and Tu (after 2001b ). Right: Another comparison of the cyclotron diffusion plateau in velocity space with proton observations. The solid lines on the left hand (right hand) side of the vertical axis represent the theoretical contours (for zero pitch-angle gradient) formed by the cyclotron resonance of protons with outward propagating (inward) left hand circularly polarised cyclotron waves. The dotted contours are measured and correspond to fractions $0.8$ , $0.6$ , $0.4$ , $0.2$ , $0.1$ , of the maximum of the VDF, respectively. The dispersion relation of a cold plasma with protons and electrons was used to calculate the phase speed. The Alfvén speed is $184 km s− 1$ Tu and Marsch (after 2002

).

The observations shown in Figure 14 suggest that Alfvén-cyclotron fluctuations propagating parallel or antiparallel to the background magnetic field influence the shape of the ion VDFs. The waves may be generated at low, non-resonant frequencies and, by propagation through the inhomogeneous coronal plasma, approach the ion-cyclotron resonances and by proton scattering cause their anisotropy. In turn, ion thermal anisotropies of sufficient magnitude can lead to growth of ion-cyclotron instabilities. The resulting enhanced Alfvén-cyclotron fluctuations scatter the ions and thereby reduce their original anisotropy.

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Figure 15:

Numerical simulation of the proton velocity distribution $fp(v∥,vy)$ at $tΩp = 56$ . The phase speeds, $ω ∕k ∥$ , are for five left hand polarised wave modes indicated on the right hand side by the five dots at the locations with $v⊥ = 0$ and $v∥ > 0$ . The five dots on the left axis represent the corresponding cyclotron resonant velocities. The related solutions of the plateau Equation (56

) are indicated by the five heavy solid lines. Note that the diffusion can even render the ions cross the $v∥ = 0$ line (after Gary and Saito, 2003

).

Gary and Saito (2003 ) have carried out particle-in-cell simulations of Alfvén-wave-scattering of protons in a magnetised, homogeneous, collisionless model plasma of electrons and one ion species to study the evolution of the VDFs in response to these scattering processes. A solar wind simulation with a spectrum of right-travelling Alfvén-cyclotron fluctuations initially imposed leads according to Gary and Saito (2003 ) to highly non-Maxwellian proton VDFs. Their computations are illustrated in Figure 15 and show that the pitch-angle scattering of left-travelling (with $v∥ < 0$ ) ions becomes weaker, as their parallel speed becomes less negative, but also that such scattering can even transport ions across the line at $v∥ = 0$ . This important numerical result confirms the basic observational features.