During the second phase, protons are heated. For negligible heat fluxes, collisions, and fluctuations, one expects the double adiabatic behavior or CGL (Chew et al. 1956; Matteini et al. 2012): the parallel and perpendicular temperatures (with respect to the magnetic field) are expected to follow 





 and 





 respectively. Tp⊥ decreases slower than 





 during the whole simulation; protons are heated in the perpendicular direction while in the parallel direction the heating lasts till about t ∼ 0.25te0, whereas afterward protons are cooled. The parallel and perpendicular heating rates could be estimated as (see Verscharen et al. 2015)
1





A more detailed analysis indicates that between t = 0.1te0 and t = 0.7te0 the parallel heating rate 





 smoothly varies from about 0.2 Qe and 





 whereas 





 is about constant at ∼0.2 Qe; here, 





 In total, protons are heated until t ∼ 0.7te0, and the heating reappears near the end of the simulation t ≳ 0.95te0. Note that the perpendicular heating rate is a nonnegligible fraction of that observed in the solar wind, where 





 (Hellinger et al. 2013); however, the proton heating in 2D hybrid simulations is typically quite sensitive to the used electron equation of state (Parashar et al. 2014) and also to the resistivity and the number of particles per cell used (Franci et al. 2015a). The turbulent heating is, however, not sufficient to overcome the expansion-driven perpendicular cooling as in the solar wind (Matteini et al. 2007). During the third phase, t ≳ 0.7te0; there is an enhancement of the parallel cooling and perpendicular heating that cannot be ascribed to the effect of the turbulent activity. For a large parallel proton temperature anisotropy, a firehose instability is expected. The presence of such an instability is supported by the fact that the fluctuating magnetic field increases (with respect to the linear prediction), suggesting a generation of fluctuating magnetic energy at the expense of protons. To analyze the role of different processes in the system, we estimate their characteristic times (Matthaeus et al. 2014). The bottom panel of Figure 2 compares the turbulent nonlinear eddy turnover time 





 at kdp = 1 (see Matthaeus et al. 2014; the expansion time te, and the linear time tl of the oblique firehose (Hellinger & Matsumoto 2000, 2001) estimated as 





 where γm is the maximum growth rate calculated from the average plasma properties in the box assuming bi-Maxwellian proton velocity distribution functions (Hellinger et al. 2006). The expansion time te is much longer than tnl at kdp = 1 (as well as at the injection scales). The expanding system becomes theoretically unstable with respect to the oblique firehose around t ∼ 0.47te0 but clear signatures of a fast proton isotropization and of a generation of enhanced magnetic fluctuations appear later t ≳ 0.7te0. This is about the time when the linear time becomes comparable to the nonlinear time at ion scales. After that, 





 slightly increases as a result of a saturation of the firehose instability, whereas 





 at kdp = 1 is about constant (note that Ωp decreases as R−2). This may indicate that the instability has to be fast enough to compete with turbulence; however, the 2D system has strong geometrical constraints. Also the stability is governed by the local plasma properties. Figure 3 shows the evolution of the system in the plane 





 During the evolution, a large spread of local values in the 2D space 





 develops. Between t ≃ 0.1te0 and t ≃ 0.65te0, the average quantities evolve in time following 





 This anticorrelation is qualitatively similar to in situ Helios observations between 0.3 and 1 AU (Matteini et al. 2007). During the third stage, when the strong parallel temperature anisotropy is reduced, both local and average values of 





 and Ap appear to be bounded by the linear marginal stability conditions of the oblique firehose (Hellinger & Trávníček 2008), although relatively large theoretical growth rates 





 are expected.
Citation: Hellinger et al. 2013
Citation function: Uses