PROTON TEMPERATURE ANISOTROPY AND MAGNETIC RECONNECTION IN THE SOLAR WIND: EFFECTS OF KINETIC INSTABILITIES ON CURRENT SHEET STABILITY

We use a two-dimensional hybrid PIC code (Matthews 1994), describing ions as particles and electrons as a massless charge neutralizing fluid. We initialize the system with a Harris equilibrium model (Harris 1962) for the magnetic field:

$$\begin{equation} B_y(x)=B_0 {\rm tanh}(x/l) \end{equation} \tag{ 1 }$$

$$\begin{equation} n(x)=n_{{\rm cs}} {\rm sech^2}(x/l)+n_b, \end{equation} \tag{ 2 }$$

with a maximum density n₀ = 1 at the current sheet. We add a background proton population with n_b = 0.2n₀ uniformly in the box. Units of space and time in the simulations are the ion inertial length c/ω_pi (ω_pi is the plasma frequency) and the inverse of the proton cyclotron frequency, Ω⁻¹_cp, respectively. The adopted simulation box is 50 c/ω_pi long in the x-direction and 200 c/ω_pi in the y-direction, with a 100×200 grid and spatial resolution Δx = 0.5 c/ω_pi and Δy = 1 c/ω_pi; we use 10³ particles per cell. The box has periodic boundaries in both directions and thus contains two current sheets of width l = 1.5c/ω_pi. Note that the spatial separation of the current sheets is such that they do not influence each other during the development of the tearing instability or of the anisotropy-driven kinetic instabilities. As expected, interactions between islands associated with distinct sheets are later observed, but only at the final stage of the nonlinear phase of the runs.

The parallel beta $\beta _\Vert =8\pi nk_BT_\Vert /B_0^2$ of the background plasma is set equal to 1 and the electron beta to 0.5. The temperature anisotropy ($T_\perp /T_\Vert$), with respect to the equilibrium magnetic field, of both populations is varied in order to test the stability of the magnetic configuration with respect to different plasma conditions. Note that in such a configuration there are regions between the current sheets of homogeneous and constant B₀ magnetic field, aligned with ±y (see top left panel of Figure 1), where fluctuations generated by small-scale processes can develop and propagate.

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We start our analysis with a simulation in which all the plasma is isotropic and focus on the development of a tearing instability from the initial configuration. In the top panels of Figure 1, we report the evolution of the current sheets at two simulation times. At the top left, at time t = 670, at the end of the linear growth a tearing mode with m = 1 is observed; at a later time, t = 780, the dynamics becomes nonlinear and a consequent strong deformation of the current is visible in the top right panel. We underline that the instability passes through a long linear phase and highly nonlinear only after t ∼ 650. This assures that, although the adopted initial conditions are not an exact kinetic equilibrium (e.g., Daughton 1999; Belmont et al. 2012), we do not observe other processes destabilizing the system on a shorter timescale than the classic tearing instability.

We consider now the case of a plasma with a temperature anisotropy. We have performed simulations using several different initial temperature anisotropies for the current sheet population. At this stage the background proton population is maintained isotropic. Our investigation qualitatively confirms the picture of Chen & Palmadesso (1984), with a faster tearing for larger perpendicular temperature, and a weaker instability if the parallel temperature is larger. Increasing the anisotropy with larger perpendicular temperatures leads to a shift of the most unstable tearing mode to larger wavenumbers with an increase of the instability growth rate. This is illustrated in the bottom panels of Figure 1 that reports the evolution of J_z for a simulation with $T_\perp =2T_\Vert$ at times t = 200 (left) and t = 290 (right). Starting with such an initial anisotropy for the proton population at the current sheet gives rise to the excitation of a higher mode with m = 6, and this occurs on a significantly faster timescale compared to the isotropic case (top panels of the figure), leading to the formation of several magnetic islands along the current sheets; subsequently, these are observed to merge giving rise to a strong nonlinear phase (right panel) already at t ∼ 290.

The top panel of Figure 2 summarizes the results obtained for various initial temperature anisotropies, reporting the reconnected flux evolution for different runs. The dotted line refers to the isotropic reference case. From right to left, the lines encode $T_\perp =0.9T_\Vert$ (dotted), $T_\Vert =T_\perp$ (dashed), $T_\perp =1.2T_\Vert$ (dash-dot-dot-dotted), $T_\perp =1.4T_\Vert$ (solid), and $T_\perp =1.5T_\Vert$ (dash-dotted). Note that in the figure the linear phase of the instability ("tearing") corresponds to the flatter initial part, and that the strong increase of the flux identifies the nonlinear phase, with large amplitude island growth and merging. We observe that when increasing the anisotropy (with $T_\perp /T_\Vert >1$), the linear slope steepens and the nonlinear stage is significantly accelerated. On the other side, when $T_\Vert >T_\perp$, the whole process is delayed. In agreement with previous studies (Chen & Palmadesso 1984; Ambrosiano et al. 1986), our results confirm that the linear growth rates also depend on the current sheet thickness; we have checked that all the growth rates reported in the figure significantly decrease when taking larger sheet widths.

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It is worth noting that when extending the investigation to large temperature ratios the plasma can become unstable with respect to kinetic instabilities driven by enhanced thermal anisotropies. In particular, an ion-cyclotron instability can develop in a $T_\perp >T_\Vert$ condition (e.g., Gary et al. 2003). This mechanism produces fluctuations that scatter particles and reduce the initial anisotropy bringing the system toward a marginal stable condition. (Note that a mirror instability can be also excited in the $T_\perp >T_\Vert$ regime (e.g., Hellinger 2007).) It is then interesting to investigate what happens in the adopted configuration when the injected anisotropy is such to destabilize this linear instability and what are its consequences on the stability of the current sheets. To deal with this aspect, we have performed simulations where the temperature anisotropy is not confined in the current sheet region, as in previous section, but it is extended to the whole background plasma. Note that when this new condition is studied in a regime of stability for the background protons (i.e., weak anisotropy), no peculiar differences are observed in the evolution of the tearing instability with respect to the previous cases; the surrounding plasma remains quiet and homogeneous, because stable, while the current is still destabilized with rates consistent with those reported in top panel of Figure 2, which correspond to an isotropic background plasma.

On the contrary, when the initial anisotropy of the background is such that the uniform regions of plasma are unstable (with $T_\perp =2T_\Vert$), we observe in the early stages of the simulation the generation of an ion-cyclotron mode, corresponding to the most unstable wavenumber, that starts to propagate along the local magnetic field. The left panel of Figure 3 reports the out-of-the-plane B_z fluctuations associated with the ion-cyclotron instability which characterizes the region of homogeneous magnetic field B_y bounded by the current sheets. As soon as these are generated, such fluctuations start to perturb the current sheet profiles (right panel); as we will discuss later, thanks to the symmetry imposed by the ion-cyclotron fluctuations at the two sides of the sheet, the forced perturbation consists of a tearing modulation of the current and since this occurs at a scale close to the most unstable current tearing wavenumber, the instability is efficiently triggered. As a consequence, this coupling leads to a fast driving of the tearing instability, as shown by the solid line in the bottom panel of Figure 2, which reports the time evolution of the reconnected flux; the dissipation of the current occurs on a shorter time than in the case when the background is isotropic (dashed line). Moreover, it is remarkable that in the linear part, the growth of the tearing is driven by the rms of the B_z fluctuations excited by the ion-cyclotron instability. This result demonstrates that unstable fluctuations generated by ion-cyclotron instability importantly influence the stability of the magnetic equilibrium and they can directly accelerate the development of the tearing instability.

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It is also important to underline that, while the unforced linear tearing depends strongly on the thickness of the current, on the other side the excitation of fluctuations by kinetic instabilities depends only on the plasma conditions outside the current sheet (temperature anisotropy and beta). As a consequence, the trigger of the tearing instability by ion-cyclotron fluctuations results approximatively independent of the sheet thickness, as shown by the dotted line in the bottom panel of Figure 2, reporting the reconnected flux for a case with a larger thickness l = 3 but same proton anisotropy in the background. Despite the increase of sheet thickness and unlike the classical linear tearing, we still recover a fast tearing, with a growth compatible to that of the thinner sheet case. Note that the only difference consists of a longer linear phase, since the tearing of the thicker current needs a longer time to become nonlinear.

An important question at this point concerns whether the presence of any fluctuations induced by a small-scale process would be sufficient to trigger such a faster destabilization of the current (Coppi 1983), and if this may occur independently of the sign of the plasma anisotropy. To address this issue, we have considered a case with the opposite temperature anisotropy, where, due to a large $T_\Vert >T_\perp$ anisotropy, protons of the background plasma are predicted unstable with respect to a fire-hose instability by the linear Vlasov theory. In this situation ($T_\Vert =2T_\perp$ and $\beta _\Vert =4$), a strong fire-hose mode is soon generated in the box: the top panel of Figure 4 reports the time history of the B_z Fourier component corresponding to this mode, showing its exponential growth in time for t < 100. The growth rate associated with this mode is in qualitative agreement with the linear Vlasov prediction for homogeneous plasma, even if a deviation is observed, probably due to the more complicated plasma configuration adopted in the simulation.

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As expected, the effect of this instability is to scatter particles and reduce their temperature anisotropy, driving the plasma to a more isotropic marginal stable state (lower panel). The saturation of the magnetic field fluctuations at t ≳ 100 corresponds to the stabilization of the plasma; after this stage, the proton anisotropy remains constant and the fire-hose fluctuations are slowly damped in time. Note that the observed behavior, which is in good agreement with the standard evolution of this instability found in simulations of homogeneous plasma (e.g., Matteini et al. 2006), is recovered here in the case of a non-homogeneous plasma, supporting the idea that such processes can be at work in real plasmas, like the solar wind, that are characterized by nonthermal distributions and small-scale coherent structures.

The activity of the fire-hose instability described in the simulation influences and perturbs the magnetic field structure; it is worth noting that the fire hose is an instability of magnetic field oscillations (e.g., Davidson & Völk 1968) and thus leads to an oscillatory modulation of magnetic field lines. As a consequence, its influence on the current sheets is to produce a kink deformation of J_z, which becomes more and more significant as the fire-hose unstable mode grows and reaches a macroscopic scale before saturation, as shown in the top panels of Figure 5. It is remarkable that this macroscopic perturbation of the current does not give rise to any tearing instability of the current sheets. On the contrary, these are observed to oscillate passively under the effect of the fire-hose activity. Such kink-type oscillations are later damped as soon as, according to Figure 4, the fire-hose mode is reabsorbed by the plasma at longer times. As shown in the bottom panels of Figure 5, at t = 1000, after the oscillatory stage, the system appears stable again and the current sheets recover their initial equilibrium configuration as at the beginning of the simulation. Starting from this late condition, a tearing instability can then eventually develop and produce the collapse of the current, but only in a significantly longer time than in the cases that are stable against the fire hose.

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To investigate in more detail the origin of the different modulation of the current sheets observed in the simulations, the color contour in Figure 5 reports the magnetic fluctuations B_x associated with ion-cyclotron (left) and fire-hose (right) instabilities in the background plasma. The superimposed black solid line contours encode the density of protons within the current sheet. Arrows identify the local direction of the electric field generated by the kinetic instability fluctuations. As shown by the figure, the modulation of the current sheet is very well correlated to these fluctuations, suggesting that the resulting deformation of the current is guided by the background activity, leading to a different evolution according to the different symmetry of the fluctuations. The spatial structure of the x-component of the magnetic field, B_x, is a relevant quantity in our analysis since, through the term J × B = c/4π(∇ × B) × B, it generates an electric field component pushing particle toward and away from the current sheet. In our case, this component of E_x is proportional to (∂_yB_x)B₀, which clearly depends on the symmetry of B_x across the discontinuity. When B_x is symmetric through the current sheet (left panel, ion-cyclotron instability case), then the associated electric field has opposite sign on the two sides (due to the orientation change in B₀), leading to diverging or converging flows. On the contrary, when B_x is antisymmetric (right panel, fire-hose case), the electric field does not change orientation across the discontinuity (both B_x and B₀ change sign) and gives rise to an oscillation along y. The local direction of this contribution to the electric field is reported by the arrows in the figure and well explains either the kink or the tearing deformation of the current under the effects of the external, anisotropy driven, unstable fluctuations.