Effects of the Background Turbulence on the Relaxation of Ion Temperature Anisotropy in Space Plasmas

Turbulence in space plasmas usually exhibits two distinct regimes separated by a spectral break that divides the inertial and kinetic ranges in which wave-wave or wave-particle interactions dominate, respectively. Large scale fluctuations are dominated by MHD non-linear wave-wave interactions following a -5/3 or -3/2 slope power-law spectrum, that suddenly ends and after the break the spectrum follows a steeper power-law $k^{-\alpha}$ shape given by a spectral index $\alpha>5/3$. Despite its ubiquitousness, few research have considered the possible effects of a turbulent background spectrum in the quasilinear relaxation of solar wind temperatures. In this work, a quasilinear kinetic theory is used to study the evolution of the proton temperatures in a solar wind-like plasma composed by cold electrons and bi-Maxwellian protons, in which electromagnetic waves propagate along a background magnetic field. Four wave spectrum shapes are compared with different levels of wave intensity. We show that a sufficient turbulent magnetic power can drive stable protons to transverse heating, resulting in an increase in the temperature anisotropy and the reduction of the parallel proton beta. Thus, stable proton velocity distribution can evolve in such a way as to develop kinetic instabilities. This may explain why the constituents of the solar wind can be observed far from thermodynamic equilibrium and near the instability thresholds.


INTRODUCTION
In many space environments the media is filled by a poorly collisional tenuous plasma. As Coulomb collisions represent an efficient mechanism for relaxing plasma populations towards a thermodynamic equilibrium state in which the particle Velocity Distribution Functions (VDFs) achieve a Maxwellian profile [1,2], when collisions are scarce Coulomb scattering becomes ineffective in establishing equilibrium. Subsequently, kinetic collisionless processes may dominate the dynamics of the system and be responsible for many of the observed macroscopic and microscopic properties of the plasma. Under these conditions the plasma VDF usually develops non-Maxwellian characteristics that can provide the necessary free energy to excite micro-instabilities that subsequently can induce changes on the macroscopic properties of the plasma. Among the fundamental problems of plasma physics belongs the understanding of the excitation and relaxation processes of these poorly collisional plasmas and the resultant state of nearly equipartition energy density between plasma particles and electromagnetic turbulence [3]. In particular, these processes play an important role in space plasma environments such as the solar wind [4,5,6,7] and the Earth's magnetosphere [8,9,10], specially at kinetic scales [11,12,13].
It is well known that in space plasmas turbulence usually exhibits two distinct regimes separated by a spectral break that divides the scales in which wave-wave or wave-particle interactions dominate. Namely, the inertial (at larger scales) and the kinetic range (at ionic and sub-ionic scales) [14,7]. Large scale fluctuations are dominated by MHD non-linear wave-wave interactions following a -5/3 or -3/2 slope power-law spectrum, that suddenly ends and after the break the spectrum follows a steeper power-law k −α shape given by a spectral index α > 5/3. The break is related with the scales in which kinetic effects and wave-particle interactions become dominant, and depending on the local plasma conditions the break can coincide with the ion inertial length or gyroradius [15,16]. Also, different plasma environments can exhibit different spectral indices. For example, considering Van Allen Probes observations Moya et al. Moya  In a magnetized plasma such as the solar wind or the Earth's magnetosphere, one of the most typical deviations from the Maxwellian equilibrium is the bi-Maxwellian distribution, representing a composed Maxwellian VDF that exhibits different thermal spreads (different temperatures) in the directions along and perpendicular to the background magnetic field. These distributions are susceptible to temperature anisotropy driven kinetic micro-instabilities that can effectively reduce the anisotropy and relax the plasma towards more isotropic states. However, in the absence of enough collisions, these instabilities are usually not able to lead the system to thermodynamic equilibrium and the plasma allows a certain level of anisotropy up to the so called kinetic instability thresholds [11,20]. From the theoretical kinetic plasma physics point of view, on the basis of the linear and quasilinear theory approximation of the dynamics of the plasma, it is possible to predict the thresholds in the temperature anisotropy and plasma beta parameter space that separate the stable and unstable regimes, and how the plasma evolves towards such states. These models are useful to study the generation and first saturation of the electromagnetic energy at the expense of the free energy carried by the plasma. To do so, in general quasilinear calculations consider initial conditions with a small level of magnetic field energy that grows as the temperature anisotropy relaxes. A comprehensive review of linear and quasilinear analysis of these instabilities considering a bi-Maxwellian model can be found in Yoon [21] and references therein.
Since the first studies by Weibel [22] and Sagdeev and Shafranov [23], the research about temperature anisotropy driven modes and the stability of the plasma have been widely studied in the last decades, and represent an important topic for space plasmas physics [24,25,26,27,28]. Predictions based on a bi-Maxwellian description of the plasma are qualitatively in good agreement with observations of solar wind protons (see e.g. Hellinger and Trávníček [29], Bale et al. [6]) and electrons (see e.g. Hellinger et al. [30], Adrian et al. [31]. However, as turbulence is ubiquitous in space environments (see e.g. Bruno and Carbone [7]), all these relaxation processes should occur in the presence of a background turbulent magnetic fluctuations spectrum. To the best of our knowledge, only a few quasilinear studies such as Moya et al. [32] or Moya et al. [28] have considered a background spectrum but nonetheless a study focused on the possible effects of a magnetic field background spectrum is yet to be done. Here we perform such systematic study by computing the quasilinear relaxation of the ion-cyclotron temperature anisotropy instability, considering different choices of the initial level of the magnetic field fluctuations, and the shape of the spectrum. We analyze their effect on the relaxation of the instability and the time evolution of the macroscopic properties of the plasma that are involved. In the next section we present the linear and quasilinear of our model. Then, in Sections 3 and 4 we show and discuss all our numerical results. Finally, in the last section we summarize our findings and present the main conclusions of our work.

QUASILINEAR TEMPERATURE EVOLUTION
We consider a magnetized plasma composed of bi-Maxwellian protons and cold electrons. The kinetic dispersion relation of left-handed circularly polarized waves, propagating along a background magnetic field B 0 is given by [33,32,34,35,36] where ω k = ω+iγ is the complex frequency that depends on the wavenumber k; v A = B 0 / 4πn p m p is the Alfvén speed, with n p and m p the density and mass of protons, respectively; Ω p = eB 0 /m p c is the proton gyrofrequency with c the speed of light; A = R − 1 where R = T ⊥ /T is the temperature anisotropy; T ⊥ and T are the proton temperatures perpendicular and parallel with respect to B 0 , respectively; ξ = ω k /ku and ξ − = (ω k − Ω p )/ku are resonance factors [37]; u = 2k B T /m p is the parallel proton thermal speed, and k B the Boltzmann constant. Z(ξ) is the plasma dispersion function [38], which is calculated numerically with the Faddeeva function provided by scipy. We also define the parallel proton β = u 2 /v 2 A . In Eq. (1), we have assumed charge neutrality (i.e. zero net charge such that the electron density n e is equal to the proton density), and v A /c 1. Numerical roots of Eq. (1) are calculated through the Muller's method [39] using our own Python code. The dispersion relation Eq. (1) supports an infinite number of solutions for ω k for each value of k, most of them being sound-like heavily damped modes with frequencies above and below the proton gyrofrequency [40,41]. Here, we focus on the quasilinear evolution of the plasma due to Alfvén-Cyclotron Wave (ACW) instabilities.  Figure 1(b) shows the effect of the anisotropy on the ACW frequency for β = 0.1. In all cases, this solution seems to approach asymptotically to ω = Ω p at large wavenumbers. This description is very similar to the solutions of the dispersion relation Eq. (1) in the cold-plasma approximation. However, for β > 0.01 [panel (a)] or , the frequency curve deviates from the cold-plasma approximation for wavelengths around the proton inertial length v A /Ω p .
Kinetic effects can damp ACWs of large wavenumbers even at low beta, and large temperature anisotropies can drive the wave unstable. Figures 1(c) and (d) show the imaginary part of the frequency for the same parameters as in Fig. 1(a) and (b), respectively. The wave is damped if its frequency satisfies Fig. 1(c) that the ACW is marginally stable (γ = 0) at a fixed wavenumber value v A k/Ω p ∼ 2.27 and fixed R = T ⊥ /T for all values of β . It can shown from Eq. (1) that this happens at (v A k/Ω p ) 2 = (R − 1) 2 /R [42]. Thus, for a lower value of the temperature anisotropy the waves are marginally stable at lower wavenumbers, as seen in Fig. 1(d), and the damping becomes stronger as the anisotropy approaches R = 1. Also, the instability decreases both with lower β and lower T ⊥ /T . It is important to mention that a semi-cold approximation of the plasma (ξ − 1) fails to describe these properties, making it inappropriate for the quasilinear evolution of the plasma temperature.
The quasilinear approximation assumes that the macroscopic parameters of the plasma evolve adiabatically, thus ω k solves the dispersion relation Eq. (1) instantaneously at all times. The quasilinear evolution of the perpendicular and parallel thermal speeds are given by [33,32] where L is the characteristic length of the plasma, |B k | 2 is the spectral wave energy satisfying such that Eqs. (1)-(4) form a closed system to address the quasilinear evolution of the ACW instability. In the next sections we explore the effects of the B k spectrum on the relaxation of the proton anisotropy.

NUMERICAL RESULTS. THE EFFECT OF A BACKGROUND SPECTRUM
In order to solve numerically the system of differential equations given by Eqs. (1)-(4) we use a fourth order Runge-Kutta method. First, here we compare the quasilinear relaxation with three different initial background spectra, namely a uniform noise |B k (0)| 2 = const., a Gaussian spectrum and a Lorentzian spectrum where the normalization constant A is adjusted depending on the initial total magnetic energy W  (d)]. Thus, we should expect that the temperatures will remain almost constant in time [41]. However, an striking feature for all the spectrum shapes, is that the anisotropy can grow in time if a sufficient level of magnetic energy is provided.
For an uniform spectrum of total level W B (0) = 0.1 (blue lines in Fig. 2), the anisotropy can grow up to high values R 14 in a small time frame. This results in a sharp increase in the perpendicular beta from β ⊥ = 0.01 to ≈ 0.014, and consistently a rapid fall in the parallel β = 0.005 towards ≈ 0.001. Afterwards, the anisotropy decreases while β rises, both steadily, towards a quasi-stationary state around R 7 and β ≈ 0.002. We note that this anisotropy growth is not as explosive for a Lorentzian (with α = 5/3, right column in Fig. 2) and a Gaussian spectrum (middle column) compared to a uniform spectrum, although they all relax to a final state around the same temperature anisotropy. This shows that high levels of a power spectrum may play a role on the regulation of the temperature anisotropies observed in different plasma environments. These results also suggests a possible mechanism to push the plasma towards the marginal stability thresholds starting from a linearly stable plasma.
The growth of the anisotropy is limited for smaller values of the magnetic field intensity, e.g. W B (0) = 0.01 (orange lines in Fig. 2) and W B (0) = 0.001 (green lines). As W B (0) is lowered to noise levels W B (0) < 10 −5 (not shown), the anisotropy and other parameters remain constant, which is consistent with the fact that the plasma is in an equilibrium state for β = 0.005, R = 2, and low levels of magnetic energy. In all cases, we observe that the total magnetic energy decreases monotonously, meaning that the quasilinear approximation is valid through the simulation runs. Figure 3 shows the time evolution of β , T ⊥ /T , and W B . A set of numerical simulations with evenly spaced initial conditions were chosen in the range 0.005 ≤ β ≤ 0.5 and 2 ≤ T ⊥ /T ≤ 7, with a uniform magnetic wave spectrum of power W B (0) = 0.1 for all cases. Following the same trend as observed in Fig. 2, the magnetic energy always decreases monotonously from W B (0) = 0.1 to values between 0.03 < W B (t f ) < 0.07. In all cases, β drops rapidly while the temperature anisotropy increases to high values above the stability thresholds. Afterwards, the magnetic wave power is not enough to supply energy to protons, so that T ⊥ /T slowly relaxes towards values where the maximum growth-rate of proton-cyclotron instability is of the order of γ/Ω p 0.001. [43] showed that the dynamical interaction between protons and electrons play a counter-balancing effect, which prevents the progression of protons toward the marginal firehose states. Similarly, Yoon [21] showed that collisional effects may drive the plasma from stable conditions towards the instability thresholds. Here, however, we show that this can happen due to a sufficient level of magnetic fluctuations. Although not shown here, the quasilinear evolution in the cases of a Gaussian or Lorentzian power spectrum are similar. They all excite some level of proton heating in the initial stage of the simulations, and then relax slowly towards the marginal stability state, with properties similar to the ones shown in Fig. 2. In the case of a Lorentzian or power-law wave spectrum, the proton heating depends on the slope of the spectrum, which is explored in the next section.

NUMERICAL RESULTS. THE EFFECT OF A TURBULENT SPECTRUM WITH A SPECTRAL BREAK
In the previous section we considered different spectrum for academic purposes, to illustrate how a different initial magnetic field background spectrum can produce different results on the evolution of macroscopic parameters of the plasma. A noise level of fluctuations can lead to the propagation of Alfvén waves, and instabilities are regulated in a quasilinear fashion towards marginal stability, normally far from thermodynamic equilibrium (or thermal isotropy with T ⊥ = T ). However, solar wind observations show that the plasma is mostly in a state below the instability thresholds, far from the isotropic state [4,29], with a non-negligible level of magnetic fluctuations [6,41], and that the magnetic field has a spectral break around the ion inertial length [15,16]. The inertial range v A k/Ω p < 1 typically shows a Kolmogorov-like spectrum B 2 k ∝ k −5/3 . For ion or sub-ions scales (in the kinetic range) the turbulent spectrum steepens to k −α , with α ≥ 2.0, arguably due to the characteristics of the dispersion relation of Alfvén or other waves in that range.
Here we compute the quasilinear relaxation considering a solar wind-like spectrum including a spectral break at the ion inertial range scale, with different slopes α in the kinetic range: W B (0) = dk|B k (0)| 2 /B 0 , with the integral calculated in the range 10 −3 < v A |k|/Ω p < 8. We tested with broader ranges in wavenumber, and we do not observe noticeable differences in all simulation runs.
In what follows, the initial anisotropy and total magnetic energy are chosen as T ⊥ (0)/T (0) = 2 and W B (0) = 0.1 for all simulation runs. For an initially low β (0) = 0.005, Fig. 4 (left column) shows that the proton distribution is cooled in the parallel direction with respect to the background magnetic field, as the parallel β decreases in time. Similarly, protons are heated in the transverse direction for all tested values of α. It is interesting to note that the magnetic energy decreases just 1% from the initial value, but causing a monotonous growth in the temperature anisotropy from T ⊥ /T = 2 to 4 for a low value of α = 5/3 1.7. For higher values of α, this parallel cooling and transverse heating is less efficient. This can be explained as a magnetic field with stepen slopes in their spectrum for v A k/Ω p > 1 do not contain enough energy to be transferred to the particles.
For β (0) = 0.05, we see in Fig. 4 (middle column) that the parallel cooling and transverse heating still occurs. It is worth noticing that the magnetic energy actually decreases more, but the parallel cooling and transverse heating is less efficient, than in cases with the same value of α and lower β (0) = 0.005. Comparing with Fig. 3, we see that both cases β = 0.005 and β = 0.05, with T ⊥ /T = 2, correspond to linearly stable plasmas. However, β = 0.05 is closer to the instability thresholds, meaning that the quasilinear evolution in this case takes shorter times to reach the a stationary state near the stability margins. For high β = 0.5, Fig. 4 (right column), the plasma is initially unstable. However, we observe that the magnetic energy is reduced in the first stages of the simulation, and it starts to grow until long times with different growth rates depending on the value of α. Since β increases in time while the temperature anisotropy is reduced, protons are effectively heated in the parallel direction and cooled in the transverse direction with respect to B 0 . However, the temperature evolution seems to be independent of α, at least in the cases we tested for high β (0). (bottom), respectively). As time goes on, the wave spectrum decreases for high values of k in all cases. For β (0) = 0.005, the spectral break is unmodified at all times of the simulation run, but the spectrum steepens for v A k/Ω p > 1.2. For β (0) = 0.05, the wave spectrum becomes smooth around v A k/Ω p = 1. In these two cases, transference of energy from the wave to protons results in a monotonous drop in magnetic energy as discussed in Fig. 4. For β (0) = 0.5, in the first stages of the simulation the electromagnetic wave loses energy at high values of k as in the previous cases. However, since the wave is unstable for this value of β , the wave energy starts to increase after some time. This results in a bump in the spectral wave energy around the wavenumber of the instability, 0.3 < v A k/Ω p 0.7. As shown in Fig. 4, the temperature anisotropy decreases and β increases. Comparing with Fig. 1, this implies that the range of unstable wavenumbers shifts towards smaller values, which in turns means that the bump in the spectral wave energy also shifts to smaller values of k. At the same time, previously unstable modes with high k become damped, thus the wave transfers energy to protons for values of v A k/Ω p > 0.7, resulting in a  steep spectrum near the initial spectral break. At larger times and since the rate at which the wave damps is negligible compared to its growth, this results in a total growth of magnetic energy as discussed in Fig. 4.

CONCLUSIONS
Starting form the Vlasov-Maxwell system of equations we have developed a quasilinear approach, in which it is possible to write differential equations that can describe the nonlinear wave-particle interaction between Alfvén cyclotron waves and protons in plasmas. Under this context We compare the quasilinear evolution of the plasma due a background magnetic field exhibiting different spectrum shapes: a uniform flat spectrum, Gaussian, Lorentzian, and a turbulent power-law spectrum including a spectral break at the proton inertial range.
A striking feature of all these spectrum shapes, is that the proton temperature anisotropy can grow in time if a sufficient level of wave intensity is provided. This happens even if protons are initially in an equilibrium state, where the proton velocity distribution is nearly isotropic and no kinetic instabilities are excited. In all cases we observe perpendicular heating and parallel cooling of protons with respect to the background magnetic field. This effect is more efficient as the intensity level of the wave spectrum is higher. For an uniform magnetic spectrum of total energy W B /B 2 0 = 0.1, the temperature anisotropy can rapidly grow from T ⊥ /T = 2 up to high values T ⊥ /T = 14 in a small time frame. Even though the plasma was considered initially in a stable state, this results in a fast decline of the parallel β = 0.005 towards β = 0.001, and the subsequent excitation of proton-cyclotron instabilities. Afterwards, the anisotropy decreases and β grows towards the instability thresholds where the temperatures stop evolving. A similar behavior is observed for a Lorentzian spectrum with power-law tail k −5/3 . On the other hand, the quasilinear evolution due to a Gaussian spectrum also results in an anisotropy growing starting from stable conditions. However, this growth is smooth over time and never exceeds the anisotropy values of the instability thresholds.
As the magnetic spectrum in space plasmas is not uniform, Gaussian, nor Lorentzian, we have laso tested with a Kolmogorov-like magnetic spectrum k −5/3 including a spectral break at v A k/Ω p , such that the kinetic range has a turbulent spectrum k −α with α > 5/3. Starting from T ⊥ /T = 2 and a total magnetic energy W B /B 2 0 = 0.1, the quasilinear equations show that the perpendicular heating and parallel cooling occurs only for low values of the initial β (0). This heating/cooling is more efficient for a spectrum with slope near α = 5/3, although the loss in the total magnetic energy is almost negligible for all α values and low beta. This happens because the magnetic field transfers energy at high values of k, a wavenumber range in which the magnetic intensity is too low, thus changes to the total magnetic energy are not appreciable.
For values of β = 0.5 and T ⊥ /T , the plasma is unstable. Independently of the value of α, protons that are initially unstable suffer parallel heating while the temperature anisotropy is reduced. In this case, the magnetic field transfers energy at high values of k, but also absorbs energy in the range of wavenumbers where the ion-cyclotron instability develops. This results in the steepening of the wave spectrum at v A k/Ω p > 1, and an effective growth of magnetic energy. This in turn results in a reduction of the proton-cyclotron instability, which can be observed as a reduction of the temperature anisotropy and heating of protons in the parallel direction with respect to the background magnetic field.
In summary, we have shown that the presence of a sufficient level of turbulent magnetic spectrum can drive an initially stable proton plasma towards higher values of the temperature anisotropy. Depending on the spectrum shape, the anisotropy can grow smoothly towards the instability thresholds, or reach a large temperature anisotropy value, far above from the kinetic instability threshold. Thus, an initially due to the presence of a turbulent magnetic field spectrum a stable plasma can evolve in such a way that instabilities can be triggered during the quasilinear dynamics. These results suggest a possible mechanism to explain why the solar wind plasma can be observed near the instability thresholds or far from thermodynamic equilibrium.