Turbulent heating and acceleration of He ++ ions by spectra of Alfvén-cyclotron waves in the expanding solar wind: 1.5-D hybrid simulations

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, , doi: /jgra.50363, 2013 Turbulent heating and acceleration of He ++ ions by spectra of Alfvén-cyclotron waves in the expanding solar wind: 1.5-D hybrid simulations Y. G. Maneva, 1,2 A. F. Viñas, 2 and L. Ofman 1,2 Received 23 August 2012; revised 13 May 2013; accepted 29 May 2013; published 21 June [1] Both remote sensing and in situ measurements show that the fast solar wind plasma significantly deviates from thermal equilibrium and is strongly permeated by turbulent electromagnetic waves, which regulate the ion temperature anisotropies and relative drifts. Thus, the ion kinetics is governed by heating and cooling related to absorption and emission of ion-acoustic and ion-cyclotron waves, as well as nonresonant pitch angle scattering and diffusion in phase space. Additionally, the solar wind properties are affected by its nonadiabatic expansion as the wind travels away from the Sun. In this study we present results from 1.5-D hybrid simulations to investigate the effects of a nonlinear turbulent spectrum of Alfvén-cyclotron waves and the solar wind expansion on the anisotropic heating and differential acceleration of protons and He ++ ions. We compare the different heating and acceleration by turbulent Alfvén-cyclotron wave spectra and by pure monochromatic waves. For the waves and the wave spectra used in our model, we find that the He ++ ions are preferentially heated and by the end of the simulations acquire much more than mass-proportional temperature ratios, T /T p > m /m p. The differential acceleration between the two species strongly depends on the initial wave amplitude and the related spectral index and is often suppressed by the solar wind expansion. We also find that the expansion leads to perpendicular cooling for both species, and depending on the initial wave spectra, it can either heat or cool the ions in parallel direction. Despite the cooling effect of the expansion in perpendicular direction, the wave-particle interactions provide an additional heating source, and the perpendicular temperature components remain higher than the adiabatic predictions. Citation: Maneva, Y. G., A. F. Viñas, and L. Ofman (2013), Turbulent heating and acceleration of He ++ ions by spectra of Alfvén-cyclotron waves in the expanding solar wind: 1.5-D hybrid simulations, J. Geophys. Res. Space Physics, 118, , doi: /jgra Introduction [2] He ++ ions, also called alpha particles, are the lightest and most abundant minor ions in the solar plasma. Their density is typically a few percent of the total electron density, but unlike the other heavy ions, their abundance is sufficient to substantially influence the density and electromagnetic fluctuations in the ambient solar wind [e.g., Cranmer, 2000],and they carry a significant fraction of the solar wind mass flux. Therefore, the He ++ ions are considered to play an important role in the open problems of coronal heating and solar wind acceleration. In situ observations of the alpha particles in the fast solar wind by Helios [Marsch et al., 1982], 1 Department of Physics, Catholic University of America, Washington D. C., USA. 2 NASA Goddard Space Flight Center, Greenbelt, Maryland, USA. Corresponding author: Y. G. Maneva, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA. (yana.g.maneva@nasa.gov) American Geophysical Union. All Rights Reserved /13/ /jgra Ulysses [Neugebauer et al., 1994], Wind [Steinberg et al., 1996], and other spacecraft [e.g., Astudillo et al., 1996] unambiguously show that the alpha particles are faster and hotter than the protons and their temperature often exceeds mass-proportionality, T (m /m p )T p ; see, for example, the early studies by von Steiger et al. [1995]. Spectroscopy and remote sensing of the solar corona reveal similar behavior of the heavy ions such as O 5+ in coronal holes [e.g., Kohl et al., 1998], which indicates that the preferential heating and acceleration of minor ions in the fast solar wind streams starts very close to the Sun, supposedly at the base of the solar corona. [3] Results from multifluid solar wind modeling show that linear low-frequency MHD waves can heat the protons, but cannot account for the observed alpha particle preferential heating and acceleration [Ofman, 2004], suggesting that one needs to invoke nonlinear and/or higher frequency waves, such as ion-cyclotron or kinetic Alfvén waves to preferentially heat and accelerate the solar wind plasma [Maruca et al., 2011]. For a review on wave-based solar wind modeling, see, for example, Ofman [2010a]. Fora 2842

2 Table 1. Summary of the Characteristic Plasma Parameters for the 1.5-D Hybrid Simulations Performed in This Study a Case # Modes ıb/b 0! 0 [ p ] k 0 [ p /V A ] Slope Exp. Param. " V p,fin [V A ] (T /T p ) fin (T?, /T k, ) fin (T?,p /T k,p ) fin [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] a The table describes some of the initial parameters like the total wave amplitude, frequencies, wave-numbers, the spectral index given by the slope of the initial transverse magnetic power spectrum, and the expansion parameter ". The rest of the parameters characterize the plasma state at the end stage of the simulations the final relative drift between the two ion species V p,fin, the temperature ratio between the alpha particles and the protons (T /T p ) fin and the final ion temperature anisotropies (T?,i /T k,i ) fin. review on resonant ion-cyclotron interaction, see Hollweg and Isenberg [2002]. Two-dimensional global multifluid modeling of the solar wind also shows that choosing a specific form of the heating function for alpha particles, which fits the observed ion anisotropies and differential flows close to the Sun (at a hundred solar radii) cannot simultaneously account for the observed solar wind measurements in situ at 1 AU near the Earth [Li and Li, 2008, 2009]. This suggests that including an ad hoc heating term in the multifluid energy equations for the minor ions is not sufficient to adequately model the ion properties in the solar wind. In this respect, kinetic modeling is needed to properly describe the plasma properties and account for the additional ion heating via wave-particle interactions in the turbulent magnetized solar wind. [4] Previous studies involving parametric instabilities of large-amplitude Alfvén-cyclotron waves and consequent wave-particle interactions have shown that the minor ions can be preferentially heated to a more than massproportional temperature ratio with larger temperatures in perpendicular direction and differentially accelerated to a substantial fraction of the Alfvén speed [Araneda et al., 2009]. In addition it has been shown that the core of the protons is preferentially heated in perpendicular direction, whereas proton beams form along the direction of the background magnetic field [Araneda et al., 2008]. Several works also consider the parametric instabilities of large-amplitude Alfvén-cyclotron waves; for instance, see Nariyuki et al. [2009] for a proton-beam system and Kauffmann and Araneda [2008] for proton-alpha particles plasma (in all the models discussed here, the electrons are considered to be a charge-neutralizing fluid). The effect of solar wind expansion on the ion drifts and anisotropies has been considered in a 1.5-D hybrid code by Liewer et al. [2001] and recently been taken into account for a plasma, consisting of two proton populations core and beam by Hellinger and Trávníček [2011], for proton-oxygen plasma by Hellinger et al. [2005], and for proton-alpha particles by Ofman et al. [2011]. Recent studies on the effect of initial wave-spectra in the expanding solar wind with 1.5-D and 2.5-D hybrid simulations have been made for pure electron-proton or proton core and beam plasmas by Matteini et al. [2010, 2012]. [5] In this paper we perform 1.5-D hybrid simulations to investigate the effect of turbulent heating and acceleration of He ++ ions in the solar wind and compare the results to previous studies of ion heating by nonlinear monochromatic waves [Araneda et al., 2009, 2011]. For both monochromatic and broadband spectrum of waves, we study the influence of the solar wind expansion within the limit of slowly expanding box model, where the transverse coordinates are stretched in time as the solar wind flows away from the Sun and expands. Although the present study includes 1.5-D simulations, preliminary results from 2.5-D hybrid simulations with quasi-parallel initial wave spectra indicate that the basic processes of preferential heating and ion beam formations are preserved, as discussed for protons by Matteini et al. [2010]. 2. Hybrid Simulations and Initial Conditions [6] The 1.5-D hybrid model we use in this study describes the electrons as a charge-neutralizing massless isothermal fluid and treats the ions fully kinetically within the particle-in-cell approach. In order to investigate the effects of the solar wind expansion in the presence of initial wavespectra, we adopted the model by Liewer et al. [2001] and incorporated it in the hybrid code by Araneda et al. [2007], based on the original hybrid code by Winske and Omidi [1993] with numerical schemes given by Matthews [1994]. All computations were done with the 1.5-D version of the code, which means that only one spatial coordinate (in the direction of the external magnetic field) is considered, but we follow the evolution of all the three components of the particle velocities, electric, and magnetic fields. We should mention that 2.5-D hybrid models of proton-helium plasma exist [Hellinger et al., 2005] and have been applied, for instance, to study the proton-helium and helium-cyclotron instabilities in the solar wind [Gary et al., 2003], ion heating by low amplitude waves and ion beams [Ofman and Viñas, 2007], and to study the effect of plasma inhomogeneities [Ofman, 2010b]. Those models, however, did not study the problem of parametric instabilities of finite-amplitude waves and their role for the differential acceleration of helium ions in the solar wind. For the 1.5-D hybrid code used here, periodic boundary conditions are assumed with 2048 grid cells and 400 particles per cell per species. The time in the simulations is normalized to the inverse proton gyro-frequency 1 p and the velocities to the Alfvén speed, defined as V A p B0 0n em p, and the simulation box has an approximate 2843

3 Figure 1. Transverse components of the initial velocity and magnetic field fluctuations representing a self-consistent initial state for electron-proton plasma. The total wave amplitude of all the modes is ıb =0.25B 0. length L 502.4V A / p. Ion velocities are advanced in time with a leap-frog algorithm, which requires a staggered grid with the electric field half a time step ahead of the velocities. The fields are advanced in time with a current advance method as described by Matthews [1994], which allows for a substepping of the magnetic field. The spatial derivatives are calculated with a fourth-order finite difference scheme. All calculations are performed in the ion center of momentum frame of reference, which allows for a nonzero constant current. The initial magnetic field is given by a constant background, B 0 = B 0 Oe z, on top of which we superimpose nonlinear transverse fluctuations, corresponding to either monochromatic or a spectrum of parallel propagating Alfvén-cyclotron waves, obeying a warm plasma dispersion relation for isotropic (Maxwellian) velocity distribution functions [Sonnerup and Su, 1967]: k 2 0 X 2 0 n i m i! 0 B 2 =0, (1) 0 1! 0 / i i e where the summation is over all ion species with number density n i and mass m i. For the monochromatic case, the summation index i covers protons and alpha particles, and the Alfvén-cyclotron wave was chosen as the solution of the Alfvén-alpha-cyclotron branch of the dispersion relation, corresponding to wave number k 0 = 0.4 p /V A. This choice of the initial wave number allows us to relate our work to previous studies on parametric instabilities of finite-amplitude Alfvén-cyclotron waves [Kauffmann and Araneda, 2008; Araneda et al., 2009] and makes the initial frequencies fall within the range of ion-cyclotron waves as observed in situ by STEREO near 1 AU and Messenger at 0.3 AU [Jian et al., 2009, 2010]. To construct the broadband spectrum, the dispersion relation for proton-electron plasma was considered, and the spectra were chosen from the Alfvén-proton-cyclotron branch with spectral slopes, range of wave-numbers k 0, and frequencies! 0 as given in Table 1. As an initial state for all simulations we consider nondrifting homogeneous isothermal Maxwellian plasma with T = T p and no relative drifts between the two ion species in parallel direction V p = 0. This means that all particles are homogeneously distributed with initially isotropic velocity distribution functions. The corresponding ion plasma beta ˇi = v 2 th /V2 A, as defined in terms of the thermal speed v th and the local Alfvén speed V A is ˇ =0.02 for the He ++ ions and ˇp =0.08for the protons. The alpha particles constitute 5% of the background electron density, n =0.05n e, and the plasma beta of the massless isothermal fluid electrons is ˇe =2 0 n e k B T e /B 2 0 =0.5. As a test study, the majority of the simulations are performed with a total initial wave-amplitude for both the monochromatic pump wave or the entire wave spectra chosen to be 25% of the magnitude of the external background magnetic field, ıb =0.25B 0.This magnitude is chosen to study moderately nonlinear effects. [7] For the case of monochromatic waves, the ion motion in phase-space in the linear stage at the beginning of the simulations is self-consistently coupled to the electromagnetic field of the waves. This results in a bulk ion drift in Figure 2. Initial magnetic field spectra in wave-number k space. The data corresponds to 31 different modes with frequencies close to the proton-cyclotron frequency and a spectral index = 1. (left) The magnetic spectra against the parallel wave number K x and (right) the power law in mode number space with m = LK x /

4 Figure 3. Power spectra of the density and magnetic field fluctuations in Fourier space in logarithmic scale. The snapshots taken at four different times describe the turbulent evolution of the daughter ion-acoustic (black lines) and Alfvén-cyclotron waves (grey lines), generated by the decay of the initial nonlinear monochromatic Alfvén-alpha-cyclotron pump with mode number m=32. The data corresponds to run 1 from Table 1. transverse direction, which depends on the wave amplitude, frequency, phase-speed, and the ion-cyclotron frequency of the given ion species [see Araneda et al., 2009]: ıv?i = (! 0/k 0 ) 1! 0 / i ıb B 0. (2) We should note that in the case of broadband spectrum, the perpendicular velocities of the particles are not entirely selfconsistent with the initial wave spectra, as the spectra were taken from the solution of the electron-proton plasma, so that the perpendicular velocity fluctuations for the protons and the alpha particles are initially considered to be the same. This inconsistency disappears within a few tens of proton gyroperiods, as the system readjusts itself in the course of evolution and the ion velocities become self-consistent with the existing wave-spectra. In this study we compare the results from 1.5-D hybrid simulations with monochromatic waves versus initial wave spectra as an energy source for plasma heating and acceleration. Most of the results presented here imply spectral index = 1for the power slope Figure 4. A snapshots of the density (black) and magnetic (grey) power spectra as displayed in Figure 3, but for simulation case 6 from Table 1, initialized with a broadband spectrum with spectral index = 1, as given by Figure

5 of the magnetic field fluctuations, as visible in Figure 2. Preliminary comparison between = 1 and white noise spectra suggest that the influence of the spectral slope on the particle heating and acceleration is less important than the choice of initial frequency range and total wave amplitude (i.e., the energy content of the spectrum). Typical relation between the perpendicular ion bulk speeds and the transverse fluctuations of the magnetic field following equation (2) for the initial simulation wave-spectra consisting of 31 modes is illustrated in Figure 1. [8] Apart from the influence of the wave-spectra, we investigate the effects of solar wind expansion on the evolution of the ion drifts, temperature anisotropies, and velocity distribution functions in the outflowing solar wind. To model the solar wind expansion, we follow the basic fluid idea of Grappin and Velli [1996], applied to hybrid simulations [e.g., Liewer et al., 2001;Hellinger et al., 2005;Ofman et al., 2011, Matteini et al., 2012]. We assume slowly expanding solar wind, outflowing in a radial direction with a constant bulk speed U 0 = U 0 Oe r at a distance R 0 from the Sun. In the expanding box model, the equations of motion and the evolution of the electromagnetic fields are modified by the expansion factor a(t) =1+"twith an assumed small expansion parameter " = U 0 /R 0, "t 1. Thus, the position of the particles in the comoving solar wind frame obeys the simple kinematic relations, modified by the expansion factor dz 0 0 = v 0 dt 0 z, dy = 1 dt 0 a(t) v0 y, 0 dx = 1 dt 0 a(t) v0 x, (3) where the ion velocities in the expanding frame and at rest are related by the magnitude of the constant solar wind bulk speed U 0 v 0 z = v z U 0, v 0 y = v y U 0 /R 0 y 0, v 0 x = v x U 0 /R 0 x 0. (4) The electric field in the expanding (prime) frame of reference is computed from the momentum equation for the massless fluid isothermal electrons n e m e dv 0 e dt = en e E 0 + (r 0 B 0 ) B 0 4 J0 i B0 c r 0 P e =0, (5) where J 0 i is the ion current in the comoving reference frame, P e = n e k B T e is the electron pressure, and the transformations for the electric field and the bulk velocities for all species are respectively E 0 = E + U 0 B and V 0 s = V s U 0.The evolution of the magnetic field is modified by the expansion as = c(r 0 E 0 ) U 0 R(t) O L B 0, (6) where OL = (2, 1, 1)ı ij is the transformation matrix, which accounts for the effect of the slow expansion. In the considerations above, all second and higher-order terms of the type O(" 2 ) have been neglected. [9] In the present hybrid model, we solve the above system of equations conserving the net charge and the total current. The charge neutrality for the three species plasma yields n e = n p +2n, and the bulk velocities in the center of momentum framework are related as follows n e V e = n p V p +4n V Figure 5. (top) Temporal evolution of the parallel and transverse components of the ion temperatures from 1.5-D hybrid simulations with a monochromatic initial Alfvéncyclotron pump wave with frequency! p, wavenumber k 0 =0.4 p /V A, and amplitude ıb =0.25B 0.The dashed lines show the results when a gradual solar wind expansion is considered with " = 10 4 p, and the solid curves show the ion temperatures when the expansion is not taken into account. (middle) Enlarged plot with the temporal evolution of the proton temperature components from Figure 5 (top). (bottom) Temporal evolution of the relative drift speed between protons and He ++ ions for this case. The solid blue line shows the numerical solution without expansion, and the red dashed line illustrates the numerical solution when slow expansion with " =10 4 p is considered. The data points correspond to runs 1 and 2 from Table 1.

6 Figure 6. (top) Temporal evolution of the parallel and perpendicular components of the individual ion temperatures from 1.5-D runs with the broadband spectra, Figures 1 and 2, consisting of 31 modes with initial wave numbers in the range k 0 2 [ ] p /V A and wave-frequencies slightly below the proton-cyclotron gyrofrequency! = [ ] p. Solid lines denote the evolution without expansion, and the dashed lines illustrate the case when slow expansion with " =510 5 p is considered. (middle) Enlarged plot with the proton temperature components from Figure 6 (top). (bottom) Temporal evolution of the relative drift speed for the same cases as described in Figure 6 (top). The solid blue line stands for the nonexpanding solar wind, and the dashed red line denotes the case of slow expansion with " =510 5 p. Contrary to the monochromatic case shown in Figure 5, where the expansion acts to slow down the alpha particles, in this case the expansion heats the He ++ ions in parallel direction and slightly increases the differential streaming. The data points correspond to runs 6 and 8 from Table Monochromatic Waves Versus Turbulent Wave Spectra: The Effect of Gradual Solar Wind Expansion [10] In this section we present the results from our 1.5-D hybrid simulations comparing the different ion heating and acceleration rates when the initial energy source is either single monochromatic nonlinear Alfvén-alpha-cyclotron wave or a broadband spectra of Alfvén-proton-cyclotron waves. The transverse velocity and magnetic field fluctuations of the initial broadband wave spectra are illustrated in Figure 1, and the initial magnetic field power spectra in k-space (or versus the corresponding wave mode numbers) is given by Figure 2. For each case we additionally study the influence of the solar wind expansion over the nonlinear wave-particle interactions and the associated energy dissipation in the system. We investigate the onset of differential streaming between the protons and the alpha particles by indirect absorption of Alfvén-cyclotron waves and discuss the effect of a gradual solar wind expansion on the preferential heating of minor ions and the generation of temperature anisotropies from initially Maxwellian velocity distribution functions. In the case of nonlinear monochromatic wave, the absorption of the original wave energy takes place in two steps. First, due to the finite temperature of the ions, the pump wave becomes parametrically unstable and couples to the thermal fluctuations caused by the ion motion. As the pump wave decays, it generates a broad spectrum of daughter Alfvén-cyclotron and ion-acoustic waves as shown, for example, by Araneda et al. [2008], Maneva et al. [2009, 2010], and Matteini et al. [2010]. [11] To study the effect of broadband wave spectra in a slowly expanding collisionless solar wind on the evolution of the plasma, we have compared the results for monochromatic nonlinear waves and initial wave-spectra, and discussed how they are further modified by the gradual expansion. In the expanding frame comoving with the solar wind, the expansion parameter was set to " =10 4 p for the monochromatic case and to " = p for the initial wave spectra (see Table 1 for details). These values represent typical expansion rates " as estimated in the solar wind and unlike other studies [e.g., Liewer et al.,2001] we do not numerically enhance them to simulate longer real time in short time intervals. The nonexpanding frame is recovered by setting " =0. Unless otherwise stated, we set the initial total wave amplitude to ıb =0.25B 0,andthe two ion species are assumed initially to be isotropic with the same temperatures, as given in section 2. The plasma parameters, which characterize the initial and the final states of the simulations are given in Table 1. In the present study, we did not vary the plasma ˇ, the mass density of the He ++ ions, nor the initial temperature anisotropy for both ion species. The results from the simulations with a single initial pump wave are similar to what was presented earlier by Araneda et al. [2009], but here we also consider slightly higher frequencies and investigate the effect of the solar wind expansion. [12] Figures 3 and 4 describe the evolution of the density and magnetic fluctuations, generated from the parametric instabilities of an initial single monochromatic pump wave and the turbulent evolution of the initial wave spectra as shown in Figure 2. The corresponding wave frequencies are! = p for the monochromatic case and the range of

7 Figure 7. Contour plots of the velocity distribution functions for (top) He ++ ions and (bottom) protons in the v k v? plane at different instants of time throughout the simulation. The abscissa denotes the parallel velocities of the particles v z, and one of their transverse components, v y, is plotted along the y-axis. The initial energy source is the low-frequency monochromatic pump wave from simulation case 1, whose early stage evolution is shown in Figure 3.! 0 = [ ] p for the initial broadband wave spectra. It is evident that in the case of initial Alfvén-cyclotron wave spectra, Figure 4, the corresponding density fluctuations also form a continuous broadband spectrum and do not show individual peaks-like structures as in the case of an initial monochromatic pump. This provides the system with a broadband spectrum of ion-acoustic waves, which can trap ions with a range of velocities as described below (see, Figure 10). Although Figure 4 refers to simulations without expansion, the nonlinear evolution of the wave-spectra in an expanding box with " = p looks very similar within the early times discussed here. [13] Figures 5 and 6 show the temporal evolution of the individual ion temperatures and the relative drift speed, resulting from simulations with an initial monochromatic wave and an initial broadband wave spectrum. The plots indicate that both the preferential heating and particularly the differential acceleration are much stronger for the case of an initial monochromatic wave as compared to the broadband spectrum case. Simulations with slightly higher frequency monochromatic wave from the Alfvénproton-cyclotron branch (case 3 from Table 1) give similar results for the final relative drifts and the final ion temperatures to the simulations with pump wave, which belongs to the Alfvén-alpha-cyclotron branch (case 1 from Table 1). The simulations with an Alfvén-proton-cyclotron pump wave are prone to similar turbulent energy cascade, and after 300 proton gyroperiods, their wave-spectrum resembles the one calculated from the case of initial alpha-cyclotron monochromatic pump, see Figure 3. Thus, although the selected frequencies of the source waves are very important, for a fixed frequency range, the ion heating and the differential acceleration strongly depend on the initial wave amplitude (compare simulation runs 5, 9, 10, and 11 from Table 1). This relates to the fact that the parametric decays for linear Alfvén-cyclotron waves take much longer, and the daughter ion-acoustic waves, which they produce, are much weaker in comparison to the parametric instabilities of finite amplitude waves (which occur much faster). Therefore, the reduced ion heating and acceleration in the case of initial wave-spectra as visible in Figure 6 can be attributed to the fact, that for the same total wave power, the individual waves in the spectrum are less energetic. Being less energetic means that, on one hand, those waves exert lower wave-pressure. On the other hand, this also implies that it would take these waves asymptotically long time to produce resonant daughter ion-acoustic or ion-cyclotron waves. Both of these effects reduce the ion heating and acceleration. [14] To better understand the onset of the ion heating, we need to consider the effect of the initial spectra on the ion motion in the cold plasma approximation, given by equations (1) and (2). In the case of a monochromatic pump wave, run 1, the magnitude of the initial velocity fluctuations of the alpha particles for the plasma parameters given in Table 1 reads m < ıv?, > 2 /2 = 0.43 in normalized units of proton mass and Alfvén speed. Together with the 2 superimposed initial thermal energy of m vth, /2 = 0.04 (as ˇ = 0.02), it gives rise to an apparent temperature k B T 0.5. This explains the observed oscillations in the initial perpendicular temperature for the alpha particles 2848

8 Figure 8. Snapshots of the ion velocity distributions in the v k v? plane ((top) He ++ ions and (bottom) protons) when the system was initialized with the broadband spectrum shown in Figures 1 and 2, whose early stage evolution is illustrated in Figure 4. The initial state is not fully self-consistent, which makes the initial Maxwellian distributions look highly distorted under the influence of the initial wave spectra, but within about 50 proton gyroperiods, the particles readjust their velocities to comply with the given spectra of Alfvén-proton-cyclotron waves. The simulations correspond to case 6 from Table 1. close to that value. The protons have the same thermal energy m p v 2 th,p /2 = 0.04, but they are hardly affected by the low-frequency waves considered here, m p < ıv?,p > 2 /2 = 0.035; therefore, their perpendicular temperature oscillations are much weaker, k B T p = As discussed above, the apparent temperature disappears, and true heating (with conversion of magnetic energy into thermal) occurs only when the pump wave is depleted by the parametric instabilities, and its amplitude becomes comparable to the amplitude of the generated daughter waves (at p t 430, cf. Figure 3). The enhanced response of the motion of the alpha particles to the initial waves is part of the reason for the observed preferential heating, together with their low number density, which facilitates the energy gain in the process of wave-particle interactions. [15] As far as the expansion is concerned, in the case of initial monochromatic wave, it acts against the parallel heating, whereas in the case of initial wave spectrum, it increases the parallel alpha temperature. For the broadband spectra considered in this study, the expansion results in 6 7%increase in the differential streaming, whereas in the case of monochromatic wave, it leads to 5% decrease in the relative drift speed. Preliminary results form 2.5-D hybrid simulations with better restricted initial wave spectra, with and without initial relative drifts, imply that the gradual expansion can significantly reduce the differential streaming for both cases of monochromatic waves and initial broadband spectra. [16] For the initial plasma parameters considered here, the time-scale for wave-particle and wave-wave interactions is much shorter than the expansion time (about 10,000 gyroperiods). Therefore, the evolution of the system is naturally dominated by the wave-particle interactions. Still, for a fully developed wave-spectra, the expansion plays an important role in modifying the resonance conditions and the wave-particle interactions with the ion-cyclotron and the ion-acoustic waves either by allowing for interactions with initially nonresonant modes, Figure 6, or by suppressing the existing resonant interactions, see Figure 5. For both monochromatic waves and initial wave spectra, the expansion significantly decreases the efficiency of the resonant ion-cyclotron interactions, cooling the ions in perpendicular direction. Nevertheless, the wave-particles interactions provide enough energy source to maintain higher perpendicular temperatures than what is expected for an adiabatic expansion [e.g., Hellinger et al.,2005;matteini et al.,2012]. [17] Figures 7 and 8 consist of four snapshots, representing the temporal evolution of the ion velocity distribution functions for He ++ ions (first row) and protons (second row). Figure 7 refers to simulations with monochromatic pump, case 1 from Table 1, and Figure 8 represents simulations with initial broadband wave spectrum with a spectral slope = 1, case 6 in Table 1. For both simulations, the same total wave power was used. In the course of evolution, the initially isotropic distribution functions deform to form ion beams due to Landau damping of the daughter ion-acoustic 2849

9 Ω p t = Ω p t = V Z /V A - alphas V Z /V A - protons Z/(V A /Ω p ) Z/(V A /Ω p ) Figure 9. Snapshots of the ion distributions in phase space at p t = and p t = The contour plots show the number of particles with a given velocity at a given position. The phase-space holes are due to the trapping of protons in the potential well of the daughter electrostatic ion-acoustic waves, born by the parametric decay of the initial finite amplitude pump wave with frequency! 0 = p.the trapping of He ++ is not visible in this and the next Figure 10, as it happens approximately inverse gyroperiods later, which causes the beam formation in their velocity distribution as shown in Figures 7 and F8. The phase space plots correspond to the power spectra presented in Figure 3 and the ion velocity distribution functions from Figure 7. The plasma parameters are from case 1 in Table 1. waves resulting from parametric decays of the pump wave or the initial wave spectra. Pitch-angle scattering by the forward and backward propagating daughter Alfvén-cyclotron waves then leads to preferential heating of the minor ions in perpendicular direction and scatters the front part of the ion beams. We should stress here that when plotted in velocity space, this perpendicular bulk velocity fluctuations relate to an apparent initial temperature anisotropy, as pointed out by Araneda et al. [2009]; Maneva et al. [2009, 2010] and discussed in Verscharen and Marsch [2011]. However, with the decay of the initial large-amplitude waves due to parametric instabilities, energy cascading, and turbulence generation, this effect disappears and true anisotropic heating takes place. During the initial stage of the broadband spectrum simulation, at t 13 1 p on Figure 8, the particles have not yet readjusted their positions and velocities to the electromagnetic field of the waves, as the initial wave spectra was calculated from the plasma dispersion relation for an electron-proton plasma, and this introduces some initial noise. It takes 50 to 100 proton gyroperiods for the particles to adjust their orbits and obtain self-consistent velocities. Afterward, proton trapping and associated beam formation sets, continuously accelerating the tail of the proton distribution till the end of the simulation. Once the proton beam is accelerated, it pulls the alpha particles due to current conservation in the center of momentum frame of reference and accelerates them. When the bulk of the alpha particles is differentially accelerated, some ions from the tail of the velocity distribution function fall in resonance with the daughter ion-acoustic waves, so that at later time p t = 1200 alpha beams are also formed and together with the proton beams remain present till the end of the simulations. In the case of monochromatic wave, Figure 7, the stronger heating of the minor ions relates to the larger volume in phase-space, which the alpha particles occupy due to their strong initial bulk velocity fluctuations imposed by the pump. Although not shown here, including a gradual solar wind expansion with " 2 [5*10 5,10 4 ] does not impede the ion beam formations, nor changes the shape of the ion distribution functions. [18] Figures 9 and 10 describe the trapping of ions in the electric field of daughter ion-acoustic waves for the case of an initial finite amplitude monochromatic Alfvén-cyclotron wave and an initial broadband spectrum, respectively. The initial pump belongs to the proton-cyclotron branch with a frequency! 0 = p and wave number k 0 =0.4 p /V A, where the rest of the parameters correspond to simulation run 1 from Table 1. The initial wave spectrum has an initial frequency and wave-number range! 0 2 [ ] p and k 0 2 [ ] p /V A, and corresponds to simulation case 6 from Table 1. Figure 9 shows clear formation of phasespace holes for the proton population, centered at 0.75V A at 2850

10 1.5 Ω p t = Ω p t = V Z /V A - alphas V Z /V A - protons Z/(V A /Ω p ) Z/(V A /Ω p ) Figure 10. Contour plots with the parallel ion velocities versus position at p t = and p t = similar to the ones in Figure 9, but for an initial broadband spectrum with frequency range! 0 2 [ ] p. The figure shows the partial trapping of protons and almost no trapping of He ++ ions in the parallel electric field of turbulent electrostatic ion-acoustic waves generated during the evolution of the initial wave spectra. The one-dimensional phase space plots correspond to the power spectra shown in Figure 4 and the ion velocity distributions presented in Figure 8. The parameters correspond to case 6 in Table 1. p t = TheHe ++ ions at that time are squashed by the waves and only partially trapped. As the proton beam gets accelerated by the daughter ion-acoustic waves, the conservation of momentum in the center of momentum reference frame accelerates the He ++ ions and causes their differential streaming. The relative drift changes their resonant condition and as a result the He ++ ions are trapped and start their own beam formations at a later time (t p ) as visible from their velocity distribution functions, shown in Figures 7 and 8. Figure 10 shows the distribution of protons and He ++ ions in the phase space of their parallel velocity versus position when a broadband spectrum of Alfvén-cyclotron waves was initially applied. The broadband electromagnetic spectrum leads to a broadband spectrum in the density fluctuations, as discussed above and visible in Figure 4. This triggers proton trapping in a spectrum of ion-acoustic waves with a wide range of phase speeds!/k 2 [ ]V A, which in return smears out the phase space holes and leads to a diffusion in phase space. Once again the trapping for He ++ ions and the resulting ion beam formations in their velocity distribution functions happens at a later stage, when the prominent proton beams have already pulled and differentially accelerated the He ++ ions in the center of momentum frame of reference. We should note that the gradual expansion only slightly shifts the phase speed of the daughter ion-acoustic waves and hence the position of the ion phase space holes, but the particles remain trapped, and the shape of their velocity distribution functions remain similar to the case where no expansion is taken into account. [19] To better understand the simulation initializations with pump waves and initial wave spectra, we should keep in mind how the plasma dispersion is modified in the presence of minor species, see equation (1). The different frequencies of the pump wave in cases 1 and 3 from Table 1 illustrate the frequency shift, which occurs in the individual branches as we add more ion species. Each additional ion species increases the order of the polynomial, representing the cold plasma dispersion relation considered here. It introduces an additional ion-cyclotron branch in the system and influences the frequencies of the previously existing ion and electron branches. Thus, in the absence of alpha particles, the proton-cyclotron branch at k = 0.4 p /V A has a frequency! = p and the alpha-cyclotron branch does not exist. Once alpha particles are included, say with a number density n = 0.05n e, for the same wave-number, the proton-cyclotron branch shifts to a higher frequency,! = p, whereas the previous low frequency solution now belongs to the alpha-cyclotron branch with! = p. Those frequencies are further modified in the presence of relative drifts. For example, for a differential flow between the protons and the alpha particles equal to just a fraction of the Alfvén speed, the frequency of the proton-cyclotron branch at high wave numbers greatly exceeds the limiting value of p. 2851

11 [20] Finally, we should note that varying the spectral index from = 0 to = 1 influences the temperature anisotropy for the alpha particles, decreasing the differential streaming, and reduces the final temperature ratio between the alpha particles and the protons. From Table 1, we can also see that the effect of the spectral index on the relative drift speeds and the temperature anisotropies decreases when the solar wind expansion is taken into account, so the initial spectral index in the power spectrum of the waves becomes less important. Although the type and slope of the initial wave spectra undoubtedly play a significant role, for the limited waveband of frequencies modeled here, the initial wave amplitude is more critical for the ion heating and acceleration. This is clearly visible if we compare the results from initial wave-spectra with a total amplitude of ıb =0.25B 0 to the results from initial wave-spectra where ıb =0.1B 0. In the latter case, the waves have 2.5 times lower amplitude (or 6.25 times lower energy) and their effect on the particles competes with the effect of the gradual solar wind expansion, which is of minor importance when nonlinear waves with larger amplitudes are considered. 4. Conclusions [21] In this paper we investigate the effects of an initial wave-spectra and a gradual solar wind expansion on the heating and acceleration of ions in a proton-he ++ ions plasma (with fluid electrons in 1.5-D hybrid model as discussed above). We answer the question whether minor ions can still be preferentially heated and differentially accelerated by nonlinear Alfvén-cyclotron waves when more than individual monochromatic modes are included in the system. We find that as in the monochromatic pump wave case, the presence of turbulent initial wave-spectra reproduces observed beam formations and temperature anisotropies and leads to preferential heating of the minor ions to a more than mass-proportional temperature ratio. The results show that for a given initial wave energy, the magnitude of the generated relative drift speed between the protons and the alpha particles decreases as the number of modes in the initial wave-spectra is increased. Higher number of modes for a fixed total wave energy means reduced amplitude for each of the constituent modes. This supports the conclusion that the origin of the differential streaming is not a stochastic process, but can be related to resonant absorption of daughter ion-acoustic waves and might require indirect wave-particle interactions with nonlinear Alfvén-cyclotron waves. Unlike the differential acceleration, the ion heating results from both resonant wave-particle interactions, related to Landau damping and cyclotron-resonance, and from nonresonant diffusion in phase-space as the imposed bulk perpendicular ion velocity fluctuations decrease with the decay (or absorption) of the pump wave (or in the course of turbulent evolution of the initial wave-spectra). Ion trapping (for protons and He ++ ) and related ion beam formations occur for both acceleration by single monochromatic waves and by initial wave spectra. In the latter case, due to the broad range of phase speeds and the lower amplitudes of the constituent waves in the initial wave-spectra, the ion trapping is not as straightforward as in the case of a single wave. In both cases, the trapping of alpha particles occurs much later than the trapping of protons, once the parametric instabilities 2852 and the turbulent evolution of the wave-spectra provide energy transfer toward high speed ion-acoustic waves, which resonate with and trap He ++ ions from the tail of their velocity distribution function. However, at the final stage of the simulations the alpha particles acquire and sustain stronger beams than the protons. Next, we have shown that the gradual solar wind expansion has different effects on the ion heating and acceleration rates for the cases of single monochromatic wave and initial wave-spectrum. In the first case the expansion cools the two ion species in parallel direction and impedes the generation of the relative drift speed. In the case of initial wave-spectra, the expansion supports the parallel heating for both ion species and slightly increases the magnitude of the final relative drift. Furthermore, the parallel heating of both ion species due to Landau damping of the resonant daughter ion-acoustic waves is enhanced in the presence of the gradual expansion. This effect can be explained by the notion that the solar wind expansion modifies the resonance conditions, allowing for additional wave-particle interactions with initially nonresonant modes. In a similar manner, the reduced parallel heating in the case of monochromatic waves can be related to suppressing the wave-particle interactions with initially resonant ion-acoustic waves. Finally, in both cases of monochromatic waves and broadband spectra, the expansion leads to perpendicular cooling for both protons and alpha particles. [22] In conclusion, we have demonstrated the effect of a spectrum of waves on the heating and acceleration of low ˇ solar wind ions. We have shown that the solar wind expansion changes the energy input required to heat the corona and accelerate the solar wind. Although both effects must be taken into account to accurately model the solar wind heating and acceleration, the present work suggests that proper modeling of the shape, amplitude and type of the initial wave-spectra as a prescribed energy source in the system is more important for the preferential heating and differential acceleration of minor ions than the effect of the gradual solar wind expansion for the low plasma ˇ conditions considered here. The 1.5-D simulations confirm that even though linear waves, ıb/b 0 1, (monochromatic waves and broadband wave spectra) can also heat the particles (to a much lower extent than the finite amplitude waves), they are not able to differentially accelerate the heavy ion species and cannot account for the observed relative drifts. [23] We should note that as we talk about Alfvéncyclotron waves, the results with initial broadband spectra discussed here include ion heating and acceleration by lowfrequency,! p, Alfvénic turbulence. Preliminary results show that higher-frequency pump waves with! 0.8 p heat and accelerate the minor ions to much higher temperatures and relative drift speeds, and we expect the minor ion preferential heating and the differential acceleration to be much stronger when higher-frequency wave spectra are considered. [24] Finally, we should add that the hybrid model used in the present work is limited to a single spatial dimension and parallel wave propagation. Hence, it omits the observed energy cascading toward perpendicular wave-numbers and can not handle the nonlinear wave-wave couplings between the parallel and oblique modes, nor any possible waveparticle interactions with oblique waves, which provide am

12 additional source for ion heating and acceleration. In this respect, performing at least 2.5-D or fully 3-D hybrid simulations is essential for the future studies of the solar wind ion heating and acceleration. [25] Acknowledgments. This work was supported by NASA, grant NNX10AC56G. The 1.5-D code used to obtain the results of this paper is an extension of the hybrid code used by Araneda et al. [2009]. Fruitful discussions with J. Araneda are highly appreciated. [26] Philippa Browning thanks the reviewers for their assistance in evaluating this paper. References Araneda, J. A., E. Marsch, and A. F. Viñas (2007), Collisionless damping of parametrically unstable Alfvén waves, J. Geophys. Res., 112, A04104, doi: /2006ja Araneda, J. A., E. Marsch, and A. F. Viñas (2008), Proton core heating and beam formation via parametrically unstable Alfvén-Cyclotron waves, Phys. Rev. Lett., 100, , doi: /physrevlett Araneda, J. A., Y. Maneva, and E. 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