The evolution of solar wind turbulence at kinetic scales

International Association of Geomagnetism and Aeronomy (IAGA) 2 nd Symposium: Solar Wind Space Environment Interaction c 2010 Cairo University Press December 4 th 8 th, 2009, Cairo, Egypt L.Damé & A.Hady (eds) DOI: 10.1012/S120027852007 The evolution of solar wind turbulence at kinetic scales Pierluigi Veltri 1, Francesco Valentini 2,1, and Francesco Califano 2 1 Dipartimento di Fisica and CNISM, Universitá della Calabria, 87036 Rende (CS), Italy 2 Dipartimento di Fisica and CNISM, Universitá di Pisa, 56127 Pisa, Italy Abstract. The understanding of the small-scale termination of the turbulent energy cascade in collisionless plasmas is nowadays one of the outstanding problems in space physics. In the absence of collisional viscosity, the dynamics at small scales is presumably kinetic in nature; the identification of the physical mechanism which replaces energy dissipation and establishes the link between macroscopic and microscopic scales would open a new scenario in the study of turbulent heating in space plasmas. We present a numerical analysis of kinetic effects along the turbulent energy cascade in solar wind plasmas which provides the first effective unified interpretation of a wide set of spacecraft observations and shows that, simultaneously with an increase in the ion perpendicular temperature, strong bursts of electrostatic activity in the form of ion-acoustic turbulence are produced together with accelerated beams in the ion distribution function. The problem of understanding which physical processes replace dissipation at small scales in a collisionless plasma, like solar wind, is of key relevance in space physics. Several spacecraft observations have shown [1 3] that the short-scale termination of the MagnetoHydroDynamics (MHD) energy spectra in solar wind is characterized by the occurrence of significant levels of electrostatic activity, consisting mostly of short-wavelength longitudinal ion-acoustic (IA) waves. The maximum intensity of these fluctuations usually occurs in regions where strong deviations from a Maxwellian distribution of particles are observed, meaning that kinetic effects presumably drive the collisionless short-scale system dynamics. The interpretation of the solar wind phenomenology at typical kinetic scales would be of far-reaching implications and would represent a significant step forward in picturing the complex scenario of turbulent heating in space plasmas. In this perspective, the impressive recent technological development of supercomputers gives now the possibility of using kinetic Eulerian Vlasov codes that solve the Vlasov-Maxwell equations in multi-dimensional phase space. The use of these zero-noise codes is crucial since Eulerian algorithms [4, 5] allow for the first time the analysis of kinetic effects in the small-scale tail of the turbulent cascade, where the energy level of the fluctuations is typically very low. In this spectral region, Lagrangian Particle In Cell algorithms [6] fail due to their intrinsic statistical noise. 43

Veltri, Valentini & Califano In this Report, we focus on physical situations where MHD turbulence evolves to a state with power predominantly stored in modes with wavevectors parallel to the ambient field (slab turbulence) [7 9]. The comparison of the numerical results with solar wind observations in slow speed streams shows that electrostatic turbulence in longitudinal IA waves represents the main physical cross-scale process of energy transport and the most efficient way to degrade the large scale energy. Fig. 1. Perpendicular (black line) and parallel (red line) temperature versus x at three different times. The mathematical model adopted to describe the evolution of the plasma is a hybrid model [5], where ion dynamics is described through Vlasov equation. Electrons are treated as a fluid and a generalized Ohm equation, that retains Hall effect and electron inertia terms, is considered. Faraday equation, Ampere equation (in which the displacement current is neglected) and an equation of state for the electron pressure close the system. Quasi-neutrality is assumed. The solutions of the above equations are obtained through a numerical hybrid- Vlasov code in 4D phase space (1D in physical space and 3D in velocity space), discretized on a uniform grid with N x gridpoints in physical space and (2Nv x + 1) (2Nv y + 1) (2Nv z + 1) in velocity space. Periodic boundary conditions are imposed in physical space. In the following discussion, times are scaled by 44

The evolution of solar wind turbulence at kinetic scales the ion cyclotron frequency Ω ci, lengths by the ion skin depth λ i = v A /Ω ci, v A being the Alfvén speed, and masses by the ion mass m i. Fig. 2. Energy spectra at four different times: magnetic energy (black line), electric energy (blue line), kinetic energy (purple line) and density (red line). We simulate a plasma embedded in a background magnetic field B = B 0 e x, x being the direction of wave propagation. The system is perturbed at t = 0 by a circularly left hand polarized Alfvén wave in the perpendicular plane. The form of velocity and magnetic perturbations [δu y (x), δu z (x), δb y (x) and δb z (x)] is found by solving the linearized two-fluid equations[5]. The initial perturbed ion distribution function reads f(x, v, t = 0) = A(x) exp [ (v δu) 2 /β ], where δu = δu y e y + δu z e z and β = 2v 2 ti /v2 A, v ti being the ion thermal speed; A(x) is such that the velocity integral of f gives the equilibrium density n 0 = 1. We choose an isothermal equation of state for the electron pressure P e = (β/2)nt e /T i, and β = 0.5 and the system dynamics is investigate for different values of the electron to ion temperature ratio. Perpendicular and parallel ion temperatures at t = 0 are T = T = β/2 = 0.25 and v ti = 0.5. The mass ratio is m e /m i = 1/1836. The length of the physical domain is L x 40.2 (the fundamental wave number is k 1 = 2π/L x 0.156), while the limits of the velocity domain in each direction are fixed at v max = 5v ti. We use 2048 gridpoints in physical space, and 51 3 in velocity space. The time step is t = 10 3. At t = 0, the first three modes in the spectrum of velocity and magnetic 45

Veltri, Valentini & Califano Fig. 3. (Top plot): parallel electric field versus x at t = 100. (Middle plot): density versus x at t = 100. (Bottom plot): x v x level lines of the reduced distribution ˆf at t = 100. perturbations are excited with amplitude ɛ = 0.5. No density disturbances are imposed at t = 0. The simulation is carried up to t = 200. In the early stage of the system evolution, ponderomotive forces give rise to compressive effects, with density fluctuations δn/n 0.07. Moreover, nonlinear three-wave coupling produces a direct energy cascade towards higher wavenumbers. At wavenumbers close to the Hall wavenumber k i = λ 1 i, proton cyclotron resonance with left-handed cyclotron waves [10] produces an increasing in the perpendicular temperature and a consequent temperature anisotropy in the ion distribution [11, 12]. As shown in Fig. 1, where T and T are plotted versus x at three different times, the perpendicular heating is not spatially homogeneous and is generally associated with a decrease in parallel temperature. Figure 2 reports the numerical spectra of particle density ( n k 2, red line), and magnetic ( Bk 2, black line), kinetic ( U k 2, purple line) and electric ( E k 2, blue line) energies. The four plots refer to different times in the simulation. From Fig. 2, one realizes that the energy transfer to small scales, across λ i, is not driven by a continuous cascade, but for t > 30 a well defined group of wavenumbers (10 < k < 100) are excited; at t = 200 the energy spectra look evidently peaked around k 30 and the magnetic energy is sensibly lower than the electric one, meaning that electrostatic activity is observed. Moreover, shortwavelength spikes of large amplitude appear in the parallel electric field E x. The top plot of Fig. 3 shows E x as a function of x at t = 100; the middle and the bottom plots show the particle density and the x v x contours of ˆf(x, v x ) = 46

The evolution of solar wind turbulence at kinetic scales fdvy dv z. Local density depressions and phase space holes are generated in correspondence of the regions of highly impulsive behavior of E x. These vortices, typical signature of particle trapping, propagate in the positive x-direction with velocity close to v ti = 0.5, indicated by a black dashed line in the figure. The signature of electrostatic activity has been recovered ubiquitous in interplanetary plasmas. The first evidence of significant levels of electric field turbulence in the solar wind is due to Gurnett and Andreson [1], who in 1977 analyzed plasma wave measurements of the solar-orbiting Helios 1 and 2 spacecrafts near 1AU. Further analyses of these observations [2, 3] showed that the typical energy spectra of the parallel electric field appear evidently peaked around 2500 Hertz, this being the same phenomenology recovered in our simulations. In Ref. [2] this electrostatic turbulence has been identified as short-wavelength IA waves propagating essentially parallel to the ambient field. These fluctuations present a highly impulsive behavior and propagate with frequency below the ion plasma frequency, Doppler-shifted upward by the motion of the solar wind. To identify the short-wavelength fluctuations observed in the simulation, we considered the k ω spectrum of the parallel electric energy. This Fourier analysis showed the presence of two acoustic-like branches of waves at different phase velocities. The upper branch (with higher frequency) has a phase velocity v φ 3.196v ti ; the numerical dispersion curve for these waves is in extremely good agreement with the dispersion relation of IA waves from Vlasov theory [13], evaluated for wavelengths larger than the Debye length λ D, ω IA = k βt e /2T i (1 + 3T i /T e ); this evidence is consistent with the identification by Gurnett et al. for the electrostatic fluctuations in solar wind. The unexpected lower branch consists of waves with v φ 0.925v ti. The analysis of the energy spectra for these two branches showed that the IA branch is dominant for 5 < k < 35, while, for k > 35, the energy is predominantly stored in the lower branch with v φ v ti. Simulations with different initial conditions [i) right hand polarized waves and no density perturbation at t = 0; ii) purely compressive disturbances at t = 0] have shown that the excitation of the short-scale IA waves is essentially independent on the polarization of the initial Alfvenic perturbations, but it is related to the presence of density fluctuations at MHD scales. On the other hand, the sonic-like branch at v φ v ti disappears for right-hand polarized perturbations at t = 0, meaning that the excitation of these waves is related to the ion cyclotron heating. The nature of these novel electrostatic waves can be investigated by looking for the roots of the electrostatic hybrid-vlasov dielectric function, in the limit of parallel propagation [14, 15]: D(k, ω) = 1 1 β T e f 0 / v x n 0 2 T i L v x ω/k dv x; (1) here, ω is the complex frequency and f 0 = f 0 (v x ) the equilibrium ion velocity distribution. The subscript L on the integral sign in Eq. (1) indicates the Landau contour [16]. 47

Veltri, Valentini & Califano Fig. 4. Spectrum of the electric energy at t = 200, for T e/t i = 10 (black line) and for T e/t i = 1 (red line). When f 0 is Maxwellian, for v φ v ti, heavily Landau damped solutions are expected, since the wave resonates with the bulk of the distribution [16]. Nevertheless, using a Bernstein-Greene-Kruskal distribution [17] for f 0, to take into account particle trapping (see Fig. 3), gives an undamped root at v φ v ti. An analogous nonlinear root (Electron Acoustic Waves) has been found with phase velocity close to the electron thermal speed [18 21]. We conclude that the excitation of the lower branch of electrostatic waves recovered in the simulations is driven by kinetic trapping effects. In 1979 Gurnett et al. [3] discussed the relationship of the IA waves to the macroscopic and microscopic characteristics of the solar wind plasma. They noted that the wave intensity is large in the low-speed solar wind regions, where compressive effects are not negligible, this being the same situation described in our simulations. Moreover, the level of the fluctuations is significantly correlated to the ratio T e /T i (the maximum intensity occurs for T e /T i 10). We then repeated the simulation for several values of T e /T i and noticed that the intensity of the IA peak in Fig. 2 is strongly correlated to T e /T i, in agreement with the results discussed by Gurnett. As an example, we found that the energy level of the fluctuations for T e /T i = 1 is ten order of magnitude lower than for T e /T i = 10. The results of this new simulation with T e /T i = 1 is shown in Fig. 4, where the electric energy spectrum is compared for T e /T i = 10 (black line) and for T e /T i = 1 (red line). Moreover, in Fig. 5 we report the time evolution of the energy level of the electrostatic peak in the spectra for three different values of T e /T i, that is T e /T i = 10 (black line), T e /T i = 3 (red line) and T e /T i = 1 (blue line), poining out that the energy level of the fluctuations at large wavenumbers increases as the electron to ion temperature ratio gets larger and larger. These 48

The evolution of solar wind turbulence at kinetic scales results are not surprising since it is well-known from Vlasov theory [13] that for cold electrons IA waves are strongly Landau damped. From the observation data analyzed in Ref. [3], the typical wavelength of IA waves at 1AU, corresponding to the frequency f exp 2500 Hertz, is λ 150m and the local Debye length is λ D 5m. For the observed ratio λ D /λ 0.03, the Vlasov dispersion relation for IA waves has an acoustic-like behavior [13]; therefore, effects of charge separation are not recovered in the data discussed in Ref. [3], this meaning that the solar wind phenomenology can be successfully reproduced within the hybrid Vlasov-Maxwell model, despite the quasi-neutrality assumption. Once the Doppler effect has been eliminated, the angular frequency of the IA peak from space observations at 1AU is ω exp/ω IA ci 1570, where Ω ci 1 Hertz is the typical ion cyclotron frequency in the solar wind. The angular frequency for the IA peak from the simulation is ω num 52.5. This value represents the IA maximum IA frequency allowed by the spatial numerical resolution of our code; indeed, a new simulation performed with a doubled number of gridpoints in the physical domain gave rise to a shift of the peak in Fig. 2 towards higher wavenumbers. Fig. 5. Time history of the maximum level of the electrostatic peak P (in semilogarithmic plot) for three different values of the temperature ratio: T e/t i = 10 (black line), T e/t i = 3 (red line) and T e/t i = 1 (purple line). As discussed in Ref. [3], another important phenomenon recovered in correspondence of the maximum intensity of electrostatic activity in solar wind is the generation of a double-stream ion distribution [3, 22, 23]. The secondary beam moves along the ambient field with velocity close to the local Alfvén speed. We then analyzed the evolution of the numerical ion distribution in velocity space. As shown in Fig. 3, x v x phase space trapping is associated with the shortwavelength activity. The top plot of Fig. 6 displays the v x v y contour lines of 49

Veltri, Valentini & Califano Fig. 6. Level lines (at the top) and surface plot (at the bottom) of the ion distribution in the perpendicular velocity plane at t = 200. 50

The evolution of solar wind turbulence at kinetic scales f at t = 200 integrated over v z and for a given point x M in the physical domain, x M being the point where the x v x trapped region has maximum velocity width. The bottom plot in the same figure shows a surface plot of the same distribution. As it is clear from these two plots, a proton beam is generated, moving along B 0 with velocity v b 1.15v A. The numerical analysis discussed in this paper shows that the electrostatic turbulence observed in space plasmas consists of parallel propagating waves with dispersion relation of the acoustic form and it is associated with the generation of ion-beam distributions. Beside the IA branch, which is in agreement with solar wind data, the k ω spectrum of the numerical signals indicates the presence of a new branch of kinetic waves propagating at v φ v ti. Unfortunately, the available measurements in space plasmas do not allow a k ω analysis at typical kinetic scales. In this view, it is desirable that future space missions could provide these crucial information, in order to test the effectiveness of the numerical model predictions. References 1. D. A. Gurnett and R. R. Anderson, J. Geophys. Res. 82, 632 (1977). 2. D. A. Gurnett and L. A. Frank, J. Geophys. Res. 83, 58-74 (1978). 3. D. A. Gurnett, E. Marsh, W. Pilipp, R. Schwenn and H. Rosenbauer, J. Geophys. Res. 84, 2029-2038 (1979). 4. A. Mangeney, F. Califano, C. Cavazzoni, P. Travnicek, J. Comput. Phys. 179, 405-538 (2002). 5. F. Valentini, P. Trávníček, F. Califano, P. Hellinger, and A. Mangeney, J. Comput. Phys. 225, 753-770 (2007). 6. C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, McGraw-Hill Book Co., Singapore, 1985. 7. M. Dobrowolny, A. Mangeney, and P. Veltri, Astron. Astrophys. 83, 26 (1980). 8. W. H. Matthaeus, M. L. Goldstein, and J. H. King, J. Geophys. Res., 91, 59 (1986). 9. V. Carbone, F. Malara, and P. Veltri, J. Geophys. Res. 100, 1763 (1995). 10. J. V. Holloweg and P. A. Isenberg, J. Geophys. Res., 107, 1147 (2002). 11. E. Marsch, X.-Z. Ao, and C.-Y. Tu, J. Geophys. Res., 109, A04102 (2004). 12. P. Hellinger and P. Travnichek, GeoPhys. Res. Lett. 33, L09202 (2006). 13. N.A. Krall and A.W. Trivelpiece, Principles of plasma physics, San Francisco Press, San Francisco, 1986. 14. F. Valentini et al., Phys. Rev. Lett. 101, 025006 (2008). 15. F. Valentini and P. Veltri, Phys. Rev. Lett. 102, 225001 (2009). 16. L.D. Landau, J. Phys. (Moscow) 10, 25 (1946). 17. I. B. Bernstein, J. M. Greene, and M. D. Krukal, Phys. Rev. 108, 546 (1957). 18. J. P. Holloway and J. J. Dorning, Phys. Rev. A 44, 3856 (1991). 19. F. Valentini, T. M. O Neil, and D. H. E. Dubin, Phys. Plasmas 13, 052303 (2006). 20. F. Anderegg et al., Phys. Rev. Lett 102, 095001 (2009). 21. F. Anderegg et al., Phys. Plasmas 16, 055705 (2009). 22. E. K. Marsch, K. H. Muhlhauser, R. Schwenn, H. Rosenbauer, W. Pilipp, and F. M. Neubauer, J. Geophys. Res., 87, 52 (1982). 23. C. -Y. Tu, L. -H. Wang, and E. Marsch, J. Geophys. Res. 107, 1291 (2002). 51