Two-dimensional hybrid modeling of wave heating in the solar wind plasma L. Ofman 1, and A.F. Viñas 2 1 Department of Physics, Catholic University of America, NASA GSFC, Code 612.1, Greenbelt, MD NASA GSFC, Code 612.1, Greenbelt, MD Abstract: We study the heating and the acceleration of protons, and heavy ions by waves in the solar wind, as well as the nonlinear influence of heavy ions on the wave structure using a 2D hybrid model. Protons and heavy ions are treated kinetically by solving their equations of motion in the self- consistent electric and magnetic fields of the waves, while electrons are treated as a neutralizing background MHD fluid. We use the 2D hybrid code to investigate more realistic obliquely propagating waves, boundary conditions, and background field structures, than previous 1D-simulation and analytical studies. Using the 2D hybrid code we consider for the first time the heating and acceleration of protons and heavy ions by a driven-input spectrum of Alfvén/cyclotron waves in the multi-species coronal plasma. We discuss the observational implication of the results to the solar wind, as well as the relation to the acceleration of stellar winds with hot magnetized coronae. Introduction In-situ observations of the solar wind plasma at 0.3AU and beyond by Helios, Ulysses, and ACE spacecraft, as well as remote sensing observation by SOHO/UVCS show that heavy ions are usually hotter than protons, and flow faster then protons in the fast solar wind streams. Interpretation of UVCS observations of O 5+ ion emission suggests that the heavy ion temperature is anisotropic with T  >T ||. The physical properties of the magnetized, heated solar wind plasma is investigated, and may be relevant to the conditions of hot stellar coronae. Ofman, Viñas, and Gary [2001] studied the evolution of the heavy ion anisotropy, and the conditions for marginal stability using 1D hybrid simulations for low-  coronal plasma. Gary et al [2001, 2003] investigated the evolution of anisotropy using 2D hybrid simulations for  ~1 conditions, and for He ++ ions. Recently Ofman, Gary, and Viñas [2004] investigated the heating and the acceleration of the solar wind plasma with heavy ions by a source f -1 spectrum of waves using 1D hybrid model. The effect of drift as well as source spectrum in the multi-ion plasma was studied by Xie et al [2004]. The observational implication of the heavy ion motion due to the effect of cyclotron waves was investigated by Ofman et al. [2005]. Here, the effect of the source wave spectrum and velocity drift on the perpendicular heating of heavy ions is investigated using 2D hybrid code. SH53A-1253 Linear Dispersion relation The linear Vlasov theory of electromagnetic fluctuations propagating in a homogeneous magnetized and collisionless plasma is well established [e.g., Gary, 1993]. The linear dispersion relation for parallel propagating electromagnetic waves in a multi-ion plasma can be written as [e.g., Davidson and Ogden, 1975; Gary, 1993]: where  =  r + i  is the complex wave frequency, k z is the wavenumber and V s is relative drift velocity along the magnetic field,  s = q s B s /(m s c) is the ion cyclotron frequency, v  s = (2k B T || s/ m s ) 1/2 is the parallel thermal speed, and z is the usual plasma dispersion function [Fried and Conte, 1961]. The temperature anisotropy is defined as  s = T  s /T ||s, the plasma frequency is  ps = (4  n s q s 2 /m s ) 1/2,  ||s = 8  n e k B T || s /B 0 2, and the Alfvén speed V A = B 0 /(4  n e m p ) 1/2. The  signs refer to positive or negative helicity that, with  r > 0 and k z > 0 correspond to right-hand or left-hand circularly polarized waves, i.e., magnetosonic/whistler or Alfvén/cyclotron modes, respectively. The plasma is assumed to be charge neutral, and to bear zero current. Numerical Results Acknowledgments: This study is supported by NASA grants NNG05GD89G, NAG , and NSF grant ATM Solution of the Vlasov dispersion relation for the right-hand magnetosonic streaming instability growing mode: (a) dispersion relation and (b) growth rates of the instability with  p =1.0,   =0.7; (c)  p =1.0,   =1.0; (d)  p =3.0,   =1.0. The dashed lines represent V  p =1.4, 1.8, and 2.2 V A in panels b, c, and d, respectively [Xie et al 2004]. 2D Hybrid model We use the 2D hybrid simulation model, in which the ions are described kinetically as particles in 2d spatial domain, and the electrons are treated as neutralizing background fluid. The three components of about half-million particle velocities are used to calculate the currents, and the fields in the 2D grid (typically 64 x 64). In several runs, driven wave spectrum is applied in a small localized region in the center of the 2d domain: where the mode amplitude  i = i -p/2, frequency  i =  1 + (i - 1) , with the frequency range  = (  N -  1 )/(N -1),  i is the random phase, and N is the number of modes. The value of p is 1, or 5/3, and N = 100. The value of b 0 is set to be The spectrum is applied in a region of area 10  x by 10  y in the center of the computational domain. 1D Hybrid 2D Hybrid Comparison of the results of 1D hybrid [Ofman et al 2001] and 2D hybrid simulations of e-p-O 5+ plasma with initial temperature anisotropy of O 5+. The initial parallel temperature was set to T p =T O =1.4 x 10 6 K, which corresponds to  ||O =0.0413, and the final  ||O = The evolution of O 5+ ion density in the 2D hybrid simulation: (a) The initial ion density. (b) The ion density structure at the end of the run with drift V d =1.5V A. (c) The final density structure in a driven case. The low density area is formed at the location of the driver. (a) (c) (b) The solutions of the Vlasov dispersion relation for e-p-He ++ plasma for left- hand polarized cyclotron mode: (a) no drift, (b) V d =0.3V A [Xie et al 2004]. The typical form of the power spectrum of the driven circularly polarized Alfvén waves given by the expressions on the left [Ofman 2002].  Davidson, R. C., and J. M. Ogden (1975), Electromagnetic ion cyclotron instability driven by ion energy anisotropy in high-beta plasmas, Phys. Fluids, 18, Fried, B. D., and S. D. Conte (1961), The Plasma Dispersion Function, Elsevier, New York. Gary, S. P. (1993), Theory of Space Plasma Microinstabilities (New York: Cambridge Univ. Press) Gary, S.P., Yin, L., Winske, D., and Ofman, L. (2001), Electromagnetic Heavy Ion Cyclotron Instability: Anisotropy Constraint in the Solar Corona, J. Geophys. Res., 106, Gary, S.P., Lin Yin, Winske, D., Ofman, L., Goldstein, B.E., Neugebauer, M. (2003), Consequences of Proton and Alpha Anisotropies in the Solar Wind: Hybrid Simulations, J. Geophys. Res., 108, Issue A2, pp. SSH 3-1, CiteID1068, DOI /2002JA Ofman, L., A. Viñas, and S. P. Gary (2001), Constraints on the O +5 anisotropy in the solar corona, Astrophys. J., 547, L175–L178. Ofman, L., S. P. Gary, and A. Viñas (2002), Resonant heating and acceleration of ions in coronal holes driven by cyclotron resonant spectra, J. Geophys. Res., 107(A12), 1461, doi: /2002JA Ofman, L., Davila, J.M., Nakariakov, V.M., and Viñas, A.F., Alfvén Waves in Multiion Coronal Plasma: Observational Implications, J. Geophys. Res., 110, A09102, doi: /2004JA010969, Xie, H., L. Ofman, and A. Viñas (2004), Multiple ions resonant heating and acceleration by Alfvén/cyclotron fluctuations in the corona and the solar wind, J. Geophys. Res., 109, A08103, doi: /2004JA References The temporal evolution of the temperature anisotropy, and the drift velocity for O 5+ and protons. (a) Driven wave spectrum with B 0 =0.03,  1 =0.25,  2 =0.4, and (b) initial p-O 5+ drift velocity V d =1.5V A. (a) (b) The perpendicular velocity distribution of the heavy O 5+ ions at t=1500  p -1 for (a) driven wave spectrum, (b) V d =1.5V A, (c) V d =2V A. (a)(b)(c) The dispersion relation obtained from the 2D hybrid simulations. (a) Driven wave spectrum, and (b) drift velocity V d =1.5V A. (a)(b) The temporal evolution of the temperature anisotropy, and the drift velocity for He ++ and protons. (a) Driven wave spectrum with B 0 =0.03,  1 =0.42,  2 =0.58; (b) initial p-He ++ drift velocity V d =1.5V A. (a)(b) Conclusions We use investigate the anisotropic heating of multi-component solar wind plasma by a source spectrum of cyclotron waves, and due to drift instability using 2D hybrid simulations. We find that the O 5+ ions are heated efficiently due to resonant interaction with driven left-hand polarized cyclotron wave spectrum. The perpendicular heating of He ++ ions due to similar spectrum that include the He ++ gyroresonant frequency is less apparent, and the proton heating is insignificant. The velocity distribution of the ions heated by broad-band spectrum of cyclotron waves can be approximated well by bi- Maxwellian distribution. When drift V d >V A is present the perpendicular velocity distribution becomes shell-like. If low-amplitude high frequency waves (i.e.,  ~  i ) are present in the corona, than the heavy minor ion perpendicular heating will be more apparent than proton and He ++ heating. The results of this study can be applied to hot-stellar coronae with multi-component plasmas. Discussion The 2D hybrid model allows the generation of oblique waves, in addition to the parallel propagating modes, and thus in principle the growth, and relaxation of ion temperature, and anisotropy can proceed at different rates than in 1D hybrid simulations. The relaxation of O 5+ ion anisotropy due to ion-cyclotron instability calculated with 1D hybrid code was compared to 2D hybrid simulation results, and good agreement was found. For the first time driven cyclotron wave spectrum was applied in 2D hybrid simulation, and perpendicular heating of heavy ions was found, in agreement with 1D results. The 2D density structure of the ions produces small scale structure (i.e., inverse cascade) as a result of ion drift relaxation. The driven spectrum produces significant resonant anisotropic ion heating, with the strongest effect on the low-abundance heavy ions. The perpendicular heating of He ++ is most efficient due to drift instability, in agreement with linear theory, and 1D hybrid models. The dispersion relation of the waves present in the 2D hybrid model is close to the linear dispersion relation, without drift. The presence of super-Alfvénic drift leads to the distortion of the dispersion relation, and to the generation of non-Maxwellian shell-like features in the velocity distribution.