Parallel Heating Associated with Interaction of Forward and Backward Electromagnetic Cyclotron Waves

J. Geomag. Geoelectr., 40, 949-961, 1988 Parallel Heating Associated with Interaction of Forward and Backward Electromagnetic Cyclotron Waves Yoshiharu OMURA1, Hideyuki USUI2, and Hiroshi MATSUMOTO1 2Department 1Radio Atmospheric Science Center, Kyoto University, Kyoto 611, Japan of Electrical Engineering, Kyoto University, Kyoto 606, Japan (Received December 28,1987; Revised April 25, 1988) We studied parallel heating of thermal particles in the presence of two forward and backward electromagnetic cyclotron waves propagating parallel to a static magnetic field in a plasma. Such two waves traveling in opposite directions are commonly excited by electromagnetic instabilities driven by a temperature anisotropy of energetic particles in a magnetized plasma. We developed a theory which gives a heating rate of thermal particles in the parallel direction and we also derived a threshold of the wave amplitude for nonlinear cyclotron trapping of the thermal particles. We performed a computer experiment using an electromagnetic hybrid code, where we assumed anisotropic hot protons as a free energy source. Forward and backward ion cyclotron waves were excited with frequencies below the helium cyclotron frequency. We observed heating of helium ions in the parallel direction. The heating rate agrees well with the theoretical estimate. 1. Introduction Two transverse electromagnetic waves propagating in opposite directions yield a longitudinal force, and they can excite electrostatic waves, such as Langmuir waves and ion acoustic waves (SCHMIDT, 1973). This mechanism is applied to plasma heating by two lasers (COHEN, 1974), and it is also studied by computer simulations (COHEN et al., 1975). Forward and backward electromagnetic waves are simultaneously excited through instabilities driven by a temperature anisotropy of high energy particles. Such examples are found for electron cyclotron (whistler) instability and for ion cyclotron instability found in the equatorial magnetosphere (GENDRIN et al., 1984). These instabilities are believed to be a cause for particle heating observed in the magnetosphere. Computer simulations were performed to study heavy ion heating by electromagnetic ion cyclotron waves excited by a temperature anisotropy of high energy protons (TANAKA,1985; OMURA et al.,1985). In these computer simulations, parallel heating of heavy ions and standing wave structure in the presence of forward and backward ion cyclotron waves were observed. However, the mechanism of the heating has not been clarified. In this paper, we clarify the mechanism of parallel heating by two oppositely 949

950 Y. OMURA et al. propagating electron/ion cyclotron transverse waves, and theoretically estimate the efficiency of the parallel heating. Applying an electromagnetic hybrid code, we perform computer simulations of the electromagnetic ion cyclotron wave instability driven by a temperature anisotropy of high energy protons. We, then, compare the heating rate of heavy ions, i.e., of helium ions with the theoretical estimation. 2. Theory We first describe vector presentations of a circularly polarized transverse electromagnetic wave propagating parallel to a static magnetic field. We assume both of the static magnetic field Bo and the wavenumber vector k are along the x-axis. The transverse wave vector in the y-z plane is represented by a complex variable, With this notation transverse waves with right-and left-handed polarizations are expressed by positive and negative ƒö's, respectively. Then, the Maxwell equation (2) gives a relation between the transverse wave vector Ew and Bw as (3) An electromagnetic wave is supported by charged particles which form a current in phase of the wave vectors. The transverse velocity of the particles with charge q and mass m should satisfy the linearized equation, (4) where we assume Bw áb0. We have from Eq. (4) Inserting Eq. (3) into Eq. (5) and solving for vw, we have (6) where ƒ 0 is defined by qb0/m and its magnitude is the cyclotron frequency of the charged particles.

Parallel Heating Associated with Interaction 951 In the following, we show that forward and backward traveling waves of the same magnitude of frequency and wavenumber can produce an effective parallel force vw ~Bw to yield a parallel acceleration of plasma particles. If ƒö/ƒ 0 ƒ1, Eq. (6) shows that vw of a forward traveling wave (ƒö/k 0) is antiparallel to the wave magnetic field Bw, and that vw of a backward traveling wave is parallel to Bw. Therefore, we represent perpendicular velocities of particles and transverse magnetic fields of the forward and backward traveling waves in the y-z plane as The phases ƒæf and ƒæb are defined as where ƒæf0 and ƒæb0 are the phase constants. The quantities Vf, Vb, Bf and Bb are the amplitudes of the velocities and wave magnetic fields for forward and backward traveling waves, respectively. We, then, obtain q(v ~B)x force along the wave vector k as where ex is a unit vector in the x (longitudinal) direction. We denote this nonlinear force in the x direction as Fx. Since Fx acts as a static force without time variation, we can define a nonlinear static potential ƒ³ as (13) From Eq. (6), Eq. (13) is rewritten as (14)

952 Y. OMURA et al. Assuming a particle moving from the top to the bottom of the potential, we can estimate the maximum parallel velocity Vmax of particles which are trapped in the nonlinear potential. Equating the kinetic energy with the potential energy as (15) we have (16) In Eq. (16), the sign of ƒö,+ or -, corresponds to the right- and left-handed polarizations, and the sign of ƒ 0, + or -, corresponds to the ion and electron gyrofrequency. For the interaction between an R-mode wave and electrons, ƒ 0 and ƒö have negative and positive values, respectively. For the interaction between an L-mode wave and ions, ƒ 0 and w have positive and negative values, respectively. Therefore, we can rewrite Eq. (16) for the R-mode whistler mode wave and for the L-mode Alfven wave as (17) where ƒ 0 is either electron or ion cyclotron frequency, and ƒö ƒ ƒ 0 is assumed. It is obvious from Eq. (17) that the parallel acceleration of particles is effective when the wave frequency is slightly less than the cyclotron frequency. Since the wave is supported by the transverse sloshing motion of the particles, the wave may collapse if the parallel acceleration is so large that the particles fall into a trapping region of cyclotron resonance. The trapping region is given by (MATSUMOTO and OMURA, 1981) VR-VT ƒv a ƒvr+vt, (18) where (19) Therefore, the critical point at which the wave collapses is when the following equation is satisfied. VR-VT=Vmax. (20) For simplicity, if we assume that the backward and forward traveling waves have

Parallel Heating Associated with Interaction 953 equal amplitudes, which is the case for an instability driven by a temperature anisotropy, we obtain the following relation from Eqs. (6), (17) and (19). VT=Vmax. (21) Substituting Eq. (21) into Eq. (20) and solving for Bw, we can estimate the maximum amplitude Bmax, above which the wave reaches the self-collapse due to the nonlinear trapping of the accelerated particles. (22) If the amplitudes of the waves growing through the wave instability exceed Bmax, the nonlinear resonant trapping of plasma particles begins, and the cyclotron resonance causes a strong particle heating in the parallel as well as the perpendicular directions. Consequently, the wave growth should stop and eventually gets damped due to cyclotron damping. Therefore, Bmax in Eq. (22) gives a saturation level of the instability if the competing quasilinear diffusion is weak. 3. Computer Experiment In this section, we examine the theory developed in the previous section by a computer experiment of an instability driven by a temperature anisotropy. In the computer experiment, we used a one-dimensional electromagnetic hybrid code in which electrons are treated as a massless fluid and ions are treated as particles moving in four dimensions (x, vx, vy, vz). We focused our attention on waves propagating parallel to the external magnetic field Bo. We assumed three species of ions, i.e., cold and isotropic protons, cold and isotropic heliums and hot protons with a temperature anisotropy. We used basically the same physical parameters as assumed in the study by OMURA et al. (1985), where heating of helium ions at the equatorial magnetosphere is studied by the same hybrid code simulation. The physical parameters are as follows. Electron beta ƒàe: 0.001 (Hot H+ ion density)/(cold H+ ion density): 0.111 (Cold He+ ion density)/(cold H+ ion density): 0.2 Thermal velocity of cold H+ ions: 0.018VA Thermal velocity of cold He+ ions: 0.0009VA Anisotropy of both cold H+ and He+ ions: T Û/T a=1 Parallel thermal velocity of hot H+ ions: 1.8VA Perpendicular thermal velocity of hot H+ ions: 2.55VA Anisotropy of hot H+ ions: T Û/T a=2, where VA is the Alfven velocity for the cold protons. The grid spacing and time step

954 Y. OMURA et al. are taken as ƒ x=va/ƒ H and ƒ t=0.05/ƒ H, respectively, where ƒ H is the proton cyclotron frequency. The simulation system consists of 256 grid points with a periodic boundary condition. The number of superparticles in the system is 8,192 for each of the cold species. As for the hot protons, which give a free energy source in the system, we have 65,536 superparticles in the system. Using the parameters stated above, we first computed the linear dispersion relation for the unstable electromagnetic ion cyclotron waves propagating parallel to the static magnetic field. As shown in Fig. 1, the left-handed circularly polarized Fig. 1. Linear dispersion relation. (a) The ƒö-k diagram. The upper and lower dotted lines indicate the cyclotron frequency of H+ and He+, respectively. The solid lines are the two branches of left-handed circularly polarized Alfven modes. (b) The ƒá-k diagram showing the linear growth rates as functions of k. The solid lines are growth rates for a temperature anisotropy of hot protorns T Û/T a=2.0, and the dashed lines are for T Û/T a=1.3. The low frequency mode (LF) has a larger growth rate than the high frequency mode (HF).

Parallel Heating Associated with Interaction 955 Alfven waves are split into two branches owing to the presence of He+ ions. The lower frequency branch in the frequency range of 0 ƒƒö ƒƒ He+ has a larger growth rate than the higher frequency branch. Based on the growth rate calculation, we performed a computer experiment to study the evolution of the instability up to ƒ Ht=409.6. The time history of the magnetic energy density in Fig. 2(a) shows that electromagnetic waves are excited and saturated around ƒ Ht=400. In Fig. 2(b) We plotted the time history of the temperature anisotropy T Û/T a of hot H+ ions, which decreases rapidly along with the wave growth. In order to apply the theory presented in the previous section to the results of the computer experiment, we need to verify the assumption of the co-existence of forward and backward traveling electromagnetic waves with the same frequency and wave number. By Fourier-transforming the wave data of the computer experiment both in Fig. 2. (a) Time history of the wave magnetic energy density. (b) Time history of the temperature anisotropy of hot H+ ions.

956 Y. OMURA et al. space and time, we can decompose the wave spectra into forward and backward traveling components as shown in Fig. 3. It is noted that forward and backward components have approximately the same amplitudes. Therefore, we can apply the theory to estimate the particle heating observed in the computer experiments. In the theoretical estimation of the parallel heating and the upper limit of the wave amplitude, we assumed a coherent monochromatic wave. To apply the theory to the simulation results, we must know the dominant coherent monochromatic wave in the system. We plotted the time evolution of the wavenumber spectra in Fig. 4. In the initial stage the modes around kva/ƒ H=0.3 are dominant in agreement with the initial growth rate for T Û/T a plotted with the solid line in Fig. 1(b). Due to the rapid decrease of T Û/T a, the maximum growth occurs at the lower wavenumbers as shown Fig. 3. The ƒö-k spectra of Bz field over the time period of ƒ Ht=0.8-409.6. (a) Spectra of forward traveling components; (b) Spectra of backward traveling components. The amplitudes of the spectra are normalized to Bn=0.007 B0.

Parallel Heating Associated with Interaction 957 Fig. 4. Time evolution of the k-spectra of the transverse wave magnetic field, which is a summation of By and Bz components. by the dashed line for T Û/T a=1.3 in Fig. 1(b). Therefore, the modes 8, 9 and 10 with kva/ƒ H=0.20, 0.22 and 0.25 are dominant in the later time. Especially, the mode 9 with kva/ƒ H=0.22 and ƒö/ƒ H=0.13 is most dominant as shown in Fig. 5(a). In agreement with the linear growth rate calculation, the frequency of the most dominant mode is below the helium cyclotron frequency ƒ He+ as shown in Fig. 3. It is obvious from Eq. (17) that the thermal helium ions are more likely to be heated than the thermal protons. We calculate the maximum parallel velocity Vmax of helium ions with Eq. (17) and compare with the results of the computer experiment. We calculate the parallel thermal velocity of the helium ions in the simulation as (23) where the pair of brackets ƒ is the average over all helium ions in the system. If we assume that the nonlinear heating makes a flat velocity distribution extending from - Vmax to+vmax, we can estimate the thermal energy density K a of the helium ions as (24)

958 Y. OMURA et al. Fig. 5. (a) Time history of the dominant modes (modes 8, 9 and 10). (b) Time evolution of the helium energy density in the parallel direction. The solid line represents the simulation result. The dashed line is the theoretical value computed from the wave amplitude of mode 9. The parallel thermal energy density of the helium ions K a in the simulation is plotted by the solid line in Fig. 5(b). The theoretical value of Eq. (24) for mode 9 is calculated from Eq. (17) using the amplitude Bw shown in Fig. 5(a), and is plotted by the dashed line. The phase diagrams of helium ions in the v a-x plane are plotted for different times in Fig. 6. The solid lines show the maximum velocities }Vmax calculated from Eq. (17), where we used BW and w of the most dominant mode at each time. It is obvious from Fig. 6 that the particles get heated in the parallel direction as time elapses. Because the nonlinear potential expressed by Eq. (14) is a function of x, the velocity modulation depends on the locations of particles. This means that all the particles are not accelerated up to the Vmax value. Particles located near the bottom of

Parallel Heating Associated with Interaction 959 Fig. 6. The phase diagrams of helium ions in the v a-x plane at different times. The solid lines represent the maximum velocity theoretically estimated from the wave amplitude of the most dominant mode. the potential well are less accelerated. Eventually, the particles are smeared out and heated in the phase diagram of v a versus x. We may well say that velocities of almost all the particles are within the theoretical value of Vmax. Similarly, we can calculate the amplitude threshold for nonlinear trapping of thermal particles from Eq. (22). Since the frequency of the most dominant mode is less than the helium cyclotron frequency, the nonlinear trapping would operate on the helium ions, if the amplitude is large enough. Inserting the helium cyclotron frequency in Eq. (22), we have B=0.115 B0 for the mode 9. However, the saturation level in the computer experiment is 0.05 Bo which is below the threshold of nonlinear trapping. This means that the saturation is caused by the quasilinear diffusion of hot

960 Y. OMURA et al. protons and not by the nonlinear trapping of helium ions. Actually the temperature anisotropy of the hot protons decreases as the waves grow as shown in Fig. 2(b). The growth rate of the waves decreases to zero and the waves stop growing. 4. Summary and Discussions We have shown that the interaction of the forward and backward electromagnetic cyclotron waves gives rise to a static longitudinal force which effectively accelerates the cold particles supporting the transverse waves. The theory we have developed gives an expression for the parallel thermal velocity of particles accelerated by the longitudinal force. To verify the theory, we performed a computer experiment of an ion cyclotron wave instability driven by a temperature anisotropy. The parallel heating of helium ions observed in the computer experiment shows a good agreement with the theoretical estimate. Using the idea of nonlinear cyclotron trapping, we also estimated the maximum wave amplitude attainable in the cyclotron wave instability driven by a temperature anisotropy. The maximum amplitude is estimated by the threshold for nonlinear trapping of thermal particles. The amplitude observed in the computer experiment is well below the threshold, indicating that the saturation of the instability is caused by the quasilinear diffusion of anisotropic hot protons. However, the idea of nonlinear trapping is important, because this gives an upper limit of the amplitude of electromagnetic cyclotron waves excited by a temperature anisotropy. We have focussed our attention on the parallel heating of particles. As for the perpendicular heating, there have been several studies for heavy ions interacting with electromagnetic ion cyclotron waves (MACK, 1982; BERCHEM and GENDRIN, 1985; TANAKA, 1985; OMURA et al., 1985). They studied perpendicular acceleration of heavy ions in the presence of relatively large amplitude waves. The perpendicular motion is basically the orderly sloshing motion in phase of the transverse wave. In the presence of both forward and backward waves, however, the effective parallel acceleration occurs as clarified in the present study. If the parallel heating is strong enough, the thermal particles supporting the wave propagation are trapped in the nonlinear potential of the cyclotron resonance. The cyclotron trapping further causes thermalization in the parallel motion as well as perpendicular motion, leading to a nonlinear collapse of the wave. Therefore, the threshold for the nonlinear cyclotron trapping derived in Section 2 is also the threshold for effective perpendicular heating. In this respect, the present study supplements the previous studies by pointing out the importance of the interaction of forward and backward electromagnetic cyclotron waves. In the present computer experiment, the wave amplitude did not exceed the threshold for the nonlinear trapping because of the quasilinear diffusion of the hot protons. If the temperature anisotropy of the hot protons is retained by a constant influx of fresh anisotropic hot protons, the nonlinear trapping and subsequent nonlinear wave collapse are expected to occur. A computer experiment of this nonlinear process is left as a future study.

Parallel Heating Associated with Interaction 961 Discussions with T. Yamada were helpful in performing the computer experiment using the hybrid code. We also thank H. Tanaka and Y. Iwane for their discussions. H. Usui acknowledges the encouragement by I. Kimura. All computations were carried out at Data Processing Center of Kyoto University. REFERENCES BERCHEM, J.R. and R. GENDRIN, Nonresonant interaction of heavy ions with electromagnetic ion cyclotron waves, J. Geophys. Res., 90, 10945-10960, 1985. COHEN, B.I., Space-time interaction of opposed transverse waves in a plasma, Phys. Fluids, 17, 496-497, 1974. COHEN, B.I., M.A. MOSTROM, D.R. NICHOLSON, A.N. KAUFMAN, and C.E. MAX, Simulation of laser beat heating of a plasma, Phys. Fluids, 18, 470-474, 1975. GENDRIN, R., M. ASHOUR-ABDALLA, Y. OMURA, and K. QUEST, Linear analysis of ion cyclotron interaction in a multicomponent plasma, J. Geophys. Res., 89, 9119-9124, 1984. MATSUMOTO, H. and Y. OMURA, Cluster and channel effect phase bunchings by whistler waves in the nonuniform geomagnetic field, J. Geophys. Res., 86, 779-791, 1981. MAUK, B.H., Electromagnetic wave energization of heavy ions by the electric "phase bunching" process, Geophys. Res. Lett., 9, 1163-1166, 1982. OMURA, Y., M. ASHOUR-ABDALLA, R. GENDRIN, and K. QUEST, Heating of thermal helium in the equatorial magnetosphere: A simulation study, J. Geophys. Res., 90, 8281-8292, 1985. SCHMIDT, G., Resonant excitation of electrostatic modes with electromagnetic waves, Phys. Fluids, 16, 1676-1679, 1973. TANAKA, M., Simulation of heavy ion heating by electromagnetic ion cyclotron waves driven by proton temperature anisotropies, J. Geophys. Res., 90, 6459-6468, 1985.