Low-freqency instabilities and the resulting velocity distributions of pickup ions at comet Halley

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. All, PAGES 21,541-21,556, NOVEMBER 1, 1994

Low-frequency instabilities and the resulting velocity distributions of pickup ions at comet Halley

H. Karimabadi, D. Krauss-Varban, and N. Omidi • Department of Electrical and Computer Engineering, University of California, San Diego

S. A. Fuselier

Lockheed Palo Alto Research Laboratory, Palo Alto, California

M. Neugebauer Jet Propulsion Laboratory, California Institute of Technology, Pasadena

Abstract. The interaction between the solar wind and newborn cometfry ions is studied using a new analytical theory as well as one- and two-dimensionM hybrid simulations. Using the observed parameters upstream of the comet Halley, a detailed study of wave excitation and the resulting particle distributions is presented. Linear theory as well as simulations show that a variety of modes such as the fast magnetosonic mode, high frequency whistlers and obliquely propagating Alfv•n ion cyclotron waves can be excited. However, parallel propagating waves are found to be dominant in the wave spectrum and to control the scattering of the pickup ions. Several features of the observed distributions of pickup protons are explained. In particular, it is shown that the observed asymmetric pitch angle distribution for the pickup protons is due to the smM1 saturation amplitude of the waves for the given parameters. Water group associated waves can lead to energy diffusion and further pitch angle scattering of protons. This effect is most likely to be important in the vicinity of the bow shock of comet HMley where the density of water group ions becomes comparable to that of protons. It is shown that the observed increase in the radius of the proton velocity shell just outside the bow shock can be due to w•ter group waves. The nearly isotropic proton pitch angle distribution observed by Neugebauer et fl. [1989] just outside the bow shock may, however, be related to the presence of a rotational discontinuity which has been identified in the magnetic field data. Just outside the bow shock, simulations show that parallel propagating water group waves can steepen with attached whistler w•ve packets. The steepening process •t p•rMlel propagation is • transient effect, in •n important contrast to the case of steepening at oblique angles. The smaller beam densities at comet Halley appears to be the main reason not only why waves at comet Halley have smaller amplitudes but also why oblique, steepening magnetosonic waves have not been detected at comet Halley, whereas they have been seen at comet Giacobini-Zinner.

1. Introduction

Newborn cometfry ions form a ring beam which is generally unstable to a variety of low-frequency waves (see review by Neugebauer [1990, and references therein]). The excited waves in turn scatter and heat the cometfry ions [e.g., Wu et fl., 1986; Gary et fl., 1986; Terasawa, 1988; Miller et fl., 1991; Yoon et fl., 1991].

1 Also at California Space Institute, University of California, San Diego, La Jolla.

Copyright 1994 by the American Geophysical Union.

Paper number 94JA01768. 0148-0227/94/94JA-01768505.00

Although the general picture of the pickup process has been known for some time, significant gaps between theory and observations remain. One of these gaps con- cerns the wave spectra at Halley and other comets. The observed wave spetrum at comet Halley is very different from that at comet Giacobini-Zinner (hereafter refered to as GZ). At comet Halley, the waves are characterized by irregular waveforms and generally have small amplitudes (ll/Bo whereas fluctuations observed at GZ are much more regular and have large amplitudes with 5lBI/Bo ~ 0.5. Another major difference is the observation of large-amplitude steepened waves with associated whistler wave packets (shocklets) at GZ and more recently at comet Grigg-Skjellerup [Glass- meier and Neubauer, 1993]. No shocklets have yet been identified in Halley observations.

21,541

21,542 KARIMABADI ET AL.: WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY

In this paper we will show that the wave characteristics, such as wave amplitude, propagation direction of the waves as well as whether they steepen or not, strongly depend on the beam density. Due to the larger gas production rate at comet Halley the mass loading and the shock formation occur at larger distances from the cometary nucleus than at either GZ or comet Grigg- Skjellerup. Using the standard model of cometary ion production [Schmidt and Wegmann, 1982] and the measured values for the production rate and the cometocen- tric distance of the bow shock for Halley and GZ, one finds that the density of the newly created protons at the cometary bow shock is roughly a factor of 4 smaller at Halley. The smaller beam densities translate into smaller growth rates of the modes in general and into a smaller ratio of the growth of oblique to parallel propagating magnetosonic mode in particular. Thus we ex- pect the waves to be mainly parallel propagating and to have smaller amplitudes at comet Halley compared to waves at GZ, consistent with the observations. We show that for the densities at GZ the oblique magnetosonic mode has a growth rate comparable to the parallel propagating mode. The large growth rate of the oblique mode is a necessary condition for the formation of shocklets.

There have been several detailed studies of velocity distributions of pickup protons [Neugebauer et al., 1989] as well as water group ions [Coates et al., 1989, 1990] observed upstream of the Halley bow shock. Neugebauer et al. [1989] showed that proton pitch angle scattering was much more rapid than energy diffusion. They also found that (1) large pitch angle anisotropies exist at all distances beyond the cometary bow shock, (2) just outside the shock, however, the pitch angle distribution was nearly isotropic and the radius of the pickup shell increased significantly, and (3) finally, for quasi- perpendicular geometries, the pitch angle distribution was very asymmetric with phase space density peaks near pitch angles of 180 ø. Here quasi-parallel and quasi- perpendicular refer to the direction between the pitch angle of injected ions and the background magnetic field. Neugebauer et al. [1989] speculated that these asymmetric pitch angle distributions may be caused by global rather than local effects. Other than these at- tempts, satisfactory explanation for conclusions 1-3 has not been forthcoming.

In this paper we address the question of wave excitation as well as the resulting pitch angle distribution of pickup ions upstream of comet Halley's bow shock. We use both one- and two-dimensional (l-D, 2-D) hybrid simulations to study the wave generation and the scattering of pickup protons. A recently developed analytical theory is then used to explain the observed scattering and energy diffusion. In order to make direct contact with observations, we have chosen the parameters of our simulations to correspond to those of Neugebauer et al. [1989]. We find that for the parameters of comet Halley, the waves with the largest amplitudes are fast magnetosonic and backward propagating Alfvdn waves, with both having maximum growth along the field direction. Even though obliquely propagating

modes (e.g., mirror) are also excited, their saturation amplitude is typically much smaller than that of the parallel propagating waves. However, the amplitude of the parallel waves is also relatively small, resulting in trapping widths that cover only a part of the shell and leading to anisotropic distributions in pitch angle. Close to the bow shock, the density of water group ions becomes comparable to that of protons. This leads to excitation of oxygen waves with larger amplitudes than the proton waves. The interaction of protons with the oxygen waves scatters them into a thicker shell in agreement with observations. Even though the oxygen waves have maximum growth at parallel propagation, they do steepen and have associated with them whistler wave packets. The details of the steepening is, however, different from that of obliquely propagating waves.

The outline of the paper is as follows. In section 2, we discuss the simulation model and the parameters. Sections 3 and 4 include the analysis of waves obtained from the simulations for quasi-parallel and quasi- perpendicular cases, respectively, as well as a compar- ison with observations. The mode generation and the resulting spectra are interpreted with the aid of linear Vlasov theory. The scattering of the pickup ions in the simulations is explained using a new theory [Karimabadi et al., 1994] based on the resonance overlap in the actual wave fields. In section 5, we discuss the conditions for which there can be significant linear growth at oblique angles, using the solar wind flow speed, the injection angle, and the beam density as free parameters. Con- sequences for wave steepening are also discussed in section 5. Further discussion and the summary are given in section 6.

2. Simulation Model and Parameters

The simulation codes used in this paper are the 1- D [Winske and Omidi, 1993] and 2-D electromagnetic hybrid codes which treat the electrons as an adiabatic fluid and ions as kinetic macroparticles. We use periodic boundary conditions for both particles and fields. Time is normalized to proton gyrofrequency (f•), x to c/wp, and velocity to Alfvdn speed V•t. Here m r is the proton plasma frequency. The simulation model consists of two ion populations: solar wind protons and the newborn protons. In order to study the effect of water group ions on the excitation of waves, we have also performed simulations with a third species: singly ionized oxygen. The simulations are typically carried out in a frame moving with a speed close to the phase velocity of the dominant modes, and the newborn ions are assumed to have an initial ring beam velocity distribution. The simulations are of the initial value type; i.e., the ring beam ions are initialized at the start of the run with a given density which is then kept fixed in time. As long as the injection rate is slow compared to the scattering rate, there is no significant difference between initial value simulations and those where the pickup ions are injected contin- uously in time [e.g., Karimabadi e! al., 1994]. In the one-dimensional simulations, the background magnetic field Bo lies in the x - z plane and makes an angle of 0

KARIMABADI ET AL.: WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY 21,543

with the x axis, whereas in the 2-D simulations Bo lies in the x- y plane.

One of the most detailed observational studies of the

velocity space distributions of pickup ions is that by Neugebauer et al. [1989]. In order to make direct com- parison with these observations, we have chosen the parameters of our simulation to correspond to those of Plate 2 (quasi-parallel interval) and Plate 6 (quasi- perpendicular) of Neugebauer et al. [1989]. We have reproduced these two plates in black and white in Fig- ures la and lb, respectively. We also calculate the mode properties from linear Vlasov theory to properly interpret our simulation results. In these linear calculations we assume a Maxwellian distribution of solar wind

electrons and protons at the appropriate temperatures [Krauss-Vatban et al., 1994]. The ring beam is modeled with a distribution that has a thermal spread Vllta in the parallel velocity and is a delta function in the perpendicular velocity component at vño [Goldstein and Wong, 1987; Omidi et al., 1994]. Such a ring beam has an effective anisotropy of A = vño2/Vllta 2. A thermal spread in vñ is unimportant; in fact, in the present con- text the dispersive properties of a ring beam are very similar to those of a drifting bi-Maxwellian with equal anisotropy A =va. ta2/Vllta2.

8. Quasi-Parallel Case In this section we present results from hybrid simula-

tions of ring beam instabilities associated with cometary ion pickup in the quasi-parallel geometry. Quasi-parallel refers to cases where the initial pitch angle of the newly created ions is near 0 ø or 180 ø. The observed velocity space distribution for the two quasi-parallel intervals shown by Neugebauer et aL [1989, Plates 1 and 2] are very similar; namely, they are one-sided distributions in pitch angle. So it suffices to explain the pitch angle distribution for one of these cases and we have chosen

Plate 2 (see Figure la). The observed parameters corresponding to Plate 2

of Neugebauer et al. [1989] are injection angle (with respect to the background magnetic field) of c• = 15204 - 11 ø, solar wind velocity of ¬,, ~ 320 km/s, Alfv•n speed (Va) of ~ 57 km/s, electron beta of/•e ~ 0.85, solar wind ion beta of/•i = 1.5, Nb/N,, ~ 0.5%. Here No and N,, are the pickup and solar wind densities, respectively. We assume that the pickup ions have a temperature equal to one hundredth of the solar wind protons. In the presence of waves the pickup ions gain energy and attain a finite temperature rapidly. So the results are not very sensitive to the initial value chosen for the initial temperature of pickup ions.

(a) soo

400

o -500

V9ar, km/s

(b) 500 400

300

.200 IO0

o -5oo

Figure 1. The observed proton phase space densities reproduced from Neugebauer et al. [1989]. Figures la and lb correspond to Plates 2 and 6 of Neugebauer et al. [1989], respectively.

21,544 KARIMABADI ET AL.- WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY

Figure 2 shows the results of linear kinetic theory for the above parameters in the electron rest frame. The dispersion curve at parallel propagation (0 = 0 ø) is shown in Figure 2a. Three modes are shown. The two dashed curves are the forward propagating (in the beam direction) magnetosonic mode (FMS +) and the backward propagating Alfv•n ion cyclotron mode (AIC-) in the absence of a beam. The beam results in modi-

fications of the normal modes of the plasma as well as additions of new linear modes supported by the beam. The beam mode is shown in Figure 2 as solid and for effective anisotropy (A) of 50, where A = V•.o•/Vll,,• , Vño - ¬•j cos(a), Vllth 2 - 2 kT/m is the beam thermal speed and ¬hi =- V,w is the solar wind velocity. In addition to the beam speed Vllo/Vllth , the effective anisotropy is a useful quantity for quantifying the free energy avail- able in a ring beam distribution. Here the beam mode for A = 10 is almost identical to that at A = 50 and

thus not shown in Figure 2. The beam mode couples to the AIC- at low wavenumber and to the FMS + at

larger k. For a ring beam with much smaller free energy (e.g., low value of A, beam density, or velocity) the growth occurs simply at the intersection of the beam cyclotron resonance with the normal modes of the plasma. For a strong ring beam, the growth occurs on the beam mode as shown in Figures 2a and 2b. In cases where the growth occurs only on the beam mode as opposed

0: 07 0: 507 (a) (c)

/ beam mode • mode-

I...F. IVi. S._i.•i • t •' ',,.'"••_ o. . ........

x161 (b) (d) A

I 1 o.o A= 10. 1

O. 0.25 0.5 O. 0.25 0.5

C k/(Op c k/(Op

Figure 2. Dispersion and growth rates in the electron rest frame for the unstable modes at directions

parallel and oblique to the background magnetic field. (a) Beam mode (solid) couples the forward propagating fast magnetosnic mode (FMS +) and the backward propagating Alfv4n ion cyclotron mode AIC-. The dispersion curves for FMS + and AIC- in the absence of the beam are shown as dashed. The beam mode shown

is for an effective anisotropy (A) of 50. (b) The growth for effective anisotropies of 10 and 50. The growth occurs on the beam mode. (c) The beam mode now couples the two AIC modes. (d) The growth for effective anisotropies of 10 and 50.

to on the normal modes of the plasma, we supersede the name of the growing mode with the letter B to indicate that we are referring to the beam mode. For instance, in Figures 2a and 2b, the first peak in the growth rate is due to the coupling of the beam mode with AIC- and we give it the name B-AIC-. Similarly, the second growth on the beam mode is referred to as B-FMS +.

Note that both AIC- and FMS + have the same helicity and hence can interact with the same resonance. The beam cyclotron resonance line (not shown) inter- sects AIC- at one point and the FMS + at two points. The second intersection with the FMS + occurs at higher frequency (whistlers) and is discussed further below. Figure 2b shows the growth resulting from the intersection of the resonance line with the AIC- and the

low frequency FMS + for two different values of effective anisotropy (A). For A-10, the two peaks in the growth rate are clearly discernible in Figure 2b, whereas they merge at large A. Since all the excited modes have the same helicity, which is positive if the beam is propagating in the direction of the background field, one can refer to the instability as a positive hellcity instability. The growth of the B-FMS + at low frequencies is also often referred to as the right-hand electromagnetic ion/ion resonant instability [Winske and Gary, 1986].

We also show in Figures 2c and 2d the dispersion and the growth rates of modes with highest growth rate at 0 - 500 . The FMS + is no longer accessible due to the fact that the term cos0 in the resonance condi-

tion w/f2 = (ck/wp)(Vll o/V,•)cosO - œ (harmonic number œ = -1) has pulled this line to frequencies too low for resonance. Also, Landau damping of the FMS + mode increases with 0. On the other hand, the helicity of the AIC + has changed from negative at 0 = 0 ø to positive at 0 = 50 ø. This change in the helicity of the AIC as a function of propagation angle has been noted by several authors [e.g., Gary, 1986; Krauss-Varban et al., 1994]. Although AIC- has changed its helicity from positive to negative due to oblique propagation angle, the coupling of the beam with AIC- changes the helicity of B-AIC- back to positive. That is, the growth occurs on a beam mode which has positive helicity for all unstable k sim- iliar to the case at 0 = 0 ø. To our knowledge, this connection between the beam mode and the normal modes

was not demonstrated in previous work [e.g., Brinca and Tsurutani, 1989; Goldstein et al., 1990]. Figure 2d shows the growth rate for two values of A. Comparing Figures 2b and 2d, it is clear that parallel propagating waves have a larger growth rate. The growth rate of B- AIC- is mostly flat as a function of propagation angle with a peak at parallel propagation. The growth of B- AIC +, on the other hand, is sensitive to the direction of propagation, having a peak at 500 and dropping rapidly for propagation angles away from 500 .

The growth of the oblique B-AIC- is particularly sensitive to A due to the fact that as the free energy associated with the ring beam becomes smaller the ability of the beam mode to alter the helicity of the AIC- becomes less effective. Since the instability is a positive helicity instability, the growth on the AIC- drops off rapidly as a function of A.


We now turn to the simulation results. As we men-

tioned earlier, the positive helicity instability at parallel direction excites B-AIC-, B-FMS +, and the whistler mode. The whistler mode instability is due to the second intersection point of the beam cyclotron resonance line with FMS and was not shown in Figure 2. It turns out that both FMS instabilities can have comparable growth rates. In order to assess the relative importance of the whistler mode to the particle scattering, we have performed 1-D simulations with high resolution and a large number of particles. The simulation box is 64c/w r and is divided into 256 cells with 1000 particles. The time step is f•At -- 0.03.

Due to the high frequency of the whistlers, it is ad- vantageous to perform the simulations in a frame where the frequency of the whistlers is Doppler shifted to lower frequencies. Here the simulations are done in a frame where the solar wind is moving with 4VA in the direction of the background magnetic field.

Figure 3 shows the time history of the magnetic field energy, effective anisotropy and the drift speed (between the solar wind and the pickup ions), respectively. The field energy reveals two growth phases. The first peak at f•t ~ 25 is due to the excitation and subsequent saturation of the whistler instability. The second peak at f•t ~ 60 is due to the saturation of the low-frequency positive helicity instability.

The effective anisotropy A is large initially. But the pickup ions are rapidly heated as waves are excited, leading to a large drop in A (note logarithmic scale). We should emphasize, however, that mode properties and even growth rates are not sensitive to initial value of A as long as it exceeds a certain threshold. After the saturation of the second instability, the value of A drops even more but it asymptotes at about 7.5, never reaching 1. As we will show shortly, this is due to the fact that the pickup ions are scattered into half a shell but no further, thus maintaining a finite A larger than unity.

x10 -3 (a)

(6B)

Bo2 o.o L--/ I I 1(/3 t I I (b)

lO o , (c)

,5.0 •

3.5 O. D t 150.

l•igure 3. Time history of the (a) magnetic field energy, (b) effective anisotropy (defined in the text) of pickup protons in the text), and (c) the relative drift between the pickup protons and solar wind. The simulation parameters correspond to Plate 2 of Neugebauer et al. [1989].

Figure 4 shows the profile of one of the transverse components (Bz) of the magnetic field at several times. Initially, the wave profile is dominated by short-wavelength modes. These are the whistlers. Whistler waves have been detected in the ion foreshock [Hoppe et al., 1982] and near comets, e.g., GZ [Tsurutani and Smith, 1986], and have been studied by various authors [e.g., Winske e! al., 1985; Akimoto and Winske, 1989; Wong and Goldstein, 1987; Kojima et al., 1989]. It is in- teresting to note that the whistlers in Figure 4 show a striking resemblance to those observed in the ion foreshock [Hoppe et al., 1982]. Recently, Mazelie and Neubauer [1993] have reported several cases of discrete wave packets at Comet Halley. These waves are observed to be at the proton cyclotron frequency and are propagating nearly parallel to the magnetic field. Mazelle and Neubauer [1993] found no systematic as- sociation of these wave packets with steepened fronts and argued that the most likely generation mechanism of these waves is a local linear instability of pickup ions. Thus these wave packets may have been generated by a process similar to that shown in Figure 4.

After fit ~ 25, a second mode with a wavelength of ~ 20c/wp appears and eventually dominates. This is the B-FMS +/B-AIC- mode. The early appearance of whistlers and subsequent generation of lower-frequency waves is consistent with the evolution of the anisotropy in Figure 3 and the growth rates from linear theory: for A > 40, the growth rate of whistlers is larger; for A ~ 10, the growth rate of the lower-frequency B-FMS + mode is a factor of 2 larger. We have performed Fourier analysis of the transverse field component (By) for f•t = 40. The power of the field in each mode is shown in Figure 5. The contour levels are 0.1, 0.25, and 0.5, respectively. We have also overplotted the dispersion curves (solid curves) as well as the growth rates (dashed curves) based on linear theory. The wave growth rate and modal properties in the simulation agree well with the linear theory.

The phase space of the pickup protons at the two saturation times as well as at the end of the run are shown

B

o. 32. 64.

x (c/%) l•igure 4. The x profile of the z component of the magnetic field at several times. The short-wavelength oscillations are whistler waves. Same parameters as in Figure 3.

21,546 KARIMABADI ET AL.' WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY

5.0

2.5

0.0

-2.5

-5.0

O. k (•p/½) 5. r)ip,:ion (oid) ,nd owth (dashed)

the unstable modes for simulation of Plate 2 of Neuge- bauer et al. [1989]. The power in each mode is shown by a combination of grey scale and contours. The contour levels are 0.1, 0.25, and 0.5, respectively.

in Figure 6. In Figure 6 (as in subsequent Figures 10 and 15), analytically derived resonance widths based on the waves present in the simulation are indicated [Karimabadi e! ai., 1994]. Inputs for this calculation are the amplitudes and polarization of the (typically 10) most dominant modes. When appropriate, only the outer envelope of overlapping resonances is shown. Figure 6a shows the phase space of pickup ions at the first growth peak. Also shown are the resonance widths due to whistlers (solid) and the B-FMS + (dashed). At this time, there are a number of discrete modes on the whistler branch in the simulation that have finite am-

plitude (Figure 5). Thus we have only plotted the outer bound of the resonance overlap region due to whistlers (solid). The dashed curves correspond to a single B- FMS + that has been excited at this time (f•t = 25). It is clear that the saturation of the whistler mode is

achieved by a rather small scattering of the pickup ions. This demonstrates the ineffectiveness of the whistler

waves in the diffusion and scattering of pickup ions as found earlier by Winske e! ai. [1985]. Had it not been for the subsequent excitation of B-FMS + waves, the scattering of the pickup ions would have been limited to that shown in Figure 6a. Whistlers do, however, result in a finite thermal width of the pickup ions. This may explain partly why observed pickup ions always have a significant thermal spread. The rare occurence of the whistler wavepackets at comet Halley [Mazelie and Neubauer, 1993] may be partially due to the fact that whistler waves are generally generated in the presence of a large effective anisotropy. This in turn implies that other modes which are more effective in scattering of the ions would also be unstable. Since the other waves would eventually scatter the particles, the whistler waves should only be discernible during early stages of the linear instability.

Later in the simulation, the B-FMS + become dominant and the pickup ions are scattered more. Figure 6b shows the scattering at f•t = 60. The region of overlap due to whistlers is again shown as solid. There are now a number of modes excited on the FMS + branch

which results in a much larger region of resonance overlap than that due to whistlers. The resonance width due to each mode (FMS +) is shown as a dashed curve. At this time, the particles have not had time to explore the whole region of overlap. In order to show the relative concentrations of particles in phase space at a given time, we show in Figure 6c the contours of pickup ions. The contour levels are 0.2, 0.4, and 0.6, respectively. The intersection of the two dashed lines indicates the

initial pitch angle of the pickup ions. The peak of the distribution remains close to this initial pitch angle in time consistent with observations for this case [Neuge- bauer et al., 1989]. As the particles interact with the waves, they are scattered more but they cannot scatter into a full shell. The boundaries of the region in phase space where resonance overlap occurs is shown in Fig- ure 6d. The particles cannot penentrate these boundaries, thereby resulting in scattering into only half a shell. Any further scattering of the particles beyond these boundaries can only be due to nonresonant interaction with the waves. Since this is a second-order

process, it is much less efficient and generally not important unless the waves have large amplitudes (see next section). The contour of phase space shown in Figure 6e shows an strong resemblance to that shown in Plate 2 of Neugebauer et al. [1989].

The above 1-D simulations do not indicate whether

oblique modes are important in the scattering of ions. We have performed two-dimensional simulations in order to assess the relative importance of oblique modes to the saturation of the instability as well as the scattering of the pickup ions. The simulation parameters are the same as in the 1-D case except now the simulation is done in the solar wind rest frame.

The simulation box is 64c/w r in x and y directions and divided into 128 cells in each direction. There are

810,000 particles for each species. Surface plot of the Bz component of the field at f•t = 75, which is shortly after sautation, is shown in Figure 7a. The wave spectrum is, as in the observations, irregular due to the excitation of various instabilities propagating at various directions to the magnetic field. The Fourier transform of Bz (the component out of the plane of k and Bo) at 0 - 500 and for the time period f•t: 22- 95 is shown in Figure 7b. The contour levels are at 0.2, 0.3, 0.4, and 0.5 of the maximum, respectively. Also shown (solid) is the dispersion curve of the beam mode in the solar wind rest frame. Both B-AIC- and B-AIC + are now excited at

oblique angles, but consistent with their smaller growth rate (Figure 2d), we find they have smaller amplitudes (by a factor of ~ 2) than the parallel propagating waves. We performed a Fourier transform on the z component of the field because obliquely propagating Alfv4n waves are mainly polarized in the plane perpendicular to that containing k and Bo. In agreement with the smaller am-

KARIMABADI ET AL' WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY 21,547

(a) 8.

v,

O.

-4. Vl I (b)

I,j II I J I J I l l I II Jl I ii I I I l I I ,, j, ', .... ;;,,

':,\ ,'......,•' .,

: ?

(d)

..

O. ",:'.•':• , ,' ,

-4. Vl I 4.

.

(c)

(e)

Figure 6. The velocity distribution of the pickup protons at several times. The solid and dashed curves in-

dicate the theoretical outer boundary of the resonance overlap region due to whistlers and lower frequency beam modes (B-FMS +, B-^IC-) modes, respectively. The initial distribution of the ions is where the two

dashed lines intersect in Figures 6c and 6e. The contour levels are 0.2, 0.4, and 0.6, respectively, in Figures 6c and 6e. Same parameters as in Figure 3.

plitudes of the oblique B-AIC waves, we find that the wave saturation and proton scattering are very similar to the 1-D simulation; i.e., the scattering is dominated by the parallel propagating waves. There are other modes such as the mirror mode which are also

excited at oblique angles. Although we can identify these in the simulation results, they have small growth rates/amplitudes and do not play a role in the scattering of pickup ions.

4. Quasi-Perpendicular Case

Of the four quasi-perpendicular cases shown by Neugebauer et al. [1989], three (their Plates 3-5) show highly anisotropic pitch angle distributions with the highest density observed near a pitch angle of 180 ø rather than at the pitch angle corresponding to the newborn injection. The fourth case, however, which is for the region just outside the bow shock of comet Halley (their Plate 6), shows a pitch angle distribution which is nearly isotropic. There is also an increase of the shell radius near the shock. Thus the observations of the

pickup ions at comet Halley have posed the following

questions for the quasi-perpendicular regime: (1) Why is the highest density observed near a pitch angle of 180ø? (2) Why are the distributions isotropic near the shock, i.e., scattered over a half shell, and one-sided further upstream of the bow shock? (3) Why does the shell radius increase near the shock?

To our knowledge, no satisfactory answers to the above questions have been given to date. While the full explanation of all of the above features of the pitch angle distribution is beyond the scope of this paper, we will provide a model that reflects many of the features of the observed distributions. To this end, we concentrate on the parameters relevant for Plate 6 of Neugebauer et al. [1989] (see Figure lb). We show under what conditions pickup protons can be scattered into a full shell and describe briefly the consequences of our findings for Plates 3-5 of Neugebauer et al. [1989].

The observed parameters corresponding to Plate 6 of Neugebauer et al. [1989] are a = 53 ø q-6 ø, V•,o '" 282 km/s, VA -- 63 km/s, fie '" 0.53, fii = 1.1, No/N,•o ,., 1.9%, and wp/f•p ,, 4782.

Even though the proton density and distribution function can be measured fairly accurately, there is no direct way to determine the density of newborn pickup ions relative to those created earlier and then scattered by waves. Thus we take the beam density to be 0.5% of the solar wind density. The effect of a larger beam density on the scattering and velocity diffusion of pickup ions is considered at the end of this section. As before, we assume the pickup ions to have an initial temperature equal to one hundredth of the solar wind protons.

An important experimental clue to the behavior of the pickup protons for this case comes from the observations of the relative number density of pickup protons and water group ions as a function of distance from comet Halley [Neugebauer et al., 1990]. Just outside the bow shock, the protons and water group ions have equal densities [Neugebauer et al., 1990], while at large distances away from the bow shock the protons dominate the number density of pickup ions. As we will show, the presence of water group ions leads to excitation of waves which can pitch angle scatter the protons and result in further energy diffusion.

Let us first ignore the effect of water group ions. Fig- ure 8a shows the time history of the magnetic field energy for a 1-D simulation with 0 = 0 ø, Nb/N,,o = 0.5%, c• - 127 ø, and for parameters of Figure lb stated above. Here we have taken c• - 127 ø, whereas in Figure la it is at 53ø; i.e., in our simulations the beam is propagating antiparallel to the magnetic field, whereas in Figure lb the beam is propagating in the same direction as the magnetic field. The direction of the beam with respect to the magnetic field is of no consequence. The simulation box is 1000c/wv long and is divided into 2000 cells with 100 particles per cell. The time step is f•vAt - 0.05 and the simulation is done in a frame moving with 1.2VA in the direction of the background magnetic field. Figure 8b shows the profile of the z component of the magnetic field at several times. Initially, the wave profile is dominated by short-wavelength B-FMS + mode, but as the instability develops, B-AIC- mode

21,548 KARIMABADI ET AL.- WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY

Figure 7a. Surface plot of Bz at •t - 75 from a 2-D simulation for the same parameters as in Figure 3.

with growth at longer wavelength becomes dominant. This is due to the fact that B-FMS + growth rate is more sensitive than B-AIC- to the value of the effective

anisotropy A, dropping more rapidly to zero as A becomes smaller. We will come back to this point shortly. Note that the presence of the two modes is reflected in the double humped saturation of Figure 8a. We have identified the two types of modes in the simulations by performing a Fourier transform on the field data for a time interval extending from the start of the simulation to fit - $5. This is shown in Figure 9, where we have displayed the wave power as a function of frequency and k. The contour levels of the power correspond to 0.15, 0.25, and 0.4, respectively. Also shown are the disper-

0.4

0.0

-0.4 ........ i ......... i ......... i ......... i ......... i

0.0 ck/(.o 0.6 p

Figure 7b. The power spectrum of Bz. The contour levels are at 0.2, 0.3, 0.4, and 0.5, respectively. Also shown is the dispersion curve of the beam mode in the electron rest frame

sion curve (solid) and the growth rate (dashed) of the beam mode. As in the quasi-parallel geometry (Figure la), the beam mode coupled to both AIC- and FMS + giving rise to two peaks in the growth rate.

The relative importance of the B-AIC- and B-FMS + to the particle pitch angle scattering is demonstrated in Figure 10. The dashed curves in Figure 10a correspond to the outer boundary of the resonance overlap region due to B-FMS +. At flpt = $0, there are Mso two B-AIC- modes which have started to grow and the trapping width due to each mode is indicated by a solid curve. At this time, the protons are starting to pen- etrate through the resonance boundaries of B-FMS +. As B-AIC- modes grow in amplitude the region of resonance overlap increases and protons are scattered more in phase space. The outer boundary of the resonance overlap region due to B-AIC- modes in indicated in Figures 10b and 10c with solid curves. Long after the saturation of the instability, the pickup protons have scattered only through hMf of a shell, as there are not enough resonances to extend the overlap region to the positive vii. The contour plot in Figure 10d (contours of 0.2, 0.4, 0.6, and 0.8) shows the relative density of protons in phase space. The initial distribution of the protons is indicated with a circle. The energy diffusion is also comparable to that in the quasi-parallel case (Fig- ure 6) and smaller than seen in the observations (see Figure lb).

The question arises as to whether the asymrnetry in the pitch angle of protons is due to our use of a small beam density and/or somehow is related to the one di- mensionality of our simulation. In order to address this concern, we have made a two-dimensional run with a higher density of pickup protons Nb/N•o= 2%. In accordance with linear theory, we found growth of AIC and mirror modes at oblique angles. But as in the quasi-


(•B) •

x10 -3 (a) (a) .

2.53 • 12.

v, 0.

o. Q t 200.

B

-i-

B-FMS

B-AIC (b) 12.

O. X (c/(•) 128. Figure 8. (a) Time history of the magnetic field energy from a 1-D simulation for parameters corresponding to Plate 6 of Neugebauer et al. [1989]. Only one population of pickup ions (protons) is included. The plasma consists of the solar wind and pickup protons. (b) The stack plot of the Bz component.

parallel geometry (Figure la), the parallel propagating modes dominated the spectrum. More importantly, the proton scattering was, as in the 1-D run, highly asymmetric. Thus we conclude that given the observed parameters for Figure lb and neglecting the presence of water group ions, the excited waves saturate at relatively small amplitudes, the scattering is only into half a shell, and there is no increase in the thickness of the shell compared to the quasi-parallel case.

Next, we include the effect of water group ions in our simulations. The simulation parameters are the same as those in Figure 8 except now we have, in addition to the pickup protons, a population of singly ionized oxygen ions. The density and beam speed of the oxygen ions are taken to be the same as the protons. Figure

0

Ow

B-FM

...... 2: ......... -" B-AIC

O. k(e)p/C) 2. Figure 9. The power spectrum of the By component of the field for the parameters of Figure 8. Also shown are the dispersion curve (solid) as well as the growth rate (dashed) of the beam mode.

O.

(c) .

.:' ...".'i: = 200 -6. Vl I 6.

(b)

ß

= 60.

(d)

,, Dt = 200

Figure 10. Time evolution of velocity distribution of pickup protons. (a)-(c) The dashed and solid curves indicate the outer boundary of the resonance overlap region due to B-FMS + and B-AIC-, respectively. (d) Contour of phase space density at fit = 200. The initial distribution function of pickup protons is shown by a circle.

11 shows the time history of various quantities for this run. The large increase in the magnetic field energy is due to excitation of B-FMS + mode of oxygen. The proton waves grow and saturate (•pt ~ 80) before the oxygen waves but since the amplitude of proton waves is much smaller (•B2/Bo 2 ~ 0.005) than that of the oxygen waves, the growth phase of proton waves is not discernible given the scales of Figure 11a. Figure 11b shows the effective anisotropy (A) of protons as a function of time. The large initial drop in A is due to excitation and subsequent saturation of proton waves. As we showed above, in the absence of oxygen, protons are only scattered into half a shell and A would never dip below ~ 8. The second and more gradual drop in A of protons, which approaches unity in time, is due to oxygen waves. Note that A --• 1, implies full isotropization of that species. As the oxygen waves grow, the A of oxygen as well as their drift speed with respect to solar wind plasma decreases.

The wave spectrum in this case is quite complex. Fig- ure 12 shows the profile of the total magnetic field at several times for the time interval •pt: 300-600. Dur- ing this interval, the wave spectrum is dominated by oxygen waves. The evolution of the waves clearly shows the steepening and subsequent formation of whistlers. We have verified that the whistlers are generated locally due to the gradients of the steepened fronts. This dearly demonstrates the possibility of steepening for parallel propagating waves, even though they are linearly noncompressional. We will say more about this steepening process in section 5.

We have performed a Fourier transform of the field data in order to identify the various modes that are excited in the simulation. Figure 13 shows the results of the Fourier transform (power) for •pt :0- 125, which is before the waves steepen. The contour levels for the


(a)

A proton

4 (C)

A xygen 10

(d)

1.5 0. 600.

Qt

Wave Steepening at Parallel Propagation

S i I total

c:) "'•whistlers

o. lOOO.

x (c/%) Figure 12. The x profile of the total magnetic field at several times for the parameters of Figure 11. The oxygen waves steepen giving rise to local generation of whistler waves.

Figure 11. Time history of the (a) magnetic field energy, (b) effective anisotropy of pickup protons, (c) effective anisotropy of pickup oxygens, and (d) the relative drift between the pickup oxygens and the solar wind. The simulation parameters are the same as in Figure 8 except that now there exists a second population of pickup ions (oxygen).

power are 0.1, 0.3, and 0.5, respectively. The two solid curves correspond to dispersion of beam modes for oxygen and proton with A - 50. Also shown (dashed) is the growth rate of the oxygen beam mode. Of the proton waves, only B-AIC- has survived, with B-FMS + being in the noise level. The reason for this becomes clear if one plots the maximum growth of each mode as a function of A of the respective species (Figure 14). The maximum growth of B-FMS + mode drops off more rapidly than B-AIC-. As a result, as the instability develops and protons are pitch angle scattered, the value of A goes down (see Figure l lb) and AIC- becomes dominant. In fact, for small values of A, B-FMS + is no longer growing at its original wave number. Small growth occurs instead at higher k on the fast mode branch. The range of anisotropy for which this hap- pens is shown as a dashed curve in Figure 14. Waves at this wavenumber cannot compete against existing waves because they have to grow from the noise level very late in the evolution. Also note that the proton AIC- waves and the oxygen FMS + waves do not directly compete, since they rely on different driving sources. This allows the oxygen waves to grow to significant amplitudes de- spite their relative smaller growth rate [see Gary et al., 1988].

Finally, we examine the scattering of protons and oxygens in the above fields. Figures 15a and 15b show the phase space distribution of protons at flrt = 60 and 600. The proton waves have long saturated and decayed away at flrt - 600. The solid curves in Figure 15a show the outer boundary of the resonance overlap region due to proton waves. We emphasize that in the absence of oxygen waves, the protons would not be scattered be-

yond what is shown in Figure 15a. In other words, had we run the simulations in Figure 10 to fit = 600, there would have been no further scattering or energy diffusion beyond what is shown in Figures 10a-10d. For the present parameters the oxygen waves cannot resonate with protons. However, due to their large-amplitude 5B/Bo ~ 0.4, they can scatter the protons nonreso- nantly. Evidence for this nonresonant scattering comes from Figure 15b which shows the proton phase space distribution at the end of the run. Comparing Fig- ures 15a and 15b shows that protons have been heated (energy diffusion) and have scattered more by oxygen waves. Some fraction of protons have even managed to scatter to the other side. Since the process is nonresonant, the total number of particles affected is smaller and the time required for scattering is longer than in the resonant case. Our results are different from that of

Gary et al. [1988], who found that protons can be scattered into a full shell due to water group ions. Their simulation parameters are different from ours and in their case waves attain much larger amplitudes. We conclude that the presence of oxygen waves can have an

0.4

0.0

-0.6

......... , ......... ! ......... • .........

..J..7 x10 + • ,, O: beam mode;

0. k 0.4 Figure 13. The power spectrum of the By component of the field for the parameters of Figure 11. Also shown are the dispersion curves (solid) of the beam modes for proton and oxygen as well as the growth rate (dashed) of the oxygen beam mode. The growth rate shown has been multiplied by a factor of 10.


0.1

O. o. 25. 5o.

anisotropy

Figure 14. The dependence of the growth rate of various modes on the effective anisotropy.

important effect on the energy diffusion and pitch angle scattering of protons. Thus the observed increase in the shell thickness of protons as one approaches the bow shock of comet Halley may partially be due to the fact that the relative density of water group ions becomes comparable to that of protons close to the bow shock, i.e., may not be due entirely to pickup at successively smaller radius shells. However, the nearly isotropic observed distribution (Figure lb) in this case cannot be explained by our simulations where as shown in Figure 15c, most of the ions are still confined to half a shell.

We have also examined the effect of a larger beam density on the proton scattering. Figure 15d shows the proton phase space density in form of a contour plot (contour levels are 0.15, 0.4, and 0.7) for a simulation similar to that in Figures 15a-15c except now the density of both proton and oxygen beams is taken to be 2% and the injection angle is taken to be 53 ø . As we men- tioned earlier, there is no difference in particle scattering and the waves generated between an injection angle of 53 ø and 127 ø . While the higher beam density has resulted in more pitch angle scattering, the distribution is still much less isotropic and with much larger energy than observed (Figure lb). Thus the measured beam density must consist of a significant number of ions that are not freshly injected. The fact that the observed distribution (Figure lb) shows somewhat higher densities on the hemisphere opposite to where the injection takes place is also consistent with this expectation. Had all ions been injected at the same pitch angle, local theory alone would not be able to explain why the phase space density would be higher on the hemisphere opposite to where the injection takes place. One way to have an accumulation of ions with different injection angles is to have a rotational discontinuity. Examination of the magnetic field data (F. Neubauer, private communication, 1993) indeed revealed the presence of a rotational discontinuity between 1914 and 1915 SCET. Based on the above, the following explanation of the distribution in Figure lb has emerged. The observed distribution consists of two populations. The first population consists of old ions which have been injected near 90 ø pitch angle, are mainly confined to the left hemisphere, and have been scattered in the waves. The second population consists of newborn ions that are injected at ..- 53 ø rather than at ..- 90 ø due to the change in the field di-

rection resulting from the rotational discontinuity. The latter ions are mainly confined to the right hemisphere even though some of them can be scattered to the other side. The presence of water group ions and the associated waves can account for the increase in the thickness

of the shell as compared to distributions further upstream from the cometary bow shock.

We have also made a run for parameters of Plate 4 of Neugebauer et al. [1989] and have found that proton waves lead to the one-sided distribution seen in Plate 4

of Neugebauer et al. [1989], similar to Figure 10. Thus we think that the most likely explanation for the one- sided distributions seen in Plates 3-5 of Neugebauer et al. [1989] is the fact that they are for intervals farther away from the bow shock and thus the water group ions are not as important. Furthermore, due to the small saturation amplitude of proton waves for those intervals the proton waves cannot scatter the protons into a complete shell. On the other hand, Neugebauer et al. [1989] argued that the one-sided distributions in the

14.

14.

14.

14.

proton

_

,

(b) r , ,

D. t -- 600

nonresonant

diffusion • ..,:...'• ..... ß 'f•;L'•. - ' •

(o) .,...,...,...,..., .........

• t = 600

(d)

D- t = 600

-•. ' Vl I 7.

oxygen (e)

D. t'= 200'

(f)

D. t = 600

.:," : '.', '-. --;,...%..., :..

..,

(g)

D. t = 600

Figure 15. (a)-(g) Time evolution of the velocity distribution of pickup protons and oxygens for parameters of Figure 11. (d) Effect of a larger beam density (changed from 0.5% to 2%) on the phase space density of protons. Also the injection angle is 53 ø in Figure 5d. There is no change in the waves or the resulting scattering whether a = 53 ø or 127 ø.


quasi-perpendicular cases are due to global rather than local effects; i.e., the pickup protons may have been in a magnetic field configuration which allowed escape in only one direction. While global effects may play a role, we have shown here that for the observed parameters proton waves scatter the protons only into half a shell consistent with the observations.

The one remaining question is why in Plates 3-5 of Neugebauer et al. [1989] the highest density is observed near a pitch angle of 180 o rather than that predicted by the newborn injection angle. Neugebauer et al. [1989] considered various possibilities, such as the reflection from the bow shock but found no plausible explanation. While we will not address this question here, we note that the location of the maximum phase density is a time-varying quantity that does not have to be at the injection angle. Its exact location in velocity space is determined by the diffusion coefficient and the ion injection rate. If the diffusion rate is too fast compared to the injection rate, the maximum phase space density will not be close to the injection point. For instance, in Figure 6 of Karimabadi et al. [1994] the protons were injected at 35 ø, while the peak in the proton phase space density was at ~ 5 ø. Thus local theory, in prin- ciple, has the potential to address this question. More work is clearly needed to clarify the actual mechanism involved.

Finally, we show in Figures 15e-15g the velocity distribution of oxygen ions at two different times. The solid curves in Figures 15e-15g indicate the outer boundary of the resonance overlap region. In both cases the scattering of the oxygen ions is described accurately by this theoretical boundary. Figure 15g shows the contours of phase space density at the end of the run 9t = 600. We have compared this distribution to the observed velocity distribution of pickup oxygen for the same interval as in Plate 6 of Neugebauer et al. [1989]. The observed distribution (not shown) shows features (A. Coates, private communication, 1993) very similar to that shown in Figures 15f-15g. The distribution shows asymmetries in both pitch angle and energy diffusion. There exists a pronounced level of energy diffusion on the left side of the shell and in the vicinity of the injection point (a- 127ø). A fraction of particles is also scattered into the other side of the shell. Since

this amount of scattering is most likely ascribed to the presence of a high level of oxygen waves at the time of observation, it is consistent with our simulation results and our interpretation of the source for the proton scattering.

5. Growth of Oblique Magnetosonic Modes' Consequences for Steepening

One of the outstanding issues in cometary physics as well as concerning the region upstream of Earth's bow shock is the observation of large-amplitude steepened magnetosonic waves with associated whistler wave packets (shocklets). The puzzle has been that while the observed shocklets propagate at oblique directions

to the magnetic field, linear theory predicts maximum growth at parallel direction. The fact that shocklets are observed at some comets, e.g., GZ and Grigg-Skjellerup [Glassmeier and Neubauer, 1993] but not at others, e.g., comet Halley, has also remained a mystery and points to a general lack of knowledge concerning the conditions for which oblique modes dominate over parallel propagating waves. In order to explain the above observations, two questions need to be addressed. One is the conditions under which oblique magnetosonic modes can have a growth rate comparable to or larger than the parallel propagating modes. The other is whether parallel propagating waves can steepen. Since shocklets are generally observed to be obliquely propagating, can one conclude that parallel propagating waves do not steepen into shocklets?

The first question was recently addressed by Omidi et al. [1994]. They reexamined the linear stability properties of a ring beam distribution and found that there are, in general, two resonant instabilities with growth on the magnetosonic branch. One is driven by the beam component of the ring beam and has maximum growth at parallel propagation. This mode is due to the œ = -1 resonance with the beam. The second instability is driven mainly by the ring component of the ring beam (i.e., it is anisotropy driven) and has maximum growth at oblique angles. This mode is due to the/•- -2 resonance [see Brinca and Tsurutani, 1989; Goldstein et al., 1990]. Depending on the parameters, higher harmonics can also have finite growth.

In order to gain insight into the conditions under which oblique modes can have large growth rates, in Figure 16a we have plotted the ratio of the maximum growth rates of/• = -2 to œ = -1 instabilities as a function of injection angle for three different solar wind flow velocities. The plasma parameters correspond to Fig- ure lb, and the ring beam parallel thermal velocity vllta is kept fixed at 32 km/s which corresponds to A = 50 for a - 530 and ¬,o- 4.5Va (injection parameters for Figure lb). The growth calculations assume an oxygen ring beam with No/N•,o= 0.5%. Although the magnetosonic mode is excited at small injection angles, the instability becomes Alfvdnic at large injection angles. The solid curves in Figure 16a show the region where the œ = -2 excites the magnetosonic mode. On the dashed curves, first the B-AIC + and then the B-AIC- is dominant at very large injection angles. The reason that the instability takes place successively on the B-FMS and B-AIC modes and (B-AIC +) propagation directions is that the inclination angle of the beam cyclotron resonance condition decreases with the injection angle.

The growth rate for the t - -1 instability varies less than a factor of 2 over the parameter regime of Figure 16. Due to the large free energy in the ring beam, it is very difficult to identify whether the mode is Alfv•nic or fast magnetosonic. We made the divi- sion between B-FMS and B-AIC based on what mode

is approached in the limit where the beam density goes to zero at the short-wavelength limit of the instability. The growth rates are calculated for a fixed propagating

KARIMABADI ET AL.' WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY 21,553

(a) 1.0

OA

1.0

0 ø injection angle a 90 ø

(b) ,

'"b =2% N•/• Nsw

........ .............. s.o 4.0 s.o 0.0 8.0

Vsw/V^

Figure 16. (a) Ratio of the maximum growth rates of œ- -2 to œ- -1 instability as a function of injection angle, for three different vlaues of solar wind flow speed. The instability is due to that of an oxygen beam with a density of 0.5% of the solar wind density. Solid curve correspond to the magnetosonic mode and the dashed curves correspond to growth of Alfvdn ion cyclotron mode. (b) Ratio of the maximum growth rates of œ - -2 to œ - -1 instability as a function of solar wind speed to two different values of beam density. Only the fast magnetosonic mode is considered and the injection angle is fixed at a = 35 ø. The growth rate calculations are performed for a fixed propagation angle of 250 with respect to the magnetic field in all cases in this figure.

angle of 25 ø. This is a reasonable simplification for the fast magnetosonic mode since the œ - -1 instability typically has a fairly fiat growth rate as a function of propagation angle. Also, 250 is a typical angle at which the œ - -2 instability maximizes if the injection angle is not too large. Figure 16, however, does not provide a complete picture of the AIC instabilities since the maximum growth of the B-AIC can be at large propagation angle, and there the growth rate of the higher harmonics can be larger than that of the fundamental. Here we are mainly interested in the oblique excitation of magnetosonic waves since they are the ones related to shocklets.

We can draw several conclusions from Figure 16a. First, for a fixed solar wind speed, the relative growth rate of œ: -2 instability increases as a function of injection angle. This is expected since the œ - -2 is driven by the ring component of the ring beam [Omidi et al., 1994]. Second, the larger the solar wind speed the larger the ratio of the growth of the œ- -2 to that of œ- -1 instability becomes. Finally, even though the œ: -2 growth rate becomes comparable to the œ- -1 instability for certain values of solar wind speed and in-

jection angle, it never becomes larger. A large growth rate of the œ = -2 magnetosonic mode is a necessary but not sufficient condition for formation of shocklets. Once

the œ = -2 mode has a growth rate comparable to the parallel propagating mode, other effects can cause the oblique mode to become dominant. For example, these waves are typically convected back by the solar wind (group velocity < V•w). Thus due to the dispersion and propagation properties of the œ - -2 mode, it may grow to larger amplitudes in a finite source region; i.e., it has a larger convective growth rate. Other effects such as nongyrotropic distributions [e.g., Neubauer et al., 1993], may also contribute to the dominance of oblique waves. In the usual linear theory, the beam has a constant density in time. Whether injection, where the beam density changes in time, can modify the result of linear theory will not be addressed here.

Next, we examine the effect of beam density on the growth of the oblique magnetosonic mode. Figure 16b shows the relative growth of the œ = -2 and œ: -1 mode as a function of solar wind speed for two different beam densities. As before, we have fixed the propagation angle at 250 in calculating the growth rates of both œ- -1 and œ - -2. Since the •? - -2 instability becomes Alfvdnic at large injection angles (Figure 16a), we have chosen a to be moderately small (a = 35 ø) in Figure 16b so as to stay on the magnetosonic branch. The important finding here is that for a beam density of 2%, the oblique mode can have a growth rate comparable to the parallel propagating mode for a wide range of the solar wind speed. The absolute growth rates of both œ = -1 and œ = -2 are also larger at larger beam densities.

From the above results, we can piece together the following explanation for the observed differences in the wave spectra at comet Halley and GZ. Since the solar wind speed was roughly the same in both Halley and GZ observations, we attribute the differences in the wave spectra to differences in the pickup density at Halley and GZ. The gas production rate at comet Halley is 25 times larger than GZ (~ 1030 molecules/s as opposed to ~ 4 x 10 as molecules/s). The smaller neutral production of GZ leads to a shock formation at an upstream distance ten times closer to the nucleus [e.g., Tsurutani, 1991] than at Halley (10 • km instead of 10 • km). An estimate for the ion injection rate and beam density can be obtained by using the standard model of cometary ion production [Schmidt and Wegmann, 1982]:

dnp Qp dt = 4•rArp2 exp(--p/•p),

where Qr and Ar are the production rate and ionization scale length (~ 10 • cm) of protons, respectively, and p is the distance from the cometary nucleus. Taking p to correspond to the location of the cometary bow shock and using the Qp values given above, we find that the injection rate at the bow shock of Halley is a factor of 4 smaller than that at GZ. We have ignored the exponential factor. Note that the steepened waves were seen at GZ at ~ 2 - 3 x 105 km from the comet,


which puts them well inside the Halley shock location of 106 km. While the exact relation between injection rate and beam density is not completely known, it is fair to say that the higher injection rate would lead to a larger effective beam density. As we showed in Figure 16, the higher beam density leads not only to a larger growth of the modes, but it also enhances the growth rate of the oblique magnetosonic mode relative to the parallel propagating mode. The latter change is a necessary condition for the formation of shocklets. Although the above calculations are suggestive and can explain the observed differences between the wave spectra at the two comets, more detailed work is necessary before a complete understanding can be achieved.

We now briefly address the second question that we posed above; i.e., can parallel propagating waves steepen into shocklets?

From linear theory, it is known that a monochromatic parallel propagating wave is linearly noncompressional and thus not expected to steepen. This expectation has also been borne out by studies based on the derivative nonlinear Schrodinger equation (DNLS), where it has been shown that a single parallel propagating wave of constant amplitude is an exact solution of MHD equations and cannot steepen [e.g., Spangler and Plapp, 1992, and references therein]. In reality, however, any instability would generate more than one wave and the superposition of several modes can lead to compression of the plasma. In fact, spatially modulated waves are not exact solutions to the MHD equations and can ex- hibit steepening [e.g., Spangler and Plapp, 1992, and references therein]. Kinetic studies have also revealed steepening of parallel propagating waves [e.g., Tera- sawa, 1988; Akimoto e! al., 1991]. The latter references dealt with parameters more relevant to those upstream of Earth's bow shock. We also showed that for parameters of Figure lb, the oxygen waves steepen, forming whistler wave packets (see Figure 12). Terasawa [1988] attributed the formation of density spikes to a feedback mechanism whereby the spatially modulated wave profile, resulting from the sum of various modes, gives rise to a mirror force. Beam ions are decelerated locally by the mirror force in regions where the magnetic field magnitude increases. The deceleration of the beam results in a local increase of the beam density. The larger local beam density then results in a further increase in the excited waves, thus forming a positive feedback loop. He found that the ions forming the spikes are not pitch angle scattered and their magnetic moment is conserved. Similar results were also found by Akimoto et al. [1991]. The mechanism leading to the steepening in Figure 12 is, however, different from that described above. Figure 17 shows the spatial variation of several quantities at the onset of the instability (see Figure 11) for the same parameters as in Figure 11. The density of oxygen beam is normalized to its initial value. The important point to note is that there are thermaliza- tions of oxygen in both Vii and Vñ components in the regions where the magnetic field has the largest peaks. There is a small change in the velocities in regions where Btot has its minima. In contrast to the case studied by Terasawa [1988], the magnetic moment is clearly not

(a)

-6.

(b)

V. L ', O.

(o)

' ,' t (d)

<v,,> i ', -2.8

(e)

N •'• o. 1 ooo.

X (c / ep) Figure 17. Spatial variation of the (a) parallel and (b) perpendicular velocities of the oxygen ring beam, (c) total magnetic field, (d) average flow speed of oxygen parallel to the magnetic field, and (e) oxygen density normalized to its initial value, at f•t - 150. The parameters are the same as in Figure 11. The waves are parallel propagating. The plasma signatures are remi- niscent to those at shocks.

conserved in this case and the particles have been scattered in regions where Bto• has its peaks. There are also deceleration of the oxygen beam and an increase in density in going from the small Btot to large B•ot. In many ways, the plasma signatures here are similar to those at fast shocks. Taking the regions where the magnetic field has its minima (maxima) as upstream (downstream), there exist heating of the beam (Figures 17a and 17b), increase in the magnetic field (Figure 17c), deceleration of the beam (Figure 17d), and finally an enhancement in beam density (Figure 17e) in going from upstream to downstream. In this case, the main density compression is in the oxygen beam, with the solar wind not being strongly affected. This is, however, a function of beam density. For a beam density of 2%, there exist large density compressions of the solar wind as well.

The connection between the above results and parallel shocks as well as differences between steepening at parallel and oblique propagation angles need to be explored in more detail. Here, it suffices to say that there are some obvious differences between the steepening of parallel and oblique waves. In Figure 12, not all cycles in the waves have steepened whereas at oblique angles each wave cycle forms a steepened front [e.g., Omidi and Winske, 1990]. Another feature of the steepening at parallel propagation is its transient nature. In our parameter regime, the steepened fronts do not maintain themselves as long as the steepened fronts for oblique modes. What determines the timescale for evolution of

steepened fronts is not understood at either parallel or oblique angles.

KARIMABADI ET AL' WAVES AND VELOCITY DISTRIBUTIONS AT HALLEY 21,555

6. Discussion and Conclusion

In this paper, we have addressed the question of wave excitation as well as the resulting pitch angle distribution of pickup ions at comets in general and for comet Halley in particular.

While the underlying physics discussed is quite general, we picked the parameters of our study to correspond to specific events described by Neugebauer et al. [1989], which remains one of most detailed studies of the observed proton pickup distribution at comets.

We showed that the pickup distribution is unstable to a variety of modes including whistlers, fast magnetosonic mode, Alfv4n, and mirror mode. For the parameters relevant to comet Halley, however, parallel propagating modes have the largest amplitudes and control the pitch angle scattering of ions. We showed that the observed large pitch angle anisotropies of the distribution that exist at sufficient distances upstream of the cometary bow shock are due to the relatively small saturation amplitude (6B/Bo < 0.2) of the waves. The dominant waves are excited due to the ion/ion resonant instability. Since for parallel propagating waves all resonances are on one side of the shell and for small wave

amplitudes the trapping widths do not extend that far onto the other side, the distribution of pickup protons is generally onesided in the vii- vñ plane. Neugebauer et al. [1989] speculated that the asymmetric pitch angle distributions for quasi-perpendicular regimes may be caused by global rather than local effects. While global effects may play a role, we have shown that for parameters of comet Halley, the one-sided distributions should be the norm rather than the exception.

Just outside the bow shock of comet Halley, however, the proton pitch angle distribution was nearly isotropic and the radius of the pickup shell increased significantly [Neugebauer et al., 1989]. We attribute at least part of this change to the increase in the relative density of water group ions to that of protons. This ratio becomes of order unity just outside the bow shock [Neugebauer et al., 1990]. Including both protons and oxygens in our calculations, we showed that in addition to protons, oxygen pickup ions are also unstable and generate waves. These waves scatter not only the oxygen ions but also the protons, resulting in considerable energy diffusion and scattering of pickup protons. However, the near isotropization of the distribution in pitch angle in this case cannot be explained by local theory. The observed distribution may consist of two populations and be related to the presence of a rotational discontinuity, which has been detected in the magnetic field data (F. Neubauer, private communication, 1993), during this interval. One population consists of newly born ions that are injected at the measured injection angle based on the local magnetic field. These ions are scattered mainly into half a shell, although a small fraction of them can scatter through 900 pitch angle. The second population consists of ions that were injected at a different pitch angle based on the magnetic field before the encounter with the RD. The latter ions have since

been scattered, filling up the other half of the shell.

We also considered the wave excitation and spectra in some detail. We showed that in the region upstream of comet Halley, whistler mode can be unstable, consistent with the recent observation by Mazelie and Neubauer [1993]. Whistlers result in a small amount of pitch angle scattering and hence are not the cause of large pitch angle scatterings seen at comet Halley. Just outside the bow shock, we found that water group ions excite parallel propagating waves. In spite of their parallel propagation, these waves steepen and generate whistler wave packets at their steepened front. The details of the steepening do, however, seem to be different from those for oblique waves and will be explored in a future publication.

Concerning the wave spectra, one of the outstanding problems in cometary physics has been the question of why the observed wave spetrum at comet Halley is so different from that at GZ. At comet Halley, the waves generally have small amplitudes (6[BI/Bo < 0.1), whereas much larger fluctuations are observed at GZ with ,•IBI/Bo ~ 0.5. Another major difference is the observation of large-amplitude steepened waves with their associated whistler wave packets (shocklets) at GZ. No shocklets have yet been identified in Halley observations. The question can be rephrased as to why oblique magnetosonic waves can grow to large amplitudes at GZ, while parallel propagating waves are dominant at comet Halley. We have examined the conditions under which obliquely propagating waves can have growth rates comparable to parallel waves. We showed that the differences in wave spectra at comet Halley and GZ may be attributed to the differences in the local beam den-

sities. Although the gas production rate is higher at Halley, its bow shock is located 10 times farther away from its nucleus than at GZ. Using a simplified model for the cometary ion production, one can show that the injection rate at Halley's bow shock is roughly a factor of four smaller than at GZ. The smaller beam densities

at comet Halley lead to dominance of parallel propagating waves whereas at the higher densities expected at GZ, the oblique magnetosonic mode can have a growth comparable to the parallel propagating mode. Finally, due to the smaller beam densities at comet Halley, the excited waves will saturate at smaller amplitudes, consistent with the observations.

Acknowledgments. The authors thank Paul Muret for developing the software package that was used for analyzing the wave spectrum. This research was supported by NASA grants NAGW-1806, NAG 5-1492 and by the IGPP at Los Alamos National Laboratory. Work at Lockheed was supported by Lockheed Independent Research. The work at the Jet Propulsion Laboratory of the California Institute of Technology was done under contract to NASA. Computing was performed on the Cray Y-MP and C-90 at the San Diego Supercomputer Center.

The Editor thanks P. H. Yoon and S. P. Gary for their assistance in evaluating this paper.

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H. Karimabadi, D. Krauss-Varban and N. Omidi, Depart- ment of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0407; (e-mail: [email protected], [email protected], nomidi @ece. ucsd.edu).

S. A. Fuselief, Lockheed Palo Alto Research Laboratory, Palo Alto, CA 94304; (e-mail: [email protected]).

M. Neugebauer, Jet Propulsion Laboratory, California In- stitute of Technology, Pasadena, CA 91109; (e-mail: [email protected]).

(Received May 19, 1994; revised June 27, 1994; accepted July 1, 1994.)

Low-freqency instabilities and the resulting velocity distributions of pickup ions at comet Halley

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