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Nonlinear dynamics of Alfvén waves: Interactions between ions and shock upstream waves
Computer Physics Communications 49 (1988) 193—200 North-Holland, Amsterdam
193
NONLINEAR DYNAMICS OF ALFVEN WAVES: INTERACTIONS BETWEEN IONS AND SHOCK UPSTREAM WAVES Toshio TERASAWA Geophysical Institute, Kyoto University, Kyoto, Japan and Institut für extraterrestrische Physik, Max-Planck-Institut für Physik und Astrophysik, Garching, Fed. Rep. Germany
In upstream regions of collisionless heliospheric shocks, large amplitude Alfvén waves are often observed. These waves are excited by the beam ions injected at the shock front. We discuss several specific topics about the nonlinear dynamics of these waves; frequency shift of the waves owing to the beam deceleration, a parametric decay process of the waves and formation of ion clumps (spatially bunched ions).
1. Introduction It is now known that a wealth of nonlinear phenomena can be found in the region around the planetary bow shocks and interplanetary shocks. The magnetic and electric fields show a turbulent behavior there. The frequency range of turbulence covers from well below the ion cyclotron frequency to about the electron plasma frequency. In this paper, I shall treat a specific topic, interaction processes of energetic ions with the upstream turbulent waves, especially with low frequency hydromagnetic (Alfvén) waves, One of the fundamental questions about physics of collisionless shocks is how to excite the upstream turbulence? It is now known that ion beam injection occurs at the shock front. This beam injection is either due to electrostatic/ magnetic reflection of the solar wind ions [1—3]or due to leakage of magnetosheath ions [4,5]. Once an ion beam is injected to the upstream solar wind, low frequency hydromagnetic waves are excited through the beam cyclotron instability [6—8].The excited waves are right-hand polarized in the rest frame of the solar wind plasma and propagating away from the shock. After exciting waves, the beam ions are pitch-angle scattered and finally become diffuse ions ref. [9] and references therein), Since the energy density of the ion beam usually exceeds the upstream magnetic energy density [10],
the amplitude of the excited waves can be quite large. This largeness of the wave amplitude opens up variety of nonlinear phenomena.
2. Ion beam cyclotron instability 2.1. Slow and fast beams The phenomena which take place in the upstream region depend critically on parameters of the beam ions. In the following, the discussion is limited to the case of excitation of wave propagating parallel to the background field. We shall discuss two extreme cases, namely cases of a slow beam and of a fast beam. First, we treat a slow beam which has a relatively slow beam velocity Vb and a relatively large beam density n b; for example, Vb/VA 3 and nb/no 0.1—0.2, where VA and n0 are the Alfvén velocity and the background plasma density, respectively. In the upstream region of several interplanetary shocks, Viñas et al. [11] found ion beams which have characteristic parameters similar to the above. Another extreme case is a fast beam which is typically found in the earth’s foreshock region: The density of the beam is relatively low and the beam velocity is much faster than the Alfvén velocity; nb/no 0.01 and Vb/ VA ~ 10. The frequency of excited waves depends on the
0010-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
194
T Terasawa
/
Nonlinear dynamics of Alfvén waves
deceleration causes a frequency shift toward lower
W
fas t b aa ~ S
frequency.
lo w b e a a
2.2. Simulation result: slow beam case For the simulation of low frequency electromagnetic phenomena, such as the upstream wave
=
k
VA
k
~/
motion, while electron response is expressed in a form of the generalized Ohm’s law. The magnetic field is advanced by solving the induction equation. Fig. 2 shows one-dimensional simulation results for a slow beam case. The + x direction is taken along the background magnetic field B0. Initially,
wkVb=Qj —
excitation process, hybrid simulation codes are used widely [8,14—18]. In these codes ions are treated as particles, whilebyelectrons fluid: ions are advanced solving are the treated equationasofa
/) I
3VA)
Fig. 1. A (w, k) diagram for the ion beam cyclotron instability, The abscissa and the ordinate show the wavenumber and the frequency (real part), respectively. Excitation of the hydromagnetic wave is expected where the cyclotron resonance line (~— kV~,= — £2.) intersects the dispersion line of Alfvén waves (w = kVA).
beam velocity. Roughly speaking, the frequency of the excited waves is determined by an intersection
beam beam Through velocity Vbthe(= beam towardions thehave + xa bulk direction. cyclotron instability, right-hand polarized waves propagating toward the same direction as the ion beam are excited. The nght panels show the evolution of the y component of the wave magnetic field. (The time scale used throughout this paper is normalized by the inverse of the ion Larmor frequency, Q~1.)In the left panels, thick and thin marks show the evolution of phase angles of the wave magnetic field and beam ions, c’~ 5and respectively, which are defined as ~,
of two lines, the linear dispersion line, w = kVA, and the cyclotron resonance condition, Co kV~,= Q~,where Q1 is the ion Larmor frequency (fig. 1). From the geometry it can be seen that the frequency becomes higher as the beam velocity becomes lower. However, as Gary et al. [12,131 pointed out,Inthis idea breaksthedown below 2.5VA. thissimple velocity regime, interaction Vb between a beam and waves becomes highly dispersive and the frequency and excited waves decreases as the beam velocity reduces. From this difference in frequency characteristic, there appears a significant difference in the nonlinear evolution of the systems between the fast and slow beam cases: When the beam energy is transferred to the waves, the beam ions are decelerated. For the fast beam, this beam deceleration causes a frequency shift toward the higher frequency. On the other hand, for the slow beam, the beam —
—
~
=
tan~
—~
and
~, = tan’
B~ At the initial stage of the wave excitation, we observe the that phase of beamfield. ionsThis are bunched that around of angles the magnetic bunching phenomenon, also seen in the simulation of a fast beam (next section), explains what is called “gyrophase bunched ions” observed in the earth’s foreshock region [19,20]. At time 60~?~1, we see a reduction of the wavenumber (equivalently, reduction of the wave frequency). This downward frequency shift is consistent with what the theory predicts for slow ion beams (section 2.1). As time further elapses, a reversal of the wave helicity appears. This can be seen as a reversal of the slopes of wave phase angle from di-
T Terasawa / Nonlinear dynamics of Alfvén waves
195
B lB
PHASE ANGLE
0 2
0 2ir t~’
,~ e
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Fig. 2. Simulation results for a slow beam case (Vb / VA = 3, nb/no = 0.2) with a periodic boundary. The abscissa is the x axis, where the scale is normalized by the ion inertia length (A VA / Q~).The length of the simulation system, L~is 51.2 A1. Left panels: the phase space plots (X— ~,) of ion distribution (thin marks). The phase angle ~ of the magnetic field is shown by thick marks. Right panels: the y component of the magnetic field (normalized by the background magnetic field B0).
196
T. Terasawa
/
Nonlinear dynamics of Alfvén waves
recting upper right to directing lower right. This is a manifestation of another nonlinear phenomenon, a parametric decay of the waves ([refs. [18,21,22] and references therein), Fig. 3 shows the evolution of wave forms more closely. The time runs from the bottom to the top, The original waveform of y component magnetic field, B~,is shown in the left panel. To see easier the wave propagation, the data are repeated twice in the x direction. Until the time 6O~2~,we observe waves propagating toward + x direction. They are waves excited by the beam cyclotron instability. After a time of 90 ~ 1, waves propagating in the opposite direction appear. A Fourier decomposition technique makes the situation clearer (ref. [18], appendix B). Original data (B~ and B~)at a given time are decomposed into two helices of different helicities. The middle and right panels of fig. 3 show the y components of these two helices. The middle panel shows a helix whose polarization is either right-hand or left-hand depending on the propagation direction (+ x or x, we shall call them as the R + or L waves for abbreviation). In this panel it is seen that the R~ waves, which are excited by the beam cyclotron —
—
~
instability, are the dominant wave component. In the right panel of fig. 3, we see a delayed evolution of the R waves (the right-hand polarized waves propagating in the x direction). This is a manifestation of a parametric decay of the waves: the mother R~wave decays into ion acoustic waves and daughter R waves. The conservation of the wave polarization is consistent with what the theory predicts [18]. Tsurutani et al. [23] observed a mixed polarization sense at 350 Re upstream of a interplanetary shock wave. The complexity of observed wave polarization may relates to the decay instability. As a result of the decay process, we expect coexistence of two wave modes, the parent R~ wave and the daughter R waves. The wave polarization in the observer’s frame is determined by the Doppler effect. After the Doppler effect, while the R + waves still have a right hand polarization, the R waves have an apparent left hand polarization. Therefore, we expect that a mixed polarization appears as a result of the decay process. However, further study is needed to identify the process responsible to produce the observed turbulent state. —
(ORIGINAL WAVE FORM)
AFTER FOURIER DECOMPOSITION
y ~
R~4
L
L~ 4
~ R
T=150
T~10O
T=50
T~O
_____________
__________________
0
Lx
__________________
2Lx
0
Lx
I
_____
2LX
0
LX
2L X
Fig. 3. Evolution of the waveforms for the slow beam case (the same case as shown in fig. 2). The middle and right panels show the result of the Fourier decomposition into two helices with different helicities. Middle: The y component of the magnetic field, B~, consisting of the L and R~waves. Right: the y component of the magnetic field, B~,consisting of the R and L~waves.
T Terasawa
/ Nonlinear dynamics of Alfvén waves
2.3. Simulation result: fast beam case Fig. 4 shows the evolution of the system for a fast beam case. Initially, we set a beam velocity Vb at 1OVA. The left panels show the evolution of the phase angle, and the middle panels of fig. 4 show the evolution of the waveform. Around time
T=35
~
T-45
2:
T-55
2~
T-60
7//A 7//A
of///A~H
T-65
:‘///A
T-70
2~
T-75
2~
T=80
_________
~
2:
:~/
~ :~ :~ :~ :~
2 -2
_________
4 0
JJ~J~J~M&~
4
~
~
EI/!112: ~ XfA~
:~ :~ DENSITY
]
~
o1/:,;J~J,~I :~i~J~ [ ~‘,: ~j~ 2 2~r
T-85
/2~A
40—65Q.~1, the phase angles of beam ions are bunched around that of magnetic field; this shows the formation of “gyrophase-bunched ions”. Around time 801, we observe a frequency shift toward the higher frequency (or the shorter wavelength). This is what is expected from the theory for fast ion beams (section 2.1).
B lB
PHASE ANGLE
197
-2
0
X/X.
_______________
IC/A
Fig. 4. Simulation results for a fast beam case (Vb/VA = 10, nb/no = 0.015) with a periodic boundary. The length of the simulation system, L~,is 256A . Left panels: the phase space plots (X— ~,) of ion distribution (thin marks). The phase angle ~ of the wave is shown by thick marks. Middle panels: the y component of the magnetic field (normalized by the background magnetic field B 0). Right panels: the spatial distribution of the beam density nb normalized by the initial beam density.
198
T Terasawa
vx
/
Nonlineardynamics of Alfvén waves
DENSITY
13. 0
idea is consistent with what is observed: Clumps
C
12
o.2. oo 00
~
13. C
1 2. CO
1
~
~ii°I
-0 C
_________________
have appeared thebut work byattracted Winske and Leroy [8] (see their fig.in 7) have no special attention. For the present case, the variation of the field
0
13. 0
~TTT~
20, ~
13.0
__________
-0.0 ____________ 13. 0
~
E~
24.O~
~
~
______
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800.
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:.
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00
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cally aand pect appear positive where where feedback the the beam field loop ions magnitude toare form decelerated clumps. reachesThis the lolocal maxima. It should be noted that the clumps
X
Fig. 5. Simulation results for a fast beam with an open boundary. The length of the simulation system is 800A 1. Left: the phase space distribution (X— J’~)of beam ions. The velocity scaleof butions is the normalized background by VA. ionsRight: (thickthe curve) spatial anddensity of the distribeam ions (thin curve). The density scale is normalized by the initial background ion density. For the beam ions, values of the density are multiplied by a factor of 10.
An interesting effect is seen in the spatial distribution of beam ion density (the right panels of fig. 4). There appears soliton-like density enhancement (“clumps”) after the time 6OQ~1.The formation of clumps can be understood in the following way. First the wave growth occurs in the system through the linear instability process. Since there is a finite width in the unstable wavenumber regime, we expect an overlap of multimode waves, Although each of the waves is circularly polarized and has a spatially uniform amplitude, their summation causes a spatial variation of the magnitude of the magnetic field. Beam ions are decelerated locally by the mirror force where the magnitude of wave magnetic field increases. This local deceleration gives a local increase of beam density. Since the beam is still active in exciting waves, a local increase of the beam density results in a local increase of the wave amplitude. Therefore we cx-
by overlapping magnitude, which of waves produces of modes the clumps, 4 and 5. is There made arises a question about the boundary condition: The formation of clumps are observed in the ber of modes periodic simulation can be system, excited. in Can whichwea finite still expect numformation of clumps in a realistic open system? lation The answer result. is Fig. positive 5(Nakagawa shows according the et result toal., a recent of simuopen system simulation in an preparation), where the solar wind is injected from the edge and the beam from the left. The density of the beam ions is 1.5% of the solar wind protons. The relative velocity 1OVA. between The left thepanels beam show and the the solar wind is set at phase space distribution (X— J/~)of beam ions. After the time 2OQ~~, a local deceleration of the beam occurs. In the right panels, thick and thin curves show, respectively, the spatial density distributions of the background ions and of the beam ions. At time 28f~1,we see the appearance of the “clumps” in the distribution of beam ion density. The clumps appear where the local deceleration occurs. It is found that the ions forming the clumps are not yet pitch-angle scattered but are gyrophase bunched (not shown). These property reminds us observations by Eastman et al. [24]: they observed upstream ions which are not only gyrophase bunched but make a cluster in space. Their observations seem quite close to what we see in the simulation. Let us see how the formation of clumps modifies the characteristic of the excited waves. Fig. 6 shows the evolution of the waveform. (We came back to the periodic system.) The original waveform of y component magnetic field, B~,is shown in the left panel. The middle and right panels show the result of the Fourier decomposition into right
T Terasawa
/
Nonlinear dynamics of Alfvén waves
they can excite both the L~and R’~waves. In this case the R~waves are still excited by the core of
Table 1 Energy sources and excited wave modes Source
Interaction mode
Excited wave
dilute beam ions dense beam ions
resonant resonant nonresonant resonant resonant
R R~ R R~,L~ R~,L
diffuse ions clumped ions
÷
two helices with different helicities. The middle panel shows B~ consisting of the R~and L waves. While the R~waves, excited by the ion beam cyclotron instability, are dominant for this helicity, we also seen the excitation of L waves, The following discussion leads us to conclude that these L waves are excited as a consequence of the instability caused by the ion clumps, The possible wave modes which can be excited by upstream particles are summarized in table 1. The resonant beam cyclotron instability excites the R + waves. If the beam density is large enough, the nonresonant excitation of the R wave becomes possible [8,25]. If the ions become diffuse, —
-
s
T—
0
the diffuse ions through the beam type instability, while the L~waves are excited through the loss .
cone type instabihty. Note that none of the above three processes results in the excitation of the L waves. Gyrophase-bunched ions, on the other hand, can excite the L— waves [26]. Although his calculation is limited to the spatially homogeneous case, his result holds qualitatively in the present case. The clumped ions, therefore, are unstable to the excitation of the L- and R~waves: For the L waves the free energy comes from the perpendicular motion of ions, while the free energy for the R~waves comes from parallel motion of ions. It is noted that the decay process is relatively unimportant for the fast beam case. In the right panel of fig. 6, the R waves, which should appear if the decay process is effective, are not excited above the noise level. Since the real frequency of the excited waves for the fast beam case is smaller than that for the slow beam case, the growth rate of the decay process is smaller accordingly. (The -
(ORIGINAL WAVE FORM)
L
199
AFTER FOURIER DECOMPOSITION
L
2L
0
L
2L
Fig. 6. Evolution of the waveforms for the fast beam case (the same case as shown in fig. 4, which is with a periodic boundary). To see easier the wave propagation, the data are repeated twice in the x direction. Left: the y component of the magnetic field (raw data from the simulation). Middle: the y component of the magnetic field, B~,consisting of the L— and R~waves. Right: the y component of the magnetic field, B~,consisting of the R— and L~waves.
200
T. Terasawa
/ Nonlinear dynamics of Alfvén waves
growth rate of the decay process is proportional to the frequency of the parent wave.) For the actual earth’s foreshock case, the geometrical effect (shock curvature) further reduces the importance of the decay process: The time required for the decay process to develop, which is several tens of minutes, exceeds the typical convection time around the curved bow shock. 3. Summary
and concluding remarks
We have presented simulation results for the upstream wave excitation process. A wide variety of nonlinear phenomena is found. For a relatively slow beam, the parametric wave decay process is found to be important. We suggest that the mixed polarization observed upstream of the interplanetary shocks could be explained by this process. For a relatively fast beam typically found in the terrestrial upstream region, we show that ion clumps are formed by the nonlinear effect of the excited waves. We propose that these ions clumps explain the observations by Eastman et al. [24]. It is found that the ion clumps excite a new wave mode (L), namely left-hand polarized waves propagating toward the downstream direction. The previous simulation works for the U~D stream wave excitation process were done in the one-dimensional space which is taken along the background magnetic field. A recent two-dimensional simulation has shown that multi-dimensional effects do not alter the basic property of the instability process [27]. However, for the parametnc decay process of the waves, the multi-dimensional effects would open new decay channels and alter the physical process significantly. A two-dimensional study for the parametric process should be done. While a periodic simulation code could only represent closed systems, the fresh beam ions are continuously supplied for the shock in reality. Further, the excited waves are conveyed back and eventually interact with the shock itself. As the final goal of the simulation study of the upstream process, the shock formation process itself should be treated self-consistently. Although we have obtamed a preliminary result from a simulation with open boundary condition, more efforts should be devoted in this direction.
Acknowledgement
The author thanks M. Scholer for valuable disCUSS~Ofl.
References [1] B.U.O. Sonnerup, J. Geophys. Res. 74 (1969) 1301. [2] G. Paschmann, N. Sckopke, I. Papamastorakis, JR. Asbridge, Si. Bame and J.T. Gosling, J. Geophys. Res. 85 (1980) 4689. [3] D. Burgess, J. Geophys. Res. 92 (1987) 1119. [4] J.P. Edmiston, CF. Kennel and D. Eichler, Geophys. Res. Lett. 9 (1982) 531. [5] M. Tanaka, C.C. Goodrich, D. Winske and K. Papadopoulos, J. Geophys. Res. 88 (1983) 3046. [6] D.H. Fairfield, J. Geophys. Res. 74 (1969) 3541. [7] A. Barnes, Cosmic Electrodyn. 1 (1970) 90. [8] D. Winske and MM. Leroy, J. Geophys. Res. 89 (1984) [9] 2763. M.F. Thomsen, in: Collisionless Shocks in the Heliosphere: Reviews of Current Research, eds. T. Tsurutani and G. Stone (American Geophysical Union, 1985, Geophysical Monograph 35) p. 253. [10] C. Bonifazi and G. Moreno, J. Geophys. Res. 86 (1981) [11] 4405, A.F. Viñas, ML. Goldstein and M.H. Acuña, J. Geophys. Res. 89 (1984) 3762. [12] S.P. Gary, C.D. Madland and B.T. Tsurutani, Phys. Fluids 28 (1985) 3691. [13] S.P. Gary, C.D. Madland, D. Schriver and D. Winske, J, 4188. Phys. Fluids 19 (1976) 126. [14] Geophys. A.S. Sgro Res. and 91(1986) C.W. Nielson, [15] J.A. Byers, B.I. Cohen, W.C. Condit and J.D. Hanson, J, Comput. Phys. 27 (1978) 363. [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
D.W. Hewett, J. Comput. Phys. 38 (1980) 378. D.S. Harned, J. Comput. Phys. 47 (1982) 452. T. Terasawa, Hoshino, J.-I. Sakai and T. Hada, J, Geophys. Res. M. 91(1986) 4171. M. Hoshino and T. Terasawa, J. Geophys. Res. 90 (1985) 57, S.A. Fuselier, M.F. Thomsen, J.T. Gosling and S.J. Bame, J. Geophys. Res. 91(1986) 91. M.L. Goldstein, Astrophys. J. 219 (1978) 700. N.F. Derby, Astrophys. J. 224 (1978) 1013. B.L. Tsurutani, E.J. Smith and D.E. Jones, J. Geophys. Res. 88 (1983) 5645. T.E. Eastman, R.R. Anderson, L.A. Frank and G.K. Parks, J. Geophys. Res. 86 (1981) 4379. D.D. Sentman, J.P. 7487. Edmiston and L.A. Frank, J. Geophys. Res. 86 (1981) R.W. Fredricks, J. Geophys. Res. 80 (1975) 7.
[27] D. Winske and K.B. Quest, J. Geophys. Res. 91 (1986)
8789.
193
NONLINEAR DYNAMICS OF ALFVEN WAVES: INTERACTIONS BETWEEN IONS AND SHOCK UPSTREAM WAVES Toshio TERASAWA Geophysical Institute, Kyoto University, Kyoto, Japan and Institut für extraterrestrische Physik, Max-Planck-Institut für Physik und Astrophysik, Garching, Fed. Rep. Germany
In upstream regions of collisionless heliospheric shocks, large amplitude Alfvén waves are often observed. These waves are excited by the beam ions injected at the shock front. We discuss several specific topics about the nonlinear dynamics of these waves; frequency shift of the waves owing to the beam deceleration, a parametric decay process of the waves and formation of ion clumps (spatially bunched ions).
1. Introduction It is now known that a wealth of nonlinear phenomena can be found in the region around the planetary bow shocks and interplanetary shocks. The magnetic and electric fields show a turbulent behavior there. The frequency range of turbulence covers from well below the ion cyclotron frequency to about the electron plasma frequency. In this paper, I shall treat a specific topic, interaction processes of energetic ions with the upstream turbulent waves, especially with low frequency hydromagnetic (Alfvén) waves, One of the fundamental questions about physics of collisionless shocks is how to excite the upstream turbulence? It is now known that ion beam injection occurs at the shock front. This beam injection is either due to electrostatic/ magnetic reflection of the solar wind ions [1—3]or due to leakage of magnetosheath ions [4,5]. Once an ion beam is injected to the upstream solar wind, low frequency hydromagnetic waves are excited through the beam cyclotron instability [6—8].The excited waves are right-hand polarized in the rest frame of the solar wind plasma and propagating away from the shock. After exciting waves, the beam ions are pitch-angle scattered and finally become diffuse ions ref. [9] and references therein), Since the energy density of the ion beam usually exceeds the upstream magnetic energy density [10],
the amplitude of the excited waves can be quite large. This largeness of the wave amplitude opens up variety of nonlinear phenomena.
2. Ion beam cyclotron instability 2.1. Slow and fast beams The phenomena which take place in the upstream region depend critically on parameters of the beam ions. In the following, the discussion is limited to the case of excitation of wave propagating parallel to the background field. We shall discuss two extreme cases, namely cases of a slow beam and of a fast beam. First, we treat a slow beam which has a relatively slow beam velocity Vb and a relatively large beam density n b; for example, Vb/VA 3 and nb/no 0.1—0.2, where VA and n0 are the Alfvén velocity and the background plasma density, respectively. In the upstream region of several interplanetary shocks, Viñas et al. [11] found ion beams which have characteristic parameters similar to the above. Another extreme case is a fast beam which is typically found in the earth’s foreshock region: The density of the beam is relatively low and the beam velocity is much faster than the Alfvén velocity; nb/no 0.01 and Vb/ VA ~ 10. The frequency of excited waves depends on the
0010-4655/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
194
T Terasawa
/
Nonlinear dynamics of Alfvén waves
deceleration causes a frequency shift toward lower
W
fas t b aa ~ S
frequency.
lo w b e a a
2.2. Simulation result: slow beam case For the simulation of low frequency electromagnetic phenomena, such as the upstream wave
=
k
VA
k
~/
motion, while electron response is expressed in a form of the generalized Ohm’s law. The magnetic field is advanced by solving the induction equation. Fig. 2 shows one-dimensional simulation results for a slow beam case. The + x direction is taken along the background magnetic field B0. Initially,
wkVb=Qj —
excitation process, hybrid simulation codes are used widely [8,14—18]. In these codes ions are treated as particles, whilebyelectrons fluid: ions are advanced solving are the treated equationasofa
/) I
3VA)
Fig. 1. A (w, k) diagram for the ion beam cyclotron instability, The abscissa and the ordinate show the wavenumber and the frequency (real part), respectively. Excitation of the hydromagnetic wave is expected where the cyclotron resonance line (~— kV~,= — £2.) intersects the dispersion line of Alfvén waves (w = kVA).
beam velocity. Roughly speaking, the frequency of the excited waves is determined by an intersection
beam beam Through velocity Vbthe(= beam towardions thehave + xa bulk direction. cyclotron instability, right-hand polarized waves propagating toward the same direction as the ion beam are excited. The nght panels show the evolution of the y component of the wave magnetic field. (The time scale used throughout this paper is normalized by the inverse of the ion Larmor frequency, Q~1.)In the left panels, thick and thin marks show the evolution of phase angles of the wave magnetic field and beam ions, c’~ 5and respectively, which are defined as ~,
of two lines, the linear dispersion line, w = kVA, and the cyclotron resonance condition, Co kV~,= Q~,where Q1 is the ion Larmor frequency (fig. 1). From the geometry it can be seen that the frequency becomes higher as the beam velocity becomes lower. However, as Gary et al. [12,131 pointed out,Inthis idea breaksthedown below 2.5VA. thissimple velocity regime, interaction Vb between a beam and waves becomes highly dispersive and the frequency and excited waves decreases as the beam velocity reduces. From this difference in frequency characteristic, there appears a significant difference in the nonlinear evolution of the systems between the fast and slow beam cases: When the beam energy is transferred to the waves, the beam ions are decelerated. For the fast beam, this beam deceleration causes a frequency shift toward the higher frequency. On the other hand, for the slow beam, the beam —
—
~
=
tan~
—~
and
~, = tan’
B~ At the initial stage of the wave excitation, we observe the that phase of beamfield. ionsThis are bunched that around of angles the magnetic bunching phenomenon, also seen in the simulation of a fast beam (next section), explains what is called “gyrophase bunched ions” observed in the earth’s foreshock region [19,20]. At time 60~?~1, we see a reduction of the wavenumber (equivalently, reduction of the wave frequency). This downward frequency shift is consistent with what the theory predicts for slow ion beams (section 2.1). As time further elapses, a reversal of the wave helicity appears. This can be seen as a reversal of the slopes of wave phase angle from di-
T Terasawa / Nonlinear dynamics of Alfvén waves
195
B lB
PHASE ANGLE
0 2
0 2ir t~’
,~ e
.~
..,.
____________________________
0
2~
T=30
~
/‘
~
P
~ ,
/
—2
2
~ 2
.~
T=60
31
______
2~//
2 T~90
‘
I /
______
~
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-2
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~
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0
~
~
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11
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~us
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~
~
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~
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~
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____________________________
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51.2
xIxi
Fig. 2. Simulation results for a slow beam case (Vb / VA = 3, nb/no = 0.2) with a periodic boundary. The abscissa is the x axis, where the scale is normalized by the ion inertia length (A VA / Q~).The length of the simulation system, L~is 51.2 A1. Left panels: the phase space plots (X— ~,) of ion distribution (thin marks). The phase angle ~ of the magnetic field is shown by thick marks. Right panels: the y component of the magnetic field (normalized by the background magnetic field B0).
196
T. Terasawa
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Nonlinear dynamics of Alfvén waves
recting upper right to directing lower right. This is a manifestation of another nonlinear phenomenon, a parametric decay of the waves ([refs. [18,21,22] and references therein), Fig. 3 shows the evolution of wave forms more closely. The time runs from the bottom to the top, The original waveform of y component magnetic field, B~,is shown in the left panel. To see easier the wave propagation, the data are repeated twice in the x direction. Until the time 6O~2~,we observe waves propagating toward + x direction. They are waves excited by the beam cyclotron instability. After a time of 90 ~ 1, waves propagating in the opposite direction appear. A Fourier decomposition technique makes the situation clearer (ref. [18], appendix B). Original data (B~ and B~)at a given time are decomposed into two helices of different helicities. The middle and right panels of fig. 3 show the y components of these two helices. The middle panel shows a helix whose polarization is either right-hand or left-hand depending on the propagation direction (+ x or x, we shall call them as the R + or L waves for abbreviation). In this panel it is seen that the R~ waves, which are excited by the beam cyclotron —
—
~
instability, are the dominant wave component. In the right panel of fig. 3, we see a delayed evolution of the R waves (the right-hand polarized waves propagating in the x direction). This is a manifestation of a parametric decay of the waves: the mother R~wave decays into ion acoustic waves and daughter R waves. The conservation of the wave polarization is consistent with what the theory predicts [18]. Tsurutani et al. [23] observed a mixed polarization sense at 350 Re upstream of a interplanetary shock wave. The complexity of observed wave polarization may relates to the decay instability. As a result of the decay process, we expect coexistence of two wave modes, the parent R~ wave and the daughter R waves. The wave polarization in the observer’s frame is determined by the Doppler effect. After the Doppler effect, while the R + waves still have a right hand polarization, the R waves have an apparent left hand polarization. Therefore, we expect that a mixed polarization appears as a result of the decay process. However, further study is needed to identify the process responsible to produce the observed turbulent state. —
(ORIGINAL WAVE FORM)
AFTER FOURIER DECOMPOSITION
y ~
R~4
L
L~ 4
~ R
T=150
T~10O
T=50
T~O
_____________
__________________
0
Lx
__________________
2Lx
0
Lx
I
_____
2LX
0
LX
2L X
Fig. 3. Evolution of the waveforms for the slow beam case (the same case as shown in fig. 2). The middle and right panels show the result of the Fourier decomposition into two helices with different helicities. Middle: The y component of the magnetic field, B~, consisting of the L and R~waves. Right: the y component of the magnetic field, B~,consisting of the R and L~waves.
T Terasawa
/ Nonlinear dynamics of Alfvén waves
2.3. Simulation result: fast beam case Fig. 4 shows the evolution of the system for a fast beam case. Initially, we set a beam velocity Vb at 1OVA. The left panels show the evolution of the phase angle, and the middle panels of fig. 4 show the evolution of the waveform. Around time
T=35
~
T-45
2:
T-55
2~
T-60
7//A 7//A
of///A~H
T-65
:‘///A
T-70
2~
T-75
2~
T=80
_________
~
2:
:~/
~ :~ :~ :~ :~
2 -2
_________
4 0
JJ~J~J~M&~
4
~
~
EI/!112: ~ XfA~
:~ :~ DENSITY
]
~
o1/:,;J~J,~I :~i~J~ [ ~‘,: ~j~ 2 2~r
T-85
/2~A
40—65Q.~1, the phase angles of beam ions are bunched around that of magnetic field; this shows the formation of “gyrophase-bunched ions”. Around time 801, we observe a frequency shift toward the higher frequency (or the shorter wavelength). This is what is expected from the theory for fast ion beams (section 2.1).
B lB
PHASE ANGLE
197
-2
0
X/X.
_______________
IC/A
Fig. 4. Simulation results for a fast beam case (Vb/VA = 10, nb/no = 0.015) with a periodic boundary. The length of the simulation system, L~,is 256A . Left panels: the phase space plots (X— ~,) of ion distribution (thin marks). The phase angle ~ of the wave is shown by thick marks. Middle panels: the y component of the magnetic field (normalized by the background magnetic field B 0). Right panels: the spatial distribution of the beam density nb normalized by the initial beam density.
198
T Terasawa
vx
/
Nonlineardynamics of Alfvén waves
DENSITY
13. 0
idea is consistent with what is observed: Clumps
C
12
o.2. oo 00
~
13. C
1 2. CO
1
~
~ii°I
-0 C
_________________
have appeared thebut work byattracted Winske and Leroy [8] (see their fig.in 7) have no special attention. For the present case, the variation of the field
0
13. 0
~TTT~
20, ~
13.0
__________
-0.0 ____________ 13. 0
~
E~
24.O~
~
~
______
_________________
~~:o:
0. 0 0. 0
800.
X
:.
0
2.
00
0.
o~
ooo.
o. 0
cally aand pect appear positive where where feedback the the beam field loop ions magnitude toare form decelerated clumps. reachesThis the lolocal maxima. It should be noted that the clumps
X
Fig. 5. Simulation results for a fast beam with an open boundary. The length of the simulation system is 800A 1. Left: the phase space distribution (X— J’~)of beam ions. The velocity scaleof butions is the normalized background by VA. ionsRight: (thickthe curve) spatial anddensity of the distribeam ions (thin curve). The density scale is normalized by the initial background ion density. For the beam ions, values of the density are multiplied by a factor of 10.
An interesting effect is seen in the spatial distribution of beam ion density (the right panels of fig. 4). There appears soliton-like density enhancement (“clumps”) after the time 6OQ~1.The formation of clumps can be understood in the following way. First the wave growth occurs in the system through the linear instability process. Since there is a finite width in the unstable wavenumber regime, we expect an overlap of multimode waves, Although each of the waves is circularly polarized and has a spatially uniform amplitude, their summation causes a spatial variation of the magnitude of the magnetic field. Beam ions are decelerated locally by the mirror force where the magnitude of wave magnetic field increases. This local deceleration gives a local increase of beam density. Since the beam is still active in exciting waves, a local increase of the beam density results in a local increase of the wave amplitude. Therefore we cx-
by overlapping magnitude, which of waves produces of modes the clumps, 4 and 5. is There made arises a question about the boundary condition: The formation of clumps are observed in the ber of modes periodic simulation can be system, excited. in Can whichwea finite still expect numformation of clumps in a realistic open system? lation The answer result. is Fig. positive 5(Nakagawa shows according the et result toal., a recent of simuopen system simulation in an preparation), where the solar wind is injected from the edge and the beam from the left. The density of the beam ions is 1.5% of the solar wind protons. The relative velocity 1OVA. between The left thepanels beam show and the the solar wind is set at phase space distribution (X— J/~)of beam ions. After the time 2OQ~~, a local deceleration of the beam occurs. In the right panels, thick and thin curves show, respectively, the spatial density distributions of the background ions and of the beam ions. At time 28f~1,we see the appearance of the “clumps” in the distribution of beam ion density. The clumps appear where the local deceleration occurs. It is found that the ions forming the clumps are not yet pitch-angle scattered but are gyrophase bunched (not shown). These property reminds us observations by Eastman et al. [24]: they observed upstream ions which are not only gyrophase bunched but make a cluster in space. Their observations seem quite close to what we see in the simulation. Let us see how the formation of clumps modifies the characteristic of the excited waves. Fig. 6 shows the evolution of the waveform. (We came back to the periodic system.) The original waveform of y component magnetic field, B~,is shown in the left panel. The middle and right panels show the result of the Fourier decomposition into right
T Terasawa
/
Nonlinear dynamics of Alfvén waves
they can excite both the L~and R’~waves. In this case the R~waves are still excited by the core of
Table 1 Energy sources and excited wave modes Source
Interaction mode
Excited wave
dilute beam ions dense beam ions
resonant resonant nonresonant resonant resonant
R R~ R R~,L~ R~,L
diffuse ions clumped ions
÷
two helices with different helicities. The middle panel shows B~ consisting of the R~and L waves. While the R~waves, excited by the ion beam cyclotron instability, are dominant for this helicity, we also seen the excitation of L waves, The following discussion leads us to conclude that these L waves are excited as a consequence of the instability caused by the ion clumps, The possible wave modes which can be excited by upstream particles are summarized in table 1. The resonant beam cyclotron instability excites the R + waves. If the beam density is large enough, the nonresonant excitation of the R wave becomes possible [8,25]. If the ions become diffuse, —
-
s
T—
0
the diffuse ions through the beam type instability, while the L~waves are excited through the loss .
cone type instabihty. Note that none of the above three processes results in the excitation of the L waves. Gyrophase-bunched ions, on the other hand, can excite the L— waves [26]. Although his calculation is limited to the spatially homogeneous case, his result holds qualitatively in the present case. The clumped ions, therefore, are unstable to the excitation of the L- and R~waves: For the L waves the free energy comes from the perpendicular motion of ions, while the free energy for the R~waves comes from parallel motion of ions. It is noted that the decay process is relatively unimportant for the fast beam case. In the right panel of fig. 6, the R waves, which should appear if the decay process is effective, are not excited above the noise level. Since the real frequency of the excited waves for the fast beam case is smaller than that for the slow beam case, the growth rate of the decay process is smaller accordingly. (The -
(ORIGINAL WAVE FORM)
L
199
AFTER FOURIER DECOMPOSITION
L
2L
0
L
2L
Fig. 6. Evolution of the waveforms for the fast beam case (the same case as shown in fig. 4, which is with a periodic boundary). To see easier the wave propagation, the data are repeated twice in the x direction. Left: the y component of the magnetic field (raw data from the simulation). Middle: the y component of the magnetic field, B~,consisting of the L— and R~waves. Right: the y component of the magnetic field, B~,consisting of the R— and L~waves.
200
T. Terasawa
/ Nonlinear dynamics of Alfvén waves
growth rate of the decay process is proportional to the frequency of the parent wave.) For the actual earth’s foreshock case, the geometrical effect (shock curvature) further reduces the importance of the decay process: The time required for the decay process to develop, which is several tens of minutes, exceeds the typical convection time around the curved bow shock. 3. Summary
and concluding remarks
We have presented simulation results for the upstream wave excitation process. A wide variety of nonlinear phenomena is found. For a relatively slow beam, the parametric wave decay process is found to be important. We suggest that the mixed polarization observed upstream of the interplanetary shocks could be explained by this process. For a relatively fast beam typically found in the terrestrial upstream region, we show that ion clumps are formed by the nonlinear effect of the excited waves. We propose that these ions clumps explain the observations by Eastman et al. [24]. It is found that the ion clumps excite a new wave mode (L), namely left-hand polarized waves propagating toward the downstream direction. The previous simulation works for the U~D stream wave excitation process were done in the one-dimensional space which is taken along the background magnetic field. A recent two-dimensional simulation has shown that multi-dimensional effects do not alter the basic property of the instability process [27]. However, for the parametnc decay process of the waves, the multi-dimensional effects would open new decay channels and alter the physical process significantly. A two-dimensional study for the parametric process should be done. While a periodic simulation code could only represent closed systems, the fresh beam ions are continuously supplied for the shock in reality. Further, the excited waves are conveyed back and eventually interact with the shock itself. As the final goal of the simulation study of the upstream process, the shock formation process itself should be treated self-consistently. Although we have obtamed a preliminary result from a simulation with open boundary condition, more efforts should be devoted in this direction.
Acknowledgement
The author thanks M. Scholer for valuable disCUSS~Ofl.
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D.W. Hewett, J. Comput. Phys. 38 (1980) 378. D.S. Harned, J. Comput. Phys. 47 (1982) 452. T. Terasawa, Hoshino, J.-I. Sakai and T. Hada, J, Geophys. Res. M. 91(1986) 4171. M. Hoshino and T. Terasawa, J. Geophys. Res. 90 (1985) 57, S.A. Fuselier, M.F. Thomsen, J.T. Gosling and S.J. Bame, J. Geophys. Res. 91(1986) 91. M.L. Goldstein, Astrophys. J. 219 (1978) 700. N.F. Derby, Astrophys. J. 224 (1978) 1013. B.L. Tsurutani, E.J. Smith and D.E. Jones, J. Geophys. Res. 88 (1983) 5645. T.E. Eastman, R.R. Anderson, L.A. Frank and G.K. Parks, J. Geophys. Res. 86 (1981) 4379. D.D. Sentman, J.P. 7487. Edmiston and L.A. Frank, J. Geophys. Res. 86 (1981) R.W. Fredricks, J. Geophys. Res. 80 (1975) 7.
[27] D. Winske and K.B. Quest, J. Geophys. Res. 91 (1986)
8789.