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Shock surfing acceleration
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Planetary and Space Science 51 (2003) 665 – 680 www.elsevier.com/locate/pss
Shock sur"ng acceleration ( cer Vitali D. Shapiro∗ , Defne U* University of California, San Diego, Physics Department, La Jolla CA 92093, USA Received 19 February 2003; accepted 19 May 2003
Abstract Analytical and numerical analysis identify shock sur"ng acceleration as an ideal pre-energization mechanism for the slow pick-up ions at quasiperpendicular shocks. After gaining su5cient energy by shock sur"ng, pick-up ions undergo di6usive acceleration to reach their observed energies. Energetic ions upstream of the cometary bow shock, acceleration of solar energetic particles by magnetosonic waves in corona, ion enhancement in interplanetary shocks, generation of anomalous cosmic rays from interstellar pick-up ions at the termination shock are some of the cases where shock sur"ng acceleration apply. Inclusion of the lower-hybrid wave turbulence into the laminar model of shock sur"ng can explain the preferential acceleration of heavier particles as observed by Voyager at the termination shock. At relativistic energies, unlimited acceleration of ions is theoretically possible; because for su5ciently strong shocks main limitation of the mechanism, caused by the escape of accelerated particles downstream of the shock during acceleration no longer exists. ? 2003 Published by Elsevier Ltd. Keywords: Shock; Wave turbulence; Accelerated particles
1. Introduction A large fraction of particle energization in astrophysical environments comes from the acceleration by shock waves. In the interstellar medium, strong shocks are formed during the mass out
Corresponding author. E-mail address: [email protected] (V.D. Shapiro).
0032-0633/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/S0032-0633(03)00102-8
leads to a shell distribution centered approximately at the solar wind velocity. The distribution has a radius of the order of solar wind speed vsw (Sagdeev et al., 1986; Lee and Ip, 1987). If the excited
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acceleration (Axford et al., 1977; Krymsky, 1977; Blanford and Ostriker, 1978; Bell, 1978). However this type of acceleration is not considered to be e6ective in quasiperpendicular shocks. In order to start di6usive shock acceleration, the accelerated particles must traverse the shock front in both directions many times. This condition is easily ful"lled at quasiparallel shocks, since the ambient magnetic "eld is parallel to the shock normal, and the ions can move freely along the magnetic "eld. This is especially true in the case of supercritical shocks (Edmiston et al., 1982; Kennel et al., 1985), where downstream heating helps ions to return to the upstream of the shock. In the case of quasi-perpendicular shocks, the particles need to be e6ectively scattered perpendicular to the ambient magnetic "eld. This is why, signi"cantly large initial velocity is needed for ions to start di6usive shock acceleration (Webb et al., 1995). In this reference it is shown that the threshold velocity to initiate di6usive shock acceleration in quasi-perpendicular shocks is vth =
3u (1 + 2 )1=2 ; r
(1)
where r is the shock compression ratio and = =rg where is the scattering mean free path along the magnetic "eld and rg is the ion gyroradius. If we assume that scattering is weak (rg ) and make a reasonable assumption that is proportional to rg , then the threshold velocity is independent of particle charge and mass and several times larger than the solar wind velocity vsw . This threshold is generally not achieved by any of pick-up ions in their incident velocity distribution, thus the observed energies in the all the previously mentioned cases of quasiperpendicular shocks are quite puzzling. The puzzle can be solved if a pre-acceleration mechanism at quasiperpendicular shocks is proposed which favors slow pick-up ions and accelerates them to the threshold velocity needed for the di6usive shock acceleration. Such mechanism of ion acceleration has been proposed long time ago by Sagdeev (1966). The idea can be explained in the following way (see also Sagdeev and Shapiro, 1973): Let us consider a periodic electrostatic wave propagating in the x-direction with phase velocity vph . The particles that have velocities close to the phase velocity of the wave can be trapped by the electrostatic potential well of the wave. Because the particle is in a symmetric well, the particle has no net energy gain. In the case when the ambient magnetic "eld is present and points in the z-direction (see caption for Fig. 1), the symmetry is broken and the particle starts to spend more time in the left half of the wave because of the additional Lorentz force. The particle is accelerated under the action of the y-component of the Lorentz force −evx B0 =c = −evph B0 =c (since the average value of vx is the phase velocity of the wave). The acceleration goes on until the Lorentz force (˙ vy ) is large enough such that the particle can no longer be re
Φ
x
Fig. 1. The sur"ng acceleration by a periodic electrostatic wave 0 (x) propagating across the magnetic "eld B0 ez . The trapped particles are moving in the x-direction with the wave phase velocity !=k and are accelerated along the wave front in the y-direction under the action of the Lorentz force Fy = −(e=c)vx B0 . The x-component of the Lorentz force Fx = (e=c)vy B0 pushes trapped ions to the left side of the potential well. Particles oscillating around non-zero wave phase gain energy from the wave.
Since the trajectory of the accelerated particles is such that they do multiple re
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u
y +
x
B Ey
Fig. 2. Schematic diagram for shock sur"ng acceleration at a perpendicular shock.
hancements at the interplanetary traveling shocks (Sarris and Van Allen, 1974; Pesses et al., 1979, 1982; Decker, 1981). In the shock drift acceleration mechanism, ion energy gain is due to ion curvature and gradient drifts in the inhomogeneous magnetic "eld at the shock front. Although the shock sur"ng acceleration relies on a strong shock potential and shock drift acceleration does not, the most important distinction between these two mechanisms is the fact that the shock drift acceleration energy gain is proportional to the initial ion energy, while the "nal energy in the shock sur"ng is greater for ions which have lower incident velocities initially (|vx0 |u where u is the shock speed). This happens because the particles with smaller initial |vx | are trapped longer at the shock front, while they are accelerated by the convective electric "eld. By this reason, shock surfing is an ideal pre-acceleration mechanism for slow pick-up ions with |v|vsw . The original idea of Sagdeev (1966) has been further developed by many authors. Sagdeev and Shapiro (1973) has considered dissipation mechanisms of non-Landau type for periodic electrostatic wave propagating across magnetic "eld considering the scenario where the sur"ng particles dissipate the energy of the wave. They also emphasized limitation on sur"ng acceleration that vy ¡ cEx =B0 , where Ex is the maximum electric "eld at the front. Sugihara and Mitzuna (1979) applied the idea of sur"ng acceleration to ion acceleration in laser fusion plasma. Katsouleas and Dawson (1983) paid attention on the analogy of this type of acceleration with a surfer’s motion and they generalized the mechanism for relativistic particles. They also showed that necessary condition for unlimited acceleration is Ex ¿ B0 . Ohsawa (1990) classi"ed the ion trajectories into those which are re
667
(1984) using both analytical calculations and numerics generalized the mechanism for quasiperpendicular periodic wave with Bx Bz and showed that acceleration in y-direction can be signi"cantly larger than that in strictly perpendicular case. This is possible by allowing ion acceleration in two directions. Using simulations, Lembege et al. (1983) pointed out that the sur"ng acceleration is an e6ective dissipation mechanism for magnetosonic wave. Sagdeev et al. (1984) have been the "rst who considered shock sur"ng acceleration at a strictly perpendicular magnetosonic solitary pulse and derived the resulting dissipation which transforms the pulse into a perpendicular shock wave. Ohsawa (1985, 1986) and Ohsawa and Sakai (1985) used simulations to investigate the formation of self-consistent shock structure from sur"ng acceleration. The idea of shock sur"ng acceleration has been applied to the acceleration of solar energetic particles by fast magnetosonic shocks in the corona (Ohsawa and Sakai, 1987), He3 ions in impulsive
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by Gloeckler et al. (1994) which indicated the acceleration e5ciency is larger for lighter particles. In order to explain the recent observations of Voyager, Shapiro et al. (2001) suggested the inclusion of lower-hybrid wave turbulence which is excited by sur"ng particles at the termination shock. Waves are responsible for particle velocity di6usion in the direction opposite to acceleration which results in detrapping of particles from the acceleration region. Since velocity di6usion is more e5cient for lighter particles, acceleration is more e5cient for heavier ions. Also Zank et al. (2001) pointed out the mass dependence of the threshold velocity vth to start di6usive shock acceleration (1). These authors "nd that vth decreases with increasing mass of the accelerated ions which means heavier ions reach the threshold velocity faster and this fact presents another possible explanation to preferential acceleration of heavy ions at the termination shocks. A numerical analysis of the shock sur"ng acceleration up to the relativistic energies has been done by Ucer and Shapiro (2001). Main limitation on sur"ng acceleration is due the fact that the acceleration along the shock front is accompanied by the growth of the energy of the bounce oscillations and "nally the particle overcomes the potential barrier in the shock front and escapes downstream. Ucer and Shapiro (2001) show that this limitation is absent at the relativistic energies, in fact the bounce energy decreases during acceleration for vy ∼ c. This brings the possibility for unlimited sur"ng acceleration of relativistic particles.
2. Basic equations We consider a plane quasi-perpendicular shock moving with u = uex . In the reference frame of the shock as shown in Fig. 2, shock is stationary and shock normal is in positive x-direction. Shock potential is (x) and upstream (x ¿ 0) magnetic "eld B = Bx ex + Bz ez with Bx ¿ 0. The plane containing vectors u and B is called the coplanar plane. By , which is the noncoplanar component, is only present within the shock front where @=@x = 0. In the frame of the shock, the non-relativistic ion equations of motion are: mi
dvx d e = −e + (vy Bz − vz By ); dt dx c
(2)
mi
dvy e = eEy + (vz Bx − vx Bz ); dt c
(3)
dvz e = eEz + (vx By − vy Bx ); mi dt c
(4)
where mi is the ion mass, e is the ion charge, E and B is the electric "eld in the shock frame. Because shock is stationary, ∇ × E = 0 and therefore Ey and Ez do not depend on x. Since the electric "eld in lab frame is zero far away from the shock E = 0, the electric
"eld in the frame moving with the shock is E=
1 u×B c
(5)
which yields Ey = −uBz =c and Ez = uBy =c. As mentioned before the noncoplanar magnetic "eld By is created only inside the shock (Jones and Ellison, 1987). This component arises from the condition that the electrons and ions should receive equal increments of velocity into z-direction passing the shock. Because otherwise there would be "nite current and this would lead to the building up of magnetic "eld By in the downstream region, and this would be against the Rankine–Hugoniot conditions. Also by using the fact that the current is dominated by electron contribution inside the shock region, the authors Jones and Ellison obtain the following relationship for By . c By d x = Bx RBz ; (6) 4en0 u where RBz is the increase of Bz inside the shock layer and n0 is electron or proton number density upstream of the shock. If L is the characteristic space scale of the shock layer then Eq. (6) can be estimated as By ∼
Bx c RBz : 4en0 u L
(7)
We will go over the analysis of ion dynamics in the normal incidence frame as shown in Fig. 2. This frame is often close to the frame in which spacecraft observations are made. In this reference frame ion equations of motion are: e d dvx =− + vy z (x) − vz y (x); dt m dx
(8)
dvy = −(u + vx ) z (x) + vz x ; dt
(9)
dvz = vx y (x) − vy x ; dt
(10)
where x = eBx =mc, y = eBy (x)=mc and z = eBz =mc. Bx is constant since ∇ · B = 0. To illustrate the creation of the e6ective potential in which accelerated particle is trapped, let’s multiply Eq. (8) by mvx . Then we can write d e [x + e(x)] = B0 vy vx ; dt c
(11)
where x = mvx2 =2 is the energy of bounce oscillations in the direction of the shock normal. Suppose we are considering the particle acceleration being adiabatic, 1 dvy !B ; vy dt
(12)
' ( er / Planetary and Space Science 51 (2003) 665 – 680 V.D. Shapiro, D. Uc vy1 vy2 vy3
the velocity of bounce oscillations increase accompanying the acceleration in the y-direction but with a slower rate. vx ˙ |vy |1=3 :
(17)
Effective Potential
Particle remains in the potential well and continues acceleration until it gains su5cient bounce kinetic energy (x ) to overcome the potential barrier in front of the shock. This corresponds to a change in vy by a factor of 3=2 vy 2e0 ; (18) 2 vy0 mi vx0 vy1
X
Fig. 3. Schematic representation of the e6ective potential well in which sur"ng particle is bouncing at three di6erent times t1 ¡ t2 ¡ t3 .
where !B is the frequency of bounce oscillations. Then we can integrate Eq. (11) and obtain the e6ective potential: e Ue6 = e + B0 |vy |x; (13) c where we also took into account that vy ¡ 0 for the geometry shown in Fig. 2. The "rst term in Ue6 is electrostatic potential at the shock front, second term is the input from the component of the Lorentz force in the x-direction which creates the turning point for ions which move away from the shock. The e6ective potential is shown in Fig. 3. Balancing the kinetic energy of bounce oscillations with the second term in Ue6 we can obtain the following estimation for the amplitude of bounce oscillations: Rx ∼
vx2 : !c |vy |
(14)
Then the period of bounce oscillations can be estimated as !B =
669
2 Rx vx : ∼ ∼ !B vx !c |vy |
(15)
During acceleration, e6ective potential, which traps the particle, changes as it is shown in the "gure. With further acceleration, Ue6 get steeper and the amplitude of bounce oscillations decreases. The corresponding energy of bounce oscillations changes as well. If condition of adiabaticity of acceleration (12) is ful"lled, the longitudinal adiabatic invariant should be conserved (Goldstein, 1980). vx d x = const; (16) where integration is taken over a closed trajectory between two turning points. We can obtain an estimation for the relation between vx and vy by substituting the amplitude of bounce oscillations Rx from (14) into (16). This yields that
where vx0 , vy0 -initial components of particle velocity. For initially slow particles, this factor is greater. Particle can also escape from the accelerated region to downstream of the shock by a strong Lorentz force. Lorentz force is pointing towards the shock front and has a detrapping e6ect. In particular, the particle is not trapped at all if dUe6 =d x ¿ 0 in the region x ¿ 0. This occurs if the Lorentz force exceeds the electric force in the shock layer. In the presence of the noncoplanar component of the magnetic "eld By this condition may be written as 1 (Bz |vy | + |By vz |) ¿ Ex ; c
(19)
where Ex ∼ 0 =L is the electrostatic "eld in the shock layer and L is the shock width. For a strictly perpendicular shock Bx = 0, (i.e. By = 0, following (7)) this condition leads to the most e6ective acceleration such that |vy | c
Ex : Bz
(20)
Together with (18), this condition provides an upper limit for particle acceleration at the shock front. The maximum acceleration factor is determined by minimum of these two estimates (18) and (20). At a quasi-perpendicular shock, with the presence of Bx , particle also acquires a possibility to escape upstream. This happens when the particle gains a positive value of vy and the Lorentz force in the x-direction starts to push the particle away from the shock. We can see this e6ect if we average Eqs. (9) and (10) over period of fast bounce oscillations. Then the oscillatory terms proportional to vx vanish and we are left with d 2 vy + x2 vy = 0; dt 2
(21)
d 2 vz + x2 vz = u x z ; dt 2
(22)
where x = eBx =mc and z = eBz =mc. Assuming that initial components of particle velocity in the shock plane satisfy the condition Bz |vy0 |; |vz0 |u ; Bx
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670
we can write the solution of Eqs. (21) and (22) in the form vy = −u(Bz =Bx )sin x t;
(23)
vz = u(Bz =Bx )(1 − cos x t):
(24)
vz ∆v
Particle escapes from the shock at x t = , when velocity vy changes sign. Therefore if particle does not satisfy conditions (18) and (20) during acceleration, it will escape upstream with the energy =
mvz2 2
= 2mu2 (Bz =Bx )2 ;
vy = vy0 + t (dvy =dt); where t = 0 is the start of the bounce. For the steep potential we can integrate Eq. (11) to evaluate the change in the energy of bounce oscillations x : dvy !B Rx = mi !c dt t vx ; (26) dt 0 where x =mi vx2 =2, and in adiabatic regime we have dvy =dt const during one bounce. The constant term proportional to vy0 vanishes in Eq. (26) because the integral is taken over the entire bounce period. Also for the steep potential, we neglected the term which is proportional to @=@x because it is insigni"cant for the main part of the bounce motion in front shock. We also can integrate Eq. (8) for the motion in the x-direction in the case of strictly perpendicular shock and we obtain (27)
where vx0 and vy0 are the velocity components at the start of the bounce. Since at t = !B =2, there is a re
2 vx0 : !c |vy0 |
(28)
Integrating Eq. (26), we obtain the following relation: Rx 1 x dy dx = ; !B dt 3 y dt
ϕ
(25)
where vy = 0, following (23), and vx = 0 as well because it is directly related to vy by (17). Ion dynamics can be easily analyzed in the case of vanishing shock width, when particle is re
vx = vx0 − |vy0 |!c t ;
vx
θ
u
(29)
where y = mvy2 =2. This relation has been derived only considering the fact that the acceleration is slow with
vy Fig. 4. Shell distribution of pick-up ions in the normal incidence frame of the shock. Ions inside the shaded region are accelerated above the threshold velocity for the di6usive shock acceleration.
respect to the bounce motion. Integration of Eq. (29) yields vx3 =vy = const which is equivalent to the conservation of adiabatic integral discussed earlier. Another important characteristic of shock sur"ng acceleration is to show that it can act as a preacceleration mechanism for di6usive shock acceleration at quasiperpendicular shocks. We will adapt the idea in Lee et al. (1996) to calculate the
vy0 = u sin ":
Acceleration is "nalized when particle overcomes the shock 2 potential with mi vxf =2 emax . It is reasonable to assume that the shock is strong enough to re
8u "5
mp mi
3=2 :
(30)
It was mentioned in the introduction that, at quasiperpendicular shocks, in order to start di6usive acceleration particles should have a threshold velocity given by (1). In order the
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671
sur"ng acceleration be e6ective as an preacceleration mechanism for di6usive shock acceleration |vy |max ¿ vth or in other words: 3=2 8u mp 5 5 : (31) " 6 "0 = vth mi
Therefore the e6ective potential for the relativistic case is
Then the
Rx
Nu 8
0
2
"0 d’
d""3 :
(32)
Taking the integral and substituting "0 from (31), the following expression is obtained for the
3. Shock Surng Acceleration at the relativistic energies An important peculiarity of sur"ng particle dynamics at the relativistic energies has been discussed by authors of the present review (Ucer and Shapiro, 2001). In relativistic case for strictly perpendicular shock (Bx = 0,By = 0) equations of motion are reduced to the following two dimensional set in shock frame.
Ue6 = e + e(u B0 x:
(38)
Now balancing the bounce kinetic energy with Ue6 we can estimate the amplitude of bounce oscillations as (vx2 : (u ! c c
(39)
Adiabatic invariant in the relativistic case takes the form px d x = (vx d x = const (40) and using estimation (39) for Rx, it is easy to see that during acceleration (with the growth of relativistic factor () velocity of bounce oscillations decreases as vx ˙
1 (2=3
:
(41)
The kinetic energy of bounce oscillations decreases during the acceleration as x ˙ (vx2 ∼ (−1=3 as it is illustrated by Fig. 5 obtained by numerical solution of Eqs. (34) and (35). It follows from this "gure that for a su5ciently large electrostatic barrier at the shock front e ¿ max ∼ 0:08 mu2 , the particle cannot overcome the potential barrier at the shock front, and it continues acceleration theoretically for an in"nitely long time. For the particle to stay in the acceleration region, as it was discussed for the non-relativistic case, the Lorentz force pushing the particle towards the shock should never exceed the repulsive force due to the shock electric "eld Ex ¿ (u B0 . The preceding analysis applies when the condition of adiabaticity of acceleration is ful"lled, in other words the typical time for acceleration is signi"cantly larger than the bounce period:
d @ e + vy (u B0 ; ((mvx ) = −e dt @x c
(34)
R( 1 (
e e d (mvy = − (u uB0 − vx (u B0 : dt c c
(35)
where R( is the change in the relativistic factor during one bounce period !B .
Here ( = (1 − v2 =c2 )−1=2 is relativistic factor. Magnetic "eld in the shock frame is B = (u B0 ez , where B0 is the ambient magnetic "eld and (u = (1 − u2 =c2 )−1=2 . Convective electric "eld in the shock frame is u E = − (u B0 ey : (36) c We assume that the particle velocity along the shock front is of the order of speed of light (vy ∼ c). Then the relativistic factor is ( ∼ (1 − vy2 =c2 )−1=2 . Multiplying Eq. (34) with vx we obtain the following equation similar to (11) in the non-relativistic case: d (mvx2 + e(x) = e(u B0 vx : (37) dt 2
!B ∼
Rx (vx : ∼ vx (u !c c
(42)
(43)
The growth of the relativistic particle energy under the action of the convective electric "eld Ey can be written as eEy d( ∼ : dt mc
(44)
Substituting convective electric "eld from Eq. (36) and bounce period from Eq. (43), we can rewrite the condition of adiabaticity in form uvx R( ∼ 2 1 ( c
(45)
Because relationship (41) holds as well, the condition of adiabaticity improves during acceleration.
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672
0.08 u/c=0.1 0.07
bounce kinetic energy
0.06
0.05
0.04
0.03
0.02
0.01
0 1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
gamma
Fig. 5. Kinetic energy of bounce oscillations ∼ (vx2 of the relativistic particle with respect to (-relativistic factor for u=c = 0:1: (vx2 reaches maximum value of :08u2 at ( ∼ 1:5, then starts to decrease (Ucer and Shapiro, 2001).
bounces
u/c=0.3
1.9 9 bounces 1.8 8 bounces 1.7 7 bounces
γ
1.6 6 bounces 1.5 5 bounces 1.4 4 bounces 1.3 3 bounces 1.2 1.1 0.3
2 bounces 0.35
0.4
0.45
0.5
0.55
0.6
vx
0.65
0.7
0.75
0.8
0
Fig. 6. The maximum energy ∼ ( that particle gains before it escapes downstream of the shock plotted with respect to its initial velocity in the shock frame vx0 (Ucer and Shapiro, 2001).
It is also signi"cant to mention that the acceleration is more e5cient for pick-up ions which are initially slow with respect to the shock same as in the non-relativistic case. Maximum energy that the particle gains ((max ) before it escapes from the acceleration region is plotted with respect to the initial velocity of the particle in shock frame (vx0 ) in Fig. 6. Final energy can be represented as a multistep function where every step corresponds to a certain number of bounces. The number of bounces as well as the "nal energy increases for initially slower particles, because these particles are trapped in front of the shock for longer time and they correspondingly gain more energy.
An application of sur"ng acceleration of ions to relativistic energies has been discussed by Ohsawa and Sakai (1988a, b). Authors analyzed proton and electron acceleration to relativistic energies by fast magnetosonic waves by means of numerical simulations. The basic motivation for this work is the observations of X- and (-ray emissions from 1982 February 8 solar
4en0 Ex ; k B0
(46)
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673
Fig. 7. Time variation of the two components of momentum (px and py ) of a particle accelerated in front of fast magnetosonic wave by shock sur"ng mechanism. Panels (a) and (b) show those of an ion, while (c) and (d) are for electrons. Taken from (Ohsawa and Sakai, 1988a, b).
√ where k ∼ !=vA and vA = B0 = 4n0 mi is the Alfven velocity and wave frequency is ! ∼ !LH ∼ !ce me =mi is the lower-hybrid frequency. Then the following estimation is obtained for the maximum velocity acquired by shock sur"ng acceleration: mi Ex |vy |max c vA : (47) Bz me A more accurate non-linear analysis, which takes into account the dependence of wave amplitude on Mach number has been done by Ohsawa (1985). The magnitude of the electric "eld at the shock front and the maximum velocity of the particle in this case is mi vA Ex B0 (48) (MA − 1)3=2 ; me c |vy |max ∼
mi vA (MA − 1)3=2 ; me
(49)
where MA is the Alfven Mach number of the shock wave. It follows from Eqs. (48) and (49), even for a weak shock MA ∼ 1 the proton can be accelerated to relativistic energies if the Alfven speed is rather large vA ¿ c me =mi which corresponds to !ce =!pe ¿ 1. To study the time evolution of the acceleration, Ohsawa and Sakai used a fully relativistic one dimensional code with electron to ion mass ratio me =mi = 10−2 , in upstream region
Ti = Te , vTi ∼ :02c, vTe ∼ :2c, !ce =!pe = 3, MA = 2:3, and ion Larmor radius 0:7c=!pe . Results of the simulation are presented in Fig. 7. They show that the electrons and ions are both accelerated to relativistic energies. Electrons acceleration is di6erent from proton acceleration since the electrons cannot be trapped by the shock potential so they cannot undergo sur"ng acceleration. However making many Larmor rotations during the passage of the shock wave, they are accelerated by the electric "eld of the shock. Ions undergo sur"ng acceleration and they escape downstream with |vy | = |vy |max ∼ vA mi =me for MA ∼ 1. Also the motional electric "eld Ey ∼ vA B0 =c. Then we can estimate the time of acceleration simply by mi vymax mi 1 tA ∼ ∼ : (50) eEy !ci me For typical parameters of corona tA ∼ 10−6 s. Typical time of electron acceleration is even shorter than the ion gyro-period. Since the acceleration is very fast for sur"ng, this mechanism can play an essential role in the particle acceleration in the impulsive phase of solar
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shock sur"ng acceleration drops fastly with the increase in the mass of the accelerated ion. This result can easily be explained since for heavier ions, it is easier to overcome potential barrier at the shock front and to escape from acceleration. However observations of anomalous cosmic rays performed by Voyagers 1 and 2 at the termination shock clearly demonstrated the preferential injection of heavier interstellar pick-up ions into acceleration (Cummings and Stone, 1996). In order to apply the mechanism of shock sur"ng to these observations, we shall consider the role of lower-hybrid waves at the shock front during sur"ng acceleration. Such wave turbulence can be either ambient, preexisting (Vaisberg et al., 1982) or self-consistently excited by sur"ng particles. The mechanism of excitation is modi"ed two stream instability caused by a crossed "eld drift of accelerated particles relative to the background of the main solar wind
f( vy)
vymax
v ymin
Fig. 8. The stationary distribution function in vy of the sur"ng particles. Particles starting acceleration at the di6erent times form the “plateau” distribution in the velocity interval (vymax , vymin ). The boundaries of “plateau” are smoothed due to the thermal dispersion of the initial shell distribution of the pick-up ions.
times t0 at which the particle starts to accelerate: 1 f (x; vx ; vy ) = !B
t
t−tA
dt0 f(t0 ; x; vx ; vy );
(51)
where tA = mi vymax =eEy is the total time of acceleration. The particles start acceleration at di6erent times, but the acceleration time is constant due to the fact that vymax is the same for all the particles. It is assumed that particles’ acceleration is "nalized when the Lorentz force towards the shock front exceeds the force from the re
ni ; |vy |max
vymax ¡ vy ¡ vymin :
(52)
It is also interesting to compare this result with the downstream distribution function obtained numerically by Zank et al. (1996) and Lipatov et al. (1998). As it was already mentioned, in the downstream region pitch angle scattering of ions and their gyration around the magnetic "eld lines lead to the isotropization of the distribution function which is strongly stretched in vy direction initially. The distribution function (52) corresponds to the energy spectrum of f(E)=E −1=2 is a reasonable "rst approximation to the spectrum that is obtained by simulations of Lipatov and Zank (1999) described in Section 5. It was mentioned earlier that the lower-hybrid waves are excited by sur"ng particles. These waves are electrostatic (in the short wavelength limit kc!pe ) and they propagate across the magnetic "eld in the direction of the
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sur"ng particles. In a su5ciently dense plasma !pe =!ce 1, 1 lower hybrid waves are quasineutral and their frequency ! ∼ !ce me =mi . Taking into account these conditions the following dispersion relation is obtained for modi"ed two stream instability excited by sur"ng particles: 2 2 !pe !pp 1 @fi 4e2 dv − 2 = ; (53) 2 !ce ! mi k kvy − ! @vy where mi is the ion mass and !pp is the proton plasma frequency. Terms in the left hand side describe lower hybrid waves in hydrodynamical approximation, and term on the right hand side describe the resonant interaction of these waves with sur"ng particles, fi (vy ) is the one dimensional distribution of sur"ng particles, vy is the component of their velocity in the direction of acceleration. On the right side of the distribution function, the number of particles increase with the absolute value of velocity. This side of the distribution is unstable with respect to wave–particle interactions, and the waves with phase speeds !=k ∼ vymin will be excited from this interaction. But this means the particles on this side will give energy to waves, which results with their velocity di6usion towards positive vy . The dispersion relation can be solved by substitution of ! = kvymin + j, where kvymin = !LH , j!LH and j|k|Rv. Then solution of the dispersion relation gives the frequency of the excited waves close to the lower-hybrid frequency: √ ! !LH = !ce !ci (54) and their growth rate as (2 =
! 3 mp ni ; 2n0 |kvymax | mi
(55)
where ni is the density of sur"ng pick-up ions at the shock front. The role of lower-hybrid waves at the shock front is determined by the so called ampli"cation factor Rx - = 0 ( d x=u, where Rx is the amplitude of bounce oscillations, i.e. it determines the region where the sur"ng particles exist. The factor - determines the ampli"cation of the wave amplitude in this region before they are convected to downstream with the plasma
this condition is typical for space plasma, e.g. at the termination shock !pe =!ce ∼ 103 .
675
waves in sur"ng acceleration, it is necessary to construct quasilinear theory describing evolution of the distribution function of sur"ng particles. This calculation can be simpli"ed if it is limited to the consideration of the
d k|Ek |2 .(kvy − !)
@f : @vy
(57)
The di6usive
|Ek |2 @f |vy | @vy
:
(58)
vy ∼vymin
Derivative of the distribution function at vy ∼ vymin can be estimated as @f f : @vy Rv Finite width of the distribution is due to the gradual deceleration of the solar wind caused by mass-loading. At the termination shock the minimum value of Rv is determined by the thermal speed of the interstellar neutrals. The velocity di6usion results in the growth of Rv, eventually extending to positive vy and creating a leak in the distribution function of sur"ng particles. To calculate the total
ni e2 |Ek |2 Rx : 2m2i vymin Rv vymax
(59)
Taking equation for the
(60)
where N is the density of pick-up ions in the incoming solar wind. Then substituting ni into (59) we can rewrite the equation for the total escaping
W NCu n0 m p u 2
mp me
mp mi
14=5 27=10 u u ; (61) max vy Rv
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676
where |ky | |Ek |2 E 2 is used as the average value of the square of the electric "eld and W=
B overshoot
2 !pe E 2 2 !ce 4
is the lower-hybrid wave energy density. In order to make conclusions about the e5ciency of the above described mechanism for scattering of particles from sur"ng acceleration by waves, it is necessary to compare the calculated
5. Numerical simulations of the shock surng acceleration Unfortunately self-consistent kinetic simulations of quasiperpendicular shocks using hybrid code (e.g. Leroy et al., 1982; Quest, 1985; Quest, 1986; Forslund et al., 1984) did not demonstrate any signi"cant acceleration of solar wind pick-up ions. The reason for this is that in hybrid
u
φ downstream
upstream
ramp
foot
Bo
x Fig. 9. Perpendicular shock geometry and "eld orientation as used in the simulations.
simulations shock front is quite broad, with the width significantly exceeding the electron skin depth, while it has been found by several authors (e.g. Lever et al., 2001) that for e5cient shock sur"ng acceleration, the shock front must be su5ciently narrow L ¡ Lcr 0:7c=!pe . Fortunately, observations reported in Scudder et al. (1986), show that narrow quasiperpendicular shocks with L ∼ c=!pe exist in the solar wind. Since it is a serious problem to obtain shock waves in fully kinetic codes, the following scheme is used in simulations. Fig. 9 shows the typical shock structure adapted in simulations. The structure includes the extended foot, ramp with width ∼ c=!pe and overshoot of the magnetic "eld. This structure is followed by extended downstream region of the magnetic "eld. Test particle method is used to analyze particle dynamics at such shocks. This approach creates the possibility to investigate the evolution of the incoming shell distribution of pick-up ions and to look into di6erent mechanisms of acceleration by running the code with di6erent shock and particle parameters. In Fig. 10, two types of ion trajectories obtained by Lever et al. (2001) are shown. Similar results are also obtained in Zilbersher and Gedalin (1997). Top panel shows the trajectory of the sur"ng particle. The formation of the vertical jet indicates that the particle makes multiple re
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677
Fig. 10. Trajectories for two di6erent types of interaction with the shock front. Vertical jet is formed for the shock sur"ng particles corresponds to multiple re
Fig. 11. The maximum velocity acquired by the particle in di6erent acceleration mechanisms plotted as a function of shock ramp width. Di6erent families of curves correspond to di6erent masses. For su5ciently narrow ramp Lcr ¡ 0:7c=!pe , the shock sur"ng leads to signi"cantly larger "nal velocities compared to shock drift acceleration (Lever et al., 2001).
mechanisms is dominant, two quantities are used; the maximum value of acquired velocity during acceleration (vmax ) and the fraction of particles that leave the acceleration region with velocity larger than vth which can be found as 1 E5ciency(%) = f(vy ¿ vth ; vx ; vz ) dv; ni where Ni is the total number of pick-up ions participating in shock sur"ng acceleration. In simulations done by Lever et √ al. (2001), the value of vth = 2 2u is used. Results of this study is presented in Figs. 11 and 12. In Fig. 11 the maximum acquired velocity is plotted with respect to the width of the shock ramp L. For L ¡ Lcrit the dominancy of shock sur"ng acceleration is evident in these "gures. Maximum acquired velocity is signi"cantly higher for sur"ng acceleration as seen in Fig. 11. E5ciency also strongly depends on the ramp width; as the shock ramp widens the e5ciency of shock drift acceleration increases signi"cantly. It is also noted that the for L ¡ Lcuto6 shock drift acceleration is no longer active. As to the mass de-
pendences, both mechanisms prove to be more e5cient for lighter ions. Fig. 13 shows the results of simulations are performed by Lipatov and Zank (1999). We can see the formation of the vertical jet in velocity space which corresponds to the predominant acceleration in the y-direction. The observed jet in vy direction extends up to velocities ∼ 10u, which is more than su5cient to start di6usive shock acceleration. These velocity distributions close to 1D analytical “plateau” in vy , discussed in the previous section. The jet observed in Fig. 13 transforms to a shell distribution by pitch angle di6usion further downstream. Similar velocity distributions of the shock sur"ng particles has been observed in numerical simulations by Zilbersher and Gedalin (1997). The simulations done by Lipatov and Zank (1999) demonstrate another important result such that shock surfing acceleration acts as dissipation mechanism for collisionless shocks. The simulations in this paper for the "rst time takes into account full electron-ion dynamics self-consistently for low -0 plasma quasiperpendicular shocks. Results of these simulations are presented in Fig. 14. This "gure clearly demonstrates formation of a shock transitional layer similar to adapted shock structure in Fig. 9. The presented structure includes a foot in the density of pick-up ions, thin ramp in the magnetic "led with thickness L = 5 × 10−2 rLi , where rLi is the ion Larmor radius. The peak in pick-up ion density corresponds to accelerated ions and it is formed inside the ramp due to the trapping and thus accumulation of pick-up ions. This idea which becomes evident with the recent simulations was also considered by Sagdeev et al. (1984). A qualitative analysis is presented in this paper which discusses how the magnetosonic solitary pulse transverse to the magnetic "eld is transformed to a perpendicular collisionless shock wave where the sur"ng particles create the dissipation mechanism. Another important outcome of the simulations is the information about the spectrum of the accelerated particles.
' ( er / Planetary and Space Science 51 (2003) 665 – 680 V.D. Shapiro, D. Uc
678
10
S
6
Mass = 1 Mass = 2 Mass = 3 Mass = 4
4
vthresh = 2√2
Efficiency (%)
8
12
8
Lcutoff
D
4 2
(a)
0 0.2
0.5
1.0
1.5
0 2.0 (b) 0.2
0.5
1.0
1.5
2.0
15
15
10
10 Vper2/Uo
Vper2/Uo
Fig. 12. E5ciency of the acceleration mechanism as a function of ramp width. Panel (a) shows the e5ciency for sur"ng particles with di6erent masses, and panel (b) shows the e5ciency for drifting particles. The sur"ng acceleration e5ciency increases signi"cantly for narrow ramp widths (Lever et al., 2001).
5 0 -5
5 0 -5
-10
-10
-15
-15
-15 -10 -5 0 5 Vper1/Uo
10 15
-15 -10 -5 0 5 Vper1/Uo
10 15
Fig. 13. The projection of the pick-up ion velocity distribution at the perpendicular shock on two directions orthogonal to the magnetic "eld (Lipatov and Zank, 1999). Right panel shows the vertical jet formed in the upstream region, left panel shows the downstream region where pitch angle di6usion is important.
6. summary As a summary we can state that both analytical and numerical analysis identify the shock sur"ng acceleration as an ideal mechanism for the injection of pick-up ions into diffusive shock acceleration at the quasiperpendicular shocks. This mechanism favors slow pick-up ions and accelerates them up to the threshold velocity needed to start di6usive acceleration. At relativistic energies the main limitation of shock sur"ng acceleration is eliminated, which means particle can undergo theoretically unlimited acceleration. Surfing acceleration of protons to relativistic energies by fast magnetosonic shocks in solar
The spectrum for shock sur"ng acceleration turns out to be
Acknowledgements V.D.S. would like to express deep gratitude to M.A. Lee with whom he discussed many of the ideas presented in this paper countless times.
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References Axford, W.I., Leer, E., Skadron, G., 1977. The acceleration of cosmic rays by shock waves. Proc. Int. Conf. Cosmic Rays 15th, 11, 132. Bell, A.R., 1978. Acceleration of cosmic rays in shock fronts. J. Mon. Not. R. Astron. Soc. 182, 147. Blanford, R.D., Ostriker, J.P., 1978. Particle acceleration by astrophysical shocks. Astrophys. J. 221, L29. Bogdan, T.J., Lee, M.A., Schneider, P., 1991. Coupled quasi-linear wave damping and stochastic acceleration of pick-up ions in the solar wind. J. Geophys. Res. 96, 161. Coates, A.J., Johnstone, A.D., Wilken, B., Jockers, K., Glassmeier, K.H., 1989. Velocity space di6usion of pick-up ions from the water group at comet Halley. J. Geophys. Res. 94, 9983 (Correction J. Geophys. Res. 95, 4343, 1990). Cummings, A.C., Stone, E.C., 1996. Composition of anomalous cosmic rays and implications to heliosphere. Space Sci. Rev. 78, 117. Decker, R.B., 1981. The modulation of low energy proton distribution by propagating interplanatery shock waves. J. Geophys. Res. 86, 4537. Edmiston, J.P., Kennel, C.F., Eichler, D., 1982. Escape of heated ions upstream of quasi-parallel shock. Geophys. Res. Lett. 9, 531. Forslund, D.W., Quest, K.B., Blackhill, J.U., Lee, K., 1984. J. Geophys. Res. 89, 2142. Galeev, A.A., Sagdeev, R.Z., 1988. Alfven waves in space plasma and its role in the solar wind interaction with comets. Astrophys. Space Sci. 144, 427. Galeev, A.A., Polydov, A.N., Sagdeev, R.Z., Shapiro, V.D., Shevchenko, V.I., Szego, K., 1987a. MHD turbulence caused by a comet in the solar wind. Sov. Phys. JETP, Engl. trans. 65, 1178. Galeev, A.A, Sagdeev, R.Z., Shapiro, V.D., Shevchenko, V.I., Szego, K., 1987b. MHD turbulence in a mass-loaded solar wind and the cometary bow-shock. The similarity and the di6erence with the cosmic ray shocks. Proceedings of the International Conference on Collisionless shocks, Szego, K. (Ed.), Centr. Res. Inst. for Physics, Budapest, Hungary, p. 237. Gloeckler, G., Geiss, J., Roelof, E.C., Fisk, L.A., Ipavich, F.M., Ogilvie, K.W., Lanzerotti, L.J., von Steiger, R., Wilken, B., 1994. Acceleration of interstellar pickup ions in the disturbed solar wind observed on Ulysses. J. Geophys. Res. 99 (17), 637. Goldstein, H., 1980. Classical Mechanics, 2nd Edition. Addison-Wesley, Reading, MA, p. 531. Hoshino, M., Shimada, N., 2002. Nonthermal Electrons at High Mach Number Shocks: Electron Shock Sur"ng Acceleration, Astrophys. J. 572, 880. Hudson, P.D., 1965. Re
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Lembege, B., Dawson, J.M., 1984. Plasma heating and acceleration by strong magnetosonic waves propagating obliquely to a magnetosonic "eld. Phys. Rev. Lett. 53, 1053. Lembege, B., Ratli6, S.T., Dawson, J.M., Ohsawa, Y., 1983. Ion heating and acceleration by strong magnetosonic waves. Phys. Rev.Lett. 51, 264. Leroy, M.M., Winske, D., Goodrich, C.C., Wu, C.S., Papadopoulos, K., 1982. The structure of perpendicular bow shocks. J. Geophys. Res. 87, 5081. Lever, E.L., Quest, K.B., Shapiro, V.D., 2001. Shock sur"ng vs. shock drift acceleration. Geophys. Res. Lett. 28, 1367. Lipatov, A.S., Zank, G.P., 1999. Pick-up ion acceleration at low 0p perpendicular shocks. Phys. Rev. Lett. 84, 3609. Lipatov, A.S., Zank, G.P., Pauls, H.L., 1998. The acceleration of pick-up ions at shock waves: Test particle-mesh simulations. J. Geophys. Res. 103 (29), 679. Ohsawa, Y., 1985. Strong ion acceleration by a collisionless magnetosonic wave propagating perpendicularly to a magnetic "eld. Phys.Fluids 28, 2130. Ohsawa, Y., 1986. Resonant ion acceleration by a oblique magnetosonic shock wave in a collisionless plasma. Phys. Fluids 29, 773. Ohsawa, Y., 1990. Conditions for ion re
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Shapiro, V.D., Lee, M.A., Quest, K.B., 2001. The role of lower hybrid turbulence in sur"ng acceleration at perpendicular shocks. J. Geophys. Res. 106 (25), 023. Shimada, N., Hoshino, M., 2000. Strong electron acceleration at high Mach number shock waves: simulation study and electron dynamics. Astrophys. J. 543, L67. Somogyi, A.J., et al., 1986. First observations of energetic particles near comet Halley. Nature 321, 285. Srivastava, R.M., Tsurutani, B.T., Goldstein, B.E., Sharma, V., Okuda, M., 1993. Acceleration of cometary H2 O group pick-up ions by obliquely propagating nonlinear magnetosonic waves. J. Geophys. Res. 98 (21), 023. Sugihara, R., Mitzuna, Y., 1979. Non-stochastic heating of a magnetized plasma by electrostatic wave. J. Phys. Soc. Japan 47, 1290. Sugihara, R., Takeuchi, E., Sakai, K., Matzumoto, M., 1984. DC acceleration of charged particles by an eelctrostatic wave propagating obliquely to a magnetic "eld. Phys. Rev. Lett. 52, 1500. Ucer, D., Shapiro, V.D., 2001. Unlimited shock sur"ng acceleration. Phys. Rev. Lett. 87 (7), 075001. Vaisberg, O.L., Galeev, A.A., Nozdrachev, M., Sagdeev, R.Z., Shapiro, V.D., 1982. Electron heating at the front of a strong collisionless shock wave. Sov. Phys. JETP 84, 1232.
Vasyliunas, V.M., Siscoe, G.L., 1976. On the
Planetary and Space Science 51 (2003) 665 – 680 www.elsevier.com/locate/pss
Shock sur"ng acceleration ( cer Vitali D. Shapiro∗ , Defne U* University of California, San Diego, Physics Department, La Jolla CA 92093, USA Received 19 February 2003; accepted 19 May 2003
Abstract Analytical and numerical analysis identify shock sur"ng acceleration as an ideal pre-energization mechanism for the slow pick-up ions at quasiperpendicular shocks. After gaining su5cient energy by shock sur"ng, pick-up ions undergo di6usive acceleration to reach their observed energies. Energetic ions upstream of the cometary bow shock, acceleration of solar energetic particles by magnetosonic waves in corona, ion enhancement in interplanetary shocks, generation of anomalous cosmic rays from interstellar pick-up ions at the termination shock are some of the cases where shock sur"ng acceleration apply. Inclusion of the lower-hybrid wave turbulence into the laminar model of shock sur"ng can explain the preferential acceleration of heavier particles as observed by Voyager at the termination shock. At relativistic energies, unlimited acceleration of ions is theoretically possible; because for su5ciently strong shocks main limitation of the mechanism, caused by the escape of accelerated particles downstream of the shock during acceleration no longer exists. ? 2003 Published by Elsevier Ltd. Keywords: Shock; Wave turbulence; Accelerated particles
1. Introduction A large fraction of particle energization in astrophysical environments comes from the acceleration by shock waves. In the interstellar medium, strong shocks are formed during the mass out
Corresponding author. E-mail address: [email protected] (V.D. Shapiro).
0032-0633/$ - see front matter ? 2003 Published by Elsevier Ltd. doi:10.1016/S0032-0633(03)00102-8
leads to a shell distribution centered approximately at the solar wind velocity. The distribution has a radius of the order of solar wind speed vsw (Sagdeev et al., 1986; Lee and Ip, 1987). If the excited
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acceleration (Axford et al., 1977; Krymsky, 1977; Blanford and Ostriker, 1978; Bell, 1978). However this type of acceleration is not considered to be e6ective in quasiperpendicular shocks. In order to start di6usive shock acceleration, the accelerated particles must traverse the shock front in both directions many times. This condition is easily ful"lled at quasiparallel shocks, since the ambient magnetic "eld is parallel to the shock normal, and the ions can move freely along the magnetic "eld. This is especially true in the case of supercritical shocks (Edmiston et al., 1982; Kennel et al., 1985), where downstream heating helps ions to return to the upstream of the shock. In the case of quasi-perpendicular shocks, the particles need to be e6ectively scattered perpendicular to the ambient magnetic "eld. This is why, signi"cantly large initial velocity is needed for ions to start di6usive shock acceleration (Webb et al., 1995). In this reference it is shown that the threshold velocity to initiate di6usive shock acceleration in quasi-perpendicular shocks is vth =
3u (1 + 2 )1=2 ; r
(1)
where r is the shock compression ratio and = =rg where is the scattering mean free path along the magnetic "eld and rg is the ion gyroradius. If we assume that scattering is weak (rg ) and make a reasonable assumption that is proportional to rg , then the threshold velocity is independent of particle charge and mass and several times larger than the solar wind velocity vsw . This threshold is generally not achieved by any of pick-up ions in their incident velocity distribution, thus the observed energies in the all the previously mentioned cases of quasiperpendicular shocks are quite puzzling. The puzzle can be solved if a pre-acceleration mechanism at quasiperpendicular shocks is proposed which favors slow pick-up ions and accelerates them to the threshold velocity needed for the di6usive shock acceleration. Such mechanism of ion acceleration has been proposed long time ago by Sagdeev (1966). The idea can be explained in the following way (see also Sagdeev and Shapiro, 1973): Let us consider a periodic electrostatic wave propagating in the x-direction with phase velocity vph . The particles that have velocities close to the phase velocity of the wave can be trapped by the electrostatic potential well of the wave. Because the particle is in a symmetric well, the particle has no net energy gain. In the case when the ambient magnetic "eld is present and points in the z-direction (see caption for Fig. 1), the symmetry is broken and the particle starts to spend more time in the left half of the wave because of the additional Lorentz force. The particle is accelerated under the action of the y-component of the Lorentz force −evx B0 =c = −evph B0 =c (since the average value of vx is the phase velocity of the wave). The acceleration goes on until the Lorentz force (˙ vy ) is large enough such that the particle can no longer be re
Φ
x
Fig. 1. The sur"ng acceleration by a periodic electrostatic wave 0 (x) propagating across the magnetic "eld B0 ez . The trapped particles are moving in the x-direction with the wave phase velocity !=k and are accelerated along the wave front in the y-direction under the action of the Lorentz force Fy = −(e=c)vx B0 . The x-component of the Lorentz force Fx = (e=c)vy B0 pushes trapped ions to the left side of the potential well. Particles oscillating around non-zero wave phase gain energy from the wave.
Since the trajectory of the accelerated particles is such that they do multiple re
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u
y +
x
B Ey
Fig. 2. Schematic diagram for shock sur"ng acceleration at a perpendicular shock.
hancements at the interplanetary traveling shocks (Sarris and Van Allen, 1974; Pesses et al., 1979, 1982; Decker, 1981). In the shock drift acceleration mechanism, ion energy gain is due to ion curvature and gradient drifts in the inhomogeneous magnetic "eld at the shock front. Although the shock sur"ng acceleration relies on a strong shock potential and shock drift acceleration does not, the most important distinction between these two mechanisms is the fact that the shock drift acceleration energy gain is proportional to the initial ion energy, while the "nal energy in the shock sur"ng is greater for ions which have lower incident velocities initially (|vx0 |u where u is the shock speed). This happens because the particles with smaller initial |vx | are trapped longer at the shock front, while they are accelerated by the convective electric "eld. By this reason, shock surfing is an ideal pre-acceleration mechanism for slow pick-up ions with |v|vsw . The original idea of Sagdeev (1966) has been further developed by many authors. Sagdeev and Shapiro (1973) has considered dissipation mechanisms of non-Landau type for periodic electrostatic wave propagating across magnetic "eld considering the scenario where the sur"ng particles dissipate the energy of the wave. They also emphasized limitation on sur"ng acceleration that vy ¡ cEx =B0 , where Ex is the maximum electric "eld at the front. Sugihara and Mitzuna (1979) applied the idea of sur"ng acceleration to ion acceleration in laser fusion plasma. Katsouleas and Dawson (1983) paid attention on the analogy of this type of acceleration with a surfer’s motion and they generalized the mechanism for relativistic particles. They also showed that necessary condition for unlimited acceleration is Ex ¿ B0 . Ohsawa (1990) classi"ed the ion trajectories into those which are re
667
(1984) using both analytical calculations and numerics generalized the mechanism for quasiperpendicular periodic wave with Bx Bz and showed that acceleration in y-direction can be signi"cantly larger than that in strictly perpendicular case. This is possible by allowing ion acceleration in two directions. Using simulations, Lembege et al. (1983) pointed out that the sur"ng acceleration is an e6ective dissipation mechanism for magnetosonic wave. Sagdeev et al. (1984) have been the "rst who considered shock sur"ng acceleration at a strictly perpendicular magnetosonic solitary pulse and derived the resulting dissipation which transforms the pulse into a perpendicular shock wave. Ohsawa (1985, 1986) and Ohsawa and Sakai (1985) used simulations to investigate the formation of self-consistent shock structure from sur"ng acceleration. The idea of shock sur"ng acceleration has been applied to the acceleration of solar energetic particles by fast magnetosonic shocks in the corona (Ohsawa and Sakai, 1987), He3 ions in impulsive
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by Gloeckler et al. (1994) which indicated the acceleration e5ciency is larger for lighter particles. In order to explain the recent observations of Voyager, Shapiro et al. (2001) suggested the inclusion of lower-hybrid wave turbulence which is excited by sur"ng particles at the termination shock. Waves are responsible for particle velocity di6usion in the direction opposite to acceleration which results in detrapping of particles from the acceleration region. Since velocity di6usion is more e5cient for lighter particles, acceleration is more e5cient for heavier ions. Also Zank et al. (2001) pointed out the mass dependence of the threshold velocity vth to start di6usive shock acceleration (1). These authors "nd that vth decreases with increasing mass of the accelerated ions which means heavier ions reach the threshold velocity faster and this fact presents another possible explanation to preferential acceleration of heavy ions at the termination shocks. A numerical analysis of the shock sur"ng acceleration up to the relativistic energies has been done by Ucer and Shapiro (2001). Main limitation on sur"ng acceleration is due the fact that the acceleration along the shock front is accompanied by the growth of the energy of the bounce oscillations and "nally the particle overcomes the potential barrier in the shock front and escapes downstream. Ucer and Shapiro (2001) show that this limitation is absent at the relativistic energies, in fact the bounce energy decreases during acceleration for vy ∼ c. This brings the possibility for unlimited sur"ng acceleration of relativistic particles.
2. Basic equations We consider a plane quasi-perpendicular shock moving with u = uex . In the reference frame of the shock as shown in Fig. 2, shock is stationary and shock normal is in positive x-direction. Shock potential is (x) and upstream (x ¿ 0) magnetic "eld B = Bx ex + Bz ez with Bx ¿ 0. The plane containing vectors u and B is called the coplanar plane. By , which is the noncoplanar component, is only present within the shock front where @=@x = 0. In the frame of the shock, the non-relativistic ion equations of motion are: mi
dvx d e = −e + (vy Bz − vz By ); dt dx c
(2)
mi
dvy e = eEy + (vz Bx − vx Bz ); dt c
(3)
dvz e = eEz + (vx By − vy Bx ); mi dt c
(4)
where mi is the ion mass, e is the ion charge, E and B is the electric "eld in the shock frame. Because shock is stationary, ∇ × E = 0 and therefore Ey and Ez do not depend on x. Since the electric "eld in lab frame is zero far away from the shock E = 0, the electric
"eld in the frame moving with the shock is E=
1 u×B c
(5)
which yields Ey = −uBz =c and Ez = uBy =c. As mentioned before the noncoplanar magnetic "eld By is created only inside the shock (Jones and Ellison, 1987). This component arises from the condition that the electrons and ions should receive equal increments of velocity into z-direction passing the shock. Because otherwise there would be "nite current and this would lead to the building up of magnetic "eld By in the downstream region, and this would be against the Rankine–Hugoniot conditions. Also by using the fact that the current is dominated by electron contribution inside the shock region, the authors Jones and Ellison obtain the following relationship for By . c By d x = Bx RBz ; (6) 4en0 u where RBz is the increase of Bz inside the shock layer and n0 is electron or proton number density upstream of the shock. If L is the characteristic space scale of the shock layer then Eq. (6) can be estimated as By ∼
Bx c RBz : 4en0 u L
(7)
We will go over the analysis of ion dynamics in the normal incidence frame as shown in Fig. 2. This frame is often close to the frame in which spacecraft observations are made. In this reference frame ion equations of motion are: e d dvx =− + vy z (x) − vz y (x); dt m dx
(8)
dvy = −(u + vx ) z (x) + vz x ; dt
(9)
dvz = vx y (x) − vy x ; dt
(10)
where x = eBx =mc, y = eBy (x)=mc and z = eBz =mc. Bx is constant since ∇ · B = 0. To illustrate the creation of the e6ective potential in which accelerated particle is trapped, let’s multiply Eq. (8) by mvx . Then we can write d e [x + e(x)] = B0 vy vx ; dt c
(11)
where x = mvx2 =2 is the energy of bounce oscillations in the direction of the shock normal. Suppose we are considering the particle acceleration being adiabatic, 1 dvy !B ; vy dt
(12)
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the velocity of bounce oscillations increase accompanying the acceleration in the y-direction but with a slower rate. vx ˙ |vy |1=3 :
(17)
Effective Potential
Particle remains in the potential well and continues acceleration until it gains su5cient bounce kinetic energy (x ) to overcome the potential barrier in front of the shock. This corresponds to a change in vy by a factor of 3=2 vy 2e0 ; (18) 2 vy0 mi vx0 vy1
X
Fig. 3. Schematic representation of the e6ective potential well in which sur"ng particle is bouncing at three di6erent times t1 ¡ t2 ¡ t3 .
where !B is the frequency of bounce oscillations. Then we can integrate Eq. (11) and obtain the e6ective potential: e Ue6 = e + B0 |vy |x; (13) c where we also took into account that vy ¡ 0 for the geometry shown in Fig. 2. The "rst term in Ue6 is electrostatic potential at the shock front, second term is the input from the component of the Lorentz force in the x-direction which creates the turning point for ions which move away from the shock. The e6ective potential is shown in Fig. 3. Balancing the kinetic energy of bounce oscillations with the second term in Ue6 we can obtain the following estimation for the amplitude of bounce oscillations: Rx ∼
vx2 : !c |vy |
(14)
Then the period of bounce oscillations can be estimated as !B =
669
2 Rx vx : ∼ ∼ !B vx !c |vy |
(15)
During acceleration, e6ective potential, which traps the particle, changes as it is shown in the "gure. With further acceleration, Ue6 get steeper and the amplitude of bounce oscillations decreases. The corresponding energy of bounce oscillations changes as well. If condition of adiabaticity of acceleration (12) is ful"lled, the longitudinal adiabatic invariant should be conserved (Goldstein, 1980). vx d x = const; (16) where integration is taken over a closed trajectory between two turning points. We can obtain an estimation for the relation between vx and vy by substituting the amplitude of bounce oscillations Rx from (14) into (16). This yields that
where vx0 , vy0 -initial components of particle velocity. For initially slow particles, this factor is greater. Particle can also escape from the accelerated region to downstream of the shock by a strong Lorentz force. Lorentz force is pointing towards the shock front and has a detrapping e6ect. In particular, the particle is not trapped at all if dUe6 =d x ¿ 0 in the region x ¿ 0. This occurs if the Lorentz force exceeds the electric force in the shock layer. In the presence of the noncoplanar component of the magnetic "eld By this condition may be written as 1 (Bz |vy | + |By vz |) ¿ Ex ; c
(19)
where Ex ∼ 0 =L is the electrostatic "eld in the shock layer and L is the shock width. For a strictly perpendicular shock Bx = 0, (i.e. By = 0, following (7)) this condition leads to the most e6ective acceleration such that |vy | c
Ex : Bz
(20)
Together with (18), this condition provides an upper limit for particle acceleration at the shock front. The maximum acceleration factor is determined by minimum of these two estimates (18) and (20). At a quasi-perpendicular shock, with the presence of Bx , particle also acquires a possibility to escape upstream. This happens when the particle gains a positive value of vy and the Lorentz force in the x-direction starts to push the particle away from the shock. We can see this e6ect if we average Eqs. (9) and (10) over period of fast bounce oscillations. Then the oscillatory terms proportional to vx vanish and we are left with d 2 vy + x2 vy = 0; dt 2
(21)
d 2 vz + x2 vz = u x z ; dt 2
(22)
where x = eBx =mc and z = eBz =mc. Assuming that initial components of particle velocity in the shock plane satisfy the condition Bz |vy0 |; |vz0 |u ; Bx
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670
we can write the solution of Eqs. (21) and (22) in the form vy = −u(Bz =Bx )sin x t;
(23)
vz = u(Bz =Bx )(1 − cos x t):
(24)
vz ∆v
Particle escapes from the shock at x t = , when velocity vy changes sign. Therefore if particle does not satisfy conditions (18) and (20) during acceleration, it will escape upstream with the energy =
mvz2 2
= 2mu2 (Bz =Bx )2 ;
vy = vy0 + t (dvy =dt); where t = 0 is the start of the bounce. For the steep potential we can integrate Eq. (11) to evaluate the change in the energy of bounce oscillations x : dvy !B Rx = mi !c dt t vx ; (26) dt 0 where x =mi vx2 =2, and in adiabatic regime we have dvy =dt const during one bounce. The constant term proportional to vy0 vanishes in Eq. (26) because the integral is taken over the entire bounce period. Also for the steep potential, we neglected the term which is proportional to @=@x because it is insigni"cant for the main part of the bounce motion in front shock. We also can integrate Eq. (8) for the motion in the x-direction in the case of strictly perpendicular shock and we obtain (27)
where vx0 and vy0 are the velocity components at the start of the bounce. Since at t = !B =2, there is a re
2 vx0 : !c |vy0 |
(28)
Integrating Eq. (26), we obtain the following relation: Rx 1 x dy dx = ; !B dt 3 y dt
ϕ
(25)
where vy = 0, following (23), and vx = 0 as well because it is directly related to vy by (17). Ion dynamics can be easily analyzed in the case of vanishing shock width, when particle is re
vx = vx0 − |vy0 |!c t ;
vx
θ
u
(29)
where y = mvy2 =2. This relation has been derived only considering the fact that the acceleration is slow with
vy Fig. 4. Shell distribution of pick-up ions in the normal incidence frame of the shock. Ions inside the shaded region are accelerated above the threshold velocity for the di6usive shock acceleration.
respect to the bounce motion. Integration of Eq. (29) yields vx3 =vy = const which is equivalent to the conservation of adiabatic integral discussed earlier. Another important characteristic of shock sur"ng acceleration is to show that it can act as a preacceleration mechanism for di6usive shock acceleration at quasiperpendicular shocks. We will adapt the idea in Lee et al. (1996) to calculate the
vy0 = u sin ":
Acceleration is "nalized when particle overcomes the shock 2 potential with mi vxf =2 emax . It is reasonable to assume that the shock is strong enough to re
8u "5
mp mi
3=2 :
(30)
It was mentioned in the introduction that, at quasiperpendicular shocks, in order to start di6usive acceleration particles should have a threshold velocity given by (1). In order the
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671
sur"ng acceleration be e6ective as an preacceleration mechanism for di6usive shock acceleration |vy |max ¿ vth or in other words: 3=2 8u mp 5 5 : (31) " 6 "0 = vth mi
Therefore the e6ective potential for the relativistic case is
Then the
Rx
Nu 8
0
2
"0 d’
d""3 :
(32)
Taking the integral and substituting "0 from (31), the following expression is obtained for the
3. Shock Surng Acceleration at the relativistic energies An important peculiarity of sur"ng particle dynamics at the relativistic energies has been discussed by authors of the present review (Ucer and Shapiro, 2001). In relativistic case for strictly perpendicular shock (Bx = 0,By = 0) equations of motion are reduced to the following two dimensional set in shock frame.
Ue6 = e + e(u B0 x:
(38)
Now balancing the bounce kinetic energy with Ue6 we can estimate the amplitude of bounce oscillations as (vx2 : (u ! c c
(39)
Adiabatic invariant in the relativistic case takes the form px d x = (vx d x = const (40) and using estimation (39) for Rx, it is easy to see that during acceleration (with the growth of relativistic factor () velocity of bounce oscillations decreases as vx ˙
1 (2=3
:
(41)
The kinetic energy of bounce oscillations decreases during the acceleration as x ˙ (vx2 ∼ (−1=3 as it is illustrated by Fig. 5 obtained by numerical solution of Eqs. (34) and (35). It follows from this "gure that for a su5ciently large electrostatic barrier at the shock front e ¿ max ∼ 0:08 mu2 , the particle cannot overcome the potential barrier at the shock front, and it continues acceleration theoretically for an in"nitely long time. For the particle to stay in the acceleration region, as it was discussed for the non-relativistic case, the Lorentz force pushing the particle towards the shock should never exceed the repulsive force due to the shock electric "eld Ex ¿ (u B0 . The preceding analysis applies when the condition of adiabaticity of acceleration is ful"lled, in other words the typical time for acceleration is signi"cantly larger than the bounce period:
d @ e + vy (u B0 ; ((mvx ) = −e dt @x c
(34)
R( 1 (
e e d (mvy = − (u uB0 − vx (u B0 : dt c c
(35)
where R( is the change in the relativistic factor during one bounce period !B .
Here ( = (1 − v2 =c2 )−1=2 is relativistic factor. Magnetic "eld in the shock frame is B = (u B0 ez , where B0 is the ambient magnetic "eld and (u = (1 − u2 =c2 )−1=2 . Convective electric "eld in the shock frame is u E = − (u B0 ey : (36) c We assume that the particle velocity along the shock front is of the order of speed of light (vy ∼ c). Then the relativistic factor is ( ∼ (1 − vy2 =c2 )−1=2 . Multiplying Eq. (34) with vx we obtain the following equation similar to (11) in the non-relativistic case: d (mvx2 + e(x) = e(u B0 vx : (37) dt 2
!B ∼
Rx (vx : ∼ vx (u !c c
(42)
(43)
The growth of the relativistic particle energy under the action of the convective electric "eld Ey can be written as eEy d( ∼ : dt mc
(44)
Substituting convective electric "eld from Eq. (36) and bounce period from Eq. (43), we can rewrite the condition of adiabaticity in form uvx R( ∼ 2 1 ( c
(45)
Because relationship (41) holds as well, the condition of adiabaticity improves during acceleration.
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672
0.08 u/c=0.1 0.07
bounce kinetic energy
0.06
0.05
0.04
0.03
0.02
0.01
0 1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
gamma
Fig. 5. Kinetic energy of bounce oscillations ∼ (vx2 of the relativistic particle with respect to (-relativistic factor for u=c = 0:1: (vx2 reaches maximum value of :08u2 at ( ∼ 1:5, then starts to decrease (Ucer and Shapiro, 2001).
bounces
u/c=0.3
1.9 9 bounces 1.8 8 bounces 1.7 7 bounces
γ
1.6 6 bounces 1.5 5 bounces 1.4 4 bounces 1.3 3 bounces 1.2 1.1 0.3
2 bounces 0.35
0.4
0.45
0.5
0.55
0.6
vx
0.65
0.7
0.75
0.8
0
Fig. 6. The maximum energy ∼ ( that particle gains before it escapes downstream of the shock plotted with respect to its initial velocity in the shock frame vx0 (Ucer and Shapiro, 2001).
It is also signi"cant to mention that the acceleration is more e5cient for pick-up ions which are initially slow with respect to the shock same as in the non-relativistic case. Maximum energy that the particle gains ((max ) before it escapes from the acceleration region is plotted with respect to the initial velocity of the particle in shock frame (vx0 ) in Fig. 6. Final energy can be represented as a multistep function where every step corresponds to a certain number of bounces. The number of bounces as well as the "nal energy increases for initially slower particles, because these particles are trapped in front of the shock for longer time and they correspondingly gain more energy.
An application of sur"ng acceleration of ions to relativistic energies has been discussed by Ohsawa and Sakai (1988a, b). Authors analyzed proton and electron acceleration to relativistic energies by fast magnetosonic waves by means of numerical simulations. The basic motivation for this work is the observations of X- and (-ray emissions from 1982 February 8 solar
4en0 Ex ; k B0
(46)
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673
Fig. 7. Time variation of the two components of momentum (px and py ) of a particle accelerated in front of fast magnetosonic wave by shock sur"ng mechanism. Panels (a) and (b) show those of an ion, while (c) and (d) are for electrons. Taken from (Ohsawa and Sakai, 1988a, b).
√ where k ∼ !=vA and vA = B0 = 4n0 mi is the Alfven velocity and wave frequency is ! ∼ !LH ∼ !ce me =mi is the lower-hybrid frequency. Then the following estimation is obtained for the maximum velocity acquired by shock sur"ng acceleration: mi Ex |vy |max c vA : (47) Bz me A more accurate non-linear analysis, which takes into account the dependence of wave amplitude on Mach number has been done by Ohsawa (1985). The magnitude of the electric "eld at the shock front and the maximum velocity of the particle in this case is mi vA Ex B0 (48) (MA − 1)3=2 ; me c |vy |max ∼
mi vA (MA − 1)3=2 ; me
(49)
where MA is the Alfven Mach number of the shock wave. It follows from Eqs. (48) and (49), even for a weak shock MA ∼ 1 the proton can be accelerated to relativistic energies if the Alfven speed is rather large vA ¿ c me =mi which corresponds to !ce =!pe ¿ 1. To study the time evolution of the acceleration, Ohsawa and Sakai used a fully relativistic one dimensional code with electron to ion mass ratio me =mi = 10−2 , in upstream region
Ti = Te , vTi ∼ :02c, vTe ∼ :2c, !ce =!pe = 3, MA = 2:3, and ion Larmor radius 0:7c=!pe . Results of the simulation are presented in Fig. 7. They show that the electrons and ions are both accelerated to relativistic energies. Electrons acceleration is di6erent from proton acceleration since the electrons cannot be trapped by the shock potential so they cannot undergo sur"ng acceleration. However making many Larmor rotations during the passage of the shock wave, they are accelerated by the electric "eld of the shock. Ions undergo sur"ng acceleration and they escape downstream with |vy | = |vy |max ∼ vA mi =me for MA ∼ 1. Also the motional electric "eld Ey ∼ vA B0 =c. Then we can estimate the time of acceleration simply by mi vymax mi 1 tA ∼ ∼ : (50) eEy !ci me For typical parameters of corona tA ∼ 10−6 s. Typical time of electron acceleration is even shorter than the ion gyro-period. Since the acceleration is very fast for sur"ng, this mechanism can play an essential role in the particle acceleration in the impulsive phase of solar
674
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shock sur"ng acceleration drops fastly with the increase in the mass of the accelerated ion. This result can easily be explained since for heavier ions, it is easier to overcome potential barrier at the shock front and to escape from acceleration. However observations of anomalous cosmic rays performed by Voyagers 1 and 2 at the termination shock clearly demonstrated the preferential injection of heavier interstellar pick-up ions into acceleration (Cummings and Stone, 1996). In order to apply the mechanism of shock sur"ng to these observations, we shall consider the role of lower-hybrid waves at the shock front during sur"ng acceleration. Such wave turbulence can be either ambient, preexisting (Vaisberg et al., 1982) or self-consistently excited by sur"ng particles. The mechanism of excitation is modi"ed two stream instability caused by a crossed "eld drift of accelerated particles relative to the background of the main solar wind
f( vy)
vymax
v ymin
Fig. 8. The stationary distribution function in vy of the sur"ng particles. Particles starting acceleration at the di6erent times form the “plateau” distribution in the velocity interval (vymax , vymin ). The boundaries of “plateau” are smoothed due to the thermal dispersion of the initial shell distribution of the pick-up ions.
times t0 at which the particle starts to accelerate: 1 f (x; vx ; vy ) = !B
t
t−tA
dt0 f(t0 ; x; vx ; vy );
(51)
where tA = mi vymax =eEy is the total time of acceleration. The particles start acceleration at di6erent times, but the acceleration time is constant due to the fact that vymax is the same for all the particles. It is assumed that particles’ acceleration is "nalized when the Lorentz force towards the shock front exceeds the force from the re
ni ; |vy |max
vymax ¡ vy ¡ vymin :
(52)
It is also interesting to compare this result with the downstream distribution function obtained numerically by Zank et al. (1996) and Lipatov et al. (1998). As it was already mentioned, in the downstream region pitch angle scattering of ions and their gyration around the magnetic "eld lines lead to the isotropization of the distribution function which is strongly stretched in vy direction initially. The distribution function (52) corresponds to the energy spectrum of f(E)=E −1=2 is a reasonable "rst approximation to the spectrum that is obtained by simulations of Lipatov and Zank (1999) described in Section 5. It was mentioned earlier that the lower-hybrid waves are excited by sur"ng particles. These waves are electrostatic (in the short wavelength limit kc!pe ) and they propagate across the magnetic "eld in the direction of the
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sur"ng particles. In a su5ciently dense plasma !pe =!ce 1, 1 lower hybrid waves are quasineutral and their frequency ! ∼ !ce me =mi . Taking into account these conditions the following dispersion relation is obtained for modi"ed two stream instability excited by sur"ng particles: 2 2 !pe !pp 1 @fi 4e2 dv − 2 = ; (53) 2 !ce ! mi k kvy − ! @vy where mi is the ion mass and !pp is the proton plasma frequency. Terms in the left hand side describe lower hybrid waves in hydrodynamical approximation, and term on the right hand side describe the resonant interaction of these waves with sur"ng particles, fi (vy ) is the one dimensional distribution of sur"ng particles, vy is the component of their velocity in the direction of acceleration. On the right side of the distribution function, the number of particles increase with the absolute value of velocity. This side of the distribution is unstable with respect to wave–particle interactions, and the waves with phase speeds !=k ∼ vymin will be excited from this interaction. But this means the particles on this side will give energy to waves, which results with their velocity di6usion towards positive vy . The dispersion relation can be solved by substitution of ! = kvymin + j, where kvymin = !LH , j!LH and j|k|Rv. Then solution of the dispersion relation gives the frequency of the excited waves close to the lower-hybrid frequency: √ ! !LH = !ce !ci (54) and their growth rate as (2 =
! 3 mp ni ; 2n0 |kvymax | mi
(55)
where ni is the density of sur"ng pick-up ions at the shock front. The role of lower-hybrid waves at the shock front is determined by the so called ampli"cation factor Rx - = 0 ( d x=u, where Rx is the amplitude of bounce oscillations, i.e. it determines the region where the sur"ng particles exist. The factor - determines the ampli"cation of the wave amplitude in this region before they are convected to downstream with the plasma
this condition is typical for space plasma, e.g. at the termination shock !pe =!ce ∼ 103 .
675
waves in sur"ng acceleration, it is necessary to construct quasilinear theory describing evolution of the distribution function of sur"ng particles. This calculation can be simpli"ed if it is limited to the consideration of the
d k|Ek |2 .(kvy − !)
@f : @vy
(57)
The di6usive
|Ek |2 @f |vy | @vy
:
(58)
vy ∼vymin
Derivative of the distribution function at vy ∼ vymin can be estimated as @f f : @vy Rv Finite width of the distribution is due to the gradual deceleration of the solar wind caused by mass-loading. At the termination shock the minimum value of Rv is determined by the thermal speed of the interstellar neutrals. The velocity di6usion results in the growth of Rv, eventually extending to positive vy and creating a leak in the distribution function of sur"ng particles. To calculate the total
ni e2 |Ek |2 Rx : 2m2i vymin Rv vymax
(59)
Taking equation for the
(60)
where N is the density of pick-up ions in the incoming solar wind. Then substituting ni into (59) we can rewrite the equation for the total escaping
W NCu n0 m p u 2
mp me
mp mi
14=5 27=10 u u ; (61) max vy Rv
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676
where |ky | |Ek |2 E 2 is used as the average value of the square of the electric "eld and W=
B overshoot
2 !pe E 2 2 !ce 4
is the lower-hybrid wave energy density. In order to make conclusions about the e5ciency of the above described mechanism for scattering of particles from sur"ng acceleration by waves, it is necessary to compare the calculated
5. Numerical simulations of the shock surng acceleration Unfortunately self-consistent kinetic simulations of quasiperpendicular shocks using hybrid code (e.g. Leroy et al., 1982; Quest, 1985; Quest, 1986; Forslund et al., 1984) did not demonstrate any signi"cant acceleration of solar wind pick-up ions. The reason for this is that in hybrid
u
φ downstream
upstream
ramp
foot
Bo
x Fig. 9. Perpendicular shock geometry and "eld orientation as used in the simulations.
simulations shock front is quite broad, with the width significantly exceeding the electron skin depth, while it has been found by several authors (e.g. Lever et al., 2001) that for e5cient shock sur"ng acceleration, the shock front must be su5ciently narrow L ¡ Lcr 0:7c=!pe . Fortunately, observations reported in Scudder et al. (1986), show that narrow quasiperpendicular shocks with L ∼ c=!pe exist in the solar wind. Since it is a serious problem to obtain shock waves in fully kinetic codes, the following scheme is used in simulations. Fig. 9 shows the typical shock structure adapted in simulations. The structure includes the extended foot, ramp with width ∼ c=!pe and overshoot of the magnetic "eld. This structure is followed by extended downstream region of the magnetic "eld. Test particle method is used to analyze particle dynamics at such shocks. This approach creates the possibility to investigate the evolution of the incoming shell distribution of pick-up ions and to look into di6erent mechanisms of acceleration by running the code with di6erent shock and particle parameters. In Fig. 10, two types of ion trajectories obtained by Lever et al. (2001) are shown. Similar results are also obtained in Zilbersher and Gedalin (1997). Top panel shows the trajectory of the sur"ng particle. The formation of the vertical jet indicates that the particle makes multiple re
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677
Fig. 10. Trajectories for two di6erent types of interaction with the shock front. Vertical jet is formed for the shock sur"ng particles corresponds to multiple re
Fig. 11. The maximum velocity acquired by the particle in di6erent acceleration mechanisms plotted as a function of shock ramp width. Di6erent families of curves correspond to di6erent masses. For su5ciently narrow ramp Lcr ¡ 0:7c=!pe , the shock sur"ng leads to signi"cantly larger "nal velocities compared to shock drift acceleration (Lever et al., 2001).
mechanisms is dominant, two quantities are used; the maximum value of acquired velocity during acceleration (vmax ) and the fraction of particles that leave the acceleration region with velocity larger than vth which can be found as 1 E5ciency(%) = f(vy ¿ vth ; vx ; vz ) dv; ni where Ni is the total number of pick-up ions participating in shock sur"ng acceleration. In simulations done by Lever et √ al. (2001), the value of vth = 2 2u is used. Results of this study is presented in Figs. 11 and 12. In Fig. 11 the maximum acquired velocity is plotted with respect to the width of the shock ramp L. For L ¡ Lcrit the dominancy of shock sur"ng acceleration is evident in these "gures. Maximum acquired velocity is signi"cantly higher for sur"ng acceleration as seen in Fig. 11. E5ciency also strongly depends on the ramp width; as the shock ramp widens the e5ciency of shock drift acceleration increases signi"cantly. It is also noted that the for L ¡ Lcuto6 shock drift acceleration is no longer active. As to the mass de-
pendences, both mechanisms prove to be more e5cient for lighter ions. Fig. 13 shows the results of simulations are performed by Lipatov and Zank (1999). We can see the formation of the vertical jet in velocity space which corresponds to the predominant acceleration in the y-direction. The observed jet in vy direction extends up to velocities ∼ 10u, which is more than su5cient to start di6usive shock acceleration. These velocity distributions close to 1D analytical “plateau” in vy , discussed in the previous section. The jet observed in Fig. 13 transforms to a shell distribution by pitch angle di6usion further downstream. Similar velocity distributions of the shock sur"ng particles has been observed in numerical simulations by Zilbersher and Gedalin (1997). The simulations done by Lipatov and Zank (1999) demonstrate another important result such that shock surfing acceleration acts as dissipation mechanism for collisionless shocks. The simulations in this paper for the "rst time takes into account full electron-ion dynamics self-consistently for low -0 plasma quasiperpendicular shocks. Results of these simulations are presented in Fig. 14. This "gure clearly demonstrates formation of a shock transitional layer similar to adapted shock structure in Fig. 9. The presented structure includes a foot in the density of pick-up ions, thin ramp in the magnetic "led with thickness L = 5 × 10−2 rLi , where rLi is the ion Larmor radius. The peak in pick-up ion density corresponds to accelerated ions and it is formed inside the ramp due to the trapping and thus accumulation of pick-up ions. This idea which becomes evident with the recent simulations was also considered by Sagdeev et al. (1984). A qualitative analysis is presented in this paper which discusses how the magnetosonic solitary pulse transverse to the magnetic "eld is transformed to a perpendicular collisionless shock wave where the sur"ng particles create the dissipation mechanism. Another important outcome of the simulations is the information about the spectrum of the accelerated particles.
' ( er / Planetary and Space Science 51 (2003) 665 – 680 V.D. Shapiro, D. Uc
678
10
S
6
Mass = 1 Mass = 2 Mass = 3 Mass = 4
4
vthresh = 2√2
Efficiency (%)
8
12
8
Lcutoff
D
4 2
(a)
0 0.2
0.5
1.0
1.5
0 2.0 (b) 0.2
0.5
1.0
1.5
2.0
15
15
10
10 Vper2/Uo
Vper2/Uo
Fig. 12. E5ciency of the acceleration mechanism as a function of ramp width. Panel (a) shows the e5ciency for sur"ng particles with di6erent masses, and panel (b) shows the e5ciency for drifting particles. The sur"ng acceleration e5ciency increases signi"cantly for narrow ramp widths (Lever et al., 2001).
5 0 -5
5 0 -5
-10
-10
-15
-15
-15 -10 -5 0 5 Vper1/Uo
10 15
-15 -10 -5 0 5 Vper1/Uo
10 15
Fig. 13. The projection of the pick-up ion velocity distribution at the perpendicular shock on two directions orthogonal to the magnetic "eld (Lipatov and Zank, 1999). Right panel shows the vertical jet formed in the upstream region, left panel shows the downstream region where pitch angle di6usion is important.
6. summary As a summary we can state that both analytical and numerical analysis identify the shock sur"ng acceleration as an ideal mechanism for the injection of pick-up ions into diffusive shock acceleration at the quasiperpendicular shocks. This mechanism favors slow pick-up ions and accelerates them up to the threshold velocity needed to start di6usive acceleration. At relativistic energies the main limitation of shock sur"ng acceleration is eliminated, which means particle can undergo theoretically unlimited acceleration. Surfing acceleration of protons to relativistic energies by fast magnetosonic shocks in solar
The spectrum for shock sur"ng acceleration turns out to be
Acknowledgements V.D.S. would like to express deep gratitude to M.A. Lee with whom he discussed many of the ideas presented in this paper countless times.
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Vasyliunas, V.M., Siscoe, G.L., 1976. On the