research report · oblique shock can reflect some electrons by a magnetic mirror effect, in spite...
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ISSN 0469-4732
INSTITUTE OF PLASMA PHYSICS
NAGOYA UNIVERSITY
PLASMA HEATING BY A MAGNETOSONIC SHOCK WAVETHROUGH RESONANT ION ACCELERATION
Yuklharu OHSAWA
(Received - Jan. 8, 1987)
IPPJ-811 Feb. 1987
RESEARCH REPORT
NAGOYA, JAPAN
PLASMA HEATING BY A MAGNETOSONIC SHOCK WAVE
THROUGH RESONANT ION ACCELERATION
Yukiharu OHSAWA
(Received - Jan. 8, 1987)
IPPJ-811 Feb. 1987
Further communication about this report is to be sent to the
Research Information Center, Institute of Plasma Physics, Nagoya
University, Nagoya 464, Japan.
PLASMA HEATING BY A MAGNETOSONIC SHOCK WAVE
THROUGH RESONANT ION ACCELERATION
Yukiharu OHSAWA
Institute of Plasma Physics, Nagoya University
ABSTRACT
Resonant wave-particle interactions in magnetosonic shock waves are studied
by theory and simulation. We evaluate the number of ions trapped by a
perpendicular laminar shock in a finite beta plasma and obtain the amount
of shock heating due to resonant ion acceleration in terms of the Mach
number and upstream plasma parameters. Some effects of trapped ions on
shock waves are also discussed. In addition, it is shown that a laminar
oblique shock can reflect some electrons by a magnetic mirror effect, in
spite of a large positive potential in the shock region. These theoretical
predictions are confirmed by a 1-2/2 dimension, fully relativistic, fully
electromagnetic particle simulation with full ion and electron dynamics.
PACS (i) 52.35.Sb. (ii) 52.35.Tc, (iii) 52.35.Mw, (iv) 96.60.Ce.
- 1 -
I. INTRODUCTION
Recently it has been found by theory and simulation that a magneto-
sonic shock wave12 can resonantly accelerate some ions3. The maximum
speed of those ions can be much larger than the Alfven speed, and its
dependence on various plasma parameters has been studied
quantitatively4"9 . Extremely strong ion acceleration has also been
observed in experiments of interplanetary shocks. For instance, accelera-
tion of energetic ions to ~ 22 Mev was detected by the low-energy charged
particle instrument on Voyager 2 in association with a quasiperpendicular
shock of the propagation angle 0^87.5° at a distance of -̂ 1.9 AU from the
sun10 . In Ref.ll, quantitative comparisons were made between the observa-
tions and the theory of resonant ion acceleration.
In addition to the prediction that resonantly accelerated ions have
huge energies, those simulations3"58 have revealed another important point;
a subcritical shock can heat ions significantly. Until 1985, the subcriti-
cal shock heating had been one of the main unresolved problems of shock
waves. In Goodrich"s review paper12 on simulations of quasiperpendicular
shock waves, he made a list of unresolved problems, and as the fourth point
he raised this problem. Since it clearly stated the problem in 1985, we
here quote it: "What is the ion heating mechanism in subcritical quasiper-
pendicular shocks? Large increases in the ion temperature are often
observed in subcritical shocks13 for which simulations predicted only
adiabatic heating." In 1985. however, surprisingly good agreement between
the simulation and experiment13 of subcritical shock heating was reported
in Ref.14.
In the resonant ion acceleration, the electric field moving in a
- 2 -
magnetic field plays an essential role15"17 . In a magnetosonic shock wave,
a large potential jump is formed in the shock region. The resulting strong
electric field normal to the wave front can trap some ions and, in combina-
tion with the Lorentz force, resonantly accelerate them in the direction
parallel to the wave front and perpendicular to the magnetic field. For a
perpendicular shock in a zero beta plasma (beta = plasma pressure / mag-
netic pressure). the maximum speed of resonantly accelerated ions is3
v - m(mi/me)1'2M-l)3'2 , (1)
where v* is the Alfven speed, mj the ion mass, m<> the electron mass, and MA
the Alfven Mach number. As the shock wave propagates, some fraction of
ions are newly trapped continuously. When those resonant ions reach the
maximum speed, they are detrapped and left behind the shock wave.
The maximum speed strongly depends on the propagation angle 0 (angle
between the wave normal and the ambient magnetic field). In a low beta
plasma, the maximum speed of resonantly accelerated ions in a quasiperpen-
dicular shock is about • m, /me) '''2 times as large as that in a quasiparallel
shock40 . This is a reflection of the fact that the width of a quasiper-
pendicular shock is about inte/m;)1''" times as small as that of a quasiparal-
lel shock: the potential jump depends on the propagation angle 6 only
weakly compared with the shock width.
Finite beta effects on the resonant ion acceleration in shock waves67
have also been studied using a finite beta theory for a nonlinear magneto-
sonic wave. One important effect of the plasma beta is that the electron
pressure raises the potential jump and increases the magnitude of the
acceleration for all propagation angles 6.
As can be seen from Eq.(l). when VA^C(rae/nij)'•/2 , we need relativistic
- 3 -
theory and simulation for the acceleration and for nonlinear magnetosonic
waves; the electron fluid velocity parallel to the wave front is also equal
to Eq.(l). It was found by a relativistic particle simulation that for
such plasma parameters a shock wave can accelerate both ions and electrons
to highly relativistic speeds8. Later, the structure of a nonlinear
relativistic magnetosonic wave was analytically studied9; from this theory
one can estimate the maximum speed of resonant ions in a relativistic
shock.
These studies now enable us to calculate, for a wide parameter regime,
the maximum energy of ions resonantly accelerated by a laminar magnetosonic
shock wave. In order to find the amount of shock heating, however, we need
to know the number of trapped ions as well as their maximum energy. In
this paper, on the basis of the theory for a nonlinear magnetosonic wave,
we will evaluate the number of ions trapped by a shock wave and obtain the
amount of shock heating due to resonant ion acceleration. We will also
study some effects of trapped ions on the shock structure. In addition, we
will discuss another interesting resonance phenomenon: electron reflection
by a laminar shock wave18 . Preliminary results have been reported in
Ref.19.
In Sec.II. we analyze a single particle orbit in a laminar shock
wave. In calculating particle orbits in a shock wave, we use the electric
and magnetic fields obtained from the finite beta theory for a nonlinear
magnetosonic wave6 ' . For simplicity, we consider a perpendicular shock.
From the analysis of a single particle orbit, we have the number of deeply
trapped ions, rtdtr • and the number n^tr that includes both the deeply and
weakly trapped ions. (Even the deeply trapped ions are not trapped for an
infinitely long time. As mentioned earlier, they are detrapped when their
- 4 -
speeds reach the maximum speed given by Bq.(l).) The number of ions
significantly accelerated by a shock wave is bounded by these two numbers.
From the number of resonant ions and the energy of those ions, we find the
amount of shock heating of ions in a collisionless plasma. This gives a
new Rankine-Hugoni->t relation, i.e., the relationship between the the Mach
number and the amount of shock heating in the presence of resonant ions.
The theory indicates that a laminar shock can increase ion perpendicular
temperature by one order of magnitude.
In Sec.Ill, we qualitatively discuss effects of trapped ions on the
potential. Since the trapped ions stay for a long time in the shock
region, the fraction of trapped ions there can be quite large.
In Sec.IV. from the analysis of an electron particle orbit in an
oblique shock, we show that, in spite of a large positive potential in the
shock region, a laminar shock can reflect some electrons by a magnetic
mirror effect. Reflected electrons go ahead of the shock along magnetic
field lines.
In Sec.V. simulation results will be presented. We use a l-2'2
dimension one dimension in space and three dimension in velocity space),
fully relativistic. fully electromagnetic particle code with full ion and
electron dynamics. First, we show shock profiles, ion distribution func-
tions, and ion temperature profile. The amount of shock heating observed
in the simulation is in good agreement with the theory. Second, we illus-
trate some effects of trapped ions on the shock wave. Third, we demons-
trate the electron reflection by an oblique shock wave.
Our work is summrized in Sec.VI.
II. TRAPPING AND HEATING OF IONS
- 5 -
We will evaluate the ion temperature in the downstream of a laminar
shock wave assuming that the heating is caused by adiabatic compression and
resonant acceleration. We consider a stationary laminar magnetosonic shock
wave propagating in the positive i direction in an external magnetic field
6 = Bo(cos6.O,sin0) . All physical quantities are assumed to be constant
along the y and z directions.
In a laboratory frame, the ion velocity may be written as
VI = Vj_ -r VU , (2)
where v± and vn are, respectively, the velocities perpendicular and
parallel to the magnetic field B. In a wave frame where the time deriva-
tives are zero, we may write the ion velocity as
U = v_-v<-vsh • (3)
where isj, is the shock velocity: in the cold plasma limit, it is NAVAOX .
with c, the unit vector in the x direction.
If we have an electric field
Ei = -d'f d x . Etv. Et~), f'4)
in a laboratory frame, the elecrtric field in a wave frame is written as
Eu = i-dy dx. £,,0. Ets;. ;'5';
Here. <p(x) is the electric potential formed in the shock, and Etv and Etz
are the y and - components of a transverse electric field Et ,
respectively. Also. £,,<> is constant in time and space and is related to Bo
and vsh= i Vsi, I as
- 6 -
£«o = -(ushBo/c) sinB . (6)
Using these quantities, we have a general expression for the energy
conservation law for an ion particle,
f-^ = Ec+ef Et2it dt ,C- •'to
(7)
where. Ec is a constant. In terms of the x, y, and z components, ifa can be
written as
i-i = (UiI+iJiicos0-us|1)2+ia!,+(ULZ+UIISXTI0)2 , (8)
where (I'ix.uu,.vx-'<=v± .
As one moves from the far upstream to the shock region, the potential
¥>(x) increases and takes the maximum value at some point. The potential
traps some ions, although most of the ions pass through the shock region
without resonant interactions. Those trapped ions will be resonantly
accelerated in the ii direction'3"1' .
We now restrict ourselves to a perpendicular shock, in which the
magnetic field has only the r component. For perpendicular shocks, i'n is
conserved, and !_r and £t; disappear. Thus, the energy conservation law,
Eq.'7'. is simplified as
e^\x)-eEvoy—^ i Ua-^ sh)2+^) = Ec(constant). (9)
Here, since v«cos8 = 0. v±x and v±y are just vx and vu . respectively.
We follow the orbit of an ion particle from a far upstream point
(xo, yo)• Without loss of generality, we can choose the point (xo, yo)
- 7 -
such that the x component of v± is zero, v±o=(0,Vyo) '< at this point
(xo> yo). the y component of the position, yo, coincides with the y compo-
nent of the guiding center position. Since the potential <p tends to zero
in the far upstream, it follows from Eq.(9) that
(mi/2)(vx-vsh)2 = (mi/2)x|h-e?)(x)+e£yo(y-yo)-(mi/2)(t|-i|o) . (10)
The quantities vz, vv, p, and (y-yo) change with the particle position x.
We assume that, until the particles are trapped by the potential, the
particle velocity at a given point (x,y) consists of the fluid velocity,
Larmor gyration, and free streaming along magnetic field lines. (Drift
approximation is not valid for ions.) The displacement Jy/i (=y-yo) due
to the fluid motion can be calculated from the nonlinear theory of a
magnetosonic wave' as
;^c;) ^ 1 (ID
where 7i| is the density perturbation and cs is the sound speed.
c; = cr - cz- '12)
with
C7=7jPio ! ni>ifi, ' . (i-3)
Ce=7ePe0 ' TtOllli) • (14)
Here. 7j is the specific heat ratio (j=x or e). and p;o the equilibrium
pressure in the upstream. On the other hand, the potential jump in a
perpendicular shock is expressed as
- 8 -
eq> = Ci»i^+(7ePeo/no)] (ni/no) . (15)
By virtue of Eqs.(6), (11), and (15), we see that
ev . (16)
Hence, we can neglect the term eEuoAyfi in Eq.(lO). Similarly, one can
readily show that the kinetic energy (mi/2)i>jj due to the fluid motion is
about iiie/'mi times samller than the potential energy ep. Therefore, for vu
and y on the right-hand side of Eq.(lO), we use the velocity and position
calculated from the Larmor gyration:
vy=v±sinip , (17)
i/-yo=PiCOST/' . (18)
where v- is the gyro-phase. We will neglect the change in the ion Larmor
radius p, before trapping.
For a perpendicular magnetosonic wave, we have a solitary wave solu-
tion in the following form0'
m-no = 3W-l)cfj sech2(ts). (19)
where .V is the Mach number, and a± is defined as
5 i i i ] "' =3/2. (20)
Here, Si =3/'2. because 7j=7e=2 for a perpendicular wave. The argument
is a stretched coordinate proportional to the quantity x-A/(i^+
(For more precise definition, see Eq.(61) in Ref.7.)
If we substitute Eqs.(15) and (17)-(20) into Eq.(10), we have
- 9 -
(mi/2)(vz~vsh)2 = (m;/2) (Ar^ifl+cf) - 4 (AM) (i%+cl)sech2(£s) ]
(21)
Here, we have neglected the last term on the right-hand side of Eq.(lO).
(mi/2) (i|-i|o) . assuming that
2Hiw+cl)w2 > v± . (22)
The inequality (22) holds if, roughly, the ion beta value is smaller than
unity. For a finite beta plasma, the perpendicular shock speed vsh is
written as
vsh = A/(ifl-i-c|)1/2 . (£3)
If the right-hand side of Eq.(21) becomes negative at some point i, then a
corresponding particle will be trapped there.
We show in Fig.l profiles of the potential and the electric field Ex
of a magnetosonic soliton propagating perpendicularly to a magnetic field.
The potential1' ' is proportional to sech2(fs) . and Ez is proportional to
seclr '.fsHanh /, . The electric field E3 has its maximum value at the
location where scclr •• fs > =2 '3 . To consider the condition for the accelera-
tion, we recall that a trapped particle coining from the upstream region
bounces off many times in a shock ramp during the resonant acceleration,
gradually shifting its reflecting points from the upstream to downstream
side. Therefore, for a strong resonant interaction, particles will have to
be reflected, at least once, before the maximum point of the electric field
Ex • On the other hand, if the first reflecting point is near the maximum
point of the potential, the resonant interaction will be rather weak.
- 10 -
Substituting the value sec/r(?s) =2/3 into Eq. (21), we see that the
ions satisfying the following inequality are deeply trapped ( i.e.,
reflected before the maximum point of the electric field):
ViCOSTp > Vdtr . (24)
with
vdtr = (W/2)(ii+ci)!/2[l-(8/3)(A/-l)Ar2(^+cf)(i|+cf)-1] . (25)
Similarly, we can obtain the critical velocity of weakly trapped ions,
which are reflected near the maximum point of the potential,
as
(^+c;)-1] . (26)
The number of deeply trapped particles, ndtr • may then be calculated
ndt,= I c/t.,/ di,,) dii/(ii,fcj,.ii:.1 . (27;
where f-Vi.vv.v=' is a velocity distribution function. For a Maxwellian
plasma.
] , (28)
we have
ndir = (no,'2) {1 - erf[Vdtr/(2]/2VTi)) } , (29)
where the error function er/(p) is defined as
erf (p. = -^[expC-t^dt . (30)
- 11 -
Also, the number of all the resonant ions, n^tr . which includes both
the weakly and deeply trapped ions, is
r w = (no/2) {1 - er/[iW(21/2uri)) } . (31)
The number of significantly accelerated ions, ntr > (i.e., accelerated to
the speeds much greater than the thermal speed) may be bounded by these two
numbers,
Tldtr < ntr < nmtr . (32)
We show in Fig.2a the dependence of the number of trapped ions on ion
beta value pi : the Mach number is chosen to be W=l.7, and the electron beta
value is &=0.1 . The solid line shows the number of deeply trapped ions,
RdtT • and the dashed line shows riu.tr • Also, the same quantities are
plotted in Fig.2b as a function of electron beta value (ie for >/=1.7 and
/3,=0.1 . These pictures show that at least a few percent of ions can
strongly interact with a laminar shock wave. The number of weakly trapped
ions is several times as large as that of deeply trapped ions.
The maximum speed of ions resonantly accelerated by a perpendicular
shock is given by'
where
- 12 -
For the electric and magnetic fields in Eq.(33), we have used average
values in the shock region, i.e., <Ex>=Ew/2, and <B>=(B,+Bo)/2 , where the
subscript m denotes the maximum value. We then obtain the kinetic energy
Kir of resonantly accelerated ions:
r "2 L Z 7 N ^ - J
Here, A is a numerical factor such that A M / 2 . This numerical factor is
needed because not all the trapped ions reach the maximum speed, Eq.(33).
In order to find an expression for .4. we have to know energy spectra of
resonantly accelerated ions. If the energy spectrum is peaked at the
maximum energy -- (.r«i/2)(c<Er>/<B>)2, the numerical factor A should be
unity. For the energy spectra roughly constant up to the maximum energy, A
will be about 1/2. In most of the cases, this factor may be well approxi-
mated by .4M "2. Also. ntr may be a few times as large as the number of
deeply trapped ions, iidt, '• we will discuss this later again, by comparing
the theory and simulation results.
We show in Fig.3a the ratio of the energy increase due to resonant ion
acceleration. At, • to the perpendicular thermal energy no7j.o of upstream
ions as a function of ion beta value (i, : the Mach number is chosen to be
W=1.7, and the electron beta value is &=0.1 . The same quantity l\tT/noT±o
is plotted in Fig.3b as a function of electron beta value &. for fii=QA .
We see from these plots that even a laminar shock can increase perpendicu-
lar ion temperature (or, more excactly, kinetic energy) by one order of
magnitude through resonant ion acceleration.
If the plasma heating is caused by adiabatic compression and resonant
ion acceleration, the ion perpendicular temperature in the downstream of a
- 13 -
perpendicular shock is
T±d = (Bd/Bo)Txo + Ktr/no . (36)
where the subscript d denotes the quantities in the downstream region.
Another possible mechanism for the collisionless shock heating of a
plasma is the dissipation of the electron kinetic energy caused by some
micro-instabilities. For a perpendicular shock, the fluid electron kinetic
energy, mei%e/2 , is nti/nte times as large as the fluid ion kinetic energy,
m,^i /2, because the electron and ion velocities parallel to the wave front
are related as3
v,ie = -cEx/B = -(mi/me)vl,i . (37)
Therefore, the fluid electron kinetic energy becomes the energy source, if
the plasma temperature rises because of micro-instabilities. We note,
however, that the maximum speed of resonantly accelerated ions is about the
same as the electron fluid velocity. Hence, the increase in the plasma
temperature due to resonant ion acceleration will dominate the temperature
increase due to micro-instabilities, if the fraction of trapped ions is
greater than •nic'«n] •
nfr/jio -> nie/ntj . (38)
III. EFFECTS OF TRAPPED IONS ON THE POTENTIAL
The number of trapped ions. njr , is determined by the ratio vtr/VT,\,
Eq.(29). One can readily see from Eq.(29) that a small change in the
critical speed for the trapping, utr - can signifcantly change the number
- 14 -
ntr , because, in many cases, the quantity vtr/vn is in the region
vtr/VTi^2 ; an increase in the quantity vtT/vri from 2.0 to 2.2 is accompa-
nied by a decrease in ntT/no by a factor of 0.6. This indicates further
that a small change in the potential significantly modifies the number of
trapped ions, because the critical trapping speed Vu is proportional to
the quantity (nii/2)i|h-ep. In this section, we qualitatively discuss
effects of trapped ions on the potential; trapped ions raise the
potential. This implies that the critical trapping speed u(r is reduced,
resulting in the increase in the number of trapped ions.
For simplicity, we consider a perpendicular shock wave in a low beta
plasma. From the upstream ions, some fraction of ions, ntr/no , will be
trapped in the shock region. Those trapped ions resonantly interact with
the shock and stay in the shock region for a long time period ta" ,
ta - :«,.'* :l/=acr
lWl(W-l)3'2 . (39)
Therefore, the number density of trapped ions in the shock region. nr, is
larger than nt, . We can calculate the n, from the continuity equation in a
wave frame:
n r<i x r : = ntrVsh , (40)
where <u x r> is the average value of t̂ of trapped ions in the shock
region. The <vIr> is given by
<vxr> - L/ta, (41)
where A is the shock width,
A -v (c/-apc)(»/-l)-|/2 . (42)
- 15 -
Thus, we have the number density of trapped ions in the shock region,
nr -\- ntrvshtah~] . (43)
The potential increase, &p, due to trapped ions can be estimated from
Poisson's equation
ecxp ~ 4;re2nrA2 . (44)
Substituting Eqs.i'42) and (43) into Eq. (44) yields
eo£ ^ (o)pi/wCi)2(ntr/no) mitfl (W-l) • (45)
As we have seen, the potential jump due to the fluid motion is e<p~m;i^ .
Hence. Eq.(45) indicates that effects of trapped ions on the potential can
become important if (&iPi/<<)c»)2(ntr/no)M .
IV. ELECTRON REFLECTION
In a shock region, positive electric potential is formed. As we have
seen in the previous sections, this potential reflects some ions. In this
section, however, we will analytically show the possibility of electron
reflection by a magnetosonic shock wave; in spite of the large positive
potential in a shock region, small fraction of electrons can be reflected
by a magnetic mirror effect. Electrons backstreaming along magnetic field
lines away from the shock wave were also observed in the experiment near
the earth's bow shock18 .
We start from the energy conservation law for the electrons in the
wave frame:
- 16 -
-e<p(x)+eEuol)+ (me/Z)-ii = Ec-ef Etzvz dt . (46)
We will analyze electron orbits in an oblique shock wave. The uniform
electric field Evo and electron speed vw in the wave frame are defined by
Eqs.(6) and (8). Let v± be the gyration speed perpendicular to the
magnetic field and V be the gyro-phase, then Eq.(46) can be rewritten as
^ - - r
- (me/2; C \V±2sin2ip-v±o2sin2wo) + (-v±cosi>cosd+ Vnsind)2
z v s h ) 2 } , - (47)
where vw is the velocity parallel to the magnetic field. The subsript 0
denotes the quantities at a certain position in the far upstream. In the
following, we assume that the electron thermal speed is much larger than
the Alfven speed, IT... ty . and that the quantities v& . x,sh • and cs are of
the same order of magnitude.
We will show that dominant components in the terms
(e<f—eEvoiy-lio-cJEt-i:~dt '•• in Eq.:'47; cancel out and that the change in
ii.2(x) becomes important. We can find the displacement, Jy-y-yo. by
integrating the electron velocity vye along the trajectory. The fluid
electron velocity is already obtained '.see Eq.(50d: in Ref.7). It is to be
noted, however, that the perpendicular fluid velocity vve is not the same
with the y component of individual particle velocity averaged over the
cyclotron period20 . It is because, in the fluid velocity. 1) the diamag-
netic velocity \j^=icTe'eBn>(yn<B) is included, and 2) we do not have the
- 17 -
Vfi-drift velocity, VB=- {c(wn?L/2.)/eB2) (fixVB) . Therefore, for vve we use
the value obtained from the equation vye=Vfive-vnu+Vbv , where Vfiue, vnu , and
Vbu are the y components of the fluid velocity, diamagnetic velocity, and
VB -drift velocity, respectively. (For the ions in a perpendicular shock,
the drift approximation is not valid, because non-resonant ions pass
through the shock region with a time period t~cocr' (me/mi)1/2 , much shorter
than the ion cyclotron period. In addition, the ion Larmor radius may not
be much smaller than the shock width.) Integrating this vye along the
trajectory, we have
^7CK In 11 -frvcos^8) vsh-v,\cos6'
»iC= sine t! [sine - ^i2k^l\ l n h _ m>(n,/no)
where upo is the phase-velocity of a linear fast magnetosonic wave propaga-
ting obliquely to a magnetic field.
° , _ . 0 0 0 9 , 0 0 0 0 1 - o .
ipO = i.l 2. - U I - C J •-!- , : . i s - c j ; - - 4 ^ c ; c o s - 6 , ' -• . • •..49;
The first term on the right-hand side of Eq.°48) is obtained from, the
integration of the fluid velocity mue. and the second term is from
(—Vny+Ubsi) • As we will seo later, qualitatively, the second term, i.e..
the correction to the fluid motion, is not important.
In Eq. (48,i. we have considered the electrons such that
vsh - vucosd > 0 , (50)
because only those electrons can enter the shock region from the far
upstream region. We will further assume that
- 18 -
0 <Vsh~ I'll COS&
because the perturbation n\/w> is assumed to be positive and small; our
theoretical analysis is not applicable to the electrons with
I U sh- V|| COS0 | /Vsh< 1 .
Also, e<p and -eJEtzvzdt are expressed as
| | | ^ ^ 2 (52)
In 11 - ^"'ffll . (53)
The quantity eg; i s always positive for a positive density perturbation,
ni>0 , and is quite large, ^mjifl . However, the quantit ies -eEvoAy and
-eJEt~v~dt are also of the order of m,wj . and the former can be negative;
by virtue of the condition ;'51), the logarithmic terms in Eq.(48) are
negative. Consequently, their sum can be much smaller than 111,15 •
As in the previous section, we choose the i n i t i a l phase wo as cos^o =
0. Then, substituting Eqs.''48). (52) and ;'53) into Eq. (47) yields
(mc '2) •'i'icost'sind- v»cosd-Vsh)2
( i pO—c;Ha |~ni (vsh~ Vvcosd). 1, vpn 1 nj ''no) \~] ->ni= tlli~—o n o—~ ' i n I 1 — —^ Q I l^THiCe—
-1 • n M T • n vi (tSn-cf)i , 1, i>po(n|/no'> 1- m,c; sine - sinO - —^—• F ,̂ In 1 E L _ L — ^
T L 2w?c vs, J vsh-VIIcose'
- 19 -
+ (me/2) {(uNocos0-ush)2+^osin2e} . (54)
Here, we have used the relation v±2/B=constant; the electron magnetic
moment may be well conserved for oblique shocks, until at least the parti-
cles are reflected by a shock, because the electron Larmor radius is much
smaller than the scale length of shock waves. (For a perpendicular shock,
drift approximation is not valid even for electrons12 .) The displacement
Ay due to finite Lamor radius is very small for the electrons and is
neglected.
The first, second, and third terms on the right-hand side of Eq.(54)
come from the sum ey-eEvohy-e\Etzvzdt . One can readily show by Taylor
expanding the logarithmic term that the first term on the right-hand side
of Eq.(54) vanishes in the lowest order of ni/no .
Now we consider the motion of electrons having small parallel veloci-
ties and large perpendicular velocities:
Vi'lJ •- Vsh \ <• I'Te .' • (55}
and
IT> < i'_ . !'56>
For those electrons, the second and the last terms on the right-hand side
of Eq. <"54) are negligibly small compared with the absolute value of the
fourth term (~m,ii2) • Also, because of the condition (56;. the third terra
on the right-hand side of Eq.(54) becomes negative.
Consequently, we see that the right-hand side of Eq.(54) can be
negative if the conditions (55) and (56) are satisfied. In other words,
the electrons having small parallel velocities and large perpendicular
- 20 -
velocities can be reflected.
V. SIMULATION RESULTS
Simulation studies of resonant ion acceleration by magnetosonic shocks
in low beta plasmas have been made in detail in Refs.3 and 5. In those
studies, we were mainly concerned with the maximum speed of resonantly
accelerated ions; its dependence on ambient magnetic field strength and
propagation angle was examined by simulation and compared with theoretical
predictions. We now further investigate, by simulation, resonant wave-
particle interactions in magnetosonic shocks. First, we will show that the
resonant ion acceleration significantly increases the ion perpendicular
temperature and will compare the simulation results with the theory.
Second, we will study effects of trapped ions on the potential. Third, we
will demonstrate that some electrons can be reflected by a laminar magneto-
sonic shock.
To study the shock wave by a simulation, we use a 1-2 2 dimension (one
dimension in space and three dimension in velocity space). fully relativ-
istic. fully electromagnetic particle code with full ion and electron
dynamics :for more detailed description of the code, see Ref.3). The
simulation parameters are taken as follows. The total grid size is
^=1024^ . where Ag is the grid spacing. All lengths and velocities in the
simulations are normalized to Ag and UpeAg , respectively. The magnetic
field and electric potential are normalized to mC"ip<>2Ag/'e and nfeUpe'Ag/e,
respectively. The total number of simulation particles is A'; =Ne =65536 .
The external magnetic field is in the (x,z) plane with
Bo-Bo(cos6, 0. sin6'; . The ion-to-electron mass ratio is 100; the light
- 21 -
velocity is c=4. The strength of the external magnetic field is chosen so
that Uec/copc =0.5. For these parameters, the Alfven speed is IM=0.2, and the
electron inertial length is c/<Dp<,=4 .
A. Trapped ions and shock heating
We show in Fig.4 the structure of a perpendicular shock (cos0=O) with
the Mach number A/=1.68; the magnetic field and potential profiles are shown
in Figs.4a and 4b. respectively, and the ion phase-space plots [(x.pxi/rnjc)
and (x,Pj/i/niiC)) are shown in Figs.4c and 4d. The ion and electron beta
values in the upstream region are ft=0.064 and /Sc=0.64, respectively; the
ion and electron thermal speeds are i>n=0.035 and iTe=l-04, respectively.
The shock is propagating in the positive x direction, and the shock front
is at x^580 at ajpct=960 . In the shock ramp, some ions are trapped and
strongly accelerated in the direction parallel to the wave front. Those
resonant ions are detrapped when they reach the maximum speed. After the
detrapping. they are left behind the shock front and continue the Larmor
gyration.
Figure 5a shows the distribution function ln./'p-Vi of ions that are
in the region 375<a<512 at f=0. The dashed line shows the distribution
function at f=0. and the solid line shows the one at u;p£.i=960 ; the dashed
line corresponds to the distribution function in the upstream region, while
the solid line represents the one in the downstream region. We see from
Fig.5a that a number of ions are strongly accelerated. The fraction of
ions that are accelerated to the energy greater than 25 m,-tTi is (4--5) %\
the kinetic energy 25 m,-iTi corresponds to the value (,Pi/i*iiC)2 0̂.4x 10~2.
The fraction of ions with the energy <r40ni;ifi (or (pi/niiC)2^0.6x \Q'2) is
- 22 -
If we substitute the observed Mach number M=\.68 into Eq.(29), we have
the theoretical number of deeply trapped ions n̂ tr/no - 1.3 %. On the other
hand, the number rwr » which includes very weakly trapped ions as well as
the deeply trapped ions, is calculated from Eq.(31) as n^r/no^SS %• As
expected, the fraction of high-energy ions observed in the simulation is
between these two theoretical values; in this case the number of high-
energy ions observed is (2-3) times as large as the theoretical number of
deeply trapped ions given by Eqs.(25) and (29).
We show in Fig.5b the variance of perpendicular momentum of ions as a
function of x-
2f«; 2/H,n(x) jz
where <p_> is the average momentum perpendicular to the magnetic field in
a small volume, and n-.x) is the number of ions in it. This quantity is a
perpendicular temperature if the ions are thermalized.
The variance of the ion momentum <6p_r> in Fig.5b is especially
large in the shock ramp. X--580 . This is because \) in the shock ramp the
y component of the momentum of resonantly accelerated ions has a sign
opposite to that of bulk ions, and 2), as discussed in Sec.Ill, the number
density of resonant ions in the shock region is larger than that of
detrapped high-energy ions in the downstream.
At £^510 , which is slightly behind the shock region, the perpendicu-
lar ion temperature is 4'-5 times as large as the upstream temperature.
This value is fairly close to the theoretical evaluation; the theory,
Eq.(36). indicates that the temperature ratio, Tld.-Tio * is about 4.6, if we
- 23 -
take the numerical factor A, the ratio Bd/Bo . and the number of trapped
ions ntT to be 4=0.5, Bd/Bo=1.6, and ntT=2n<jtr . respectively. This shows
that the theoretical estimation for the temperature rise is in good agree-
ment with the simulation results, if we set ntr to be a few times as large
as the number of deeply trapped ions calculated from Eq.(29).
It should be noted that the temperature rise depends on the mass
ratio. The mass ratio in the simulation is mi/m<>=100 . In a real plasma
with the mass ratio equal to or greater than 1836, the amount of shock
heating through resonant ion acceleration should be larger than that in the
simulation. It is because the ratio of the maximum speed of resonantly
accelerated ions to the ion thermal speed, v,ar/uri > increases with the mass
ratio (see Eq.(33)): in the cold plasma limit, the ratio i w A T i is propor-
tional to (Wi'nic)1''-.
B. Effects of trapped ions on the potential
According to the nonlinear theory of a magnetsonic wave in a finite
beta plasma based on a two-fluid model, the potential jump is related to
the magnetic field jump as'
e? = »i(Vj 1 - 7e/Jc.2;("B./Bo - 1) , .'58.;
for perpendicular waves. We now examine this relationship by a
simulation.
We show in Fig.6 the relationship between the potential jump and the
magnetic field jump of ,je;pendicular shocks observed in our simulations.
The electron anH ion beta values are taken to be ft> =0.064 and fa=0.064 in
Fig.6a. /Sc=0.64 and (I, =0.064 in Fig.6b. The solid lines show the theoreti-
- 24 -
cal relation, Eq.(58). In all the cases, the theory and observations are
in good agreement, especially for small-amplitude waves. For large-
amplitude waves such that (B»-Bo)/Bo~l , the observed potential jump is
slightly larger than the theoretical one obtained from the two-fluid
model. This is because, as discussed in Sec.Ill, trapped ions further
raise the potential jump in the shock region.
In connection with this, we point out that for some cases a shock
profile has a small hump in front of the major peak of the wave; see Fig.7,
which shows the magnetic field and potential profiles of a perpendicular
shock with the Mach "number M=1.8 (upstream plasma parameters are the same
with those in Fig.4). The small hump is indicated by the arrow in Fig.7.
The formation of the small hump seems to be caused by trapped ions. We do
not go into details of this phenomenon in this paper, because its effects
on the acceleration are rather small. We only note that this is more
likely to occur when the amplitude is large, and when the ratio v^/c is
small. However, when the ion Larmor radius is very large because of high
temperature, such a hump is smoothed out.
C. Electron reflection
We have shown analytically in Sec.IV the possibility of electron
reflection by a laminar shock wave. We here demonstrate by a simulation
that the electron reflection does occur in spite of the large positive
potential in the shock region.
Figure 8 represents the potential profile p(x) and the electron phase
space plot (x,pze) of an oblique shock with the Mach number A/=2.4; 0=67.5°,
/3j=0.13. and /Sc=l .3. with other parameters same with those in Fig.4. The
- 25 -
shock is propagating in the positive i direction, and the shock front is at
1^550 at 4)pet=720 . In the upstream region (x<T550 ), there are electrons
with large positive pze . They are reflected electrons backstreaming along
magnetic field lines away from the shock wave. They have large velocities
along the magnetic field lines and hence large positive pze .
VI. SUMMARY
From the analysis of single particle orbits of ions in a laminar shock
wave propagating perpendicularly to a magnetic field in a finite beta
plasma, we have evaluated the number of trapped ions and the amount of
shock heating due to resonant ion acceleration. Our formula for the shock
heating gives a new Rankine-Hugoniot relation in the presence of resonant
ions: the ion temperature behind the shock is expressed in terms of the
Mach number and upstream plasma parameters. The theory indicates that the
perpendicular ion temperature ;or kinetic energy; can increase by one order
of magnitude by a laminar shock through resonant ion acceleration.
Some effects of trapped ions on the shock wave were also discussed.
Since the trapped ions stay for a long time (t--GiCi"'•.us;/tiic)''2) in the shock
region, the fraction of trapped ions there can be quite large. We have
shown the parameter regime in which effects of trapped ions on the poten-
tial can become important.
In addition, we have shown that a laminar oblique shock can reflect
some electrons by a magnetic mirror effect, in spite of the fact that a
large positive potential jump is formed in the shock region. The reflected
electrons go ahead of the shock along magnetic field lines.
These theoretical predictions have been examined by a 1-2/2 dimension.
- 26
fully relativistic, fully electromagnetic particle simulation with full ion
and electron dynamics; the electron dynamics is important in determining
the shock structure. The theory and simulation results have been compared
in detail, and it was found that they are in good agreement.
- 27 -
References
1). R.Z. Sagdeev, in Reviews of Plasma Phvsics. edited by M.A. Leontovich,
(Consultants Bureau, New York, 1966), vol.4, pp.23-91.
2). D.A. Tidman and N.A. Krall, in Shock Waves in Collisionless Plasmas.
(Wiley, New York, 1071), chap.3.
3). Y. Ohsawa, Phys. Fluids, 28, 2130 (1965).
4). Y. Ohsawa, J. Phys. Soc. Jpn. 54, 1657 (1985).
5). Y. Ohsawa, Phys. Fluids, 29, 773, (1986).
6). Y. Ohsawa, J. Phys. Soc. Jpn. 54, 4073 (1985).
7). Y. Ohsawa, Phys. Fluids, 29, 1844 (1986).
8). Y. Ohsawa, J. Phys. Soc. Jpn. 55. 1047 (1986).
9). Y. Ohsawa. Phys. Fluids. 29, 2474 (1986).
10). E.T. Sarris and S.M. Krimigis, Astrophys. J. 298, 676 (1985).
11). Y. Ohsawa. Geophys. Res. Lett, in press.
12. C.C. Goodrich, in Collisionless Shocks in the Heliosphere: Reviews of
Current Research, edited by B.T. Tsurutani and R.G. Stone, (AGU.
Washington D.C.. 1985). pp.153-168.
13;. M.F. Thomsen. J.T. Gosling, S.J. Bame. and M.M. Mellott. J. Geophys.
Res. 90. 137 ;1985;.
14). Y. Ohsawa and J-I. Sakai. Geophys. Res. Lett. 12, 617 (1985).
15.;. R.Z. Sagdeev and V.D. Shapiro, Pisma Zh. Eksp. Teor. Fiz 17, 387
(1973, [ JETP Lett. 17, 279 (1973)] .
16). R. Sugihara and Y. Midzuno, J. Phys. Soc. Jpn. 47, 1290 (1979).
17). B. Lembege. S.T. Ratliff. J.M. Dawson, and Y. Ohsawa, Phys. Rev. Lett.
51, 264 (1983;..
18). W.C. Feldman. R.C. Anderson. S.J. Bame, S.P. Gary, J.T. Gosling,
- 28 -
D.J. McComas, M.F. Thomsen, G. Paschmann, and M.M. Hoppe, J. Geophys.
Res. 88, 96, (1983).
19). Y.Ohsawa, J. Phys. Soc. Jpn. in press.
20). L. Spitzer, in Physics of Fullv Ionized Gases. (Wiley, New York,
1962), chap.2.
- 29 -
Figure Captions
Fig.l Profiles of the potential <p and the electric field Ex of a perpen-
dicular magnetosonic soliton. Here, £s is a stretched coordinate
proportional to the quantity z-M(ifl+c|)l/2t.
Fig.2 Dependence of the number of trapped ions on ion beta value, (a), and
on electron beta value, (b). The solid lines show the number of
deeply trapped ions, ndtr > while the dashed lines show the number
riutr , which includes both the weakly and deeply trapped ions. (a) :
ft>=0.1. (b): tfi=0.1. The Mach number is 1.7 for both cases.
Fig.3 Energy increase Atr due to resonant ion acceleration as a function
of ion beta value, (a), and of electron beta value, (b). The Mach
number is A/=1.7 for both cases, and A'tr is normalized to perpendicu-
lar thermal energy no7j.o .
Fig.4 Structure of a perpendicular shock with the Mach number A/=1.68.
Here. ;.a- is the magnetic field profile, (b) the potential profile.
:.c) the ion phase-space plot (x.Pxi/mic) , and id) the ion phase-
space plot 'x,pVi/JII;C) . The shock is propagating in the positive x
direction, and the shock front is at x^580 at apct=9G0 .
Fig.5 Ion distribution function ln(/(p2)), (a), and profile of the
variance of perpendicular ion momentum, (b). In (a), the dashed
line shows the ion distribution function in upstream region, while
the solid line shows the one in downstream.
Fig.6 Potential jump versus magnetic field jump. White circles are values
observed in the simulation, while the solid lines show the theoreti-
cal curves. The electron beta value is /3e=0.064 in (a) and 0e=O.64
in (b).
- 30 -
Fig.7 Profiles of the magnetic field and potential in a perpendicular
shock with A/=1.8. Small hump in the shock front is indicated by the
arrow.
Fig.8 Structure of an oblique shock with 6=67.5° and M=2.4. The potential
profile <p and electron phase-space plot (x,p2e/i«eC) are shown in (a)
and (b), respectively. Electrons backstreaming along magnetic field
lines away from the shock are clearly shown in (b).
- 31 -
Fig. 1
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