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 -

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