Pitch angle diffusion of low energy protons due to gyroresonant interaction with hydromagnetic waves

Join'hal of Atmospheric and Terrestrial Physics, 1968, Vol. 30, pp. 1313-1330. Pergamon Press. Printed in IgorthemIreland

Pitch angle diffusion of low energy protons due to gyroresonant interaction with hydromagnetic waves R. GENDRIN Groups de Recherches Ionosphdriques, 4, Avenue :Neptune, 94-Saint Maur des Fossds, :France (Received 10 _February 1967; in revlsedform 15 1Vovember 1967)

Abstract--Using a hydromagnetic approximation for the two circularly polarized waves that can be propagated longitudinally in a cold magneto-plasma, we derive the frequencies that a r e emitted by gyroresonanco processes involving low energy proton beams (E ~ 100 keV). The reaction of the generated waves on the particles is evaluated and the resulting trajectories of these particles in their velocity space are computed; they differ greatly from circles centered at the origin. The pitch angle diffusion coefficient that we can then deduce is different also from the simple one that is generally used: it is greater for (p, L) interactions, and lower for (p, _~) interactions. The importanco of the energy diffusion in such interaction processes is stressed, and orders of magnitude are given. In an AppendLx, a general diffusion equation for this type of interaction is established. 1. INTRODUCTION THE gyroresonant interactions occurring in the magnetosphere between waves and particles have been extensively studied during the last three years (see, for instance: ]3RICE, 1964 and Gv,ND~r~, 1967a, b). The phenomenological approach has been fully worked out (B~IcE, 1964; GALLET, 1964; STE:FANT and VASSEUR, 1965; GENDRIN, 1965a) even for waves propagating obliquely with respect to the constant magnetic field B (Bv.T,T,, 1964; ETCHETO and GENDRr~, 1966). I t leads to a classification of the interactions, according to the nature of the interacting particles (electrons or protons) and to the polarization of the generated waves (right or left). The amplification coefficients for such collective interactions were computed with various different assumptions regarding the relative parts played by the longitudinal and transverse components of the beam velocity (NEUFELD and WRIGHT, 1963, 1964, 1965; BELL and BUNE~N, 1964; CORNWALL, 1965; SUDAN, 1965; GENDRIN, 19655 ; FU~O, 1966 ; JACOBSand WATANABE, 1966 ; HRU~I~_~, 1966). In some works (CORNWALL, 1966; LIEMOHN, 1967), pitch-angle distribution functions of the energetic particles have been introduced. I t soon appeared, however, t h a t the generated wave, in its turn, must act upon the particles and change their movement. When the velocities of the particles do not possess any phase coherence among themselves--and there is, at the moment, no experimental evidence t h a t they do--this reaction will not lead to a systematic change of the energy or of pitch angle of the particles, but it will produce, as a consequence, a 'diffusion' in E and ~ space. The importance of such processes has been stressed for m a n y years by DRAGT (1961) and DUNOEY (1962, 1963). Quantitative estimations have been worked out principally by CORNWALL (1964, 1966) and DUNOE¥ (1965); more recently, KENNEL and PETSCEE~ (1966) have 1313

1314

R. GE~D~nV

shown that the limiting particle fluxes that are observed in the magnetosphere are the consequence of such a diffusion mechanism. However, the conclusions of these papers were arrived at by using, for the diffusion coefficient, an approximate expression which is not valid for interactions involving low energy protons. The purpose of the present work is to study in more detail these types of interactions, and to derive correct expressions for the diffusion coefficients. First, we set out our assumptions and notations (Section 2). Then we give the algebraic formulae involved in the theory and compute the emitted frequencies (Section 3). Trajectories of the interacting particles in velocity space are then derived, and some remarks are made about the differences between (p, L) and (p, ~) interactions (Section 4). Diffusion coefficients are computed for the full ranges of energy and pitch angle that are of interest (Section 5). I n Section 6, we show t h a t energy diffusion is also an important mechanism for these interactions. Orders of magnitude are given in Section 7 together with a discussion of the validity of the use of two coefficients, in relation with the general diffusion equation which is derived in an Appendix. 2. NOTATIONS The medium is assumed to contain only one ion 'species' (protons) F B and F o being their gyro- and plasma frequencies respectively. In such a plasma, the A]fv6n velocity V a is given by

Vo

= cFB/F0

(1)

c being the velocity of light, provided that the plasma is moderately dense, t h a t is to say if (F0/FB) 2 >> 1. (2) The inequality (2) holds throughout most of the magnetosphere. All frequencies will now be normalized with respect to the proton gyrofreqnency FB, and all velocities with respect to the Alfv6n velocity V~ (f stands for f/_FB,and v for v~V~). Energy will be thus expressed in units of Eo = ½M~V~ ~ (3) in which M , is the proton mass. In order to distinguish between R and L waves, we will denote the former by positive frequencies, and the later by negative freqnencies. The longitudinal, transverse and total velocities of the interacting particles are denoted by vii, v . and v. The positive direction will be taken along the longitudinal beam velocity (vii > 0). With this assumption, it can be shown t h a t (p, L) interactions generate waves of negative phase velocity (v¢ < 0; 'backward emission'), and (:p,/~) interactions, waves of positive phase velocity (re > 0 but vt < vii; 'anomalous Doppler effect'). As was stated in Section 1, the major differences between the usual formulae and the correct ones arise for gyroresonant interactions involving low energy protons. During these interactions, ultra low frequency waves are generated. We will restrict ourselves, accordingly, to the 'hydromagnetic' part of the spectrum , that is, to normalized frequencies less than about 20. I n this frequency range,

Pitch-angle diffusion of protons due to interaction with hydromagnetic waves

1315

interactions with electrons occur only for such electrons as are relativistic; t h e y will not be studied here. We will consider only (p, 15) and (p,/~) interactions involving waves with frequencies such that 1 < f < 20 (4) and with strictly longitudinal propagation. 3. EMITTED FREQUENCIES With conditions (2) and (4), the phase velocity of the wave is given b y (OBAYASHI, 1965; GENDRIN, 1966) v, = ~(1 + f ) . 2 (5) = q-1 having the same sign as f. On the other hand, the gyroresonance condition, for interaction with protons co = --f2 B q- kvil becomes, b y transformation into the normalized units.

vu v* = 1 d-

1If

(6)

This represents the usual expression for the phase velocity of a cyclotron wave propagating along a beam (GALLET, 1964; GENDR~, 1965a). When combining conditions (5) and (6), one obtains a relation between the emitted frequency and the longitudinal beam velocity.

~(1 + f)3/2 vii

--

f

(7)

Relation (7) is represented on Fig. 1. One can see that, for (p, R) interactions, there is no emission ff the longitudinal velocity is less than a minimum value u 1. The threshold velocity u 1 and the corresponding emitted frequency are (GENDRIN, 1965b) f~ = 2 (8)

~I

=

3%/3/2

(9)

These expressions can be deduced directly from equation (7). Numerical applications of this equation to different magnetospheric regions, and to different energy ranges of the particles, are given elsewhere ( G E N D m N and DE LA PORTE DES VAUX, 1965). In the following sections we will need the relation between vii and v~. From conditions (5) and (6) we obtain v÷3 vll ~

v~ 9 --

(i0) I

In fact, v÷ is the most convenient parameter to use in carrying out this study. As a consequence of equations (4) and (7), we can fix a limit to the energy of the particles that we are speaking about. We are interested in protons with velocity v ~ 5 that is with energy E ~ 25. A typical value of E 0 in the magnetosphere

1316

:R. GENDRnV

J

/

I

h

%

-%6

i

I I z u.J

4

-

-

IR

uL_ u.J i--

~- 2 - F1

waves

i.....

r

~z u.J

0

i

-1 S

IL

I I

I

i

waves 2

1

l

5

LONOffUD,NAL BEAM VELOCITY LV,,/'V"a,)

Fig. I. Relation between the emitted frequencies and the longitudinal component of b e a m velocity (normalized units). I: (p, L) interaction; 2 and 3: (p, R) interactions.

being 5 keV, what we mean by 'low energy protons' are protons with energy less than ~ I 0 0 keV. 4. T R A J E C T O R I E S

OF T H E P A R T I C L E S

ll~ V E L O C I T Y

SPACE

Suppose that we have a hydromagnetic wave propagating along the constant magnetic fieldB, and a particle moving in the same direction with a velocity equal to the phase velocity of the wave. It can be shown that the force acting upon the particle, due to the electromagnetic fieldof the wave, is zero (seefor instance BRICE, 1964). Thus the energy of any particle measured in a frame of reference moving with the phase velocity of the wave, is constant, and we can write ½mv_L ~ --}- ½re(vii -- v÷)2 ---- constant

(11)

v ±dv ± --}- (vii -- v¢) dvli = 0.

(12)

so that Now, consider a set of values v±, vii and %, the last two quantities being related by equation (10). In velocity space, the particle is represented b y a point P(vll , v±) and the wave b y a point W ( % , 0). Equation (12) means that the differential increase of the particle velocity, dv is perpendicular to WP. In a frame fixed in space, the curves of constant energy are circles, centred at the origin (Fig. 2). Thus, in a (p, L) interaction, for which v~ < 0, the pitch angle must decrease ff we wish to take kinetic energy away from the beam (Fig. 2a). This is the reason why no emission can take place in such an interaction, ff there is no initial perpendicular component

Pitch-angle diffusion of protons due to interaction with hydromagnetic waves

o ---o//"

j

®

E_cons .

/

w

1317

o

| ' /t

(c)

v//

vl

-

."'"

/

./

w

E = const.. >

vH

Fig. 2. Movement in velocity space of a gyroresonant particle. (a) (p, L) interaction, vii and v~ are opposite in sign. The pitch angle of the particle must decrease if it is to loose energy. (b) (p, R) interaction, vii and v~ have the same direction. If the particle radiates energy, its pitch angle increases. of velocity, a conclusion drawn already b y N~.UFF.LD and WgmET (1963). ~or (p, /¢) interactions (v~ > 0) the pitch angle must increase (Fig. 2b). An algebraic demonstration of such properties has been also given b y BRINE (1964) in the case of the whistler mode (R wave). The trQectories of the particles in velocity space are obtained b y integrating equation (12) v÷ being a function of vii given b y equation (10). I t is easier to keep v± and v~ as independent variables, to integrate and then to go back to the original set of variables v±, v I. This is done easily b y writing equation (12) in the following manner v x d v ± + (vii - - v÷)(dvll -- dv,~) + (vii - - v÷) dv,~ = 0 (13) which becomes, with the help of equation (10) d v ± ~ "t- d(vll -- v4,)~ -t- 2

v~ dv,~ : O. v 4,~ - - 1

The equation of the trajectories is thus given b y equation (10) and v± ~ + Ivll -- v~[9 ~- Ln (v~2 -- 1) = constant.

(14)

(15)

We have kept vii -- v÷ as a variable, instead of v~/(v÷ ~" - - 1), in order to show that the integration of equation (12) does not give equation (11) again. This difference is due to the fact that, as vii changes, the wave with which the particle interacts no longer has the same frequency, and thus no longer the same phase velocity: v÷ cannot be kept constant during the integration of equation (12).

1318

R. GE~TI)RI~T

'"-.,

i',,

",,

"~

",

2

1

',,-

-~-

.

3

LONGITUDINAL BEAM VELOCITY(v~,/Va)

Fig. 3. Trajectories of the particles in velocity space. 1: (p, L) interactions. 2: (p,/~) interactions in which the emitted frequency is less than 2F/~. 3: (p,/~) interactions in which the emitted frequency is greater than 2_~B. The integration constant in equation (15) can be obtained by putting vlt ------u 0 and v± ----0. We thus obtain the system of curves drawn in Fig. 3. One sees t h a t these curves differ very much from circles centred at the origin, especially for (T, ~) interactions, and for (p, L) interactions involving very low energy protons. For higher energies (E ~ Eo), t h e y become similar to those given by KENNEL and PETSCHEK (1966), that is to say circles centred at the origin. 5. PITCH-ANGLE DIFFUSION COEFFICIENT

As the particle moves along its trajectory, its pitch angle changes. Starting from the equation tg~ = V_L/Vll (16) and by using equation (12), it is easy to show t h a t d=

=_ 1v± (1

~*~"~d~,.

-

-~-/

(17)

I n the low energy range, this is different from the value dec - -

dvll

v±

(18)

used by DU~GEY (1965), :K_E~EL and PETSCHEK (1966) and others, because, for these low energies, the phase velocity of the wave is no longer negligible, as compared to the velocity of the beam. For computing the pitch angle diffusion coefficient, we will follow the method given by KENNEL and PETSCKEK (1966). The reader is referred to this work for some detailed demonstrations t h a t will not be given here.

Pitch-angle diffusion of protons due to interaction with hydromagnetic waves

1319

Taking the general expression dv

m~-/ = e { E + v × ( B + b ) }

in which b is the magnetic field of the wave, one can see that the electromagnetic force, exerted by the wave upon the particle produces a change in its longitudinal velocity dr, m~ = ev. x b But ff we want to keep the normalization conventions adopted in Section 2, the unit of time must be taken equal to 1/~ B (the angular gyrofrequency ~B = 2~rFB) and the unit of wave number is ~B/v,. The computations will be made in this system of units, so t h a t one has dt -- V± x

(19)

which means t h a t b

(20)

IAvlll = v ± sin B At

0 being the phase angle between the magnetic component of the wave and the instantaneous perpendicular velocity of the particle. One sees that, depending upon the sign of 0, v, can be increased or decreased, but the mean squared value of Av, is not zero. Making use of equation (17), one obtains the pitch angle diffusion coefficient D~:

D~ -

At

--2

B

I--

v~_l At.

(21)

At must be taken roughly equal to the time during which a particle at distance Ak/2 out of resonance changes its phase 0 by 1 rad (KENN~,L and PETSCHEK, 1966). At ~.~ 2/vllA]c.

(22)

So t h a t

vii b~~ = b~lAk being the magnetic spectral density per unit wave number, at the wave

number k of the generated wave. D~ differs from the value given by Kennel and Petschek just by the last term, which is important only when v÷ is not negligible as compared to v. We will study the quantity r =-

1

[

1-

v,v,,]~-

(24)

which represents, for a given spectral magnetic density b k, the variation of the diffusion coefficient as a function of the position of the particle in velocity space,

1320

R . GENDRI~

uF,

,

a.

lc

IC

]

LONGITUDINA

BEAM

5 4 VELOCITY(~dva)

Fig. 4. P i t c h anglo diffusion coefficient as a f u n c t i o n of t h e longitudinal c o m p o n e n t of v e l o c i t y (normalized quantities). T h e indices 1, 2, 3 h a v e t h e s a m e m e a n i n g s as on Fig. 3. The t h i c k curve corresponds to t h e simple expression (25).

and which must be compared to the expression

Y x p = 1~vii. (25) On Fig. 4, the quantity Y is plotted as a function of Vii, for different values of the initial longitudinal velocity %. The quantity Y is also plotted as a function of ~ with the same parameter u 0 (Fig. 5). Figm'e 6 shows the variation of Y (~ ---- 0) as a function of u 0. One can see from Fig. 4 that the diffusion coefficient for (p, L) interactions is greater than YKP. The ratio between these two quantities can be as ga'eat as 20. On the other hand, pitch angle diffusion due to the effect o f / ~ waves is much less than that predicted b y equation (25). This result means that left hand waves do affect strongly the dynamics of the low energy protons, but that right hand waves haw practically no influence on it (in the ultra low frequency range at least). For (1o, L) interactions however, there is a value of ~ for which D~ is minimum as shown on Fig. 5. This implies that those particles for which ~ is equal to this value will diffuse slower than the others. The quantities ~min and Y ~ n are plotted as functions of the energy of the particles in Fig. 7. We have also plotted ~m~ and Ym~x as a function of E for interactions taking place with right hand waves (Fig. 8). I f the pitch angle diffusion were the only diffusion mechanism, the angular distribution function should present a minimum for ~ x and a maximum

P i t c h - a n g l e diffusion o f p r o t o n s d u e t o i n t e r a c t i o n w i t h h y d r o m a g n e t i c w a v e s

1321

t~

u_

2 u0

10"

t3a

10"

10-

0

10

20

30 40 50 60 PITCH ANGLE IX

70

80

90

Fig. 5. P i t c h a n g l e diffusion coefficient as a f u n c t i o n of c¢ for different v a l u e s of u 0. T h i c k lines: e q u a t i o n (25); t h i n lines: e q u a t i o n (24).

for ~min (asm, ming t h a t the spectral density bk is the same over the whole frequency range of possible interactions). 6. ENERGY DIFFUSION One of the most important facts which was obtained in Section 4 is that the trajectories of the particles in their velocity space are very much different from circles centred at the origin. Therefore, the energy of the particle is changing during the interaction. There is an important energy diffusion and the coefficient which charaeterises it can be computed on the same basis as was the pitch angle diffusion coefficient. Starting again with equation (12), the change in energy is expressed as dE

=

d(v ~)

=

2(v±dvj_ + vudvll )

=

2vtdvll.

(26)

Putting DE = "

<(~E)9> At

(27)

and using equations (20), (22) and (26) one obtains DE = 4

. v4'Uv±2 vii

(28)

1322

R. GENDRIN

10~ ~-

,

Approxtrn,t.ion (25)Ir~ Eluat.ion (24) 1 ........

"', i

10"1~

t

Y1

r I0"2~_

\

10-3i T

q

d

K

2

~ L L

I

o

J 3

,

i

4

F i g . 6. P i t c h a n g l e d i f f u s i o n coefllcient for a l m o s t l o n g i t u d i n a l b e a m s (~ --~ 0). F o r u o -~ ul, y ( ~ = 0) - * 2 / 2 7 ~-3, w h i c h is 9 t i m e s less t h a n YxP (~ = 0).

I

I

I

I

7O

Z -Jr

60 "U X

so r-

40 ~

1C

rr~

',(° \

~0 R 20

\ \

10

I I I 0 2 3 4 ENERGY Of THE PARTICLE5 Fig. 7. Mi~fimum pitch angle diffusion coefficient Ym*n and the corresponding value of pitch angle, ~min as a function of the energy of the pargicles (for (p, L) °o~

I 1

interactions).

Pitch-angle diffusion of protons due to interaction with hydromagnetic waves × 0~3

I

t

I

60

-~ C

C~ mix

~ z

1323

0,2

50

J,/./~

Y max

40 c n~

3 c~ C G~

z

30::: G')

o,1

2o E,

c}

3

o

lO =

z

O

0

I

I

5

O

I

10 15 20 ENERGY OF THE PARTICLES Fig. 8. Maximum. pitch angle ~ i o n coefficient, Ymax, and the corresponding value of pitch angle, ~max, as a function of the energy of the particles (for (p, R) interactions). T h e value of g = %2v±2 (29) vii is p l o t t e d in Fig. 9, as a function of u. One sees t h a t t h e e n e r g y diffusion is negligible w h e n c¢ --* 0 (or u --~ %) b u t increases v e r y r a p i d l y as soon as a is different from zero. 7. I)ISCUSSION Before discussing the real meaning of t h e two coefficients t h a t we h a v e i n t r o d u c e d it is worthwhile to give some orders of m a g n i t u d e . To do this, we will t r a n s f o r m t h e spectral density per u n i t w a v e l e n g t h into a spectral d e n s i t y per u n i t f r e q u e n c y band, a n d t a k e t h e o p p o r t u n i t y to come b a c k to the usual s y s t e m of units (except for e n e r g y which will be always expressed in units o f M~V,,2/2). One has bk~ = ~ 1

bI~ 1 " ~ve" ~2B

SO t h a t (A~> = ~

L BJ

LV~J

= ~ L-B-A L~_i

Y At

Z ~t

or

~ 1 . 6

× 10 -3

Y At

~ 6 × 10 -a Fb'-I [ % 1 z At

L~fJLV2

if one expresses the power spectral d e n s i t y of t h e wave in units of 7~.ttz -1.

(30) (31)

R. G~.~roRr~

1324 l

=

102

5

LLo=5

I -/"

5'", ".....

,0,5',,

"'"',,

"""

-

4

,,

"2""

!

1

2 3 LONGITUDINAL BEAM VELOCITY

4

(v,,/Va)

Fig. 9. Energy- diffusion coefficient as a f u n c t i o n o f t h e l o n g i t u d i n a l c o m p o n e n t of b e a m v e l o c i t y , for different v a l u e s of u 0. T h e indices 1, 2, 3 h a v e t h e s a m e m e a n i n g as o n Fig. 3.

These expressions show that the diffusion--for a given value of b2/Af--does not depend on the magnetic line of force which is involved because v, is always of the order of V=. Thus, if the hydromagnetic noise level is approximately constant throughout the magnetosphere between 3 ~ L ~ 6, the diffusion will roughly not be L-dependent. From Figs. 4 and 9, it can be seen that typical values for Y and Z are 1 and 5 respectively. Satellite measurements made with e G O I (HoLz~R, McL~oD and S ~ T ~ , 1966.) show that the hydromagnetic spectral density is between 10 -9 and 1 7~-Hz -1 for frequencies around 1 Hz. Choosing the highest of these two values, and assuming a time of interaction of 10 sec, we will obtain characteristic orders of magnitude for the total changes in pitch angle and energy (one must remember that strong hydromagnetic noise can be present in the magnetosphere during many hours). With these assumptions, one obtains %/((Aoc)2} ~ 10°;

t/(~)

~) ~-~ 2 k e y

(32)

which are very high values. Diffusion due to whistlers will give numbers very much lower because the corresponding values of b2/Af at 2 kHz are between 10-1° and 10 -8 72-Hz -z aeeording

Pitch-angle diffusion of protons due to interaction with hydromagnetic waves

1325

to the measurements made with I N J U N I I I (Gu~Nv.TT and O'BRI~.N, 1964). Of course v,fl V a is of the order of 30, but this is not sufficient to give a strong pitch angle diffusion in one event as it can be the case for any individual pulsation train of a Pc 1 event for instance. Moreover there is, in this case, no energy diffusion. Now we will discuss the validity of having introduced two coefficients D~ and D E. Due to the interaction with the wave, the particles are moving in their velocity space. Their representative points must follow the curves (C) defined in Section 4. The speed at which t h e y are moving along these lines depends upon the spectral intensity of the wave at the resonant frequency. But, if we let the interaction take place during a sufficiently long" time, the density distribution of the particles along these curves will be the same. I n other words, the distribution function of the particles will be a constant along these lines.

af dr± -t- Of dVll = 0 Or. ~vtt

along (C)

(33)

or, introducing equation (12) v.

Of

-

(v, -

Of = 0

along (C).

v,) Or.

(34)

This suggests (DwcoEY, private communication) that the diffusion equation should contain differential operators of the form 0 2v

-

v.

0 -

(vii -

v,) 0v.

More precisely, the diffusion equation should be

0--}----J ~v

D

(36)

in which D is the diffusion coefficientand J the determinant of the metric of the system in the v coordinate system which is used for expressing the differential operator. That the diffusion equation should take this form can be partially explained by the following arguments. (I) Any function which is constant along a curve (C) should be a stationary solution of equation (36), so that the last operator ~/~v should have the form of equation (35). (2) The first o p e r a t o r ~ / ~ v e x p r e s s e s t h e divergence of a flux, so that the Jacobian J must appear in the formula in the way in which it is written (for this and related problems, see for instance ~-TAERENDEL,1967). The validity of equations (35) and (36) will be demonstrated in the Appendix but one must notice t h a t D is the only diffusion coefficient that we can rigorously use in such a problem. The diffusion proceeds along the lines (C). As a consequence, there are changes in a and E, but D~ and D~ are just 'images' of what these changes are. They do not represent any specific coefficient which could be introduced into two separate diffusion equations.

1326

R. GENDRIN 8. C O N C L U S I O N

The gyroresonant plasma-beam interaction in the magnetosphere leads to the generation of an electromagnetic wave. The electromagnetic field of the wave reacts upon the particles and changes their pitch angle and energy distribution functions. We have studied these effects in detail for low energy protons (E ~ 100 keV) interacting with ultra low frequency waves (f ~ 100 Hz), and we have given the trajectories of the particles in their velocity space for various conditions. Semiempirical diffusion coefficients have been computed both for pitch angle diffusion and for energy diffusion. It has been shown that the diffusion effects are more pronounced than foreseen, especially for interactions with left-hand waves. Orders of magnitude have been computed, which show that gyroresonant interactions with the hydromagnetic noise (which is always present in the magnetosphere), could be a determinant process in the dynamics of low energy protons. A general diffusion equation for these processes has been worked out and its consequences will be discussed in the future. Aclc~owledgements---I would like to t h a n k Professor J. ~V. DUNGEY (Imperial College, London), who called m y attention on the interest of the velocity space considerations both during the lecture he gave at a summer school (DuNGEY, 1963), and in private discussions. I am also greatly indebted to Drs. LAVAL and PELAT (Commissariat & l'Energie Atomique, Paris) for m a n y valuable suggestions in the derivation of the Appendix. REFERENCES BELL T. F. and BUNEI~LAN O. BRICE N. M. C O R N W A L L J. M. C O R N W A L L J. M. C O R N W ~ J. M. D R A G T A. J. D U N G E Y J. W .

1964 1964 1964 1965 1966 1961 1963

Phys. l~ev. 133, 1300. J. Geophys. t~es. 69, 4515. J. Geophys. Res. 69, 1251. J. Geophys. Res. 70, 61. J. Geophys..Res. 71, 2185. J. Geophys. l~es. 66, 1641. Geophysics: the Earth's

environment

GENDRIN R . GUHNETT D. A. and O'BRIEN B. J. ]~'A~2RENDEL G.

1967b 1964 1967

HOLLER R. E., MCLEOD M. G. and SMITH E.J. HRUSKA A. JACOBS J. A. and WATAI~ABET. K E N N E L C. F. and PETSCHEK H. E. LrEMoHN H. B. NEuFu'.LD J. and W R I G H T H. NEUFw.LD J. and WRIGHT H. NEUFELD J. and ~VRIGHT H. OBAYASHI T. STEFANT R. and VASSEbU~G. S U D A N R . N.

1966

(edited by C. de Witt, J. Hieblot and A. Lebeau), Gordon and Breach New York, pp. 503-550. Space Sci. Rev. 7, 314. J. Geophys. t~es. 69, 65. Earth's Particles and Fields, Freising, Germany. J. Geophys. l~es. 71, 1481.

1966 1966 1966 1967 1963 1964 1965 1965 1965 1965

J. Geophys. l~es., 71, 1377. J. Atmosph. Terr. Phys. 28, 235. J. Geophys. l~es. 71, 1. J. Geophys. tees. 72, 39. Phys. Res. 129, 1489. Phys. l~ev. 135, 1175. Phys. Rev. 137, 1076. J. Geophys. Res. 70, 1069. Compt.-Rend. 260, 1465. Phys. Bey. 189

Pitch-angle diffusion of protons due to interaction with hydromagnetic waves

1327

Reference is also made to the following unpublished material BELL T.F. 1964 Technical Report No. 3412-5, University of Standford. DRVmVm~rI)W. E. and P I ~ S D. 1961 Prec. Conf. Plavma Phys. Control Nuclear tPusion Res. (Austria). GENDRINR. and D~.LAPORTEDESVAUX 1965 Technical Report GRI/NT]51, Oroupe de B. Recherches Ionosph~riques. APPENDIX Derivation of equation (36) Each varying quantity is assumed to be proportional to exp [i(a~t --kz)], in which z is the direction of the constant magnetic field B. Thus we are interested only with strictly longitudinal propagating waves. One starts with the equations of the quasi-linear theory (DRuM~m~D and PriEs, 1962) : [ 0 + v. V + ~q (v x B) .--~1 f~ =

0 + m--q ( v × B ) . ---~1 f o =

mq [ E + v x b ] .

mq[E* + v

afz xb*].--Ov

Of° a-v-

(A1)

(A2)

in which E, b correspond to the perturbation field; fl to the varying part of the distribution function, f0 to the homogeneous part of it (Vf0 = 0). q and m are the charge and mass of the particles of interest. From now on, we will restrict ourselves to protons, so t h a t f0 and fl are the distribution functions for protons (q > 0). E* and b* are the complex conjugates of E and b, so that equation (A2) corresponds to the zero-order Fourier component of the Boltzmann-Vlassov equation. Equation (A1) is the first-order Boltzmann-Vlassov equation. When used in conjunction with Maxwell's equations, it gives the usual dispersion relation, including the effects of amplification or damping of the waves (see for instance B E ~ and BUN~M~¢, 1964). We begin to change from the Cartesian coordinate system: %, %, %, to a cylindrical coordinate system: vii, v±, O, 0 being the phase angle of the velocity vector. We have 0 a ~- COS 0 - 0% Or± 0 a

av% = s i n

0 OVa

o

sin0 0 v± 00 cos 0 a +--v± 00

0 OVlt

so t h a t

a.f

(v x B ) . K =

- B af a~"

(A3)

1328

R. GENDRI~

Using Maxwell's equation --k

×

E

--oJb

=

(A4)

for strictly transverse waves (E, b _]_k), one obtains

af

= ½[(E+ e-i°(..~ ' -- i ~ ) "t- E_ e+~°(A ÷ iB)] f

(E + v × b) • ~

(A5)

in which

d= .~ =

~-

~+---¢0

__I I--

9vll

(A6)

----r~90

vi

(AS)

E ± = Ex ± iE~.

Equation (A2) thus becomes [a - - ~ B ~ a I f o - - - - - - ( E * ÷ v × b * ) ~afl .

(A9)

The distribution function fo is a slowly varying function. This means that we are interested in variations of fo in times very much larger than 1 / ~ , so that, to a first-order approximation

af,,

-

a0

o

(A~0)

and, by use of equations (A5) and (A7): (E ÷ v

×

b ) - ~9fo = ½[E+

e_iO ÷

E _ e +~°] d f o .

(All)

Now, turning back to equation (A1), and solving for fl = fl(O) exp [i(wt --kz)], with the condition that f 1(0 + 2~) = fl(0), one obtains

f~(o)

i q F" 2~ kw

E+e -i° --

kvLi ÷ aB

÷

E_:__e+~° _]

0)

__:kvll ~a.j~'fo

(A12)

Knowing fl as a function off0 , it is now possible to put it in equation (A2) and to obtain the diffusion equation for fo. But the expression for (E* ÷ v × b*) • Of/Ov must be written

al = ½[E+* e +i° (~' + i ~ ) + E_* e - i ° ( d -- i~)] f (E* + v × b*) • -~

(A13)

with E:e* ---- E~* :h iE~*.

(A14)

Pitch-angle diffusion of protons due to interaction with hydromagnetic waves

1329

Using equation (A12), it is easy to show that

iq I ~ f l ---- 2 ~ ~¢ ~fl--q

E+ e-~° o) - - kVll +

[1--~][

~B

+

E

-

e+'°

]

o~ - - kVll - -

dfo

E+ e-'° E-e+~°Bl ¢o -- kVll + £2B -- co -- kVll -- ~

2mv--~

(A1 5)

~fo

(A16)

so that the right-hand part of equation (A2) can be computed, using equations (A13), (A15) and (A16). The left-hand part of equation (A2) is reduced to arc/at , because of equation (A10), so that an integration over the phase angles can be made: 2~r - = -Of° - = f2,, dO x (right-hand side). at o Only the terms which include the product E+* • E_* = IE+l 2 will contribute to the integral, the other products leading to terms in exp (%2i0) or exp (--2i0) which give no contribution to the integral. Finally, one obtains af° 1 4m q2 9-[A + G] I at --- 2~ri" co

--

IE+I~ kv31 + ~ B +

BI

IE-I2 co

--

kVll

--

[A]fo

(A17)

in which the operators A and G have been defined in equations (A6) and (A7). I f we restrict ourselves to diffusion due to a single t y p e of circularly polarized waves, the whistler mode for instance (IE_ I = 0), we see that the interaction is important only with waves for which the wave number satisfies the gyroresonance condition (o~ = - - ~ n + kvil in this example). B u t the total change of f0 must be computed b y integrating over all wave numbers

afo

1

e

t

a---t = 2wi" 4m 2 [~¢ + W] t&-k d~o-ak/2 o~ -- kvll + ~B)[zdJf°"

(A18)

Here we have admitted that the electromagnetic noise is restricted to a finite bandwidth Ak centered around some vahie k 0. Assuming a small imaginary part in the denominator of the integrand, and performing the integration of the Cauehy integral*, it comes

0fo = 1. e 2 at

2 ,~m

+

V!E+(k°)t3 Ed]/0. L vilAk J

(A 9)

Due to the definition of E+, IE+I~ is equal to 4 times the square of modulus of the real electric field of the wave. Moreover, using equations (Ad), (A6) and (A7), another rearrangement can be made, so that we obtain 3fo

[ __3 _

[I

3 ]]{I 9b~.B (~

~"v,.

3

av~] (A20)

bk 2 = b~/Ak having the same meaning than in Section 5. * We have assumed a negative imaginary part for to, e.g. a wave which is amplified. When the imaginary part of (o is positive (damped waves), care must be taken when performing the integration of equation (A18). We will not discuss this case here.

1330

R. GENDRII~

The first differential operator can be written

{ °°,,a --

[E

~ - -

Y±

oo,°

--

} (A21)

so that equation (A20) can be p u t in the form suggested b y Dungey

0fo

1 3[

at -- v ± ~ v

v±

D~l

(A22)

~v J

with a .~v-- = v± ~v I D --

a (v I - v~) av± 1 ~'~B2 bk~ 2 v I Be

(A23) (A24)

and v± -~ J being the expression of the volume element (dv=dv~dv, = vidvldv±dO ) Equations (35) and (36) have thus been demonstrated. It is of some interest to see in which conditions, the diffusion equation (3.5) from KENNEL and PETSCHEK'S paper is valid. In order to do this, we will change the variable from v±, v I to v and 0

a ~Ov

a

0

av---~----sin

cos ~ 0 ]

3- ~

(A25) sm ~ 0 ) " •

OVll -- COS 0¢ av

v

0-~

Performing the transformation, and assuming that v~ ~ vii

(A26)

the terms in a/Ov disappear, v± also and one has

Of Ot

1

a [

sin~a~

Dsin~¢

ofl

.

(A27)

Apart from a factor of 2, the formulas (A27) and (A24) give the same diffusion equation and the same diffusion coefficient as were given b y KEN~V.L and PETSOHEK (1966), on the basis of a semi-phenomenological treatment. The equation (A27) can be used only when condition (A26) is fulfilled, which is not the case for interactions involving ultra low frequency waves in the magnetosphere.