Shortly after a strongly anisotropic beam of charged particles is injected along a guiding magnetic field on which is superimposed a small random conponent, the particle density can be represented by a Gaussian profile whose center moves with the coherent velocity and whose width increases with time at a rate controlled by the coefficient of dispersion. Both parameters depend upon the mean free path, which characterizes scattering by the random fields, and the focusing length, which characterizes spatial variations of the guiding field. These dependencies are known explicitly for the coherent velocity. Formulae for coefficient of dispersion are available only in the limits of very weak and very strong focusing. A new expression for coefficient of dispersion, which spans this gap, is presented

Earl, J. A.

English

NASA Technical Reports Server

872 SH4.1-2
THE DISPERSIVE EVOLUTION OF CHARGED-PARTICLE BUNCHES
IN RANDOM MAGNETIC FIELDS
J.A. Earl
Dept. of Physics & Astronomy, Unlv. of MD, College Park, MD 20742
ABSTRACT
Shortly after a strongly anlsotropic beam of charged
particles is injected along a guiding magnetic field on which
is superimposed a small random component, the particle
density can be represented by a Gaussian profile whose center
moves with the coherent velocity V, and whose width increases
with time at a rate controlled by the coefficient of
dispersion D,. Both parameters depend upon the mean free
path I, which characterizes scattering by the random fields,
and the focusing length L, which characterizes spatial
variations of the guiding field. These dependences are known
explicltly for V,. _ormulae for D, are available only in the
limits of very weak and very strong focusing. This paper
presents a new expression for D_, whlch spans thls gap.
i. Introduction. The equation which describes particle transport
along a guiding field under the comblned influences of scatterlng
by magnetic turbulence and focusing by spatial inhomogenieties of
the guiding field is
_f _f i G _ -G _f
_-_+ _V_ = _ e -_e -_, (I)
where f is particle density in phase space, z is distance along
the guiding field, _ is the pitch-angle cosine and V is particle
velocity. If the coefficient of pitch-angle scattering is given
by the standard form
• 3(V/X) (i_ 2)l_lq-I
= (2-q)(4-q) ' (2)
where q is an index that measures the anisotropy of scattering,
then the odd function G is given by
G{z,_} = V _B _ i-} (4-q) 1
B _z _ +_7_} dv = 3 L _l_ll-q, (3)
where L is the focusing length and B is the guiding field.
Throughout this paper both I and L will be assumed to be constant.
Even with these assumptions, it is not posslble to solve
the transport equation in closed form. However, there are
approximations that provide considerable insight into phenomena,
such as those outlined in the abstract, that do not fit into the
familiar picture of purely diffuslve transport. A first-order
analysis (Earl, 1981; Kunstmann, 1979) leads to simplified
transport equations which describe coherent propagation in terms
of infinitely narrow pulses, but which do not include the
dispersive spreading of these disturbances. This paper presents a
372 SH4.1-2 
THE DISPERSIVE EVOLUTION OF CHARGED-PARTICLE BUNCHES 
IN RANDOM MAGNETIC PIELDS 
J.A. Earl 
Dept. of Physics & Astronomy, Un1v. of MD, Coll ge Park, MD 20742 
ABSTRACT 
Shortly after a strongly an1sotropic beam of charged 
particles is injected along a gu ding magnetic field on which 
is super1mposed a small random component, the particle 
density can be repr sented by a Gaussian profile whose center 
moves with the coh rent velocity V* and whose width increa es 
with time at a rate controlled by the coefficient of 
dispersion D*. Both p rameters d pend upon the mean free 
path A, which ch racterizes scattering by the random f1elds, 
and the focusing length L, which ch racterizes spat1al 
variations of the gu ding field. Th se d pendences are known 
explic1tly for V*. Tormulae for D* are available only in the 
limits of very weak and very strong focusing. This aper 
presents a new expression for D~, wh1ch spans th1s gap. 
1. Introduction. The equation which describes particle transport 
along a gu ding field under the comb1ned influences of scatter1ng 
by magnetic turbulence and focusing by spatial inh mog ni ties of 
the guiding field is 
1£ + V~ = 1. e G ~"",-G ~ (1) 
at ]..I Clz 2 Cl]..l 'I"" Cl]..l' 
where f is particle density in phase space, z is distance along 
the gu ding field, ]..I is the pitch-angle cosine and V is particle 
velocity. If the coefficient of pitch-angle scattering is given 
by the standard form 
</J = 
3(V/A) 2 q-l (2-q)(4-q) (1-]..1 ) I ]..II , (2) 
where q is an index that measures the anisotropy of scattering, 
then the odd function G is given by 
2 
G{ } V ClB r ]..II-v d (4-q) A Il l - q (3) z,]..I = - B Tz 0 <Hz, 'J} V = --3- L ]..1]..1 , 
where L is the focusing length and B is the gu ding f1eld. 
Throughou  this aper both A and L will be assumed to be constant. 
Even with th se assumptions, it is not poss1ble to solve 
the transport equation in closed form. However, th re are 
approximations that provide considerable insight into phenomena, 
such as those outlined in the abstract, that do not fit into the 
familiar picture of purely diffus1ve transport. A first-order 
nalysis (Earl, 1981; Kunstmann, 1 79) leads to simplified 
transport equations which describe coh rent prop gation in terms 
of inf nitely narrow pul es, but which do not include the 
dispersive spreading of th se disturbances. This aper pr sents a 
https://ntrs.nasa.gov/search.jsp?R=19850026515 2020-03-20T17:01:16+00:00Z
378 SH4. I-2
second-order analysis which includes this effect.
In the context of new possibilities for numerical solutions
opened up by advances in computer technology, the phenomenon of
dispersion takes on special slgniflcance, for there is an
artifact, which I call "numerical dispersion", and which can
obscure the physical effect. (See Paper SH 4.1-4) Numerical
dispersion is an artifact of the Boltzmann operator that appears
on the left hand side of equation (I). Consequently, it is not
affected by focusing. To avoid inaccuracies, it is necessary to
specify sufflaiently fine spatial resolution in the numerical
implementation so that the numerical dispersion coefficient is
much smaller than the physlcal one. This paper provides the
quantitative knowledge of D, needed to make this specification.
2. The Coefficient of Dispersion. The second approxlmation to
the distribution function takes a form
BeGH_ _F_ _ BeGw{__} _Hf = F_{z,t} + {z,t} + W{_} _z _z#, (4)
in which the first two terms represent supercoherent and
pseudodiffuslve components that appeared in the first-order
approximation and the second two are anisotropic components that
are proportional to the spatlal gradients of F_ and H_. Both of
these additions involve the same function W{_}, but the sign of
its argument is different in the two new terms. This relationship
reflects a fundamental symmetry of focused transport that I call
the "principle of complementarity". When equation (4) is
substituted in the transport equation, and when the first-order
transport equations are satisfied, the following equation for W is
obtained:
i _ e-G _W{_} V#K V_K -G V#K
---- W_-U} = _uV + --) e , (5)
2 _ _ L(K2_I) (K2_I) (K2_I)
where
i r +leG d_, (6)K =_ -i
is a normalization constant that goes from K = 1 in the limit of
weak focusing (I/L << i) to K = _ in the limit of strong focusing,
and
V 1+1 Gv# =ii (7)
is a characteristic velocity that I call the pseudodiffuslve
velocity. Note that this is not the coherent velocity V,, which
is given by
V#K
V, (K2_I)I/2 • (8)
Although the presence of the term in W{-B} complicates
equation (5), it can be solved for W with the aid of the method of
373 SH4.1-2 
second-order analysis which includes this effect. 
In the context of new possibilities for numerical solutions 
opened up by advances in computer technology, the phenomenon of 
dispersion takes on special s1gnif1cance, for there is an 
artifact, wh1ch I call "numerical dispersion", and which can 
obscure the physical effect. (See Paper SH 4.1-4) Numerical 
dispersion is an artifact of the Boltzmann operator that appears 
on the left hand side of equation (1). Consequently, it is not 
affected by focusing. To avoid inaccuracies, it is necessary to 
specify suff1~iently fine spatial resolution in the numerical 
implementation so that the numerical dispersion coefficient is 
much smaller than the phys1cal one. This paper provides the 
quantitative knowledge of D* needed to make this specification. 
2. The Coefficient of Dispersion. The second approx1mation to 
the distribution function takes a form 
G aF G aH f = F1r {z,t} + Be Hf {z,t} + W{)J} azfP - Be W{-)J} az~' (4) 
in which the first two terms represent supercoherent and 
pseudodiffus1ve components that appeared in the first-order 
approximation and the second two are anisotropic components that 
are proportional to the spat1al gradients of F~ and H~. Both of 
these additions involve the same function W{)J}, but the sign of 
its argument is different in the two new terms. This relationship 
reflects a fundamental symmetry of focused transport that I call 
the "principle of complementarity". When equation (4) is 
substituted in the transport equation, and when the first-order 
transport equations are satisfied, the following equation for W is 
obtained: 
(5) 
where 
K=.!.r+lGd 2 -1 e )J, (6) 
is a normalization constant that goes from K = 1 in the limit of 
weak focusing (AIL « 1) to K = 00 in the limit of strong focusing, 
and 
(7) 
is a characteristic velocity that I call the pseudodiffus1ve 
velocity. Note that this is not the coherent velocity V*, which 
is given by 
V1,K 
V = (8) 
* (K2_1)1/2· 
Al though the presence of the term in W{ -)J} complicates 
equation (5), it can be solved for W with the aid of the method of 
874 SH4.1-2
eigenfunctions. In fact, because this term is relatively small, W
can be evaluated fairly accurately by a slmple double integratlon.
When the additional terms in f are taken into account, the
second-order transport equatlons take the forms
_F_ _ B _H§ _2F_
_--_---V# _+ K _ - D, (9)
_z2 '
and
_I_ _H_ 1 _FT _21_
_--_--+V# _+ B-K_t-- = D, -- (i0)
_z2 '
in which the left hand sides reproduce the first-order transport
equations and the right hand sides are second-order terms• Here,
the coefficient of dispersion, given by
1 +i V#K 2 V#K
--- _i d_ ((_V +-- -G _ __) W{_}, (Ii)
D, = 2K K2-I) e K2-I
determines the size of the new terms, which involve the second
spatial derlvatives of the supercoherent and pseudodlffuslve
components. They describe the dispersive aspect_ of coherent
propagation, in which the delta function pulses of the first-order
description are more accurately described as Gaussian disturbances
whose centers move with the coherent veloclty V, while thelr
widths increase with time at a rate controlled by the coefficlent
D •
In the I.O .......... ..•..._
• I I|_ I I I I I II
figure, separate
curves show how D, "'" q =1.5
depends upon (I/L) D# "'..
when the function -- t
W{_} is evaluated D
in three different
ways. The dashed
curve gives the O.I
exact result, in
which the term in
W{-_} is retained ---.... __..____
in equation (5). _ "_
The solid curve
gives a much simp-
ler result obtain-
ed from an approx-
imate evaluation O.O1-
in which this term
is ignored. For
most applications,
in which its _20% .00_I , , , i ,,,,I i , , , , l,,
deviation from the O.I ;.0 I0.0
exact result is _/
not important,
3 4 S 4.1-2 
eigenfunctions. In fact, because this term 1S relatively s all,  
can be evaluated fairly accurately by a s1mple double integrat10n. 
hen the additional terms in f are taken into account, the 
se - r er tran rt e t10  take the form  
 
aF, B aH§ a2FII' 
II az + at = * -2-' 
az 
a2H J 
--2-' 
az 
(9) 
(10) 
in hich the left hand sides reproduce the first-order transport 
equations and the right hand sides are second-order terms. ere, 
th  effi i nt  i r i , iv   
(11) 
i es h      er s, hi  n   ec  
spatial der1vatives of the supercoherent and pseudod1ffus1ve 
co ponents. hey describe the is ersi e aspect_ of coherent 
r agation, i  hi  t  lt  f ti  l  f t  first-or r 
scri ti   or  rat l  s ri   aussi  ces 
hose ce ters ove ith the c ere t el c1t  * hile t e1r 
idths increase ith time at a rate controlled by the coeffic1ent 
  
* 
In the 
i r , arat  
curves show how D* 
depends upon (AIL) 
hen  ct  
{ lJ}  al ated 
  i t 
ays.  as ed 
curve gives the 
exact result, in 
hic   t  i  
{-lJ} is retained 
i  uati n ). 
 li  r  
i es a uch si p-
 sult btain-
 fr   prox-
at  aluati n 
 hich t i  t  
 red. or 
ost pplicati ns, 
i  hich it  ~  
eviati n f  t  
exac t resul t is 
not i portant, 
I. a ,------,,..,.., .... 7 •• =-=.-:-. -•• -.T ••TT,----,--,--,--,rrTTn 
'  
....... q=1.5 
0.1 
0.01 
. 30.1 /.0 
AIL 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
. 
1 .  
S75 SH4. I-2
this approximation is adequate. In contrast, the dotted curve, which
gives a result obtalned by Kunstmann (1979) and by Bieber (1977_
deviates by a factor of ~20 at small values of (%/L). These results
grew out of an analysis which describes correctly the strong focuslng
regime, but which does not take into account the coupling between
components embodied in equations (9) and (i0). Unless focusing is very
strong, this neglect leads to a large overestimate of the dispersive
effect.
3. Summary. The coefficlent of dispersion given by equatlon (ii) is a
basic parameter needed to implement numerical solutions of the transport
equation. In this context, where great accuracy is not required, the
limitations of the present analysis to constant % and L are not
important.
4. Acknowledgements. This research was supported by the Natlonal
Aeronautics and Space Administration through grant NGR-21-002-066.
References
Bieber, J.W. 1977, Ph.D. thesis, University of Maryland.
Earl, J.A. 1981, Ap.J., 251, 739.
Kunstmann, J. 1979, Ap.J., 229, 812.
o
375 SH4.1-2 
this approximation is adequate. In contrast, the dotted curve, which 
gives a result obtal.ned by Kunst ann (1979) and by Bieber (1977, .. 
deviates by a f ct r of "'20 t s all al es f (AiL). hese r sults 
grew out of an analysis which describes correctly the strong focusl.ng 
regi e, but hich does not take into account the coupling bet een 
co ponents e bodied in equations (9) and (10). nless focusing is very 
strong, this neglect leads to a large overesti ate of the dispersive 
ffect. 
3. Su mary. The coeffic1ent of dispersion given by equat10n (11) is a 
basic para eter needed to i ple ent nu erical solutions of the transport 
equation. In this context, where great accuracy is not required, the 
li itations of the present analysis to constant A and L are not 
i portant. 
4. cknowledgements. This research as supported by the at10nal 
eronautics and Space dministration through rant GR-21-002-066. 
eferences 
ieber, J. . 1977, Ph. . thesis, niversity of aryland. 
Earl, J. . 1981, p.J., 251, 739. 
Kunstmann, J. 1979, Ap.J~229, 812. 
 


The dispersive evolution of charged-particle bunches in random magnetic fields

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