It is not trivial to solve the equations that describe charged particle transport with the aid of computers, for instabilities, inaccuracies, and subtle artifacts are well known afflictions of numerical analysis. Two specific points are discussed. First to avoid inaccuracies, pitch angle scattering must be treated with great care. In particular, slightly inappropriate numerical formulations give rise to mean free paths that are in error by large factors. Secondly, A previously unrecognized artifact, numerical dispersion, is very similar to the physical phenomenon of dispersion. To avoid misinterpretations arising from this similarity, the spatial increment of the finite difference grid must be a small fraction of the mean free path. These points are illustrated by calculations based upon finite difference approximations to the transport equation

Earl, J. A.

English

NASA Technical Reports Server

380
SH 4.1-4
NUMERICAL AND ANALYTIC DESCRIPTIONS OF COSMIC-RAY TRANSPORT
J.A. Earl
Univ. of MD, Dept. of Physics & Astronomy, College Park, MD 20742
ABSTRACT
It is not trivial to solve the equations that describe
charged particle transport with the aid of computers, fo_
instabilities, inaccuracies, and subtle artifacts are well
known afflictions of numerical analysis. Two specific points
are:
i. To avoid gross inaccuracies, pitch-angle scattering must
be treated with great care. In particular, slightly inappro-
priate numerical formulations give rise to mean free paths
that are in error by large factors.
2. A previously unrecognized artifact, "numerzcal disper-
sion", is very similar to the physical phenomenon of dis-
persion. To avoid misinterpretations arising from this
similarity, the spatial increment of the finite-difference
grid must be a small fractzon of the mean free path.
These points are illustrated by calculations based upon
finite-difference approximations to the transport equation.
I. Introduction. The diffusive idealization, which has been almost
universally invoked in discussions of cosmic-ray transport is easy to
treat analytically. However, many observed phenomena give clear evi-
dence for the presence of non-diffusive effects. One example is the so-
called "scatter-free" propagation of kilovolt solar electrons whzch is
inconsistent with diffusion, but which can readily be interpreted in
terms of the coherent mode of propagation. Although the qualitative
features of these effects have been outlzned (Earl, 1974a, 1976), the
theory is very complicated. Consequently, there is a need for reliable
numerical computations which bypass these complexities and yield con-
crete results that are well suited for comparison with observations.
This paper explores such methods within the Izmited context of
rectilinear propagation of cosmic-ray along a uniform guiding field on
which are superimposed random fzelds.
Under these circumstances, transport is described by
_f _f _ _f
(i)
in which f is particle density in phase space, _ is the pitch-angle co-
sine, and z is distance parallel to the gulding fzeld. _e parameter
s = Vt, where V is particle velocity, plays the role of a temporal vari-
able. The coefficient of pltch-angle scatterzng is given by
_i_° (3/2%) (I-_2) E_Iq-l, (2)where % _eq)_4a_ ) free path, and q Is an index that measures the
anisotropy of scattering.
 
SH -4 
E I   TI  I  OF I R   
J  rl 
i . f , ept. of si s  stron y, lle  r ,   
 
It is t tri i l to l  th  e ti  th t ri  
r  rti l  tra s rt it  the i  f puters, ~ 
t iliti s, in r cies,  t  f   ell 
known afflictions of nu erical analysis. Two specific points 
: 
1. o a i  r ss in racies, itc - l  s tt ri  ust 
be treated ith great care. In particular, slightly inappro-
priate nu erical for ulations give rise to ean free paths 
t at are i  err r by large fact rs. 
2. A previously unrecognized artifact, "nu erl.cal disper-
sio ", is er  si ilar t  the sical pheno enon f is-
persion. To avoid isinterpretations arisi  from t is 
i ilarit , t  ti l i re ent f th  i ite i ce 
ri  ust be a s all fractl.on of the ean free path. 
ese i ts r  illustr t   l l tl. ns ase  upon 
fi ite- iff r e a r xi ations t  the tr s ort ati n. 
1. t cti n.  i i  li ti , hi    l ost 
universally invoked in discussions of cos ic-ray transport is easy to 
tr t alytically. owever, any ser ed pheno ena i e l r e i-
dence for the presence of non-diffusive effects. ne exa ple is the so-
ll  att r-fr e" r pagati n f il olt l r l t s hl.c  i  
i c nsistent ith iff si n, but hich can rea il  be i ter rete  i  
ter s of the coherent ode of propagation. lthough the qualitative 
features of these effects have been outll.ned ( arl, 1974a, 1976), the 
theory is very co plicated. onsequently, there is a need for relia le 
nu erical co putations hich bypass these co plexities and yield con-
crete res lts that are ell suited for co parison ith observations. 
his er plores  ethods ithi  t  ll. it  ntext  
rectilinear propagation of cos ic-ray along a unifor  guiding field on 
hich  r posed  l.el s. 
nder these circu stances, transport is described by 
1E. + af a I af 
as llaz = ~ 111 1Jj (1) 
in which f is particle density in phase space, ~ is the pitch-angle co-
sine, and z is distance parallel to the gUl.ding fl.eld. TIle para eter 
s = Vt, where V is particle velocity, plays the role of a temporal vari-
able. The coefficient of pitch-angle scatter1ng is given by 
here 
1/1= (3/2;) (l-l)I~lq-l, (2) 
A is <t~-eq) ~eag) free path, and q 1S an index that easures the 
nisotr py f att ri g. 
https://ntrs.nasa.gov/search.jsp?R=19850026517 2020-03-20T17:01:25+00:00Z
381
SH 4.1-4
Diffusion refers to a configuration of slow temporal variations
of the anisotropy and weak spatial inhomogenities of the isotropi¢ den-
sity Fo (Earl, 1974b), which act as the source of a diffusive anisotropy
given by
@FO 1.2 '''l ' ' ' ' ' '''
F1 = -V _ g, (3) . q=I0
where I0 .... _--"'_-w_e •
g{u} = [(4-q)_/3] .lul1-q (4) 0.8-
is a solution of 0.6
¢ _ = -.. (s) " q:1.90.4
2. Finite-Difference Approximations _./"
to the Scattering Operator. In the .2 -
discrete formulation, the continuous
variables are replaced by a three-
dimensional grid whose spacings are O , ,,I
Az, A_, and As = VAt, and the deriv- 6 lO 30 lO0
atlvesappearingin equation(I) are [V_m_x
replaced by their flnlte-dlfference
analogs. A test of the relationships implied by diffusion can be made
without extensive calculations by replacing the differential operator
in equation (5) by its finite version and solving numerically for the
diffusive anisotropy. This procedure should give an answer consistent
with equation (4), and if it does not, the effective mean free path is
given by
+i
= (3/2) '-i _ du. (6)
Kota et el. (1982) gave an explicit flDite-difference form for the scat-
tering operator. Although this derivation seems eminently reasonable
from the standpoint of both mathematics and physics, the resulting scat-
tering operator does not pass this test near the threshold at q = 2 for
pure coherent propagation.
Mean free paths derived from this operator are given in the
figure above, where the ratio of the actual value of I obtained from
equation (6) to the nominal value that appears in equation (2) is
plotted against the number M = 2/A_ of increments in pitch-angle. The
correspondence is clo_e for q = I and fairly close for q = 1.5, but for
q = 1.9, I is a factor of ~2.5 too small and shows little indication of
converging to the nominal value as M increases. These inaccuracies are
a consequence of the fact that weak scattering near _ = 0 gives rise to
large deviations from the analytic behavior. To avoid these deviations,
I use a finite operator constructed so that equation (5) is satisfied
exactly when g is described by the analytic expression. Far from B = O,
this operator behaves the same as Kota's operator, but it also gives the
correct diffusive aniso_tropy and makes the actual mean free path
identical to the nominal one.
381 
SH 4.1-4 
Diffusion refers to a configuration of slow temporal variations 
of the anisotropy and weak spatial inhomogenities of the isotropic den-
sity Fo (Earl, 1974b), which act as the source of a diffusive anisotropy 
given by 
(3) 
1.0 
q= 1.0 
~e_~_~ __ =_=_.e __________ e ___ 
where 
g{l1} = [(4-q)A/3] 111111 1- q (4) 
is a solution of 
(5) 
0.8 
0.6 
0.4 
__ ---e-e 
_  ---e-
__ e- q=I.5 
q=I.9 
2. Finite-Difference Approximations ______ e--
-
...--e 
~~ the scfatterfngi oper~t
or. lin the .2 
screte ormu at on, t e cont nuous 
variables are replaced by a three-
dimensional grid whose spacings are 0 6 bz, bJ,J, and b.s = V bt, and the deriv-
atives appearing in equation (1) are 
replaced by their fini e-difference 
10 30 100 
Mmax 
analogs. A test of the relationships implied by diffusion can be made 
without extensive calculations by replacing the differential operator 
in equation (5) by its finite version and solv~ng numerically for the 
diffusive anisotropy. This procedure should give an answer consist nt 
with equation (4), and if it does not, the effective mean free path is 
given by 
r +l A = (3/2) '-1 ).g dJ,J. (6) 
Kota et ale (1982) gave an expl~cit f~nite-difference form for the scat-
tering operator. Although this derivation seems eminently r a onable 
from the standpoint of both mathematics and physics, the resulting scat-
tering operator does not pass this test near the threshold at q = 2 for 
pure coherent propagation. 
Mean free paths derived from this operator are given in the 
figure above, where the ratio of the actual value of A obtained from 
equation (6) to the nominal value that appears in equation (2) is 
plotted against the number M = 2/f:,.J,J of increments in pitch-angle. The 
correspondence is close for q = 1 and fairly close for q = 1.5, but for 
q = 1.9, A is a factor of ~2.5 too small and shows little indication of 
converging to the nominal value as M increases. These inaccuracies are 
a consequence of the fact that weak scattering near J,J = 0 gives rise to 
large deviations from the analytic behavior. To avoid these deviations, 
I use a finite operator constructed so that equation (5) is sati fied 
exactly when g is described by the analytic expression. Far from J,J = 0, 
this operator behaves the same as Kota's operator, but it also gives the 
correct diffusive aniso~tropy and makes the actual mean free path 
identical to the nominal one. 
382
SH 4.1-4
3. Numerical Dispersion. In a configuration of weak scattering, parti-
cle bunches propagate coherently in a moving Gaussian profile given by
F = I/(4_D,t_/2exp{-(z - V,t)2/4D, t} (7)o
where V, is a characteristic velocity, which corresponds to an angular
distribution that is nearly isotropic in one hemisphere and zero in the
other, and D, is the coefflcient of dlspersion, which describes the
broadening of the Gausslan that arises from statistical fluctuations of
individual velocities within this distribution. (Earl, 1974a).
The finite-difference implementation leads to a very similar
effect which can be described by equation (7) with D, replaced by
The total dispersion is a superpositlon of the physical effect described
by D, and the numerical effect described by D_. To ensure that physics
dominates, the condition D, >> D_ must be satisfied. In the examples
presented below, where q = 1.8, this condition translates into a re-
quirement that Az < _/40. This condition is similar in concept to the
condition derived by Kota et al. for the validity of calculations of
diffusion, which is Az < 3%/4. However, the required spatial resolution
is much finer in the coherent regime that it is in the diffusion regime.
4. A Specific Example. To illustrate the points made above, two calcu-
lations were made which had identical values of % and AB = 0.22, but
which had values Az = %/21 (plotted as solid lines) and Az = k/210
(plotted as dotted lines) that are, respectively, well below and well
above the required spatial resolution. In the figure to the left,
snapshots of the density profiles at the instant s = 0.19%, display the
familiar coherent pulses followed by small but perceptlble wakes. At
this time, the dotted peak has moved a distance of 0.I05% to the right
of the point indicated by a vertical dotted llne at which a localized
coherent angular distribution was injected at s = 0. This corresponds
382 
SH 4.1-4 
3. Numerical Dispersion. In a configuration of weak scattering, parti-
cle bunches propagate coherently in a moving Gaussian profile given by 
1/ 2 Fo = 1/(41f0*t) 2 exp {-(z - v*t) /4D*t} (7) 
where V* is a characteristic velocity, which corresponds to an angular 
distribution that is nearly isotropic in one hemisphere and zero in the 
other, and 0* is the coeff1cient of d1spersion, which describes the 
broa~ening of the Gaussian that arises from statistical fluctuations of 
individual velocities within this distribution. (Earl, 1974a). 
The finite-difference implementation leads to a very similar 
effect which can be described by equation (7) with D* replaced by 
I I Dolo = -V( t.z - -/$) 
I + ~ (8) 
The total dispersion is a superposition of the physical effect described 
by 0* and the numerical ffect descrtbed by Dt. To ensure hat physics 
dominates, the condition D* » Of must be sat sfied. In th  examples 
presented below, where q = 1.8, this condition translates into a re-
quirement that t:.z < A!40. This condition is similar in concept to the 
condition derived by Kota et ale for the validity of calculations of 
diffusion, which is t.z < 3A/4. However, the required spatial resolution 
is much finer in the coherent regime that it is in the diffusion regime • 
• 48 
f /"'. of 
.sa 
: : 
.213 ('\': 
.113 J \A 
.1313 _.?' ., 
. 
• 75 1.aa 1.25 1.513 
z/}. 
.613 
.513 
.413 
.sa 
.213 
.113 
.1313 
-1.1313 -.513 
A : 
I 
i 
! 
I 
/./ 
/' 
,// 
/~ •.. // .... 
.aa .513 1.ea 
4. A Specific Example. To illustrate the points made above, two alcu-
lations were made which had identical values of A and t:.~ = 0.22, but 
which had values t:.z = A!21 (plotted as solid lines) and 6.z = A/210 
(plotted as dotted lines) that are, resp~ctively, well below and well 
above the required spatial resolution. In the figure to the left, 
snapshots of the density profiles at the instant s = O.19A, display the 
familiar coherent pulses followed by small but percept1ble wakes. At 
this time, the dotted peak has moved a distance of O.105A to the right 
of the point indicated by a vertical dotted line at which a localized 
coherent angular distribution was injected at s = O. This corresponds 
388
SH 4.1-4
to a coherent velocity V_: 0.55V that is in good agreement with the
corresponding velocity of 0.53V tabulated by Bieber (1977). Simzlarly,
the FWHM width of the pulse 0.135% is in good agreement with the wzdth
0.125% expected from a combination of physical dispersion with
D, = 0.0064_V and numerical dispersion with D_= 0.001_V.
The coherent velocity of the solid line peak is less than that of
the dotted one and its dzspersion, which is dominated by the numerical
effect, is larger. To the right of point A, slgniflcant solid density
appears in regions where the dotted density is zero. This discrepancy
is relevant to descriptions of the early onset phase of solar partzcle
events, for it means that the numerical dispersion can bring particles
from the sun faster than the physzcs can, and thzs could lead to signi-
ficant errors in the timing of flare events.
The angular distributions in the figure that appears at the right
above add weight to this point. They refer to point A where the two
profiles have equal densities and, consequently, can be directly com-
pared without normalization. The dotted distribution is much more
collimated in the forward direction near B = i than the solzd one is.
It is clear that any attempts to interpret observed angular distribu-
tions of the first few particles arriving in a flare event should take
rigorous measures to exclude the effect of numerical dispersion.
5. Conclusions. Before non-diffusive transport can even be considered,
the scattering operator must be carefully tailored to give valid results
in the diffusion regime. Once this has been done, calculations that
take proper account of numerical dispersion give results in good agree-
ment with those expected from the theory of coherent propagation, but
not with those calculated with inadequate spatial resolution. This sit-
uation can lead to insidious errors, for the invalid results appear to
be qualitatively reasonable, and can be detected only by quantitative
tests.
6. Acknowledgements. This research was supported by NASA under Grant
NGR 21-002-066.
References
Bieber, J.W. 1977, Ph.D. thesis, Unzverszty of Maryland.
Earl, J.A. 1974a, Ap.J., 188, 379 (Paper II).
Earl, J.A. 1974b, Ap.J., 193, 231 (Paper III).
Earl, J.A. 1976, Ap.J., 205, 900 (Paper IV).
Kota, J., et al. 1982, Ap.J., 254, 398.
3 
H  
to a coherent velocity V~ = 0.55V that is in good agreement with the 
rr s nding el cit  f O.S  t l t   i er ( 7). ~  
the FWHM idth of the pulse 0.13SA is in good agree ent ith the 1dth 
O.125A expected from a co bination of ysical is ersi  ith 
D* = 0.0064AV and numerical dispersion with Dt = O.OOlAV. 
The coherent velocity of the solid line peak is less than that of 
the dotted one and its d1spersion, hich is do inated by the nu erical 
ff ct, is l r r. o t e ri t f i t , s1 nif1cant s li  e sit  
ears i  r i s here t  tt  nsit  i  r . his is r  
is rele a t to escri ti s of the earl  onset phase of s lar part~cle 
ents, f r  eans t t t  erical i rsi   ri  rti l s 
fro  the sun faster than the phys1cs can, and th1s could lead to signi-
fi t rr rs i  t  ti i  f fl r  ents. 
The angular distributions in the figure that appears at the right 
  eight t  t i  oint.  r t  i t  here th  tw  
r fil s  al ensiti s , s quently,   i tl  -
r  it ut r alization.  tt  i t i ti  i  uc  or  
l at d   r  i t  ar 11  1   11    
 i  l r t t  tte pts t  i t r r t s r  ular i tri -
tio s f t e fir t few arti l s rri i  i  a fl r  e e t s l  ta e 
i s easures t  l e t  t  erical i ersi n. 
5. onclusions. efore non-diffusive transport can even be considered, 
t e scatteri  erat r ust be caref ll  tailore  t  give ali  res lts 
i  the diffusion regi e. nce t is has been done, calculations that 
take proper account of nu erical dispersion give res lts in good agree-
ent ith those expected fro  the theory of coherent propagation, but 
t it  t  l l t  it  i uate ti l l ti n. i  
uation can lead to insidious errors, for the invalid results appear to 
be ualitati ely reasonable, and can be detected only by antitati e 
t . 
6. Acknowledgements. This research was supported by NASA under Grant 
 - 2-066. 
eferences 
ieber, J. . 1977, Ph. . t esis, Un1vers~ty of aryland. 
arl, J. . 1974a, p.J., 188, 379 (Paper II). 
arl, J. . 1974b, p.J., 193, 231 ( aper III). 
arl, J. . 1976, p.J., 205, 900 (Paper I ). 
Kota, J., et al. 1982, Ap.J., 254, 398. 


Numerical and analytic descriptions of cosmic-ray transport

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