Inhomogeneities of the interplanetary magnetic field can be divided into small scale and large scale ones as may be required by the character of the problem of cosmic ray (CR) propagation. CR propagation in stochastic magnetic fields is of diffusion character. The main contribution into the scattering of CR particles is made by their interaction with inhomogeneities of the magnetic field H which have characteristic dimensions 1 of the order of Larmor radius R=cp/eH of particle (p is the absolute value of particle momentum, e is particle charge, c is velocity of light). Scattering of particles on such inhomogeneities leads to their diffusion mostly along a magnetic field with characteristic dimensions of variation in space exceeding the mean free path

Dorman, L. I.

Fedorov, Y. I.

Katz, M. I.

Nosov, S. F.

Shakhov, B. A.

English

NASA Technical Reports Server

413 SH 4.1-20
OOS_IO RAY MODULATIC_ IN A RANDOM ANISOr_ROPIO _A_ETIO
FIELD
L .I.Dorman
Institute of Terrestrial _agnetism, Ionosphere and Radio
Wawe Propagation of the US_ Academy of Sciences, 142092
Troitsk, Moscow Region, USSR
Yu.I.Fedorov, M.I.Katz, S.F_osov, B.A.Shakhov
The Astronomical Observatory of the Ukrainian Academy of
Sciences, 252127, Kiev, Goloseevo
Inhomogenelties of the interplanetary magnetic field
can be divided into small-scale and large-scale ones/I,2/,
as may be required by the character of the problem of cos-
mic ray (OR) propagation. GR propagation in stochastic
magnetic fields is of diffusion character. The main contri-
bution into the scattering of GR particles is made by their
interaction with inhomogeneities of the magnetic field
H(r,t) which have characteristic dimensions 1 of the order
of Larmor radius R=cp/eH of particle (p is the absolute
value of particle momentum, e is particle charge, c is
velocity of light). Scattering of particles on such inhomo-
geneities leads to their diffusion mostly along a magnetic
field with characteristic dimensions of variation in space
exceeding the mean free path _ of particles before their
scattering. In view of this fact the inhomogeneities of the
interplanetary field can be divided into small-scale (1 )
and large-scale (i<_) ones, and the random magne_.ticfield
can be represented as a sum of the small-scale _ (r,t)
and the large-scale _ (r,t) fields. The boundary value l_
of the inhomogeneities can be determined by several length_
of the mean free path of particles and i_ can be taken
as the principal scale of the small-scale turbulence. The
choice of the scale L_ of the large-scale turbulence de-
pends on a concrete fSrmulation of the problem.
It is necessary to divide a random field into small-
scale and large-scale parts because one cannot describe the
whole of the magnetic field turbulence spectrum on a single
basis in terms of the local isotropy. As the scale of fluc-
tuations increases, the statistic distribution of these
fluctuations becomes, apparently, nonisotropic in space.
The diffusion coefficient along the large-scale field
H for R_<l o can be written in the form /2/
3 SH 4.1-20 
C MIC  ODULATI N N   I TR C MAGNETIC 
I I  
I.Dorm  
I stit te of errestrial Magoetis , Ionosphere and adio 
a e r pagation f t  SR c y f i ces,  
roitsk, osco  egion, SS  
u.I. edorov, .I. atz, . .Nosov, . . akhov 
The stronomical bservatory of the krainian cade y of 
ci nces, 127, iev, ol s o 
I ogeneities f t  i t r l etary agnetic i l  
can be i i e  i t  s all-scale and lar e-scale es/I,2/, 
as ay e re ire   t e c aracter f t e r le  f c s-
i  r  ( ) r pagation. O  r agati n i  t astic 
agnetic fiel s is of iff si  c aracter. he ain c ntri-
bution into the scattering of OR particles is ade by their 
i t r ti  it  i ogeneities f t e agnetic fi l  (r,t) hich  aracteristic i ensions  f t  r r 
f ar or r i s =cp/e  f rti l  (  i  t  s l t  
value of article o entu , e is article charge, c is 
el city f li t). catteri g f rti l s on s c  in o-
geneities leads to their diffusion ostly along a agnetic 
fi l  it  aracteristic i ensions f ri ti  i  s ace 
ceeding t  ea  fr  t ./\ f rti l s f r  t ir 
att ri g. I  i  f t i  t t  i ogeneities f t  
interplanetary fi.l  can be divided into s all-scale (1 ) 
a  l r -s ale (l A) es, a  t e ran  agnjltic fi l  
c~n be represented as a sum of the s all-scale ~1 (r,t) 
and the large-scale JjC (r, t) fields. The boundary value 1 
f t  i ogeneities ca  e eter i ed  s eral le t 2 
-A f t e ean free at  f arti l s and 1 can be ta e  
as the ri cipal scale of the s all-scale turBUlence. he 
choice of the scale LS of the large-scale turbulence de-
pends on a concrete f r ulation of the proble . 
It is necessary to divide a rando  fiel  into s all-
~  c  ~ ~e rt  ause ce ccot scri e t  
hole of the agnetic fiel  turbulence spectru  on a single 
asis i  ter s f t  l al is tr . s t e s l  f fl -
t ti s i r ses, t  t ti ti  i tri ti  f t  
fl t ati ns beco es, a parently, nisotr pic in s ace. 
he iff si n effi i nt al  t e lar e-scale fi l  
H for R<'<lo can be written in the form /2/ "'-Iv 
vll v(Vt9v)r(~) ,/ ~-YOY-l H .~>1 
ef,,= 3") J1/1=31{:iT(V+j)r(,,/~/CP/€) 1tD ( 1Jlt) , (1) 
https://ntrs.nasa.gov/search.jsp?R=19850026526 2020-03-20T17:00:48+00:00Z
414
SH 4.1-20
where H= l_ =IH @ a regular magnetic field,
is the index _ the z eque cy spectrum of the small-scale
turbulence, (x) is gamma-function, V is particle veloci-
ty, _lh is the mean free path of e particle along the field
H before scattering. N
The transport equation in the case of the field H
has the form /2,3/
where g(r,p,t) is CR particle concentration, u is solar
wind velocity, __ _ __&_is tensor diffusion coeffi-
cient which for a strong field _><_ R<<./I H , canbe _ritten as
(3)
where _= _/_._ If the field H is represented by a sum
of the random _ end the regular H o fields, the expression(3) for<_><< _oca_bere_ittenas
W,4.
-- "" " If r,,,-'t)
:h.:o:=HoIHo, a,/ H°+"
The,,_coefficient_ in (2) depends on the random
field _(r,t) and, hence, the function N_ (r,t)] ;
(r,p!t)]itself is a functional of the random field, and if
e q. (2) is averaged over a large-scale random field, we
can use the standard averaging procedure /%/. As a result,
we are led to the following equation for the mean OR con-
centration <II>
where
414 
SH 4.1-20 
'" --
where H.: 1m =IH +ffll, H is a regular magnetic field, ~ is the index o~ the fre~uency spectrum of the small-scale 
turbulence, (x) is gamma-function, V is particle veloci-
ty, ~\I is the mean free path of a particle along the field 
H before scattering. ,...., 
-The transport equation in the case of the field H 
has the form 12,31 tV 
~l:~~?i -CUV)N j.(VU)~~/O (2) 
where N(r,p,t) is OR ~ticle concentration, u is solar 
wind velocity, Vorv =.'0(0 ~g(", J afo\,~is tensor diffusion coeffi-
cient which for a strong field 'H~ »< '<Jl"Q.> R« J\" ' can 
be wr tten as 
ef",,1. == 2f" k,J 1 (3) 
--- --- ;:;-
where h = 1i' IR'.", If the field H i~represented by a sum 
of the random j{ and the regular Ho fields, the expression 
(3) for <1Jt,'l.>« Ho can be rewritten as 
cf.;, ~:=. ~~ ( VL-".}'~ Pi + ilL-<i. f!Lj. J.I;~ I-t ~ YJi", I-I:~ (V -.2 ) itA 1< f-/~( 4) 
where 0 = l\, IRo' Bf~' = ~I H -~ / H~ .. 
The~oefficient ~~ in (2) depends on the rando~ 
field ~ (r,t) and, hence, the function N(..tJl (r,t)j ; 
(r,p,t)] itself is a functional of the random field, and if 
eq. (2) is averaged over a large-scale random field, we 
Can use the standard averaging procedure /4/. As a result, 
we are led to the following equation for the mean OR con-
centration <N> -
(i) <N).. =- 'V 'qo v<N >- ciZV-><tV> + EVUro <Nl ,<5l 
ro i ~ oIv~ ~ 3 ro f 
where 
415
SH 4.1-20
(8)
Here _o_iC_c) is a Fourier-transform of the correlation
tensor of a random magnetic field. When deriving (5), we
used the hypothesis of frozen turbulence.
For an isotro_ic probability distribution of fluctua-
tions of the field.(r ,t) we have
Here Lo is the maximum dimension of large-scale inhomoge-
neities.
The coefficient _L describes particle transport
across a regular magnetic field under the action of a large-
scale random field. Its expression for the isotropic case
agrees with the analogous expression obtained in the papers
by Toptygin /2/ in the averaging of the kinetic equation
for _,_=8/11 . In our case, however g_j, depends on the
random field _ , and to compare it with the value of _o
one should average _i/ over an ensemble of a large-scale" ,,
ranom ield.Asaresult wehave< -#,?
/Ho ). Such a renormalization of the diffusion coefficient
will not affect noticeably the values of _ and _'t _
in which _ is replaced by <_u> without an essential _
change in their values, but the coefficient of longitudinal
diffusion changes the form. So, for the expression (9)
we have
from which one can conclude that an isotropic large-scale
turbulence changes particle diffusion along the regular
field if the spectrum of small-scale turbulence does not
fall very rapidly, namely, if the index of the spectrum
is not larger than _64 _.8. Usually V = I - 2/2/.
Large-scale turbulence of a magnetic field in interpla-
netary space is apparently anisotropic, i.e. spatial probabi-
lity distribution of a random magnetic field has a distin-
guished direction e. As working models of anisotropic tur-
15 
S  4.1-20 
iJ'.l -= of 0 (!.l Vt-ll J1, ~ + 6. Vi" fb /tt~ ') \ o!I J6 tb i' c f") (8 ) 
0. ~ ,~cIv ~ 0 ~~ I .) I tJ 
Here .. ~,/).(..tc) is a Fourier-transform of the correlation 
tensor of a rando  agnetic fiel . hen deriving (5), we 
used the hypothesis of frozen turbulence. 
For an isotro.ec probability distribution 
tions of the field~ (I', t) we have 
~ =10@+(»'-1)\~~) 
1/ II 3 1-1 ~ , 
~l. = 19/0 <YJL~> 6. °u ~f~L-=O 
dv~ 3 II 1-1; r{ oV~' O(,~ ) 
Ii <~((l>==2J;- SeUckOv:Jo-c~) ~ 
Lo 
Here Lo is the maximum dimension of large-scale 
eities. 
of fluctua-
(9) 
(10) 
(11) 
inhomoge-
he efficient ~f~de s arti l  tr s ort 
across a regular magnetic field under the action of a large-
s l  rando  fi l . It  e pressi  f r t e is tr i  case 
agrees ith the analogous expression obtained in the papers 
by Toptygin 121 in the averaging of the kinetic equation 
f or at,~ -= 8/" • In our case, however ~, depends on the 
random field ~ , and to compare it wl.th the value of ~o 
one should average ~II over an ensemble of a large-scale /I 5t 
random field. As a result we have < fi,,). =<I,~ (1+ V < -YttJ.>/1{ 
IRo ). Such a renormalization of the diffusion coefficient 
will not affect noticeably the values of 9/;:1 and Bf"Jl-
in whichef,~ is replaced by e(,/') ithout an essentialc:;oV~ 
change in their values, but the coefficient of longitudinal 
iff si  changes the for . o, f r the expression (9) 
we have _ 4 <1JlCJ.> ~ 
<t" -= <9f,,>-{1-"5 IJ;' (3V-CV-i)) i) (12) 
ro  hi    l    s i  a l  
turbulence changes particle diffusion along the regular 
fi l  if the s ectru  f s all-scale turb le ce es t 
fall very rapidly, na ely, if the index of the spectru  V 
is not larger than ...J ~ 4.8. sually V = I - 2121. 
ar e-scale turbulence of a agnetic fiel  in i ter la-
netary space is apparently anisotropic, i.e. spatial probabi-
lity distribution of a random magnetic field has a distin-
guished direction e. As orking odels of anisotropic tur-
416
SH 4 1-20
bulence we take its two limiting cases: two-dimensional
and one-dimensional fluctuation distribution. In the case
of two-dimensional turbulence, the correlation tensor_will be written in the form
and for the one-dimensional model we have
where _I and_It are corresponding spectral functions.
For two-dimensional turbulence with a distinguished
direction e'[ln- the values of the diffusion coefficient are
L_oiS t o e direc-
In the model of one-dimensional turbulence with a dis-
tinguished direction along the solar wind velocity, _'II_,
we have
_ _c_,= C,_ ) _ _ o/
where u_, and u_ are a parallel and a perpendicular compo-
nents of the solar wind velocity vector relative to the re-
gular magnetic field direction Lu is the or" _in_1 _n_1_
...... , ,, ._ in __ .......
o_- inaomogenel_ies along, u. In t_is case the cross diffu-
szon coefficient _J_ _s nonzero, whereas in all the
rest of the consid_*ed cases it is equal to zero. The latter
model of large-scale turbulence is apparently a most realis-
tic reflection of the situation in interplanetary space
with fluctuations going radially from the Sun, with systems
of discontinuities and waves following one another in the
far regions of solar wind.
REFERENOES
I. Toptygin I.N. Fast particles interaction with a magneto-
hydrodynamic turbulence. Astrophys. Space Sci , vol.2_,
1973, p. 329-350.
2. Toptygin I_. Cosmic rays in interplanetary fields.M.:
aul a, 1983.
3. Dorman L.I° Exper.and Theoret.bases of OR astroph. M. :
Nauka, 1971.
#. Rytov S._.,Kravtsov Yu.A.,Tatarsky V.I.Introduction to
statistical radiophysics .P.2.Random Fields .M. "_auka, 1971.
4  
SH 4.1-20 
b len  w  take its two limitin  ca : tw - i ensional 
and o ensi nal fluctu io  d strib . In the case 
of two-dimensional turbulence, the correlation tensor ,1)0\,:::\ 
ill be ritte  in the form L1c1. ._ _ ~:; ck) -= 9/-( -let) (~~- tk}'J )6(k,,)~ (13) 
 fo  the o ensional el  have 
90 11 (.[ =_Q...II L9.. 1.9.. _ •. -:- - _ -_-- (14) 
oIt1 ) )0 (1(;" )&C1CL ) (~~ eo<.ej )J~;-ke) Jc[IK !c,,' 
er  !f:/ and 1>" are c resp i  s t l fu t s. 
or tjlo-SLimensional turb le ce it  a isting is e  
 f/  h    th  ffu o  ff t a e 
CJI, :: <Df. > {{- ))<flJ(I.> 3' all. =-<~> <%J{I;'>~ (15) II 1/ ~ /1 Ho:J..) o(,jj'PO I~I I. ~} .. taL ~{/J 
ct.:.? -::: 0., < 1fl,!2 > -=?Jr L;/ VfA(; 1. 4<-1 "d (1(;. 1. ) 
La is the principal sca~e of inhomogeneities in the direc-
t10n transverse to Ho• 
 h  odel  - i ensional u b  it   
tinguished direction along the solar ind velocity, e 1/ \i, 
we have ~ ~ Q... 1. .L ~II ~ {I '<)/)) ,\Q,U:)~l>} . ~ L _< ,'J!l.L >{'L _~&} 
OJ -<dI,,> ",lI\. -1} Ui j:jo:- ) ~1- ~/). 1-/1 3~~ VL:l. ) 
o 0 ~ 9!/~L =<9f). <.'Jli> (Y._1)U/J (ULn, -rn L-~J...). (RJ£~>=-rzJT( dkll:i~Ic/~); oIv~ II H; U~ _~ 1 01. J.) L (tux' 
4c II ': (-Ie u ) U ) 0 
here u/' and UL are a arallel and a perpendicular co po-
nents of the solar ind velocity vector relati e to the re-
ular agnetic  ir cti n, U   principal scale 
of inhomogeneities along u. In tRis case the cross diffu-
si  c efficient OIl'lL is nonzero, hereas i  ll t e 
st f t  nsi~~ d s s  i  ual t  r .  l tt r 
odel of lar e-scale turbulence is apparently a ost realis-
tic reflection of the situation in interplanetary space 
with fluctuations going radially from the Sun, with systems 
f i continuities  aves f ll i g  ot er i  t  
r r i ns f lar ind. 
EFERENCES 
I. ~opt.ygin I. . ast articles interaction ith a agneto-
hydrodynamic turbulence. Astrophys. Space Sci., vol.20, 
973, . 29-350. 
2. ~optygin leN. Cos ic rays in interplanetary fields. .: 
Na ka, . 
. or an .I. Exper.and ~heoret.bases of OR astr ph. .: 
 auka, 971. 
4. ytov .M.,Kravtsov Yu.A.,~atarsky .I.l tr duction t  
statistical radiophysics.1' .2. ando  Fields . . :N auka, 1971. 


Cosmic ray modulation in a random anisotropic magnetic field

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