An approach is presented for deriving a diffusion equation for charged particles in a static, random magnetic field. The approach differs from the usual, quasi-linear one, in that particle orbits in the average field are replaced by particle orbits in a partially averaged field. In this way the fluctuating component of the field significantly modifies the particle orbits in those cases where the orbits in the average field are unrealistic. The method permits the calculation of a finite value for the pitch angle diffusion coefficient for particles with a pitch angle of 90 rather than the divergent or ambiguous results obtained by quasi-linear theories. Results of the approach are compared with results of computer simulations using Monte Carlo techniques

Birmingham, T. J.

Jones, F. C.

Kaiser, T. B.

English

NASA Technical Reports Server

NASA: :,,1,
A NEW APPROACH
DIFFUSION
X-641-.73-177 T
66263
TO COSMIC RAY
THEORY
(NASA-TM-X-66263) A NEW APPROACR TO.*
COSMIC RAY DIFFUSION THEORY (N1SA) 4
SRC $3~. 1 CSCL 03B
N73-25848
Unclas
G3/29 06236
FRANK C. JONES
THOMAS B.
THOMAS J.
KAISER
BIRMINGHAM
JUNE 1973
GODDARD SPACE FLIGHT CENTER
GREENBELT, MARYLAND
N
N
l
https://ntrs.nasa.gov/search.jsp?R=19730017121 2020-03-11T19:12:20+00:00Z
A NEW APPROACH TO COSMIC RAY DIFFUSION THEORY
Frank C. Jones
Theoretical Studies Branch
NASA Goddard Space Flight Center
Greenbelt, Maryland 20771
Thomas B. Kaiser*
Department of Physics and Astronomy
University of Maryland
College Park, Maryland 20742
Thomas J. Birmingham
Theoretical Studies Branch
NASA Goddard Space Flight Center
Greenbelt, Maryland 20771
*This work was supported by the National Aeronautics and
Space Administration under Grant NASA NGL 21-002-033
A NEW APPROACH TO COSMIC RAY DIFFUSION THEORY
We have investigated a new approach to deriving
a diffusion equation for charged particles in a
static, random magnetic field. Our approach
differs from the usual, quasi-linear one, in that
we replace particle orbits in the average field
by particle orbits in a partially averaged
field. In this way the fluctuating component
of the field significantly modifies the particle
orbits in those cases where the orbits in the
average field are unrealistic. This method
allows us to calculate a finite value for the
pitch angle diffusion coefficient for particles
with a pitch angle of 90 rather than the
divergent or ambiguous results obtained by
quasi-linear theories. Results of this new
approach are compared with results of computer
simulations using Monte Carlo techniques.
We propose a new scheme for deriving a kinetic equation
for the one particle cosmic ray distribution function <f>
averaged over an ensemble of static, random magnetic fields B
The essence of our new method is that the zeroth order partic
orbits partially contain the effects of the fluctuating mag-
netic component 6B. The result of our theory is that <f>
satisfies a diffusion equation in p=cos 0, where 8 is the
particle pitch angle measured with respect to the average
field . So long as p is not too small, the diffusion co-
efficient D(P) is the same as that derived from quasi-linear
theories 1 - 5 . When p o (8 = 7/2), a regime in which consid
erable controversy has existed 5 '6 , we obtain a D which is
finite and markedly different from previous incorrect result
The reason for this difference near 08 = /2 is that our thec
-2-
adequately describes the motion of such particles over a
coherence time of the fluctuations, while quasi-linear theory
follows the motion of particles as if only existed and
is hence a very poor approximation to their actual motion.
The correctness of our theory is substantiated by comparison
with the results of a Monte Carlo analysis.
For computational simplicity we consider the slab model4
A Ain which B = e + e 6B(z), being spatially homogen-
ous and 6B depending only on the single spatial variable z.
The method can be generalized to more complex geometries.
The theory begins from the continuity equation for F,
the cosmic ray distribution function in the phase space whose
dimensions are z, a, speed v, and gyrophase 4. It proceeds
by a formalism analogous to Weinstock's7 plasma turbulence
2T
theory to a diffusion equation for <f> = (2f) lfd%<F>. The
0
assumptions made in the derivation are a) 6F(t=o) E o,
b) <F>has only a weak ¢-dependence and c) <f> has a slow
phase space evolution so that the usual adiabatic approxima-
tion is valid. The diffusion coefficient D(p,t) is given by
q 27r t 1/2
D(jsn < zt)= (1-P 2 ) 1/2 dosing c <6B(z)U(t,T)(l-p)
o o (1)
sin46B(z)>
The operator U operates on everything to its right. In
its exact form7, U is UA, a complicated non-linear operator
-3-
which has no physical interpretation nor algorithm for con-
structing it.
Quasi-linear theory approximates UA by Uo, a propagator
which propagates particles along helical trajectories in the
field . 86=/2 particles thus execute nearly circular
orbits and remain in a correlated region of field for arbi-
trarily long times. Hence the true orbits are incorrectly
described to an arbitrarily large degree.
In our theory we approximate UA by Up, a propagator
which propagates along trajectories in a partially averaged
magnetic field. The partial averaging is over a subset of
realizations of the full ensemble, the subset being all those
realizations which have a given value of 6B(z) at the field
point z. We assume that 6B is a Gaussian process so that Up
propagates along particle trajectories in the partially
averaged field Bp (z',z)= z + ex6B(z)C(z-z'). Here C is
the normalized correlation function for the fluctuations.
Addition of effects of the partially averaged field
removes 8O=/2 particles from the correlation region in a time
short enough that deviations from their true trajectories are
not catastrophic. Further, our method of partial averaging
accounts for the effects of the fluctuations most accurately
at the spatial point where they are most important, viz. at z.
-4-
We have thus far considered the guiding center limit,
where 6B(z)rg/zc<<l,rg being the particle gyro-radius in
B and z the correlation length of the fluctuations. In thisp c
limit the z-motion of the guiding center is that of a particle
on a potential hill of height v [6B(z)/]2 . So long as the
guiding center has any (even infinitesimal) speed at z=z', it
moves a distance z c in a finite time interval of order
zc/6B(z)v in magnitude. The particle motion in Bp is
reasonably described by the approximate form
z(T)= z+v(p 2 +a6B 2 (z)/2 )1 /2(t-T)Ez+v(t-T)G(p,B) (2)
with a a numerical constant of order unity.
The orbit given by Eq. (2) is used in evaluating Eq. (1).
We further assume a) the effect of 6B in propagating p and 4
can be neglected and b) a spatially homogeneous ensemble. The
diffusion coefficient which we obtain
2 2 2 2-1 2 +coD(,t)=q(1-2) (4m2 c 2)-l<6B(z) dkP(k)sin[(<w>+kvG)t]
-co
(<w>+kvG) - > (3)
is exactly of the form of Jokipii's diffusion coefficient but
with G appearing in place of simply p. (Here P(k) is the power
spectrum corresponding to C and <w> is the gyro-frequency
q/mc.) D(p,t) consists of a transient term which decays
with the characteristic time zc/v<G> and is thus negligible
after at most a few deflection times in the rms fluctuating
field plus a resonant time independent term. Because of the
presence of G rather than p in Eq. (3) the resonance is broad-
-5-
ened in a fashion similar to that found in the theory of
strong plasma turbulence8
For an exponential correlation function, the resonant
contribution is
1-[Iicor2(<w 2 2 -1>2 2 221/2D(p,-)=(21a)-l/2[2(<2>Zc2+I2v2)] -P2(l-1-2)<w2>vz <6B2>1/2
exp 2 2 K 1 2(4)exp (
2 2 ) >22
4<6B >a 4<6B2>a
where K1 is the modified Bessel function. When
2 2/4<6B2 >a>>l, Eq. 4 approaches the quasi-linear result.
In the opposite extreme, however, we differ markedly from
quasi-linear diffusion theory: D(O,®) = O in quasi-linear
theory; we find D(O,-) = (2a)l/2v6B2>3/2 l/2 z - Z 3 Our
theory allows free diffusion through 900 pitch angles.
In the Monte Carlo analysis we integrate on a computer
Newton's equations of motion for a single particle in 200
A A
realizations of the random field B = e + 6B(z)ex. In eachZ x
realization the particle starts at t=o with random phase 4
but at the same z=zo and with the same p=1o' The statistics
of 6B are Gaussian with an exponential correlation function.
Absorbing boundaries are placed at p1<o and p2>O. The prob-
+00
ability distribution <f>(p,ti ) =_idz<f>(z,p,ti ) is examined
stroboscopically at regular intervals ti . We also follow the
time development of the time summed function
<f> (,ti ) = <E f (,tj)j=O
-6-
<f> is identified as the distribution function corres-
ponding to steady (in time) injection at zo, .Po This identi-
fication is based on the fact that <f> corresponds to impul-
sive injection and thus is the Green's function corresponding
to arbitrary injection.
Because of the absorbing boundaries there ultimately re-
sults a steady state in which there are constant fluxes j
and jr away from po and toward p1 and p2 respectively. Our
computation routine evaluates jR and jr as well as <f> 's(,).
Since
j =-D(p,) <f>s(H,/)/aP (5)
we are thus able to evaluate D from the Monte Carlo method.
The histogram in Figure 1 shows <> s(p,-) for a simulation
with the typical values po=0.2,p1 =-0.2, and p2 =0.6. The sta-
tistical fluctuations in <f>s result from using a finite
ensemble. Note that particles move freely through p=o. The
curves result from integrating Eq. 5 using the known values
of Jr and jR and D as given by our theory, Eq. 4 with a=l.
The boundary conditions used here are that <f> =o a mean
9free path in p outside the absorbing walls9
Figure 2 shows plots of D(,-)z c/v. The solid curve is
again obtained from Eq. 4 with a=l. A smaller <6B2>1/2/
would result in a larger (in p) region of agreement between
our theory and quasi-linear theory. Shown also are values of
-7-
D(p,o) obtained from the computer work and Eq. 5, using the
known j's and measured values of the slope 3<f>s (P,)/aP.
Error diamonds reflect our estimate of the uncertainties in
these "experimental" quantities.
The agreement between theory and numerical simulation
shown in Figures (1) and (2) clearly illustrates our conclu-
sion. For small i, quasi-linear theory is grossly inadequate.
One must develop a theory which adequately describes the
physics of cosmic rays in all p-regimes. The theory which
we propose evidently does so.
y*Work supported by the National Aeronautics and Space Admin-
istration under Grant NASA NGL 21-002-033.
1. D.E. Hall and P.A. Sturrock, Phys. Fluids 10, 2620 (1967).
2. K. Hasselman and G. Wibberenz, Zs.f. Geophys. 34, 353
(1968).
3. R. Kulsrud and W.P. Pearce, Astrophys. J. 156, 445 (1969).
4. J.R. Jokipii, Rev. Geophys. and Space Phys. 9, 27 (1971);
Astrophys. J. 172, 319 (1972).
5. A. Klimas and G. Sandri, Astrophys. J. 169, 41 (1971);
Astrophys. J. 180, 937 (1973).
6. T.B. Kaiser, F.C. Jones, and T.J. Birmingham, Astrophys.
J. 180, 239 (1973); F.C. Jones, T.J. Birmingham, and
T.B. Kaiser, Astrophys. J. 181, L139 (1973).
7. J. Weinstock, Phys. Fluids 12, 1045 (1969).
8. T.H. Dupree, Phys. Fluids 9, 1773 (1969).
9. A.M. Weinberg, and E.P. Wigner, The Physical Theory of
Neutron Chain Reactors, (Univ. of Chicago Press, Chicago,
1959), Chap. IX.
FIGURE CAPTIONS
Figure 1. Histogram of <f> s(p,) for jo=0.2,p1=-0.2, and
12=0.6 produced by computer "experiment". Smooth
curves result fror. integrating Eq. 5 with D given
by Eq. 4 and jr and jA taken from "experiment".
Figure 2. Pitch angle diffusion coefficient D(p,-) computed
from Eq. 4 as compared with that given by quasi-
linear theory. "Data" points are obtained from
the computer simulation of <f> s(p,) shown in
Figure 1 by means of Eq. 5.
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A new approach to cosmic ray diffusion theory

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