E F F E C T O F D R I F T - R E S O N A N C E B R O A D E N I N G O N

R A D I A L D I F F U S I O N I N T H E M A G N E T O S P H E R E

MICHAEL SCHULZ Space Physics Laboratory, The Aerospace Corporation, El Segundo, Calif., U.S.A.

(Received 10 December, 1974)

Abstract. In the quasilinear theory of magnetospheric radial diffusion caused by fluctuating electro- static (E) or magnetic (B) fields, the diffusion coetficient DLL is proportional to the spectral density of E or B at the particle drift frequency Oa/2n. Since 03 varies with L at fixed M and d (adiabatic in- variants), the drift resonance co=~23 can be maintained only transiently, and therefore is not perfectly sharp. Its bandwidth Aog* is approximately (16nDLL/L2123)~I3Q3. In magnetospheric radial diffusion caused mainly by electrostatic fluctuations, the value of Aco*/Q3 typically exceeds 0.4 for particle energies E<40 keV. However, the numerical value of DLL is correctly given (within 1% in all cases) by quasilinear theory because the spectrum of E is rather flat at resonance frequencies for which the bandwidth is an appreciable fraction of 12a. (Numerical conclusions are based on a quasilinear model for DLL used successfully by Cornwall in 1972.)

1. Introduction

The purpose of this work is to address a conceptual problem inherent in the theory of magnetospheric radial diffusion. In the usual formulation (e.g., F~lthammar, 1968),

one considers the radiation belts to be immersed in randomly fluctuating electrostatic

and electromagnetic fields (superimposed on the Earth's time-averaged B field). The

fluctuations are considered adiabatic on the gyration and bounce time scales, but de-

scribable by smooth spectra that are nonvanishing at the particle drift frequency

t23/2n. The radial-diffusion coefficient DLL is then proportional to the spectral density at ~0=123.

Overlooked in the usual formulation is the fact that, by virtue of the radial diffusion thus described, a particle's drift frequency t23/2~ is itself a stochastic function of time.

The 'quasil inear ' derivation (e.g., Ffilthammar, 1968) leading to a perfectly sharp

resonance at 09 = 03 requires that 03 be time-independent. Where I23 varies randomly with time, the classical uncertainty principle defines a nonvanishing bandwidth At23 over which the fluctuation spectrum must be averaged in order to define DLL.

The magnitude of At23 can be estimated straightforwardly. Since the radial diffusion

is presumed to conserve the first two adiabatic invariants (M and J), one obtains (e.g., Schulz and Lanzerotti, 1974)

(At'23) 2 = 2DzL[(Sg23/OL)u,s] 2t (1)

after an interaction time t between a particle and the fluctuation spectrum. On the other hand, the classical uncertainty principle limits description of the spectrum to a resolution zfco = 2z~/t. Setting At23 = 21o9, one obtains an estimate of the time t* during

Astrophysics and Space Science 36 (1975) 455-458. All Rights Reserved Copyright �9 1975 by D. Reidel Publishing Company, Dordrecht-Holland

456 MICHAEL SCHULZ

which the interaction involves (for all intents and purposes) a single Fourier component of the fluctuation Spectrum. The result corresponds (in the nonrelativistic limit) to a

diffusion-induced bandwidth

zlco* = 2zc/t* = (4rtDLL)l/s(eQs/OL)~S,s = (167tDzL/L292s)~lsf2s (2)

that characterizes the drift resonance between a particle and the fluctuation spectrum.

The structure of (2) reminds one of a bandwidth calculated more formally by Dupree (1966) in the context of strong plasma turbulence (the spatially homogeneous case).

Application of his methods to the magnetosphere would have led to a result having the same form as (2), up to factors of order unity.

2. Application

I t is interesting to consider the ratio zlco*/s s as a measure of the ' strength' of radial

diffusion for various magnetospheric particle populations. Thus, for nonrelativistic protons having J = 0, Cornwall (1972) has proposed a diffusion coefficient of the form

Dr.L = {2 + 0.4[(M/Mo) 2 + lO-6L4]-I}(L/lO)~~ day -~, (3)

where M o ~ 1 GeV/G. The same coefficient theoretically should apply to nonrela-

tivistic electrons. Using parameters appropriate to the geomagnetic dipole field, viz.,

L2s s -~, one thus obtains Aco*/t2s~0.0323 for M = 0 . 1 GeV/G at L = 4 , i.e., for values of L and M representative of the outer-zone protons studied by

Cornwall (1972). Other 'quasil inear ' estimates for DLL and Aco*/f2s are given in

Table I. The form of (3) is based on fluctuation spectra (of B and E, respectively) proportional

to m - 2 for magnetic impulses and to (o~ 2 + v 2)- 1, with v = 0.739 x 10- s s - i, for electro-

static impulses. The spectral forms correspond to randomly occurring magnetic im-

pulses that are step-like on the drift time scale, and to electrostatic impulses that decay as exp [ - v ( t - t l ) ] after sharp onset at randomly occurring times ti. However, the quasilinear derivation leading to (3) is illicit unless the result implies that Aco*~g23 .

TABLE I

Lowest-order values of DLL from (3) and Aco*/f23 from (2), respectively

Value of M

GeV/G

10-s 10-2 10-1 10 o 101

'Quasilinear' value of DL~., day-1 Lowest-order estimate of Ato*/g23

2.41x I0 -s 1.63x10 -~ 1.88x 10 ~ 1.07x 10 ~ 0.1238 0.5048 1.1390 2.0346 3.53x 10 -4 1.18x 10 -1 1.74x 10 ~ 1.05x 101 0.0303 0.2102 0.5159 0.9370 4.29x10 -6 4.30x 10 -s 2.26• 10 -1 3.26x10 ~ 0.0032 0.0323 0.1212 0.2950 2.46x 10 -7 2.52x 10 -4 1.45x 10 -2 2.58 x 10 -1 0.0006 0.0058 0.0225 0.0587 2.05x 10 -7 2.10x 10 -4 1.21 x 10 -2 2.15 x 10 -1 0.0003 0.0025 0.0098 0.0257

L=2 L=4 L=6 L=8 L=2 L=4 L=6 L=8

EFFECT OF DRIFT-RESONANCE BROADENING ON RADIAL DIFFUSION

TABLE II

Self-consistent values of DLL and Aco*/I2a derived from (4)

457

Value of M

GeV/G

Self-consistent value of DLL, day-1 Self-consistent value of Aog*/fga

L=2 L =4 L =6 L=8 L = 2 L = 4 L=6 L=8

10 -a 2.41x 10 -3 1.63x10 - t 1.88x10 ~ 1.07x101 0.1238 0.5048 1.1390 2.0345 10 -z 3.53• 10 -4 1.18x 10-1 1.74x 10 o 1.04x10 t 0.0303 0.2102 0.5158 0.9365 10 -1 4.29x10 -6 4.30• -a 2.27• -1 3.29x10 ~ 0.0032 0.0323 0.1213 0.2959 100 2.46x10 -7 2.52x10-* 1.45x10 -2 2.58x10-1 0.0006 0.0058 0.0225 0.0587 101 2.05 • 10 -7 2.1Ox 10-* 1.21 • 10 -2 2.15 x 10 -1 0.0003 0.0025 0.0098 0.0257

The effect of drift-resonance broadening is to average the fluctuation spectrum over a

bandwidth Aco*. Dupree (1966) approximated an analogous effect by introducing a

rectangular ' l ine-shape ' funct ion equal to (1/Aco*)0(Aco*- 21o9- s ). In the present

application one thereby obtains

DLL = 2 • 10-1~176 day -1 +

+ 4 x 10-11(I22/vAco*)(0+ - O-)(Mo/M)2L ~~ day -1 (4)

with tan 0 +- -= (t2a/v)[1 + (Aco*/2~2a)]. Self-consistent solutions o f (4) for Aco*/f2a and

D f r are given in Table I I for v=0.739 • 10 -3 s -1. The results were obtained numeri-

cally by iteration. The most severe discrepancy ( ~ 1% in DLL) between Table I and

Table I I occurs for M = 0.1 GeV/G at L = 8.

3. Discussion

Drift-resonance broadening is thus surprisingly unimpor tant as an influence on the

numerical magnitude o f DLL, especially where Aco*~ t23. The quasilinear procedure

for evaluating DLL might be questioned at low values o f M, for which Aco*/f2a ~ 1

in the outer zone. However, the numerical value o f DLL is little affected by drift-

resonance broading there, because it happens that 123 < v for this range o f parameters.

Thus, the spectrum of E at co ~ s is relatively flat for parameters that yield Aco*/123 ~ 1.

N o legitimate purpose would be served by carrying out extensive numerical itera-

tions here. Once it has been established that drift-resonance broadening is present in a

given practical application, the numerical consequences for DLL are (of course) model-

dependent. For example, the value o f v (=0 .739 • 10 -3 s -1) used here was tacitly

chosen by Cornwall (1972) so as to simplify the numerical constants that appear in (3). It corresponds physically to a decay time ~ 20 min for electrostatic impulses, i.e., to a

realistic time scale for the typical magnetospheric substorm.

I t is not difficult to express (2) in relativistic form for application to outer-zone electrons. 2 The drift frequency satisfies the relation 7L t2a~ 0.739(M/Mo) s - 1 in the rela-

458 MICHAEL SCHULZ

tivistic case, where 72= 1 +(2MB/moc2). Since B is proportional to L -a for J = 0 , one

obtains

Aco*/g23 = (16zcDLLly4L2s + (MBl2moc2)] 2/3. (5)

Moreover, the factor MIMo must be replaced by M/~,Mo in (3) and (4). The numerical outcome (as in the nonrelativistic treatment) is that the quasilinear and self-consistent values of DLL agree within 1% for all cases examined.

The particte populations for which Aco*/s is largest in Tables I - I I are also the particle populations for which static E and B fields in the magnetosphere most signifi- cantly distort the adiabatic trajectories. Such static distortions alter the functional form of DLL in their own right, and introduce drift-harmonic resonances in the quasi- linear diffusion formalism (e.g., Schulz and Eviatar, 1969). In this sense, the present expressions for DLL are incomplete. However, the results summarized in Tables I - I I above should at least enable one to conclude that very little error is made in failing to consider the effect of drift-resonance broadening on radial diffusion in the magneto-

sphere.

Acknowledgements

The author is pleased to thank Dr J. B. Blake for computational advice and Dr J. M. Cornwall for helpful comments. Dr F. V. Coroniti has proposed further calculations that might supplant the approximation of a rectangular line shape. This work was conducted under U.S. Air Force Space and Missile Systems Organization (SAMSO)

contract F04701-74-C-0075.

ReferenCes

Cornwall, J. M.: 1972, J. Geophys. Res. 77, 1756. Dupree, T. H.: 1966, Phys. Fluids 9, 1773. F/ilthammar, C.-G. : 1968, in B. M. McCormac (ed.), Earth's Particles andFields, Reinhold, New York,

p. 157. Schulz, M. and Eviatar, A. : 1969, Y. Geophys. Res. 74, 2182. Schulz, M. and Lanzerotti, L. J. : 1974, Particle Diffusion in the Radiation Belts, Springer, Heidelberg,

pp. 57, 58, 83.