Form Approved REPORT DOCUMENTATION PAGE OMB No. 0704-01-0188 The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions searching existing data sources gathering and maintaining the data needed, and completing and reviewing the collection of information Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden to Department of Defense. Washington Headquarters Services Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302 Respondents should be aware that notwithstanding any other provision of law no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 2. REPORT TYPE 1. REPORT DATE (DD-MM-YYYY) 3. DATES COVERED (From - To) REPRINT 9-03-2010 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Diffusion by one wave and by many waves 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 62101F Q- O 5d. PROJECT NUMBER 6. AUTHORS o J. M. Albert 1010 o 5e. TASK NUMBER RS F Ci 5f. WORK UNIT NUMBER Al 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER Air Force Research Laboratory /RVBXR AFRL-RV-HA-TR-2011-1017 29 Randolph Road Hanscom AFB. MA 01731-3010 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITORS ACRONYM(S) AFRL/RVBXR 11. SPONSOR/MONITOR'S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for Public Release; distribution unlimited. 13. SUPPLEMENTARY NOTES Reprinted from Journal of Geophysical Research. Vol. 115. A00F05. doi: 10.1029/2009JAO14732. 2010 14. ABSTRACT Radiation belt electrons and chorus waves are an outstanding instance of the important role cyclotron resonant wave-particle interactions play in the magnetosphere. Chorus waves are particularly complex, often occurring with large amplitude, narrowband but drifting frequency and fine structure. Nevertheless, modeling their effect on radiation belt electrons with bounce-averaged broadband quasi-linear theory seems to yield reasonable results. It is known that coherent interactions with monochromatic waves can cause particle diffusion, as well as radically different phase bunching and phase trapping behavior. Here the two formulations of diffusion, while conceptually different, are shown to give identical diffusion coefficients, in the narrowband limit of quasi-linear theory. It is further shown that suitably averaging the monochromatic diffusion coefficients over frequency and wave normal angle parameters reproduces the full broadband quasi-linear results. This may account for the rather surprising success of quasi-linear theory in modeling radiation belt electrons undergoing diffusion by chorus waves. 15. SUBJECT TERMS Cyclotron resonant Diffusion Radiation belt Chorus wave Wave-particle interactions 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON ABSTRACT OF a. REPORT b. ABSTRACT c. THIS PAGE James I. Metcalf PAGES 19B. TELEPHONE NUMBER (Include area code) UNCL UNCL UNCL UNL Standard Form 298 (Rev. 8/98) Prescnbed by ANSI Std Z39 18 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, A00F05, doi:10.!029/2009JA014732, 2010 Click Here tor Full Article Diffusion by one wave and by many waves 1 J. M. Albert Received 5 August 2009; revised 27 October 2009; accepted 30 October 2009; published 19 March 2010. Radiation belt electrons and chorus waves are an outstanding instance of the important [I] role cyclotron resonant wave-particle interactions play in the magnetosphere. Chorus waves are particularly complex, often occurring with large amplitude, narrowband but drifting frequency and fine structure. Nevertheless, modeling their effect on radiation belt electrons with bounce-averaged broadband quasi-linear theory seems to yield reasonable results. It is known that coherent interactions with monochromatic waves can cause particle diffusion, as well as radically different phase bunching and phase trapping behavior. Here the two formulations of diffusion, while conceptually different, are shown to give identical diffusion coefficients, in the narrowband limit of quasi-linear theory. It is further shown that suitably averaging the monochromatic diffusion coefficients over frequency and wave normal angle parameters reproduces the full broadband quasi-linear results. This may account for the rather surprising success of quasi-linear theory in modeling radiation belt electrons undergoing diffusion by chorus waves. Citation: Albert, J. M. (2010), Diffusion by one wave and by many waves, J. Geophys. Res.. 115, A00F05, doi:10.1029/2009JA014732. [e.g., Inan et al, 1978; Albert, 1993; Omura et al, 2008]. 1. Introduction The relevant regime also depends strongly on the particle [2] Cyclotron resonant wave-particle interactions play a energy and pitch angle, so all three types of behavior may key role in both the acceleration and loss of radiation belt occur under the same conditions. Albert [1993, 2000, electrons. Chorus waves, in particular, are believed to be hereafter Papers I and II, respectively] derived analytical key to both the energization and loss of energetic electrons expressions for the changes in pitch angle and energy for all in the outer zone [Chen et al, 2007; Home, 2007; Bortnik three types of motion, using a Hamiltonian formulation, and Thome, 2007]. Chorus waves propagate in the whis- though frequency drift was neglected. Similar considera- tler mode and are observed, with sufficient time resolution, tions also apply to large amplitude electromagnetic ion to be coherent, with well-defined frequencies that drift cyclotron waves [Albert and Bortnik, 2009]. In the diffusive during their growth to large amplitude [Santolik et al, 2003; regime, a key quantity is the effective interaction time, Breneman et al, 2009]. The wave growth is intimately which is controlled by how long (or far) the particle has to connected to the linear [Li et al, 2008, 2009] and nonlinear move in the varying background field before the resonance [e.g., Nunn, 1974; Katoh and Omura, 2007] behavior of condition is violated. resonant electrons with energy in the keV range. MeV range [4] The large-scale effects of chorus waves on the radia- electrons are also subject to nonlinear behavior induced by tion belts have also been modeled using quasi-linear theory the developed waves, but their motion can be considered in one, two, and three dimensions (see Albert [2009] for a "parasitic," i.e., not feeding back to the development of the brief review). This framework assumes a continuum of waves. uncorrelated, small amplitude waves, with wide distribu- [3] Coherent cyclotron resonant interactions of test elec- tions in frequency and wave normal angle, in a constant trons with individual whistler mode waves has been treated background magnetic field. Here, the diffusion can be by many authors, and yields three distinctly different kinds considered limited by the relative parallel velocity of the of particle behavior, namely diffusion, phase bunching, and particle and the group velocity of a nearly resonant wave phase trapping. Both phase bunching (without trapping) packet [Albert, 2001]. The resulting local pitch angle and and phase trapping are favored by large amplitude waves energy diffusion coefficients are computed locally and then and low inhomogeneity of the background magnetic field; a bounce averaged, which finally introduces variation of the quantitative criterion has been developed by many authors background magnetic field. Recently, the expressions for broadband quasi-linear diffusion coefficients were expressed in a relatively transparent form [Albert, 2005], which turned Air Force Research Laboratory. Space Vehicles Directorate, Hanscom out to be convenient for isolating single waves within the Air Force Base. Massachusetts. USA. broad frequency and wave normal angle distributions. Such single waves, suitably chosen, are perhaps surprisingly well This paper is not subject to U.S. copyright. able to represent the entire distributions, leading to accurate Published in 2010 by the American Geophysical Union. A00F05 1 of 8 20110509005 ALBERT: DIFFUSION BY ONE WAVE AND BY MANY WAVES A00F05 A00F05 lated from each approach, and explicitly demonstrates their approximations [Albert, 2007, 2008]. These may be con- equivalence. This is followed by a brief discussion. sidered a generalization of the parallel propagation approx- imation [Summers et al., 2007]. [s] Thus diffusion emerges from both quasi-linear and nonlinear treatments, but the underlying pictures are quite 2. Quasi-Linear Diffusion Coefficients different. Since the quasi-linear diffusion approach seems to [s] The condition for gyroresonance between a particle model the actual particle behavior fairly well [Albert, 2009], and a wave is it is of great interest to relate the two sets of diffusion coefficients. This was done by Albert [2001], working with w-*||V|j = n», S2„ = snilj-y, (1) quasi-linear expressions for whistler mode waves in the high-density, low-frequency limit [Lyons et al., 1972], where n is an integer, s = ±\ is the sign of the charge of the which invoked considerable simplifications of both the particle, Sl = \q\ Blmc is its local nonrelativistic gyrofre- c whistler mode dispersion relation and the resonance condi- quency, and 7 is its relativistic factor. The local pitch angle tion. It was concluded that the narrowband limit of the of the particle is a, the index of refraction is p = kcluj, and quasi-linear pitch angle diffusion coefficient was approxi- the wave normal angle is 8. The underlying mechanism of mately equal to the Hamiltonian-derived pitch angle diffu- quasi-linear diffusion can be thought of as involving con- sion coefficient for monochromatic waves. Here, the tinuous resonance: even as the particle diffuses in a and 7, it comparison of the two analytical frameworks is recon- is always able to find an instantaneously resonant wave sidered in much greater generality, using the full description within the u> and 0 distributions. of stationary cold plasma waves. It is shown that the nar- [9] Albert [2001] considered whistler waves, using ex- rowband limit of bounce-averaged quasi-linear theory and pressions based on the approximations uj/Q <g 1 < uipjfll e the diffusive regime of the Hamiltonian analysis yield [Lyons et al., 1972; Lyons, 1974b], but here any cold plasma exactly the same pitch angle, energy, and cross diffusion mode is considered, without any such approximations. coefficients. Furthermore, averaging the monochromatic 2.1. Local Expressions results over distributions of frequency and wave normal angle, which statistically models a sequence of resonant [10] The local diffusion coefficients in a spectrum of interactions with individual waves, recovers exactly the full waves were given by Lyons [1974a, 1974b], as derived from broadband quasi-linear diffusion coefficients. This seems a the Vlasov equation [Kennel and Engelmann, 1966; Lerche, meaningful step toward reconciling the behavior expected 1968], although it can also be obtained by considering from coherent dynamics, in the diffusion regime, with the motion of a single particle acted on by single wave, for an apparent utility of bounce-averaged quasi-linear theory for interaction time related to the wave packet bandwidth [e.g., modeling of radiation belt electrons. Albert, 2001]. In either case, the spatial variation of the [6] Any possible coupling between changes in a and p background magnetic field and all other parameters is ig- Q with changes in L will be ignored. This is usually justified nored for the local calculation, and accounted for later by by the wide separation of time scales associated with the bounce averaging. first two adiabatic invariants compared to that of the third, [11] The derivation is fairly involved (see also the pre- i.e., the drift period compared to the cyclotron and bounce sentations by Walker [1993] and Swanson [1989]), but the periods. Such coupling, which would lead to cross diffusion results for pitch angle a and momentum p can be expressed terms involving D, and D , has only been considered as XoL pL occasionally, usually in the context of so-called drift shell splitting [Roederer, 1970; Schulz and Lanzerotti, 1974], ^-Ef/s^-*!-*) although Brizard and Chan [2004] recently formulated the (2TT) "full" matrix of diffusion coefficients generated by an ar- bitrary wave spectrum in axisymmetric geometry. The 2 2 2 2 x |B | |«,,| (-«n o + «./w) «r 2 k resulting diffusion equation could be solved numerically by v iT- p- cos- a an algorithm based on stochastic differential equations [Tao D" p sin a cos a et al., 2008] or the layer method described by Tao et al. pp (2) [2009]. 2 D"„ •sin Q + !2 /w i?„, B [7] Section 2 exhibits the local quasi-linear diffusion coefficients and their monochromatic limit, following Albert D,„ has dimensions of Mt, because of the explicit division v 2 [2007], and carries out the bounce average following Albert byp . B is the Fourier transform of the wave magnetic field k [2001], leading to closed form expressions with no re- taken over the plasma volume V (which is effectively infi- maining integrals. Section 3 presents the diffusion coeffi- nite), and |$„| , as given by equation (9) of Lyons [1974b], cients of Albert [1993, 2000] resulting from coherent is the result of resonance averaging the geometric details of interactions with a single, monochromatic wave, which are the particle motion in the electromagnetic field of an oblique found to be identical to the final results of section 2. Section plane wave. The ratios of the diffusion coefficients were 4 then considers the coherent diffusion coefficients suitably interpreted by Kennel and Engelmann [1966] in terms of averaged over wave frequency and wave normal angle single-wave characteristics of a quasi-linear diffusion op- parameters, reproducing the full quasi-linear expressions. erator, and were further discussed by Lvons [1974a] and Section 5 presents some numerical examples of diffusion Summers et al. [1998]. coefficients for a model of nightside chorus waves, calcu- [12] The expressions get more involved after transforming the integration variables from (k , fci) to (w, 9), and mod- L 2 of 8 A00F05 ALBERT: DIFFUSION BY ONE WAVE AND BY MANY WAVES AO0FO5 2 9 and z. As the 9 distribution is narrowed, *j becomes a eling |B | /F as a function of 9), which brings in nor- (OJ, Ki k well-defined function of z. And as the w distribution is malization integrals. As expressed by Albert [2005], and narrowed, lf(uj) approaches a <5 function of u>. Assuming 9 similarly by Glauert and Home [2005], the resulting form m and are compatible with resonance at some location z , of the diffusion coefficients can be written as the sum over n UJ,„ m n the bounce average and G combine to give of terms D given by x fdz VcBl / — G\{uj (zJ„)) •• n 6(w (r) - uO 'jf stn&/0A„G,G , «L = (3) m c TO 2 «• J V| (7) in*" with \duj/dz\ 2 secfl , (-sin a + n„/u) •K The full wave intensity, 5 , is now considered to be wave :* A„= = 3 2| ,/c| - |l - (aw/s*,,),/*., concentrated at the single pair (9 , uj ). V| m m [i6] The derivative of w is evaluated using the resonance 2 n B {u) c G, = condition, and it is important to note that k\\ is a function of both z and u), as specified by the dispersion relation. G = Therefore implicit differentiation of the resonance condition 2 Qdff sinffg {0)T' u gives 2 (4, T =M )"'I('. (8) + «.)• [13] The refractive index /i is a known function of (ui, 9) for the given cold plasma wave mode [e.g., Stix, 1962]. [n] The factors of (AJT)/\duj/dz\ containing partial deri- 2 B ^) describes the frequency distribution of wave power, vatives combine and simplify: and is nonzero only between lower and upper cutoffs, u^c < < uiuc- Similarly, the distribution of wave power with k\v UJ 1 du> 1 + (9) I - Vll wave normal angle 9 is described by g^(9), which is nonzero du. ~ v dk~, 2 only for (9 < 0 < # . Both B ^) and g ,(tan9) are usually mjn max u modeled as truncated Gaussians, peaked at u and 9 , Putting everything together gives m m respectively. The quantities G and G are explicitly nor- x 2 2 malized versions of 5 (u>) and g (9), and are discussed 2 2 2 «,, B Jil <J> u *^wavc n D" further in Appendix A. 2 2 2 2 2 2T B |v|||v 7 fi B cos n h 0 (10) a 2 a 2.2. Narrowband Limit x I - sin •(Vii + n.) As shown by Albert [2007, 2008], the integral in [H] equation (3) may be approximated as a weighted average, where B and a are equatorial values but all other quantities eq 0 which becomes exact as gjjf) becomes narrowly peaked. In are evaluated at the resonance location. This is the mono- that limit, chromatic limit of the bounce-averaged, broadband quasi- linear diffusion coefficient for each n. 2 (l B A„ G, c mw D" = (5) [is] The bounce-averaged coefficients D" and Ef are 2 oP pp 2 i r ' B derived similarly, and in the monochromatic limit are related to D^ by evaluated at some resonant pair (us, (?) within the specified distributions. For the purposes of Albert [2007, 2008], ui LC IT D" p sin ao cos Qo B and tjJuc were used to find 9 ranges containing resonances, PI< 'JfK (ii) 2 and Ef „ was approximated using representative values from Dr., £>!U -sin ft+ft,,/wV D" n m within these ranges. In section 2.3, equation (5) is evaluated at 8„„ with u taken to be the corresponding resonant value at for each n. Albert [2004] discussed the role of these ratios in each location. 2 enforcing the condition £>„„„„ D > D . pp aP 2.3. Bounce Averaging 3. Coherent Interactions [is] The bounce-averaged diffusion coefficient for the equatorial pitch angle, is given by the sum over n of [19] A quite different scenario is that of a particle inter- Q , 0 acting with a single wave in a spatially varying magnetic field, so that the resonance condition of equation (1) is only CT (6) D".„ = -/-( do satisfied at discrete, isolated locations through which the particle passes. As mentioned, analytical estimates of the where z is distance along the magnetic field line (and is resulting particle motion were obtained in Papers I and II. 2 easily converted to latitude). In equation (3), 5 (u>) is For large amplitude waves and small background inhomo- evaluated at the resonant frequency, which depends on both geneity, nonlinear behavior (phase bunching and phase 3 of 8 ALBERT: DIFFUSION BY ONE WAVE AND BY MANY WAVES A00F05 AOOF05 (16) into equation (15) yields the first major result of this trapping) can occur, but here the opposite limit is consid- paper: the coherent interaction versions of I>" „ , D", , and ered, which leads to random walks, or diffusion. 0 0 aP Dp work out to be exactly the same as in equations (10) and [20] Papers I and II write out the full equations of motion P (11) for the narrowband limit of the bounce-averaged quasi- in Hamiltonian form, transform to gyroresonance variables, linear expressions. expand to first order in B JB, and appropriately average wm away nonresonant terms. For n t 0, this leads to two con- 3.2. Landau Resonance stants of motion which can be used to reduce the number of [23] For the special case n = 0, Paper II gives variables to a single action-angle pair, (/, Q. To lowest order, / is proportional to the familiar first adiadatic invariant, and 2TT 2 2 £ is the usual wave-particle phase which is stationary at (6r) =M (17) 2 COS + Tr d M /dzdt ^ ' 4 can De 0 resonance. The evolution equations for / and £ expressed in terms of a reduced Hamiltonian, K = K (I, z) + 0 and cr-f is the sign of d^Mo/dzdY at resonance. Here a is 0 Ki(I, z) sin£, with z playing the role of time. The adiabatic just a„ with n = 0, but now motion is described by K , while K\ captures the effects of 0 the resonant wave. For n = 0, a similar treatment yields a 2 2 2 2 2 M =alp cos e = 4cos 6-^^-^-^ . (18) reduced Hamiltonian M = M (Y, z) + M|(T, z) sin£, where 0 2 T = 7 . The reduced Hamiltonians can be used to derive analytic approximations to the resonant changes in the adi- The Hamiltonian equation of motion for £ yields abatic invariants / or T. An "inhomogeneity parameter" R, proportional to (dB/dzyB^^^, delineates diffusion from the 2 2 2 d (d£\ c d . d M d (d(,\ - 0 (19) nonlinear regimes involving phase bunching and/or phase dzdT trapping. Here we only consider the case \R\ 3> 1, which indicates diffusion. where again u> can be omitted in the z derivative. [24] The diffusion coefficients are now 3.1. Cyclotron Resonance 2 [21] At an isolated resonance n •£ 0. according to Papers I ^ ((6T) ) (dap 0 (20) and II, 2r \dX h (M)2=K s2 + (,2) and so on. Using 'wikdrr (^ <)- 2 2 : 2 dao c dp m c (21) Again, z is distance along the field line, and 07 is the sign of = dr ^-^ip* dr ~2p~ 2 d KcJBzdl at resonance. Averaging over £ , which depends res on the gyrophase and is randomized between bounces, from Paper II, the resulting coherent interaction expressions yields 1/2. Papers I and II also give the perturbation Ham- for A"v A"J°. and Dpp° again agree exactly with equations V iltonian K\ in terms of a,„ which describes the wave field (10) and (11) from the narrowband limit of bounce-averaged components. The relation between K a and $>„ noted by quasi-linear theory. u m Albert [2001] holds for general cold plasma waves 2 2 2 2 4. Average Over Wave Distributions n v H B c wave xJ. * K =• (13) 2 2 2 4 (p /mc) S v» J B -• [25] It has just been shown that the monochromatic limit ]: of bounce-averaged, broadband quasi-linear theory is well The Hamiltonian equation of motion for £ yields behaved, and reduces to the results of a Hamiltonian anal- ysis of a resonant interaction with a single wave (in the 2 &K d (di\ c d . diffusive regime). Conversely, pitch angle diffusion by a 0 single, coherent wave can be expressed in terms of the quantities defined for quasi-linear diffusion where u> is the constant frequency of the single wave and 2 can be omitted in the z derivative. n B c i/£>**»—> <=> D" =• •T(£) [22] Diffusion coefficients are constructed from 2 B for either n ± 0 or n = 0. We now consider the result of many I "o""' '""" "J 2r )\dl ) ' 81 dl' coherent interactions with individual waves all with ampli- A tude B but with frequency and wavenormal angle sta- Wivc 115) 2 tistically distributed according to B (J) and gjfi). [26] The appropriate average is where ( ) denotes the average over £ . From Paper II, rcs /D lB |Vk 2 2 2 dcto _ — sin <\+Q„/ui B m tr 7 dp _ np-c 7 k k eq (23) (16) 2 dl sin a cos <* B p sn' dl p sn 0 0 The corresponding ratio dp!da is closely related to the ra- where D refers to the single-wave equation (22). The Q k tios in equation (11). Substituting equations (12>—(14) and denominator of (23) is just Bl, . Converting from d\ to 3vc 4 of 8 ALBERT: DIFFUSION BY ONE WAVE AND BY MANY WAVES A00F05 A00F05 [2005] and Albert [2008], the wave normal angle distribu- 1 D (day" ) aa tion is modeled with 8 = 0,66 = 30°, 6 = 0, and 8 = 45°. m mm max r 100.0000 [28] Figure 1 shows the local quasi-linear pitch angle diffusion coefficients for 1 MeV electrons for several values 10.0000 [ X= 0° of latitude, calculated from equation (3). Only contributions 5° by n = -1 are shown. For each wave normal angle in the 1.0000 distribution, the resonant frequency is found; if both lie 15° within the model distributions, a contribution is made to the diffusion coefficient integrals. The 'usual' quasi-linear 0.1000 results [e.g., Horne et al., 2005; Albert, 2005] consist of just such calculations, converted from a to a and bounce 0 0.0100 - averaged, as in equation (6), and summed over n. [29] Figure 2 shows equatorial pitch angle diffusion coefficients for individual waves with 8 = 0 = 0 and var- 0.0010 r m ious frequencies between and u> , calculated according UJ uc LC to the coherent formulation, equations (12) and (15), with 0.0001 n = -1. Related calculations were previously presented by 0 15 30 45 60 75 90 Albert [1993, 2000, 2002] and Albert and Bortnik [2009]. a As mentioned in section 3, integration along the field line is Figure 1. Local quasi-linear pitch angle diffusion coeffi- inherent in the formulation. The curve for 6 = 6 ,u = u> , is m m cient for 1 MeV electrons interacting with a broadband spec- emphasized by the dashed curve. Figure 3 is similar, but trum of chorus waves at L = 4.5 at different latitudes. At the shows the results holding u = u fixed and varying 0 from m equator, the spectrum is peaked at 8 - 0, w /f2<. = 0.35. m m Only the lowest harmonic (n = -1) term is shown. [30] Figure 4 shows, as solid curves, the quasi-linear diffusion coefficients after carrying out the bounce averages of the local results illustrated in Figure 1. The sum of contributions from n = -\ and n = +\ are shown in the top du)dO in the numerator, using the results of Appendix A, row, and just n = 0 is shown in the bottom row. Also shown, gives as red squares, are the results of numerically averaging the diffusion coefficients for monochromatic waves, from 2 sin 6gU0)T S M (24) (D" ) Figures 2 and 3, weighted according to equation (24). It is duid8D • k 2 1 • jB {J)dJ jdff sinffgUS )^' -/ apparent that, allowing for numerical accuracy, the com- Then, schematically, I duj I d8 I dz <5(u> (r) - u ) rcs n -1 D o (day ) (25) a0a 100.0000 F => / dz j dO I duj 6{ui - u> ) => f d: I d6. m aj/n. =o 05 q 0.10 10.0000 r 0.15 which yields 0.20 1.0000 r 025 0 30 0.35 0.1000 r 0.40 This is the second major result of this paper: the coherent interaction diffusion coefficient, suitably averaged, is iden- 0.0100 r tical to the full quasi-linear result given by equation (6). The analogous relations hold for {£>"„,,) and (Dp ). P 0.0010 r 0.0001 . 5. Numerical Example 15 30 45 60 75 90 [27] For illustration, we consider the model of Li et al. [2007] for nightside chorus during a magnetic storm main phase, at L = 4.5 with uJ JQ. = 3.8 at the equator. They Figure 2. Equatorial pitch angle diffusion coefficient for p e computed quasi-linear diffusion coefficients for waves with 1 MeV electrons interacting with monochromatic chorus = Swavc 50 pT, with the equatorial frequency distribution waves at L = 4.5, treated as a coherent interaction. Results specified by ui = 0.35 il , 6ui = 0.15 Q„ uj = 0.05Q , and are shown for a fixed value of wave normal angle and sev- m e LC e uiuc - 0.65 Q ,. The waves are considered present only for eral fixed values of frequency; the dashed line indicates 8 = c latitude A < 15°. In that work the waves were all taken to 8 = 0, uj = tj = 035Q (at the equator). Only the lowest m m e propagate with 8 = 0, but here, following Horne et al. harmonic (n = -1) term is shown. 5 of 8 A00F05 ALBERT: DIFFUSION BY ONE WAVE AND BY MANY WAVES A00F05 interactions. Generalizing a previous study, it has been -1 D„o o (day ) a shown analytically that taking the narrowband limit of 100.0000 [ bounce-averaged, broadband quasi-linear diffusion coeffi- cients agTees exactly with the diffusive limit of coherent e= o° 10.0000 r interactions with a monochromatic wave. Moreover, con- 5° 10- sidering the individual waves to be drawn from specified 15" 1.0000 r frequency and wavenormal angle distributions, and aver- 20° aging diffusion coefficients accordingly, reproduces the full 25° quasi-linear expressions. 0.1000 r [32] It has been a puzzle why global simulations using 40° quasi-linear theory [Li et al., 2007; Albert, 2009] are at least 45' 0.0100 r moderately successful in reproducing the observed effects of chorus waves, which upon close examination are discrete 0.0010 and coherent [Santolik et al., 2003]. Parameters used to model chorus waves as a population which are based on 0.0001 wave measurements with coarse time resolution [Meredith _i_ et al., 2003] should reflect the distribution of the underly- 15 30 45 60 75 90 ing individual waves. As just shown, multiple interactions «o with this distribution of waves will be well described sta- Figure 3. Same as Figure 2 but showing results for fixed tistically by the quasi-linear approach, as long as the indi- vidual waves are not large enough to induce nonlinear frequency and several values of wave normal angle. particle behavior [Cattell et al., 2008; Cully et al., 2008]. [33] It should be noted that in all cases, the wave para- putational evaluations verify the analytical result that the meters (amplitude, frequency, wave normal angle) have been two formulations are the same. treated as constant during each individual wave-particle interaction. Although the quantities can vary significantly, indeed, frequency drift is a characteristic feature of chorus 6. Summary and Discussion waves, the duration of an isolated interaction is brief in the [31] This paper has investigated the relationship between diffusive regime. This would not apply to phase-trapped two seemingly different formulations of wave-particle particles, which experience an extended resonant interaction Figure 4. Bounce-averaged quasi-linear diffusion coefficients (solid curves) and diffusion coefficients for coherent interactions with monochromatic waves, averaged over the same frequency and wave normal angle distributions (red squares), (top) The contributions from n = ±1 and (bottom) the contributions of just n = 0 are shown. As predicted analytically, calculations using the two approaches agree. 6 of 8 ALBERT: DIFFUSION BY ONE WAVE AND BY MANY WAVES A00F05 A00F05 which corresponds to equation A8 of Lyons, and which time, and which are believed to be key for the self-consistent, 2 satisfies equation (Al) above for any choice of B ^) and nonlinear growth of chorus waves. gjfi). In the notation of equation (4), [34] For computing diffusion coefficients, there is no apparent major advantage to either viewpoint; the same 2 |B | 4TTV number of integrals must be done either way. However, the k (A7) G G Btmc l l< coherent interaction approach has the large benefit of indi- cating when the diffusion approach becomes invalid, and which is used in section 4. nonlinear effects must be considered. Estimates of these effects have the form of velocity space advection, and may [37] Acknowledgments. This work was supported by the Space Ve- be included in a combined diffusion-advection equation hicles Directorate of the Air Force Research Laboratory and by UCLA by [Albert, 1993, 2000, 2002]. The refinement of these esti- NSF grant ATM-0903802. [38] Amitava Bhattacharjcc thanks the reviewers for their assistance in mates, and their use in global simulations, is the subject of evaluating this manuscript. ongoing work. References Appendix A: Parameterization of the Wave Albert, J. M. (1993), Cyclotron resonance in an inhomogencous magnetic Distribution field. Phys. Fluids B. 5. 2744. Albert, J. M. (2000), Gyrorcsonant interactions of radiation belt particles [35] The Fourier transform of the squared wave magnetic with a monochromatic electromagnetic wavc, J. Geophys Res., 105. field is 21,191, doi:10.1029/2000JA000008. Albert, J. M. (2001). Comparison of pitch angle diffusion by turbulent and 2 3 |B ] d k 2 monochromatic whistler waves, / Geophys. Res.. 106. 8477. s - f k (Al) wave / Albert, J. M. (2002), Nonlinear interaction of outer zone electrons with v 3 (2*) VLF waves, Geophys. Res. Lett.. 29(8). 1275, doi:IO.I029/ where 2001GL013941. Albert, J. M. (2004), Using quasi-linear diffusion to model acceleration and loss from wavc-particlc interactions. Space Weather. 2. S09S03, doi: 10.1029/2004SW000069. Albert, J. M. (2005). Evaluation of quasi-linear diffusion coefficients for whistler mode waves in a plasma with arbitrary density ratio, J. Geophys. Res.. 110. A03218, doi:10.1029/2004JA010844. Albert, J. M. (2007). Simple approximations of quasi-linear diffusion coef- ficients. J. Geophys. Res.. 112, A12202. doi: 10.1029/2007JA012551. Albert. J. M. (2008), Efficient approximations of quasi-linear diffusion = =• / dui I d8 k J, (A2) coefficients in the radiation belts, J. Geophvs Res.. 113. A06208. 2 (2n) Jo Jo doi:10.1029/2007JA012936. Albert, J. M. (2009), Three-dimensional simulation of outer radiation belt for any cold plasma mode. electrons during the 9 October 1990. magnetic storm. J. Geophvs. Res., A09214, doi:10.1029/2009JA014336. 1 •J sin# Albert, J. M.. and J. Bortnik (2009), Nonlinear interaction of radiation belt r, (A3) J = electrons with electromagnetic ion cyclotron waves, Geophvs Res. Lett.. d(uj.8) 36. L121I0, doi:10.1029/2009GL038904. Bortnik. J.. and R. M. Thome (2007), The dual role of ELF/VLF chorus and T(u;, 0) is (j. \fi + u>{dfj./duj)\ as in equation (4). waves in the acceleration and precipitation of radiation belt electrons. J. Atmos. Solar Terr. Phys.. 69, 378. doi:10.1016/j.jastp.2006.05.030. [36] Lyons [1974b] explicitly assumed that the wave dis- Brcncman, A. W., C. A. Klctzing, J. Pickctt, J. Chum, and O. Santolik tribution was independent of both 4>k and the sign of k\\, so (2009), Statistics of multispacccraft observations of chorus dispersion that the integrals could be restricted to 0 < k^ < oc or 0 < 6 < and source location, J. Geophys. Res.. 114. A06202, doi: 10.1029/ 2008JA013549. nil, with an additional factor of 2. However, to connect to Brizard, A. J., and A. A. Chan (2004). Rclativistic quasilincar diffusion in single-wave results, it is more natural not to assume sym- axisymmctric magnetic geometry for arbitrary-frequency electromagnetic metry with respect to ±k\, and to take 0 integrals from 0 to jr. fluctuations. Phys. Plasmas. 11. 4220. doi: 10.1063/1.1773554. Then Cattcll. C. ct al. (2008), Discovery of very large amplitude whistler-mode waves in Earth's radiation belts. Geophys. Res. Lett., 35. L01105, j;r¥- doi: 10.1029/2007GL032009. jdujde. (A4) 2 Chen, Y., G. D. Reeves, and R. H. W. Fricdcl (2007), The energization of B = • rclativistic electrons in the outer Van Allen radiation belt. Nat. Phys.. 3, (27T) 614. Following Lyons, we now write Cully, C. M., J. W. Bonncll. and R. E. Ergun (2008). THEMIS observa- tions of long-lived regions of large-amplitude whistler waves in the inner magnctosphcrc, Geophys. Res Lett., 35, L17SI6, doi: 10.1029/ 2 (A5) £.« = (^ B (uj)dcj 2008GL033643. ,/o Glaucrt, S. A., and R. B. Home (2005), Calculation of pitch angle and en- ergy diffusion coefficients with the PADIE code, / Geophys. Res., 110, 2 2 and also factor B {ui) out of |B ' /K. This leads to A04206, doi: 10.1029/2004JA010851. k Home, R. B. (2007), Plasma astrophysics: Accclcrarion of killer electrons, 2 2 Nat. Phys.. 3. 590. |B | 4;rV _, B {ui) 4rrV . t Home. R. B.. R. M. Thome. S. A. Glaucrt. J. M. Albert. N. P. Meredith, wavc ur 2 and R. R. Anderson (2005), Timcscalc for radiation belt electron accel- / B Lj)du eration by whistler mode chorus waves. J. Geophvs. Res.. 110. A03225, doi: 10.1029/2004JA010811. gM Inan, U. S„ T. F. Bell, and R. A. Hclliwcll (1978). Nonlinear pitch angle (A6) scattering of energetic electrons by coherent VLF waves in the magncto- Ig^Y(u^) sin S'd9' sphcrc, J, Geophys. Res.. 83. 3235. 7 of 8 AOOF05 ALBERT: DIFFUSION BY ONE WAVE AND BY MANY WAVES A00F05 Katoh. Y., and Y. Omura (2007), Rclativistic particle acceleration in the Omura, Y.. Y. Katoh, and D. Summers (2008). Theory and simulation of process of whistlcr-modc chorus wave generation, Geophys. Res. Lett., the generation of whistlcr-modc chorus, J. Geophys Res., 113, 34, L13102, doi:10.1029/2007GL029758. A04223, doi:10.I029/2007JAOI2622. Rocdcrcr, J. L. (1970), Dynamics of Geomagnetwally Trapped Radiation, Kennel, C. F., and F. Engclmann (1966), Velocity space diffusion from weak plasma turbulence in a magnetic field, Phys. Fluids, 9, 2377. Springer. New York. Santolik. O., D. A. Gumctt J. S. Pickctt. M. Parrot, and N. Comilleau- Lcrchc, I. (1968). Quasilincar theory of resonant diffusion in a magneto- Wchrlin (2003), Spatio-temporal structure of storm-time chorus. J. Geo- active, rclativistic plasma, Phys. Fluids, II, 1720. phys. Res., /0«(A7), 1278, doi:10.1029/2002JA009791. Li, W., Y. Y. Shprits, and R. M. Thome (2007), Dynamic evolution of en- Schulz, M., and L. J. Lanzcrotti (1974), Particle Diffusion in the Rudiahon ergetic outer zone electrons due to wavc-particlc interactions during Belts, Springer, New York. storms,/ Geophys. Res., 112, A10220, doi:I0.I029/2007JA012368. Sox. T. H. (1962), 77ie Theory of Plasma Waves. McGraw-Hill. New York. Li, W„ R. M. Thome, N. P. Meredith. R. B. Home, J. Bortnik, Y. Y. Summers, D„ R. M. Thome, and F. Xiao (1998). Rclativistic theory of Shprits, and B. Ni (2008), Evaluation of whistler mode chorus amplifica- wavc-particlc resonant difusion with application to electron acceleration tion during an injection event observed on CRRES, J. Geophys. Res.. in the magnctosphcrc, J. Geophys. Res., 103, 20.487. 113, A09210, doi:10.1029/2008JA013129. Li, W., R. M. Thome, V. Angclopoulos, J. W. Bonncll, J. P. McFaddcn, Summers, D„ B. Ni, and N. P. Meredith (2007), Timcscalcs for radiation belt electron acceleration and loss due to resonant wave-particle interac- C. W. Carlson, O. LcContcl. A. Roux, K. H. Glassmcicr, and H. U. Austcr (2009), Evaluation of whistlcr-modc chorus intensification on tions: 1. Theory, J. Geophys. Res., 112, A04206. doi: 10.1029/ the nightsidc during an injection event observed on the THEMIS space- 2006JA011801. craft. J Geophys. Res., 114, A00CI4, doi: 10.1029/20O8JA013554. Swanson. D. G. (1989), Plasma Waves. Academic, San Diego, Calif. Lyons, L. R. (1974a), General relations for resonant particle diffusion in Tao. X., A. A. Chan, J. M. Albert, and J. A. Miller (2008), Stochastic mod- cling of multidimensional diffusion in the radiation belts, J. Geophys. pitch angle and energy, / Plasma Phys., 12, 45. Lyons, L. R. (1974b), Pitch angle and energy diffusion coefficients from Res.. 113. A07212, doi:10.1029/2007JA012985. resonant interactions with ion-cyclotron and whistler waves, J. Plasma Tao, X., J. M. Albert, and A. A. Chan (2009), Numerical modeling of multidimensional diffusion in the radiation belts using layer methods. Phys., 12, AM. Lyons, L. R., R. M. Thome, and C. F. Kennel (1972), Pitch angle diffu- J. Geophys. Res.. 114, A02215. doi:10.1029/2008JA013826. sion of radiation belt electrons within the plasmasphcrc, J, Geophys. Walker, A. D. M. (1993), Plasma Waves in the Magnetosphere, Springer, New York. Res.. 77, 3455. Meredith, N. P.. R. B. Home, R. M. Thome, and R. R. Anderson (2003), Favored regions for chorus-driven electron acceleration to rclativistic J. M. Albert, Air Force Research Laboratory. Space Vehicles Directorate. energies in the Earth's outer radiation belt, Geophys. Res. Lett., 30(\6), 29 Randolph Rd., Hanscom Air Force Base. MA 01731-3010. USA. (jay. 1871. doi:10.1029/2003GL017698. [email protected]) Nunn, D. (1974), A self-consistent theory of triggered VLF emissions. Planet. Space Sri.. 22(12), 349. 8 of 8