Numerical modeling of stably and transiently confined energetic heavy ion radiation in the Earth's magnetosphere

Radiation Measurements, Vol. 26, No. 3, pp. 309--320, 1996

Pergamon PH: S1350-4487(96)00059-5

Copyright © 1996 ElsevierScienceLtd Printed in Great Britain. All fights reserved 1350-,1487/96 $15.00+ 0.00

NUMERICAL MODELING OF STABLY AND TRANSIENTLY CONFINED ENERGETIC HEAVY ION RADIATION IN THE EARTH'S MAGNETOSPHERE W. N. SPJELDVIK Physics Department, Weber State University, Ogden, UT 84403-2508, U.S.A.; and NASA Headquarters--Code SSM, 300 E-Street, SW, Washington, DC, 20546, U.S.A. Abstract--The Earth's radiation belts contain substantial fluxes of electrons and ions of hydrogen, helium, carbon, nitrogen, oxygen, silicon, iron and other ion species. In the early space exploration era it was thought that hydrogen ions (protons) were the dominant ion species with only minor contributions of helium and heavier ions. Sophisticated instrumentation flown on modem spacecraft have shown that the heavier ion species can be very important and even be the dominant contributor to the Earth's trapped particle environment when relative comparison is viewed in terms of total ion energy rather than energy per nucleon. In such comparisons it is found that helium ions can compete favorably with hydrogen in relative abundance, and that at MeV energies oxygen and even iron ions can be very significant. In contrast, comparison of ionic composition generally tends to favor protons in the upper keV and MeV energy ranges when energy per nucleon is considered. Copyright © 1996 Elsevier Science Ltd

Numerical modeling of geomagnetically confined particles has been carried out by many authors in Russia, in the U.S.A. and elsewhere, and in particular stably trapped particle modeling has been developed for equatorially mirroring particle species both for time stationary conditions and for specific event periods (for magnetic storm injections and for man-made nuclear events in geospace). The radiation belt pitch angle scattering process has been studied for energetic electrons (but not very much for ions) at specific geomagnetic L-shells. The full coupled problem of simultaneous radial and pitch angle diffusive transport has only recently been worked on with useful results yet to be forthcoming. The charge exchange process turns out to be a prime physical process for geomagnetically confined ions below a few MeV, and energy degradation due to Coulomb collisions and non-adiabatic transport are important at the higher energies. The present workshop report briefly describes the status of energetic ion analysis and energetic heavy ion modeling from the perspective of stochastic adiabatic invariant violation theory using prescribed boundary conditions (steady state or time variable) and given diffusive transport coefficients in adiabatic invariant space (also steady state or time variable). The predictions from these models entail relative ion abundances, ion charge state distributions, energy spectra and radial distributions. Representative graphical modeling results are presented for ion species that have not received exposition in the accessible scientific literature; in the

interest of brevity, graphical results that are accessible in the literature are not reproduced here and only pointed out and discussed. So far no fully comprehensive numerical modeling has taken into account the effects of a dynamically variable geomagnetic field, of a time variable exosphere, and of an evolving plasmasphere. The source process for trapped radiation belt particles is also a current research topic, and some literature is cited for relevant aspects of the radiation belt source and for dynamics problems.

1. INTRODUCTION The radiation belts of the Earth have been observed and studied since in situ space exploration began. This is an area where direct in-situ measurements, theoretical analysis and modeling have been quite successful in providing insight into many of the governing physical processes. Since the first reports on the discovery of geomagnetically confined radiation (high fluxes of very energetic protons and electrons) came to our attention, a multitude of spacecraft have carried increasingly sophisticated instrumentation into what is now known as the Earth's radiation belts which is a fairly persistent, although also a strongly variable energetic particle confinement region. Over the space era there have been important developments in spacecraft 309

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detector instrument technology: Geiger counters of the early space era have been replaced by solid state energetic particle detectors, and these have now been extended with elaborate time-of-flight techniques using foils, solid state detectors and high time resolution microchannel plates particle 'optics' and imaging detectors. From the analysis of Wentworth et al. (1959) and Liemohn (1961) it was known that radiation belt electrons and protons had finite residence times within the geomagnetic confinement region. The existence of geomagnetically trapped ions heavier than protons was subsequently reported more than two decades ago (e.g. Krimigis and Van Allen, 1976; Van Allen et al., 1970; Krimigis et al., 1970). With current spacecraft instrumentation technology we are now able to distinguish between ions of hydrogen, helium, lithium, carbon, nitrogen, oxygen, the neon/silicon group, the iron/nickel group, and ions even heavier than iron. These instruments have been flown on several spacecraft including ISEE-1 (an early TOF design), AMPTE/CCE and AMPTE/IRM (later improved TOF designs). Some of the results from the AMPTE mission have been summarized by Gloeckler and Hamilton (1987). Russian instruments have been flown on a multitude of spacecraft including the Molniya-series, the Cosmos-series and many other spacecraft. Similar and further developed instruments are expected to be utilized on Earth orbiting spacecraft within the International Solar Terrestrial Physics (ISTP) program in the middle 1990s and beyond. Already launched is the U.S./ Japanese GEOTAIL spacecraft and the NASA SAMPEX spacecraft. Several advanced instruments for measuring ions and electrons, and instrumentation for plasma and fields measurements will be carried on the Global Geospace (GGS) spacecraft POLAR and WIND; on the FAST spacecraft in the NASA/SMEX program; and on the European EQUATOR-S spacecraft. Soon to be launched are the four Russian research spacecraft, collectively known as INTERBALL, for detailed investigation of the polar and far magnetotail parts of the Earth's magnetosphere. The great advances in experimental instrumentation and observations call for matching advances in theoretical analysis and modeling. Indeed, there have also been significant theoretical advances since the earliest explanations of the Earth's radiation belts were presented (e.g. Rosenbluth et al., 1957; Kellogg, 1959; Parker, 1960). It it now well known that the characteristics of much of the trapped radiation can be described in terms of certain approximate constants, called adiabatic invariants (Alfven and Falthammar, 1963), and their stochastic violation. The fundamental dynamics of geomagnetically confined radiation is summarized in reports and treatises by Tverskoy (1964, 1965, 1969, 1971), and in the books of Roederer (1970) and of Sehulz and Lanzerotti (1974). Summaries also exist in various

review papers (e.g. Walt, 1971; Spjeldvik and Rothwell, 1983) and in practical handbooks such as the U.S. Air Force Handbook of Geophysics and the Space Environment (Spjeldvik and Rothwell, 1986). Today we understand some of the main reasons why the electron component of the Earth's radiation zone is split into two 'belts' by the effects of wave-particle interactions (e.g. Lyons et al., 1972; Lyons and Thorne, 1973), while the fairly stable confined ionic component, at an earlier time thought to be principally protons, appears as a single principal broad zone structure when observed at a given ion energy (e.g. Nakada and Mead, 1965; Cornwall, 1972; Spjeldvik, 1977; Spjeldvik and Fritz, 1978a,b). A 'third intermittent radiation belt' has also been predicted and recently discovered, consisting exclusively of ions heavier than hydrogen and appears to originate in the 'anomalous' cosmic rays that temporally permeate the solar system from an acceleration region assumed to be at the heliopause. Its characteristics are discussed elsewhere in these proceedings. For the geomagnetic stable trapping region, and from particle source considerations, it was first thought that electrons and protons should be the dominant constituents of the magnetically trapped radiation while ions of helium, carbon, nitrogen, oxygen etc. should be present only in trace amounts. In an overall sense this may indeed hold true, but seen within certain energy ranges some surprises have emerged. Fritz and Wilken (1976) reported that MeV helium ions were more abundant than MeV protons at the geostationary altitude, and some years earlier, Shelley et al. (1972) had reported oxygen to hydrogen ion flux ratios well exceeding unity at a few keV total ion energies in the deep interior of the trapping region. These early findings have subsequently been confirmed and extended both to L-shells below the geostationary altitude and to other ion species. There have been many advances in experimental studies (e.g. review by Spjeldvik and Fritz, 1982), and today we know that ions are supplied both from an extraterrestrial source (mostly the sun) as well as from a terrestrial source (mostly the topside ionosphere). The importance of the terrestrial particle source at the lower (eV to several keV) energies is now well established (e.g. Lennartsson et al., 1985; Chappell et al., 1987; Lennartsson, 1989, 1991) while evidence from hundreds of keV to MeV energy iron ions point to a solar source, most likely from acceleration of solar wind constituents, and from the high energy tail of the energy distribution in the solar wind flow and the intermittent flux of solar energetic particles arriving in the vicinity of the Earth. In particular, large fluxes of carbon and iron ions stably confined in the radiation belts are unlikely to be of terrestrial origin (e.g. Blake, 1973; McEntire, personal communication, 1993).

HEAVY ION RADIATION IN THE EARTH'S MAGNETOSPHERE 2. EARLY RADIATION BELT ION CONTENT MODELING EFFORTS Tverskoy (1964) in Russia and Nakada and Mead (1965) in the United States developed descriptive proton diffusion models that physically portrayed the radiation belts in an overall sense. Farley et al. (1970) and Cornwall (1972) computed proton and helium ion fluxes in the equatorial trapping region, and Spjeldvik (1977) modeled the proton fluxes using realistic boundary conditions obtained from a geostationary spacecraft. Figures 2-4 in the earlier work of Spjeldvik (1977) show the computed hydrogen ion content of the radiation belt region from this research. Within a limited energy range where the comparison was possible (roughly 100 keV to a few MeV) there was a good match between quiet time theory and observation. At tens of keV and below the theoretical proton flux computations tend to fall short of observed ion fluxes. One may possibly conclude that either the ions at these low energies are not protons or the assumed crossfield diffusive transport rate was taken to be too low. These aspects are addressed in recent research (e.g. Sheldon and Hamilton, 1993). In analysis of radiation belt heavy ions, Spjeldvik and Fritz (1978a,b), also applied realistic outer radiation zone boundary condition to helium and oxygen ions and predicted the radiation belt content of these species. Within the very restricted energy range of 1.2-3.2 MeV total helium ion energy where experimental data were available (in two helium ion passbands) a good match between helium ion theory and observation was found, but a critical intercomparison between theory and observation at lower energies was then not possible. Some details are found in Fig. 5 in the work of Spjeldvik and Fritz (1978a). A fair match between theory and observation was also obtained for 1.8-4.8 MeV total oxygen ion energy (with only a single oxygen ion passband available), and this is contained in Fig. 11 in the work of Spjeldvik and Fritz (1978b). As in the case of helium ions, a more critical comparison between theory and observation extending over a greater energy range is needed for radiation belt oxygen ions. 3. EQUATIONS OF ENERGETIC PROTON MODELING The hydrogen ion steady state transport equation for equatorially mirroring particles has the form (e.g. Cornwall, 1972; Spjeldvik, 1977; Spjeldvik and Rothwell, 1983; Spjeldvik, 1988): L20/OLIDL, pUOfdOt.] + Gp(L)#-'/'Ofp/O#

-fp/Tp - A,0fp = 0

(1)

where fp is the proton distribution function, L is geomagnetic L-shell, D,,p is the proton radial diffusion coefficient at the equator, Gp(L) is the

311

proton energy loss factor (Cornwall, 1972), la is magnetic moment, Tp is the effective time dependent scattering loss rate for energetic protons away from the geomagnetic equatororial plane, and A~0 is the proton to fast neutral hydrogen charge exchange frequency per unit distribution function (e.g. Spjeldvik, 1979). The most common form of the radial diffusion coefficient, DLLi (with ion label ~=p for protons) applied to equatorially mirroring energetic charged particles has been developed by Falthammar (1968) implemented by Cornwall (1972) and applied by Spjeldvik (1977) in the form: BEt. i = CmL ~°+ C~LI°/[L 4 -]- {/2/(Zi/~0)} 2]

(2)

where ~ = 1 MeV/G, Z~ = ionic net charge state number ( = 1 for protons), and the most common values of the effective radial diffusion amplitude sub-coefficients have been taken as: C m = l to 2 0 0 x 10-15s-1 and Ce= 1 to 200× 10-~°s-~

(3)

although these values have generally been arrived at either by fitting the descriptive equation to particle data and solving for the transport coefficients (as is most often the case for the Cm-values), or from considerations of geomagnetic field fluctuations at the geostationary altitude (e.g. Arthur et al., 1978; Lanzerotti et al., 1978). From the latter approach there is a quasi-empirical relation between the geophysical Kp-index and the magnetic radial diffusion coefficient amplitude sub-coefficient: Cm = 10 -a where t2 = 9.60-0.07 × 2;Kp

(4)

with the summation extended over Kp-values for the 12 h preceding the point of time in question. The Co-values are empirically related to the strength of the azimuthally averaged geoelectric field fluctuations experienced by the drifting particles (e.g. Cornwall, 1972; Holzworth and M ozer, 1979), and these electric fields can either be globally induced (e.g. dawn-dusk directed cross-magnetosphere electric fields) or charge separation fields (Falthammar, 1965), or even locally generated electric fields associated with the Earth's atmosphere and ionosphere. Crowley et al. (1976) have empirically analyzed radial diffusion of inner zone protons. Mozer (1971), Kelley et al. (1979), Fejer (1986), Earle and Kelley (1987) and Fejer et al. (1990) have studied time fluctuations of ionospheric and magnetospheric electric fields, and Sheldon and Hamilton (1993) and Sheldon (1993) have presented evidence in favor of a large electric field fluctuation generated radial diffusion coefficient at middle to low geophysical L-shells. Thus it appears that the effective values of the C0-parameter should be subject to further studies with the aim of possibly linking it with standard geophysical indices. The charge exchange coefficient for charge exchange from charge state i to charge state j is given

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by the azimuthal drift phase averaged exospheric density p, the charge exchange cross section in question, o~j, and the speed v~of the incident energetic ion: A~j = oov, p

(5)

where o~j is the charge exchange cross section from ionic state i to state j, v, is the ion kinetic speed, and p is the mean density of neutal hydrogen atoms in the Earth's exosphere that the azimuthally drifting energetic ion experiences. The proton diffusive transport equation written in the above form (1) is strictly valid only for equatorially mirroring particles. There is a more general form needed when off-equatorially mirroring particles are considered, involving the conservation of both the first adiabatic invariant and the (non-zero) second adiabatic invariant, and for such particles it is also necessary to carry out a combined bounce and drift averaged value of p. Restricting the attention to the geomagnetic equator, the coefficient for the Coulomb collision energy degradation term attains a particularly simple mathematical form: Gp(L) = 5 0 Z 2 r c e 4 p 2 x / ~ / m , B 3/2

(6)

where m, is the electron mass, M, is the ion mass (i = p for protons), e = the unit electric charge, p is the effective drift phase averaged exospheric density, and B is the azimuthally averaged magnetic field induction value (in gaussian units) at the driftshell of the particles [for details, see Cornwall (1972) for the case of ~t defined in terms of mass per nucleon and Spjeldvik (1977) for the derivative case of ~t defined in terms of total ion mass]. A more recent and quite detailed analysis of the Coulomb collision processes in the magnetosphere has been given by Fok et al. (1991) taking into account realistic thermal plasma parameters appropriate for the plasmasphere and terrestrial exosphere region. There is currently no reliable information on the proton pitch angle scattering life time scale, Tp, and the term containing this quantity has usually been ignored by setting Tp = infinity. This is in sharp contrast to radiation belt electron studies where the corresponding wave-induced pitch angle scattering effect plays a dominant role. 4. EQUATIONS OF ENERGETIC HELIUM ION MODELING Magnetospheric in situ measurements of ions heavier than protons were made by space scientists both in the United States and in Russia (e.g. Krimigis and Van Allen, 1976; Van Allen et al., 1970; Krimigis et al., 1970; Shelley et al., 1972; Mogro-Campero, 1972; Panasyuk personal communication, 1977), and follow-up work gave many details about the confined ion populations (e.g. Hovestadt et al., 1972a, 1972b, 1978a, 1978b, 1981; Fritz and Williams, 1973; Blake, 1973, 1976; Blake et al., 1973, 1980; Sharp et al.,

1974a, 1974b, 1976a, 1976b, 1977a, 1977b; Fennell et al., 1974; Shelley et al., 1974, Shelley et al., 1976a, 1976b, 1977; Johnson et al., 1974, 1975, 1977, 1978;

Fritz and Spjeldvik, 1978, 1979; Spjeldvik and Fritz, 1978a, 1978b, 1978c, 1978d, 1981a, 1981b, Panasyuk, 1980; Lundin et al., 1980; Blake and Fennell, 1981; Panasyuk and Vlasova, 1981). In attempts to match this wealth of experimental information, transport modeling of hundreds of keV and MeV helium ions have been carried out using a pure diffusion approximation for equatorially mirroring ions that considers azimuthal averages around the Earth. The 'standard' descriptive diffusive equatorially mirroring helium ion transport equations are most often formulated as:

L2d/OL[DLL,L2af~/,:3L]+ G,(L)~- ':20A/O# - AIT~ - - A l O f l - - A l 2 f l + A21f2 = 0

(7)

and

L20/OL[DLL2L2Of2/OL] + G2(L)la-'/20A/OU--f2/T2 - .42,f~ + A , ~

= o

(8)

where f~ and f2 are the helium ion distribution functions for helium ion charge states 1 and 2 respectively, and the other parameters are the helium ion equivalent of the proton transport parameters and coefficients discussed above. 5. RADIATION BELT CARBON ION MODELING

The general heavy ion transport equations for a given elemental species mirroring at the geomagnetic equator form a coupled set of transport equations and can be given in the form (Spjeldvik and Fritz, 1978b; Spjeldvik, 1979): 63fJt;3t = Lec3/OL[D<
n

ou - £ / r ~ - A~£ - E ,4~£ + Z A~jf~ k=l

(9)

k-I

where t is time, j and k is the positive ionic charge state numbers, n is the maximum number of available charge states of a given elemental ion species, f~ is the electron distribution function, t is time, L is geomagnetic L-shell, DLU is the ion radial diffusion coefficient at the equator, Gj(L) is the electron energy loss factor (Cornwall, 1972), Ix is magnetic moment, Tt is the effective time dependent scattering loss rate for energetic electrons away from the geomagnetic equatororial plane, and A~ is the charge exchange frequency per unit distribution function from ionic charge state j to ionic charge state k (e.g. Spjeldvik, 1979). In case of carbon ions, there are six possible positive charge states and thus six simultaneous second order partial differential equations to solve. Unlike results for radiation belt protons and helium ions which are generally available in the scientific literature, there is a scarcity of information

HEAVY ION RADIATION IN THE EARTH'S MAGNETOSPHERE available on confined energetic carbon ions. For this reason Fig. 1 includes a display of model predictions of radiation belt carbon ions under geomagnetically quiescent conditions, presented as a collection of radial profiles of theoretical carbon ion phase space densities (distribution functions) for a range of magnetic moments from 1.6 MeV/G to 16400 MeV/ G. The model computations utilize carbon ion boundary conditions at a high geomagnetic L-shell (L = 8) as recorded with the Medium Energy Particle Analyzer (MEPA) instrument (i.e. McEntire et al., 1985) on the AMPTE/CCE spacecraft (i.e. Spjeldvik, 1991). The 'moderate' radial diffusion sub-coefficients applied here are:

313

Undoubtedly, the model results have much more limited validity than the large range of magnetic moments indicated in Fig. 1 might suggest. Presumably the results are less than valid for L-shells and energies where the carbon ion energy is below roughly 100 keV per ion (where electric field effects and convection becomes important) and above 10 MeV per ion (where finite gyroradius effects and non-adiabatic ion motion may well invalidate the adiabatic invariant approximation). One of the prominant features of this theoretical calculation is a diminishing phase space density with higher magnetic moments in the deep interior of the radiation belt confinement region, for iron ions up to about 300 MeV/G, and a disappearance of the loss controlled features at higher ~t-values caused by the low interaction cross sections at the higher ion energies.

Cm=2.3 x 10-~Ss-J and C , = 2 . 3 x 10-1°s-~ (10)

Carbon ion radial diffusion theory 10-26 10-27 10-28

i

f

~

10-29

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~

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5 6 7 8 9 L-shell Fig. I. Results from carbon ion radial diffusiontheory. The phase space densitydistribution function is computed for moderate radial diffusion conditions. Notice the strong variation of carbon ions effective penetration depth into the radiation belt confinement region with ion magnetic moment and the qualitative break in the features around 400 keV/G.

314

W. N. SPJELDVIK

Charge state distributions of carbon ions were also computed along with the phase space densities, and the results indicate that there is a transition from solar wind and solar energetic particle dominance of charge states 5 and 6 in the outermost parts of the carbon ion confinement region to a relaxation-controlled charge state distribution at lower L-shells. Spjeldvik (1991) showed that charge state 1 is dominant below about L = 4.5 for carbon ions of magnetic moment 16.4 MeV/G, a more complicated structure of multiple charge state dominance with successive lower L-shells below L = 6 for 164 MeV/G ions, and a predominance of the higher charge states for greater than 1640 MeV/G carbon ions. The radial profiles of relative charge state distributions are quite different when viewed in terms of constant magnetic moment as opposed to being viewed in terms of constant ion energy (as would be the case for space observations). This calls for careful juxtapositions of observations and theory to insure compatible comparisons.

6. RADIATION BELT OXYGEN ION MODELING In the case of oxygen ions, there are eight possible positive charge states and thus eight simultaneous second order partial differential equations to solve. Because of at least partial availability of oxygen ion charge exchange cross sections early on, radiation belt oxygen ions were the first 'heavy' ion species to be studied (i.e. Spjeldvik and Fritz, 1978b). In a series of computer simulations it was demonstrated that the availability of higher charge states resulted in effectively longer residence times for MeV oxygen ions within the confinement region and thus higher fluxes of those ions. The descriptive equations are basically the same as those stated above for carbon ions, except that n = 8 for oxygen and the charge exchange cross sections and ion mass are those appropriate for oxygen ions. As with carbon ions, oxygen ions also have a radiation belt charge state distribution which favors the higher charge states at high (MeV) energies and low charge states at low (keV) energies. The total (charge state summed) oxygen ion fluxes become competitive with the proton fluxes above about 1 MeV total ion energy, The predicted oxygen ion spectra exhibit spectral 'holes' (or depletions) centered around 800 keV total ion energy deep within the interior of the Earth's radiation belts. The detailed oxygen ions results from this modeling are generally available in the scientific literature and are not repeated here. 7. RADIATION BELT IRON ION MODELING

In the case of iron ions, there are 26 possible positive charge states and thus 26 simultaneous

second order partial differential equations to solve. Modeling work carried out to date has included the first 12 positive charge states for iron with the justification that the innermost 14 electrons in the iron atom are very tightly held by the atom with the effect that stripping (further ionization) charge exchange cross sections are likely to be very small, perhaps of the order 10-22cm 2 or smaller. Figure 2 shows theoretically computed phase space density of radiation belt iron ions based on assumed moderate radial diffusion coefficients (the same as those applied to the carbon ion study noted above), the best available iron ion charge exchange cross sections based on laboratory measurements and their extrapolations. Qualitatively these distributions are similar to the ones computed for carbon ions, namely with diminishing phase space density with higher magnetic moments in the interior of the confinement region, for iron ions up to about lO00 MeV/G, and a disappearance of the loss controlled features at higher ~t-vahies. In future comparison with spacecraft data it is of interest to convert the phase space distribution functions to differential fluxes at constant iron ion energy. The reason for this is the way the spacecraft detectors work. The result of this conversion is illustrated in Fig. 3 which depicts radial profiles of iron ion fluxes at energies from below 100 keV to 100 MeV. Again it is cautioned that the computations should not be considered entirely valid (because of a lack of inclusion of relevant physics) below 100 keV and above about 10 MeV. The figure nevertheless shows interesting radial profiles with features that could be looked for in available experimental data. Finally, when theoretical iron ion flux energy spectra are computed, the prominent feature is the spectral 'hole' that develops at a few MeV total iron ion energy. As noted above, this is a feature that is also computed for other ion species. For example, for protons it appears at tens of keV energies, for helium ions around a hundred keV, for carbon ions just below an MeV. Thus the theoretical computations indicate a systematic shift in a spectral deficit with ionic mass.

8. TIME DEPENDENT CONFINED PARTICLE MODELING Time dependent modeling of the radiation belt ion content is yet in its embryonic stage. It is known from many spacecraft observations that enhanced fluxes of protons and heavy ions occur in the outer radiation zone during geomagnetic disturbances. Sudden 'injections' or in-situ accelerations are also often reported. The hydrogen ion time dependent transport equation for equatoriaUy mirroring particles has the

HEAVY ION R A D I A T I O N IN THE EARTH'S M A G N E T O S P H E R E

315

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10-26 10-27 5.33MeV/G 16.4MeV/G

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L-shell Fig. 2. Results from iron ion radial diffusion theory. The phase space density distribution function is computed for moderate radial diffusion conditions. Notice the strong variation of iron ions effective penetration depth into the radiation belt confinement region with ion magnetic moment and the qualitative break in the features around 1 MeV/G.

form (e.g. Cornwall, 1972; Spjeldvik, 1977; Spjeldvik and Rothwell, 1983; Spjeldvik, 1988):

Ofpldt = L2t? /c~L[DLL~L 2c~fplt~L] + Gp(L)/t- '/2Op/co/a --fp/Tp -- Ai0fp

(11)

where j~ is the proton distribution function, t is time, and the other parameters are defined in the foregoing description. Based on this equation and time dependent boundary conditions, Spjeldvik (1988) carried out a time dependent radiation belt proton simulation study which emphasized the effects of the duration of solar energetic particle events. Figure 4 shows an example of the theoretical results obtained for an event which was modeled to have increased the outer radiation zone boundary

proton fluxes by a factor of 1000, lasting for a day, and with radial diffusion coefficient a factor of 100 higher than for quiescent conditions. Depicted are proton phase space distribution functions (in arbitrary units) computed for proton magnetic moments of la = 1.1, I1, 110 and 550 MeV/G for eight different times during the simulated event. It is clear that the radial profiles of protons are highly variable with the fastest time variations occurring for the lower la-values, corresponding to ring-current energies. This simulation does not include effects of convection and changing magnetic field topology, and merely represents a first attempt at time-dependent computation of the global trapped proton dynamics at the geomagnetic equator.

316

W. N. SPJELDVIK

Other time dependent modeling including heavier ion species is under way at this time, and there is indeed a wealth of data available to be used as boundary conditions for such modeling from many spacecraft ( e . g . R . McEntire, personal communication, 1990; Belian et al., 1992). 9. TOWARDS PREDICTIVE QUALITY MODELING It is understood that the diffusion approximation to physical transport is not the only process operating on geomagnetically confined energetic ions. There are clearly other processes that also operate. However, with the stated limitations on applicability, diffusion modeling nevertheless forms a framework for a first

coarse study of the steady state and dynamically evolving radiation belt region. There is a hope that the present modeling can be extended and further developed towards radiation belt predictive capability, to eventually also include off-equatorially mirroring particles (e.g. Smith and Bewtra, 1978) and to perhaps also take into account the simultaneous radial and pitch angle diffusion processes that appear to be of great importance for radiation belt electrons, and may be of significance for ions at some energies. This will require further theoretical and computer programming efforts in transport modeling in the time ahead. There is a transition energy range below which a convection description adequately simulates the behavior of geomagnetically contrained charged

Iron ion radial diffusion theory: 102

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particle a n d a b o v e w h i c h a diffusion d e s c r i p t i o n is the most expedient simulator. This was considered by B i r m i n g h a m (1969), a n d a r e c e n t c o m p a r i s o n o f the t w o a p p r o a c h e s is given by Riley a n d W o l f (1992). It

is f o u n d t h a t the t w o a p p r o a c h e s c a n give similar results for t h e t r a n s i t i o n r a n g e energies, typically tens o f k e V p e r ion, a n d t h a t diffusion a p p e a r s as a n a d e q u a t e d e s c r i p t i o n for e n s e m b l e a v e r a g e s o f

318

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disturbed periods. Progress towards predictive quality modeling of geomagnetically confined particles requires critical testing and comparisons using more than one approach to the problem. The boundary conditions appropriate to heavy ion modeling must properly take into account both the heavy ion energy spectra and their charge state distributions. Information on these quantities are now becoming more available from interplanetary observations and from monitoring platforms in the outer parts of the terrestrial magnetosphere (e.g. Luhn et al., 1984; McGuire et al., 1986; Zwickl and Kunches, 1989; Reames et al., 1990; Reeves et al., 1992; Torsti et al., 1992a, 1992b). The years ahead should be a rich time for sophisticated computer modeling and simulation of the terrestrial charged particle confinement region. This is a time where advances in computer technology (computation speed, automatic vectorizing and massively parallel processing capabilities) provides tools of unprecedented power and versatility for the modeler. At the same time there is a growing body of particle data on the appropriate boundary conditions, in-situ particle observations that form the reality test for the models, an increasing knowledge of the necessary interaction cross sections and transport coefficients, and a growing maturation of theoretical analysis tools. Progress is now made towards an eventual goal of an 'international standard magnetosphere' along the lines of existing intenational standard atmosphere and ionosphere parameters, and towards the development of predictive capability in magnetospheric modeling.

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