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A Review of ULF Interactions with Radiation Belt Electrons
Scot R. Elkington
Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado, USA
Energetic particle fluxes in the outer zone radiation belts can vary over orders of
magnitude on a variety of timescales. Power at ULF frequencies, on the order of a
few millihertz, have been associated with changes in flux levels among relativistic electrons comprising the outer zone of the radiation belts. Power in this part of
the spectrum may occur as a result of a number of processes, including internallygenerated waves induced by plasma instabilities, and externally generated processes
such as shear instabilities at the flanks or compressive variations in the solar wind.
Changes in the large-scale convective motion of the magnetosphere are another
important class of externally-driven variations with power at ULF wavelengths.
The mechanism for interaction between ULF variations and the radiation belts
may result in (or require) pitch angle scattering, or may conserve the first two
adiabatic invariants of particle motion. Of the latter class of interactions, radial
diffusion describes the result when ULF variations lead to stochastic motion
among the particle populations, and has been studied extensively as a description
of radial transport within the belts. Rates of radial diffusion depend strongly on
the characteristics of the driving ULF waves. This work is intended as a nonexhaustive review of radiation belt interactions with ULF waves, outlining the current theories and methods in studying the interaction, and describing pertinent
wave properties.
1. THE RADIATION BELTS
where perturbations have characteristic frequencies of a few
millihertz.
The outer zone radiation belts are largely field-aligned
structures, with equatorial crossing distances of approximately 3–7 Earth radii (RE). This region of space is of particular significance due to the large number of spacecraft
operating at these altitudes, and global society’s increasing
reliance on space-based platforms for communications, navigation, weather prediction, and a variety of other economic
and geopolitical purposes. Spacecraft operating within the
belts have sometimes been found susceptible to “anomalies”
related to changes in the local radiation environment [e.g.,
Wrenn, 1995; Baker et al., 1994, 1998a]. For this reason, considerable effort has been put into observing and understanding the basic physics driving changes in the outer zone belts.
Energetic electron fluxes in the outer zone show substantial variation over a variety of time scales. There is a noted
association between solar wind speed and relativistic electron enhancements [Paulikas and Blake, 1979; Li et al.,
The outer zone radiation belts consist of energetic electrons trapped in the geomagnetic field. The dynamics of the
belts are dictated by the global and local electric and magnetic fields which guide the individual particles’ motion.
The plasmas and magnetic fields comprising the geomagnetic environment support a large variety of waves resulting
from a range of internal and external processes. This paper
reviews what is currently understood of the interactions
between energetic electrons in the outer zone and electric
and magnetic field variations occurring on ULF time scales,
Solar Eruptions and Energetic Particles
Geophysical Monograph Series 165
This paper is not subject to U.S. copyright. Published in 2006 by
the American Geophysical Union
10.1029/165GM12
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A REVIEW OF ULF/RADIATION BELT INTERACTIONS
1997; Baker et al., 1997]. Rapid enhancement of the belts, on
time scales of a few minutes, has been observed during sudden commencement of storms [Blake et al., 1992b,a]. More
frequently discussed are the flux variations associated with
the main phase and recovery phase of geomagnetic storms,
which may occur over a range of a few hours to 1–2 days [Li
et al., 1998; Reeves et al., 2003]. These variations occur outside of the flux dropout and recovery normally associated
with the large scale changes in the geomagnetic field during
storms, the so-called “Dst effect” [e.g., Kim and Chan, 1997].
While storms are often associated with enhancements in the
radiation belts, Reeves et al. [2003] showed that they can in
fact produce a variety of outcomes in terms of enhancing or
depleting the belts. Indeed, there are a variety of processes
capable of affecting the belts in terms of the transport, heating, and loss of particles [e.g., Friedel et al., 2002; O’Brien
et al., 2003], and the outcome of any substantial geomagnetic
activity is a result of the complex interplay among the range
of active processes.
The motion of the trapped particles comprising the belts
can be decomposed into three essential types, consisting of a
gyromotion about the field line, a periodic motion parallel to
the field lines as the particles bounce between magnetic mirror points, and a gradient-curvature drift about the Earth.
Time scales associated with each type of motion for particles
trapped in the radiation belts typically differ by 1–2 orders of
magnitude, ranging from kHz time scales associated with the
gyromotion, to mHz time scales corresponding to the drift.
Particle dynamics may be ordered in terms of an “adiabatic
invariant” associated with each of these types of motion
[Northrop, 1963; Schulz, 1996], which will be conserved so
long as the fields that guide the particle motion do not
change significantly on temporal or spatial scales commensurate with the particle motion. The invariant associated with
the gyromotion is given by M = p2⊥/2m0B, where p is the relativistic particle momentum, m0 the rest mass, and B the
magnetic field measured at the guiding center of the gyromotion. The second invariant is based on the parallel motion
evaluated along the guiding center trajectory of the particle
bounce motion, and the third invariant results in a form equal
to magnetic flux enclosed by the guiding center drift path
about the Earth. Typically the third invariant is restated as L*,
a quantity related to the inverse of the magnetic flux enclosed
by the particle drift [Roederer, 1970]. In the special case of a
dipole magnetic field, L* is often denoted simply as “L”, and
is numerically equivalent to the radial distance (measured in
RE) at which a drifting particle crosses the equatorial plane
and frequently. Because the dipole L is so directly related to
the L* as defined by Roederer [1970], the use of the variable
L in the remainder of this review will refer to the radial
dimension defined through the third invariant, rather than the
simple radial distance commonly implied by the dipole limit.
The widely differing time scales associated with each
motion mean that field variations that lead to violation of one
invariant may not necessarily perturb the other invariants. In
the case of variations on drift time scales that violate the third
invariant while conserving the first and second invariants, the
resulting radial motion and change in guiding center field
strength will lead to a change in the bounce-averaged perpendicular momentum, and hence, energy. In general, the
energy change of an electron conserving the first invariant is
given by
dW
∂B
,
= qE ⋅ v d + M
dt
∂t
(1)
where q is the particle charge, E is the background electric
field, and vd is the total particle drift velocity [Northrop,
1963].
2. MAGNETOSPHERIC ULF VARIATIONS
There are a variety of geomagnetic disturbances that may
produce power at ULF frequencies. Field line resonances represent a particular class of ULF variations. Waveforms that
have quasisinusoidal signals that exist for several periods are
classified as continuous, and are further broken down into
five subcategories, Pc-1–5, depending upon their frequency
[Jacobs et al., 1964]. Pc-5 waves have frequencies in the
range of a few mHz. A simple mathematical description of
these waves was developed by Dungey [1967], and uses the
idea of standing Alfvén waves on geomagnetic field lines. In
the “toroidal mode” of this theory, magnetospheric shells
oscillate coherently with perturbations in the azimuthal
direction, and is obtained in the low-m limit where m is the
global azimuthal mode structure of the oscillations. Toroidal
ULF waves have an associated induced electric field in the
radial direction. The mode in which the magnetic field oscillations are constrained to the meridional plane is called
poloidal, and is obtained in the high-m limit. Poloidal waves
are additionally characterized by compressional magnetic
perturbations, and electric fields induced in the azimuthal
direction. Numerous ground- and space-based observations
as well as modeling efforts have confirmed the basics of this
description [Hughes, 1994, and references therein].
The broader class of ULF variations may be either broadband or quasisinusoidal in nature, and may originate in a
variety of internal (magnetospheric) or external (solar winddriven) processes. As an example of internal generation
mechanisms, Hasegawa [1969] developed a theory for the
generation of ULF waves by invoking a mirror instability in
the cold plasma population drifting in the inner magnetosphere. Southwood et al. [1969] similarly considered a drift
instability as a possible internal source of the ULF waves.
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More recently, Takahashi et al. [1985b] proposed anisotropy
in the perpendicular ring current as source region.
Shear flow instabilities along the magnetopause provide
one means of generating ULF energy in the magnetosphere
[Cahill and Winckler, 1992; Mann et al., 1999]. Miura [1992]
used a two-dimensional magnetohydrodynamic (MHD)
simulation to show that the velocity shear along the magnetopause along the tail flanks is unstable to the KelvinHelmholtz effect regardless of the sonic Mach number of the
magnetosheath flow. This research showed that a ULF wave
could be generated by energy and momentum transfered into
the LLBL. Mathie and Mann [2000b] used observations of
Pc-5 waves with Common Azimuthal Phase Speeds (CPS) as
a diagnostic for the generation of ULF waves by a shear flow
instability along the magnetopause. Figure 1, taken from
their paper, schematically shows these waves occurring preferentially in the morning and afternoon sectors. The speed of
the “ripple” generated by the shear instability along the
flanks couples into the field line resonances present on high
latitude fields, with each resonance having the same phase
speed due to the common source. This figure implies an
equal likelihood for wave occurrence on the dawn and dusk
sides of the magnetosphere; the work of Greenstadt et al.
[1981], by contrast, suggests that because the dawnside of
magnetopause forms a quasiparallel shock configuration, a
larger seed population of perturbations is present here, resulting in more ULF waves on the dawnside than on the duskside.
A second category of external driver for the generation of
ULF waves within the magnetosphere is related to the transmission of variations in solar wind pressure through the
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magnetosphere, allowing the flow of energy in the form of
compressional waves [Kivelson and Southwood, 1988; Lysak
and Lee, 1992] which may couple with toroidal- or poloidalmode field line resonances in the inner magnetosphere
[Mann et al., 1995]. Spacecraft observations of strong
dynamic pressure variations upstream of the Earth’s bow
shock were shown to cause wavelets along the magnetopause
and drive quasi-periodic variations within the magnetosphere
[Sibeck et al., 1989a,b]. Wright and Rickard [1995] proposed
that discrete variations in the solar wind driving would create
field line resonances with independent phase speeds (IPS)
across a wide range of local times, with maximum being
centered around noon as illustrated in Figure 1. Kepko et al.
[2002] and Kepko and Spence [2003] completed a spectral
analysis for a series of events during which ULF waves were
seen at a series of discrete frequencies in the Pc-5 range. A
subsequent analysis of the solar wind density driving the
magnetosphere during the same intervals revealed significant
wave power at the same set of discrete frequencies as
observed in the geomagnetic pulsations. They proposed that
the driving of the ULF waves was not accomplished by excitation of a cavity mode, but by the slow modulation of the
entire magnetospheric cavity as it changes size to maintain
pressure balance under slowly varying driving conditions.
A third type of magnetospheric field variation that may
have power at ULF frequencies is related to changes in the
large-scale convective motion of the magnetosphere as a
result of changes in the global magnetospheric reconnection
rate and the flow of solar wind around the magnetopause. In
the Earth’s frame, the quasi-steady convective motion of
fields and plasmas may be described in terms of a global
potential electric field, generally pointing from dawn to dusk.
The magnitude of Earth’s convection electric field may
respond directly to variations in the direction and strength of
the interplanetary magnetic field and the solar wind electric
field [Ridley et al., 1997, 1998; Ruohoniemi and Greenwald,
1998]. While not often considered a “ULF wave” in the sense
of the compressional variations and field line resonances
described above, ULF variations in the Earth’s convection
electric field occur on a global scale, and may have a direct
impact on trapped energetic particles in the magnetosphere.
This impact is described in more detail in Section 4.
3. OBSERVATIONS: THE ULF/RADIATION
BELT CONNECTION
Figure 1. Schematic [Mathie and Mann, 2000b] showing the shear
generation of ULF waves along the dawn and dusk flanks, while
ULF waves generated by stationary pulses in the solar wind are
predominant in the noon sector.
Much of the recent attention on the connection between
ULF waves and radial transport has resulted from radiation belt
flux increases occurring on time scales ranging from a few
hours to 1–2 days, commonly observed following onset of geomagnetic storms. For example, in a comparison of the May 27,
1996, magnetic cloud event with that of January 10-11, 1997,
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Baker et al. [1998c] found that large-amplitude oscillations
in the Pc-5 frequency range were associated with the relativistic electron event of the 1997 storm, while the 1996
storm, which did not exhibit extensive Pc-5 activity, had no
comparable increase in electron fluxes. Likewise, an analysis
of the May 15, 1997, electron event also showed large
increases in Pc-5 activity [Baker et al., 1998b]. Nakamura et al.
[2002] used in situ wave measurements from Equator-S in
combination with particle measurements from POLAR
during the March 10, 1998, storm. Observations indicated a
rapid enhancement of electrons in association with increased
ULF wave activity during the main phase of the storm, during
which the electrons exhibited a pitch angle distribution
strongly peaked toward 90°. A subsequent, more gradual
period of electron acceleration was observed, this having a
more isotropic pitch angle distribution. Green and Kivelson
[2001] examined ULF wave power and electron response for
nine storms in 1997, noting an association between ULF
wave power and electron fluxes in most of the cases. Mathie
and Mann [2000a] carried out an epoch study of 17 storms in
1995, examining ULF waves observed by ground magnetometers and electron flux measurements from geosynchronous. Here storms were subcategorized according to
energetic electron response during the storm, which indicated
that significant electron flux increases were only observed in
response to ULF wave power which is sustained at high levels over a number of days following storm onset.
A study by Rostoker et al. [1998] examined a 90 day
period in 1994, comparing GOES-7 >2 MeV electron flux
with ULF wave activity observed by a ground magnetometer
at Gillam in Manitoba, Canada. The results are shown in
Figure 2, and visually indicate a strong correlation between
levels of observed ULF activity and energetic electron flux.
Throughout the period under study, increases in energetic
electron fluxes are seen to be preceded by increases ULF
wave activity by a period of 1–2 days. More quantitative studies of the correlation between ULF waves and energetic electron dynamics have been undertaken. O’Brien et al. [2001]
used a cross correlation to determine which parameters in the
solar wind and magnetosphere might most influence energetic electron fluxes. Similar to the results epoch study conducted by Mathie and Mann [2000a], they found sustained
solar wind speeds in excess of 450 km/s to be a strong external indicator of increasing magnetospheric electron fluxes,
and long-duration Pc-5 activity during the recovery phase of
a storm to be the best discriminator between those storms that
produced relativistic electrons and those that did not. This
study was extended by O’Brien et al. [2003] to compare electron flux increases with both ULF waves and VLF chorus
emissions to suggest that ULF waves are generally the
Figure 2. ULF wave power observed by a ground magnetometer plotted together with energetic electron fluxes observed at geosynchronous orbit. Throughout the 90 day period there is a strong association between energetic electron fluxes and ULF wave activity
(from Rostoker et al. [1998]).
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geoeffective driver of radiation belt dynamics outside of 5 RE
geocentric. More recently, Mann et al. [2004] used an array
of magnetometers to determine a rank-order correlation
between intervals of high solar wind speed, increased ULF
wave power, and subsequent increases in MeV electron
fluxes, for a period spanning a complete solar cycle
(1990–2001). This work indicated that the vsw and ULF wave
power had highest correlation with MeV electron flux late in
the declining phase of the solar cycle, when electron fluxes
were highest.
4. ULF/PARTICLE INTERACTIONS:
THEORY AND MODELING
There have been a range of possible interactions suggested
to explain the observed relation between ULF waves and energetic electron fluxes. They may be broadly categorized according to whether the mechanism features or requires violation of
the first or second adiabatic invariants, or whether the interaction acts through violation of the drift invariant only.
4.1. ULF Interactions with Pitch Angle Scattering
Based on the observations of Rostoker et al. [1998]
depicted in Figure 2, Liu et al. [1999] proposed a “magnetic
pumping” acceleration mechanism, whereby acceleration
was accomplished by ULF waves in combination with pitch
angle scattering via whistler-mode turbulence. In this mechanism, the electrons are acted on by a global, symmetric
(m = 0) magnetic compression, with attendant induced electric field. Since symmetric compressions by themselves cannot energize particles [Fälthammar, 1965], the mechanism
required scattering by whistler waves in order to break particle adiabaticity. The ULF waves provide the energization
through the E ⋅ vd component of equation (1); in a symmetric
perturbation, the time integral of this function is zero.
However, since the drift velocity is pitch angle-dependent
[Roederer, 1970], random scattering by the whistler waves
can lead to a net change in energy over time. Acceleration of
the particles in aggregate is attributed to the “zero energy
barrier”, where an ensemble of particles undergoing a random walk in energy space will experience a net acceleration
because particles cannot be decelerated past a “zero energy”,
a point where no more energy can be lost; the potential for
energy acceleration, on the other hand, is unbounded. This
mechanism predicts the most efficient acceleration at small
pitch angles, and is most effective when the wave period and
scattering time scales are approximately equal.
Summers and Ma [2000] offered an alternate mechanism
for accelerating particles via interaction with ULF waves,
including pitch angle scattering as a fundamental element of
the interaction. In the process of “transit time acceleration”,
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electrons with a particle velocity exceeding the local Alfvén
velocity interact gyroresonantly with oblique fast-mode
MHD turbulence. This interaction is a resonant form of
Fermi acceleration, so named because the interaction
requires λ||/v|| ≈ T, where v|| is the parallel particle velocity
and T and λ|| are the wave period and parallel wavelength,
respectively. Under geomagnetically active conditions,
Summers and ma [2000] suggest that transit-time acceleration can lead to heating of keV electrons to MeV energies in
timescales of a few hours.
4.2. ULF Interactions Conserving the First
and Second Invariants
Since the minutes-long periods associated with ULF waves
are commensurate with electron drift timescales in the radiation belts, but orders larger than the gyro or bounce
timescales, many theories on the relation between electron
fluxes and ULF waves assume only violation of the third
invariant. During the January 9-11, 1997, geomagnetic
storm, Hudson et al. [1999, 2000] noted large-scale ULF
waves occurring concurrently with a rapid rise in relativistic
electron fluxes. The waves were largely monochromatic, with
a dominant toroidal polarization. The observation of these
waves, in conjunction with the energetic electron response,
led to the suggestion that electrons could be accelerated via a
drift-resonant interaction with global toroidal-mode field line
resonances. The mechanism was quantified by Elkington
et al. [1999, 2003], and depends essentially on the noonmidnight asymmetry that results from the dynamic interaction of the solar wind with the Earth’s magnetosphere. The
basic mechanism is depicted in the leftmost panel of
Figure 3. Here an equatorial electron drifting in a compressed
dipole experiences an interaction with a low mode number
global toroidal wave [e.g., Takahashi and McPherron, 1984].
On the dawn side, the electron has a component of motion in
the radially outward direction; at dusk, radially inward. This
permits energization through interaction with the radial electric field associated with toroidal-mode field line resonances
via the first term in equation (1). Those electrons drifting
with a frequency ωd satisfying
ω = (m ± 1)ωd
(2)
where ω is the wave mode number and frequency, respectively, will see an electric field of constant sign throughout
their motion, and thus will experience a continual acceleration. Further observational evidence for electron acceleration
via toroidally-polarized field line resonances was reported in
Tan et al. [2004] for an event occurring in August 1991.
Since the largest component of a drifting electron’s motion
is in the azimuthal direction, interaction with waves with an
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associated azimuthal electric field is likely to be of generally
more importance to radiation belt dynamics than interaction
relying on the radial motion of the electrons as described
above. Any of the wave generation mechanisms described in
Section 2 may include azimuthal components of the electric
field. The idealized toroidal mode described above represents
a low-mode number limit; for finite mode number the Alfvénic
disturbances will occur in mixed modes with both toroidal
and poloidal signatures and radial and azimuthal electric
fields [Ukhorskiy et al., 2005; Loto’ aniu et al., 2006]. The
resonant acceleration for an electron drifting in a dipole
magnetic field and interacting with an azimuthal electric
field is depicted in the center frame of Figure 3. The resonance condition for this azimuthally symmetric interaction is
given by
ω = mωd.
(3)
An asymmetric resonance exists also for the poloidal component of a ULF disturbance, and is depicted in the rightmost
panel of Figure 3. Here the resonance condition is the same
as for the toroidal mode discussed above. A particle drifting
in an asymmetric magnetic field will be subject to the combined effects of each of the resonances outlined in the figure,
the degree of interaction depending on the wave power at frequencies corresponding to the asymmetric or symmetric resonance conditions described in equations (2) or (3).
As an aside, we note that the internally-generated poloidal
mode ULF waves discussed in Section 2 [Southwood et al.,
1969; Takahashi et al., 1985b] are typically high-m in character [Takahashi et al., 1985a]. Since wave power in the ULF
spectrum is typically a decreasing function of frequency
[Bloom and Singer, 1995; Brautigam et al., 2005; Elkington
et al., 2004], the resonance conditions (2) and (3) indicate
that these internally-generated sources of wave power will be
of less importance to overall radiation belt dynamics.
The use of Poincaré plots to map the dynamics of a particle in phase space provides a powerful tool for analyzing the
resonant interactions described above [Elkington et al., 2003;
Degeling et al., 2006; Loto’aniu et al., 2006]. An example is
shown in Figure 4, showing the resonant interaction of an
electron with a poloidal-mode ULF wave. The plot indicates
the dynamics in a space consisting of the third invariant L and
Figure 3. Sketches indicating resonant interaction for electrons with
the toroidal component of a global field line resonance in a magnetic
field compressed by the solar wind (a), and for electrons interacting
with the poloidal component of the ULF waves in a dipole (b) and
compressed (c) magnetic field. Top panels in each case show resonant
fields at particular locations along representative electron orbits;
bottom panels indicate the drift velocities, fields, and instantaneous
energy gain throughout a resonant trajectory [Elkington et al., 2003].
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Figure 4. Poincaré plot indicating the resonant interaction of
an electron with the poloidal component of a ULF wave [Loto’aniu
et al., 2006].
the corresponding drift angle variable, here the particle drift
phase measured relative to the phase of the propagating resonant ULF wavefront. Particles move through this space on
trajectories constrained by the indicated lines; those with trajectories inside the separatrix of the resonant island centered
at ∼3.5 MeV are resonant, and are able to experience energy
changes dictated by the extent of their possible motion along
the energy (or L) axis.
4.3. ULF Variations and Radial Diffusion
Magnetospheric power at ULF frequencies may be either
largely monochromatic, or it may be broadband in nature
[e.g., Brautigam et al., 2005]. For the general case of finite
bandwidth variations, the drift resonant interactions
described above will lead to stochastic motion among individual particles in the third invariant, L. This can be seen in
Figure 5, where the separate interaction of two independent
toroidal-mode waves at differing frequencies leads to diffusive motion in the regions of resonant island overlap. In the
case that the frequency spectrum approaches a continuum
over some range, as shown in the bottom panel of Figure 5,
the motion becomes fully stochastic in energy, or equivalently, L.
A distribution of particles undergoing this type of stochastic motion in L can be shown to satisfy a diffusion equation
of the form
∂f
∂  DLL ∂f  f
= L2
− ,
∂t
∂L  L2 ∂L  τ L
(4)
where f is the phase space density averaged over all phase
angles, a function of M, L, and t [Roederer, 1970; Schulz and
Lanzerotti, 1974], and τL is the characteristic particle lifetime. The radial diffusion coefficient DLL describes the average rate of radial transport; analytic forms for the diffusion
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coefficient may be derived starting with either the drift
equations describing the influence of the perturbing waves
[Fälthammar, 1965, 1968; Schulz and Eviatar, 1969; Schulz
and Lanzerotti, 1974; Ukhorskiy et al., 2005], or starting
from a Hamiltonian formulation [Brizard and Chan, 2001].
In general, the diffusion coefficient will be an explicit function of L, the wave power and mode structure of the driving
waves, and will contain either an implicit or explicit dependence on particle energy. Further, the relevant form of the
diffusion coefficient will depend on whether the waves are
electromagnetic in nature, or whether the wave can be
ascribed to variations in a large-scale potential field [Cornwall,
1968].
Diffusion resulting from interaction with the symmetric
resonance (3) may take one of two forms, depending on the
nature of the waves. If the field perturbations driving the
diffusion results from variations in a potential electric field,
or the magnetic field perturbations in equation (1) are otherwise zero, the diffusion coefficient for particles drifting with
frequency ωd takes the form
E , Sym
DLL
=
1
L6 ∑ PmE ( L, mω d ).
8 BE2 RE2
m
(5)
Here BE and RE are the Earth’s dipole moment and radius,
respectively, and PmE is the power spectral density of the perturbing electric fields at the resonant frequency mωd, where
m describes the global azimuthal structure of the waves.
Variations in the dawn-dusk electric field associated with
global magnetospheric convection is one example of such
field perturbations [Cornwall, 1968]; the ideal toroidal field
line resonance acting on an equatorially-bound particle
(where the associated magnetic perturbation has a null) provides a second example [Loto’aniu et al., 2006]. If, however,
the perturbation is electromagnetic in nature, where magnetic
and electric perturbations are related by Faraday’s law, then
the diffusion can be described by the expression
B , Sym
DLL
=
M2
L4 ∑ m 2 PmB ( L, mω d ).
8q γ 2 BE2 RE4
m
2
(6)
In this expression, γ is the Lorentz factor, M the relativistic
first adiabatic invariant, and PmB is the power spectral density
of the compressional wave magnetic field at frequency mωd.
The electric diffusion coefficient (5) has an L6 dependence,
plus the L dependence in P Em. The L dependence of the magnetic diffusion coefficient is more complicated since γ also
depends on L. In the ultra-relativistic limit γ 2 ∝ L−3, so for
radiation belt electrons L4/γ 2 is approximately proportional
to L7, not including the L dependence implicit in P Bm. Note
that equation (6) does not exhibit the same explicit L10
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Figure 5. Poincaré plot showing the onset of radial diffusion for electrons interacting with toroidal-mode ULF waves spanning a range
of frequencies [Elkington et al., 2003].
dependence as that of Fälthammar [1965], which is nonrelativistic, has only m = 1 terms, and assumes an explicit radial
dependence in the perturbing fields. The effects of both the
magnetic perturbation and the induced electric perturbation
contribute to the diffusion described by equation (6).
The asymmetric diffusion coefficients have different resonance frequencies, dictated by equation (2), and are also proportional to the square of the global asymmetry in the Earth’s
magnetic field [Elkington et al., 2003; Perry et al., 2005;
Ukhorskiy et al., 2005], here described by the factor b1(r)/BE.
For variations in a potential electric field, we find
2
E , Asym
LL
D
2 b 
= 2 2  1  L12
9 BE RE  BE 
× ∑ m 2 [ PmE ( L,(m + 1)ω d ) + PmE ( L, (m − 1)ω d )], (7)
m
whereas if the perturbations are electromagnetic in nature,
we get
2
B , Asym
LL
D
b 
2M 2
= 2 2 2 4  1  L10
9q γ BE RE  BE 
× ∑ m 2 [ PmB ( L,(m + 1)ω d ) + PmB ( L,(m − 1)ω d )]. (8)
m
The asymmetric modes have steeper L dependence, indicating they may be more important at larger L-shells.
Specifically, the electric asymmetric diffusion coefficient has
an explicit L12 dependence, and in the ultrarelativistic limit
B , Asym
the magnetic asymmetric diffusion coefficient, DLL
, has
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approximately an L dependence. In each case there may be
an additional L dependence in the b1(r)/BE term and in the
power spectral density, P B,E
m .
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The total diffusion coefficient is the sum of equations (5)
to (8), including asymmetric and symmetric diffusion coefficients and all m modes:
Asym , m −1
Sym , m
Asym , m +1
DLL = ∑ ( DLL
+ DLL
+ DLL
)
(9)
m
Note that in expressions (5) through (8), the stochastic
motion of the particles is a result of the intrinsic randomness
of superposed disturbance fields with a range of differing
frequencies and corresponding phases, as seen by the particles. However, the needed stochasticity may arise as a result
of extrinsic variations of the driver as well. For example,
Ukhorskiy et al. [2005] conducted a study that included an
analysis of the effects of monochromatic electric waves
occurring over a limited extent in local time in an asymmetric magnetic field, but driven by a randomly-varying external
driver. Similar to equation (5) and equation (7), an L6 dependence is found for azimuthal electric fields in a dipole, and
an L12 dependence is found for radial electric fields effective
as a result of the asymmetry in the magnetic field. Degeling
et al. [2006] likewise considered waves occurring in a limited
local time extent, here using a wave model describing a scenario of a fast compressional wave with a single discrete frequency coupling to a field line resonance at a particular L. As
depicted in Figure 6, they show that particles may undergo a
transition from ordered to stochastic behavior if acted upon
by monochromatic waves with an amplitude that depends on
local time, without the need for a random external driver.
Figure 6. Poincaré plot showing the onset of radial diffusion when
acted on by monochromatic ULF waves with increasing azimuthal
asymmetry (from Degeling et al. [2006]).
9
This change in dynamics is attributed to the appearance and
eventual overlap of additional resonance islands produced by
the spectral components of the azimuthal wave structure.
5. EMPIRICAL QUANTIFICATION
OF THE EFFECTS OF ULF WAVES
There has been considerable effort to determine diffusion
rates based on observational evidence, using a number of different techniques. A summary of a few of these results is
indicated in Table 1, taken from Elkington et al. [2003]; more
extensive compilations can be found in Lanzerotti et al.
[1970], Tomassian et al. [1972], Holzworth and Mozer
[1979], Riley and Wolf [1992], and elsewhere. Two of the
works listed in the table, Lanzerotti and Morgan [1973] and
Holzworth and Mozer [1979], assumed that diffusion rates
were correctly given by equations (5) or (6) and used measured magnetic and electric field spectra to arrive at the relevant diffusion coefficients. The remainder of those listed in
the table arrived at diffusion rates without an assumed
L-dependence.
The characteristic time scales exhibited in Table 1 range
from ∼1 day at L = 6 to several years for electrons in the inner
belt. Except in the case of Lanzerotti and Morgan [1973],
who found that the diffusion coefficient at L = 4 could vary
by orders of magnitude with magnetic activity, the observations used in these analyses were generally made during quiet
geomagnetic periods.
Two of the works listed in Table 1, Newkirk and Walt
[1968b] and Selesnick et al. [1997], made an attempt to
determine the radial dependence of outer zone diffusion coefficients based on particle measurements. The approach used
by Selesnick et al. [1997], for example, examined electron
dynamics over a 3-month period in 1996. Assuming a radial
diffusion coefficient of form DLL = D0 (L/4)n and a characteristic particle lifetime of form τ = τ0 (L/4)−m, they were able to
fit observations from three clear electron injections to a timedependent radial diffusion equation with losses [Schulz and
lanzerotti, 1974] and deduce a radial diffusion coefficient
varying as L(11.7 ± 1.3), where the error bars were based on statistical rather than systematic considerations. Newkirk and
Walt [1968b] similarly found a radial diffusion coefficient
of L(10 ± 1).
In addition to the works listed in Table 1, two recent works
deduce relevant diffusion time scales based on field
measurements seen by CRRES, a spacecraft in an equatorial
geotransfer orbit that precessed over a range of local times
during its lifetime. Brautigam et al. [2005] assumed all the
observed electric power was in potential electric fluctuations
(rather than electromagnetic), and parameterized their observations according to local time and geomagnetic activity
measured by Kp. By applying the results to the diffusion
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Table 1. Examples of of empirically-determined radial diffusion coefficients (from Elkington et al.[2003])
DLL(day−1)
L
−7
2.0 × 10
4.0 × 10−6
10−8 L(10 ± 1) a
4 – 8 × 10−10 L10 a
2.7 × 10−5 M −0.5L7.9
10(0.75ΚFR – 10.2) b
(2.23 ± 0.67)ωd−1.1 ± 0.15 c
∼0.2 – 5.0
W or M range
Reference
L = 1.20
L = 1.16
1.76 ≤ L ≤ 5.0
3.0 ≤ L ≤ 5.0
1.7 ≤ L ≤ 2.6
L=4
L = 6.0
L = 5.3, L = 6.1
W > 1.6 MeV
Newkirk and Walt [1968a]
W > 1.6 MeV
W > 0.5 MeV
13.3 – 27.4 MeV/G
350 – 750 MeV/G
50 keV – 1.2 Mev
100 keV – 1.6 Mev
Newkirk and Walt [1968b]
Lanzerotti et al. [1970]
Tomassian et al. [1972]
Lanzerotti and Morgan [1973]
Holzworth and Mozer [1979]
Chiu et al. [1988]
3.0 ≤ L ≤ 6.0
3 – 8 MeV
Selesnick et al. [1997]
11.7 ± 1.3
L
2.1 × 10−3  
4
aD based on an assumed L10 radial dependence.
0
bBased on equation (6) and measured magnetic power
cBased
spectrum; KFR ≡ Fredricksburg magnetic index (see also Brautigam and Albert [2000]).
on equation (5) and measured electric power spectrum; ωd in h−1.
E that varied by
coefficient (5), they found a Kp-dependent DLL
up to two orders of magnitude between Kp = 1 and Kp = 6
depending on the L shell. Ukhorskiy et al. [2005] similarly
used CRRES electric field measurements in their analysis of
the diffusive effects of monochromatic ULF variations.
Diffusion resulting from these monochromatic waves at L ∼ 6
had characteristic time scales on the order of a day.
In spite of the strong explicit L dependence of any of the
the forms (5) through (8), at times, radial diffusion may act at
low L on very fast time scales. For example, Loto’aniu et al.
[2006] combined ground magnetometer measurements of
ULF waves with in situ measurements of energetic electrons
during the October 31, 2001, storm, calculating diffusion
E by mapping the wave observations from the
coefficients D LL
ground to the equatorial plane using the methods of Ozeke
and Mann [2004]. They suggested that radial diffusion time
scales were sufficient to account for the two-order of magnitude, ∼24 hour increase in >2 MeV electrons within the slot
region ( L 2 − 2.5) .
6. QUANTIFYING DIFFUSION:
MODELING EFFORTS
As alluded in previous sections, numerical modeling can
provide considerable insight into the nature of radial diffusion, both in terms of the basic physical characteristics of the
diffusive process, as well as its overall applicability in radiation belt dynamics.
Test particle simulations using idealized geomagnetic and
wave fields has provided a method of examining the effects
of ULF waves on radiation belt dynamics in a controlled way.
Elkington et al. [1999, 2003] examined the dynamics of
equatorially-trapped particles in a simplified magnetic field
model of a compressed dipole. Applying an electric field
model of global-scale ULF waves, they were able to examine
the nature of the asymmetric resonances (2) for waves with
both toroidal and poloidal polarizations, and take a first look
at the nature of radial diffusion that results from these higherorder resonances. Perry et al. [2005] extended this work in an
examination of the effect of the poloidal mode on particles
with motion out of the equatorial plane. The ULF wave
model used in the Perry et al. effort was based on a vector
potential, and included the effects of the compressional
magnetic perturbations necessary to examine both terms in
equation (1). This work also incorporated variations of wave
power with frequency and radial location, and concluded that
radial diffusion was most effective for particles with large
equatorial pitch angles and at higher L values. Test particle
simulations by Ukhorskiy et al. [2005] and Degeling et al.
[2006], described in Section 4.3, examined the physical
effects of waves occurring over a finite local time. Ukhorskiy
et al. [2005] determined the rates of diffusion expected from
externally-driven random waves of a single frequency, while
Degeling et al. [2006] determined that finite local time effects
by themselves could induce radial diffusion.
A useful aspect of these controlled simulations is their
ability to determine relevant time scales for a given wave
interaction, important both as a means of verifying the functional forms of the resulting diffusion, and as a means of
determining the relative effects of diffusion for different particle populations in different regions of the magnetosphere.
An example from Ukhorskiy et al. [2005] is shown in
Figure 7. Here an ensemble of particles starting from a particular L value is allowed to interact with waves occurring in
the noon sector of the magnetosphere. The dynamics of individual particles are indicated by the grey lines; in aggregate,
the mean spread of the particles in time is given by the dotted line. The diffusion coefficient may be easily determined
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Figure 7. Calculation of rates of radial diffusion of an ensemble of
particles as considered by Ukhorskiy et al. [2005]. Grey lines indicate the motion of individual particles in the ensemble, dotted line
indicates the mean-squared deviation of the ensemble, and the solid
line is a fit which allows calculation of DLL.
from the slope of a line fit to this mean spread, using the definition DLL ≤ ∆L2 > / 2t [Schulz and Lanzerotti, 1974].
Test particle simulations carried out using physical field
models driven by realistic solar wind conditions do not offer
the same controlled environment for examining diffusion as
the empirical models described above, but do allow examination of the effect of low frequency waves under more realistic
conditions. Using MHD/particle simulations of the magnetosphere, Hudson et al. [1999, 2000, 2001] and Elkington
et al. [2002] examined the effects of ULF waves during a
series of geomagnetic storms, and suggested that the resulting
diffusive effects were an important driver of particles during
the main phase of storms.
Direct solution of the radial diffusion equation (4) provides
another means of looking at the global effects of radial diffusion induced by ULF variations. Selesnick et al. [1997]
solved a lossy radial diffusion equation during an extended
quiet period occurring in 1996, fit to data to empirically
determine the relevant rates of radial diffusion and loss.
Several works have compared the results of radial diffusion
simulations, using presumed diffusion coefficients, to data,
in an effort to determine contrasting effects of radial diffusion
vs. local heating [e.g., Summers et al., 1998]. Brautigam and
Albert [2000] solved the radial diffusion equation in an
analysis of a storm occurring in October of 1990. They
considered the effects of both electromagnetic and potential
electric fields, using symmetric forms for DLL that varied
with Kp to capture the effects of increased ULF activity during geomagnetically-active times. Comparing the results to
CRRES observations, they concluded that radial diffusion
was the dominant process among electrons at keV energies,
11
but that radial diffusion alone was not enough to account for
the dynamics of >MeV electrons. Shprits and Thorne [2004]
used the diffusion coefficients corresponding to the electroB , Sym
magnetic DLL
coefficients of Brautigam and Albert, in a
study of the competing effects of diffusion and loss in the
outer zone, and similarly concluded that radial diffusion was
not the only process at work in the radiation belts.
The Salammbô diffusive radiation belt code [Beutier and
Boscher, 1995; Bourdarie et al., 1996] attempts to simulate
not only the effects of radial diffusion, but also include the
effects of high-frequency interactions that lead to pitch angle
scattering and local heating. Rates of radial diffusion are handled in an activity-dependent manner, while pitch angle diffusion is modeled considering plasmaspheric hiss at the
plasmapause. Albert and Young [2004] noted that simulating
the effects of processes with such disparate characteristic
time scales (particle drift vs gyromotion) can be problematic,
leading to large and rapidly varying cross terms in the full
diffusion tensor. They suggest a particular coordinate transformation as a means of avoiding this problem and allowing
simulation including the range of high- and low-frequency
effects.
Insight can be gained through a combination of both test
particle simulations and solution of the radial diffusion equation. Riley and Wolf [1992] used a time-dependent, observation-based model electric field for the storm event of August
1990, and used it to compare results of a drift calculation to
results of a radial diffusion simulation, the latter using the
power spectrum of the fluctuations in the electric field model
to fix the symmetric electric diffusion coefficient considered
in the study. They concluded the diffusion formalism used
gave only roughly right answers for a single real storm but
did much better for an average over a statistical ensemble of
storms. Ukhorskiy et al. [2006] used a model of the electric
fields induced by solar wind pressure variations during a
period with near-zero Dst to drive a kinetic simulation of the
radiation belts. They found that the induced fields resulted in
effective radial transport of energetic electrons, but similarly
found that the resulting particle distribution was not well
described in terms of simple radial diffusion. Fei et al. [2006]
conducted a study comparing a radial diffusion simulation
with the results of an MHD/particle simulation of the
September 1998 storm. This study conducted a global analysis of the ULF wave power occurring in the MHD model,
including azimuthal mode structure using the methods of
Holzworth and Mozer [1979]. They applied power appropriately to the range of symmetric and asymmetric magnetic and
electric diffusion coefficients, equations (5) through (8), and
used boundary conditions based on the results of a test particle simulation using the same fields. Plate 1 shows an example of the comparison between the radial diffusion and test
particle results. Fei et al [2006] were able to find much better
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Plate 1. Radial diffusion solution (left) compared with an MHD/particle solution (right), for the September 1998 geomagnetic storm
[Fei et al., 2006]. They find effective solution of the radial diffusion equation requires detailed knowledge of the driving diffusion
coefficients, and hence, ULF activity.
Plate 2. keV electrons in the plasmasheet may be convectively injected into the inner magnetosphere, gaining energy through conservation of the first invariant in the process [Elkington et al., 2004]. The plasmasheet population acts as a boundary condition on the
trapped particles undergoing diffusion in the inner magnetosphere.
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agreement between the two simulation methods when using
all the appropriate symmetric and asymmetric diffusion coefficients, than using any single or simplified diffusion coefficient alone.
7. DOES RADIAL DIFFUSION ACT
AS A SOURCE OR LOSS?
As with any stochastic process acting on a distribution of
particles, radial diffusion will generally act to smooth out
gradients in the space describing that distribution. Therefore,
for radial diffusion to act alone in populating the inner
regions of the radiation belts, a source of particles at high L
must be present. That is, the phase space density of the particle must generally increase out to the particle trapping
boundary. Lacking this source at high L, the effect of diffusion driven by ULF variations will be to deplete the belts as
particles are transported from regions of higher phase space
density through the outer trapping boundary.
Figure 8 illustrates three scenarios by which the distribution function of the radiation belts may evolve [Green and
Kivelson, 2004]. In the leftmost panel, radial diffusion acts
alone on a distribution with a monotonically increasing density. By Fick’s Law [Walt, 1994], this outward gradient will
result in an inward flux of particles from the high-L boundary. In the center panel, a local heating mechanism acts to
increase the phase space density at a limited range of L values, producing a local maximum in the phase space density
relatively low in the magnetosphere. In this case, radial diffusion will act to move particles outward in the high-L
regions where the gradient is negative, and inward in the
inner regions where the gradient is positive. A similar situation will result in case illustrated in the rightmost panel,
where a sudden drop in phase space density at the boundary
again results in a peak at relatively low L. In either of the two
13
latter cases, those particles with a net outward drift will presumably be lost to the magnetopause or outer stable trapping
boundary.
Significant observational effort has been made to determine the evolving nature of the phase space density profiles,
similar to the illustrations in Figure 8, as a means of determining those times when radial diffusion associated with
ULF activity may be dominant, vs. those times when local
acceleration processes may be active. Typically, these studies
are made difficult by uncertainties in the global magnetic
field configuration that dictates the particle adiabatic invariants. Recent studies include those by Green and Kivelson
[2004], who used observations from the polar-orbiting spacecraft Polar to suggest that local heating processes were frequently active during the recovery phase of geomagnetic
storms, forming peaks in the phase space density in the inner
magnetosphere over the course of a few days following storm
onset. During these times, radial diffusion driven by ULF
variations would act as a loss as particles were driven out
through the outer boundary. Onsager et al. [2004] used two
geosynchronous spacecraft at different magnetic latitudes to
calculate the phase space density gradient at L ∼ 6.6, and
found at various times both positive and negative gradients
with respect to L.
Taylor et al. [2004] used equatorial measurements from the
Cluster constellation of spacecraft, in conjunction with geosynchronous and other observations of the inner magnetosphere, and showed that there was often sufficient phase
space density to allow for inward radial transport from the
plasmasheet to populate the belts. Simulations by Elkington
et al. [2004] for the March 2001 storm, depicted in Plate 2,
provide insight into the possibility of keV plasmasheet electrons being transported inward and trapped in the radiation
belts. They find that during times of strong global convection
associated with geomagnetic storms, a sufficient number of
Figure 8. Radial phase space density profiles as related to various processes acting on the radiation belts [Green and kivelson, 2004].
In the left panel, radial diffusion acts alone on a distribution with a monotonically-increasing distribution function. Following panels
indicate phase space density bumps that result from internal acceleration (center) and a change in the outer boundary condition (right).
In either of the second two cases, radial diffusion at high L will act as a loss process.
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energetic electrons may be trapped in the inner magnetosphere to contribute to the overall distribution of radiation
belt electrons at MeV energies. However, during times of
northward IMF (last frame of Plate 2), higher energy plasmasheet electrons do not always have access to the inner
magnetosphere.
8. SUMMARY AND OPEN QUESTIONS
Understanding (to the point of being able to quantify) the
relevant effects of ULF waves on energetic particle dynamics
in the radiation belts requires knowledge of a number of specific properties of the driving perturbations and global fields.
Therefore, we conclude this paper with a few of the outstanding questions related to ULF-radiation belt interactions,
and a review of ULF wave properties necessary to establish
and quantify their effect on the radiation belts.
A basic question related to the transport of radiation belt
electrons is the extent to which the dynamics may be considered in a diffusive framework, versus the extent to which
kinetic effects must be taken into account. The modes of
interaction described by Liu et al. [1999] and Summers and
Ma [2000] explicitly involve violation of the first adiabatic
invariant, and therefore cannot be considered purely in the
framework of radial diffusion. Further, Riley and Wolf [1992],
Elkington et al. [2003], and Ukhorskiy et al. [2005, 2006] all
noted that radial diffusion only provided an adequate description of particle transport when averaged over a statistical
ensemble of driving conditions. In consideration of single
events, diffusion gave only a rough description of the electron
dynamics, with considerable variation from the diffusive formalism that may have been attributable to the details of the
driving fields and only examined from a kinetic point of
view. A closely-related question is the relative importance of,
and differences between, intrinsic vs. extrinsic stochasticity;
that is, diffusion resulting from randomness inherent in the
external driving perturbations (e.g., the solar wind), as
opposed to diffusion resulting from the intrinsic spectral
[Elkington et al., 2003] or spatial [Degeling et al., 2006]
properties of the waves themselves.
In addition to the physical properties of the ULF variations
affecting the transport of radiation belt electrons in the inner
magnetosphere, the global configuration of the background
magnetic field may prove to have a significant effect on the
interaction of ULF power with energetic particles. Elkington
et al. [2003] demonstrated that global distortion resulting
from Chapman-Ferraro currents at the magnetopause can
lead to additional modes of interaction between particles and
waves. The influence of global, steady-state convection can
similarly alter basic particle drift paths and change the energy
gains and losses associated with interaction with ULF waves,
as can the storm-time evolution of the ring current and partial
ring current. The effect of these latter interactions have not
yet been examined in detail.
The efficiency of the interaction between ULF power and
energetic electrons depends on the power spectrum at frequencies including ω = mωd, where ωd is the (energy-dependent)
particle drift frequency and m is the azimuthal mode structure of the interacting waves [Fälthammar, 1965; Schulz and
Lanzerotti, 1974; Hudson et al., 2000; Elkington et al., 1999,
2003; Perry et al., 2005; Summers and Ma, 2000] Resonances
higher-order in m may be induced by global asymmetries in
the guiding background magnetic field [Elkington et al.,
2003].
The radial extent and azimuthal extent over which the
waves globally occur determines the rate at which particles
will be transported/energized through interaction with ULF
waves in different regions of the magnetosphere [Mathie and
Mann, 2000b; Elkington et al., 2003; Tan et al., 2004;
Ukhorskiy et al., 2005; Perry et al., 2005]. For example, if
waves from a particular source occur over only some
azimuthal fraction of a drift shell, the total effect of the ULF
waves on the particles on that shell will be reduced in proportion to the fractional azimuthal occurrence of the waves.
Degeling et al. [2006] shows that local time asymmetries in
the power itself is enough to induce diffusion.
In accordance with the resonance condition suggested
above, the direction of propagation of the ULF disturbances
determines whether the particles may efficiently interact with
the waves. In particular, only waves with positive mode numbers will effectively transport counterclockwise-drifting electrons [Ukhorskiy et al., 2005]; variations in a global
convection electric field, which can be described in terms of
“standing” waves consisting of counterpropagating waves
of equal magnitude and phase, will interact with trapped
electrons at a level dictated by the magnitude of only the
positive-m component, E0/2.
Further, simulations and theoretical considerations have
suggested that electromagnetic vs. potential ULF variations
have explicitly different effects in L and energy among
trapped particle populations in the magnetosphere [Cornwall,
1968; Fälthammar, 1968; Schulz and Lanzerotti, 1974; Perry
et al., 2005]. The difference between equations (5) and
(6) illustrates this point. This has particular significance,
for example, when ULF variations in a given region are
dominated by convective rather than compressional waves.
The depth to which particles may be effectively transported
in the radiation belts via interaction with ULF waves therefore depends not only on the radial penetration of the ULF
waves, but explicitly on the characteristic nature of the waves
as well.
The need for more detailed understanding of the global
nature and occurrence of the waves driving energetic particle
dynamics has been underscored by recent research works by
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Perry et al. [2005], Brautigam et al. [2005], Ukhorskiy et al.
[2005], and Fei et al. [2006]. For example, the observational
and modeling efforts undertaken in Brautigam et al. [2005]
and Ukhorskiy et al. [2005] were forced to make an m = 1
assumption for all wave power considered when estimating
the effect of ULF waves on energetic particles. Perry et al.
[2005] attempted to apply realistic radial and spectral
envelopes to the particle simulations conducted in that study,
which had to be constructed based on a limited combination
of measurements [Bloom and Singer, 1995] and global MHD
results [Elkington et al., 2004]. Fei et al. [2006] used a
model-model comparison between a radial diffusion formulation of the radiation belts and the evolution suggested by
global MHD/particle simulations of the September 1998
storm, and found that the best agreement was only obtainable
when the diffusion simulation incorporated global properties
such as mode structure and radial and spectral profiles of
ULF activity into the appropriate transport coefficients.
Indeed, ultimately disentangling the relative effectiveness of
various processes acting in the radiation belts from among
the zoo of possible high- (VLF) [e.g., Summers et al., 1998;
Horne and Thorne, 1998] and low-frequency interactions
will require an extensive knowledge of the global characteristics of the waves involved in each of these processes.
Acknowledgments. Work has been supported by NSF grant
ATM-0303209 and NASA grants NNG05-GG27G and NNG05GK04G at the University of Colorado.
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