Particle Transport in the Magnetosphere: A New Diffusion Model
Robert B. Sheldon^{1} and Timothy E. Eastman^{2}
^{1}Boston University Center for Space Physics
725 Commonwealth Av, Boston, MA 02215
^{2}University of Maryland, College Park, MD
18 May 1997
Previous transport models of the plasmasphere, ring current and
radiation belts have considered either diffusive or convective
transport, but not both. Since the grad B drift speed is
proportional to energy, analyses of particles with E > 100 keV have
generally used only grad B drift and diffusive
transport. Conversely, particles of E < 30 keV have been treated with
only ExB drift and convective transport. The ring current,
which lies between these two regimes, cannot be described by either
mechanism alone, so that these standard models, though widely used and
otherwise applicable, fail in detail to describe the trapped ion
population as observed with modern instrumentation. We develop a
formalism that can describe both diffusive and convective transport in
a completely general way, including UT, LT, radial, and pitch angle
dependence, with arbitrary magnetic and electric field models. The
formalism is general enough to rederive the electric or magnetic
diffusion coefficients for trapped particles without the simplifying
assumption of axial symmetry or a vanishing convection electric field,
thus improving on the standard coefficients derived nearly 30 years
ago. More importantly, we include the previously neglected diffusion
term due to a localized, nonglobal perturbation of the fields, and
show that under some circumstances this term may dominate over the
resonant diffusion caused by global perturbations.
Two types of models have been used to describe the ring current
(1 < E < 300 keV), diffusion models (e.g., Sheldon [1994a]) and convection models (e.g., Fok et
al. [1995]). Both models are very sophisticated yet have
deviations of an order of magnitude from the data, showing that
neither approach alone can describe the ring current. A variant on the
second approach used MonteCarlo simulations of convective transport
but with the impulsive bursty characteristics of the stormtime
electric field (Chen [1994]), and found that quasidiffusive
transport, faster than driftresonant (Schulz [1974]) diffusive
transport was needed in such simulations to describe the ring current
ions. Therefore, not only do we need a formalism for including
diffusion in a convection model, but we need a robust definition of
diffusion as well.
In addition to simulations, there is compelling empirical evidence
that no present convection or diffusion model can adequately explain
the data. One example from the AMPTE mission will suffice. On days 138143
of 1987, the CHEM instrument observed the quietest ring current in its
fouryear mission, as determined by both ground based indices and
in situ measured densities. A plot of those six radial passes (see
Figure 1) shows a distinct peak around L=2.8 that
surfaced as the magnetosphere quieted (Days 139142), but appears
diffused during more active periods (Day 138). Now this peak is not
the result of a timedependent transport due to electric fields or
currents, since the magnetosphere was in an extremely quiescent state,
but arises from the convective access of plasmasheet ions to the inner
ring current, which can be observed only in a realistic convection
model (Sheldon [1994b]). Although one might include such ad
hoc source terms in either the diffusive or the convective models
above, the data show other discrepancies not so easily included.
FIGURE 1: Phase space density of 1020 MeV/G H^{+}
on six successive AMPTE/CHEM passes.
Timedependent convection models can only move particles from the
plasmasheet on open drift orbits, and use timedependence to get
particles onto closed, ring current drift orbits. In any quasistatic
timeindependent magnetosphere, however, the particles on closed
orbits would decay with time, producing sharp boundaries between open
and closed drift orbits. Thus the ring current should be essentially
empty for any protracted period of magnetospheric quieting. Since the
loss times typically quoted for the ring current are about 10 hours,
the five days of quiet shown here should have produced an empty ring
current out to about L=6. Clearly an additional transport mechanism is
operating in our data.
The standard diffusion model implements such a
quiettime radial transport, but typically with a steep power law
dependence on Lshell. So even if we include a lowL source to mimic the
convective access, there would not be the enhanced diffusion observed
around L=4. That is, the diffusion models predict a steep flux gradient
at L~5 when the diffusion rate drops below the
chargeexchange loss rate. Now the flattening of the flux gradient at
L=4 on day 138 can only occur if the diffusion rate is again greater
than the loss rate. Since the loss rate increases monotonically as one
approaches the atmospheric source of neutral hydrogen, this implies
that the diffusion rate show a local maxima at L=4. But the standard
diffusion models all have diffusion coefficients which monotonically
decrease with L in disagreement with the data (West [1981], Sheldon [1993a]).
As before, we can again add a second internal source of diffusion to
account for the nonmonotonic shape at L=4 (Sheldon [1994a]), which
does not change the character of the solution at L > 5. Yet even with
this highly ad hoc model tuned to explain L < 5 fluxes, we
cannot explain the change in the slope of the phase space density seen
around L=5 for days 138140 to days 141142. Since the slope depends
not on the amplitude of the diffusion coefficient but its power law
exponent, this change in slope implies a temporal change in the power
law exponent of the electric diffusion coefficient. This is only
possible in current diffusion theory if the spectral index of the
perturbations of the solar wind driver should change in exactly the
same manner. In this case, we would require a solar wind perturbation
spectral index change from f^{2} to about f^{6}, which appears
rather unsupported by solar wind observations. Thus it appears
difficult to account for temporal changes in the ring current using
current diffusion models whenever long time averages, which tend to
introduce substantial numerical diffusion, are not performed.
We propose that the above theoretical and observational problems can
be solved, first by transforming the convection problem into more
intuitive coordinates and then considering the diffusion of these
convection drift orbits using recent descriptions of the diffusion
process. In Section 2 we describe the coordinate transformation that
enables us to separate convection from diffusion. In Section 3 we
develop a standard approach to diffusion in energyspace and show how
we can derive diffusion coefficients for random perturbations in the
fields. In Section 4 we describe the differences between resonant
diffusion and our derivation, and when this additional diffusion is
important.
Several approaches have been used to solve the problem of ion
trajectories in the earth's magnetosphere. Most rely on integrating
the forces along a particle trajectory, a Lagrangian approach. Our
approach uses conservation of energy and the first two adiabatic
invariants along a bounceaveraged trajectory so that the orbit is
determined by the isoenergy contours for a prescribed electric and
magnetic field model. This Hamiltonian approach has the advantage
that errors do not propagate and that trajectories are found quickly
and accurately. The relevant equation is given by (Whipple [1978]),
Ho = K.E. + P.E. = M Bm + q U
where Ho is the unperturbed Hamiltonian consisting of the sum of
kinetic (KE) and potential (PE) energies. At the mirror point, all
the KE is in the perpendicular motion, so that the magnetic moment,
M, multiplied by the mirror point magnetic field strength, Bm,
gives the ion KE. The PE is assumed to be well
described by a scalar field, U, times the charge, q.
The Hamiltonian method outlined above is completely general for all
pitch angles. The electrostatic potential U is assumed to be
constant along a field line, and therefore is independent of pitch
angle. However Bm is clearly pitch angle dependent, and must be
known for every value of the second adiabatic invariant J or K
(Sheldon [1993b]).
(Birmingham [1984] discusses the conditions under which the first two
invariants are not conserved, which for a "realistic" model field,
e.g. Tsyganenko89, occurs near midnight at L=810 for 100 keV
particles (private communication, B. Anderson 1996). Time dependence
is also possible if changes in Bm and U are "slow enough", or
quasistatic, a property met by all models previously described. Thus
the boundary conditions under which the above formulation is
applicable are the ionosphere, the magnetopause, and the
energydependent "Birmingham boundary" around 10 Re at midnight.
In the adiabatic regime of a quasistatic magnetosphere, ions neither
gain nor lose total energy so that we can write the derivative,
0=dHo/dt= M dBm/dt + q dU/dt ==> dU/dBm = M/q
i.e., if we convert from real (x,y)space into (U,Bm)space, then
the ion trajectories are straight lines with the slope M/q. Not
only does this greatly enhance our intuition for magnetospheric
convection, but it clearly separates the competing transport
mechanisms of diffusion and convection. Although diffusion is
occurring in all directions, it is only apparent when no other
transport mechanism, such as convection, overwhelms its small
displacement. Thus we can identify convection with the motion along a
drift orbit line, and diffusion as the motion transverse to drift
lines.
This definition of diffusion overcomes most of the difficulties
encountered in the diffusion models discussed above. They all assume
that the drift orbits are circular after removing the effect of a
nonaxisymmetric magnetic field so that diffusion then corresponds to
radial transport. This is clearly not true for ring current ions,
which because of oppositely directed grad B and ExB drifts, orbit on distorted ellipses, bananas, and other unnamed
figures. Thus we are unable to compute ring current diffusion
coefficients from previous theory and instead rederive the diffusion
coefficient in a Hamiltonian formalism.
Let us write a simple extension of the above quasisteady state
Hamiltonian including a small perturbation term,
H = Ho + H1 = (M Bm + qU) + M b + q u
We assume that the average value of this Hamiltonian is the steady
state, Ho, which is to say that fluctuations in the magnetosphere
neither energize nor damp the total energy of the system, e.g. a
steady ramp up of the electric field would violate this
assumption. Near the drift resonance, < H1> > 0, unless
we also average over drift phase. This average over drift phase is
potentially the thorniest issue in calculating both resonant and
nonresonant diffusion, as well as in distinguishing irreversible,
entropyincreasing diffusion from reversible "migration", but in the
following discussion we assume that a timeaverage is sufficient.
If we consider an ensemble of nearly coincident orbits,
with nearly the same
total energy, then we can describe their variance as
H^{2}rms = <(H  < H>)^{2}>
= < M^{2} b^{2}> + < q^{2} u^{2}>
where we assume that b and u are
uncorrelated. This is probably not true in general, though we believe
that the cross term < b u> will be smaller than the
other terms, which can be seen by noting that changes in U will
occur only in the ionosphere, which because of its conductivity will
"freeze in" any fluctuation of magnetic foot points.
Now diffusion in energy space is nothing but broadening
of the variance that we have just defined (Reif [1965]),
2D = d/dt(H^{2}rms) =
d/dt(< M^{2} b^{2} >
+ < q^{2} u^{2} >)
This broadening is in the direction perpendicular to the convection
velocity since convective velocities are generally an order of
magnitude faster than diffusive broadening so that we do not observe
the component of diffusion parallel to convection. This direction can
be calculated from the slope, tan(\theta)= M/q, where \theta
is the angle wrt B, the xaxis. The diffusive direction requires
that a fluctuation in B be multiplied by sin(\theta), and
conversely that a fluctuation in U be multiplied by cos(\theta).
Thus we have,
D= (sin(\theta)/2) d/dt(< M b>^{2} 
q/M < q u>^{2})
There are several features of this expression that indicate
selfconsistency and coherence of our approach. For high energies
(M>>1) the expression is dominated by magnetic fluctuations, as
is well known for radiation belt ions. For low energies (M<<1)
the expression is dominated by electric field fluctuations, as
expected for plasmasheet ions. Thus we have spanned the entire range
of ring current energies with this one expression.
The diffusion coefficient derived above can be converted from energy
space to real space by using spatial derivatives of H
for a specified magnetic and electric field configuration. For example,
a fluctuating VollandStern electric field where
the fluctuations occur not in the first, corotation term, but in the
second, convection term, can be written,
U = A/L + C L^{2} sin(LT)
u = L^{2} sin(LT)\delta C
dH/dL = q (A/L^{2} + 2C L sin(LT))
where LT is the Local Time measured from midnight, and A,C are the coefficients
for corotation and crosstail field. Then at low energy (M<<1),
D_{LL} =cos(\theta)L^{8} sin^{2}(LT) / 2( 2C L^{3} sin(LT)  A)^{2}} d/dt(\delta^{2}C)
Thus for A<<2C sin(LT)L^{3}, D_{LL}~L^{2}, whereas for A>>2C sin(LT)L^{3}, D_{LL}~L^{8}. That is, active Kp periods have
a smaller radial gradient for diffusion than quiet periods, which is
precisely the observed temporal deviation shown in Figure 1. We can
now explain timedependence in the powerlaw exponent of the diffusion
coefficient.
There are significant differences between the standard derivation of
the resonant diffusion coefficient and that for our Hamiltonian
diffusion coefficient as shown above. The resonant diffusion
derivations of Parker [1960], Falthammar [1965], and
Nakada [1965], assume that the major contribution to the diffusion
coefficient is the resonant component. Since the magnitude and
direction of the resonant component are phase dependent, changing sign
for day and night sides of the magnetosphere, a second assumption is
that magnetospheric fluctuations are phase mixed in time so that the
average effect is diffusive. This classical approach makes two
critical assumptions: the perturbations are incoherent in time and
coherent in space. That is, the perturbations are resonant, having a
coherence time that is short with respect to the system, and global,
causing all drift paths to shift in phase together. But there are also
situations in which violation of the second assumption, coherence in
space, might lead to a larger diffusion coefficient than the resonant
model. That is, under certain circumstances, turbulent spatial
diffusion should be more effective than temporal, phasemixing
diffusion.
Arguments were presented nearly 30 years ago in the context of
magnetized solar wind plasma, (e.g., Taylor [1971],
Matthaeus [1995]) and references therein), that resonant, "slab"
diffusion depends on the second power of the fluctuations
(D_{LL}~(b/Bo)^{n}) where n=2, whereas
nonresonant "2D" diffusion depends only on the first power,
n=1. If this be the case for the magnetosphere, and crucial
assumptions have yet to be proven, then in the limit of b/Bo
==> 0, (i.e. for quiet magnetospheric conditions) this
incoherent, nonresonant diffusion may dominate over the resonant
contribution. Since these are exactly the conditions under which the
data were taken, it is possible that we are perhaps seeing a time
period dominated by nonresonant diffusion. At the very least, we
have demonstrated that the superiority of resonant diffusion over
nonresonant diffusion cannot be taken for granted, it must be proven.
In a future paper, we shall apply this model to the data set described
in the introduction.
We have attempted to show how the use of a Hamiltonian formalism can
help us to calculate diffusion in the magnetosphere and separate it
from convective effects. This means that we can now treat ring
current and plasmasheet ions with a single consistent theoretical approach
that incorporates both diffusion and convection. We go on to show that
this treatment can also improve the standard diffusion coefficients
both by generalizing magnetic impulses to any type of electromagnetic
perturbation and by incorporating recent insights into the nature of
"2D" and "slab" diffusion. We believe that this reformulation of the
diffusion paradigm will be of great value to modellers attempting to
fit modern data sets. Not only has the basis set of diffusion models
been enriched, but convection has been incorporated in a way that
permits a fluid description of magnetospheric particles from 1 eV100
MeV.
Acknowledgements
This study was supported by NASA contract
NAG51558. The authors gratefully acknowledge D. Hamilton and the
AMPTE/CCE team for the excellent data analysis tools.

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