An Empirical Phase-Space Analysis of Ring Current Dynamics:
Solar Wind Control of Injection and Decay
T. Paul O’Brien and Robert L. McPherron
Institute for Geophysics and Planetary Physics, University of California
Los Angeles, California 90095
This empirical analysis of the terrestrial ring current, as measured by Dst, uses conditional probability density in Dst phase space
to determine the evolution of the ring current. Our simple model, with 7 nontrivial parameters describes the dynamics of 30 years of
hourly Dst with solar wind data provided by the OMNI database. The solar wind coupling is assumed to be determined by VBs. We
conclude that the Burton Equation [Burton et al., 1975] correctly describes the general dynamics with a slight correction. We show
that the ring current decay lifetime varies with VBs, but not with Dst, and we relate this variation to the position of convection
boundaries in the magnetosphere. Convection boundaries closer to the Earth result in shorter charge exchange decay times due to
the higher neutral density near the Earth. The decay time in hours varies with VBs in mV/m as  = 2.40exp[9.74/(4.69+VBs)]. We
also show that the energy injection function as derived by Burton et al. [1975] is essentially correct. The injection Q is zero for
VBs < Ec = 0.49 mV/m, and it is Q = -4.4(VBs-Ec) for VBs > Ec. In addition, we derive the correction for magnetopause
contamination: Dst* = Dst -7.26P1/2 + 11 nT, where P is solar wind dynamic pressure in nPa. Finally we apply the model to a
moderate storm and to an intense storm. We demonstrate that, in spite of the fact that spacecraft observe compositional changes in
the ring current at intense Dst, the dynamics of the two storms are not obviously different in the context of our model. We
demonstrate that the generally observed dependence of the decay parameter on Dst is actually an alias of the coincidence of intense
Dst and intense VBs.
INTRODUCTION
The ring current is a toroidal current that flows in the Earth’s
magnetosphere between 2 and 10 Earth radii (RE) [Gonzalez et
al., 1994]. The balance of the current is carried by energetic
ions, mostly protons with energies between 30 and 200 keV.
During magnetic storms, the ring current is greatly enhanced,
and its energy budget contains a significant contribution from
Oxygen ions [Daglis et al., 1998]. The exact mechanism that
adds particles to the ring current is not completely understood
[Daglis et al., 1998], but the loss mechanisms are fairly well
documented. The primary loss mechanism is charge exchange,
wherein an energetic ion exchanges an electron with a cold
neutral and, having lost its charge, escapes the magnetic field of
the Earth, carrying most of its kinetic energy with it [Daglis et
al., 1998]. Although we understand the mechanism well, there is
considerable uncertainty in the specific features of this decay
mechanism during different phases of a magnetic storm. In this
study, we will attempt to develop an empirical model of the
dynamics of the storm-time ring current, including both
injection and decay.
Our study of ring current dynamics will be focused on an
analysis of the Dst index. While directly measuring the
magnetic field of the toroidal current flowing in the
magnetosphere, Dst is also a measure of the kinetic energy E of
the particles that make up the ring current. This is stated
formally in the Dessler-Parker-Sckopke relation [Dessler and
Parker, 1959; Sckopke, 1966]:
2/6/2016 2:42:00 AM v.2
Dst * (t ) 2 E (t )

B0
3E m
(1)
Here B0 and Em represent the magnetic field at the surface of the
Earth and the magnetic energy in the Earth’s field above the
surface, respectively. From there, assuming the particle kinetic
energy is a combination of a source U and proportional loss,
dE (t )
E (t )
 U (t ) 
dt

(2)
we arrive at the Burton Equation [Burton et al., 1975]:
dDst *
Dst * (t )
 Q( t ) 
dt

(3)
This particular equation performs quite well for various
definitions of Dst*, Q, and  [Burton et al., 1975; Feldstein et
al., 1984; Vasyliunas, 1987; etc.]. Dst* is one of various
corrections to Dst which will be discussed below. Q is an
injection term, which we will treat as being uniquely determined
by VBs, the interplanetary electric field in Geocentric Solar
Magnetospheric (GSM) coordinates. We define VBs to be
positive in terms of the solar wind parameters in GSM:
 VB
VBs   z
 0
Bz  0
Bz  0
(4)
The decay time  is presumed to be a result of charge
exchange loss through collisions with the neutral geocorona
[Daglis et al., 1998]. Based on the observation that this recovery
time is much shorter during very intense storms, as identified by
large, negative Dst, it has been postulated that  depends on Dst.
O’Brien and McPherron Phase Space Analysis of Dst
Page 1
Essentially three mechanisms have been suggested to explain
how  could vary during a storm. Assuming  is related to
charge exchange lifetimes, we would write an effective decay
time:
  (n H v) 1
(5)
Where nH is the density of Hydrogen in the geocorona,  is an
effective charge exchange cross section, and v is an effective
particle velocity [Simth and Bewtra, 1978]. The geocorona
density falls off with distance from the Earth L, in RE, giving
the following approximate dependence to nH [Smith and Bewtra,
1978]:
n H  e  L / L0
(6)
Here L0 is a scale height determined by the mass and
temperature of the atmosphere, by the gravitational pull of the
Earth, and by the approximate location of the quiet time
convection boundary.
The first mechanism for varying  assumes that there are two
ring currents, an inner one and an outer one. The inner one
would have a larger effective nH, and would therefore decay
faster. As the inner current decayed, the apparent decay rate
would seem to slow as the outer current became the dominant
contributor to the Dst index [Akasofu et al., 1963]. Satellite
experiments have shown that there are not two spatially distinct
ring currents [Hamilton et al., 1988], but the idea of radial
dependence on decay time will be used below.
The second mechanism for varying  assumes that wave
activity is very intense at the storm peak. During enhanced wave
activity, particles are constantly being scattered into states that
allow them to travel very close to the Earth during their bounce
motion [Daglis et al.1998]. By traveling closer to the Earth, the
particles experience higher effective nH, and therefore shorter
Smith and Bewtra, 1978.
The final mechanism, and by far the most popular, assumes a
significant change in the composition of the ring current during
the most intense part of the storm. Specifically, satellite
measurements show that during intense storms, a significant
fraction, sometimes over 50%, of the ring current energy is
carried by O+ ions [Hamilton et al., 1988]. Having a relatively
larger cross section  at the same energy, these ions have a
much shorter lifetime to charge exchange than H+ ions, which
are usually the dominant contribution to the ring current
[Hamilton et al., 1988]. By shifting the balance of energy
among various species, it is possible to change the effective
decay time of the ring current.
There have been numerous other corrections to the Burton
Equation. Some authors [Gonzalez 1989; Prigancova and
Feldstein, 1992] have suggested that  drops to an hour or less
during very intense activity. Other authors have been concerned
with a contamination of Dst by the tail current [Alekseev et al.
1996]. Vassiliadis et al. [1997] have suggested that the Burton
equation be replaced with a second order equation:
d 2 Dst
dDst

  2 Dst (t )  Q(t )
2
dt
dt
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(7)
Recently Kamide et al. [1998] have suggested that a storm is
actually composed of a two-step development where the sudden
injection of Oxygen provides a second intensification. There are
so many models that any new model is obliged not merely to fit
some data but also to explain how other models would also be
able to fit the data. We will derive our model from the data, then
we will analyze the deviations from the model, and finally we
will use our model to reproduce the observed dependence of 
on Dst.
METHOD OF ANALYSIS
Typically, the analysis of any dynamic system begins with a
look at the system’s phase space. For Dst this is a non-trivial
task because hourly changes in Dst are often lost in the noise
background. Therefore, we have chosen to analyze the
probability density in phase space. Specifically, we are
interested in the probability of a given hourly change in Dst,
signified by Dst, for a given Dst and Q; we write this
probability P(Dst|Dst,Q). Recall that Q represents the injection
of new energy into the ring current and that Q, being unknown,
is assumed to be uniquely determined by VBs. The discrete
form of the Burton Equation suggests that the probability
density in phase space will be focused on a line:

Dst * 
Dst  Q(t ) 
t
 

*
(8)
The slope of such a line would be related to , and the offset
would be related to Q, according to:
offset  Qt
(9)
slope  
t

(10)
In phase space, we plot P(Dst|Dst,Q) versus Dst and Dst for
a specified VBs. This approach is completely general, where the
only assumptions about the dynamics are: a) Q is uniquely
determined by VBs, with Q being zero when VBs is zero, b) the
spread of the probability density is caused by processes which
are completely random with respect to Dst and VBs, and c) the
state of the system is otherwise entirely determined by Dst and
Q. We do not assume a priori that the system follows the Burton
Equation. Assumptions a and b will be discussed in more detail
below. Assumption c allows us to generalize statistics on 30
years of hourly Dst transitions to the dynamics of individual
storms.
For this analysis, we will use the OMNI database [King, 1977]
of hourly solar wind and Dst measurements. These solar wind
measurements are calculated as hourly averages from spacecraft
near the Earth, but outside the magnetosphere. No propagation
is performed from the spacecraft location to the magnetopause.
We use the years 1964-1996; during this interval, there are
many times when no solar wind data are available. We must,
therefore, omit these times from our statistical analysis. In total,
the database contains nearly 300,000 hours of Dst data. After
removing times when no solar wind data are available, we are
left with 130,000 hours when VBs is available, and 125,000
O’Brien and McPherron Phase Space Analysis of Dst
Page 2
hours when VBs and dynamic pressure Psw are available. At
final accounting, our dataset includes hundreds of storms of
various sizes and many hours of quiescent behavior taken from
various stages of the solar cycle.
Since we will be computing probability densities from discrete
samples, we will perform some binning of the data. In phase
space, this binning amounts to calculating average quantities in
a neighborhood of Q and Dst values. We will denote this type of
average of a variable X as <X>|Q,Dst. In cases where we have a
large enough sample size, we will use medians rather than
means to eliminate the influence of large outliers. This
averaging will be useful in generalizing with regard to the
standard technique for removing the magnetopause contribution
to Dst.
PRESSURE CORRECTION
It is common when studying Dst to remove the contribution of
the magnetopause current system. When the solar wind dynamic
pressure is enhanced, the magnetopause moves closer to the
Earth, and the currents associated with it contaminate Dst. The
formal equation for this correction is given by Burton et al.
[1975]:
Dst *  Dst  b Psw  c
(11)
Here, c contains not only an adjustment for the quiet time
magnetopause, but it may also contain an offset in the
calculation of Dst itself. This definition of Dst* gives rise to a
modification of the Dst equation:
Dst  Dst  b Psw
*
(12)
Various values of b and c have been calculated [Burton et al.,
1975; Feldstein et al., 1984; Pudovkin et al., 1985, etc.], but
disagreement exists over the precise values. In the statistical
averaging that we will perform, the correction to Dst becomes a
constant:
Dst *
Q , Dst
 Dst
Q , Dst
 Dst
Q , Dst

 c  b

 c'
Psw
Q , Dst


(13)
This constant c’ will be defined as the value of Dst for which
Dst is zero when Q is zero.
As there is no reason to believe that the hourly changes in P sw
should be correlated with the magnitude of Dst nor with VBs,
we can reasonably assume that the effects of hourly changes in
Psw should average to zero in our statistical analysis:
 Psw
Q , Dst
0
(14)
We can therefore use uncorrected Dst for statistical averaged
dynamics:
Dst *
Q , Dst
 Dst
Q , Dst
(15)
For these reasons, it is not necessary to know either b or c a
priori. It should be noted that these are largely technical details
provided here to satisfy the skeptical reader, but that the
analysis is not appreciably affected by this correction.
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ANALYSIS IN PHASE SPACE
Having laid the statistical groundwork, we will now turn to the
actual analysis of data. Figure 1 shows the phase-space
probability density for 3 values of VBs. It is of supreme
importance that in each case the probability density is clustered
around a linear trajectory. This tells us explicitly that the Burton
Equation is the correct description of the dynamics. It also tells
us that, for a given VBs, , as given by (10), does not change
with Dst because the slope of the trajectory does not change
with Dst. We do see that the slope and offset change for
different values of VBs. The most general conclusion of this
observation is that  and Q depend on VBs, but not on Dst.
Before we begin to analyze the forms of these dependencies on
VBs, we should take some time to address some sources of error
and to test their randomness.
ERRORS
There are numerous sources of error, most of which are
believed to be small and random with respect to the influence of
injection and decay. One source of error is the calculation of Dst
itself. Since Dst is calculated as the weighted arithmetic average
of H values measured at 4 stations around the Earth [Sugiura
and Kamei, 1991], there is some inherent uncertainty in Dst.
Assuming that this uncertainty is spread in a Gaussian around
the true value of Dst, we can characterize the uncertainty in Dst
as:
 Dst (t )  ASY (t ) / N ~ VBs(t )
(16)
Where N is the sample size, 4, ASY is the asymmetry index,
which measures the range of H values contributing to Dst, and
asymmetry is related to VBs according to [Clauer et al., 1983].
This suggests that more intense VBs corresponds to more
uncertainty in Dst.
Another source of error is the hour averaging performed in the
construction of the OMNI database. First, this contributes to an
error in the computation of VBs from hour averaged V and hour
averaged Bz:
(17)
V Bz s  VBs
These quantities are not exactly equal and are particularly
unequal when Bz is nearly zero. It is simple to understand that at
least 30 minutes of southward Bz may be lost in this
approximation.
Next, we must consider rapid dynamic pressure changes that
occur on a scale of less than one hour. These can give rise to
unexpected behavior in the Dst index due to a similar inequality
between hour averaged quantities. A rapid change in dynamic
pressure may not be evident in the hour averages of solar wind
density and velocity, which contribute to dynamic pressure, but
may be evident in the hour averages of the H values which are
used in computing Dst.
Finally, we must consider hour boundaries in the data points.
Since the OMNI database does not account for propagation of
the solar wind, it is easy to see how a change in the solar wind
may occur earlier or later in the database than its effects appear
O’Brien and McPherron Phase Space Analysis of Dst
Page 3
in the Dst index. Taken together, these sources of error should
be random and small relative to the influence of injection and
decay. However at small values of Dst or for weak injection,
these errors may likely dominate hourly changes in Dst.
If the errors described above represent the sum of random
deviations from the actual values of Dst, they should result in a
Gaussian distribution of Dst values for a given Dst and Q
[Hogg and Craig, 1995]. There will be some difficulty in
measuring the distribution of Dst values because Dst is
measured on an integer scale, in nT, and this rounding will mask
the precise distribution of the errors.
For a given set of preconditions, Dst and Q, we observe a
nearly Gaussian distribution of values of Dst. We would like
to constrain the likelihood that this distribution of Dst values
does indeed come from a Gaussian source distribution. That is,
we would like to know if the deviations from the mean Dst can
be explained as the superposition of many small deviations
which are not uniquely determined by Dst and Q. We wish to
test the hypothesis that our measured distribution of Dst comes
from a Gaussian whose mean and standard deviation are the
same as the distribution we measured. Unfortunately, statistical
tests can only tell us the certainty with which a sample does not
come from a hypothetical distribution. The statistical measure is
called the significance, and it represents the probability that the
differences between the Dst distribution and the Gaussian
distribution are the result of an underlying difference between
the two distributions rather than simply the result of finite
sample size. A significance of 1 tells us that the Dst values did
not come from a Gaussian distribution, and a significance of 0
tells us that we cannot prove that the Dst values did not come
from a Gaussian distribution.
To test the Gaussian hypothesis, we will use a direct
comparison of the measured Dst distributions to those of
integer-rounded Gaussian distributions. We will employ the
Kolmogorov-Smirnov (KS) test [Press et al., 1986], which
measures the similarity of two distributions based on the
maximum separation of their cumulative distribution functions.
The cumulative distribution function F(x) is the probability that
a sample measured from the distribution will have a value less
than or equal to x. The KS test is accurate to significances of
0.01 or smaller for sample sizes of 20 or larger [Press et al.,
1986]. Figure 2 shows that for VBs = 2 mV/m the difference
between the measured Dst distribution and a Gaussian is
significant for small values of Dst and not clearly significant for
large values of Dst. This result is typical for all values of VBs.
With such high significances for rejection of the Gaussian
hypothesis at small Dst, we conclude that the distribution of
Dst stems from the additive combination of errors that are not
Gaussian [Campbell, 1996]. That is, at small values, Dst evolves
from one value to the next according to a deterministic equation
with a finite amount of uncertainty introduced by non-randomwalk noise. We will revisit the importance of this result when
we perform the pressure correction.
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HOW INJECTION AND DECAY DEPEND ON VBs
From the phase-space plots it is clear that the errors are not
systematic; therefore, we can safely say that the systematic
features of the ring current are described by Q and . We have
seen that these vary with VBs, so we must characterize this
variation. To calculate precise values of Q, we must
accommodate the fact that a given calculation of the phasespace trajectory is biased because of binning in VBs. In Figure
1, 1 mV/m bins were used. This is a fine approximation for
values of VBs below 5 mV/m, but above that threshold the
sparseness of VBs values requires us to enlarge our VBs bins.
Large VBs bins will be heavily biased toward the smaller VBs
values because of the rapid occurrence drop-off with larger
VBs. Therefore, we must accommodate this drop-off by
correcting the bin bias. We will use the slope and offset of the
phase-space fits to estimate Q and  according to (9) and (10).
We represent the measured value of Q as:
VBs0  / 2
Q(VBs0 )

 Q( x) N ( x)dx

(18)
VBs0  / 2
where values with |VBs-VBs0| <  were used in calculating
Q(VBs0), and N(x) is the fraction of data points at a given VBs
such that the integral of N over the entire bin is 1. If we expand
Q(VBs) to first order about VBs0, we have:
Q(VBs0 )


VBs0  / 2
 Q(VBs )  Q'(VBs )( x  VBs ) N ( x)dx
0
0
(19)
0
VBs0  / 2
We can now remove Q(VBs0) and Q’(VBs0) from the integrand:
VBs0  / 2
Q(VBs0 )

 N ( x)dx
 Q(VBs0 )
VBs0  / 2
VBs0  / 2
 Q' (VBs0 )
(20)
 ( x  VBs ) N ( x)dx
0
VBs0  / 2
Since N integrates to 1, this becomes:
Q(VBs0 )

 Q(VBs0 )
 Q' (VBs0 ) VBs

 VBs0 
(21)
This equation has two unknowns, Q(VBs0) and Q’(VBs0). If we
repeat this process for several values of , we will have an overdetermined system and we can perform a least squares
regression to find Q and Q’. Recall that each <Q>  is measured
as the offset of a linear fit to the medians of the Dst
distributions for a range of Dst with Q constrained by
|VBs - VBs0| < . By using this technique, we can extend our
estimation of Q to values of VBs up to about 9 mV/m. It is
difficult to extend beyond that threshold because of scarcity of
data. However because Q turns out to be very linear in VBs, we
are able to trust values slightly beyond that limit. As well as
extending the region of VBs over which we can measure the
phase-space fit parameters, this technique has also given us
O’Brien and McPherron Phase Space Analysis of Dst
Page 4
much more stable measurements compared to using the uniform
bin size of 1 mV/m.
Figure 3 shows the estimates of Q we obtain from this
process. It is clear that Q is a linear function of VBs. Since a
positive Q is non-physical, we define Q in terms of VBs as:
a (VBs  E c ) VBs  E c
Q(nT / h)  
0
VBs  E c

(22)
where a is -4.4 nT/h(mV/m)-1, and Ec is 0.49 mV/m. These
values are similar to those provided by Burton et al. [1975],
-5.4 nT/h(mV/m)-1 and 0.5 mV/m respectively.
We can also obtain from the value of Q(0) the correction
parameter c introduced in (11). Since Dst should be defined
such that no input results in Dst = 0, the corrected Dst should be
adjusted according to (11) and (13) by a constant offset c’ of
0.1 nT. This offset is so small, however, that it can be taken to
be essentially zero, since Dst is only measured on a 1 nT
resolution. Because c’ is essentially zero, there is no appreciable
difference in Q or  whether calculated from Dst or Dst* by our
method.
We can apply the same technique to the slope of the phasespace fits to obtain  as a function of VBs. Figure 4 shows that
this relationship is by no means linear. The value of 7.7 hours
calculated by Burton et al. [1975] corresponds roughly to a VBs
of about 4 mV/m, a typical storm-time value. In a later section
we will discuss one way of fitting this relationship to an analytic
function. For now it is sufficient to note that Q is linear in VBs
and that  is nonlinear in VBs.
CALCULATION OF PRESSURE CORRECTION
We now return to the issue of magnetopause contamination of
Dst and the associated pressure correction. We argued above
that ignoring pressure fluctuations was legitimate because the
influence of those fluctuations averages to zero. However, if we
choose to fix pressure fluctuations to a particular range and then
recalculate the linear fits in phase space, we can estimate the
pressure correction parameter b introduced in (11). We extend
the definition of the offset to be:
offset  Qt  b  Psw
(23)
Q , Dst
A linear fit of the residual phase-space offset after the removal
of Q to the change of the square root of the dynamic pressure
yields a pressure correction term b of 7.26 nT(nPa)-1/2. Burton et
al. [1975] provided 16 nT(nPa)-1/2. Figure 5 shows how the
dynamic pressure affects the phase-space offset by fitting the
residual to the hourly change in the square root of the dynamic
pressure. Now that we have obtained b and c’, we can recover c,
as defined in (11) and (13). We obtain a value for c of 11 nT,
whereas Burton et al. [1975] obtained 20 nT.
Now that we know how to correctly remove the magnetopause
contamination, we can reevaluate the randomness of our model
errors. We have performed the KS analysis after the pressure
correction. The resulting distribution of Dst appears to be
significantly less Gaussian. This suggests that after removal of
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the magnetopause contamination the remaining deviations from
the deterministic equations we have described are not simply the
superposition of many small random deviations. This new
information does not tell us what kind of deviations we are
seeing, but it does tell us that the assumption of Gaussian
random noise is not valid for all values of Dst, and this can
significantly affect the tools we use to measure how our models
make errors.
HYPOTHESIS ON VBs CONTROL OF DECAY
The interesting shape of the decay parameter  versus VBs
suggests that an analytical function could fit much of the
variation. A mathematical fit, however, would have two
shortcomings. First, it would tell us nothing about the
underlying physical mechanism which governed the control of
decay by VBs. Second, we could not reliably extrapolate a
simple mathematical fit beyond VBs = 9 mV/m. If we can use
some physical insight to provide the functional form of the
-VBs relationship, we could both gain some insight into the
mechanism for  variation and also have some confidence when
extrapolating beyond the region of fit.
Pudovkin et al. [1988] derived a relation which roughly
matches the -VBs curve in Figure 4, but they were not able to
adequately fit the transition from long to short recovery time. To
explain the shape of the -VBs curve, we suggest that VBs
controls the position of the ring current by controlling the
positions of convection paths and boundaries. In the
approximation that there is a global electric field from dawn to
dusk throughout the magnetosphere, the hot ions that make up
the ring current will set up a convection pattern [Southwood and
Kaye, 1979]. At a given energy and equatorial pitch angle, the
convection pattern will have both closed and open trajectories.
Nominally the ring current is made up only of the particles on
closed trajectories. The position of the boundary between open
and closed drift orbits defines the outer edge of the ring current.
This position depends on the strength of the dawn-dusk electric
field and on particle energy and equatorial pitch angle
[Southwood and Kaye, 1979]. This is consistent with in situ
observations, which showed that the energy density peaks closer
to the Earth during the most intense part of the February 1986
great storm [Hamilton et al., 1988]. If the ring current is closer
to the Earth, the particles that make up the ring current will be
exposed to a higher neutral density. This higher neutral density
will increase the effective charge exchange rate, or decrease the
effective charge exchange lifetime in (5), of the ring current.
The final piece of the puzzle is the fact that the dawn-dusk
electric field is related to VBs [Reiff et al., 1981]. We can
follow this line of reasoning to determine the functional form of
the -VBs relationship, and then we can fit that form to the
curve we have derived from our data analysis.
The following derivation roughly follows that of Southwood
and Kaye [1979]. We begin the derivation by assuming that a
given particle convects on contours of constant total energy, a
combination of electrostatic potential energy and kinetic energy.
O’Brien and McPherron Phase Space Analysis of Dst
Page 5
We will perform this derivation for an equatorially mirroring
particle with kinetic energy given by:
W  W  B
(24)
Here,  is the dipole moment of the gyrating particle and B is
the local magnetic field strength. We use the dipole magnetic
field approximation:
B
B0
L3
(25)
B0 is the magnetic field at the surface of the Earth and L is
radial distance measured in Earth Radii. We employ a simple
electric field E according to Volland [1979] and Stern [1977]:
E   E 0 RE Lp sin 
(26)
E0 is the field strength,  is the angular local time, and p is a
shielding parameter. When there is no electric field shielding by
the plasmasphere, p is 1. A value for p of 2 may also be
appropriate. We will continue the derivation without specifying
p in order to maintain generality.
We now calculate an effective electrostatic potential , which
is the total energy per charge q:
   E 0 RE Lp sin  
B0
qL3
(27)
Equatorially mirroring particles drift on contours of constant .
In order to calculate the drift boundary, we calculate the critical
point Ls of  with the magnetic moment held constant:
3B0
  
0

  pE 0 R E L p 1 
 L  
qL4
(28)
Solving this equation for Ls, after substituting in for , we have:
 3W 
Ls  

 qpE 0 R E 
1/ p
(29)
Physically Ls is a point on the dawn side of the Earth at which
equatorially mirroring ions do not drift. This is known as the
stagnation point, and it is part of the convection or trapping
boundary. We will use this point as a representative of the entire
trapping boundary. Next, we must replace E0 with some relation
to VBs. We can assume that the convection electric field is
proportional to the polar-cap potential drop, and then use the
relation from Reiff et al. [1981] that relates the polar cap
potential drop to VBs:
E 0  PC  a 0  a1VBs
(30)
Normalized by a1, this gives rise to an overall dependence of:
Ls  (a'VBs) 1/ p
(31)
Now we can combine this result with (5) and (6) to achieve an
equation for  as a function of VBs:
  e ( a ' VBs)
1/ p
(32)
Here,  is a constant resulting from the combination of the
preceding equations. We have performed a least-squared-error
fit of the measured  for VBs values between 0 and 9 mV/m.
The database was too sparse to construct phase-space density
plots with median VBs values above 9 mV/m, therefore the
2/6/2016 2:42:00 AM v.2
points above 9 mV/m are suspect due to extreme extrapolation
in the estimation technique described in (18)-(21) above. We
performed the fit for various values of the shielding parameter p
and found p = 1, no shielding, provides the best fit to the data.
This fit is provided in Figure 6:
 (hours)  2.40e 9.74 /( 4.69VBs)
(33)
We are quite satisfied with the quality of this fit. Statistically we
can say that the likelihood that this fit occurred by chance is less
than 0.01%. However, the statistical argument does not remove
the possibility that a different physical mechanism creates the
observed relationship between  and VBs. In this fit, VBs is
given in mV/m. The value a’ = 4.69 mV/m we have achieved
through our fit disagrees with the value of 1.1 provided by Reiff
et al. [1981]. One physical implication of our fit relative to that
of Reiff et al. [1981] is a weaker transmission of the
interplanetary electric field into the magnetosphere than into the
polar cap region.
COMMENTS ON VBs CONTROL HYPOTHESIS
We have suggested a physical mechanism that fits the
observed data. However, we have not provided any collateral
physical evidence that the mechanism is correct. We would like
to suggest ways to test this hypothesis in more detail and we
would also like to discuss the physical implications of the fit we
have obtained.
First, we address the issue of testability. The mechanism we
have described depends specifically on the variation of nH with
altitude. We mentioned that the scale height L0 varies with the
temperature of the exosphere. The temperature of the exosphere
varies with solar cycle. We can, therefore construct a test of
how the fit parameters change with solar cycle. Another method
of testing our hypothesis is Energetic Neutral Atom, ENA,
imaging. ENAs are the byproducts of charge exchange, and can
be used to create images of charge exchange loss throughout the
magnetosphere [Roelof, 1987; Jorgensen et al., 1997]. If the
charge exchange lifetime of ring current particles is indeed
shorter for increased VBs because the ring current is closer to
the Earth, we should see the effects in the ENA image.
Specifically, more ENAs should originate from closer to the
Earth at times of enhanced VBs compared to times of weak
VBs. This correlation would, however, be influenced by Dst
because there would be more charged source particles at larger
Dst. These tests may be within reach of our current data
resources.
Second, we would like to address some of the immediate
consequences of the fit we have performed. Spacecraft have
observed that the ring current composition changes during
storms [Hamilton et al., 1988]. Specifically, Oxygen makes up a
larger fraction of the total energy budget in the ring current for
the most intense values of Dst. This leads us to conclude that,
since we do not have to account for this compositional change
in our fit of  to VBs, the effective charge exchange cross
section  does not change significantly with composition. This
conclusion is certainly counterintuitive, and leaves the door
O’Brien and McPherron Phase Space Analysis of Dst
Page 6
open for a compositional, rather than convective, mechanism to
explain the dependence of  on VBs.
Scientists have observed that the decay time  is shorter at
larger Dst, but we do not find any  dependence on Dst. We will
show in the next section that the apparent dependence of  on
Dst is actually an alias of the typical concurrence of large VBs
and large Dst. This result will affect all alternate models for the
-VBs relationship.
Finally, we note that a dependence of the decay time on the
cross-tail electric field suggests that our calculation of the offset
might be influenced by the quiet time ring current. That is, the
quiet time ring current is created by the quiet time cross-tail
electric field: its intensity is determined by the balance between
quiet time input and charge exchange decay. The quiet time ring
current is removed in the calculation of Dst, which is why,
generally, Q does not include the quiet time term a’ above.
However, if the decay lifetime suddenly drops due to a non-zero
VBs, the quiet time ring current will decay away from its value
for VBs = 0, while simultaneously new energy is being injected
by the nonzero VBs. The decay of the quiet time ring current
and the new injection both contribute to the phase-space offset,
and therefore our measured Q-VBs relation. As it turns out, the
reciprocal of  is nearly linear for VBs < 5 mV/m, and so it just
adds another small linear term to the measured Q. This term, in
turn contributes to the observed offset Ec in the Q-VBs relation.
We do not know if this term entirely explains E c or merely
contributes to it.
RECREATION OF DECAY DEPENDENCE ON DST
We now turn to directly assessing why we observe an
apparent -Dst relationship when our analysis suggests that
there is indeed no such relationship. Having demonstrated that
the Burton Equation is correct if the -VBs relationship is
accommodated, we will use it to demonstrate how a -Dst
relationship might arise.
If we solve the discrete Burton Equation for , we have:

Dst * 
  Dst Q 
t 

*
1
(34)
This equation is disastrously unstable, considering that typical
Dst values are 3 nT in an hour but that the uncertainty in Dst is
typically at least 8 nT. This error gives rise to values of  which
are infinite and negative. It is impossible to accommodate the
divergence of this equation through direct averaging. A more
stable technique for calculating  is linear regression fitting of
Dst to Dst as in (8). The slope of this fit is given by (10).
However, if this method is not performed for restricted values of
VBs, then the calculated value of  will depend on the range of
VBs values that were used in the fit. This gives rise to the
typically observed  dependence on Dst because of the
coincidence of large VBs and large Dst. In other words, 
appears to depend on Dst because large Dst is correlated with
large VBs, and  does depend on VBs. Our phase-space analysis
2/6/2016 2:42:00 AM v.2
performed above is highly analogous to this least-squares fitting
technique, except that by fixing VBs, we were able to remove
the aliasing created by the coincidence of large VBs and large
Dst. Figure 7 demonstrates the effects of using various ranges of
Dst in calculating . This analysis requires the use of very large
bins in Dst because we are fitting to Dst in each bin, and the
trend would be lost in the noise if small bins were used. Panel a
shows the usual result: if larger values of Dst are used, a shorter
decay time is calculated. However in Panel b, it is clear that
when VBs is restricted, the same  is calculated regardless of
what range of Dst values are used. This proves that the
dependence of  on Dst is an alias effect.
Formally, the conceptual error is seen in the partial derivatives
that make up the -Dst relationship:
d
  
    VBs 

 
 

dDst  Dst  VBs  VBs  Dst  Dst  
(35)
The total derivative is what we typically measure, and it is
typically assumed that the right-hand side of the equation is
dominated by the partial with fixed VBs. However, we have
shown that this term is actually zero, and that the measured
variation is due to the second term. The partial with fixed Dst is
clearly not zero in Figure 7: when the same range of Dst are
used to calculate  but VBs is fixed to a different value, we
calculate a different . The partial with fixed  is an odd
quantity, since the relationship between Dst and VBs is given in
terms of the time change of Dst rather than Dst itself. However,
because Dst and VBs generally change smoothly over several
hours, there is a residual correlation between Dst and VBs. In
our dataset, the statistical correlation coefficient of Dst and VBs
is, in fact, -0.42. This analysis clarifies how  appears to depend
on Dst when it actually only depends on VBs.
APPLICATION TO TWO SAMPLE STORMS
At this point we have completed a thorough analysis of the
qualitative and quantitative aspects of our model in a statistical
sense. However, it is necessary at this time to turn to modeling
storms as they occur—individually. In the interest of brevity, we
will model only two sample storms, one very large storm and
one small storm. It is important to note that our model is only a
single-step model of Dst, and that it makes no corrections for
missing solar wind data. We have chosen two storms for which
there is good solar wind coverage. We have chosen the
moderate storm that occurred on October 11, day 285, of 1980,
and the large storm that occurred on March 2, 1982. Figure 8
compares our model (solid line) to the standard Dst index (dots)
for these two storms. It is obvious that our model does quite
well for one-hour steps: for the smaller storm, the prediction
efficiency is 97.2%, and 97.7% for the larger storm. It should be
noted, however, that these measures of prediction efficiency are
misleading because persistence, a model that assumes Dst does
not change at all in an hour, provides upwards of 95%
prediction efficiencies. The skill score, which measures how
much more of the variation in Dst is provided by our model
relative to persistence, gives a better idea of the performance of
a one-hour step model. The skill scores are 36.7% and 31.6%
O’Brien and McPherron Phase Space Analysis of Dst
Page 7
for the smaller and larger storms, respectively. Effectively, our
model accounts for about a third of the hourly change in Dst.
The remaining portion is assumed to be related to the error
sources described above.
At the peak of the large storm, there is presumably a larger
component of O+ in the ring current energy budget. If the
compositional change caused a change in decay rate, we would
see our model consistently overestimating the decay time, which
would result in one-hour step model values of Dst too far from
zero. However, there is no such consistent error, indicating that
intense Dst is not the source of rapid decay time.
We provide additional details of our model performance in
Figure 9, which shows how the model errors relate to Dst. It is
clear that the model commits a few large errors for each storm,
committing more large errors for the larger storm. However, the
large errors do not occur consistently at large Dst, indicating
that the model is performing well in that region. Again, this
shows that the presumed compositional change at large Dst does
not significantly alter the evolution of the Dst index.
In an operational setting, ground-based estimates of Dst are
typically not available until hours, if not days, behind real-time.
Solar Wind data, however, are now available in real time
through the Advanced Composition Explorer Real-Time Solar
Wind service from the Space Environment Center
(http://sec.noaa.gov/ace/ACErtsw_home.html). Therefore, an
operational application of our model would require the use of
multi-step evolution from the last available Dst measurement
forward in time to the latest available solar wind measurements.
That is, from an initial Dst value provided from ground
measurements taken several hours in the past, we could use
more recent hourly spacecraft measurements of solar wind to
advance Dst to the present time. In this case, after the last hour
when ground-based Dst data is available, we would have to use
the model output from the previous hour as the starting Dst
value. This recursive use of our model can be a significant
source of error. We have simulated this process in Figure 8 as
the multi-step model (dashed line). It is apparent that the trace
of the model in multi-step mode only separates from Dst when
either Dst experiences new injection without an accompanying
increase in VBs or there is a data gap. The resulting prediction
efficiencies are 80.7% for the smaller storm and 87.6% for the
larger storm. The viability analysis of our model in an
operational setting has not been performed, but the improved
definition of  should yield improved performance over existing
models that do not include such a -VBs dependence.
CONCLUSIONS
We have shown that the Burton Equation, with only slight
modification, does accurately describe the dynamics of the ring
current index Dst. We have calculated the injection function Q
in terms of the interplanetary electric field, VBs. Q is linear in
VBs with a cutoff for small VBs. We have also calculated a
pressure correction coefficient and a small offset correction for
Dst. Our model provides values similar to those of Burton et al.
[1975].
2/6/2016 2:42:00 AM v.2
We have shown that the deviations from the deterministic
behavior are probably not Gaussian. Non-Gaussian error
distributions suggest that the standard techniques for handling
noise, specifically least-squared-error fitting, may not be
appropriate. We have used medians whenever possible as a
more stable alternative to means.
We have demonstrated that the decay time of Dst does not
explicitly depend on Dst itself. Instead, it depends on VBs, a
proxy for either the injection rate or the convection electric
field. We have been able to reproduce the observed dependence
of  on Dst as a consequence of the coincidence of large VBs
and Dst.
We have suggested one mechanism that can produce the
observed functional dependence of  on VBs. This mechanism
stipulates that the recovery rate is increased for larger VBs
because the ring current is confined to lower altitudes by the
convection electric field. The increased neutral density at these
lower altitudes gives rise to a shorter effective charge exchange
lifetime. Although compositional changes are observed by
spacecraft [Hamilton et al., 1988], it has not been necessary to
explicitly account for such changes in our model. This suggests
that the effective charge exchange cross section of the ring
current does not depend significantly on composition in the
context of our mechanism. We have also suggested some ways
in which our mechanism can be independently tested.
Over the years, models of Dst have become increasingly
complex. Some of this may be attributable to the problem of
determining the general dynamics of a noisy system from a
small set of events. This study, unlike many of its predecessors,
uses a large database of ring current and solar wind parameters,
covering hundreds of storms. Any study of individual storms is
highly susceptible to the uncertainty inherent in the Dst index.
We have shown, however, that all available Dst data can be
organized according to our slight modification of the Burton
Equation. That is, allowing the decay time  to vary with VBs is
the only modification necessary. This study marks a rare step
backward in the complexity of ring current modeling, and we
have done so without employing any highly sophisticated
mathematics.
In the interest of full disclosure, we must admit that the
simplicity of our results and of our analysis technique is
somewhat unintended. When we began our study we intended to
apply some sophisticated probabilistic modeling techniques to
the time evolution of Dst. However, even from the beginning of
our analysis, it was apparent that complexity is unwarranted.
Simplicity has clearly won the day, and the sophisticated
mathematics will have to wait for another problem.
ACKNOWLEDGEMENTS
This research was graciously funded by NSF grant
ATM 96-13667. We would like to thank Larry Kepko, Chris
Russell, Vytenis Vasyliunas, and our colleagues at UCLA for
their helpful insights and discussions in preparing this work.
O’Brien and McPherron Phase Space Analysis of Dst
Page 8
REFERENCES
Akasofu, S.-I., S. Chapman, and D. Venkatesan, The main
phase of great magnetic storms, J. Geophys. Res., 68, 33453350, 1963.
Alexeev, I.I., E.S. Belenkaya, V.V. Kalegaev, Y.I.Feldstein, and
A. Grafe, Magnetic storms and magnetotail currents, J.
Geophys. Res., 101, 7737-7747, 1996.
Burton, R.K., R.L. McPherron, and C.T. Russell, An empirical
relationship between interplanetary conditions and Dst, J.
Geophys. Res., 80, 4204-4214, 1975.
Campbell, W.H., Geomagnetic storms, the Dst ring-current
myth and lognormal distributions, J. Atmos. Terr. Phys., 58,
1171-1187, 1996.
Clauer, C.R., R.L. McPherron, and C. Searls, Solar wind control
of the low-latitude asymmetric magnetic disturbance field, J.
Geophys. Res, 88, 2123-2130, 1983.
Daglis, I.A., R.M. Thorne, W. Baumjohann, and S. Orsini, The
Terrestrial Ring Current: Origin, Formation, and Decay, Revs.
Geophys.(in press), 1998.
Dessler, A.J., and E.N. Parker, Hydromagnetic theory of
magnetic storms, J. Geophys. Res., 64, 2239-2259, 1959.
Feldstein, Y.I., V.Y. Pisarsky, N.M. Rudneva, and A. Grafe,
Ring current simulation in connection with interplanetary
space conditions, Planet. Space Sci., 32, 975-984, 1984.
Gonzalez, W.D., J.A. Joselyn, Y. Kamide, H.W. Kroehl, G.
Rostoker, B.T. Tsurutani, and V.M. Vasyliunas, What is a
geomagnetic storm?, J. Geophys. Res., 99, 5771-5792, 1994.
Gonzalez, W.D., B.T. Tsurutani, A.L. Gonzalez, E.J. Smith, F.
Tang, and S.-I. Akasofu, Solar wind magnetosphere coupling
during intense magnetic storms (1978-1979), J. Geophys.
Res., 94, 8835-8851, 1989.
Hamilton, D.C., G. Gloeckler, F.M. Ipavich, W. Studemann, B.
Wilken, and G. Kresmer, Ring current development during
the great geomagnetic storm of February 1986, J. Geophys.
Res., 93, 14,343-14,355, 1988.
Hogg, R.V., and A.T. Craig, Introduction to Mathematical
Statistics, 564 pp., Prentice Hall, Englewood Cliffs, New
Jersey, 1995.
Jorgensen, A.M., H.E. Spence, H.E. Henderson, M.G. Reeves,
M. Sugiura, and T. Kamei, Global energetic neutral atom
(ENA) measurements and their association with the Dst
index, Geophys. Res. Lett, 24, 3173-3176, 1997.
Kamide, Y., N. Yokoyama, W.D. Gonzalez, B.T. Tsurutani, I.A.
Daglis, A. Brekke, and S. Masuda, Two-step development of
geomagnetic storms, J. Geophys. Res., 103, 6917-6921, 1998.
King, J., Interplanetary Medium Data Book, NASA GSFC,
Greenbelt, MD 20771, 1977.
Press, W.H., B.P. Flannery, S.A. Teukolsky, and W.T.
Vettering, Numerical Recipes, Cambridge University Press,
New York, 1986.
Prigancova, A., and Y.I. Feldstein, Magnetospheric storm
dynamics in terms of energy output rate, Planet. Space Sci.,
30, 581-588, 1992.
Pudovkin, M.I., A. Graffe, S.A. Zaitseva, L.Z. Sizova, and A.V.
Usmanov, Calculating the Dst-variation field on the basis of
2/6/2016 2:42:00 AM v.2
solar wind parameters, Gerlands Beitr. Geophys., 97, 525533, 1988.
Pudovkin, M.I., S.A. Zaitseva, and L.Z. Sizova, Growth rate and
decay of magnetospheric ring current, Planet. Space Sci., 33,
1097-1102, 1985.
Reiff, P.H., R.W. Spiro, and T.W. Hill, Dependence of polar
cap potential drop on interplanetary parameters, , 86, 76397648, 1981.
Roelof, E.C., Energetic neutral atom imaging of a storm-time
ring current, Geophys. Res. Lett, 14, 652-655, 1987.
Sckopke, N., A general relation between the energy of trapped
particles and the disturbance field over the earth,, , 71, 31253130, 1966.
Smith, P.H., and N.K. Bewtra, Charge exchange lifetimes for
ring current ions, Space Sci. Revs., 22, 310-318, 1978.
Southwood, D.J., and S.M. Kaye, Drift Boundary
Approximations in Simple Magnetospheric Convection
Models, J. Geophys. Res., 84, 5773-5780, 1979.
Stern, D.P., Large-scale electric fields in the earth's
magnetosphere, , 15, 156-194, 1977.
Sugiura, M., and T. Kamei, Equatorial Dst index 1957-1986,
ISGI Publications Office, Saint-Maur-des-Fosses, France,
1991.
Vassiliadis, D., A.J. Klimas, and D.N. Baker, Models of D st
geomagnetic activity and of its coupling to solar wind
parameters, in Symposium on Solar-Terrestrial Coupling
Processes, , Paros, Greece, 1997.
Vasyliunas, V.M., A method for evaluating the total
magnetospheric energy output independently of the epsilon
parameter, 14, 1183-1186, 1987.
Volland, H., Semiempirical models of magnetospheric electric
fields, in Quantitative Modeling of Magnetospheric
Processes, edited by W. P. Olson, American Geophysical
Union, Washington, D.C., 1979.
O’Brien and McPherron Phase Space Analysis of Dst
Page 9
Captions
Figure 1. Panels a-c show the probability density in phase space for 3 values of VBs. The function is normalized so that horizontal
bands integrate to 100%. It is clear that the trajectories are lines, but the slope and offsets of these lines change with VBs. The
rectangle in the upper right-hand corner indicates the bin size. The stability line Dst = 0 is also provided. A linear fit to the median
values (x) is provided.
Figure 2. The KS test is used to determine if the distribution of Dst is Gaussian. The significance level is the probability that the
observed difference between the Dst distribution and a Gaussian is real rather than an artifact of finite sample size. The asterisks *
indicate occasions when the sample size for the KS test was smaller than 20, where very low significances are poorly determined.
Figure 3. Injection Q versus VBs. Q is calculated from a set of linear fits to the phase-space trajectories indicated by the probability
density P(Dst|Dst,Q). Q is clearly linear in VBs beyond the injection cutoff E c. Some data were not used in determining the best
fit: low values of VBs were removed due to the injection cutoff, and high values of VBs were removed due to the scarcity of data
beyond VBs = 9 mV/m.
Figure 4. Decay time  versus VBs.  is calculated from a set of linear fits to the phase-space trajectories indicated by the probability
density P(Dst|Dst,Q). is clearly nonlinear in VBs.
Figure 5. Residual phase-space offset is plotted against changes in the square root of the solar wind dynamic pressure P. Here we
have removed the part of the phase-space offsets due to Q. We perform a linear fit of the residual to P1/2. This reveals the coupling
parameter b.
Figure 6. The least squares fit of decay time (VBs) according to the trapping boundary hypothesis.
Figure 7. Panels a and b show the decay time  calculated for various ranges of Dst using least-squared-error fits of Dst to Dst.
Panel a shows  calculated without specification of VBs. Panel b shows the same calculation with VBs fixed at three different
values. In Panel a, the often-observed -Dst dependence is reproduced because calculations which include larger values of Dst result
in shorter decay times. However, in Panel b, it is clear that fixing VBs obliterates that dependence because at any fixed value of VBs
the same  is calculated for any range of Dst. The endpoints of the lines indicate the Dst range used and the point in the middle of
each line indicates the median Dst value in that range.
Figure 8. Panels a and b show the Dst track and VBs for two storms of different sizes. Storm 1980-285 is a moderate storm and
storm 1982-061 is a large storm. The dots indicate actual Dst measurements. The solid line indicates the one-hour step model. The
dashed line is obtained by using the model recursively starting with Dst at epoch time zero: standard Dst at epoch time zero is used,
but thereafter Dst is calculated only from model results and the solar wind data. Gaps in the solar wind data are indicated by the
absences of Dst dots. It is clear that the model does quite a good job matching Dst for both magnitudes of storms. The model
performs similarly on other storms for which solar wind data is generally available.
Figure 9. Panels a and b show the errors of the one-hour step model. The solid line connects points that are adjacent in time. The
fact that the errors are clustered around zero with a few outliers indicates that the model is performing well. The fact that subsequent
errors alternate in sign indicates that the hour boundary problem is important. There is no evident systematic error for the intense
portion of storm 1982-061; this contradicts the notion that large storms have special dynamics at large Dst.
P(Dst|Dst,Q) for Dst transitions (VBs = 0)
0
2.8%
88.6%
-50
0.9%
Dst
3.5%
-100
Dst = (-0.06)Dst+(0.1)
-150
-40
-30
-20
-10
a)
0
10
20
Dst(t+1hrs)-Dst(t)
30
40
50
P(Dst|Dst,Q) for Dst transitions (VBs ~ 2 +/- 1.0 mV/m)
0
35.8%
67.9%
Dst = (-0.07)Dst+(-4.2)
-50
Dst
1.0%
20.7%
-100
6.6%
-150
-30
-20
-10
b)
0
10
Dst(t+1hrs)-Dst(t)
20
30
40
P(Dst|Dst,Q) for Dst transitions (VBs ~ 4 +/- 1.0 mV/m)
0
11.5%
44.6%
4.8%
-50
Dst
8.3%
-100
9.0%
Dst = (-0.12)Dst+(-13.1)
-150
-50
c)
-40
-30
-20
-10
0
10
Dst(t+1hrs)-Dst(t)
20
Figure 1
30
40
K-S Test that  Dst is Gaussian (VBs ~ 2 +/- 1.0 mV/m)
1
0.9
Significance of K-S Test
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-160
-140
-120
-100
-80
-60
Dst (nT)
Figure 2
-40
-20
0
20
Injection (Q) vs VBs
10
0
Offsets in Phase Space
Points Used in Fit
Q = (-4.4)(VBs-0.49)
Injection (Q) (nT/h)
-10
-20
-30
-40
-50
-60
Ec = 0.49
-70
-80
0
2
4
6
VBs (mV/m)
Figure 3
8
10
12
Decay Time () vs VBs
20
18
16
 (hours)
14
12
10
8
6
4
2
0
2
4
6
VBs (mV/m)
Figure 4
8
10
12
(Phase-Space Offset) - Q vs  [P1/2]
6
4
(PS Offset) -Q (nT/h)
2
0
-2
-4
(PS Offset) - Q
Best Fit ~ (7.26) [P1/2]
-6
-8
-10
-12
-0.6
-0.4
-0.2
0
0.2
 [P1/2] (nPa1/2/h)
Figure 5
0.4
0.6
0.8
Decay Time () vs VBs
20
 from Phase-Space Slope
18
Points Used in Fit
 = 2.40e9.74/(4.69+VBs)
16
 (hours)
14
12
10
8
6
4
2
0
2
4
6
VBs (mV/m)
Figure 6
8
10
12
 for various ranges of Dst (without specification of VBs)
 for various ranges of Dst (with specification of VBs)
20
20
All VBs
VBs = 0
VBs = 2
VBs = 4
18
16
16
14
14
 (hours)
 (hours)
18
12
12
10
10
8
8
6
6
4
4
-200
-150
-100
-50
Dst Range (nT)
0
-200
a)
-150
-100
-50
Dst Range (nT)
b)
Figure 7
0
Dst Comparison for storm 1980-285
Dst Comparison for storm 1982-061
20
50
0
0
-50
Dst (nT)
Dst (nT)
-20
-40
-60
Dst
Model (1hr step)
Model (multi-step)
VBs
-80
-100
Dst
Model (1hr step)
Model (multi-step)
VBs
-100
-150
-200
-120
-250
0
50
100
150
0
6
20
40
60
80
100
120
140
160
180
15
4
VBs mV/m
VBs mV/m
5
3
2
5
Ec = 0.49 mV/m
1
Ec = 0.49 mV/m
0
0
0
50
100
150
0
Epoch Hours
a)
10
b)
Figure 8
20
40
60
80
100
Epoch Hours
120
140
160
180
Dst Transitions for 1982-061
Dst Transitions for 1980-285
50
20
0
Error
VBs > E c
VBs > 5
0
-20
-50
Dst
Dst
-40
Error
VBs > E c
VBs > 5
-100
-60
-150
-80
-200
-100
-120
-50
a)
-40
-30
-20
-10
0
10
Error: Model-Dst (nT)
20
30
40
-250
-50
50
b)
Figure 9
-40
-30
-20
-10
0
10
Error: Model-Dst (nT)
20
30
40
50