The Validity of the Frozen-in Condition for the Inner Magnetosphere
Poster: XY681
Abstract: EGU2012-13181
Session: ST2.3
1,2
3
R. J. Strangeway, J. Raeder
1Institute of Geophysics and Planetary Physics, UCLA, 2Department of Earth and Space Sciences, UCLA
3Space Science Center, University of New Hampshire
(4)
If we assume that the flow is barotropic, i.e.,
hand side of (4) vanishes. Further, defining vorticity as
, then the last term on the right, then
(5)
If
then the fluid vorticity can be ignored. This condition can be rewritten as
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Energy
(eV)
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Acknowledgements
This work was supported by the RBSP mission, NASA contract NAS5-01072, through
a subcontract from the University of Minnesota and J. Wygant RBSP/EFW Principal
Investigator. Simulation results have been provided by the Community Coordinated
Modeling Center at NASA Goddard Space Flight Center.
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Parker, E. N., Newtonian development of the dynamical properties of ionized gases of
low density, Phys. Rev., 107(4), 924-933, doi:10.1103/PhysRev.107.924, 1957.
Le, G., C. T. Russell, and K. Takahashi, Morphology of the ring current derived from
magnetic field observations, Ann. Geophys., 22(4), 1267-1295, 2004.
Siscoe, G. L., in Solar-Terrestrial Physics, Principles and Theoretical Foundations,
edited by R. L. Carovillano and J. M. Forbes, pp. 11-100, D. Reidel, Norwell, Mass.,
1983.
Strangeway, R. J., Comment on “Pressure gradient effect on a particle velocity
distribution” by A. T. Y. Lui, Geophys. Res. Lett., 37, L10105,
doi:10.1029/2009GL042001, 2010.
Wolf, R. A., The quasi-static (slow-flow) region of the magnetosphere, in SolarTerrestrial Physics, Principles and Theoretical Foundations, edited by R. L.
Carovillano and J. M. Forbes, pp. 303-368, D. Reidel, Norwell, Mass., 1983.
dB_fac
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fieldaligned
current
(mA/m2)
References
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ALT
ILAT
MLT
Y (Re)
-5
-10
X (Re)
-5
-10
-15
0
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Pressure and Electric Field
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X (Re)
-5
0
Pressure and Modified Electric Field
5
0
-5
-10
-15
0
-5
b
o
e
04:00
3008.4
-72.5
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04:02
3124.9
-70.7
0.5
04:04
04:06
3231.9
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-68.5
-66.3
1.0
1.3
Hours from 2008-12-23
04:08
3415.7
-64.0
1.6
04:10
3492.0
-61.8
1.9
Figure 1. FAST data acquired in the southern hemisphere on December 23, 2008.
-10
X (Re)
-5
0
-10
-15
-10
X (Re)
-5
0
Figure 4. Electric field vectors for frozen-in ions (left-hand side) and frozen-in electrons
(right-hand side). A vector of length 1 Re corresponds to an electric field of 1mV/m.
Electron
current
(mA/m2)
Ions
Angle
(deg)
The colors show the common terms in the two equations. For example, the magenta
term in (1) corresponds to the magenta term in (2). Importantly, the grad-B dependent
terms (shown in red) in (2) cancel. Furthermore, if the pressure is isotropic, then the
curvature terms cancel.
The current in the plasma is related to the pressure gradient, not the gradient and
curvature drifts. Le et al. [2004] show that the current at the inner edge of the ring current
is eastward, while the guiding center drifts are westward.
FAST Orbit 49871
0
Figure 3. Inner magnetosphere pressure contours (at left) and density contours (at right),
overlaid with plasma flow velocity vectors, as derived from the MHD simulation. The
time interval corresponds to the simulation results shown in Figure 2.
It is clear that there are some significant differences in both the divergence and curl of
the electric field depending on whether the magnetic field is frozen to the ion or electron
fluid, but it should be remembered that this is a 2-dimensional cut of a 3-dimensional
system. Nevertheless, these differences could affect the evolution of the system.
Differences in the divergence may change the current pattern in the ionosphere, and so in
turn modify the currents in the magnetosphere. Differences in the curl will directly affect
the temporal evolution of the magnetic field, through Faraday's law.
Log
eV/cm2
/s/sr/eV
(1)
and it is this quantity that is frozen-in to the fluid.
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Density and Velocity
5
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Ion Versus Electron Frozen-in Condition
Figure 4 shows the electric field derived from the fluid flow velocity (i.e., the classical
frozen-in approximation) on the left-hand side, and the electric field derived from the
electrons, assuming they are cold (i.e., including the Hall term). For comparison with
Figure 3 the pressure contours are also shown.
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We note that sometimes the flows are diverted by the pressure gradient and sometimes
they appear to be unaffected by the pressure gradient. In addition, the plasma and density
contours are not always aligned. The flows are not barotropic, suggesting that the frozenin condition may have to include either pressure gradient or Hall terms.
Log
eV/cm2
/s/sr/eV
Parker [1957] first showed the equivalence between particle orbit theory and fluid
theory. Equation (1) show the current density per species based on guiding center drifts
plus the diamagnetic current from the curl of the magnetic moment. The first row of (2)
gives the electric field, gradient, and curvature drifts. The second row shows the drifts
from D/Dt of the electric and magnetic fields. The last row shows the magnetization
current.
This leads to the concept of generalized vorticity, or generalized magnetic induction (cf.
Siscoe [1983],
We can also use the simulation results to explore the validity of the frozen-in condition.
Electrons
Electrons Energy
Angle
(eV)
(deg)
2. Equivalence Between Fluid Theory and Drift Theory
The simulations can guide our interpretation of the data. Figure 2 shows radial currents
flowing into and out of the ionosphere, as color, with the corresponding electric potential
contours showing the flow pattern. The white line marks the open-closed field line
boundary. The polar ionosphere flows, currents and open-closed boundary show
significant structure. In particular there are regions near midnight where flows pass from
the polar cap to a region of closed field lines, back into the polar cap, and then onto
closed field lines. This could explain the structure in the FAST observations, including
the disappearance of the plasmasheet between the two regions of upward current.
Figure 3 shows simulation results covering the range -15 Re £ xGSE £ 0 Re, -10 Re £
yGSE £ 10 Re in the zGSE = 0 plane. On the left hand side the color contours show plasma
pressure, with red showing ≥ 0.8 nPa. The white arrows show flow velocity, scaled so
that an arrow with a length of 1 Re in the plot corresponds to 100 km/s. The z = 0 plane
does not correspond to the plasmasheet, as the plasmasheet is distorted. The data in the
upper left quadrant of the figure are on lobe field lines. The rest of the data are in the
plasmasheet. The right-hand figure shows flow velocity and density, with red
corresponding to 1 cm-3.
Pressure and Velocity
Y (Re)
From (3)
FAST was in the southern hemisphere and observed a double auroral oval, as can be
seen in Figure 1, where there are two bands of electron precipitation, one at ~ 04:02 UT,
and the other at 04:07 UT. FAST is clearly in the polar cap at 04:00 UT, and clearly in the
plasmasheet after 04:06 UT. But what happens in between?
Pressure Versus Density
Y (Re)
If the terms in the red rectangle are ignored, then (3) is equivalent to the ideal MHD
frozen-in condition. Even though the frozen-in condition comes from the electron
momentum equation, it is tempting to state that MHD does not include the last two terms
when considering the drift of the ions. But these terms are present in the single fluid
plasma momentum equation. Within the MHD context the frozen-in condition is used to
derive E from U, and further for MHD it is only
that is important.
10
Y (Re)
(3)
For isentropic flows P/r is a constant following the fluid flow. This is usually assumed
within MHD, and also within drift-physics models, strongly suggesting that flows are
also barotropic. This seems reasonable if we can assume that adjacent flux-tubes have
similar histories, but we shall nevertheless explore the validity of the frozen-in condition
with an MHD simulation. For this purpose we will use results from a simulation used to
explain some features in the structure of precipitating electron as observed by the FAST
spacecraft.
5. Simulation Results
Log
eV/cm2
/s/sr/eV
We show that provided flows are barotropic, then it can be shown that the frozen-in
condition is valid if the flow vorticity is much less than the ion gyro-frequency. But
comparisons with simulations show that flows may not be barotropic. This suggests that
the next step in reconciling MHD and models such as RCM may be to add the Hall term
to the frozen-in condition (i.e., frozen-in electrons).
The ion fluid momentum equation can be rewritten as an equivalent drift equation (3).
4. OpenGGCM Simulation and FAST data - Case Study
Log
eV/cm2
/s/sr/eV
It is often stated that ideal magnetohydrodynamics (MHD) ignores “drift physics,”
making it invalid in the inner magnetosphere where particle drifts can be large. However,
as we show here, there is a direct equivalence between fluid theory and drift theory, a
result that goes back to Parker [1957] (see also Strangeway [2010]). We consider that the
difference may instead be in the interpretation of the frozen-in condition. In MHD the
frozen-in condition is used to derive the electric field (E) given the fluid flow velocity
(U). But if we use the frozen-in condition to derive U given E then the additional drift
terms are required. Thus reconciling MHD and drift-physics models, such as the Rice
Convection Model (RCM, Wolf et al. [1983]) may be related to the accuracy of the
frozen-in condition.
3. MHD, Diamagnetic Drift and the Frozen-in Condition
Thu Apr 16 19:06:26 2009
1. Introduction
Figure 2. Simulation results showing field-aligned currents and flows for
comparison with the FAST data.
6. Summary and Conclusions
MHD does include drift physics in the plasma momentum equation. The guiding center
drift is exactly cancelled by the magnetic-field gradient term in the magnetization current,
with only the pressure gradient term remaining.
If flows are barotropic then the classical frozen-in condition applies unless the flow
vorticity is large.
Preliminary analysis indicates that flows may not be barotropic in the inner
magnetosphere. Why is this? Reconnection?
It may be possible to improve the MHD-RCM type of model comparisons by using the
frozen-in electron condition as this could affect both current closure in the ionosphere and
the temporal evolution of the magnetic field.