Numerical methods in Solar Wind Simulations

1
Numerical methods in Solar Wind Simulations
FENG Xueshang1 , WEI Fengsi1 ,S.T Wu2 , FAN
Quanlin1
Presented at WSEF 2002, Adelaide
1?Laboratory of Space Weather, Center for
Space Science and Applied Research, Chinese
Academy of Sciences 2?CSPAR, The University of
Alabama In Huntsville, AL 35899 USA
2
Contents

• Numerical MHD Models in Solar Wind modeling
• Numerical Methods in Solar Wind Simulations
• Conclusions

3
1. Numerical MHD Models in Solar Wind Simulations
In the past three decades, solar-terrestrial
physicists have introduced many kinds of
numerical schemes in computational fluid
mechanics to MHD system in order to simulate
various phenomena of solar-terrestrial physics.
Various works has help us understand the coronal
process, mechanism of CME,its propagation in
interplanetary space and the interaction between
solar wind and magnetosphere.
? Dryer, M., Multi-dimensional MHD simulation of
solar-generated disturbances space weather
forecasting of geomagnetic storms, AIAA Journal,
1998, 36 365 ? Wu, S.T., Andrews, M.D. and
Plunkett, S.P., Numerical MHD modeling of coronal
mass ejections(CMEs), Space Sci. Rev., 95(2001),
191-213
4
Numerical models used for transient phenomena may
consist

• Flare-bomb exploding models
• Coronal helmet-streamer models
• Flux-rope models
• Photospheric magnetic shear Model
• Beakout Model
• Flare-bomb exploding models by assuming the
  initial corona(i.e., pre-event state) be static
  with potential(i.e., current free) or force-free
  field
• Coronal helmet-streamer models uses the coronal
  helmet-streamers as the pre-event coronal state.
  This is a MHD equilibrium atmosphere
  configuration by advancing the ideal MHD through
  numerical relaxation.
• Flux-rope modelsThese models combine
  helmet-streamer and flux-rope as a pre-event
  configuration such that magnetic energy in the
  form of a detached magnetic field with
  cross-field currents can be used to fuel a CME.

5
Breakout CME model use a quadrupolar system as
initiation, where the inner part of the central
arcade are sheared by antiparallel footpoint
motions near the neutral line(equator). A runaway
eruption follows the bulge and reconnection
process of the magnetic field.
Typical Works about Flare-bomb exploding
models Nakagawa, Y., Wu, S.T. and Han, S.M.,
APJ, 1978, 219 314-323 Nakagawa, Y., Wu, S.T.
and Han, S.M., APJ, 1981, 244 331-339 Wu, S.T.,
Dryer, M., Nakagawa, Y., Han, S.M., APJ, 1978,
219 324-335 Wu, S.T., Nakagawa, Y., Han, S.M.
and Dryer, M., APJ, 1982, 262 369-376 Dryer, M.,
Wu, S.T., Steinolfson, R.S. and Wilson, R.M.,
APJ, 1979, 227 1059 As in the early stage of
such models, the numerical domain usually is put
in the supersonic superAlfvenic range of
solar-terrestrial system!
6
Typical Works about Coronal Streamer Models
Steinolfson, R.S. and Hundhausen, A.J., J.
Geophys. Res., 1988, 93 14267 Steinolfson,
R.S., J. Geophys. Res., 1992, 97 10811 Mikic,
Z., Barnes, D.C. and Schnack, D.D., Astrophys.
J., 1988, 328 830 Mikic, Z., Phys. Fluids B,
1990, 2 1450-1454 Linker, J.A., Van Hoven, G.
and Schnack, D.D., GRL, 1990, 17 2281 Wang,
A.H., Wu, S.T., Suess, S.T. et al., Solar
Physics, 1995, 161 365
7
Typical works about Flux rope models

• Chen, J. et al., Astrophys. J., 1997, 490
  L191-L194
• Chen, J. et al., Astrophys. J., 2000, 533
  481-500
• Low, B.C., Plasma Physics, 1994, 1 1684-1690
• Low, B.C. and Smith, D.S., Astrophys. J., 1993,
  410 412-425
• Guo, W.P. and Wu, S.T., Astrophys. J., 1998,
  494 419-429
• Wu, S.T., Guo, W.P. et al., Solar Physics, 1997a,
  175 719-735
• Wu, S.T., Guo, W.P. and Dryer, M., Solar Physics,
  1997b, 170 265
• Wu, S.T., Guo, W.P., Michels, D.J. and Burlga,
  L.F., JGR, 1999,
• 104 14789
• Wu, S.T., Guo, W.P. and Wang, J.F., Solar
  Physics, 1995, 157 325

8
Typical works about Photospheric shear model

• Lionello, R.Z., Mikic, Z. and Linker, J.A.,
  Magnetohydrodynamics of solar coronal plasmas in
  cylindrical geometry,Journal of Computational
  Physics, 1998, 140 172-201
• Mikic, Z., Linker, J.A., Schnack, D.D., Lionello,
  R. and Tarditi, A., Magnetohydrodynamic
  modeling of the global solar corona, Phys.
  Plasmas, 1999, 6 2217-2224
• Riley, P., Linker, J.A., Mikic, Z. and Lionello,
  R., MHD modeling of the solar corona and
  heliosphere Comparison with observations,
  preprint

Works about Breakout Model Antiochos, S.K., C.R.
Devore, and J.A. Klimchuk, A Model for solar
coronal mass ejections, APJ, 510(1999), 485-493
9
By using the above mentioned numerical models,
there are also many works on space weather
events/observational studies. To say a few,
A) S.T. Wu and his co-workers selected three CME
events to numerically reproduce their
observations by LASCO/SOHO by introducing sound
mechanisms of the initiation processes with
two-dimensional MHD simulations.

• Wu, S.T., Guo, W.P. and Dryer, M.,Dynamical
  evolution of a coronal streamer-flux-rope system,
  II. A self-consistent non-planar
  magnetohydrodynamic simulation, Solar Physics,
  1997b, 170 265
• Wang, A.H., Wu, S.T., Suess S.T. and Poletto, G.,
  Global model of the corona with heat and momentum
  addition, J. Geophys. Res., 1998, 103 1913-1922
• Wu, S.T., Wang, A.H., Plunkett, S.P. and
  Michels, D.J.,Evolution of global-scale coronal
  magnetic field due to magnetic reconnection the
  formation of the observed blob motion in the
  coronal streamer belt, Astrophys. J., 2000a, 545
  1101-1115
• Wu, S.T., Guo, W.P., Plunkett, S.P., Schmider,
  B. and Simnett, G.M., Coronal mass
  ejections(CMEs) initiation models and
  observations, Journal of Atmospheric and
  Solar-Terrestrial Physics, 2000b, 62(16)
  1489-1498

10
B) Linker, Mikic and their co-workers simulated
the global solar corona by using observed
photospheric magnetic fields as a boundary
condition and interpret some solar observations,
including eclipse images of the corona, Ulysses
spacecraft measurements of large-scale
interplanetary magnetic field, and extreme
ultraviolet images from SOHO.

• Linker, J.A., Mikic, Z., Bisecker, D.A., et al.,
  J. Geophys. Res., 1999, 1049809-9830
• Mikic, Z. and Linker, J.A., The large-scale
  structure of the solar corona and inner
  heliosphere, in Solar Wind Eight, edited by D.
  WInderhalter et al., AIP Conf. Proc., 382(1996),
  104
• Mikic, Z., Linker, J.A., Schnack, D.D.,
  Lionello, R. and Tarditi, A.,, Phys. Plasmas,
  1999, 6 2217-2224
• Riley, P., Bame, S.J., Barraclogh, B.L., et al.,
  Adv. Space Res., 1997, 20 15
• Riley, P., Gosling, J.T., McComas, D.J., et al.,
  J. Geophys. Res., 1999, 104 9871-9879

11
C)IMF Bz
In space weather event studies, the prediction of
IMF Bz is very important since the long duration
southward interplanetary magnetic field(IMF) in
solar magnetospheric coordinate system(GSM),
usually called -Bz or Bs play a crucial role in
determining the amount of solar wind energy to be
transferred to the magnetosphere. Thus,
understanding the causes of and predicting the
length and strength of the large southward IMF Bz
are key goal of knowing the occurrence of intense
geomagnetic storms.
12
Shi Yong, Fengsi Wei and Xueshang Feng.
Numerical study of Bz of interplanetary magnetic
fields in the period of January 1997 event.
Science in China (A), 3061-64, 2001.
Here, a procedure for modelling the southward
IMF Bz by using 3D-time dependent MHD equations
with McCormack difference scheme is proposed by
using near real initial-boundary conditions
constructed from the source surface magnetic
field observation. As an example, the
propagation and evolution of the January 1997
interplanetary CME are numerically studied using
this 3-D MHD model. The numerical results show
that the parameters obtained near the earth are
in agreement with observations of WIND satellite,
especially the temporal behavior Bz near the
earth.
13
Other quantitative studies of southward IMF Bz
are made by Wu, C. C., M. Dryer, S.T. Wu, L.H.
Lyu. Recipe for predicting the IMF Bz polarity's
change of direction following solar disturbances
and at the onset of geomagnetic storms. Journal
of Atmospheric and Terrestrial Physics,
581805-1812, 1996a. Wu, C.C., M. Dryer.
Predicting the IMF Bz polarity's change at 1AU
caused by shocks that precede coronal mass
ejections. Geophys. Res. Lett, 231709-1712,
1996b. Wu, C.C., M. Dryer. Three-dimensional MHD
simulation of interplanetary magnetic field
changes at 1AU caused by a simulated disturbance
and a tilted heliospheric curent/plasma sheet.
Solar Physics, 173391-408, 1997. A recent review
of previous attempts for predicting large IMF Bz
based on some numerical methods and
Hakamada-Akasofu scheme is given by J K Chao and
H H Chen, Prediction of southward IMF Bz. In Song
P., H.J. Singer, and G.L. Siscoe, editors, Space
weather, pages 143-158. Geophysical Monograph
125, AGU Washington, 2001.
14
Models of describing the process of CME
formation, interplanetary propagation and
interaction with magnetospheric-ionosphere have
also made a progress. We may refer to the recent
work by

• Groth, C.P.T., De Zeeuw, D.L., Gombosi, T.I.
  and Powell, K.G., Space Sci. Rev., 1999, 87
  193-198
• Groth, C.P.T., De Zeeuw, D.L., Gombosi, T.I.
  and Powell, K.G., Adv. Space Res., 2000, 26(5)
  793-800
• Groth, C.P.T., De Zeeuw, D.L., Gombosi, T.I. and
  Powell, K.G., J. Geophys. Res., 2000, 105
  25053-25078
• Gombosi, T.I., De Zeeuw, D.L., Groth, C.P.T.
  and Powell, K.G., Adv. Space Res.,2000, 26(1)
  139-149
• Gombosi, T.I., De Zeeuw, D.L., Groth, C.P.T.,
  Powell, K.G. and Stout, Q.F., JASTP, 2000,
  62(16) 1515
• T.I. Gombosi, D.L. De Zeeuw, C.P.T. Groth, K.G.
  Powell, and P. Song, Phys. Space Plasmas,
  15(1998), 121
• Song, P., De Zeeuw, D.L., Gombosi, T.I., Groth,
  C.P.T. and Powell, K.G.,J. Geophys. Res., 1999,
  104 28,361
• Song, P., Gombosi, T.I., De Zeeuw, D.L. and
  Powell, K.G., Planet. Space Sci., 2000, 48 29-39
• SEE A SUMMARY
• Gombosi, T.I., De Zeeuw, D.L., Groth, C.P.T.,
  Powell, K.G, C. Robert Clauer, and Paul Song,
  From Sun to Earth Multiscale MHD simulation of
  space weather, SPACE WEATHER, ED. By Song P.,
  H. J. Singer, and G. L. Siscoe, Geophysical
  Monograph 125, AGU Washington 2001, pp169-176

15
MHD simulation has also been for long used to
simulate the global magnetospheric configuration
and to investigate the response of
magnetosphere-ionosphere system to changing
solar wind conditions.
Fedder, J.A., S.P. Linker, J.G. Lyon, and R.D.
Elphinstone. Global numerical simulation of the
growth phase and the expansion onset for a
substorm observed by Viking. J. Geophys. Res. ,
100 19083-19093, 1995. Fedder, J.A., S.P.
Linker, and J.G. Lyon. A comparison of global
numerical simulation results to data for January
27-28, 1992, Geospace Environment Modeling
Chanllenge Events. J. Geophys. Res., 103A
14799-14810,1998. Raeder, J., J. Berchem, and
M. Ashour-Abdalla. The geopsace environment
modeling and chanllenge results from a global
geospace circulation model. J. Geophys.Res.,
103 14787-14797, 1998.
16
2. Numerical Schemes in Solar Wind modeling
The Numerical Schemes in the former MHD code may
include

• Full-Implicit-Continuous-Eulerian(FICE)
• Upwind scheme for flow Lax-Wendroff scheme for
  magnetic induction equations (S.T. WU and his
  Co-workers)
• Finite volume TVD scheme of Roe type by
  decomposing the magnetic field into a potential
  part and an non-potential one(Tanaka et al.)
• Lax-Wendroff scheme (Han et al., Steinolfson,
  etc))
• MacCormack Scheme (Steinolfson and his
  coworkers)
• Two-step Lax-Wendroff scheme with flux-corrected
  transport technique(Zhang and Wang)
• UpwindSpectral Method(Mikic et al)
• So-Called Eight Wave Method(Powell et al. )
• Modified Lax-FriedrichsMacCormack Scheme (Feng
  et al., 2002)

17
http//www.saic.com
UpwindSpectral Method
P. Riley, J. Linker, Z. Mikic, and R. Lionello,
MHD modeling of the solar corona and inner
heliosphere comparison with observations, SPACE
WEATHER, ED. By Song P., H. J. Singer, and G. L.
Siscoe, Geophysical Monograph 125, AGU
Washington 2001, 159-168

• Mikic, Z., Barnes, D.C. and Schnack, D.D.,
  Astrophys. J., 1988, 328 830-847
• Mikic, Z., Phys. Fluids B, 1990, 2 1450
• Mikic, Z. and Linker, J.A., Astrophys. J., 1994,
  430 898-912
• Mikic, Z. and Linker, J.A., Astrophys. J., 1994,
  430 898-912
• Mikic, Z., Linker, J.A., Schnack, D.D.,
  Lionello, R. and Tarditi, A., Phys. Plasmas,
  1999, 6 2217
• Linker, J.A., Mikic, Z., Bisecker, D.A., et al.,
  J. Geophys. Res., 1999, 1049809-9830

18
In this Model

• Resistive MHD System
• In the radial and meridional directions, a
  finite-differencing approach of upwind type is
  used in the Phi direction, the derivatives are
  calculated pseuospectrally
• At the lower boundary are specified the radial
  component of the magnetic field, Br , based on
  the observed line of sight measurements of the
  photospheric magnetic field and uniform,
  characteristic values for the plasma density and
  temperature. This initial solution is advanced in
  time by a leapfrog time integration scheme with a
  semi-implicit method until a steady state is
  reached.
• This scheme is particularly efficient for
  simulating the solar evolution with long
  wavelengths.

19
Eight Wave formulation for MHD system of
conservative form in Cartesian (X,Y,Z) Coordinates
This scheme uses Special Treatment of MHD system
according to Godunovs symmetric formulation for
MHD in (x,y,z) coordinates. Powell et al.(AIAA
Paper 95-1704-CP, 1995) applied approximate
Riemann solvers based on the waves associated
with the full MHD system. Godunov S K,
Symmetric form of magnetohydrodynamics(in
Russian), In Numerical methods for mechanics of
continium medium, Vol. 1 pp26-34, Siberian Branch
of USSR Acd. of Sci., 1972 This scheme adopts a
cell-centered upwind finite-volume discretization
procedure and uses approximate Riemann solvers,
and explicit multi-stage time stepping to solve
the MHD equations in divergence form. A parallel
adaptive solution method is employed! The
numerical scheme is, due to the high resolution
approach, second-order accurate in smooth
regions, and locally first order in discontinuous
regions.
20
This code is further developed by the Michigan
research group to model the process of CME
formation, interplanetary propagation and
interaction with magnetospheric-ionosphere! Powel
l, K.G., Roe, P.L., Linde, T.L., Gombosi, T.I.
and De Zeeuw, D.L., JCP, 1999, 154 284 De
Zeeuw, D.L., Gombosi, T.I., Groth, C.P.T.,
Powell, K.G. and Stout, Q.F., IEEE Transactions
on Plasma Science, 2000, 28 1956 Groth, C.P.T.,
De Zeeuw, D.L., Gombosi, T.I. and Powell, K.G.,
JGR, 2000, 105 25053
21
In summary, although both upwind and symmetric
TVD schemes in the framework of the
shock-capturing approach are thoroughly
investigated and applied with great success to a
number of complicated multidimensional fluid
problems, the extension of these schemes to MHD
equations is not a simple task.
First, the exact solution of the MHD Riemann
problem is too multivariant to be used in regular
calculations.
On the other hand, the extensions of Roe's
approximate Riemann problem solvers for MHD
equations in general case are nonunique and need
further investigation.
Kulikovskiy, A. and Lyubimov, G.,
Magnetohydrodynamics, Addison-Wesley, Reading,
MA, 1965 Barmin and Kulikovskiy, JCP, 1996
22
Which is better ?Conservative or
Non-conservative Shock-capturing capability of a
numerical scheme in conservative form is superior
to that of the numerical scheme in
non-conservative form with the same
accuracy. Lax, P.D. and Wendroff, B., Systems of
conservation laws, Comm. Pure Appl. Math., 1960,
13 217-237 But, in MHD simulation we are
facing another problem about how to keep
Div(B)0 numerically in order to avoid the
non-physical flow caused by the numerical
non-zeroness of Div(B).
23
Former experiences have told us 1)when
numerical schemes in conservative form are
considered, we have to deal with Div(B)
numerically, which is of course a little
time-consuming.
Tanaka, T., Journal of Computational Physics,
1994, 111 381-389 Tanaka, T., Journal
Geophysical Research, 1995, 100
12057-12074 Toth, G., Journal of Computational
Physics, 2000, 161 605-652 Brackbill, J.U. and
Barnes, D.C., Journal of Computational Physics,
1980, 35 426-430 Powell, K.G., Roe, P.L.,
Myong, R.S., Gombosi, T. and De Zeeuw, D.L.,
AIAA Paper 95-1704-CP, 1995 Powell, K.G., Roe,
P.L., Linde, T.L., Gombosi, T.I. and De Zeeuw,
D.L., Journal of Computational Physics, 1999,
154 284-309
24
Div(B) cleaning-up procedure
Projection Method
BB0B1 with B1 time-dependent
Use to modify B
Directly solve this Poisson equation by
Bi-conjugate Method
van der Vorst, H.A., Bi-CGSRAB a fast and
smoothly converging variant of Bi-CG for the
solution of nonsymmetric linear systems, SIAM J.
Sci. Statist. Comput., 1992, 13 631-644
Time-relaxation method to keep Div(B)0
numerically
25
Former experiences have also told us 2)However,
the numerical scheme in non-conservative form for
MHD system can lead to an tolerable error in
keeping Div(B)0 numerically and thus reduce the
numerical error.
Taking the above consideration, a new combined
numerical scheme is proposed for MHD system. In
this method, the modified Lax-Friedrichs scheme
is used for the fluid equations of the MHD
system and the well-known MacCormack II is
employed for the magnetic induction equations in
spherical coordinates. (Feng et al., Chin
Journal of Space Science, 22318-323, 2002)
26
Conclusions
In this presentation, we just briefly mention
some current used numerical methods in solar wind
simulation. For more complete discussion of the
observational properties of CMEs and theory of
CMES, we can refer to the following papers
St Cyr, O. C., et al., Properties of coronal
mass ejections SOHO/LASCO observations from
January 1996 to June 1998, JGR,
105(2000)12493-12506 Webb, D. R., U. N.
Crooker, S. P. Plunkett, and O. C. St. Cyr, The
solar sources of geoeffective structures,
p123-142 James A. Klimchuk, Theory of Coronal
mass ejections, pp143-158, SPACE WEATHER, ED. By
Song P., H. J. Singer, and G.L. Siscoe ,
Geophysical Monograph 125, AGU Washington 2001
27
Taking account of the above mentioned
discussions we still need to construct some
simplified approaches for A MHD Code by
considering the following points (I) satisfy the
TVD property by using the conservative form of
MHD system in order to exactly catch the
discontinuity, (II) keep the Div(B)0
constraint numerically in order to reduce the
non-physical numerical flow as much as possible,
and (III) be enough economical and robust(such
as to avoid the splitting of associated Jacobian
matrices and the calculation of corresponding
eigenvalues and eigenvectors), (VI) be able to
reproduce the typical characteristics of the
solar wind if used in solar wind simulations.
28
From the point of view of space weather numerical
prediction, an operable numerical MHD model of
solar-interplanetary transient phenomena should
bear the following 1) robust GLOBAL MHD code of
quick convergence with high resolution 2)reasonabl
e triggering mechanism(How to drive or fuel a
CME)/input(Initial-Boundary Conditions ) based on
the global structures of solar mass
output,magnetic field and velocity map,based on
operational, real-time solar (3-D) observations
(optical, x ray, radio, etc.), must be derived
and used to generate physical parameters to
initialize, and to update, background and
transient solar wind conditions for a
three-dimensional MHD code, in order to get the
basic solar wind parameters near the earth orbit,
such as the solar wind velocity, temperature and
interplanetary magnetic field
29
3)Testing Validating framework of the numerical
results spacecraft data at the earth(outside the
bowshock) or at the liberation point L1 (such as
ACE) must be used as ground truth to validate the
output of the code 4)The output of the code
near the earth orbit must be used as inputs to a
suitably-established global numerical
magnetosphere-ionosphere-thermosphere model or
other empirical models,in order to predict the
geophysical effects. 5)be able to reproduce
observation give the associated interpretation,
furthermore predict CMEs solar-terrestrial
phenomena, 6)Other Properties
30
It has been believed that three-dimensional,
numerical, MHD modeling must play a crucial role
in a seamless forecasting system. This system
refers to space weather originating on the sun
propagation of disturbances through the solar
wind and interplanetary magnetic field(IMF), and
thence, transmission into the magnetosphere,
ionosphere, and thermosphere. This role comes
as no surprise to numerical modelers that
participate in the numerical modeling of
atmospheric environments as well as the
meteorological conditions at Earth.
31
?Up to now, in event studies we have no enough
observational data available as input! USUALY, At
the lower boundary are specified the radial
component of the magnetic field, Br, based on the
observed line of sight measurements of the
photospheric magnetic field and uniform,
characteristic values for the plasma density and
temperature. As usual, an initial estimate of the
field and plasma parameters are found from a
potential field model and a Parker transonic
solar wind solution then this initial solution
is advanced in time by time-relaxation method.
?How to determine the disturbed parameters needs
further study! It is expected that high
performance computer, application of modern
computational fluid methods and advanced
observation technologies will make this kind of
prediction practical in the foreseeable? future.
Scientists from some communities such as
CISMCenter for Integrated Space Weather Modeling
(CISM located at Boston Univ.), CCMCCommunity
coordinated modeling center, are in a position
to make their efforts To this end!
32
Thank you for your patience!
33
Center For Integrated Space Weather Modelling
http//www.bu.edu/cism/index.html
CISM's Vision To Understand Our Changing Sun And
Its Effects on the Solar System, Life, and
Society CISM, the Center for Integrated Space
Weather Modelling, is a National Science
Foundation (NSF) Science and Technology Center
(STC). CISM is in the final stages of the
approval process, and is set to officially
commence operations on August 1, 2002. ITS
GOAL To create a physics-based numerical
simulation model that describes the space
environment from the Sun to the Earth.
34

• Current Models
• Model boundary conditions based on solar
  observations (UCB, Stanford)
• Global coronal models (SAIC)
• Global solar wind models (U of Colorado/NOAA-SEC)
  
• Active region models (Stanford, UCB)
• Solar energetic particle models (UCB)
• Coronal transient (CME) models (SAIC)
• CME propagation models (U of Colorado/NOAA-SEC)
• Solar energetic particle models (UCB)
• CISM Solar/Solar Wind tasks include
• Determining boundary conditions for the models
  from solar observations.
• Coupling active region and global coronal models.
  
• Coupling coronal and solar wind models.
• Coupling coronal and solar wind models with
  energetic particle models.
• Joining magnetospheric modelers in coupling solar
  wind, energetic particle, and magnetospheric
  models.

35
Thank you for your patience!
36
For more complete discussion of the observational
properties and theory of CMES, we can refer to
the following papers.
Dryer, M. Multi-dimensional MHD simulation of
solar-generated disturbances space weather
forecasting of geomagnetic storms. AIAA
Journal,36365,1998. Wu, S.T., M.D. Andrews, and
S.P. Plunkett.Numerical MHD modeling of coronal
mass ejections(CMEs),Space Sci. Rev., 95191-213,
2001. St Cyr, O.C., et al. Properties of coronal
mass ejections SOHO LASCO observations from
January 1996 to June 1998. JGR, 10512493,
2000. Webb et al.The solar sources of
geoeffective structures. pages 123-142.
Klimchuk, James A. Theory of Coronal mass
ejections. pages 143-158. In Song P., H.J.
Singer, and G.L. Siscoe, editors,Space weather,
Geophysical Monograph 125, AGU Washington, 2001.