Flux Rope Modeling of an Interplanetary Coronal Mass Ejection Observed at WIND and NEAR

T. Mulligan1, C.T. Russell1, B.J. Anderson2, D.A. Lohr2, B.A Toth2, L.J. Zanetti2,
M.H. Acuna3, R.P. Lepping3, J.T. Gosling4, and J.G. Luhmann5


1Institute of Geophysics and Planetary Physics and the Department of Earth and Space Sciences University of California Los Angeles
2Applied Physics Laboratory, John Hopkins University, Laurel, Maryland
3Goddard Space Flight Center, National Aeronautics and Space Administration, Greenbelt, Maryland
4Los Alamos National Laboratory, Los Alamos, New Mexico
5Space Science Laboratory, University of California Berkeley


Originally Published in: Solar Wind Nine, pp 689-692,
American Institute of Physics, 1999



Interplanetary magnetic field measurements made as NEAR approached Earth in December 1997 have allowed us to compare a magnetic cloud like event seen by the NEAR and WIND spacecraft. The magnetic cloud seen by WIND and NEAR has temporal signatures both in magnitude and direction that indicate a similar geometry and orientation. The magnetic helicity of the cloud structure seen at NEAR is the same as at WIND. The magnitude of the field in the cloud event is consistently lower at NEAR than at WIND as expected for radial convection plus expansion. We model the field of this cloud structure at each spacecraft, obtain fits to the cloud axis as well as rates of expansion, and find that the predicted spacecraft separation distance is consistent with observations.


The NEAR spacecraft (1) was launched on February 17, 1996 and is scheduled for a January 1999 rendezvous with asteroid 433 Eros (2). In order to bend NEAR's trajectory out of the ecliptic plane toward its rendezvous with 433 Eros, the spacecraft performed an Earth flyby maneuver on January 23, 1998. During the period of September 1997 through December 1997, NEAR encountered four magnetic structures that resembled magnetic-cloud type ICMEs. We have examined the WIND magnetic field and plasma data and have found five magnetic-cloud events that were observed at Earth during this period. Four of these events correspond to those seen at NEAR. As illustrated in Figure 1, the spacecraft and Earth were separated radially by 0.18 AU to 0.63 AU and azimuthally by 1.2o 33.4o. In the plot the Earth is fixed and NEAR moves along an arc towards it. Both the Earth and NEAR are in the ecliptic plane during this period.

Figure 1: NEAR positions in the ecliptic plane during four events observed at NEAR and at Earth. Radial distances from the sun are given in AU, azimuthal separation in degrees. The December event corresponds to an azimuthal separation of 1o (0.02 AU).

The clouds are modeled as flux ropes using gaussian representations of both axial and poloidal field strengths. The effect of flux rope expansion with radial distance is included in the model and follows a general B R3/2 radial decrease of the field. The Downhill Simplex Method developed by Nelder and Mead (3) is used to invert the eight-parameter model. The method requires the minimization of the sum of the squares of the residuals defined by the difference between the observed magnetic field and the model field components. The model describes a cylindrically symmetric divergence free magnetic field in which the axial field strength has a gaussian distribution strongest at the center of the rope and the poloidal field has an inverted gaussian distribution strongest at the rope edges. Contours of equal total field strength are on cylindrical shells. The advantage of using this type of flux rope model is that it does not require a force-free field configuration and therefore allows more accurate modeling of high beta structures. The model also accounts for a depression of the magnetic field through a factor that controls the ratio of field strengths at the entry and exit points of the rope. The depression factor adds extra flexibility when fitting the compression region often observed at the leading edge of high speed magnetic clouds and also when modeling expanding magnetic clouds.

Figure 2: Magnetic field data for the December 1997 event. Top panel contains WIND data, bottom contains NEAR data. The shock and ICME interior is labelled in each case. Flux rope model fits are shown as smooth black lines overlaying the data in the ICME interior.

The parameters used in the model affecting flux rope topology are azimuthal and polar angles describing the orientation of the rope axis, the impact parameter (i.e. distance between the axis of the flux rope and the closest point of observation to that axis), magnetic field magnitudes of both the axial and poloidal fields, the width of the gaussian field distribution for both axial and poloidal fields (associated with the ``pitch'' of the helical windings of the flux rope), and the depression factor determined by the ratio of the fields at the entry and exit points of the rope. The Nelder and Mead downhill-simplex method requires only function evaluations along each point during the minimization. The initial set of parameter values found by forward modeling of the data is used to define the vertices of a mathematical object called a ``simplex.'' The simplex is a geometrical figure consisting in N dimensions of N+1 points (or vertices) and all their interconnecting lines and polygon faces, etc. In two dimensions the simplex is a triangle. In three dimensions it is a tetrahedron. In this case the parameters of the model define the simplex as an eight-dimensional polyhedron with nine vertices. A downhill-simplex algorithm (4) is then used to proceed through a series of steps moving the point of the simplex where the function is the largest through the opposite face of the simplex to a lower point. At all times the algorithm conserves the volume of the simplex to ensure non-degeneracy. Termination of the algorithm occurs when the decrease in the function value in the terminating step is on the order of the machine precision.


In this study we concentrate on the interplanetary coronal mass ejection (ICME) observed on December 10, 1997 at the WIND spacecraft, when NEAR is located at a radial distance from the Sun of 1.18 AU and 1.2o (.02 AU) to the east relative to the Earth's position (see Figure 1). At WIND upstream of the shock the plasma is flowing at 290 km/sec. A shock passes over WIND and the observed solar wind speed jumps to 365 km/sec. Assuming radial propagation and conservation of mass the speed of the shock is calculated to be 405 km/sec. By the time of the arrival of the leading edge of the ICME the observed speed has fallen to 359 km/sec. At this speed the same ICME should be observed at NEAR approximately a day later. Figures 2a and b show the magnetic field data for the ICME observed by both WIND and NEAR. The top part of the figure contains data from WIND while the bottom contains data from NEAR. The first black line in each panel marks the shock seen at the corresponding spacecraft. The next two lines in each panel indicate the approximate start and stop times of the ICME determined by the magnetic field jump and at WIND the proton temperature depression (not shown, see (6)). These lines also mark the interval used to determine the model fits. Preliminary analysis of the plasma and magnetic characteristics of this ICME are found in (5).


Modeling the December 1997 ICME as a magnetic cloud results in two fits for the axis of the cloud as seen at each spacecraft. The cloud axis at WIND is estimated to have a clock angle of 200o and a cone angle of 35o (i.e., close to the ecliptic pole and 35o from the radial direction). At NEAR the cloud has a similar orientation with respect to the ecliptic pole, having a clock angle of approximately 215o and a cone angle of 65o. The estimated circular radius of the cloud at WIND and NEAR is 0.045 AU and 0.067 AU respectively indicating expansion of the cloud with radial distance. Assuming the cloud expands symmetrically as it propagates from WIND to NEAR so that the impact parameters obtained by the fits at WIND and NEAR can be easily compared, we calculate the azimuthal separation between the two spacecraft based on the model results to be approximately 0.030 AU. Returning to Figure 1, the actual azimuthal separation of the spacecraft is 1.2o (0.021 AU). The apparent disrepancy of 0.009 AU can be accounted for by correcting for the aberrated solar wind direction as the cloud travels from WIND to NEAR. This correction results in an extra 0.014 AU indicating that the model correctly predicts the spacecraft separation to 0.005 AU. The model also indicates the cloud at both WIND and NEAR has left-handed helicity. The similarity between the model fits and observed values demonstrates the accuracy of this particular flux rope model.

Discussion and Conclusions

In this study we have presented one of the four ICME events observed by WIND at 1 AU that was later observed by NEAR at 1.18 AU in December 1997. Comparison of the WIND and NEAR observations indicates that when the separation distance between the spacecraft is small the magnetic profiles at the two spacecraft show strong similarities. Fits to the observed axial and poloidal magnetic fields in the December 10-11, 1997 cloud result in impact parameter distances consistent with the azimuthal spacecraft separation distance between WIND and NEAR demonstrating the accuracy of the gaussian flux rope model. The cloud has the same helicity at each spacecraft. Coronal field plots related to this December event indicate the axis of symmetry of the cloud was aligned along the neutral line of the streamer belt (6). If the orientation of magnetic clouds is determined by that of the helmet streamer belt, (6,7,8), it is reasonable to consider the neutral line as defining the orientation of the axis of symmetry. This cloud may have resulted from reconnection in sheared helmet streamer arcades (9), which is consistent with the streamer belt association with transients (10).


We are grateful to the many members of the NEAR team who successfully built and launched this first Discovery mission that has enabled varying baseline studies of ICMEs. We also thank C. Smith for making ACE data available for study. This work was supported in part by the National Aeronautics and Space Administration through a grant from the John Hopkins University #730607 and supported at LANL by internal funding.


1. Cheng, A.F., R.W. Farquhar, R.E. Gold, K.J. Heeres, J.A. Landshof, S.C. Lee, and A.G. Saato, Near Earth Asteroid Rendezvous: Mission overview, Space Science Reviews, 82, (1,2), 3-29, (1997.)

2. McFadden, L.A., C.T. Russell, and A.F. Cheng, NEAR-Earth asteroid mission to travel to 433 Eros, EOS Trans. Am. Geophys. Union, 77, 73-79, (1996.)

3. Nelder, J.A., and R. Mead, Computer Journal, 7, 308-313, (1965.)

4. Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., Press Syndicate of the University of Cambridge, New York, pp 408-412, (1997.)

5. Mulligan, T., C.T. Russell, B.A. Anderson, D. Lohr, D. Rust, B.A. Toth, L.J. Zanetti, M.H. Acuna, R.P. Lepping, and J.T. Gosling, Intercomparison of NEAR and WIND ICME observations, J. Geophys. Res., in review, (1999.)

6. Mulligan, T., C.T. Russell, and J.G. Luhmann, Solar cycle evolution of the structure of magnetic clouds in the inner heliosphere, Geophys. Res. Lett., 25, 2959-2962, (1998.)

7. Crooker N.U., G.L. Siscoe, S. Shodhan, D.F. Webb, J.T. Gosling, and E.J. Smith, Multiple heliospheric current sheets and coronal streamer belt dynamics, J. Geophys. Res., 98, 9371, (1993.)

8. Zhao, X. and J.T. Hoeksema, Effect of coronal mass ejections on the structure of the heliospheric current sheet, J. Geophys. Res., 101, 4825-4834, (1996.)

9. Gosling, J.T., Coronal mass ejections and magnetic flux ropes in interplanetary space, in Physics of Magnetic Flux Ropes, ed. by E.R. Priest, L.C. Lee, and C.T. Russell, AGU Geophysical Monograph, 58, 343-364, (1990.)

10. Crooker N.U., and A.H. McAllister, Transients associated with recurrent storms, J. Geophys. Res., 102, 14041-14047, (1997.)

Back to CT Russell's page More On-line Resources
Back to the SSC Home Page