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Numerical methods in Solar Wind Simulations

Numerical methods in Solar Wind Simulations. FENG Xueshang 1 , WEI Fengsi 1 , S.T Wu 2 , FAN Quanlin 1. Presented at WSEF 2002, Adelaide. 1、 Laboratory of Space Weather, Center for Space Science and Applied Research, Chinese Academy of Sciences

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Numerical methods in Solar Wind Simulations

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  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 models:These 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, 104:9809-9830 • 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), 30:61-64, 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, 58:1805-1812, 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, 23:1709-1712, 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, 173:391-408, 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) • Upwind+Spectral Method(Mikic et al) • So-Called Eight Wave Method(Powell et al. ) • Modified Lax-Friedrichs+MacCormack Scheme (Feng et al., 2002)

  17. http://www.saic.com Upwind+Spectral 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, 104:9809-9830

  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 methoduntil 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 Godunov’s 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! Powell, 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 B=B0+B1 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, 22:318-323, 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)reasonable 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 CME’s 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 CISM[Center for Integrated Space Weather Modeling (CISM located at Boston Univ.)], CCMC[Community 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,36:365,1998. Wu, S.T., M.D. Andrews, and S.P. Plunkett.Numerical MHD modeling of coronal mass ejections(CMEs),Space Sci. Rev., 95:191-213, 2001. St Cyr, O.C., et al. Properties of coronal mass ejections: SOHO LASCO observations from January 1996 to June 1998. JGR, 105:12493, 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.

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