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Proceedings Cluster/Double Star Conference 1 MAGNETOPAUSE CUSP INDENTATION: AN ATTEMPT FOR A NEW MODEL CONSIDERATION M. Kartalev1, S. Savin2, E Amata 3, P. Dobreva1, G. Zastenker2, and N.Shevyrev2 1Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bontchev Str., Block 4, 1113, So a, Bulgaria 2Space Research Institute, Russian Academ of Sciences, Profsousnaya, 84/32, 117811 Moscow, Russia 3Istituto di Fisica dello Spazio Interplanetario C.N.R. Via del fosso del cavaliere 100 00133 ROMA Italy ABSTRACT culiarities could be useful for understanding of important details of the process of energy transfer from solar wind to magnetosphere and polar ionosphere. In this short We report an attempt for numerical modeling of the mag- communication we focus our attention on the problem of netopause indentation, formed by the magnetosheath - the speci c cusp associated magnetopause shape. cusp interaction. Our attempt is based on a new numeri- cal magnetosheath - magnetosphere model, developed in It is well known that most of the widely used data based the Institute of Mechanics, So a. The model includes, magnetopause models besides assuming general simpli - in a self-consistent modular approach, the models of the cations like axial symmetry, also do not take into account magnetosheath and the magnetosphere. The latter is a so called magnetopause cusp indentations (e.g. (Shue et hybrid between data-based magnetospheric current sys- al., 1997; Roelof and Sibeck, 1993)). Some recently de- tem and numerically obtained magnetopause shielding veloped data based models (Boardsen et al., 2000; East- current system. A simpli ed gasdynamic approach is ap- man et al., 2000), making use of very extensive polar plied for the magnetosheath. The positions and the shapes orbit measurements of Hawkeye satellite, provide more of the bow shock and the magnetopause are determined precise prediction of the high latitude antisunward of the self-consistently as a part of the numerical procedure, cusp part of the magnetopause,as well of its nose subso- based on the pressure balance. The substantial differ- lar part, but the authors especially underline that they do ence of the processes, responsible for the pressure forma- not model the cusp indentation itself. Thus, some kind tion inside and outside magnetopause indentations over of “asymptotic” behavior of the cusp associated region is cusps, poses dif culties of the problem solution for the estimated nevertheless by this model. The de nite deter- closed magnetosphere. We discuss the considered prob- mination of the magnetopause cusp indentation however lem in the light of CLUSTER and Interball-1 data. Both is avoided by this model too. magnetopause normals and plasma decelerations into the over-cusp indentation can be taken into account in the The situation is quite similar regarding the existing the- frame of the model, along with the magnetopause shape oretical approaches dealing with magnetopause shape dependence on the geomagnetic dipole tilt. Our attention and position. The majority of the used in the literature addresses the speci c challenging problem, arising in de- models, from “classic” numerical magnetosheath mod- termining the magnetopause indentations around the cusp els (e.g. Spreiter and Stahara (1994)) to most of the regions. sophisticated 3D global MHD models of the solar wind- magnetosphere interaction, are not designed for treatment of the magnetopause cusp indentations. Key words: magnetopause; magnetosheath; cusp; inden- tation. One of the rst attempts for deriving more realistic magnetopause geometry utilizing magnetosheath / mag- netosphere pressure balance was done by Mead and 1. INTRODUCTION Beard (1964), but relying on too simpli ed approach for the magnetospheric magnetic eld. Recently Sotirelis (1996); Sotirelis and Meng (1999) solved analogical The knowledge of the cusp associated magnetopause re- problem using more realistic data based Tsyganenko’96 gions are of great importance for understanding of very magnetospheric eld model (Tsyganenko, 1995). The basic aspects of the solar wind in uence on the Earth’s magnetopause shape obtained by these authors includes environments. The understanding of such details of these cusp indentations. The used there magnetosheath pres- regions as magnetopause structure and its geometry pe- sure on the magnetopause however is taken by idealized _________________________________________________________________ Proceedings Cluster and Double Star Symposium – 5th Anniversary of Cluster in Space, Noordwijk, The Netherlands, 19 - 23 September 2005 (ESA SP-598, January 2006) 2 Newton approach, which is quite inadequate in particular use of this e xibility of the scheme. It is worth em- especially over the geometric peculiarities of the magne- phasizing that all the ”data based shielding elds” from topause cusp indentations. Tsyganenko model, adjusting that model to some data based, but quite idealized (symmetric in particular) mag- In the present work we implement a new magne- netopause, are omitted here. Respectively, the data based tosheath/magnetosphere numerical model, in which the magnetopause, used in Tsyganenko model, doesn’t par- magnetopause shape is determined again satisfying pres- ticipate in our consideration. Our numerically obtained sure balance, but utilizing the real gasdynamic “outside” shielding eld corresponds to our numerically obtained pressure on magnetosheath. This model is described magnetopause. brie y in Section 2. In Section 3 we implement this model for a comparison with the measurements from two The implemented Gas-dynamic numerical magne- satellite magnetosheath crossings. tosheath model utilizes a slightly modi ed grid - char- acteristic numerical approach, developed by Magome- dov and Holodov (1988) and applied previously in mod- 2. SHORT DESCRIPTION OF THE IMPLE- elling of the magnetosheath problem in axially symmet- MENTED MAGNETOSPHERE - MAGNE- ric approach, as well as in other astrophysical problems TOSHEATH MODEL (Kartalev et al., 1996, 2002; Keremidarska and Kartalev, 1998, 1999) . Essentially 3D Euler gasdynamic equations are applied in appropriate curvilinear coordinates. The used here new model of the system magnetosphere - magnetosheath (Kartalev et al., 1995, 1996; Dobreva The problem domain is divided into two sub-domains for et al., 2004; Kartalev et al., 2005) comprises in a self- better description of arbitrary elongated tail region of the consistent way two “sub-models”: a 3D numerical data magnetosheath. Spherical coordinates x1 = θ, x2 = r, based model of the magnetosphere with included data x3 = ϕ are applied for the dayside magnetosheath (stan- based magnetospheric magnetic eld systems and ar- dard de nition, where ϕ is the azimuthal direction), and bitrary magnetopause, and a 3D gasdynamic numeri- in the tail region the equations are applied in cylindrical cal model of the magnetosheath with “ e xible” magne- coordinates x1 = −z, x2 = r, x3 = ϕ (again standard topause. The shapes and positions of the shock wave and de nition where z is direction to the sun). In these coor- of the magnetopause are self - consistently determined as dinates the equations for the shock wave and for the mag- a part of the solution. netopause (tangential discontinuity) could be presented as: x2 = Rs(t, x1, x3) and Rm = Rm(t, x1, x3), The new 3D finite element magnetosphere magnetic where t is the time. An additional coordinate transforma- field model is a generalization of the earlier developed tion is applied and practically used in the computational 2D (Kartalev et al., 1995) and simpli ed 3D (Kojtchev scheme: et al., 1998) models. The shielding eld is obtained nu- merically solving the Chapman-Ferarro problem in these x2 − Rm ξ = x1, ζ = x3, η = , models. The solution utilizes the dipole eld and data Rs − Rm based cross-tail, Birkeland and ring currents. In the con- sidered case we make use of the Tsyganenko data based in which the shock wave and the magnetopause are co- (T 96 and T 01) models (Tsyganenko, 1995, 2002a,b) (de- ordinate surfaces. In the implemented time marching tails in (Dobreva et al., 2004)). The 3D Chapman Ferarro scheme, for the points of the boundaries, the Rankine- problem is solved for an arbitrary magnetopause, consid- Hugoniot relations are used at each time forward step. ering the whole magnetospheric eld B as a sum of the The computed on the magnetopause magnetosphere pres- sought shielding eld Bs, the dipole eld Bd, the elds, sure essentially participates in these steps for the magne- produced by the cross-tail current system (Bt), the Birke- topause points. The algorithm determines the new posi- land currents (Bb), and the ring current (Br). As usual, it tion of each boundary point of the next time layer. is supposed that div Bs = 0, rot Bs = 0 and, therefore, a eld potential U exists: The leading procedure in the self-consistent model of the whole system magnetosheath-magnetosphere is the external (magnetosheath) problem. Current magne- ∇U = Bs; 4U = 0 topause shape and position are known at each time step of the procedure. The magnetosphere problem is solved us- Neumann boundary condition is posed on the magne- ing this current magnetopause geometry. The Neumann topause with local normal n (here we consider only problem for this shape is solved then, obtaining somed in- closed magnetosphere): side pressure distribution on the magnetopause. This cur- rent inside pressure distribution is essential in the work of ∂U the magnetosheath part of the algorithm, which not only (B , n) = − = −[(B , n)+(B , n)+(B , n)+(B ,
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