ON THE SELF-CONSISTENT DETERMINATION OF DAYSIDE MAGNETOPAUSE SHAPE AND POSITION.



M.D.Kartalev,V.I. Nikolova,I.P. Mastikov
Institute of Mechanics, BAS,
Acad. G.Bontchev Str, block 4,
1113 Sofia, Bulgaria
e-mail : geospace@bgcict.acad.bg

V.F. Kamenetsky
Copmuting Center, RAS, Moscow, Russia

Accepted for publication in Planetary and Space Science (1996).

Acknowledgements
This research was partialy supported by the Bulgarian National Foundation "Scientific Research" under contracts: NZ 522.

[Abstract] [Introduction] [Mathematical model] [Numerical procedure] [Numerical implementations] [Summary and discussion] [References] [Figures]

Abstract

Numerical procedure is developed for joint solution of models for dayside magnetosheath and dayside magnetosphere. It results in a modification of the numerical scheme for the transitional region behind the bow shock in a way, permitting self-consistent determination of the magnetopause as a detail of the solution. The choice of simple models allows comparisons with well studied in the literature results for magnetosheath solutions with preliminary determined magnetopause. A new for the astrophysical tasks grid- characteristic gasdynamic numerical scheme is proposed for simulation the axially symmetric dayside magnetosheath. Magnetosphere analytical model of Stern, 1985, as well as Beard's, 1960 approach for the inside magnetopause magnetic field are utilized in the numerical experiments. The results point out that the idea for joining solutions of two regional models is worthwhile, providing some specifications of the magnetopause shape and position even in the frame of the used quite simple regional approaches. Obtaining of additional new details could be expected if more sophisticated regional models will be applied.

Introduction

Two basic strategies in the magnetogasdynamic modelling of solar wind interaction with planetary magnetosphere or ionosphere have been developed in the literature: Despite of the impressive success reached recently in the global simulation, there are important aspects of the problem, which could be analyzed better in the frame of the regional approach. The determination of the shape and position of the magnetosheath boundaries (bow shock and magnetopause) is an example of such an aspect. The shock wave shape and position is determined self-consistently in the frame of the usually used '' shock fitting`` technics for magnetosheath numerical consideration. The magnetosheath shape and position however are not determined self- consistently in the widely used magnetosheath models.
We develop in the present study a numerical scheme for a gasdynamic (as a first step of gasdynamic convected field approximation) modeling of the magnetosheath. An essentially new moment in this scheme is the self-consistent determination of the magnetopause shape and position, performed by equalizing pressures on both sides of this surface. The pressure on outer side is that obtained by gasdynamic computation. The inside pressure could be given by any magnetospheric model. For elaborating and testing the numerical scheme, a vacuum magnetosphere analytical model of Stern, 1985 [Stern D.P. J. Geophys. Res., 90, 10851] for closed and open magnetosphere is utilized here. The simplified Beard's magnetopause field approach [Beard D.B. J. Geophys. Res., 65, 3559, 1966], used by now for preliminary magnetopause determination, could be also applied in the context of the proposed here approach i.e. for self-consistent magnetopause determination.
A grid characteristic numerical scheme is used for computation of the magnetosheath region. This is a gasdynamic numerical method not earlier applied to astrophysical problems.

Mathematical model

A system of two geospace regions: dayside magnetosphere and dayside magnetosheath is considered. Simple models, describing these regions are accepted and these models are solved self- consistently. This self-consistency is reached also in a quite simplified manner in the present consideration. This is because of the models' simplicity (gasdynamic outside and vacuum inside) as well as because of the acceptance for the magnetopause to be a tangential discontinuity.
The models' choice permits to reveal explicitly the differences and similarities with well studied results, based on analogical approaches.

Magnetosheath model

Unsteady gasdynamic Euler equations are utilized for describing the dayside magnetosheath. The problem study here is limited to axially symmetric case only (shock wave and the magnetopause are supposed to be rotational surfaces) and correspondingly- the system of equation is reduced to 2D one. The system of equations is represented in a form suitable for applying the grid-characteristic numerical scheme [Magomedov and Holodov, Moscow, Nauka, 1988]. For more information see Plate 1.
The domain under consideration is limited by:
  1. shock wave
  2. magnetopause surface
  3. additional boundary, limiting computational region
The usual jump conditions are imposed on this surface.

Magnetosphere magnetic field model

A simple variant of the analytical magnetosphere model of Stern, 1985 is implemented here for computing the dayside magnetosphere magnetic field. This field (especially on the magnetopause) is needed for specifying the pressure equilibrium condition.
In this model the magnetopause is supposed to be a fixed rotational paraboloid. The magnetosphere magnetic field B is considered as a sum of assigned internal fields (dipole field B_d, ring current field, tail plasma sheet and so on) and a sought divergent-free and curl-free field B_s with potential G. This potential in the considered case have to be harmonic.
The searched field B_s is thought to be arising from currents on the magnetopause or beyond it. It is supposed that B_s compensates the contributions of the internal field ingredients on the magnetopause in such a way that the normal component of the total field B_n is zero (closed magnetosphere), or some non-zero function (open magnetosphere).
The model magnetopause is given by a paraboloid. The dipole is placed in the coordinate origin.

Pressure equilibrium on the magnetopause

The goal here is obtaining such a shape and position of the magnetopause in solving the magnetosheath problem, that the pressures, computed from models on both sides of this surface to be maximally equalized. In the frame of the used analytical magnetosphere model the achieving of exact pressure balance in each magnetopause point is not possible. This is because of the model restriction for the magnetosphere boundary- to be paraboloid with two free parameters only. Instead of introducing similar restriction for the magnetopause shape, we are preferring in the present consideration to accept some reasonable scheme for approximate pressure balancing.
The computed rotational magnetopause surface in the present scheme is correlated with the parabolic rotational surface, introduced in Stern's model. Its generating parabola is accepted to be the least-square best- fit approximation to the computed magnetopause cross-section. The axially symmetric magnetospheric magnetic pressure distribution on the parabolic surface is accepted here to correspond to the equatorial cross- section of the Stern's paraboloid. The dynamic pressure in the point of computed on each time step magnetosheath surface is equalized to the magnetic pressure of the best-fit to this surface paraboloid in its point, placed on the same radius.

Numerical procedure

A discontinuity- fitting approach is accepted for treating both magnetosheath boundaries. An appropriate transformation of independent variables is applied, transforming the computational domain (magnetosheath region ) to a fixed rectangular domain.
An explicit first - order non-conservative difference scheme, determined in the grid-characteristic method is used for internal nodes of the computational domain .
The developed numerical scheme appears to be quite stable relatively to the given initial distribution of the parameters. In the numerical examples, presented in the next section, a linear initial distributions is accepted along radiuses between their values inside the shock and values, obtained by reasonably applied Bernully integral on magnetopause stream line.
Time- marching numerical iterations are repeated until reaching stationary solution. The stationary state with accuracy approximately 0.01 is usually achieved for 1000-1200 iterations.
The solver's performance was tested, repeating numerous analogical implementations of the numerical scheme of Stahara et al., 1977 [Stahara S.S., Chaussee D.S., Truddinger B.C., and Spreiter J.R., NASA Contract. Rep. , CR 2924]. A close coincidence of the results was obtained.
An implementation of our solver for the case with firmly determined magnetopause shape and position is presented in Figure 1. The obtained bow shock and sonic line locations for various Mach numbers match exactly those, obtained by Stahara et al., 1977 (Fig. 7 there).
The shock-capturing capability of the solver is examined again in comparison with the same well known computational scheme. The presented in Figure 2 result for solar wind flow past the rotated principal meridian of the magnetopause exactly repeats the analogical picture in the paper of Spreiter and Stahara, 1980 (Fig. 4 there).

Numerical implementations

As expected, the solar wind flow past a magnetic planet, computed using the proposed in this work scheme of magnetopause determination as a component of the problem solution, differs in general from those, obtained on the basis of preliminary determined magnetopause. Correspondingly, there are differences between magnetopause and bow shock shapes and positions. A comparison between the two different approaches is demonstrated in Figure 3
Computational results, analogical to those in Figure 1 , but received here for self- consistent solar wind flow interaction with Stern's closed magnetosphere, are shown in Figure 4. The dashed line traces the fixed Beard's magnetopause .
One can perform numerical test, in which the Stern's magnetosphere is approached rotating its principal meridional trace. This test is analogical to that of Spreiter and Stahara, 1980 [Spreiter J.R. , and Stahara S.S. J. Geophys. Res., 85, 6769] . The result of such an experiment is shown in Figure 5 . Despite of the appearing similarity with Figure 2 , one can observe differences between the numerical results in both cases.

Numerical results obtained by applying Beard's magnetopause field approach self-consistently

The developed scheme for self-consistent magnetopause determination is applied here using the Beard's, 1960 approach as an inside magnetopause tangential magnetic field. The procedure for numerical realization of the pressure equalizing in this case becomes formally even simpler than that for the case of Stern's model. The Beard's formula is directly applicable in each point of the magnetopause, obtained in certain time step. A special care is required in applying this procedure near the cusp in the meridional magnetopause trace inasmuch as the approximate Beard's formula does not work over arbitrary curved magnetopause shape. Particularly this formula is incorrect when the local bend of the boundary becomes considerable.
The result, obtained in rotating the principal meridian is shown in Figure 6.

A comparison of the flow geometries, obtained in the above described three analogical computational experiments are presented in Figure 7.

Pressure distributions on the magnetopause, obtained in some of considered numerical implementations are presented in Figure 8.

Numerical examples with simulation of open magnetosphere The introduced here scheme permits a simplified simulation of the effect of an open magnetosphere. We make use of the idea of Stern, 1985 (or Voigt, 1981 [Voigt G.H., Planet. Space Sci., 29, 1] ). ``Open'' boundaries are simulated there allowing some fraction of the incident magnetic flux to cross them. In our approach such a simulation results in some modification of the boundary condition, see Plate 2.
Note that this consideration is not an adequate modelling of the magnetic field reconnection on the open magnetopause - such a modelling is not possible in the frame of our limited approach here. The applied modification of the internal problem however is expected to demonstrate the tendency, which appears in considering the open magnetopause in comparison with considering the closed one.

An example of open magnetosphere simulation is shown in

Summary and discussion

An idea for self-consistent numerical solution of models of dayside magnetosheath and dayside magnetopause is examined. The goal is to find out the magnetopause shape and position in the process of this solution . The used regional models are the possibly simplest. Gasdynamic description is utilized in the magnetosheath. Magnetosphere model of Stern, 1985, as well as simple approximate representation of Beard, 1960 for magnetic field on the magnetopause are used in the numerical testing.

Grid - characteristic gasdynamic numerical scheme, not earlier used in astrophysics tasks, is applied in solving the Euler equations for perfect gas in axially symmetric (rotational) magnetosheath. This scheme could be used as discontinuity- capturing solver. Its generalizations are possible for treating 3D approaches, as well as for considering more complicated models. The numerical scheme performance is checked out in repeating some key results of the numerical scheme of Stahara et al, 1977.

In all of the implemented numerical examples the shape and the position of the bow shock and those of the tangential discontinuity simulating dayside magnetopause, are obtained in terms of discontinuity-fitting approach.

A common expectable conclusion could be done, that the implementation of preliminary determined dayside magnetopause position and shape, usually used by now, is in general reasonable enough approximation. The observed deviations of 10-15% in comparison with more precisely self- consistent magnetopause determination are really acceptable in the frame of the whole models' precision.

The significance of the differences, obtained practically in all of the numerical examinations should be analyzed in the context of each specific problem.

It could be expected that complicating the magnetosphere models and removing the restrictions for solving them only analytically could lead to essential changes in the magnetopause shape and position specification. Thus the further efforts in developing self- consistent solutions of the transitional region behind the bow shock and the magnetosphere seem to be a reasonable task.

References

Figures

Figure 1. Bow wave and sonic line locations for fixed magnetopause shape and position and Mach numbers 3.0, 6.0, 12.0, 25.0; the case of rotated equatorial trace with gamma 5/3. A complete coincidence with the analogical Figure 7 in Stahara et al., 1977 is observed here.

Figure 2. Shock wave and embedded shock location for solar wind flow with Mach number 5 and gamma 5/3 past the rotated principal meridian of the magnetopause. Sonic lines are drawn by dashed lines. The position of the embedded shock, outlined here by condensing of the drawn Mach-number iso- lines, is in good coincidence with the corresponding result of Spreiter and Stahara, 1980 (Fig. 4 there).

Figure 3. Shock, magnetopause and sonic line, obtained in applying the developed in the present paper grid characteristic numerical scheme (solid curves). Parameters for the case, corresponding to curves 1: Mach number 12, gamma 5/3. Dashed curves present analogical result, obtained for fixed magnetopause- equatorial trace of Beard's magnetopause.

Figure 4. Bow wave, magnetopause and sonic line locations for various supersonic flows past the rotational Stern's, 1985 closed magnetosphere, making use of its equatorial trace. The upstream flow parameters are as in Figure 1. The fixed magnetopause, used in the results of the Figure 1 is drawn here by dashed line.

Figure 5. Axially symmetric solar wind flow past magnetopause with inside magnetic pressure, based on the magnetic field distribution along the principal meridian of the Stern's magnetopause. Parameters: Mach number 5,gamma 5/3 . The character dimple on the magnetopause arises almost at the same position, while the subsonic ``pocked'' with Mach number smaller than 1 in the self- consistent case is moved slightly downstream.

Figure 6. Result of self-consistent magnetosheath computation of solar wind flow past a rotational (around the principal meridian trace) closed Beard's, 1960 magnetosphere. The flow parameters are as in Figure 5.

Figure 7. Comparison between magnetopause and bow shock shapes and positions for the computed flows: with Stern's magnetosphere model (from Figure 5) - solid lines; with Beard's magnetosphere magnetic field approach (Figure 6) - dashed lines; and with fixed Bear's magnetopause - dotted lines.

Figure 8. a) Pressure distribution line 1, as derived by Beard's, 1960 approach over the magnetopause shape, computed self- consistently and shown in Figure 5. Line 2 presents analogical pressure distribution over preliminary fixed ``meridional'' rotational magnetopause with magnetosheath, shown in Figure 2. Line 3 is the analogical distribution for the case with Stern's model.
b) The distributions of the corresponding pressure coefficients .

Figure 9. Simulation of solar wind flow past an open Stern's magnetosphere. The picture, presented by solid lines 2 corresponds to parameters' values: C_E*= 0; C_F*=-0.3, imitating southward IMF; The picture, drawn by solid lines 1, corresponds to the parameters: C_E*= 0; C_F*= 0.3. The remaining parameters for both cases are as in Figure 7. Dashed lines present the corresponding result for the closed magnetosphere.

Last modified: June 19, 1996