DAYSIDE MAGNETOPAUSE MODELS

Transcription

1 DAYSIDE MAGNETOPAUSE MODELS A.V. SUVOROVA, A. V. DMITRIEV and S. N. KUZNETSOV Skobeltsyn Institute of Nuclear Physics, Moscow State University, , Moscow, Russia, ABSTRACT - A review of empirical data -based models of the magnetopause and a comparative analysis are given with special attention to the dynamics of the dayside boundary. Recently different research groups have presented new magnetopause models as an alternative to the model of Roelof and Sibeck (1993). All models have a greater parametric extent than the model of Roelof and Sibeck and allow prediction of the magnetopause location during extreme solar wind and IMF conditions. The models of Shue et al. (1997) and Kuznetsov et al. (1998), developed using classic multi-factor regression analysis are two-dimensional and bivariate. The model of Dmitriev et al. (1998) created using artificial neural networks (ANNs) is three-dimensional and contains multiple parameters. A statistical study of Kuznetsov et al. confirmed by the ANN modeling of Dmitriev et al. has shown that the shape of dayside magnetopause has dawn-dusk asymmetry. The uncertainty in the determination of the dayside magnetopause position is practically the same for these models in spite of some discrepancies of the model results caused by different data sets, different assumptions and functional forms, different treatment methods of the models. 1. INTRODUCTION The problem of the Earth s magnetopause (MP) shape and size was solved theoretically (Ferraro, 1960; Mead and Beard, 1964; Spreiter et al., 1966; Tsyganenko, 1

2 1989) and studied empirically (Fairfield, 1971; Sibeck et al., 1991; Roelof and Sibeck, 1993; Kuznetsov et al., 1992,1994; Kuznetsov and Suvorova, 1996,1997,1998a,b; Petrinec and Russell, 1993, 1996). In this paper we compare three new empirical magnetopause models (Shue et al., 1997; Kuznetsov et al., 1998; Dmitriev et al., 1998) with the model of Roelof and Sibeck (1993) (here and after RS model). The magnetopause model can be used both for modeling the realistic magnetospheric magnetic field and for comparisons with numerical simulations or theoretical models. In most empirical magnetopause models an axial symmetry of the surface was assumed, which offers a reasonably good first approximation. But this assumption can lead to certain errors because realistic magnetopause surface is asymmetrical (Sibeck et al., 1991; Kuznetsov et al. 1992; Petrinec and Russell, 1995; Kuznetsov and Suvorova, 1997). Simple parabolic, elliptical or trigonometric funtions provide a good fit of the MP shape at distances as far as 20 R E in the tailward direction. At larger distances on the nightside two or more separate functional forms were used, then the solutions were attached to each other in a piecewise continuous manner (Petrinec and Russell, 1996; Kuznetsov and Suvorova, 1994). In previous statistical studies a number of external (solar wind plasma and magnetic field) and internal (geomagnetic activity indices) parameters were examined to evaluate the physical quantities controlling the magnetopause size and shape. It was established that the main parameters controlling the dayside MP position are the dynamic pressure of the solar wind (SW) and the negative B z component of the interplanetary magnetic field (IMF). 2

3 According to the classical Chapman-Ferraro theory the magnetopause position can be derived from the pressure balance between the SW dynamic pressure, p, and the pressure of the geomagnetic field, B 2 /2µ 0, where µ 0 is permeability of vacuum. Therefore the MP distance, at least at the dayside, should vary as p -1/6. The important problem of the quantitative description of B z influence on the MP position is quite successfully solved with empirical modeling only. 2. THE BIVARIATE MAGNETOPAUSE MODEL The RS model is the first complete empirical model based on the largest statistics of MP crossings (795 of crossings have available SW and IMF hourly averaged data) by high-apogee satellites during The MP shape was presented as an ellipsoid, the three parameters of which are second-order bivariate expansions in logarithm of p and B z. The effective parameter range of the RS model is limited by the moderate SW condition (0.5<p<8 npa; B z <7 nt). Since the RS model becomes physically unreasonable immediately outside the parametric ranges and can not be extrapolated to extreme SW conditions, under which geosynchronous satellite crossings at 6.6 R E (in Earth s radius) are often observed, it was necessary to develop a new model (Petrinec and Russell,1993,1996; Kuznetsov and Suvorova,1994,1996). 3. ALTERNATIVE MAGNETOPAUSE MODELS Two such models (Shue et al., 1997; Kuznetsov et al., 1998) are bivariate like the RS model with p and B z IMF input parameters and were developed using the standard regression analysis method. The model of Dmitriev et al. (1998) is multi-parameter and besides p and B z IMF includes B y IMF as an additional input parameter. This model was developed using a nonlinear analysis method - artificial neural networks (ANNs). 3

4 Shue et al. model A new functional form that has a sufficient flexibility to fit the MP shape was used in this model. Shue et al. used different data sets of crossings than in RS model including several geosynchronous crossings and also 5-min averaged IMF B z and p corresponding to each individual crossing. The data set contains 553 crossings acquired near the equatorial region. The pressure varies from 0.5 npa up to 8 npa and B z varies from -20 nt to +15 nt for the MP data set. It was obtained that the subsolar point relates with the dynamic pressure by a power law of -1/(6.6±0.8) and with the B z by a linear function. The model results are presented by rather simple function. Kuznetsov et al. model. The model developed in (Kuznetsov and Suvorova 1994,1998b; Kuznetsov et al., 1998) is based on the main data set of the RS model, with added geosynchronous crossings, and on the hour averages of SW data and IMF. Below we consider some physical aspects of the model. Fig. 1 illustrates the SW statistical distribution in the space of control variables: the dynamic pressure p and B z (GSM) for the time period of The rectangle denotes magnetopause crossings from the RS data set. Within this range, p and B z have the most probable values and correspond to quiet or moderate SW conditions when the magnetopause size is 'normal'. In Fig. 1 the points indicate the individual events of large SW disturbances, when the nearest MP distances to the Earth (6.6 R E and 5.2 R E ) were observed. One can see that the dynamic ranges of both variables at the moment of MP crossings cover nearly the whole range of physically reasonable p and B z values. The 4

5 pressure varies from 0.5 npa up to 50 npa and B z varies from -28 nt to +20 nt for the MP data set. Certain preliminary studies permitted to use the geosynchronous magnetopause crossing (GMC) data set in the model most efficiently (Kuznetsov and Suvorova, 1997,1998a). Complete list of 172 GMCs ( ) is present now on the www-page From the total number of GMCs the non-sporadic events were chosen and also those crossings at given local time, for which the SW pressure value was minimal. Thus the data array of the model (with GMCs added to the RS main data set) included 842 MP crossings with hour averages of p and B z varying in a greater ranges than in Shue et al. model (from 0.5 npa to 33 npa and from -28 to +20 nt, respectively). In these papers it has been concluded that two modes of the magnetopause formation exist for B z >0 and B z <-6 nt. This means that equilibrium MP position is determined exclusively by the pressure balance but with smaller pressure value for B z <- 6 nt than for B z >0, and practically doesn t depend on the B z value. In the intermediate mode (-6>B z >0) the magnetopause depends both on p and B z. On the basis of this conclusion the type of the model function describing dependence on B z was chosen. Finally, a few experimental facts serve as evidence of the asymmetry of the dayside equatorial magnetopause for the disturbed magnetosphere both for B z >0 and B z <0 (Rufenach et al., 1989; Kuznetsov and Suvorova, 1997,1998a). A simplified representation of this phenomenon by a parabola shifted duskward (Kuznetsov and Suvorova, 1997,1998a) allows to easily incorporate it with the MP model. 5

6 Note, that two versions of the model were developed: the magnetopause shape fitted by two parabolas and by one parabola with inclusion of the additional angle dependence. Here we present the calculation formulas for the last model. The formula for calculating the magnetopause distance R at a fixed angle θ is the following: cos θ + 4D E( θ) sin 2E( θ) sin θ θ cosθ 2 2 R = 2 (1) Coefficients D and E(θ) are expressed as function of p and B z : 8.51 D = p E( θ) = 3.45 exp 0.22 p 2 ( B B ) z 200p z 0.15 β { ( 1 cosθ) } p ( B B ) p 0.15 z z (2) where β=4.75/p 0.5. For B z >0 it is better to use a more simplified expressions for the coefficients : D = 0.19 p E( θ) = ( cosθ) β p 0.19 (3) For B z >0 the subsolar point position, the coefficient D, in the model varies as p to the power 1/(5.3±0.3). To take into account the y-asymmetry of the dayside equatorial MP the following analytical semi-empirical approximation by a paraboloid of revolution with shift ρ 0 =y 0 towards the dusk side is suggested (Kuznetsov and Suvorova,1998b): 6

7 X = D E( θ) (?? ) 0 2 ρ Re for D>6.6 Re, B z >0 ρ 0 2Re for D 6.6 Re, B z <-6 nt, where X=Rcosθ and ρ= Rsinθ. Dmitriev et al. model The Artificial Neural Networks (ANNs) are powerful non-linear method widely used for geophysical data analysis (Lundstedt, 1992,1997). ANNs have not been used yet for development of the Earth's magnetopause models. The ANN model of the magnetopause was developed in (Dmitriev and Orlov,1997;Dmitriev et al., 1998) by means of ANNs from package NeuroShell 2 (1996). It was assumed that the MP shape has a mirror symmetry relatively to the ecliptic plane, because insufficient statistics at high latitude in south hemisphere resulted in misrepresented MP shape. 516 of dayside MP crossings from (Kuznetsov et al.,1998) along with various hourly averages of SW plasma and IMF parameters and some physical parameters (dynamic pressure p and some others) were used in the study. The output parameter of the ANN model is the MP distance R. The optimal input parameter set consists of latitude ϕ and longitude λ of MP location, B z and B y IMF components in GSM and dynamic pressure p. The model solution is represented as an analytical expression of R(ϕ, λ, B y, B z, p) in the form of a 3 rd order polynomial. The ANN modeling has showed that the known features on the dayside magnetopause surface and its dynamics can be reproduced quite well. The model dayside MP surface has magnetic cusp regions at latitudes ϕ ±45 and is oblate. The 7

8 MP shape is asymmetrical relatively to the X-axis for all solar wind conditions. The magnetospheric magnetic field erosion due to strong negative B z is observed near the equatorial plane. The R depends on p -1/α (α varies from 1/4.5 for B z >10 to 1/10 for B z <- 12). The contribution of the B y is quantitatively defined. It is shown that there is no dependence of the magnetopause shape on the sign of B y (positive or negative), it has been concluded that magnetopause asymmetry can not be explained by the direction of IMF along Y-axis. 4. DISCUSSION Fig.2a shows the subsolar point position in (p,b z ) coordinate for three models: RS, Kuznetsov et al. and Shue et.al. The Kuznetsov et al. and Shue et al. models are more advanced in comparison with the RS model. The Kuznetsov et al. and Shue et al. models are quite similar in the range of B z >-10 nt and moderate pressure, though the different data sets with different averaging time scale of the SW and IMF parameters were used. The models differ somewhat for the extreme p-b z condition. During positive B z the Shue et al. model predicts the pressure value of 37 npa to move the subsolar magnetopause at 6.6 R E, while the Kuznetsov et al. model predicts 22 npa and fairly well agrees with the value of about 20 npa, derived from the pressure balance. In Fig.2b the subsolar point distance in (p,b z ) coordinates for the neural network model is presented. The results are shown for B y =0 (solid lines) and for B y =20 nt (dashed lines). The difference of the subsolar point location under different B y absolute value is increased with dynamic pressure from -0.5 R E when p<0.8 npa to 0.4 R E when p>10 npa. Also one can see the poor dependence of the subsolar point location on B z in the range of positive and strong negative values B z <-12 nt, this is in good 8

9 agreement with the model of Kuznetsov et al. Nonmonotonous B z dependence was explained by using linear and logarithmic functions on B z as one of the input parameters. Earlier statistical investigation of the shape (Fairfield,1971; Kuznetsov and Suvorova,1998b) did not show significant asymmetry of the magnetopause near the equatorial plane at moderate B z and p values, but asymmetry has been found for large negative B z <-7 nt or large p >20 npa (Kuznetsov and Suvorova, 1997). The neural network modeling shows that the degree of the asymmetry depends on both p and B z and under any solar wind conditions the dimension of the magnetopause at X=0 for Y>0 is larger than for Y<0. To compare the accuracy of the models we use the standard deviation defined as? = (R exp R mod ) 2 N, where R exp is the observation, R mod is the calculation, and N is the total number of data points. For ANN model σ=1.44 R E, for Kuznetsov et al. model σ=1.55 R E and for Shue et al. model σ=1.23 R E. Poorer accuracy of the Kuznetsov et al. and ANN model in comparison to the Shue et al. model could be caused by the use of hourly averaged parameters. 5. CONCLUSION The considered models may be applied in an extended range of solar wind and IMF parameter values. The Shue et al., Kuznetsov et al. and Dmitriev et al. models are in good agreement in most part of the parametric range of (p, B z ). The Kuznetsov et al. (1998 b) model is based on a new physical concept that magnetopause magnetic field on day side does not increase under the influence of negative B z. ANN modeling permits to investigate the influence of many more factors than traditionally used p and B z and to 9

10 reconstruct the real three-dimensional geometry of the magnetopause surface. Also the ANN model defines the influence of IMF B y component on the magnetopause size and shape. The effect of the asymmetry of the magnetopause surface is evaluated in the Kuznetsov et al model and duly represented only in the Dmitriev et al. model. Both models are now presented on the www-page The ANN model will be further developed and improved by using more suitable functional forms and minute averages of the SW and IMF parameters, and extending to the magnetotail locations. Acknowledgement This work was particialy supported by the Russian Foundation of Fundamental Research under grant N

11 REFERENCES Dmitriev, A.V. and Orlov Yu.V. (1997) Neuro Shell and multi-factor analysis coupling in the solution of some space physics problems. In: Book of abstracts of StatPhys- Taipei-1997 International Workshop, Taipei, 41. Dmitriev, A.V., Orlov Yu.V., Persiantsev I.V. and Suvorova A.V. (1998) 3D dayside magnetopause model with artificial neural networks. Submitted to Geomagnetizm. i aeronomia. Fairfield, D.H. (1971) Average and unusual locations of the Earth's magnetopause and bow shock. J. Geophys. Res. 76, Ferraro, V.C.A. (1960) An approximate method of estimating the size and shape of the stationary hollow carved out in a neutral ionized stream of corpuscles impinging on the geomagnetic field. J. Geophys.Res. 65, Kuznetsov, S.N., Zastenker G.N. and Suvorova A.V. (1992) Correlation between interplanetary conditions and the dayside magnetopause, Cosm.Res. (USA) 30, N6, Kuznetsov, S.N. and Suvorova A.V. (1994) Influence of solar wind to some magnetospheric characteristics. In: Proc. 3rd Conference of Doctoral Students, Part II, Charles University, Prague, Kuznetsov, S.N., Suvorova A.V., Zastenker G.N. and Sibeck D.G. (1994) Solar wind control of the geomagnetopause position. In: Proc STEP Symposium. Cospar Colloguium Series. 5, Kuznetsov, S.N. and Suvorova A.V. (1996) Solar wind control of the magnetopause shape and location. Radiation measurement 26, N3,

12 Kuznetsov, S.N. and Suvorova A.V. (1997) Magnetopause shape near geosynchronous orbit. Geomagnetizm i Aeronomia, 37, N3, Kuznetsov, S.N. and Suvorova A.V. (1998a) Solar wind magnetic field and plasma during magnetopause crossings at geosynchronous orbit. To be published in Adv.Space Res. (COSPAR 96). Kuznetsov, S.N. and Suvorova A.V. (1998b) An empirical model of the magnetopause for broad ranges of solar wind pressure and Bz IMF. In: Polar cap boundary phenomena, eds. by J.Moen, A.Egeland, M.Lockwood, Kluwer Academic Publishers, Dordrecht, Kuznetsov, S.N., Suvorova A.V. and Dmitriev A.V. (1998) Magnetopause shape and size. Relation with parameteres of the interplanetary medium. To be published in Geomagnetizm. i aeronomia. Lundstedt, H. (1992) Neural networks and prediction of solar-terrestrial effects. Planetary Space Science Lundstedt, H. (1997) Solar wind magnetosphere coupling: predicted and modeled with intelligent hybrid systems. Phys. Chem.Earth. 22, No7-8, Mead, G.D. and Beard D.B. (1964) Shape of the geomagnetic field solar wind boundary. J. Geophys.Res. 69, NeuroShell 2 User Manual. Fourth edition (1996) Ward Systems Group, Inc., Frederick (MD, USA, June 1996). Petrinec, S.M. and C.T.Russell (1993) An empirical model of the size and shape of the near-earth magnetotail. Geophys.Res.Lett. 20,

13 Petrinec, S.M. and C.T.Russell (1995) An examination of the effect of dipole tilt angle and cusp regions on the dayside magnetopause. J. Geophys.Res. 100, Petrinec, S.M. and C.T.Russell (1996) Near-Earth magnetotail shape and size as determined from the magnetopause flaring angle. J. Geophys. Res. 101, 137 Roelof, E.C. and Sibeck D.G. (1993) The magnetopause shape as a bivariate function of IMF Bz and solar wind dynamic pressure. J. Geophys. Res. 98, Rufenach, C.L., Martin R.F.,Jr. and Sauer H.H. (1989) A study of geosynchronous magnetopause crossings. J.Geophys. Res. 94, Shue, J.-H., J.K.Chao, H.C.Fu, C.T.Russell, P.Song, K.K.Khurana and H.J.Singer (1997) A new functional form to study the solar wind control of the magnetopause size and shape. J.Geophys. Res. 102, Sibeck, D.G., Lopez R.E. and Roelof E.C. (1991) Solar wind control of the magnetopause shape, location, and motion. J.Geophys.Res. 96, Spreiter, J.R., Summers A.L., Alksne A.Y. (1966) Hydromagnetic flow around the magnetosphere. Planet.Space Sci. 14, Tsyganenko, N.A. (1989) A solution of the Chapman-Ferraro problem for an ellipsoidal magnetopause. Planet.Space Sci. 37,

14 FIGURE CAPTIONS Fig.1 Contour plot for the 2D distribution function in Log(p)-B z variable space, derived from the statistics of the hourly averaged parameters of the solar wind and IMF during The rectangle denotes the range of p and B z values matched with the magnetopause crossings from the data set of Roelof and Sibeck (1993). Dark circles indicate the individual events of the magnetopause crossings by various geosynchronous satellites. Fig.2 Contour plot for the subsolar point distance as a function of p and B z. (a)- comparison of the RS (dushed curves), Kuznetsov et al. (solid curves) and Shue et.at. (dushed-dotted curves) models; (b) - ANN model calculation for B y =0 (colid curves) and for B y =20 nt (dashed curves). 14

15 L o g ( P ) (n P a ) Bz (nt) Fig. 1 15

16 Xo(Re) Bz IM F, nt SW Pressure,nPa (a) 5 B z ( n T ) Fig log(p) (npa) (b) 16