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A parametric study of magnetosphere–ionosphere coupling in the paleomagnetosphere
Advances in Space Research 38 (2006) 1707–1712 www.elsevier.com/locate/asr
A parametric study of magnetosphere–ionosphere coupling in the paleomagnetosphere B. Zieger
a,*
, J. Vogt a, A.J. Ridley b, K.-H. Glassmeier
c
a School of Engineering and Science, International University Bremen, Campus Ring 1, D-28759 Bremen, Germany Space Physics Research Laboratory, University of Michigan, 2455 Hayward Street, Ann Arbor, MI 48109-2143, USA Institute fu¨r Geophysik und extraterrestrische Physik, Technische Universita¨t Braunschweig, Mendelssohnstr. 3, D-38106 Braunschweig, Germany b
c
Received 29 October 2004; received in revised form 9 February 2005; accepted 27 April 2005
Abstract Geomagnetic polarity reversals last about 1–8 thousand years, while the intensity of the EarthÕs dipole field may drop as much as 1 order of magnitude. In this paper, we aim at modeling the paleomagnetosphere, in particular, the magnetosphere–ionosphere coupling during such polarity transition epochs, using the BATS-R-US global magnetohydrodynamic (MHD) simulation code. We study the variation of the transpolar potential and the total field aligned currents with changing dipole moment, ionospheric Pedersen conductance, and interplanetary magnetic field. The simulations remarkably well reproduce the transpolar potential saturation predicted by the analytical Hill model and confirmed by observations. Estimates of transpolar potential and region 1 field aligned currents are given for paleomagnetospheres with strongly decreased dipole moments. In general, a fairly good agreement was found between the simulation results and the predictions of the Hill model, however, MHD simulations give a somewhat steeper drop of transpolar potential as the dipole moment decreases. Some corrections to the dipole scaling relation in the Hill model are suggested. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Magnetosphere–ionosphere coupling; Paleomagnetosphere; Global MHD simulation; Transpolar potential; Field aligned currents
1. Introduction Paleomagnetic measurements clearly show that the EarthÕs internal magnetic field has two stable modes, a quasi-axial dipole with normal and reversed polarity. The transitions between these two stable modes of the geodynamo are referred to as geomagnetic polarity transitions. The exact temporal development of a geomagnetic polarity transitions is not known because of the lack of adequate paleomagnetic data, but we have, at
*
Corresponding author. Permanent address: Geodetic and Geophysical Research Institute, Hungarian Academy of Sciences, P.O. Box 5, H-9401 Sopron, Hungary. Tel.: +49 4212 003 258; fax: +49 421 200 3229. E-mail addresses: [email protected], [email protected] (B. Zieger), [email protected] (J. Vogt), [email protected] (A.J. Ridley), [email protected] (K.-H. Glassmeier).
least, good estimates of the paleointensity and the average duration of such polarity transition epochs. Geomagnetic polarity transitions last about 1–8 thousand years, while the paleointensity may drop as much as 1 order of magnitude (Merrill and McFadden, 1999). Different transition scenarios can be conceived, e.g., strongly reduced axial dipolar, equatorial dipolar or higher order multipolar internal fields. The first MHD simulations of such paleomagnetospheres have been published recently by Zieger et al. (2004) and Vogt et al. (2004). In this paper, we discuss only axial dipolar paleomagnetospheres for simplicity, and focus on the issues of magnetosphere–ionosphere (M–I) coupling. The magnetosphere is formed through the coupled interaction of the solar wind, the EarthÕs internal magnetic field and the conductive ionosphere. It is generally accepted that field aligned currents play a key role in M–I coupling. According to MHD simulations, region 1 field
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.04.077
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aligned currents tend to close on the magnetopause, forming current loops quasi-perpendicular to the solar wind flow, while region 2 field aligned currents close in the near-Earth equatorial magnetotail. In this paper, we carry out a parametric study of M–I coupling, comparing results obtained from analytical (Section 2.1) and numerical (Section 2.2.) models. First, we study the saturation of the transpolar potential for the present-day magnetosphere (Section 3.1) in order to validate the models. Thereafter, we apply the models for paleomagnetospheres, and derive the dipole moment dependence of the transpolar potential and field aligned currents (Section 3.2). Conclusions are given in Section 4.
tential saturation is an intrinsic feature of the Hill model. If the first term in the denominator in Eq. (1) becomes negligible with respect to the second one, Esw drops out from the equation, yielding the so-called saturation potential 4=3 =ðnR0 Þ. US ¼ 4610p1=3 sw D
ð4Þ
The transpolar potential saturation is discussed in more details in Section 3.1. Since Eq. (1) includes the dipole moment dependence and the expected scaling relation of the Pedersen conductance (Eq. (3)), it can be used to estimate the transpolar potential in axial dipolar paleomagnetospheres with reduced dipole moments. Related results are presented in Section 3.2.
2. Models of M–I coupling 2.2. Numerical model: M–I coupling in the BATS-R-US MHD code
2.1. Analytical model: the Hill model We use a simple analytical model of M–I coupling invented by Hill et al. (1976) and further developed and tested by Siscoe et al. (2002a,b). In the Hill model, the solar wind–magnetosphere–ionosphere coupling is a nonlinear process that results from the feedback of the magnetic field generated by two idealized region 1 current loops. This field reduces the EarthÕs dipole field at the magnetopause where field line reconnection occurs, thus limiting the reconnection rate. Eventually, this negative feedback leads to the saturation of the transpolar potential. Without more detailed discussion of the derivation of the Hill model (see Siscoe et al., 2002a), we adopt the following expression: 1=3 4=3 Upc ¼ 57.6Esw psw D F ðhÞ=ðp1=2 sw D þ 0.0125nR0 Esw F ðhÞÞ;
ð1Þ where Upc is the transpolar potential in kV, Esw is the solar wind motional electric field in mV/m, psw is the solar wind ram pressure in nT, D is the dimensionless normalized dipole moment (D = M/M0, where M is the dipole moment for the paleomagnetosphere and M0 is the present-day dipole moment), F(h) describes the interplanetary magnetic field (IMF) clock angle dependence of magnetopause reconnection (F(p) = 1), and n is a geometry coefficient of current flow in the ionospheric OhmÕs law: I 1 ¼ nRUpc ;
ð2Þ
where I1 is the total region 1 field aligned current, and R is the height integrated ionospheric Pedersen conductance. Furthermore, the following dipole scaling relation for the Pedersen conductance (Rassbach et al., 1974; Glassmeier et al., 2004) have been assumed in Eq. (1): R ¼ R0 D1 ;
ð3Þ
where R0 is the present-day ionospheric Pedersen conductance. One can easily realize that the transpolar po-
The BATS-R-US code is a self-consistent global fully three-dimensional MHD code that solves the ideal MHD equations with properly selected boundary conditions using finite volume upwind schemes and a highly efficient adaptive mesh refinement technique (Powell et al., 1999). It has been successfully used to simulate the EarthÕs magnetosphere, the heliosphere, and different planetary magnetospheres (Bauske et al., 1998, 2000; Hansen et al., 2000; Gombosi et al., 2000; Israelevich et al., 2001; Roussev et al., 2003; Kabin et al., 2003; Rae et al., 2004; Manchester et al., 2004). Recently the code has been adapted to simulate dipolar and higher order multipolar paleomagnetospheres as well (Zieger et al., 2004; Vogt et al., 2004). The M–I coupling model has been validated through a comparison of simulated and observed ground-based magnetograms (Ridley et al., 2001), and it was concluded that the global structure of the region 1 field aligned currents are reproduced accurately, whereas region 2 field aligned currents are poorly modeled. Region 2 field aligned currents can be simulated more accurately through coupling an inner magnetospheric convection model, e.g., the Rice Convection Model (Wolf et al., 1982), to the MHD simulation. Since we aim at comparing MHD simulation results with the Hill model, which is based on the feedback of region 1 field aligned currents, it is reasonable to use the standard ionospheric module in this study. The M–I coupling is implemented through an inner boundary condition that represents a two-dimensional resistive ionosphere. This allows the field aligned currents generated by the MHD code to close at the inner boundary. Different ionospheric models can be used with different distributions of the height integrated Pedersen and Hall conductivities. Here, we use the simplest ionospheric model with a uniform Pedersen conductance. Field aligned currents are calculated at Rcurrents (typically 4 RE) near the inner boundary of
B. Zieger et al. / Advances in Space Research 38 (2006) 1707–1712
Rbody ¼ 3L=10; and
ð5Þ
Rcurrents ¼ 4L=10;
ð6Þ
where L is the actual standoff distance of a simulated paleomagnetosphere. The numerical simulations discussed in this paper were run in a simulation box of 64 · 64 · 64 RE, using 18,000 blocks of 4 · 4 · 4 cells with a resolution of 0.125 RE near the inner boundary. Steady-state solutions were calculated with a Boris factor of 0.011, and the iterations were continued until both the magnetospheric and the ionospheric solutions have converged. The transpolar potential and the total region 1 current were obtained from each simulation, and the corresponding geometry coefficient n was calculated using Eq. (2). This geometry coefficient was then inserted into Eqs. (1) and (2) to plot theoretical reference curves based on the Hill model. The value of n was close to three in all the simulations.
3. Results 3.1. Saturation of the transpolar potential As a first step we wanted to test whether our MHD simulations can reproduce the saturation of transpolar potential as predicted by the Hill model. Therefore, we ran simulations with the present dipole moment, gradually increasing the solar wind electric field from 2 to 20 mV/m through increasing the magnitude of the southward IMF Bz component, while keeping the solar wind ram pressure constant (1.3 nPa). Three sets of simulations were completed with three different values of the ionospheric Pedersen conductance, namely 5, 10, and 20 S. The simulated transpolar potentials are plotted in Fig. 1 together with the corresponding curves of the Hill model (Eq. (1)). The geometry coefficient was obtained for each simulation through inserting the simulated transpolar potential and total region 1 current into Eq. (2). As no significant trend was found in n in
300 5S
250 Transpolar Potential (kV)
the MHD simulation and mapped down to the height of the ionosphere (approximately 1 RE), where the ionospheric potential equation is solved. The potential is then mapped up along the field lines to the inner boundary at Rbody (typically 3 RE), where the corresponding electric field and plasma velocities are calculated (v = E · B/B2). Since the standoff distance of a paleomagnetosphere with strongly reduced dipole moment is comparable to 3 RE, we had to adjust the inner boundaries to the actual size of the paleomagnetosphere. In order to eliminate the possible numerical and/or geometric effects related to the choice of the inner boundary, we applied the following scaling relations for Rbody and Rcurrents in all simulations presented here:
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200
150
10 S
100 20 S 50
0 0
5
10 15 Solar Wind Electric Field (mV/m)
20
Fig. 1. Simulated (symbols) and theoretically predicted (curves) transpolar potentials as a function of the solar wind electric field for 5, 10, and 20 S Pedersen conductance, respectively. The IMF Bz component was kept constant at 5 nT.
these sets of simulations, a constant, the median of the simulated n values (2.97), was used in the Hill model. The simulated transpolar potential curves follow the Hill model remarkably well. The higher the Pedersen conductance is, the sooner the saturation value is reached. Among the three sets of simulations, the best fit was found for the Pedersen conductance of 10 S. However, one can see in Fig. 1 that the Hill model slightly underestimates the simulated transpolar potential for lower values of Pedersen conductance, and slightly overestimates it for higher Pedersen conductance. This may imply that the ionospheric conductance dependence of M–I coupling is somewhat different in the MHD simulations from that in the Hill model. Numerical effects related to geometry and grid resolution are excluded in this case, as the size of the magnetosphere and the resolution of the ionosphere hardly changes, if it changes at all, between simulations with different Pedersen conductances. Another possible numerical effect could be a different convergence rate of the ionospheric solutions. The convergence rate is indeed slower for higher conductance, but as mentioned above, we ran our steady-state simulations until the solutions have fully converged. Anyway, a perfect agreement between the simulated transpolar potentials and the transpolar potentials predicted by the Hill model cannot be expected, as the Hill model is based on a very simple two-dimensional current loop representation of region 1 currents, whereas numerical MHD models simulate the three-dimensional field aligned current systems in a fully self-consistent manner. The total region 1 field aligned currents for the three sets of simulations with different Pedersen conductances are shown in Fig. 2. The corresponding current curves of
B. Zieger et al. / Advances in Space Research 38 (2006) 1707–1712 5 20 S 4.5
10 S
Region 1 FAC (MA)
4
5S
3.5 3 2.5 2 1.5 1 0
5
10
15
20
Solar Wind Electric Field (mV/m)
Fig. 2. Simulated (symbols) and theoretically predicted (curves) total region 1 field aligned currents as a function of the solar wind electric field for 5, 10, and 20 S Pedersen conductance, respectively. The IMF Bz component was kept constant at 5 nT.
the Hill model (also plotted in this figure) were calculated from Eq. (2), using the theoretical transpolar potential from Eq. (1). The geometry coefficient n was taken from the simulations, as described above. It should be mentioned that the simulated total region 1 currents were calculated through integrating current densities on an ionospheric grid of 65 grid points in latitude and 257 grid points in longitude for each hemisphere, which might have introduced a small discretization error in the simulated current curves. The simulated region 1 current curves follow the predicted ones reasonably well. Region 1 currents increase with increasing solar wind electric field and eventually approach a saturation value, as one would expect from the linear current–voltage relation of Eq. (2). The saturation current is reached faster, i.e., at lower solar wind electric field, for higher Pedersen conductance. Like in case of the potentials, the Hill model tends to overestimate the simulated total region 1 currents for higher Pedersen conductance and underestimate it for lower Pedersen conductance. The best fit is again at 10 S Pedersen conductance. This discrepancy in the Pedersen conductance dependence cannot be attributed to numerical effects, as discussed above.
spheric simulations, we used the dipole scaling relation of the ionospheric Pedersen conductance assumed in the Hill model (Eq. (3)), where we set the value of R0 to 5 S. Moreover, we had to introduce scaling relations for the inner boundary of the simulations (Eqs. (5) and (6)) as the size of the simulated paleomagnetospheres varied on a large scale, from a standoff distance of 3.6RE to 10.4RE. The simulated dipole scaling curves of the transpolar potential are plotted in Fig. 3, along with the corresponding theoretical curves of the Hill model. The simulated geometry coefficient n showed a linear trend in the function of the dipole moment in these sets of simulations, but the absolute value changed only slightly from 3.2 to 2.9. Therefore, we used the best fitting linearly changing n in the transpolar potential curves of the Hill model. The dipole moment dependence of the transpolar potential is similar in the two models regarding the quasi-linear decrease with decreasing dipole moment and the convergence of the potential curves with different Bz, but the simulated transpolar potentials show a significantly steeper drop with decreasing dipole moment. This steeper drop of the transpolar potential can be attributed to several factors. Dayside reconnection controlled by southward IMF (negative Bz) leads to magnetic flux erosion decreasing the standoff distance. This Bz dependence of the size of the magnetosphere is not taken into account in the Hill model. The dipole scaling relation of the standoff distance is Bz dependent, too. For example, our simulations yielded a D0.39 dipole scaling of the standoff distance for Bz = 30 nT, whereas the Hill model assumes a scaling relation of D1/3. It means that the Hill model overestimates the size of the magnetosphere for low dipole mo-
250 –30 nT 200 Transpolar Potential (kV)
1710
–20 nT
150 –10 nT 100 –5 nT 50
3.2. Transpolar potential in paleomagnetospheres After validating our M–I coupling models for the present-day magnetosphere, we investigated the dipole scaling relations of transpolar potential and region 1 field aligned currents in paleomagnetospheres. We ran four sets of simulations with Bz of 5, 10, 20, and 30 nT, respectively, gradually changing the normalized dipole moment from 0.1 to 1. In these paleomagneto-
0 0
0.2
0.4 0.6 0.8 Normalized Dipole Moment
1
Fig. 3. Simulated (symbols) and theoretically predicted (curves) transpolar potentials as a function of the dipole moment for 5, 10, 20, and 30 nT IMF Bz component, respectively. The Pedersen conductance was scaled with the dipole moment according to (Eq. (3)), where R0 had been set to 5 S.
B. Zieger et al. / Advances in Space Research 38 (2006) 1707–1712
ments, which results in a higher predicted transpolar potential. The second factor influencing the slope of the theoretical transpolar potential curve is the Pedersen conductance, which increases with decreasing dipole moment according to the dipole scaling relation of Eq. (3). We learned in Section 3.1 that the Hill model overestimates the transpolar potential for higher Pedersen conductance for some yet unknown physical reason. As a third possible factor we should mention the possible effect of the resolution of the MHD simulations. As we moved the inner boundary of the simulation (and the surface where the field aligned currents are calculated) closer to the Earth in the paleomagnetic simulations, the resolution of the ionosphere, i.e., the number of mapping points to the ionosphere, decreased significantly for paleomagnetospheres with smaller dipole moments. The lower ionospheric resolution can result in smaller peaks in the ionospheric potential pattern, which yields a smaller simulated transpolar potential for smaller dipole moments. In addition, the Alfve´n speed can get higher close to the Earth, which limits the local time step of the simulation, delaying the convergence of the numerical solution. This effect is, however, more or less compensated by the reduced dipole field of the paleomagnetosphere. In our paleomagnetospheric simulations the ionospheric potential solution converges reasonably well, thus this effect must be of minor importance. The dipole moment dependence of the simulated and predicted region 1 currents is plotted in Fig. 4. Here, the difference between the simulated and theoretically predicted curves is more striking than in case of the transpolar potential. While the simulated region 1 currents
4 3.5 –30 nT
Region 1 FAC (MA)
3
–20 nT
2.5 2
–10 nT
1.5 –5 nT 1 0.5 0 0
0.2
0.4
0.6
0.8
1
Normalized Dipole Moment
Fig. 4. Simulated (symbols) and theoretically predicted (curves) total region 1 field aligned currents as a function of the dipole moment for 5, 10, 20, and 30 nT IMF Bz component, respectively. The Pedersen conductance was scaled with the dipole moment according to (Eq. (3)), where R0 had been set to 5 S.
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decrease systematically with decreasing dipole moment, the current curves predicted by the Hill model show a slow increase and a subsequent decline as the dipole moment decreases. We attribute this behavior mainly to the different Pedersen conductance dependence of M–I coupling in the Hill model and the MHD simulations.
4. Conclusions It has been demonstrated in Section 3.1 that MHD simulations reasonably well reproduce the characteristic features of the Hill model for the present-day magnetosphere. The two M–I coupling models show very similar Bz dependence, i.e., the saturation of the transpolar potential with increasing solar wind electric field, which has been validated by observations (Russell et al., 2000; Siscoe et al., 2002a). The Pedersen conductance dependence of the MHD simulations is slightly different from that implemented in the Hill model. We are convinced that this difference comes from the different physical approach of M–I coupling in the theoretical and numerical models, and not from artificial numerical effects. The theoretical and numerical models yield comparable dipole scaling relations for the transpolar potential (see Section 3.2), but the transpolar potential drops faster in the simulations as the dipole moment decreases. We explain this difference with a cumulative effect of several factors. For example, the scale size of the magnetosphere is not modeled properly in the Hill model. We suggest that the Bz dependence of the dipole scaling of the magnetospheric scale size should be taken into account in the Hill model. Such a correction to the dipole scaling relation has been suggested recently on purely theoretical basis (Vogt and Glassmeier, 2001). Furthermore, it has been proven earlier by in situ satellite observations that the shape of the magnetopause depends not only on the solar wind dynamic pressure but also on Bz (Roelof and Sibeck, 1993). We are going to investigate the Bz dependence of the dipole scaling of paleomagnetospheric scale size by means of MHD simulations in a subsequent paper. The Pedersen conductance dependence of the Hill model needs to be revised as well. We believe that a greater part of the discrepancy between simulated and theoretically predicted transpolar potentials (or region 1 field aligned currents) is related to this parameter. The grid resolution effect in the MHD simulations of paleomagnetospheres should be further quantified and controlled in future simulations. It has been shown that MHD simulations can be successfully applied in modeling the M–I coupling in paleomagnetospheres. The comparison of the analytical Hill model and our numerical MHD model reasonably well confirmed the validity of the physics involved in these models. In addition, our study revealed some weak
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points in both models that should be considered in further theoretical models or numerical simulations of M–I coupling in paleomagnetospheres or other planetary magnetospheres.
Acknowledgements B.Z., J.V. and K.H.G. gratefully acknowledge support by the German Research Council (DFG) under Grants Nos. VO-855/1 and GL-142/12. Simulation results were obtained using BATS-R-US, developed by the Center for Space Environment Modeling at the University of Michigan with funding support from NASA ESS, NASA ESTO-CT, NSF KDI and DoD MURI. Simulation runs were carried out on high-performance computing facilities of the International University Bremen. References Bauske, R., Nagy, A.F., Gombosi, T.I., De Zeeuw, D.L., Powell, K.G., Luhmann, J.G. A three-dimensional MHD study of solar wind mass loading processes at Venus: effects of photoionization, electron impact ionization, and charge exchange. J. Geophys. Res. 103 (A10), 23625–23638, 1998. Bauske, R., Nagy, A.F., De Zeeuw, D.L., Gombosi, T.I., Powell, K.G. 3D multiscale mass loaded MHD simulations of the solar wind interaction with Mars. Adv. Space Res. 26 (10), 1571–1575, 2000. Glassmeier, K.-H., Vogt, J., Stadelmann, A., Buchert, S. Concerning long-term geomagnetic variations and space climatology. Ann. Geophys. 22 (10), 3669–3677, 2004. Gombosi, T.I., De Zeeuw, D.L., Groth, C.P.T., Powell, K.G. Magnetospheric configuration for parker-spiral IMF conditions: results of a 3D AMR MHD simulation. Adv. Space Res. 26 (1), 139–149, 2000. Hansen, K.C., Gombosi, T.I., De Zeeuw, D.L., Groth, C.P.T., Powell, K.G. A 3D global MHD simulation of SaturnÕs magnetosphere. Adv. Space Res. 26 (10), 1681–1690, 2000. Hill, T.W., Dessler, A.J., Wolf, R.A. Mercury and Mars: the role of ionospheric conductivity in the acceleration of magnetospheric particles. Geophys. Res. Lett. 3, 429–432, 1976. Israelevich, P.L., Gombosi, T.I., Ershkovich, A.I., Hansen, K.C., Groth, C.P.T., De Zeeuw, D.L., Powell, K.G. MHD simulation of the three-dimensional structure of the heliospheric current sheet. Astron. Astrophys. 376 (1), 288–291, 2001. Kabin, K., Rankin, R., Marchand, R., Gombosi, T.I., Clauer, C.R., Ridley, A.J., Papitashvili, V.O., De Zeeuw, D.L. Dynamic response
of the EarthÕs magnetosphere to by reversals. J. Geophys. Res. 108 (A3). Art. No.1132. Manchester, W.B., Gombosi, T.I., Ridley, A.J., Roussev, I., De Zeeuw, D.L., Sokolov, I.V., Powell, K.G., To´th, G. Modeling a space weather event from the Sun to the Earth: CME generation and interplanetary propagation. J. Geophys.Res. 109 (A2), 2004. Art. No. A02107. Merrill, R.T., McFadden, P.L. Geomagnetic polarity transitions. Rev. Geophys. 37, 201–226, 1999. Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I., DeZeeuw, D.L. A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys. 154, 284–309, 1999. Rae, I.J., Kabin, K., Rankin, R., Fenrich, F.R., Liu, W., Wanliss, J.A., Ridley, A.J., Gombosi, T.I., De Zeeuw, D.L. Comparison of photometer and global MHD determination of the open-closed field line boundary. J. Geophys. Res. 109 (A1), 2004. Art. No. A01204. Rassbach, M.E., Wolf, R.A., Daniell Jr., R.E. Convection in a Martian magnetosphere. J. Geophys. Res. 79, 1125–1127, 1974. Ridley, A.J., De Zeeuw, D.L., Gombosi, T.I., Powell, K.G. Using steady-state MHD results to predict the global state of the magnetosphere–ionosphere system. J. Geophys. Res. 106 (A12), 30,067–30,076, 2001. Roelof, E.C., Sibeck, D.G. Magnetopause shape as a bivariate function of interplanetary magnetic field Bz and solar wind dynamic pressure. J. Geophys. Res. 98, 21,412–21,450, 1993. Roussev, I.I., Gombosi, T.I., Sokolov, I.V., Velli, M., Manchester, W., DeZeeuw, D.L., Liewer, P., To´th, G., Luhmann, J.G. A threedimensional model of solar wind incorporating solar magnetogram observations. Astrophys. J. 595, L57–L61, 2003. Russell, C.T., Lu, G., Luhmann, J.G. Lessons from the ring current injection during the September 24, 25, 1998 storm. Geophys. Res. Lett. 27, 1371–1374, 2000. ¨ , Maynard, N.C., Siscoe, G.L., Erickson, G.M., Sonnerup, B.U.O Schoendorf, J.A., Siebert, K.D., Weimer, D.R., White, W.W., Wilson, G.R. Hill model of transpolar potential saturation: comparisons with MHD simulations. J. Geophys. Res. 107 (A6), 2002a. Art. No. 1075. Siscoe, G.L., Crooker, N.U., Siebert, K.D. Transpolar potential saturation: roles of region1 current system and solar wind ram pressure. J. Geophys. Res. 107 (A10), 2002b. Art. No. 1321. Vogt, J., Glassmeier, K.-H. Modelling the paleomagnetosphere: strategy and first results. Adv. Space Res. 28 (6), 863–868, 2001. Vogt, J., Zieger, B., Stadelmann, A., Glassmeier, K.-H., Gombosi, T.I., Hansen, K.C., Ridley, A.J. MHD simulations of quadrupolar paleomagnetospheres. J. Geophys. Res. 109 (A12), A12221, 2004. Wolf, R.A., Harel, M., Spiro, R.W., Voigt, G.-H., Reiff, P.H., Chen, C.K. Computer simulation of inner magnetospheric dynamics for the magnetic storm of July 29, 1977. J. Geophys. Res. 87, 5949– 5962, 1982. Zieger, B., Vogt, J., Glassmeier, K.-H., Gombosi, T.I. Magnetohydrodynamic simulation of an equatorial dipolar paleomagnetosphere. J. Geophys. Res. 109 (A7), 2004. Art. No. A07205.
A parametric study of magnetosphere–ionosphere coupling in the paleomagnetosphere B. Zieger
a,*
, J. Vogt a, A.J. Ridley b, K.-H. Glassmeier
c
a School of Engineering and Science, International University Bremen, Campus Ring 1, D-28759 Bremen, Germany Space Physics Research Laboratory, University of Michigan, 2455 Hayward Street, Ann Arbor, MI 48109-2143, USA Institute fu¨r Geophysik und extraterrestrische Physik, Technische Universita¨t Braunschweig, Mendelssohnstr. 3, D-38106 Braunschweig, Germany b
c
Received 29 October 2004; received in revised form 9 February 2005; accepted 27 April 2005
Abstract Geomagnetic polarity reversals last about 1–8 thousand years, while the intensity of the EarthÕs dipole field may drop as much as 1 order of magnitude. In this paper, we aim at modeling the paleomagnetosphere, in particular, the magnetosphere–ionosphere coupling during such polarity transition epochs, using the BATS-R-US global magnetohydrodynamic (MHD) simulation code. We study the variation of the transpolar potential and the total field aligned currents with changing dipole moment, ionospheric Pedersen conductance, and interplanetary magnetic field. The simulations remarkably well reproduce the transpolar potential saturation predicted by the analytical Hill model and confirmed by observations. Estimates of transpolar potential and region 1 field aligned currents are given for paleomagnetospheres with strongly decreased dipole moments. In general, a fairly good agreement was found between the simulation results and the predictions of the Hill model, however, MHD simulations give a somewhat steeper drop of transpolar potential as the dipole moment decreases. Some corrections to the dipole scaling relation in the Hill model are suggested. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Magnetosphere–ionosphere coupling; Paleomagnetosphere; Global MHD simulation; Transpolar potential; Field aligned currents
1. Introduction Paleomagnetic measurements clearly show that the EarthÕs internal magnetic field has two stable modes, a quasi-axial dipole with normal and reversed polarity. The transitions between these two stable modes of the geodynamo are referred to as geomagnetic polarity transitions. The exact temporal development of a geomagnetic polarity transitions is not known because of the lack of adequate paleomagnetic data, but we have, at
*
Corresponding author. Permanent address: Geodetic and Geophysical Research Institute, Hungarian Academy of Sciences, P.O. Box 5, H-9401 Sopron, Hungary. Tel.: +49 4212 003 258; fax: +49 421 200 3229. E-mail addresses: [email protected], [email protected] (B. Zieger), [email protected] (J. Vogt), [email protected] (A.J. Ridley), [email protected] (K.-H. Glassmeier).
least, good estimates of the paleointensity and the average duration of such polarity transition epochs. Geomagnetic polarity transitions last about 1–8 thousand years, while the paleointensity may drop as much as 1 order of magnitude (Merrill and McFadden, 1999). Different transition scenarios can be conceived, e.g., strongly reduced axial dipolar, equatorial dipolar or higher order multipolar internal fields. The first MHD simulations of such paleomagnetospheres have been published recently by Zieger et al. (2004) and Vogt et al. (2004). In this paper, we discuss only axial dipolar paleomagnetospheres for simplicity, and focus on the issues of magnetosphere–ionosphere (M–I) coupling. The magnetosphere is formed through the coupled interaction of the solar wind, the EarthÕs internal magnetic field and the conductive ionosphere. It is generally accepted that field aligned currents play a key role in M–I coupling. According to MHD simulations, region 1 field
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.04.077
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aligned currents tend to close on the magnetopause, forming current loops quasi-perpendicular to the solar wind flow, while region 2 field aligned currents close in the near-Earth equatorial magnetotail. In this paper, we carry out a parametric study of M–I coupling, comparing results obtained from analytical (Section 2.1) and numerical (Section 2.2.) models. First, we study the saturation of the transpolar potential for the present-day magnetosphere (Section 3.1) in order to validate the models. Thereafter, we apply the models for paleomagnetospheres, and derive the dipole moment dependence of the transpolar potential and field aligned currents (Section 3.2). Conclusions are given in Section 4.
tential saturation is an intrinsic feature of the Hill model. If the first term in the denominator in Eq. (1) becomes negligible with respect to the second one, Esw drops out from the equation, yielding the so-called saturation potential 4=3 =ðnR0 Þ. US ¼ 4610p1=3 sw D
ð4Þ
The transpolar potential saturation is discussed in more details in Section 3.1. Since Eq. (1) includes the dipole moment dependence and the expected scaling relation of the Pedersen conductance (Eq. (3)), it can be used to estimate the transpolar potential in axial dipolar paleomagnetospheres with reduced dipole moments. Related results are presented in Section 3.2.
2. Models of M–I coupling 2.2. Numerical model: M–I coupling in the BATS-R-US MHD code
2.1. Analytical model: the Hill model We use a simple analytical model of M–I coupling invented by Hill et al. (1976) and further developed and tested by Siscoe et al. (2002a,b). In the Hill model, the solar wind–magnetosphere–ionosphere coupling is a nonlinear process that results from the feedback of the magnetic field generated by two idealized region 1 current loops. This field reduces the EarthÕs dipole field at the magnetopause where field line reconnection occurs, thus limiting the reconnection rate. Eventually, this negative feedback leads to the saturation of the transpolar potential. Without more detailed discussion of the derivation of the Hill model (see Siscoe et al., 2002a), we adopt the following expression: 1=3 4=3 Upc ¼ 57.6Esw psw D F ðhÞ=ðp1=2 sw D þ 0.0125nR0 Esw F ðhÞÞ;
ð1Þ where Upc is the transpolar potential in kV, Esw is the solar wind motional electric field in mV/m, psw is the solar wind ram pressure in nT, D is the dimensionless normalized dipole moment (D = M/M0, where M is the dipole moment for the paleomagnetosphere and M0 is the present-day dipole moment), F(h) describes the interplanetary magnetic field (IMF) clock angle dependence of magnetopause reconnection (F(p) = 1), and n is a geometry coefficient of current flow in the ionospheric OhmÕs law: I 1 ¼ nRUpc ;
ð2Þ
where I1 is the total region 1 field aligned current, and R is the height integrated ionospheric Pedersen conductance. Furthermore, the following dipole scaling relation for the Pedersen conductance (Rassbach et al., 1974; Glassmeier et al., 2004) have been assumed in Eq. (1): R ¼ R0 D1 ;
ð3Þ
where R0 is the present-day ionospheric Pedersen conductance. One can easily realize that the transpolar po-
The BATS-R-US code is a self-consistent global fully three-dimensional MHD code that solves the ideal MHD equations with properly selected boundary conditions using finite volume upwind schemes and a highly efficient adaptive mesh refinement technique (Powell et al., 1999). It has been successfully used to simulate the EarthÕs magnetosphere, the heliosphere, and different planetary magnetospheres (Bauske et al., 1998, 2000; Hansen et al., 2000; Gombosi et al., 2000; Israelevich et al., 2001; Roussev et al., 2003; Kabin et al., 2003; Rae et al., 2004; Manchester et al., 2004). Recently the code has been adapted to simulate dipolar and higher order multipolar paleomagnetospheres as well (Zieger et al., 2004; Vogt et al., 2004). The M–I coupling model has been validated through a comparison of simulated and observed ground-based magnetograms (Ridley et al., 2001), and it was concluded that the global structure of the region 1 field aligned currents are reproduced accurately, whereas region 2 field aligned currents are poorly modeled. Region 2 field aligned currents can be simulated more accurately through coupling an inner magnetospheric convection model, e.g., the Rice Convection Model (Wolf et al., 1982), to the MHD simulation. Since we aim at comparing MHD simulation results with the Hill model, which is based on the feedback of region 1 field aligned currents, it is reasonable to use the standard ionospheric module in this study. The M–I coupling is implemented through an inner boundary condition that represents a two-dimensional resistive ionosphere. This allows the field aligned currents generated by the MHD code to close at the inner boundary. Different ionospheric models can be used with different distributions of the height integrated Pedersen and Hall conductivities. Here, we use the simplest ionospheric model with a uniform Pedersen conductance. Field aligned currents are calculated at Rcurrents (typically 4 RE) near the inner boundary of
B. Zieger et al. / Advances in Space Research 38 (2006) 1707–1712
Rbody ¼ 3L=10; and
ð5Þ
Rcurrents ¼ 4L=10;
ð6Þ
where L is the actual standoff distance of a simulated paleomagnetosphere. The numerical simulations discussed in this paper were run in a simulation box of 64 · 64 · 64 RE, using 18,000 blocks of 4 · 4 · 4 cells with a resolution of 0.125 RE near the inner boundary. Steady-state solutions were calculated with a Boris factor of 0.011, and the iterations were continued until both the magnetospheric and the ionospheric solutions have converged. The transpolar potential and the total region 1 current were obtained from each simulation, and the corresponding geometry coefficient n was calculated using Eq. (2). This geometry coefficient was then inserted into Eqs. (1) and (2) to plot theoretical reference curves based on the Hill model. The value of n was close to three in all the simulations.
3. Results 3.1. Saturation of the transpolar potential As a first step we wanted to test whether our MHD simulations can reproduce the saturation of transpolar potential as predicted by the Hill model. Therefore, we ran simulations with the present dipole moment, gradually increasing the solar wind electric field from 2 to 20 mV/m through increasing the magnitude of the southward IMF Bz component, while keeping the solar wind ram pressure constant (1.3 nPa). Three sets of simulations were completed with three different values of the ionospheric Pedersen conductance, namely 5, 10, and 20 S. The simulated transpolar potentials are plotted in Fig. 1 together with the corresponding curves of the Hill model (Eq. (1)). The geometry coefficient was obtained for each simulation through inserting the simulated transpolar potential and total region 1 current into Eq. (2). As no significant trend was found in n in
300 5S
250 Transpolar Potential (kV)
the MHD simulation and mapped down to the height of the ionosphere (approximately 1 RE), where the ionospheric potential equation is solved. The potential is then mapped up along the field lines to the inner boundary at Rbody (typically 3 RE), where the corresponding electric field and plasma velocities are calculated (v = E · B/B2). Since the standoff distance of a paleomagnetosphere with strongly reduced dipole moment is comparable to 3 RE, we had to adjust the inner boundaries to the actual size of the paleomagnetosphere. In order to eliminate the possible numerical and/or geometric effects related to the choice of the inner boundary, we applied the following scaling relations for Rbody and Rcurrents in all simulations presented here:
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200
150
10 S
100 20 S 50
0 0
5
10 15 Solar Wind Electric Field (mV/m)
20
Fig. 1. Simulated (symbols) and theoretically predicted (curves) transpolar potentials as a function of the solar wind electric field for 5, 10, and 20 S Pedersen conductance, respectively. The IMF Bz component was kept constant at 5 nT.
these sets of simulations, a constant, the median of the simulated n values (2.97), was used in the Hill model. The simulated transpolar potential curves follow the Hill model remarkably well. The higher the Pedersen conductance is, the sooner the saturation value is reached. Among the three sets of simulations, the best fit was found for the Pedersen conductance of 10 S. However, one can see in Fig. 1 that the Hill model slightly underestimates the simulated transpolar potential for lower values of Pedersen conductance, and slightly overestimates it for higher Pedersen conductance. This may imply that the ionospheric conductance dependence of M–I coupling is somewhat different in the MHD simulations from that in the Hill model. Numerical effects related to geometry and grid resolution are excluded in this case, as the size of the magnetosphere and the resolution of the ionosphere hardly changes, if it changes at all, between simulations with different Pedersen conductances. Another possible numerical effect could be a different convergence rate of the ionospheric solutions. The convergence rate is indeed slower for higher conductance, but as mentioned above, we ran our steady-state simulations until the solutions have fully converged. Anyway, a perfect agreement between the simulated transpolar potentials and the transpolar potentials predicted by the Hill model cannot be expected, as the Hill model is based on a very simple two-dimensional current loop representation of region 1 currents, whereas numerical MHD models simulate the three-dimensional field aligned current systems in a fully self-consistent manner. The total region 1 field aligned currents for the three sets of simulations with different Pedersen conductances are shown in Fig. 2. The corresponding current curves of
B. Zieger et al. / Advances in Space Research 38 (2006) 1707–1712 5 20 S 4.5
10 S
Region 1 FAC (MA)
4
5S
3.5 3 2.5 2 1.5 1 0
5
10
15
20
Solar Wind Electric Field (mV/m)
Fig. 2. Simulated (symbols) and theoretically predicted (curves) total region 1 field aligned currents as a function of the solar wind electric field for 5, 10, and 20 S Pedersen conductance, respectively. The IMF Bz component was kept constant at 5 nT.
the Hill model (also plotted in this figure) were calculated from Eq. (2), using the theoretical transpolar potential from Eq. (1). The geometry coefficient n was taken from the simulations, as described above. It should be mentioned that the simulated total region 1 currents were calculated through integrating current densities on an ionospheric grid of 65 grid points in latitude and 257 grid points in longitude for each hemisphere, which might have introduced a small discretization error in the simulated current curves. The simulated region 1 current curves follow the predicted ones reasonably well. Region 1 currents increase with increasing solar wind electric field and eventually approach a saturation value, as one would expect from the linear current–voltage relation of Eq. (2). The saturation current is reached faster, i.e., at lower solar wind electric field, for higher Pedersen conductance. Like in case of the potentials, the Hill model tends to overestimate the simulated total region 1 currents for higher Pedersen conductance and underestimate it for lower Pedersen conductance. The best fit is again at 10 S Pedersen conductance. This discrepancy in the Pedersen conductance dependence cannot be attributed to numerical effects, as discussed above.
spheric simulations, we used the dipole scaling relation of the ionospheric Pedersen conductance assumed in the Hill model (Eq. (3)), where we set the value of R0 to 5 S. Moreover, we had to introduce scaling relations for the inner boundary of the simulations (Eqs. (5) and (6)) as the size of the simulated paleomagnetospheres varied on a large scale, from a standoff distance of 3.6RE to 10.4RE. The simulated dipole scaling curves of the transpolar potential are plotted in Fig. 3, along with the corresponding theoretical curves of the Hill model. The simulated geometry coefficient n showed a linear trend in the function of the dipole moment in these sets of simulations, but the absolute value changed only slightly from 3.2 to 2.9. Therefore, we used the best fitting linearly changing n in the transpolar potential curves of the Hill model. The dipole moment dependence of the transpolar potential is similar in the two models regarding the quasi-linear decrease with decreasing dipole moment and the convergence of the potential curves with different Bz, but the simulated transpolar potentials show a significantly steeper drop with decreasing dipole moment. This steeper drop of the transpolar potential can be attributed to several factors. Dayside reconnection controlled by southward IMF (negative Bz) leads to magnetic flux erosion decreasing the standoff distance. This Bz dependence of the size of the magnetosphere is not taken into account in the Hill model. The dipole scaling relation of the standoff distance is Bz dependent, too. For example, our simulations yielded a D0.39 dipole scaling of the standoff distance for Bz = 30 nT, whereas the Hill model assumes a scaling relation of D1/3. It means that the Hill model overestimates the size of the magnetosphere for low dipole mo-
250 –30 nT 200 Transpolar Potential (kV)
1710
–20 nT
150 –10 nT 100 –5 nT 50
3.2. Transpolar potential in paleomagnetospheres After validating our M–I coupling models for the present-day magnetosphere, we investigated the dipole scaling relations of transpolar potential and region 1 field aligned currents in paleomagnetospheres. We ran four sets of simulations with Bz of 5, 10, 20, and 30 nT, respectively, gradually changing the normalized dipole moment from 0.1 to 1. In these paleomagneto-
0 0
0.2
0.4 0.6 0.8 Normalized Dipole Moment
1
Fig. 3. Simulated (symbols) and theoretically predicted (curves) transpolar potentials as a function of the dipole moment for 5, 10, 20, and 30 nT IMF Bz component, respectively. The Pedersen conductance was scaled with the dipole moment according to (Eq. (3)), where R0 had been set to 5 S.
B. Zieger et al. / Advances in Space Research 38 (2006) 1707–1712
ments, which results in a higher predicted transpolar potential. The second factor influencing the slope of the theoretical transpolar potential curve is the Pedersen conductance, which increases with decreasing dipole moment according to the dipole scaling relation of Eq. (3). We learned in Section 3.1 that the Hill model overestimates the transpolar potential for higher Pedersen conductance for some yet unknown physical reason. As a third possible factor we should mention the possible effect of the resolution of the MHD simulations. As we moved the inner boundary of the simulation (and the surface where the field aligned currents are calculated) closer to the Earth in the paleomagnetic simulations, the resolution of the ionosphere, i.e., the number of mapping points to the ionosphere, decreased significantly for paleomagnetospheres with smaller dipole moments. The lower ionospheric resolution can result in smaller peaks in the ionospheric potential pattern, which yields a smaller simulated transpolar potential for smaller dipole moments. In addition, the Alfve´n speed can get higher close to the Earth, which limits the local time step of the simulation, delaying the convergence of the numerical solution. This effect is, however, more or less compensated by the reduced dipole field of the paleomagnetosphere. In our paleomagnetospheric simulations the ionospheric potential solution converges reasonably well, thus this effect must be of minor importance. The dipole moment dependence of the simulated and predicted region 1 currents is plotted in Fig. 4. Here, the difference between the simulated and theoretically predicted curves is more striking than in case of the transpolar potential. While the simulated region 1 currents
4 3.5 –30 nT
Region 1 FAC (MA)
3
–20 nT
2.5 2
–10 nT
1.5 –5 nT 1 0.5 0 0
0.2
0.4
0.6
0.8
1
Normalized Dipole Moment
Fig. 4. Simulated (symbols) and theoretically predicted (curves) total region 1 field aligned currents as a function of the dipole moment for 5, 10, 20, and 30 nT IMF Bz component, respectively. The Pedersen conductance was scaled with the dipole moment according to (Eq. (3)), where R0 had been set to 5 S.
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decrease systematically with decreasing dipole moment, the current curves predicted by the Hill model show a slow increase and a subsequent decline as the dipole moment decreases. We attribute this behavior mainly to the different Pedersen conductance dependence of M–I coupling in the Hill model and the MHD simulations.
4. Conclusions It has been demonstrated in Section 3.1 that MHD simulations reasonably well reproduce the characteristic features of the Hill model for the present-day magnetosphere. The two M–I coupling models show very similar Bz dependence, i.e., the saturation of the transpolar potential with increasing solar wind electric field, which has been validated by observations (Russell et al., 2000; Siscoe et al., 2002a). The Pedersen conductance dependence of the MHD simulations is slightly different from that implemented in the Hill model. We are convinced that this difference comes from the different physical approach of M–I coupling in the theoretical and numerical models, and not from artificial numerical effects. The theoretical and numerical models yield comparable dipole scaling relations for the transpolar potential (see Section 3.2), but the transpolar potential drops faster in the simulations as the dipole moment decreases. We explain this difference with a cumulative effect of several factors. For example, the scale size of the magnetosphere is not modeled properly in the Hill model. We suggest that the Bz dependence of the dipole scaling of the magnetospheric scale size should be taken into account in the Hill model. Such a correction to the dipole scaling relation has been suggested recently on purely theoretical basis (Vogt and Glassmeier, 2001). Furthermore, it has been proven earlier by in situ satellite observations that the shape of the magnetopause depends not only on the solar wind dynamic pressure but also on Bz (Roelof and Sibeck, 1993). We are going to investigate the Bz dependence of the dipole scaling of paleomagnetospheric scale size by means of MHD simulations in a subsequent paper. The Pedersen conductance dependence of the Hill model needs to be revised as well. We believe that a greater part of the discrepancy between simulated and theoretically predicted transpolar potentials (or region 1 field aligned currents) is related to this parameter. The grid resolution effect in the MHD simulations of paleomagnetospheres should be further quantified and controlled in future simulations. It has been shown that MHD simulations can be successfully applied in modeling the M–I coupling in paleomagnetospheres. The comparison of the analytical Hill model and our numerical MHD model reasonably well confirmed the validity of the physics involved in these models. In addition, our study revealed some weak
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points in both models that should be considered in further theoretical models or numerical simulations of M–I coupling in paleomagnetospheres or other planetary magnetospheres.
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