# Solar cycle dependence of quiet-time magnetospheric currents and a model of their near-Earth magnetic fields

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## Abstract

Satellite measurements from high-resolution magnetic field mapping missions over almost a complete solar cycle have been used to investigate the variability of major magnetospheric currents during quiet times. As judged from near-Earth magnetic field observations, the magnetotail current system exhibits no long-term trend. Activity-dependent enhancements decay within hours. A suitable quantity to parameterize the temporal variation is the merging electric field, delayed by 60 minutes with respect to its value at the bow shock. Conversely, the ring current strength shows pronounced solar cycle dependences. We show for the first time that a solar cycle dependent bias has to be added to *D*_{ST} to represent the actual ring current strength. It ranges from −15 nT at solar maximum to zero at solar minimum. The scaled solar flux index, *F*_{10.7}, delayed by 20 months, is found to be a suitable proxy for predicting the bias. This empirical result reflects no physical relation between *F*_{10.7} and ring current bias. A spherical harmonic model valid at Earth’s surface is presented that reflects the quiet-time magnetospheric fields.

## Key words

Geomagnetism magnetic field modelling magnetospheric fields magnetospheric currents## 1. Introduction

For high-resolution geomagnetic field modeling from data of low-Earth orbiting satellites, uncertainties in characterizing contributions from ionospheric and magnetospheric currents can be regarded as the prime limiting factor for the achievable accuracy. For satellites, the ionospheric currents generally represent internal sources. Large-scale magneto-spheric currents flow on the magnetopause. They include the Chapman-Ferraro (CF) currents on the sunward side, and they are part of the magnetotail current system on the night side (e.g., Hughes, 1995). Furthermore a magneto-spheric ring current circles the Earth, aligned with the geomagnetic equatorial plane. Large-scale field-aligned currents connect the various magnetospheric currents with the high-latitude ionosphere (e.g., McPherron, 1995).

If vector field measurements are available over the complete surface of a sphere, then it is possible to uniquely distinguish between (1) internal poloidal fields from sources below the satellite (e.g. core, crust, *E*-region currents), (2) toroidal fields due to field-aligned currents passing through the orbital plane, and (3) poloidal fields flowing entirely above the satellite orbit (e.g. magnetospheric currents). We make use of this important property of spherical harmonic (SH) analysis to derive a model of the external poloidal field, which reflects typical characteristics of magneto-spheric current systems. Our analysis is limited to quiet times (*K*_{p} < 3) because of their relevance for modeling Earth’s internal magnetic field.

It has been recognized for quite some time that currents in the inner magnetosphere (i.e. the ring current) are well described in the solar-magnetic (SM) frame. In that frame the *z* axis is parallel to the geomagnetic dipole axis, pointing northward, the *y* axis is perpendicular to the plane defined by dipole axis and the Sun-Earth line, points toward dusk and *x* completes the triad pointing sunward. Currents in the outer magnetosphere are better organized in geocentric-solar-magnetospheric (GSM) coordinates. In that case the *x* axis points to the sun, the *y* axis is perpendicular to the plane defined by the *x* axis and the dipole axis, pointing duskward, and *z* completes the triad, pointing northward. For an efficient description of the external magnetic features it has recently been proposed to separately expand the contributions in SM and GSM coordinates (Maus and Lühr, 2005; Olsen et al., 2005). A reliable separation between field variations in the GSM and SM frames is not simple but, it is achievable. The orientation of the two frames differs by up to ±34° during the solstice seasons. For a reliable separation, at least one year of data has to be analyzed.

*n*= 1 and order

*m*= 0 (subsequently termed coeff (1,0)) describes a uniform field anti-parallel to the

*z*axis of the coordinate system. The geometries of higher degree terms are described elsewhere (e.g. Langel, 1987). The salient features of that model are

- 1)
a constant field (coeff (1, 0)) in GSM coordinates of about 13 nT pointing in opposite direction to the

*Z*_{GSM}component, - 2)
a field contribution in the

*Y*_{GSM}direction that varies at a quarter of the strength of the interplanetary magnetic field (IMF)*B*_{ y }component, - 3)
a constant field (coeff (1, 0)) in SM coordinates of about 7.6 nT pointing in opposite direction to the ZSM component,

- 4)
a field component along the

*Z*_{SM}axis varying as 0.8 ∙*E*_{ST}, the external part of the*D*_{ST}index.

In addition, all the magnetic effects due to induction in the Earth have been considered. Presently this kind of parameterization is widely accepted in the main field modeling community, and it is employed, for example, in model families such as CHAOS (CHAMP, Ørsted, SAC-C) (e.g., Olsen et al., 2006) and Pomme (e.g., Maus et al., 2006).

In recent years it has become increasingly evident that the model of ML (2005) has to be updated in order to better represent the present magnetospheric field under solar minimum conditions. It has been suggested that the constant terms in items 1 and 3 (above) may show some secular variation. In this study we make use of 9 years of CHAMP magnetic field measurements to check the time dependence of the magnetospheric contributions. In the second step suitable proxies are investigated that can be used for parameterizing the temporal changes of the magnetospheric fields.

## 2. Data Set and Processing Approach

Prime data sources for this study are the CHAMP vector and scalar magnetic field measurements. They are fully calibrated and available under product identifier CH-ME-2-FGM-NEC and CH-ME-2-OVM, respectively, from the Information and Science Data Center (ISDC) at GFZ in Potsdam. The employed data interval starts immediately following the launch of CHAMP, 15 July 2000, and lasts until the end of August 2009. Therefore the entire time span, from the peak of the solar cycle to the present deep minimum, is covered. Ørsted data are used for validation purposes. Magnetic field vector data are considered from July 2000 until November 2005 sampled by that spacecraft. Since vector data become progressively sparse after July 2002, Ørsted scalar data from July 2002 to June 2009 are also taken into account.

For main field modeling, data is commonly selected from periods of low magnetic activity. We therefore exclude storm times from our study. Data were selected from periods when the magnetic activity index, *a*_{m}, was less than 15 (*K*_{p} < 3) and the previous 3-hour interval had *a*_{m} < 18. Rapid variations of magnetospheric currents on time scales of the orbital period (1.5 h) are excluded from such a dataset. Furthermore, data from orbits showing abnormal signatures were discarded. No selection with respect to solar wind, IMF or *D*_{ST} conditions was performed. The index *a*_{m} is an alternative version of *a*_{p} (for details see Menvielle and Marchaudon (2007)).

*F*region driven by plasma pressure gradients are corrected for, as proposed by Lühr et al. (2003), making use of the electron density and temperature measurements of CHAMP and an

*E*

_{st}imate of the ion temperature,

*T*

_{i}, following Kohnlein (1986)where the resulting temperature is in Kelvin (K) and LT represents the local time in hours.

To *E*_{st}imate the coefficients describing the magneto-spheric field, we make use of a spherical harmonic expansion by solving simultaneously for coefficients in the GSM and SM frames. Since satellites sample the sphere more densely near the poles than at the equator, an equal-area data weighting is performed prior to the least-squares inversion. Subsequently we make use of the SHA property to derive the coefficients of an external poloidal field.

## 3. Results

The magnetospheric field model derived from CHAMP data has the same structure as the one presented by ML (2005). Due to the longer data interval, time dependences can be investigated. We have used two different approaches to describe the temporal variation of the external field. First we derived the mean magnetospheric contribution from an independent inversion for every year. Subsequently we fitted all data in a single run, accounting for temporal variations over the 9 years by an appropriate parameterization.

### 3.1 Temporal variations of the GSM and SM fields

_{GSM}component. They vary around 8.5 nT without exhibiting any trend.

*Z*

_{sm}component are calculated. Results are presented in Fig. 2 as plus signs with error bars. This quantity shows major solar cycle dependence. Large annual means up to 17 nT are obtained during the years 2001 through 2005; subsequently they decay to 2 nT in 2009. An interpretation in terms of magnetospheric currents will be given in Section 4.

### 3.2 Modeling of the external fields

A widely accepted approach is the parameterization of the various external field sources by suitable parameters. In the case of the SM field, Maus and Weidelt (2004) proposed the *E*_{st} index (external part of *D*_{st}) as a suitable proxy for describing the temporal changes of the ring current effect. In addition, a bias value has to be considered, reflecting the magnetic effect of the quiet-time ring current (for *E*_{st} = 0). The annual means shown in Fig. 2 suggest that the bias can no longer be considered as constant when analyzing a 9-year interval.

In order to obtain a continuous description of the nearEarth external field, we have introduced a number of controlling parameters for the SM and GSM models. Scaling and temporal shift of these parameters are optimized by minimizing the difference between model prediction and satellite observations. The SM model (mainly coeff (1, 0)) is controlled by three different entries. Time variations (order of days) are accounted for by a parameterization with the *E*_{st} index. The strength and variation of the *E*_{st}-related part is plotted in Fig. 2 as a thin line. This curve is systematically below the annual means. To reconcile the model with the observations, we tested several solar cycle dependent functions that could represent the time-variable bias. Promising results were obtained using the solar flux (*F*_{10.7} index) delayed by about 20 months, for parameterizing the second part (dashed curve). The third part is a constant bias (dashed-dotted curve).

*F*

_{10.7a}) has been determined empirically by testing the RMS discrepancy between model prediction and observation. As can be seen from Fig. 3 (top frame), best results are achieved when the field lags behind

*F*

_{10.7a}by 18–22 months. We obtain consistent delay times from independently processed CHAMP and Ørsted data. For our final model we selected a time lag of 20 months, to be applied to the

*F*

_{10.7a}readings. Taking all these effects into account, the SM field in nT is modeled (coeff (1,0)) aswhere the solar flux values have been centered about their mean value,

*F*

_{10.7}= 120 sfu.

*E*

_{m}, as suggested by Kan and Lee (1979). According to these authors

*E*

_{m}is calculated aswhere

*v*

_{SW}is solar wind velocity,

*B*

_{ y }and

*B*

_{ z }are IMF components in GSM frame and Θ is the clock angle of the IMF. For strong solar wind conditions, the transpolar potential goes into saturation. Ober et al. (2003) derived saturation curves for the transpolar potential from DMSP measurements. We make use of their results for computing an effective electric field, Open image in new window where

*E*

_{m}is in mV/m.

*E*

_{m}was validated by comparison with observations. As can be seen in Fig. 3 (bottom frame), an optimal agreement between the data sets is obtained for an Open image in new window delayed by 60 minutes with respect to solar wind data at the bow shock. The uniform GSM field in nT is calculated as followswhere Open image in new window is in mV/m. The temporal evolution of the modeled GSM coeff (1, 0) is plotted in Fig. 1 as a full line. No trend emerges.

In this study we also confirm the partial penetration of the IMF *B*_{ y } field to the Earth’s surface. The scaling factor 0.26 confirms earlier results (e.g., Lesur et al. (2005); ML (2005)). A new element is the optimal delay of 35 minutes with respect to the IMF reading at the bow shock (see Fig. 3, bottom).

## 4. Discussion and Conclusion

The evaluation of continuous magnetic field recordings from satellite over 9 years has revealed new features of magnetospheric field contributions. These features are of concern for main field modelers. In this section we interpret the modeled magnetic fields in terms of currents. In ML (2005), variations in GSM coordinates had been related to the far-field effect of the tail lobe field, and field contributions in the SM frame were related to the ring current plus the Chapman-Ferraro currents.

An important result is the time-dependent bias of the *D*_{st} (*E*_{ST}) index. This means that the *D*_{st} value underestimates the actual field produced by the ring current by a varying amount, ranging from 0 to 15 nT depending on solar cycle phase. The observatory baseline, which is used for *D*_{st} estimation, is obviously contaminated by the slowly varying quiet-time ring current field. Previous modeling studies based on observatory data may have suffered from this imperfection. According to the Dressler-Parker-Schopke theorem, the total energy of the particles in the ring current is proportional to the magnetic field effect at the Earth’s surface. According to Eq. (2), at very low solar flux (e.g. *F*_{10.7a} = 60) and vanishing *E*_{st} (*D*_{ST}) value the ring current practically disappears. Satellite missions such as Cluster and/or THEMIS should be used to verify the ring current decay during this deep solar minimum.

To parameterize the solar cycle dependence of the *E*_{st} (*D*_{ST}) bias, the solar flux index *F*_{10.7a} was used here. No direct causal dependence between the two quantities exists. Therefore, *F*_{10.7a} is probably not the most suitable choice. Significant differences between the observations and the model curve (heavy line in Fig. 1) appear for the years 2002 and 2003. The obtained relationship is probably only valid for solar cycle 23. Thus, the search for a more appropriate parameter reflecting the quiet-time ring current field should be continued.

Another important finding of this study is the long-term stability of the contribution in GSM coordinates. Variations of the field strength aligned with the *Z*_{GSM} component occur on time scales of hours. We identified the merging electric field as a suitable parameter for describing the short-term variability outside of magnetic storms. The derived time-delay for *E*_{m} of 60 minutes with respect to conditions at the bow shock, when parameterizing the *Z*_{GSM} field variations, supports the suggestion of ML (2005), namely, that the field in the GSM frame is primarily caused by tail currents. This implies that no solar cycle dependence of the average tail lobe field could be deduced. The magnetic field generated by Champman-Ferraro currents likely ends up as part of the SM component.

From Eq. (5) we see that close to Earth a GSM field of 8.3 nT is found for *E*_{m} = 0.5 mV/m. This result is smaller than the stable GSM value of 12.9 nT mentioned by ML (2005). Part of the difference can be explained by the higher merging electric field during the years 1999–2004, but some 3 nT remain unexplained. Olsen et al. (2005) also presented a decomposition of the external magnetic field in SM and GSM parts. Their reported value of 8.3 nT for the GSM field agrees fully with the value presented here.

In summary, important features of magnetospheric fields, as observed close to Earth, have been inferred in this study. Solar cycle dependences of the magnetospheric ring current and tail currents were deduced. A new time-dependent SHA model characterizing the different quiet-time magnetospheric contributions has been presented and is available as part of the software of the Pomme-6 model at http://geomag.org/models/pomme6.html. The European Space Agency’s upcoming multi-satellite mission Swarm should be used to develop improved magnetospheric current indices, complementing *D*_{st}.

## Notes

### Acknowledgments

The CHAMP mission is sponsored by the Space Agency of the German Aerospace Center (DLR) through funds of the Federal Ministry of Economics and Technology, following a decision of the German Federal Parliament (grant code 50EE0944). The data retrieval and operation of the CHAMP satellite by the German Space Operations Center (GSOC) is acknowledged. The Ørsted Project was made possible by extensive support from the Danish Government, NASA, ESA, CNES and DARA.

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