Empirical Magnetopause Models Research Articles

Relation between magnetopause position and reconnection rate under quasi-steady solar wind dynamic pressure

The lunar environment heliospheric X-ray imager (LEXI) and solar wind–magnetosphere–ionosphere link explorer (SMILE) will observe the magnetopause motion in soft X-rays to understand dayside reconnection modes as a function of solar wind conditions after their respective launches in the near future. To support their successful science mission, we investigate the relationship between the magnetopause position and the dayside reconnection rate by utilizing super dual auroral radar network (SuperDARN) observations and widely used empirical models of magnetopause position (Shue et al. in J Geophys Res 103:17691–17700. https://doi.org/10.1029/98JA01103, 1998 and Lin et al. in J Geophys Res 115:A04207. https://doi.org/10.1029/2009JA014235, 2010). We select three cases when the interplanetary magnetic field rotates during periods of quasi-steady solar wind dynamic pressure. We first estimate the dayside reconnection rate by calculating the electric field along the open–closed magnetic field boundary (OCB) in the OCB moving reference frame. Then, we estimate the magnetopause position near the local noon by inputting NASA OMNI solar wind data into the empirical magnetopause models. The reconnection rate shows anti-correlation with the magnetopause position that it generally increases as the magnetopause located closer to Earth and vice versa. Our result also confirms that the reconnection rate increases as the empirical coupling efficiency between solar wind and the magnetosphere increases.Graphical abstract

Magnetopause location modeling using machine learning: inaccuracy due to solar wind parameter propagation

An intrinsic limitation of empirical models of the magnetopause location is a predefined magnetopause shape and assumed functional dependences on relevant parameters. We overcome this limitation using a machine learning approach (artificial neural networks), allowing us to incorporate general, purely data-driven dependences. For the training and testing of the developed neural network model, a data set of about 15,000 magnetopause crossings identified in the THEMIS A-E, Magion 4, Geotail, and Interball-1 satellite data in the subsolar region is used. A cylindrical symmetry around the direction of the impinging solar wind is assumed, and solar wind dynamic pressure, interplanetary magnetic field magnitude, cone angle, clock angle, tilt angle, and corrected Dst index are considered as parameters. The effect of these parameters on the magnetopause location is revealed. The performance of the developed model is compared with other empirical magnetopause models. Finally, we demonstrate and discuss the inaccuracy of magnetopause models due to the inaccurate information about the impinging solar wind parameters based on measurements near the L1 point. This inaccuracy imposes a theoretical limit on the precision of magnetopause predictions, a limit that our model closely approaches.

Extremely Distant Magnetopause Locations Caused by Magnetosheath Jets

AbstractMagnetopause position is controlled mainly by the solar wind dynamic pressure and north‐south interplanetary magnetic field component and these quantities are included in different empirical magnetopause models. We have collected about 50,000 of dayside magnetopause crossings observed by THEMIS in course of 2007–2019 and compared the observed magnetopause position with model prediction. The difference between observed and predicted magnetopause radial distance, Robs − Rmod is used for quantifying the model‐observation agreement. Its median values are well predicted for cases up to Robs ≈ 12 RE for all models but higher positive deviations are found for larger magnetopause distances, mainly under a nearly radial field and low dynamic pressure. The analysis reveals their connection with transient magnetopause displacements caused by strong sunward flows in the magnetosheath. We discuss the possible origin of the observed magnetosheath flow switching in terms of the interaction of magnetosheath jets with the magnetopause.

The Relationship between Solar Wind Charge Exchange Soft X-ray Emission and the Tangent Direction of Magnetopause in an XMM–Newton Event

With the advent of soft X-ray imaging enabling global magnetopause detection, it is critical to use reconstruction techniques to derive the 3-dimensional magnetopause location from 2-dimensional X-ray images. One of the important assumptions adopted by most techniques is that the direction with maximum soft X-ray emission is the tangent direction of the magnetopause, which has not been validated in observation so far. This paper analyzes a magnetospheric solar wind charge exchange (SWCX) soft X-ray event detected by XMM–Newton during relatively stable solar wind and geomagnetic conditions. The tangent direction of the magnetopause is determined by an empirical magnetopause model. Observation results show that the maximum SWCX soft X-ray intensity gradient tends to be the tangent of the magnetopause’s inner boundary, while the maximum SWCX soft X-ray intensity tends to be the tangent of the magnetopause’s outer boundary. Therefore, it is credible to use the assumption that the tangent direction of the magnetopause is the maximum SWCX soft X-ray intensity or its gradient when reconstructing the 3-dimensional magnetopause location. In addition, since these two maxima tend to be the inner and outer boundaries of the magnetopause, the thickness of magnetopause can also be revealed by soft X-ray imaging.

Geosynchronous Magnetopause Crossings and Their Relationships With Magnetic Storms and Substorms

AbstractThe paper investigates the strengthening of magnetospheric activity related to geosynchronous magnetopause crossings (GMCs). We make a list of GMC events using the empirical magnetopause model (Lin et al., 2010, https://doi.org/10.1029/2009ja014235) and hourly averaged OMNI data and find which solar wind and magnetospheric conditions accompany and follow the GMCs. The GMCs are mostly caused by the impact of interplanetary coronal mass ejections (ICMEs) and/or interplanetary shocks often with a strong increase in the density and a moderate increase in velocity. The average solar wind density during the first GMC hour is higher than 20 cm−3 in 70% cases, while the velocity is higher than 500 km/s in 56% cases. The hourly interplanetary magnetic field (IMF) BZ is negative in 87% cases. The average over all events SMU (SML), Kp, and PC indices reach maxima (minima) in 1 h after the GMC beginning, while the delay of the minimum of the Dst index is usually 3–8 h. These average time delays do not depend on the strength of the storms and substorms. The SML (Dst) minimum is less than −500 nT (−30 nT) in the next 24 h in 95% (99%) cases, that is, the GMC events are mostly followed by magnetic storms and substorms. We compare solar wind and magnetospheric conditions for GMCs connected with ICMEs and stream interaction regions (SIRs). Our study confirms that the ICME‐related events are characterized by stronger ring current and auroral activity than the SIR‐related events. The difference might be explained by the different behavior of the solar wind velocity.

Long‐Term Variations in Solar Wind Parameters, Magnetopause Location, and Geomagnetic Activity Over the Last Five Solar Cycles

AbstractWe use both solar wind observations and empirical magnetopause models to reconstruct time series of the magnetopause standoff distance for nearly five solar cycles. Since the average annual interplanetary magnetic field (IMF) Bz is about zero, and the annual IMF cone angle varies between 54.0° and 61.2°, the magnetopause standoff distance on this timescale depends mostly on the solar wind dynamic pressure. The annual IMF magnitude well correlates with the sunspot number (SSN) with a zero time lag, while the annual solar wind dynamic pressure (Pdyn) correlates reasonably well with the SSN but with 3‐year time lag. At the same time, we find an anticorrelation between Pdyn and SSN in cycles 20–21 and a correlation in cycles 22–24 with 2‐year time lag. Both the annual solar wind density and velocity well correlate with the dynamic pressure, but the correlation coefficient is higher for density than for velocity. The 11‐year solar cycles in the dynamic pressure variations are superimposed by an increasing trend before 1991 and a decreasing trend between 1991 and 2009. The average annual solar wind dynamic pressure decreases by a factor of 3 from 1991 to 2009. Correspondingly, the predicted standoff distance in Lin et al.'s (2010, https://doi.org/10.1029/2009JA014235) magnetopause model increases from 9.7 RE in 1991 to 11.6 RE in 2009. The annual SSN, IMF magnitude, and magnetospheric geomagnetic activity indices display the same trends as the dynamic pressure. We calculate extreme solar wind parameters and magnetopause standoff distance in each year using daily values and find that both extremely small and large standoff distances during a solar cycle preferably occur at solar maximum rather than at solar minimum.

On the Influence of the Earth's Magnetic Dipole Eccentricity and Magnetospheric Ring Current on the Magnetopause Location

AbstractWe investigate the influence of two typically unconsidered parameters—Earth's magnetic dipole eccentricity and ring current—on the location of the magnetopause. Although empirical magnetopause models generally assume the Earth's magnetic field to be axially symmetric, the terrestrial magnetic dipole is shifted by as much as 500 km out of the Earth's center. Additionally, the magnetic field at the magnetopause is further modified by magnetospheric currents, most importantly by the ring current. In order to quantify the effects related to these phenomena on the magnetopause location, we compare observed magnetopause distances with model distances calculated using a model, which does not take them into account. International Geomagnetic Reference Field is used to describe the dipole eccentricity and relevant multipoles, which results in a statistically observable magnetopause displacement (≈0.2RE) at locations where the magnetic field strength increases/decreases as compared to the dipole field. Additionally, the magnetic field at the magnetopause is modified by the ring current, becoming stronger at the times of a stronger ring current. We use the corrected Dst* index to describe the ring current strength, and we demonstrate that its variation results in a change of the magnetopause stand‐off distance as large as about 0.8RE. We further use the T96 magnetic field model to estimate the total contribution of magnetospheric currents to the magnetic field at the magnetopause and their effects on the magnetopause location. We suggest empirical relations that can be used to incorporate the obtained dependences into existing empirical models of magnetopause location.

Shape of the equatorial magnetopause affected by the radial interplanetary magnetic field

The ability of a prediction of the magnetopause location under various upstream conditions can be considered as a test of our understanding of the solar wind-magnetosphere interaction. The present magnetopause models are parametrized with the solar wind dynamic pressure and usually with the north-south interplanetary magnetic field (IMF) component. However, several studies pointed out an importance of the radial IMF component, but results of these studies are controversial up to now. The present study compares magnetopause observations by five THEMIS spacecraft during long lasting intervals of the radial IMF with two empirical magnetopause models. A comparison reveals that the magnetopause location is highly variable and that the average difference between the observed and predicted positions is ≈+0.7RE under this condition. The difference does not depend on the local times and other parameters, like the upstream pressure, IMF north-south component, or tilt angle of the Earth dipole. We conclude that our results strongly support the suggestion on a global expansion of the equatorial magnetopause during intervals of the radial IMF.

Solar cycle variations of magnetopause locations

The magnetopause location is generally believed to be determined by the solar wind dynamic pressure and by the sign and value of the interplanetary magnetic field vertical (BZ) component. The contribution of other parameters is usually considered to be minor or negligible near the equatorial plane. Recent papers have shown a magnetopause expansion during intervals of a nearly radial IMF but our ability to predict the magnetopause location under steady or slowly changing upstream conditions remains rather weak even if the effect of radial magnetic field is considered. We present a statistical study based on more than 10,000 magnetopause crossings identified in the THEMIS data in the course of the last half of the solar cycle. The observed magnetopause locations are compared with an empirical magnetopause model of Shue et al. (1997) and the sources of differences between observations and model predictions are analyzed. This analysis reveals that the magnetopause location depends on the solar activity being more compressed during the solar maximum. Furthermore, we have found that, beside the solar wind dynamic pressure and vertical magnetic field component, the solar wind speed and ionospheric conductivity (F10.7 used as a proxy) are important physical quantities controlling this compression.

The dayside magnetopause location during radial interplanetary magnetic field periods: Cluster observation and model comparison

Abstract. The present work has investigated the midlatitudinal magnetopause locations under radial interplanetary field (RIMF) conditions. Among 262 (256) earthward (sunward) RIMF events from years of 2001 to 2009, Cluster satellites have crossed the magnetopause 22(12) times, with 10 (7) events occurring at midlatitudes. The observed midlatitudinal magnetopause positions are compared with two empirical magnetopause models (Shue et al., 1998; Boardsen et al., 2000) (hereafter referred to as the Shue98 model and the Boardsen00 model). The observation–model differences exhibit local time asymmetry. For earthward RIMF cases, the Shue98 model underestimates the magnetopause positions in the postnoon sector, while it overestimates the magnetopause positions in the dawn and dusk sectors. The Boardsen00 model generally underestimates the magnetopause after 6 MLT (magnetic local time), with larger deviations in the postnoon sector as compared to those in the prenoon. For sunward RIMF cases, the selected events are mainly clustered around the dawn and dusk sectors. The comparison with the Shue98 model indicates contractions in the dawn and expansions in the dusk sector, while the comparison with Boardsen00 indicates general expansions, with larger expansions in the later local time sectors. The local time variations in the differences between observations and the Shue98 and the Boardsen00 models indicate that the real magnetopause could be asymmetrically shaped during radial IMF periods, which should be considered by magnetopause models. The observation–model differences in the magnetopause positions (Δ RMP) during RIMF periods correlate well with the solar wind dynamic pressure, with larger Δ RMP for larger Pd. The southern magnetopause expands further outward relative to the model prediction when the dipole tilt angle is more negative (local summer in the Southern Hemisphere). For earthward RIMF cases, the generally good correlations between Δ RMP and the IMF cone angle are consistent with the previous hypothesis (Dušík et al., 2010) that, with more radial IMF, the subsolar magnetopause will expand further outward, owever, this is not the case for the comparison with Boardsen00 during sunward IMF periods, as it shows less dependence on the IMF cone angle.

On the performance of global magnetohydrodynamic models in the Earth's magnetosphere

We study the performance of four magnetohydrodynamic models (BATS‐R‐US, GUMICS, LFM, OpenGGCM) in the Earth's magnetosphere. Using the Community Coordinated Modeling Center's Run‐on‐Request system, we compare model predictions with magnetic field measurements of the Cluster, Geotail and Wind spacecraft during a multiple substorm event. We also compare model cross polar cap potential results to those obtained from the Super Dual Auroral Radar Network (SuperDARN) and the model magnetopause standoff distances to an empirical magnetopause model. The correlation coefficient (CC) and prediction efficiency (PE) metrics are used to objectively evaluate model performance quantitatively. For all four models, the best performance outside geosynchronous orbit is found on the dayside. Generally, the performance of models decreases steadily downstream from the Earth. On the dayside most CCs are above 0.5 with CCs for Bx and Bz close to 0.9 for three out of four models. In the magnetotail at a distance of about −130 Earth radii from Earth, the prediction efficiency of all models is below that of using an average value for the prediction with the exception of Bz. Bx is most often best predicted and correlated both on the dayside and the nightside close to the Earth whereas in the far tail the CC and PE for Bz are substantially higher than other components in all models. We also find that increasing the resolution or coupling an additional physics module does not automatically increase the model performance in the magnetosphere.

A new three‐dimensional magnetopause model with a support vector regression machine and a large database of multiple spacecraft observations

We present results from a new three‐dimensional empirical magnetopause model based on 15,089 magnetopause crossings from 23 spacecraft. To construct the model, we introduce a Support Vector Regression Machine (SVRM) technique with a systematic approach that balances model smoothness with fitting accuracy to produce a model that reveals the manner in which the size and shape of the magnetopause depend upon various control parameters without any assumptions concerning the analytical shape of the magnetopause. The new model fits the data used in the modeling very accurately, and can guarantee a similar accuracy when predicting unseen observations within the applicable range of control parameters. We introduce a new error analysis technique based upon the SVRM that enables us to obtain model errors appropriate to different locations and control parameters. We find significant east‐west elongations in the magnetopause shape for many combinations of control parameters. Variations in the Earth's dipole tilt can cause significant magnetopause north/south asymmetries and deviation of the magnetopause nose from the Sun‐Earth line nonlinearly by as much as 5 Re. Subsolar magnetopause erosion effect under southward IMF is seen which is strongly affected by solar wind dynamic pressure. Further, we find significant shrinking of high‐latitude magnetopause with decreased magnetopause flaring angle during northward IMF.

An MHD simulation study on the location and shape of magnetopause in equatorial plane

The determination of the varying position and shape of magnetopause is one of the important Gordian knots in geophysics and space physics. According to the solar wind-magnetosphere-ionosphere coupling global magnetohydrodynamic (MHD) simulation, and with the maximum electric current criterion, we study the position and shape of the magnetopause under several solar wind dynamic pressure (Dp) and interplanetary magnetic field conditions. The simulation results show that the subsolar position (r0) of the magnetopause is controlled mainly by Dp with the significant decrease of r0 as Dp increases. At a certain Dp, when southword Bz (Bz0) decreases to zero, then shifts to northward (Bz0) and increases, the subsolar position r0 keeps increasing. For all cases studie here, the flare angle () of the magnetopause experiences small changes. This provide an evidence for the structural self-similarity of magnetopause in equatorial plane. Compared with the empirical low-latitude magnetopause model of Shue98, MHD simulation can reproduce the dependence of the subsolar point r0 on Dp, while the saturation effect of r0 varying with Bz in empirical model is represented only with slow solar wind. As to the flare angle , although the difference between MHD simulation and empirical model is less than 2.5, the variation of with Bz in MHD simulations is nonlinear and different from the linear trend in empirical model.

IMF cone angle control of the magnetopause location: Statistical study

We investigate the dependence of the magnetopause location on the interplanetary magnetic field (IMF) cone angle (the angle between the IMF and solar wind velocity vectors) in a statistical study based on ≈6500 magnetopause crossings observed by the five THEMIS spacecraft, both at the dayside and flanks. To remove other well‐known effects, we analyze the difference between observed magnetopause radial distances and those predicted by an empirical magnetopause model (scalable by the solar wind dynamic pressure and IMF BZ component). The results demonstrate a systematic increase of the magnetopause distance for radial IMF directions, from ≈0.3 RE at 90° to ≈1.7 RE at 0° or 180° cone angle. Moreover, a stronger dependence of the magnetopause location on the solar wind dynamic pressure than predicted by the current models was observed.

A three‐dimensional asymmetric magnetopause model

A new three‐dimensional asymmetric magnetopause model has been developed for corrected GSM coordinates and parameterized by the solar wind dynamic and magnetic pressures (Pd + Pm), the interplanetary magnetic field (IMF) Bz, and the dipole tilt angle. On the basis of the magnetopause crossings from Geotail, IMP 8, Interball, TC1, Time History of Events and Macroscale Interactions during Substorms (THEMIS), Wind, Cluster, Polar, Los Alamos National Laboratory (LANL), GOES, and Hawkeye, and the corresponding upstream solar wind parameters from ACE, Wind, or OMNI, this model is constructed by the Levenberg‐Marquardt method for nonlinear multiparameter fitting step‐by‐step over the divided regions. The asymmetries of the magnetopause and the indentations near the cusps are appropriately described in this new model. In addition, the saturation effect of IMF Bz on the subsolar distance and the extrapolation for the distant tail magnetopause are also considered. On the basis of this model, the power law index for the subsolar distance versus Pd + Pm is a bit less than −1/6, the northward IMF Bz almost does not influence the magnetopause, and the dipole tilt angle is very important to the north–south asymmetry and the location of indentations. In comparison with the previous empirical magnetopause models based on our database, the new model improves prediction capability to describe the three‐dimensional structure of the magnetopause. It is shown that this new model can be used to quantitatively study how Pd + Pm compresses the magnetopause, how the southward IMF Bz erodes the magnetopause, and how the dipole tilt angle influences the north–south asymmetry and the indentations.

Estimation of magnetopause motion from low‐energy neutral atom emission

A new method for deriving the position of the dayside equatorial magnetopause directly from the measured intensity of energetic neutral atom (ENA) emissions is presented. This approach makes it possible to track the position of the magnetopause using data observed by the low‐energy neutral atom (LENA) imager on board the Imager for Magnetopause‐to‐Aurora Global Exploration (IMAGE) spacecraft. The model is applied to data recorded during a period of high solar wind dynamic pressure on 13 April 2001. In this interval, significant ENA flux was observed originating from the dayside low‐latitude magnetosheath. This ENA flux is primarily the result of enhanced charge exchange between the increased solar wind plasma and exospheric hydrogen neutrals. The temporal variation in the estimated magnetopause position is compared with in situ measurements of magnetopause crossings by the LANL‐01A spacecraft in geosynchronous orbit and the results of a recent empirical magnetopause model. It is demonstrated that the subsolar distance of the magnetopause was successfully tracked for a period of more than 1 h. In this particular case example, the dayside magnetopause is closer to the Earth and fluctuates on a shorter timescale than predicted by the previous empirical model based on in situ data. It is also revealed that the subsolar magnetopause can move with speeds of 100–200 km s−1 in response to marked dynamic pressure changes, and during periods of stable dynamic pressure can fluctuates with speeds of up to 50 km s−1.

The magnetospheric boundary motion during the main phase of the moderate magnetic storm (case study)

As a result of ICME interaction with magnetosphere the negative sudden impulse Si and substorm event were followed. The simultaneous plasma and magnetic field measurements near the magnetospheric boundary (INTERBALL-1), in the nearest magnetotail (GOES 8, 9), in the middle tail (GEOTAIL), in solar wind (WIND), and ground based data were compared to trace the time sequence of events. It was shown that negative sudden impulse Si was observed by ground stations and at geosynchronous altitudes. The character of magnetic field variations that were observed by GOES 9 permit to suggest that compression/expansion wave was propagated inside the magnetosphere. The time sequence of events observed during substorm development pointed that substorm has begun from auroral breakup. The cross current disruption in the nearest tail and neutral line formation in the middle tail occurred almost simultaneously, but some time after auroral breakup. It is possible that both processes associated with releasing of stored energy during substorm were occurred independently in this substorm event. By using an empirical magnetopause model [Shue et al., 1998. Magnetopause location under extreme solar wind conditions. J. Geophys. Res. 103, 17691], the amplitude of magnetopause motion was evaluated. During substorm evolution the amplitude of the magnetopause motion was ∼ 2 –3 Re. The excitation of Kelvin-Helmholtz instability can explain only a few magnetopause crossings.

Finding the Lyon‐Fedder‐Mobarry magnetopause: A statistical perspective

The determination of the location and global structure of the magnetopause is not a well‐defined problem, either in satellite data or in magnetohydrodynamic (MHD) models. Assessing the accuracy of MHD models by comparing single‐point crossings of the magnetopause cannot provide a complete picture of the accuracy of the model. We evaluate the accuracy of magnetopause location in the Lyon‐Fedder‐Mobarry (LFM) MHD model by examining LFM behavior on a statistical, rather than single event basis. Empirical magnetopause models seek to fit a functional form to large data sets of magnetopause crossings parameterized by controlling factors such as solar wind ram pressure and interplanetary magnetic field orientation. By comparing the LFM magnetopause with the empirical magnetopause models in their region of greatest accuracy, we determine the overall accuracy of the LFM magnetopause position in the dayside equatorial plane. We find the LFM magnetopause to be earthward of the empirical models by, on average, 0.5 to 1.0 RE at noon and 1.0 to 2.0 RE at dusk. This observed discrepancy is shown to be, at least in part, due to the insufficient ring current in the LFM.

Report from scientific sessions at the 35 th COSPAR scientific assembly, Paris 2004

As a result of ICME interaction with magnetosphere the negative sudden impulse Si and substorm event were followed. The simultaneous plasma and magnetic field measurements near the magnetospheric boundary (INTERBALL-1), in the nearest magnetotail (GOES 8, 9), in the middle tail (GEOTAIL), in solar wind (WIND), and ground based data were compared to trace the time sequence of events. It was shown that negative sudden impulse Si was observed by ground stations and at geosynchronous altitudes. The character of magnetic field variations that were observed by GOES 9 permit to suggest that compression/expansion wave was propagated inside the magnetosphere. The time sequence of events observed during substorm development pointed that substorm has begun from auroral breakup. The cross current disruption in the nearest tail and neutral line formation in the middle tail occurred almost simultaneously, but some time after auroral breakup. It is possible that both processes associated with releasing of stored energy during substorm were occurred independently in this substorm event. By using an empirical magnetopause model [Shue et al., 1998. Magnetopause location under extreme solar wind conditions. J. Geophys. Res. 103, 17691], the amplitude of magnetopause motion was evaluated. During substorm evolution the amplitude of the magnetopause motion was ∼2–3 Re. The excitation of Kelvin-Helmholtz instability can explain only a few magnetopause crossings.

The shape and location of the high-latitude magnetopause

Recent empirical magnetopause models are bi-variate with respect to both solar wind dynamic pressure and B Z component of the interplanetary magnetic field (IMF). Models use various functional forms to represent the shape and location of the magnetopause. Moreover, the models have different ranges of validity because, among other things, the data sets used for their development were usually different. A majority of these models were created for magnetopause crossings regardless of latitude. For the shape of the near-Earth high-latitude magnetopause, a new empirical model was published by Boardsen et al. [Boardsen, S.A., Eastman, T.E., Sotirclis, T, Green, J. L. An empirical model of the high-latitude magnetopause. J. Geophys. Res. 105, 23,193, 2000]. This model is parameterized by solar wind dynamic pressure, IMF B Z and dipole tilt angle. However, this model covers only a high-latitude part of the magnetopause in a limited range of the X GSE coordinate (∼±8 R E). We have used a new set of high-latitude magnetopause crossings and compared their coordinates with predictions of several magnetopause models with motivation to analyze differences between measurements and models. Our study reveals that the effect of the near-cusp magnetopause indentation can be reflected by an addition of a simple geometrical formula to the generally accepted model. This approach provides an universal model which describes the whole magnetopause surface with sufficient accuracy regardless of latitude.