Impulse-driven oscillations of the near-Earth’s magnetosphere

. It is argued that a simple model based on magnetic image arguments sufﬁces to give a convincing insight into both the basic static as well as some transient dynamic properties of the near-Earth’s magnetosphere, accounting in particular for damped oscillations being excited in response to impulsive perturbations. The parameter variations of the frequency are given. Qualitative results can be obtained also for heating due to the compression of the radiation belts. The properties of this simple dynamic model for the solar wind – magnetosphere interaction are discussed and compared to observations. In spite of its 5 simplicity, the model gives convincing results concerning the magnitudes of the near-Earth’s magnetic and electric ﬁelds. The database contains ground based results for magnetic ﬁeld variation in response to shocks in the solar wind. The observations also include satellite data, here from the two Van Allen satellites

the magnetospheric oscillations and overall changes of the magnetosphere in response to abrupt changes in the solar wind. The present study addresses this question using data from space observations obtained using "in situ" data acquired by spacecraft 25 and also ground based observations. The case illustrated here assumes a strong compression of the magnetosphere by a solar wind pressure pulse by taking the distance to the magnetopause to be 7.8 RE. This value has relevance for data to be shown later. Note the formation of two cusp-points.
As the impulse from an ICME-shock event arrives at the vicinity of the stagnation point of the solar wind at the magnetopause its perturbation propagates along the magnetosphere with velocity depending on the direction with respect to the magnetic field or the magnetopause. As an order of magnitude we can use where V A = B/ √ µ 0 ρ is the Alfvén speed for a plasma mass density ρ, and c the speed of light in vacuum. For vacuum or dilute plasmas we have ϑ ≈ c, for dense plasmas ϑ ≈ V A . We assume the velocity ϑ to be sufficiently large to allow the motion of the magnetopause at all relevant points to be assumed nearly instantaneous for the present problem.

A simple model 30
In its original form, the basic model (Børve et al., 2011) assumed a plane interface between the solar wind and the near-Earth's magnetosphere. An equilibrium state is found when the solar wind ram-pressure balances the magnetic field pressure at the stagnation point of the solar wind flow as argued by Chapman and Bartels (1940) and Alfvén (1950). The solar wind gives up all its parallel momentum as in an inelastic collision and flows with a reduced velocity along the interface, i.e. the magnetopause, in a boundary layer with an otherwise unspecified thickness and plasma density. The model predicts static parameters such as 35 the distance between the Earth and the magnetopause (stand-off distance), as well as some dynamic features, in particular the frequency and damping of magnetospheric oscillation in response to an impulsive perturbation in the solar wind. For describing the Earth's magnetic field we here ignore the small tilt of the magnetic axis with respect to the rotation axis. For generalizing the model to other planets it is straightforward to include such a tilt of the magnetic axis (Børve et al., 2011). The model can be generalized as shown in Appendices A and B. These changes will, however, only have small consequences for the results.

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In the following we use the simplest version of the model.

Static limit
For the present formulation of the problem the total magnetic field resulting from the Earth's dipole and the Chapman-Ferraro current can be found by a simple method of images with details as well as figures presented by Børve et al. (2011). The spatial variations of the magnetic field in the near-Earth's magnetosphere predicted by the model are illustrated here in Fig. 1. 45 In particular, the model predicts the distance from the Earth to the stagnation point of the solar wind. The analysis can be generalized to account also for the curvature of the magnetosheath in the vicinity of the stagnation point, see Appendix A.
The surface currents consistent with Fig. 1 are shown in Fig. 2. Near the stagnation point at the magnetic equator the radius of curvature, κ, increases and ∇B decreases as compared to the value for a magnetic dipole field in free space.
The equilibrium position R for the stand-off distance from the Earth to the magnetopause is found (Børve et al., 2011) by 50 equating the magnetic field pressure (Ferraro, 1952) from the Earth's magnetic dipole moment Q E = 8.0 × 10 22 A m 2 to the solar wind ram-pressure to give the relation in SI-units, with µ 0 = 4π10 −7 H m −1 being the permeability of free space, and nM is the mass density of the solar wind in terms of number density n and average ion mass M . The mass density of the solar wind is distinguished from the magnetopause 55 mass density ρ. The contribution to the pressure from the weak solar wind magnetic field is assumed to be negligible. A numerical coefficient in (1) is a result of the analysis and not a free adjustable parameter. Expressions similar to (1) can be found in the literature (Walker and Russell, 1995). The scaling with the solar wind dynamic pressure nM U 2 −1/6 is generally accepted (Southwood and Kivelson, 1990). In fact, apart from a numerical factor, it can be derived from basic dimensional reasoning as shown in Appendix C. The predictions of the model for the distance from the Earth to the magnetopause are  (Chapman and Bartels, 1940).
The distance is measured in units of the Earth radius RE shown for varying momentum flux density nM U 2 (expressed in nP a) in the solar wind.
The surface current that models the Chapman-Ferraro current at the interface between the Earth's magnetosphere and the solar wind at x = 0 in Fig. 1 induces a small correction to the magnetic field at the surface of the Earth. A change of the stand-off-distance R in Fig. 3 will give rise to a change in this correction as illustrated in Fig. 4

Dynamic features
In response to an impulse change in the solar wind ram-pressure, the near-Earth's magnetosphere is set into motion. The model of Børve et al. (2011) accounts for this by moving the image dipole in the simple plane interface model as well as in its 75 generalization summarized in Appendix A.
To find oscillating features a physical system needs inertia or its equivalent (Smit, 1968;Cairns and Grabbe, 1994;Freeman et al., 1995). The model is not able to predict this inertia, and it is here quantified by a thickness D and a mass density ρ which has to be determined by observations (Song et al., 1990;Phan and Paschmann, 1996) or numerical simulations that are also available (Spreiter et al., 1966). On the other hand, analytical models have been proposed (Cairns and Grabbe, 1994) for the width D.

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Both these parameters vary depending on whether the magnetopause is open (so it has high density solar wind matter inside) or closed. Densities of ρ = 5 − 25 cm −3 are considered to be typical values. To discuss a finite amplitude nonlinear case, we write Newton's second law for the position of the interface in the form where ∆(t) is the time varying displacement of the interface from its equilibrium value R from (1). The left side of (2) is 85 the product of the mass and the acceleration of a volume element of the moving magnetopause. The first term on the right hand side is the ram-pressure of the solar wind using at any time the relative velocity between the solar wind and the moving interface. The solar wind is assumed to interact with the magnetopause as an inelastic collision. Reflection as in ideal elastic collision does not apply here. The second term on the right hand side is the counteracting magnetic pressure B 2 /µ 0 due to the dipolar magnetic field of the Earth taken at the magnetic equator. This force is also derived at the position of the moving 90 magnetopause. We take the sign-convention so that ∆ > 0 when the magnetopause boundary moves in the direction of the Earth (this definition differs from the one used by Børve et al. (2011)). The equilibrium solution of (2) with ∆ = 0 gives (1).

Oscillation frequencies and damping
If we linearize (2) we can derive a scaling law for the characteristic oscillation period as Apart from the numerical factor, also this result can be found by dimensional reasoning, see Appendix C. A small amplitude damping coefficient can be found as nM U/Dρ. Large inertia ρD gives a long oscillation period and a reduced damping. This is intuitively reasonable since it reduces the velocity of the magnetopause. In a related study (Smit, 1968), a drag force was introduced "ad hoc". Here, the damping is caused by an asymmetry in the solar wind ram-pressure: when the magnetopause is approaching (i.e., moving away from the Earth) the magnetopause is doing work on the solar wind, while in the receding phase 100 it is opposite. The two cases are not symmetric since the effective solar wind ram-pressure depends on the relative velocity between the solar wind and the magnetopause. In the approaching phase this force is large, while it is smaller in the receding phase. The work done in the two oscillation phases is different. Integrated over an oscillation period 2π/Ω, the oscillations lose energy to the solar wind so the net result is a damping of the oscillations. The initial transient time interval is different: here the solar wind pulse or shock arrives at an interface at rest, and the oscillations are initiated to reach full amplitude.

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Magnetopause velocities in the range 10−20 km/s along the normal to the magnetopause have been reported (Paschmann et al., 1993;Phan and Paschmann, 1996). The velocities depend on plasma parameters, the magnetic shear in particular. The larger of the values quoted refers to high-shear (Phan and Paschmann, 1996) although it also seems that the observed speeds have a large statistical scatter. Since the magnetopause is accelerated upon impact from the perturbation in the solar wind (Freeman et al., 1995), these values are only representative, i.e., a large velocity is indicating a large acceleration. The results refer to the 110 satellite frame which is here moving with a velocity much smaller than the magnetopause. Thereby only large magnetopause velocities can be determined unambiguously.
The analysis using (2) and its extensions can readily give also the time variations of the velocity dZ(t)/dt as well as the acceleration d 2 Z(t)/dt 2 . These results are not shown here since we have no access to relevant data for magnetopause velocities, nor accelerations, for comparison. Concerning the time variation of the position Z(t) we have for comparison indirect results 115 in terms of oscillations in, for instance, the magnetic fields that are induced by the moving magnetopause and thereby the Chapman-Ferraro current systems.
There are alternative and more complicated mechanisms that can give rise to damping, i.e, field aligned currents (FACs) that flow into the polar ionosphere and further to the global ionosphere, where the energy is consumed by the Pedersen currents (Kikuchi et al., 2021). The damping suggested in the present study is of a different nature. The basic dynamic equation (2) can be rewritten in normalized form (Børve et al., 2011) as with Z ≡ ∆/R where R is the equilibrium solution (1), while time is normalized by T 0 from (3). The basic equation (4) is strongly nonlinear and the solutions are characterized by significant harmonic generation. Equation (4) can be solved for different conditions, the standard one being where Z is slightly displaced from the equilibrium position to perform damped 125 oscillations, eventually reaching Z = 0 as illustrated in Fig. 6. Alternatively, as shown in Fig. 7, we can assume the interface at its equilibrium position until a reference time τ = 0, where there is a sudden and lasting change in the solar wind conditions, changing the equilibrium position. The differential equation has to be modified slightly to account for this case (Børve et al.,  These transient oscillations will modulate those conditions in the Earth's magnetosphere and ionosphere that are induced by changes in the magnetopause. We estimate the average speed of the magnetopause after it was subject to an impact from a shock-like disturbance in the solar wind by V = 1 2 (R 0 −R 1 )/T 0 , where its initial position is R 0 at time t = 0 and the first local maximum the magnetopause 135 displacement is R 1 at t = T 0 , see Fig. 7. We can write this velocity as V = 1 2 V 0 1 − (n 1 U 2 1 /n 0 U 2 0 ) 1/6 by using (1) where a representative speed is V 0 = R 0 /T 0 , while the parenthesis is a numerical factor where 1 2 comes due to the averaging from initial time to T 0 . Realistic values T 0 ≈ 10 min and R 0 − R 1 ≈ 4R E give V ≈ 2 × 6 × 10 3 /6 × 10 2 = 20 km/s. This is a large velocity, but it agrees with observations of magnetopause speeds better than an order of magnitude.
The reference calculations in Fig. 7 use RnM = Dρ/4, i.e. a relatively large inertia associated with the moving magne-140 topause. To illustrate the nonlinear character of the oscillations, we show solutions for both positive and negative changes in the solar wind momentum density. For a linear system, the positive and negative parts of Fig. 7 should be mirror images with respect to the horizontal axis. We expect, however, generally a different nonlinear response to a sudden increase and a sudden rarefaction in the solar wind. This may also occur, albeit not as often as compression by a shock. Details of the derivation of the results summarized here are given by Børve et al. (2011), in particular also discussing the simplified linearized limit of the 145 equations.
The physical mechanism causing the damping of the oscillations is found to be an asymmetry in the forcing and the displacement of the magnetospheric boundary. The momentum transfer depends on the solar wind velocity relative to the moving boundary and this is different for an approaching and a receding magnetopause. The damping is thus not due to direct dissipation.

Time variations of the magnetic fields
The motion of the Chapman-Ferraro current system induces temporal variations in the magnetic field detected at the surface of the Earth. These are illustrated in Fig. 8. The asymptotic limits t → ∞ correspond to Figs. 4 and 5. The analytical expression for the B-field perturbation at the magnetic equator as found by use of the image dipole is The nonlinear features of ∆(t) are magnified by the analytical form of dB, the oscillation period depending, in particular, on the perturbation amplitude as well as its sign. In Fig. 8 we introduced the normalizing by B RE ≡ µ 0 Q E /(4πR 3 E ) being the B-field at magnetic equator at r = R E for t = 0; we have a representative value of B RE = 30 µT . A characteristic value for the perturbation of the magnetic field deduced from Fig. 8 is thus dB ≈ 30 nT for the given parameters.

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The time-varying model magnetic field has a straightforward analytical expression in terms of the moving image dipole.
The induced electric field can be derived by Faraday's law, as illustrated here in Fig. 9 for the case where the Chapman-Ferraro current system moves with constant velocity. This is here modeled by moving the image magnetic dipole. Note that a calculation starting from the moving Chapman-Ferraro current system would be complicated, while the result for a moving image dipole is simple. Oscillations in ∆ seen in Figs. 6 and 7 give corresponding time variations in the magnetic field. A 165 change in the sign of ∂B/∂t gives rise to a corresponding change in the sign of the induced electric field in Fig. 9.
The discussion so far assumed that the density of matter, plasma in particular, is negligible between the Earth and the magnetopause. We discuss next some of the effects on the radiation belts and the Earth's near ionosphere.

Motions of the radiation belts.
The moving Chapman-Ferraro current system induces E × B/B 2 -motions of the magnetic field lines (in the MHD sense 170 (Pécseli, 2012)) in the radiation belts as illustrated in Fig. 10, here shown in the horizontal plane of Fig. 9. Details of individual particle motions are then introduced as corrections to this, e.g. as polarization drifts. In the analysis we assumed initially quiet conditions with the stagnation point at a large distance from the Earth, see Fig. 1, so that the initial boundary of the radiation belts can be assumed circular. The Chapman-Ferraro current system is then allowed to move the stagnation point from 11R E to a distance of 7.8R E from the Earth. We note that the inner boundary is hardly affected since the Earth's magnetic field 175 is too strong there. The deformation of the outer boundary is asymmetric: at the magnetotail side the electric fields are too weak to induce a motion of any significance, see Fig. 9. The outer boundary on the sun-ward side, is on the other hand compressed because the magnetic field is relatively weak while the induced electric field has a sufficient magnitude to give a noticeable E × B/B 2 -velocity of the magnetic field line motion. The velocity does not matter for a closed adiabatically compressed system: only the initial and final positions of the magnetopause are relevant. For the large magnetopause velocities 180 quoted before (Paschmann et al., 1993;Phan and Paschmann, 1996) we can ignore interactions with the surrounding plasma and use the adiabatic model. The conclusion is that the moving Chapman-Ferraro current system gives rise to an asymmetric compression of the outer radiation belt, which for the given case amounts to approximately 10%.
The discussion and derivation of the results of Fig. 10 assume that the motion is solely described by the E×B/B 2 -motion of the radiation belts, with the electric field derived from the motion of the Chapman-Ferraro current system. The sudden impulse-185 like compression of the radiation belts will act as a "piston" and excite compressional Alfvén waves propagating in the Earth's direction. These waves will give rise to a heating of the plasma that will penetrate deeper into the radiation belts (Zong, 2022).
The asymmetry of the day and night sides will however be well represented by results like those shown in Fig. 10.
The sudden accelerated compression illustrated in Fig. 10 gives rise to polarization drifts (Chen, 2016;Pécseli, 2012) of the heavy component, here the plasma ions. For the given geometry, the drift velocity is to a first approximation U D = 190 Ω −1 ci ∂(E/|B|)/∂t, with time varying electric and magnetic fields where Ω ci is the local ion cyclotron frequency. At the beginning of the shock compression, the associated currents will be in the dusk-dawn direction as described by Araki (1994), and confined to the compressed region. Strong accelerations during the compression give rise to strong currents. Excess charges will build up at the dawn and dusk boundaries of the compressed regions, see Fig. 11, and these charges can expand only along magnetic field lines or be canceled by ions or electrons flowing up from the ionosphere along the same magnetic field 195 lines. The polarization drift act as generator for these field aligned currents (Araki, 1994). The imposed current density (as modeled by the current generator in Fig. 11) is given as a product of the charge density and the imposed polarization drift, qnU D . The corresponding generator is modeled best by an ideal current generator (Garcia et al., 2015), in contrast to the ideal voltage or potential generator usually assumed for studies of field aligned currents (Knight, 1973), see also Fig. 11. The ideal current generator has infinite inner impedance while the voltage generator (ideal battery) has vanishing internal impedance 200 (Scott, 1959). It is known (Garcia et al., 2015) that the distinction has important consequences. Numerical simulations of, for instance, ionospheric double layers (Smith, 1982) demonstrated the importance of the generator impedance. Realistic generator models have finite internal resistances and the two generators are related by Thevenin's and Norton's theorems (Scott, 1959).
The potential variations in the circuit, i.e., along magnetic field lines and in the ionosphere, develop in response to the imposed currents. The compression of the radiation belt plasma is modulated by the damped oscillations of the magnetopause. These 205 oscillations in turn modulate the field aligned currents and their time variation will be recognized also in the magnetic fields they give rise to on Earth.
In response to a change in energy density of the ring current, the Dessler-Parker- Sckopke relations (Dessler and Parker, 1959;Sckopke, 1966) predict a detectable perturbation of the magnetic field as measured by e.g. ground-based stations, but these theorems refer to symmetric conditions. The asymmetric perturbation illustrated in Fig. 10 will take some time to re-210 lax and thermalize at a rotational symmetry (Summers et al., 2012), i.e. of the order of 4-6 h for a localized distribution of 3 MeV electrons to transform into a uniformly distributed ring. Some details concerning the dynamics of the radiation belts are summarized in Appendix D.
Shock-induced relativistic electron acceleration in the inner magnetosphere have been observed by instruments on spacecrafts (Foster et al., 2015;Tsuji et al., 2017). As the radiation belts are compressed in our model, the plasma will be adiabat- particles in the radiation belt. Charged particles in the MeV-range will pass through the compressed region in a time that is negligible compared to the compression time and will not be affected. In our model the particle energization is due to the conservation of the magnetic moment (Chandrasekhar, 1960) and therefore a bulk plasma heating, the only constraint being that the plasma particles spend more than a few gyro-periods in the compressed region. For protons this time will be approximately 220 2π/Ω ci (r) = (r/R E ) 3 2.2 × 10 −3 s, taken at the magnetic equator at a distance r from the Earth's center.
The space-time varying electric and magnetic fields generated by the dynamic variations in the position and intensity of the Chapman-Ferraro current system induces currents in the Earth's near ionosphere, the E and F-regions. A simple model for idealized conditions is outlined in Appendix B.

Comparison with observations 225
The model predictions concerning R and T 0 , as well as the damping of the oscillations received numerical confirmation (Børve et al., 2011). The agreement was even better than stated by the authors due to an incorrect velocity used for normalization in the simulations of their Figs. 9, 10 and 13. In reality the agreement was close to perfect. The numerical model used for the analysis is however in two spatial dimensions and the steady state conditions depended on numerical resistivity and viscosity that dominate model viscosity and resistivity (Børve et al., 2014). The importance of viscosity is different for numerical 230 simulations in 2 and 3 spatial dimensions. Although this does not affect the dynamic features, a more general test would be worthwhile. Later fully 3 dimensional numerical Magneto Hydrodynamic (MHD) simulations (Desai et al., 2021) have given Figure 11. Diagram for illustrating polarization currents (Araki, 1994) generated by an asymmetric compression of the radiation belts, indicates a current generator. The full circuit is discussed by Araki (1994).
more detailed results supporting the restricted solutions found by Børve et al. (2014). There is also a slight difference between the 2 and the 3 dimensional versions of the analytical expressions used in the present work.
The predictions of the model discussed in the foregoing can be compared also to space observations as done in the following.

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Here we distinguish steady state and dynamic observations.

Steady state conditions
Inserting typical numbers as U ≈ 3 × 10 5 m/s, n ≈ 5 × 10 6 m −3 , and an average mass equaling the hydrogen mass, M = 1.66 × 10 −27 kg, we find R ≈ 7.2 × 10 7 m, or R ≈ 11.2 R E in terms of the Earth radius R E = 6.4 × 10 6 m. The estimate for R are comfortably within the generally accepted range of R ∼ 10 − 15 R E (Kivelson and Russell, 1995). The model equation

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(1) implies a closed scaling law for the distance to the magnetosheet boundary in terms of the solar wind velocity U and the solar wind mass density nM . Note that there are no free parameters to fit in equation (1), i.e. all are measurable quantities.

Time varying conditions
Solar wind disturbances such as interplanetary (IP) shocks induce significant variations of solar wind parameters during a short time interval, introducing perturbations in the geospace environment, in particular sudden variations of the magnetic 245 fields both in the magnetosphere and on the ground as measured also by ground-based magnetometers (Araki, 1994;Sun et al., 2015). Fluctuations on the minute time scales are often observed in the magnetosphere in response to strong perturbations in the solar wind. Consistent with a model using nonlinear oscillators (Børve et al., 2011), harmonics of the magnetospheric oscillations are often observed (Kepko and Spence, 2003). These are consistent with the strong nonlinear harmonic generation features of the basic model (4). Details of other predictions of the model will here be compared with two sets of observations In Fig. 12 we show plasma and magnetic field data from the Wind spacecraft illustrating the propagation of a shock in the solar wind, seen at ∼18:40 UT. Wind is located at {197.5, −53.5, −8.8} R E upstream of Earth. From top to bottom, the plot shows 255 the proton density, bulk speed, and temperature, the dynamic pressure, the components of the magnetic field in geocentric solar magnetospheric (GSM) coordinates, and the storm-time SymH index. Parameters relevant to the shock are given in Table 1.
The shock is being driven by an ICME (Richardson and Cane, 2010). The magnetic field upstream of the shock (average over 3 min) is {−0.13, 8.24, −4.02} nT, and the shock normal, using the magnetic coplanarity theorem (Colburn and Sonett, 1966) is {−0.58, −0.79, −0.21}, both in GSE coordinates. This gives an angle between the upstream field and the shock normal, 260 Θ BN = 52.7 • , so the shock is quasi-perpendicular. The shock speed is 344.6 km/s, based on Rankine-Hugoniot relations (see Abraham-Shrauner and Yun (1976) and references therein). The Mach number of the shock is ∼4.
In Fig. 13 we show data from the CARISMA magnetometer network in Canada for a 30 min period. Signals from a few other stations are shown in Fig. 14. An abrupt rise in the magnetic field intensity, followed by some damped oscillations can be seen  The radius in the small circles give the relative variation of the tangential component of the magnetic field perturbation. An intense magnetic field perturbation with a large vertical component and simultaneously a small horizontal component will thus be shown with a circle having a small radius. The color is red if the vertical component is into the ground, and blue if it is in the opposite direction. The stereographic mapping of the globe is chosen to make the circles having approximately the 280 correct relative magnitudes. When plotting these results we took the first peak maximum after onset of the signal. Note that in general the magnitudes of the horizontal components is larger than the vertical components. We find an overall tendency for positive B z -values in the northern hemisphere and small or negative B z -values in the southern hemisphere. The variation across Northern America appears uniform, in particular. The results are in fair agreement with the model, although not perfect.
The strongest deviations are found near the magnetic poles.

Event of March 17, 2015.
In Fig. 16 we show data for 1 day (March 17, 2015) from the Wind spacecraft, illustrating the propagation of a shock in the solar wind at ∼5 UT. See also Table 2. The field and plasma data are analyzed in the same manner as for previous shock. The magnetic field upstream of the shock is {1.75, 4.59, 8.70} nT and the shock normal, using the magnetic coplanarity theorem (Colburn and Sonett, 1966 (Gjerloev, 2012). The first data-point is at 2014-12-21, 19:00 UTC. We note some heavily damped oscillations in all figures, where oscillations with 5 -10 min periods are discerned.  (Gjerloev, 2012 The shock speed is 601.3 km/s, based on Rankine-Hugoniot relations, see Abraham-Shrauner and Yun (1976) and references therein. The speed of plasma along shock normal is 405.2 km/s. In the last panel of Fig. 16 we plot the temporal profile of the SymH index over a 2-day period. SymH is a measure of the strength of the ring current. In this case it has a two-dip structure, indicating that we have a 2-dip storm (Kamide et al., 1998).

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The weaker dip occurs at ∼9:15 UT, March 17. This caused a major storm already. The SymH then recovers for ∼14 hrs, only to decrease again and reach a new and deeper minimum at ∼21 UT, March 17. The first dip is caused by the shock compressing B z < 0 (GSM) fields. The second one is caused by the long (∼10 hrs) B z < 0 phase inside the ICME itself. Both strengths correspond to major geomagnetic storms, but the second one almost reaches "superstorm" values (SymH < −250 nT). The same thing holds for Dec. 21, 2014 event, only the SymH dips are here much weaker (∼35 and ∼60 nT) and they are separated The bipolar signal seen seen in our Fig. 17 at auroral latitudes corresponds to the DP-type perturbations described by Araki 310 (1994). The interpretation of this signature given (auroral zone, morning local time) is in terms of an M-I-coupling illustrated in Fig. 12 in that work. The signatures described shown in the present work at Svalbard latitudes are explained in terms of lobe-cell polar cap convection with an associated Hall-current.
Our focus in the present study is on the impulse/oscillation at lower latitudes during the interval 04:46-04:50 UT, which is more directly related to the IP shock. After a short transition we see small-amplitude, damped oscillations of periods 4-6 min 315 at 4:46 UT. The signal obtained by selected ground stations at various local times is illustrated in Fig. 18. Here the MLTs are: at GUA,14:30 MLT,at DLT,about 12 MLT and at M08 about 22 MLT. Note the vertical scales are larger than those of Fig. 14, consistent with the shock intensity. From Figs. 12 and 16 we estimate the solar wind pressure in the event of March 17-18, 2015 to be approximately twice as large as in the event of December 21, 2014, implying that the characteristic frequency Ω in the former case is ∼ √ 2 larger than in the latter case. The oscillation period is readily estimated visually in Figs. 14 and 18 but 320 the corresponding local frequencies can also be estimated by a wavelet transform (Kaiser, 1994). Illustrative results are shown in Fig. 19. A full wavelet analysis of the signals from a large representative dataset fall outside the scope of the present study, but we note that the wavelets reveal oscillations of 0.5 − 2 mHz trailing the step-like magnetic field enhancement originating from the solar wind shock. This numerical value agrees with our model. The signal obtained by ground stations at various local times is illustrated in Fig. 20. As in Fig. 15, we show the component 325 normal with respect to ground with a color code. Positive values point into the Earth also here. The results near the magnetic poles have magnitudes typically up to 2-3 times larger than the average of the values shown, and also the time variations found there can be more irregular. The same comments apply also to Fig. 15 and the values for these regions are not shown. These polar features are believed to be due to field aligned currents (Knight, 1973;Lühr and Kervalishvili, 2021), not accounted for in the present model. Ground magnetic disturbances at auroral and sub-auroral latitudes can be induced by both ionospheric 330 and magnetospheric currents (Araki et al., 1997). At middle and low latitudes, the cause of magnetic field disturbances are dominated by magnetopause currents superimposed by weak ionospheric currents. At the dayside equator, the ionospheric Cowling currents are the major source for the equatorial Sudden Commencements (SC). The spatial variations in the magnetic field perturbations seen in Fig. 20 are larger and more non-uniform compared to those found in Fig. 15. We take this as an indication of a stronger influence of the ionospheric currents in the latter case where the solar wind shock is strongest.

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For the dynamics in the radiation belts we have data from the Van Allen Probes (formerly known as the Radiation Belt Storm Probes (RBSP)), with the relevant positions of the two satellites deep inside the magnetosphere shown in Fig. 21. The satellites measure the electric fields as shown in Fig. 22. From (1) we have for this case R ≈ 6 × 10 7 m, and from (3) Fig. 22. Due to the compression of the sunward part of the Earth's magnetic field, the estimate of charged particle velocities based on a magnetic dipolar field as in Appendix D will only serve as a guideline.

Conclusions
A simple model for illustrating the near-Earth magnetospheric static as well as dynamic features has been presented. The model predicts the distance between the Earth and the magnetopause (stand-off distance) without introducing free parameters. Some    Fig. 18. The data were obtained by SuperMAG (Gjerloev, 2012).
dynamic features, in particular the frequency of magnetospheric oscillations in response to an impulse in the solar wind is derived as well. The parameter variations of the oscillation frequency is expressed analytically. A damping of the oscillations is predicted, in particular also its variation with solar wind parameters. This damping is not caused by dissipation but is an inherent feature of phase relations in the model. For testing the predictions of the model we considered two events, i.e., two geomagnetic storms: one moderate for the December 21, 2014 with SymH ∼ −70 nT and a strong one occurring at March 17, 365 2015 with SymH ∼ −237 nT. The magnetospheric response on the impact of them was similar, but with significant differences in the details. The agreement with our model was best for the moderate shock. The magnetic field perturbations in Fig. 18 are significantly larger than those shown in Fig. 14 as expected for the stronger shock. The observed oscillations are consistent with results reported by other studies (Plaschke et al., 2009;Farrugia and Gratton, 2011). Considering the simplicity of our model, we find its overall agreement with observations to be satisfactory. The basic ideas apply for other magnetized planets 370 as well.
The main conclusion of the present study can be summarized in few words: "Three dipoles suffice for the lowest order modeling of the near-Earth magnetosphere", one, Q E , for the Earth's magnetic field and two image dipoles, where one, Q I , is placed in the solar wind, the other, Q S , in the Earth's interior see Appendix B. We have Q I and Q E to be parallel so their magnetic field contributions add at the Earth's surface. The interior dipole Q S is anti-parallel to Q E so that the radial 375 component of the magnetic field cancels in the the ionosphere at a radius here taken to be R E to sufficient accuracy.
One partial result of the present analysis is an emphasis of the strongly nonlinear features and damping of the magnetospheric oscillations. These are explained here by the basic properties of a simple physical model. The observed frequencies and damping rates seen in, e.g., the CARISMA data in Fig. 13 and in part also the IMAGE data in Fig. 17  The oscillations observed in figures like 13 or 18 are thus natural for the system and not due to some external forcing. The model also predicts the magnitudes of magnetic and electric fields detected by ground stations and satellites to better than an order of magnitude. We consider this to be satisfactory. The ideas put forward in the present study can be applied to any magnetized planet like the Earth, orbiting a star like the Sun. The main limitations of the model are found in: its inability to account for the far magnetotail conditions, the dynamics in particular. A cross-tail current is not included in the model.
-Magnetic field line reconnection (Califano et al., 2009) is missing in the model. This is important when the interplanetary magnetic field has a downward component. Field aligned currents (FAC) following reconnection are consequently not 390 accounted for.
-Surface eigenmodes of the dayside magnetopause (Hwang, 2015;Hartinger et al., 2015;Archer et al., 2019) are not accounted for. These will give rise to additional, presumably small amplitude, oscillations to the modes described in our work.
-The model gives only a schematic account for the excitation of Alfvénic waves and the particle accelerations associated 395 with them.  We believe the present simple model deserves scrutiny. The predictions can be compared to other related data which can be classified according to details such as SymH for the observed shocks.
ments. CJF was supported by NASA Wind grant 80NSSC19K1293 and 80NSSC20K0197. Figure 14 presented in this paper rely on the data collected at the GUA, DLT, and M08 stations. We thank Institut de Physique du Globe de Paris and United States Geological Survey for supporting their operation and INTERMAGNET for promoting high standards of magnetic observatory practice (www.intermagnet.org). Data from the Van Allen probes were accessed at the Science Gateway, maintained by the Johns Hopkins University Applied Physics Laboratory.
Author contributions. All authors contributed to the data collection, analysis and manuscript preparations. HS, PES, and CJF mostly to the 405 data collection, HLP and JKT mostly to the analysis and the preparation of the manuscript and figures.
Competing interests. The authors declare that they have no conflict of interest.
in terms of the angle λ measured from the magnetic equator. We introduced the magnetic dipole moment Q. For the Earth we Figure A1. Illustration of coordinate system for a modified model of the Magnetospheric interface with QE being an equivalent dipole for the Earth's magnetic field. The interface follows the magnetic field lines for small Ψ.
Assume that the cut in interface between the solar wind and the Earth's magnetosphere can be approximated locally by a circle with radius R 0 , see cases (Ferraro, 1952;Spreiter and Summers, 1965), a plane boundary being one of them.
To obtain an approximation for the magnetic field between the Earth and a curved magnetosphere (Spreiter and Summers, 1965) we here take two parallel dipoles, the Earth's magnetic dipole Q E and an image dipole Q I at positions x E and x I , respectively. Introducing the magnetic field from a dipole (A1) and the given radius of curvature for the ideally conducting surface we find that to lowest approximation Q I ≈ Q E (R 0 /x E ) 3 placed on the x-axis at the position x I ≈ (R 0 /x E ) 2 x E . Near the stagnation point, the total magnetic field becomes a result that can be derived from the related problem for an electric dipole near a conducting sphere. The first term gives the Earth's magnetic field, the second is the image field. With the notation of Fig. A1, we have r = R 2 0 + x 2 E − 2R 0 x E cos Ψ and 540 sin λ = (R 0 /r) sin Ψ. Introducing (A3) in (A2) it is convenient to define a characteristic scale C L ≡ 2(4π) 2 M nU 2 /(µ 0 Q 2 E ) having the physical dimension length 6 . The origin of the coordinate system is not specified but here determined through R 0 and x E . We could choose to place the origin at the Earth, but the analytical expressions will become more complicated, and we would then have to determine R 0 as well as the position for the radius of curvature. For the stagnation point (standoff-distance) of the solar wind at (x, z) = (R 0 , 0) the expression (A3) with (A2) gives the relation nM U in particular. This is consistent with the result of Børve et al. (2011), since R 0 −x E is the distance between the Earth and the interface between the solar wind and the magnetopause. A plane interface is a good approximation at (R 0 , 0). Given the parameters we use (A2) to determine the radius R 0 that eliminates the Ψ dependence, at least to lowest approximation. A stronger solar wind pressure gives a smaller radius of curvature. As the solar wind pressure decreases, i.e.
nM U 2 → 0, we have C −1/6 L → ∞ and R 0 → ∞. We find these latter results to be intuitively reasonable. The approximation 550 works best when Ψ is small. We find the approximate result R 0 = γx E where γ ≈ 1.2 − 1.5 with the given definition of parameters. This gives x E = (4/C L ) 1/6 /(γ − 1) and R 0 = (4/C L ) 1/6 γ/(γ − 1). An example of the modified model with a curved interface between the solar wind and the Earth magnetic field is shown in Fig. A2. The ideal or desired result would have been a parabolic form for the magnetopause. The method of images is, however, not well developed for such problems.
We can postulate a solution with a parabolic shape, where the curvature at the stagnation point is given through the foregoing 555 analysis.
An impulse in the solar wind, be it in velocity or density or both, will give rise to a reduction in the distance R 0 − x E , but at the same time it will induce also a change in the curvature R 0 of the part of the magnetosphere facing the sun. Within the present model there is no anisotropy in this curvature: it is the same in the plane parallel and perpendicular to the direction of the Earths magnetic dipole. The modified model can not account for the formation of the magnetotail.

Appendix B: Currents induced in the ionosphere
The space-time varying electric and magnetic fields generated by the dynamic variations in the position and intensity of the Chapman-Ferraro current system induces currents in the Earth's near ionosphere. The ionosphere has a significant altitude variation in the Pedersen and Hall resistivities as well as in the magnetic field aligned conductivity. The problem can be solved only by considering strongly idealized conditions, but these can be helpful by giving insight into some general features.

565
For conditions with large plasma parameters, we have a high plasma conductivity ξ, but it will never be super-conducting conditions, so it will be penetrated by a steady magnetic field. For dynamic conditions with large magnetic Reynolds number U L/ξ where L is a characteristic scale size and U some characteristic velocity, we can assume that the ionosphere acts passively for time-stationary magnetic conditions, but responds as an ideally conducting "shell" to rapid temporal changes in electric and magnetic fields (Davidson, 2001;Pécseli, 2012). This limit has an exact analytical solution when we assume that the moving 570 image dipole field imposes a locally homogeneous time varying magnetic field at the Earth. In this case we can formulate the question as: "what secondary image dipole is needed to make the boundary conditions at the conducting shell to be fulfilled?", the boundary condition being that the normal component of the magnetic field vanishes at the conducting shell. For the simple limit mentioned before the answer is readily found. We let B I (t) = µ 0 Q I /2π(2R(t)) 3 be the locally homogeneous magnetic field originating from the moving image dipole at a distance of 2R(t) from the Earth, see Figs. 1 and 3. We now introduce one 575 more image magnetic dipole with dipole moment Q S placed at the Earth's center. For the radial and angular variations of the total magnetic field we have where it is also here most convenient to measure the angle λ from the equator of the dipole. With the given choice of polarities 580 we find from (B1) that the normal component of the magnetic field at the conducting shell with radius R E vanishes for Q S (t) = 2πB I (t)R 3 E /µ 0 . At r = R E we then find from (B2) the angular magnetic field component B θ (t) = 2B I (t) cos λ. The corresponding surface current density at the bottom of the ionosphere is then K S (t) = 2(B I (t)/µ 0 ) cos λ in the direction perpendicular to Q S in the azimuthal direction ⊥ B, thus contributing to the electrojet current. From Fig. 4 we note that the assumption of a locally homogeneous magnetic field imposed by the image dipole representing the Chapman-Ferraro 585 current system can be questioned when the magnetosphere is strongly compressed. In such a case we can obtain a slight improvement of the previous result by displacing the image dipole Q S slightly in the sun-ward direction. An illustrative result is shown in Fig. B1. An ideally conducting ionosphere would thus shield ground stations completely from temporal variations of the magnetic field. It seems a safe conclusion that a partially conducting ionosphere will reduce the effects of the electric and magnetic field variations as detected on ground. The salty waters of the oceans also act as a conductor, albeit poor in 590 comparison to the ionosphere. The time varying electric fields will induce currents also in the oceans, and the resulting (weak) magnetic field variations might be detectable by ground stations. The time variation of the magnetic field at r > R E follows the variation in B I (t) directly within the given model, see Fig. 8.

Appendix C: Dimensional arguments
Some results can be derived from simple dimensional arguments (Buckingham, 1914). Consider for instance the distance 595 between the Earth and the magnetopause. The basic parameters are here the solar wind ram-pressure nM U 2 where we note that the solar wind density and velocity will always appear in this combination so there is no generality gained by taking the variables separately. Similarly, µ 0 and Q will also appear together, but since µ 0 also enters the magnetic pressure it has to be included explicitly as well. The mass loading will be here ρD. The dimension matrix for the problem is given by the table For a time stationary problem where the magnetopause is at rest we have time T 0 in the sixth column to vanish from the problem, and similarly the inertia term ρD can not have any effect either. Then the first and third rows are proportional. We write from left to right in terms of the variables on the top in the dimension matrix , and determine the exponents α j is such a way that the exponents of mass, of time, of length and of current are each equal to zero. Evidently this requires α 1 = α 4 , α 2 = 6α 4 and α 3 = −2α 4 . We arrive at the combination of parameters nM U 2 R 6 µ 0 (µ 0 Q) 2 α4 = 1.
Choosing α 4 = 1 we arrive at the result found in (1), apart from a numerical constant that can not be recovered by dimensional analysis. Given the parameters entering the combination in (1) is thus the only possible one (except a numerical constant) for 605 the stationary problem with the given assumptions. Experimentally verifiable deviations from the scaling will thus indicate that there are missing parameters in (1). One possibility could be the solar wind resistivity: the magnetic Reynolds number there (Davidson, 2001;Pécseli, 2012) is large but still finite so the assumption of ideal conductivity in the application of the method of images can be challenged. We believe that a systematic investigation of this problem is worthwhile.
The dynamic problem is somewhat more complicated. Here we retain also the two last columns in the dimension matrix, and note that any dimensionally correct combination of parameters can be multiplied by the left side of e.g. (C1), or by (nM U 2 T 2 0 R −1 (ρD) −1 ) α0 to an arbitrary power α 0 . We can thus decide that some parameters are kept constant, and determine the dimensionally correct combination of the rest. To derive a characteristic period of oscillation T 0 we first note that by Newton's second law ρDd 2 ∆/dt 2 = F orce for the displacement ∆ of the magnetopause, we expect the product ρD T −2 0 to appear, rather than these quantities individually. As long as the solar wind pressure is kept constant the variation of the force with varying displacement ∆ will be due to the variations of the magnetic pressure with varying distance. We ignore the first column. From the dimension matrix we then have We find α 1 = 7α 4 , α 2 = −2α 4 , α 3 = α 4 , giving Taking again α 4 = 1, this result is consistent with (3) apart from a numerical factor.

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A damping factor arises by a phase difference between the magnetopause displacement ∆ and the velocity d∆/dt, where it is taken into account that it is the relative velocity between the solar wind and d∆/dt that matters. A dimensional analysis of this problem will be lengthy.

Appendix D: Radiation belt details
In this appendix we summarize some details concerning the radiation belt heating due to the asymmetric compression caused