# Interplanetary Shock-induced Magnetopause
Motion: Comparison between Theory and Global Magnetohydrodynamic Simulations

R. T. Desai1 , M. P. Freeman2, J. P. Eastwood1, J. W. B. Eggington1,
M. O. Archer1 Y. Y. Shprits3,4, N. P. Meredith2, F. A. Staples5, I. J. Rae6,
H. Hietala1, L. Mejnertsen1, J. P. Chittenden1, R. B. Horne2 corresponding
author<EMAIL_ADDRESS>
###### Abstract

The magnetopause marks the outer edge of the Earth’s magnetosphere and a
distinct boundary between solar wind and magnetospheric plasma populations. In
this letter, we use global magnetohydrodynamic simulations to examine the
response of the terrestrial magnetopause to fast-forward interplanetary shocks
of various strengths and compare to theoretical predictions. The theory and
simulations indicate the magnetopause response can be characterised by three
distinct phases; an initial acceleration as inertial forces are overcome, a
rapid compressive phase comprising the majority of the distance travelled, and
large-scale damped oscillations with amplitudes of the order of an Earth
radius. The two approaches agree in predicting subsolar magnetopause
oscillations with frequencies 2–13 mHz but the simulations notably predict
larger amplitudes and weaker damping rates. This phenomenon is of high
relevance to space weather forecasting and provides a possible explanation for
magnetopause oscillations observed following the large interplanetary shocks
of August 1972 and March 1991.

1Blackett Laboratory, Department of Physics, Imperial College London, London,
UK

2British Antarctic Survey, Cambridge, UK

3GFZ German Research Centre for Geosciences, Potsdam, Germany

4Department of Earth, Planetary, and Space Sciences, University of California,
California, USA

5Mullard Space Science Laboratory, University College London, Surrey, UK

6Department of Mathematics, Physics and Electrical Engineering, Northumbria
University, Newcastle Upon Tyne, UK

##  1 Introduction

The Earth’s magnetopause exists in a delicate balance between forces exerted
between the impinging solar wind and the Earth’s intrinsic magnetic field. The
subsolar magnetopause is typically located approximately ten Earth radii (RE)
upstream but, during periods of enhanced solar wind forcing, this can be
compressed to half this distance and inside the drift paths of radiation belt
electrons and protons (Shprits et al., 2006) and the orbits of geosynchronous
satellites (Cahill and Winckler, 1999). Moreover, magnetopause motion can
drive gloabl ultra-low-frequency (ULF) pulsations (Li et al., 1997; Green and
Kivelson, 2004) and intense ionospheric and ground induced current systems
(Fujita et al., 2003; Smith et al., 2019). The dynamics and location of the
magnetopause are therefore of wide relevance to the understanding of planetary
magnetospheres and to space weather forecasting.

The location and shape of the magnetopause was initially theoretically
predicted to depend on the pressure exerted by a stream of charged particles
from the Sun (Chapman and Ferraro, 1931) and its three dimensional geometry
was derived based on solar wind dynamic pressure alone (Mead and Beard, 1964).
Measurements with in-situ spacecraft broadly confirmed these predictions and
were then used to derive a large suite of empirical models of the magnetopause
location (e.g. Shue et al., 1998, and references therein) based on elliptical
and parabolic functions. These empirical studies revealed additional
influences from the Interplanetary Magnetic Field (IMF) orientation, which
modulates magnetic reconnection and the Dungey (1961) cycle, solar wind
magnetic pressure and dipole tilt (Lin et al., 2010), IMF cone angle (Merka et
al., 2003), and ionospheric conductivity and solar wind velocity (Němeček et
al., 2016). These best-fit models are, however, static and can deviate when
compared to specific observations (Samsonov et al., 2020), particularly during
extreme solar wind conditions with discrepancies of $>$1 RE observed when
located less than 8 RE upstream (Staples et al., 2020).

Satellite observations have revealed that the magnetopause boundary exists in
a perpetual state of motion (Bowe et al., 1990). Solar wind pressure
variations drive the magnetopause response which results in fast magnetosonic
waves that can couple to poloidal and toroidal Alfvén modes of the large-scale
magnetospheric fields (Southwood, 1974; Kivelson et al., 1984). Bow shock- and
magnetosheath-generated phenomena, including; hot flow anomalies (Burgess,
1989), magnetosheath jets (Hietala et al., 2009), foreshock cavities (Sibeck
et al., 2002) and bubbles (Omidi et al., 2010), similarly produce pressure
fluctuations which elicit magnetopause motion.

Only four studies have formally examined the directly driven response of the
magnetopause to upstream pressure variations. Smit (1968) initially formulated
magnetopause motion as a simple harmonic oscillator consisting of inertial,
damping, and restoring forces. Freeman et al. (1995) and Freeman et al. (1998)
subsequently used the Newton-Busemann approximation to develop a formal
consistent theory of the magnetopause as an elastic membrane which could be
applied locally. Børve et al. (2011) similarly modelled the magnetopause
response to solar wind pressure pulses and found qualitative agreement with
2–D MHD simulations. Freeman et al. (1995) and Børve et al. (2011) notably
predict magnetopause oscillations to be strongly damped.

These studies, however, focussed on small perturbations in solar wind dynamic
pressure. Fast-forward inter-planetary (IP) shocks, as occur at the front of
Interplanetary Coronal Mass Ejections and corotating-interaction-regions, can
rapidly compress the magnetosphere in just a few minutes (Smith and Wolfe,
1976; Araki, 1994), and present a further regime for studying magnetopause
motion. Global magnetohydrodyanmic (MHD) codes are able to self-consistently
model the dynamic solar wind-magnetosphere interaction for a wide variety of
solar wind conditions and, in this study, we test theoretical predictions
using global MHD simulations to constrain nonlinear magnetopause behaviour
across extreme scenarios for which spacecraft observations are limited or
unavailable.

This letter is organised as follows: Section 2 describes the Gorgon Global MHD
model, the simulation parameters including the IP shocks considered and theory
of the magnetopause. Section 3 then describes the simulations conducted and
the comparison to theory. Sections 4 concludes with a summary discussed in
relation to space weather forecasting.

##  2 Method

###  2.1 Global-MHD

Gorgon is a 3-D simulation code with resistive MHD and hydrodynamic
capabilities, originally developed to study high-energy-density laboratory
plasmas (Chittenden et al., 2004; Ciardi et al., 2007). Gorgon has been
adapted and applied to planetary magnetospheres in several contexts,
including: the inclined and rotating Neptunian magnetosphere (Mejnertsen et
al., 2016), the variable motion of the terrestrial bow shock (Mejnertsen et
al., 2018), and the effects of dipole-tilt on terrestrial magnetopause
reconnection and ionospheric current systems (Eggington et al., 2020).

The MHD equations are implemented to represent a fully ionised quasi-neutral
hydrogen plasma on a 3-D uniform Eulerian cartesian grid. A second order
finite volume Van Leer advection scheme uses a vector potential representation
of the magnetic field on a staggered Yee (1966) grid which maintains a
divergence free magnetic field to machine-precision. The system is closed
assuming an ideal gas and stepped forward with a variable time-step using a
second order Runge Kutta scheme. These numerics conserve the internal energy,
rather than total energy, which negates negative pressures. A split magnetic
field is implemented (Tanaka, 1994) where the curl-free dipole-component is
omitted from the induction equation which reduces discretisation errors within
the magnetosphere. A Boris (1970) correction is used to limit the Alfvén speed
in the presence of a reduced speed of light, and a Von Neumann artificial
viscosity is applied to accurately capture shock physics and improve energy
conservation (Benson, 1992). Due to its heritage in simulating laboratory
plasmas, Gorgon also includes individual pressure terms for protons and
electrons, Ohmic heating based upon the Spitzer resistivity, optically thin
radiative loss terms, and electron-proton energy exchange. These are, however,
vanishingly small within collisionless magnetospheric plasmas. Magnetic
reconnection therefore develops through numerical diffusion alone.

The simulation domain extends from -20 to 100 RE in X and -40 to 40 in Y and
Z, with a uniform grid spacing of 1/2 RE, and which corresponds to GSM
coordinates with –X. An inflow boundary condition is located on the sun-ward
edge (–X) where the solar wind propagates into the domain, and outflow
boundary conditions are used at the tailward X, and Y and Z boundaries. The
dipole is located at the origin and the inner ionospheric boundary is located
at $\mid$3$\mid$ RE with a 370 cm-3 fixed density of cold 0.1 eV plasma which
diffuses outward to form a rudimentary plasmasphere. The ionosphere at the
inner boundary (Eggington et al., 2018) is represented by a thin conducting
shell, upon which the generalized Ohm’s law is solved for a given ionospheric
conductance profile to obtain an electrostatic potential (Ridley et al.,
2004). The corresponding electric field then modifies the plasma flow via the
associated drift velocity. The simulation is initialised with a dipole field
with an exponentially decreasing low plasma density through the domain and
with a mirror dipole within the solar wind to produce a Bx=0 surface (Raeder,
2003). Constant solar wind conditions of n0 = 5 cm-3, Bz = -2 nT, Ti = Te = 5
eV, vx = 400 km s-1, as shown in Table 1, are run for two hours with
geomagnetic dipole moment Mz = 7.94$\cdot$1022 Am2 to produce a fully formed
magnetosphere.

###  2.2 Interplanetary Shocks

Interplanetary shocks are produced at the interface of plasma regimes in the
solar wind when the relative speed of the shock structure to the ambient solar
wind exceeds the magnetosonic velocity (Kennel et al., 1985). Fast-forward
shocks are characterised by an increase in velocity, density, pressure and
magnetic field strength, as produced at the leading edge of impulsive
phenomena such as interplanetary coronal mass ejections (Burlaga, 1971) and
between fast and slow solar wind streams as these boundaries steepen into
corotating interaction region-driven shocks (Smith and Wolfe, 1976).

Table 1: Rankine-Hugoniot jump conditions for four fast-forward perpendicular IP shocks corresponding to four Gorgon simulations with the same initial solar wind conditions.  |  | n
---
[cm-3]
|  vx
---
[km s-1]
|  Dp
---
[nPa]
| T
---
[eV]
| B
---
[nT]
|  vshock
---
[km s-1]
Solar Wind | 5 | 400 | 1.34 | 5.0 | [0, 0, -2] | -
Shock I | 7.5 | 500 | 3.14 | 210.1 | [0, 0, -3] | 700
Shock II | 10 | 600 | 6.03 | 416.3 | [0, 0, -4] | 800
Shock III | 15 | 800 | 16.1 | 830.1 | [0, 0, -6] | 1000
Shock IV | 20 | 1000 | 33.5 | 1244.3 | [0, 0, -8] | 1200

Four perpendicular fast-forward shocks of varying strengths are injected into
the solar wind within four separate Gorgon simulations in order to
characterise the magnetospheric response to impulsive events of varying
magnitude. Perpendicular shocks denote shock geometries where the magnetic
field is orthogonal to the shock normal. The jump in solar wind conditions
therefore manifests as a spatially uniform front. The shocks are calculated in
accordance with the Rankine-Hugoniot conditions (Priest, 1982) with the four
jumps from the same initial solar wind, as shown in Table 1. Shock I shows a
modest jump in all parameters representative of the median southward IP shock
properties pbserved at 1 au during solar minimum (Echer et al., 2003). The
solar wind number density, n, jumps from 5 to 7.5 cm-3, southward IMF, Bz,
from -2 to -3 nT and solar wind velocity, vx, from 400 to 500 km s-1. Shocks
II, III and IV represent increasingly stronger cases up to the maximum
possible four-fold increase in the solar wind density and magnetic field for
Shock IV with a solar wind velocity jump of 400 to 1000 km s-1 . All shocks
are travelling at 200 km s-1 in the solar wind frame which is at the upper
bound of the 50–200 km s-1 range typically observed at 1 au (e.g. Berdichevsky
et al., 2000).

The parameters of Shock IV are judged as an estimate (Hudson et al., 1997) of
the extreme IP shock of 24 March 1991 which rapidly compressed the
magnetosphere over the course of minutes and promptly formed a new radiation
belt in the slot region (Blake et al., 1992; Horne and Pitchford, 2015). Space
weather events of this extremity are rare (Riley, 2012; Meredith et al.,
2017), but not unique as there are other examples where the magnetopause has
been observed inside geosynchronous orbit as low as 5.2 RE. (Cahill and
Skillman, 1977). It is also important to note that greater shock velocities of
over twice that of Shock IV are possible. For example, on 23 July 2012, the
STEREO-A spacecraft observed a non-Earth directed fast-forward shock with a
velocity of $\approx$ 2250 km s-1 (Russell et al., 2013) and theoretical
studies have highlighted the possibility of shock velocities over 3,000 km/s
emerging from the solar corona (Yashiro et al., 2004; Gopalswamy et al., 2005;
Tsurutani and Lakhina, 2014) with corresponding velocities of up to $\approx$
2,750 km s-1 manifesting at 1 AU (Desai et al., 2020).

###  2.3 Theory

To understand the motion of the subsolar magnetopause in response to an IP
shock, it is useful to consider the forces acting upon it. The following is
based on the theory of Freeman et al. (1995) and Freeman et al. (1998), and
references therein. In steady state, the geocentric distance to the subsolar
magnetopause, R, is well approximated by a balance between the pressure
exerted on the magnetopause by the shocked solar wind and the magnetic
pressure of the compressed dipole magnetic field of the Earth

$s\rho u^{2}=\frac{f^{2}B^{2}_{e}R^{6}_{E}}{2\mu_{0}R^{6}},$ (1)

where $\rho$ and $u$ are the solar wind density and speed, respectively, and
$s$ = 1 in the Newtonian approximation. Beq = 31100 nT is the equatorial
magnetic field strength at $1$ RE, and $\mu_{0}$ is the permeability of free
space. $f\approx 2.44$ is the typical dipole compression factor but this can
theoretically vary between $f=2$ for a plane magnetopause to $f=3$ for a
spherical magnetopause (Mead and Beard, 1964).

$m\frac{d^{2}R}{dt^{2}}=\frac{f^{2}B^{2}_{eq}R^{6}_{E}}{2\mu_{0}R^{6}}-s\rho(u_{\infty}+\frac{dR}{dt})^{2},$
(2)

where the final term is the Newtonian pressure applied to the now-moving
magnetopause and the subscript $\infty$ denotes the constant post-shock solar
wind values. The inertial mass m is expected to be that of the subsolar
magnetosheath column. Writing m = c $\rho_{\infty}$ R∞, where R∞ is the final
equilibrium position, we estimate c $\approx$ 1.2 in this case. Also rewriting
the magnetic pressure term using the final equilibrium version of Equation 1,
Equation 2 becomes

$\frac{d^{2}R}{dt^{2}}+\frac{s}{cR_{\infty}}\left[\left(u_{\infty}+\frac{dR}{dt}\right)^{2}-u_{\infty}^{2}\left(\frac{R_{\infty}}{R}\right)^{2}\right]=0.$
(3)

Linearising Equation 3 by substituting R(t) = R∞ \+ r(t), assuming r $<<$ R∞,
and retaining only first-order terms, the equation of motion becomes:

$\frac{d^{2}r}{dt^{2}}+\left(\frac{2}{K\tau}\right)\frac{dr}{dt}+\left(\frac{6}{K\tau^{2}}\right)r=0$
(4)

where $\tau$ = R∞ / u∞ is the characteristic system time scale, and K = c/s.
The homogeneous second-order ordinary differential Equation 4 is that of a
damped simple harmonic oscillator whose solution is an exponentially-decaying
sinusoid,

$r=Ae^{-bt}cos(\omega t+\phi),$ (5)

where b = 1 / (K $\tau$) and $\omega$ = b $\sqrt{(}6K-1)$. For a stationary
pre-shock magnetopause at position, R0, we have tan($\phi$) = -b / $\omega$
and A cos($\phi$) = R0 \- R∞.

##  3 Results

###  3.1 Shock-Magnetosphere Interaction

Figure 1 shows the Gorgon pressure at six stages during the simulation of
Shock IV, starting within the upstream solar wind, then at four stages within
the magnetosphere, and then sometime after when the system has reached a new
compressed steady state. Selected magnetic field lines are depicted in white
and the shock moves through the domain shown in just over 200 seconds.

Figure 1: Gorgon pressure in the x–y plane at six instances corresponding to
before, during, and after, IP Shock IV impacts the simulated magnetosphere.
Selected magnetic field lines are depicted in white and the shock parameters
are listed in Table 1.

The IP shock slows down upon passing through the bow shock and panel (b) shows
it develops a curved front as it propagates through the dense magnetosheath
(Samsonov et al., 2006; Andreeova et al., 2011). The subsequent impact on the
magnetopause disrupts the pressure-balanced equilibrium which initiates the
commencement phase associated with geomagnetic storms (Smith et al., 1986;
Araki, 1994). The initial magnetospheric state shows pressures below 1 nPa and
the enhanced solar wind pressure consequently produces magnetosheath pressures
over an order of magnitude higher. The tailward propagating magnetosonic
pulse, panels (d–e), subsequently produces enhanced plasma sheet pressures,
thinning of the tail current sheet and induces near-Earth tail reconnection
(Oliveira and Raeder, 2014). The enhanced dynamic pressure in the solar wind
compresses the magnetopause boundary from its initial position near –10 RE to
its final position near –6 RE.

###  3.2 Subsolar Magnetopause

The magnetopause can be characterised as possessing a finite thickness from
several ion gyroradii of several hundred kilometres (Le and Russell, 1994) to
over half an Earth radius (Kaufmann and Konradi, 1973). The different plasma
conditions on either side results in this being an asymmetric structure. Along
the sub-solar line the shocked solar wind first slows and diverts at the
fluopause (Palmroth et al., 2003) which, based on the gradient of the velocity
stream lines, is initially determined at $\approx$ –10.9 RE. The southward
oriented magnetic field then passes through zero at –10.75 RE as it tends to
the significantly larger positive magnetospheric fields. Further inward, the
peak in the magnetopause current density is located at –10.45 RE, which is
then followed by a local depletion in the plasma density at –10.1 RE. In this
study we determine the magnetopause position using the Bz = 0 condition, which
provides a consistent measure for southward IMF regardless of solar wind
conditions and stand-off distance.

Figure 2: Simulated subsolar magnetopause stand-off distance compared to
linear and nonlinear theoretical predictions for the four Shocks listed in
Table 1. For reference the location of geosynchronous orbit is annotated with
an arrow.

Figure 2 shows traces of the Gorgon subsolar magnetopause stand-off distances
(solid lines) over time for the four shocks simulated. The motion of the
magnetopause appears as three distinct phases. The first involves an
acceleration as the inertia of the magnetosheath is overcome. The second
appears as a rapid compressive phase which comprises the majority of the
change in standoff distance. The end of this rapid compression marks the third
stage of large-scale oscillatory motion with amplitudes of the order of an
Earth radius before the magnetopause reaches pressure-balanced equilibrium.

Shock I has the smallest compressive phase as the final oscillations around
pressure balance appear of a comparable magnitude to the total stand-off
distance travelled. For increasing shock strengths, the duration of the
compressive phase increases and the amplitudes and also frequencies of the
oscillations appear to decrease. The underlying position about which the
oscillations occur shifts Earthward as the oscillations are damped away which
may be attributed to changing conditions within the sheath, see Figure 1. The
oscillations also appear more strongly damped for the stronger shocks with
Shock IV producing magnetopause oscillations for approximately 300 seconds
compared to Shock I which produces oscillations which last four times as long.
Shocks I and II also feature more oscillations than III and IV, indicating
that they have a weaker damping ratio.

Also shown in Figure 2 are the subsolar magnetopause motions of the four
shocks predicted by the nonlinear numerical solution of Equation 3 (dashed
lines) and the linear solution given by Equation 5 (dotted lines), using
c=1.2, s=1, and values for $\rho_{\infty}$, u∞, R0, and R∞ taken from the
simulations. As expected, the linear solution is most similar to the nonlinear
numerical solution for Shock I, where the approximation r$<<$R∞ is most valid.
The difference increases with shock strength, especially in the initial phase
when the second-order (dR/dt)2 term in Equation 3 is not negligible.
Nevertheless, the linear theory is instructive in explaining the qualitative
response characteristics of a finite magnetopause response time, overshoot,
and decaying oscillation. For the nonlinear theory solution, the oscillation
period of the simulation produces good agreement in all cases but the initial
response time and oscillation damping rate are both progressively
overestimated compared to the simulation with weakening shock strength. This
suggests that the second term in Equation 3 may be an oversimplification in
the weak shock limit, and particularly the (dR/dt)2 term within it because the
linear theory that neglects (dR/dt)2 actually captures the initial simulation
response better than the nonlinear theory for Shocks I and II. It should also
be noted that the initial response is very sensitive to the initial conditions
in the magnetosheath (not shown) which may differ in the simulation from those
assumed in the theory.

The linear theory is instructive in understanding the underlying physics of
the magnetopause response. The nonlinear theoretical solutions of Equations 3
provide a means to extend this to larger peturbations but the solutions are
still necessarily dependent on the choice of coefficients and the assumptions
behind these such as the shape of the magnetopause surface and constant sheath
thickness. Further effects such as magnetic reconnection, magnetosheath
heating, finite solar wind mach numbers and wave speeds, and the reflection at
the pulse of the inner boundary back onto the magnetopause (Li et al., 1993;
Samsonov et al., 2007), are also not accounted for. The time-dependent and
self-consistent numerical solutions to the MHD equations, as solved by Gorgon,
instead provide the means of testing the the theory outlined in Section 2.3
for realistic nonlinear system-scale scenarios of strong fast-forward IP
shock-induced magnetopause motion. Large-scale periodic magnetopause motion,
consistent with those described here, have been observed following the arrival
of strong fast-forward IP shocks. During the impact of the August 1972 ICME,
when the sub-solar magnetopause was compressed to less than 5.2 RE upstream,
the Explorer 45 satellite experienced multiple magnetopause crossings in rapid
succession (Cahill and Skillman, 1977). Similarly, during the extreme event of
March 1991, the GOES-6 satellite experienced six inward-outward periodic
movements of the magnetopause over a 30 minute period (Cahill and Winckler,
1992). The lack of an upstream solar wind monitor does, however, complicate
further direct comparison to these events.

###  3.3 Frequency Analysis

The response of the magnetopause in the Gorgon simulations requires time-
frequency analysis suitable for non-stationary and nonlinear processes. Figure
3 uses ensemble empirical mode decomposition (EEMD) (Wu and Huang, 2004;
Torres et al., 2011) to derive the statistically significant modes associated
with the magnetopause motion and a Hilbert transform spectrum shows the
associated characteristic frequencies.

Figure 3: (a) Shows the original modes shown in Figure 2, (b) shows these
decomposed into their empirical modes and (c) shows a Hilbert transform of
their instantaneous frequencies.

These show the oscillations as a function of up to four statistically
significant modes, the primary of which exhibit frequencies between 2–13 mHz
with the frequencies of the dominant modes increasing with shock strength. The
instantaneous frequencies initially increase from zero as the inertial phase
begins and then plateaus somewhat during the compressive phase. They then
rapidly increase during the first magnetopause rebound before relaxing back to
values between 2–5 mHz. The instantaneous frequency is the time-derivative of
the phase at each moment and this distinct peak is therefore interpretted as
evidence of nonlinear phase steepening. Due to the strong damping, the
instantaneous frequencies don’t always provide a good handle on the overall
periodicity of the oscillations in each mode. Taking the auto-correlation of
each mode and finding the peak, we find slightly higher overall frequencies
of: 3.3, 4.1, 5.4 and 5.8 mHz for Shocks I–IV, respectively. The primary
empirical modes appear strongly damped with Shocks I and II inducing
approximately three total periods of diminishing amplitude whereas Shocks III
and IV induce less than two such periods. The eventual periods of the
oscillations are much longer than the first few oscillations for both Shocks I
and II. These are being picked up by the secondary mode and the frequencies
appear in the range of magnetopause surface eigenmodes being reflected between
the northern and southern ionospheres (Chen and Hasegawa, 1974) as seen in
high-resolution global MHD simulations (Hartinger et al., 2015) at 1.8 and 2.3
mHz respectively. With Shocks III and IV these are not apparent, possibly due
to the grid resolution not sufficiently resolving field-aligned currents near
the inner boundary and magnetopause reconnection at the subsolar point from
the strong southward driving prohibiting a surface eigenmode forming (Plaschke
and Glassmeier, 2011; Archer et al., 2019). Further modes are apparent
extending up to 0.1 Hz but these likely correspond to nonlinear higher order
terms.

The simulated magnetopause frequencies at the subsolar point lie where the
natural frequencies of the magnetopause fall according to the theory outlined
in Section 2.3. These oscillations notably occur at the lower end of the ULF
range observed throughout the magnetosphere (Menk, 2011) and Freeman et al.
(1995) point out that the linear theory predicts that the magnetopause acts as
a low pass filter of compressional waves due to solar wind dynamic pressure
variations and resonances may thus be selectively enhanced at the natural
eigenfrequency and suppressed at higher frequencies. Higher frequency waves,
however, could well exist further within the magnetosphere, for example via
field line resonances excited by the fast magnetosonic pulse during the
compression phase. The reproduction of ULF waves in global-MHD simulations
can, however, be sensitive to numerical effects (Claudepierre et al., 2009)
and an exploration of the magnetospheric ULF counterparts to IP shocks is
therefore left for a future endeavour.

##  4 Conclusions

This study has examined the magnetospheric and magnetopause response to four
synthetic IP shocks of varying magnitudes using Global-MHD simulations. While
previous studies (Smit, 1968; Freeman et al., 1995; Børve et al., 2011)
focussed on small-scale dynamic pressure changes in the upstream driver, we
developed nonlinear theory suitable for large peturbations and compared these
to self consistent global MHD simulations. This approach enabled the
characterisation of magnetopause motion for extreme scenarios representative
of fast-forward shocks striking the magnetosphere, as occur at the forefront
of coronal mass ejections.

In response to the IP shocks, the simulated magnetopause notably featured
large-scale oscillatory motion of the order of an Earth radius, prior to
reaching pressure balance. This was readily explained when considering the
driving, inertial and restoring forces associated with theory of the
magnetopause as a forced damped simple harmonic oscillator. The frequencies of
the oscillations occurred in the range of 2–13 mHz, predominantly occurring
between 2–5 mHz. The response times and oscillation periods seen in the
simulations were quantitatively consistent with the nonlinear theory, and the
damping time of the oscillation was also quantitatively consistent with
nonlinear theory for the stronger shocks but underestimated by theory for the
weaker shocks. The initial magnetopause response was also best predicted by
linear theory for the weaker shocks and by nonlinear theory for the strongest
shock, which is consistent with the assumptions beyond deriving the linearised
solutions.

These large-amplitude oscillations provide an explanation for periodic
magnetopause motion observed following the impact of strong interplanetary
shocks during the extreme space weather events of August 1972 (Cahill and
Skillman, 1977) and March 1991 (Cahill and Winckler, 1992). The time-delay in
the magnetopause response due to the inertia of the magnetosheath, combined
with the large-scale oscillatory motion, also helps to understand why static
models of the magnetopause break down during periods of strong solar wind
driving (e.g. Staples et al., 2020). Furthermore, the varying structure
throughout a given Earth-bound coronal mass ejection, combined with the
dynamic magnetopause response, could well mean that the magnetopause rarely
settles into highly compressed equilibrium states, which would also introduce
a significant bias to in-situ measurements of its locations.

Rapid inward motion of the magnetopause has been observed to consistently
produce enhancements and dropouts in the radiation belt phase space
distributions (Reeves et al., 2003; Schiller et al., 2016) and to drive an
abundance of global ultra-low-frequency wave activity (Li et al., 1997; Green
and Kivelson, 2004) and enhance ionospheric and ground-induced currents
(Fujita et al., 2003; Smith et al., 2019). These phenomena therefore present
further observables which could be affected by and tested (e.g. Wang et al.,
2010) for the large-scale magnetopause oscillations described herein.

## Acknowledgements

RTD, JPE and JPC acknowledge funding from NERC grant NE/P017347/1 (Rad-Sat).
MPF was supported by NERC grant NE/P016693/1 (SWIGS). JWBE is funded by a UK
Science and Technology Facilities Council (STFC) Studentship (ST/R504816/1).
MOA holds a UKRI (STFC / EPSRC) Stephen Hawking Fellowship EP/T01735X/1.
Research into magnetospheric modelling at Imperial College London is also
supported by Grant NE/P017142/1 (SWIGS). NM and RH would like to acknowledge
the Natural Environment Research Council Highlight Topic grant NE/P10738X/1
(Rad-Sat) and the NERC grants NE/V00249X/1 298 (Sat-Risk) and NE/R016038/1.
IJR and FAS acknowledge STFC grants ST/V006320/1 and NE/P017185/1. This
project has received funding from the European Union’s Horizon 2020 research
and innovation programme under grant agreement No. 870452 (PAGER). This work
used the Imperial College High Performance Computing Service (doi:
10.14469/hpc/2232).

## Data Availability Statement

The simulation data used in this paper is openly available on the UK Polar
Data Centre (UK PDC):
https://doi.org/10.5285/3774fa5b-f2fb-42c3-9091-5b11ac9744ea

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