11institutetext: Department of Physics, University of Helsinki, Helsinki,
Finland
11email<EMAIL_ADDRESS>22institutetext: Department of Physics
& Astronomy, University of Turku, Turku, Finland

# Modelling the interaction of Alfvénic fluctuations with coronal mass
ejections in the low solar corona

Chaitanya Prasad Sishtla 11 Jens Pomoell 11 Rami Vainio 22 Emilia Kilpua 11
Simon Good 11

###### Abstract

Context. Alfvénic fluctuations of various scales are ubiquitous in the corona;
their non-linear interactions and eventual turbulent cascade result in an
important heating mechanism that accelerates the solar wind. These
fluctuations may be processed by large-scale, transient, and coherent
heliospheric structures such as coronal mass ejections (CMEs). In this study
we investigate the interactions between Alfvénic solar wind fluctuations and
CMEs using magnetohydrodynamic (MHD) simulations.

Aims. We study the transmission of upstream solar wind fluctuations into the
CME leading to the formation of CME sheath fluctuations. Additionally, we
investigate the influence of the fluctuation frequencies on the extent of the
CME sheath.

Methods. We used an ideal MHD model with an adiabatic equation of state. An
Alfvén pump wave is injected into the quiet solar wind by perturbing the
transverse magnetic field and velocity components, and a CME is injected by
inserting a flux-rope modelled as a magnetic island into the quasi-steady
solar wind.

Results. The upstream Alfvén waves experience a decrease in wavelength and
change in the wave vector direction due to the non-radial topology of the CME
shock front. The CME sheath inhibits the transmission of long-wavelength
fluctuations due to the presence of non-radial flows in this region. The
frequency of the solar wind fluctuations also affects the steepening of MHD
fast waves causing the CME shock propagation speed to vary with the solar wind
fluctuation frequencies.

###### Key Words.:

coronal mass ejections (CMEs), Alfvén waves, MHD,

##  1 Introduction

The turbulent fluctuations in velocity, magnetic field, electric field, and
density are ubiquitous in the solar wind and corona (Coleman Jr 1968; Belcher
& Davis Jr 1971; Bale et al. 2005). The convective motions of the dense
photospheric plasma, which contains the solar magnetic field, are considered
to be the primary source of energy for these fluctuations (Cranmer & Van
Ballegooijen 2005; Kato et al. 2016). These fluctuations have been observed
both in situ (Belcher & Davis Jr 1971; D’Amicis & Bruno 2015) and remotely
(Tomczyk et al. 2007). In the solar wind, the power contained in Alfvénically
polarised fluctuations dominates over the power in compressive fluctuations
(Tu & Marsch 1995; Chen 2016). Additionally, the solar wind exhibits broad-
band Alfvénic fluctuations, which can then non-linearly interact to initiate
an energy cascade leading to dissipation via heating on smaller spatial
scales. In this view of a turbulence cascade, the inertial range is the
spatial scale of the fluctuations exhibiting a power-law behaviour between the
energy injection and dissipation scales. This inertial-range turbulence is
often studied within the framework of reduced magnetohydrodynamics (RMHD), in
which Alfvén waves are the linear wave modes (Zank & Matthaeus 1992;
Schekochihin et al. 2009; Perez & Chandran 2013). Previous studies (Matthaeus
et al. 1984; Gershman et al. 2019; González et al. 2021) have also discussed
the role of Alfvén wave propagation and reflection-driven Alfvénic turbulence
for particle acceleration in planetary radiation belts, in MHD reconnection
sites, and at interplanetary discontinuities.

In this study we investigate the interaction of Alfvénic perturbations with a
coronal mass ejection (CME) in the low corona. A CME is a transient plasma and
magnetic field eruption from the solar corona; it exhibits complex magnetic
substructures. They are one of the primary drivers of geomagnetic activity
near Earth (Kilpua et al. 2013, 2015; Kalliokoski et al. 2020, 2022). In
coronagraph images CMEs often exhibit a three-part structure with a bright
loop of compressed coronal plasma enclosing a dark low-density cavity,
corresponding to a flux rope (FR), which contains a high-density core (Gibson
& Low 2000; Kilpua et al. 2017b). A spacecraft encountering a CME typically
observes a shock, followed by a turbulent sheath and the ejecta. Only part of
the ejecta at 1 AU shows clear FR signatures due to interaction and evolution.
The internal structure of the CME is of significant interest as FR can cause
strong and sustained southward magnetic fields influencing the Earth (Kilpua
et al. 2017a). In addition, the turbulent and compressed CME sheath is highly
geoeffective (Kilpua et al. 2017b, 2019). Sheaths are known to exhibit an
extensive range of inertial and kinetic range spectral indices (Kilpua et al.
2020, 2021), embed multi-scale structures (Ruohotie et al. 2022), and
contribute to the acceleration of solar energetic particles (Kilpua et al.
2021). The fluctuations in the CME sheath have been seen to exhibit turbulence
characteristics often observed in the slow solar wind (e.g. higher
compressibility), and yet they are still dominated by non-compressible
Alfvénic fluctuations (Moissard et al. 2019). Additionally, compared to the
predominantly anti-sunward fluctuations in the solar wind preceding CMEs near
1 AU, sheaths are found to exhibit a more balanced distribution of sunward and
anti-sunward fluctuations (Good et al. 2020, 2022; Soljento et al. 2023).
There is currently a wide range of models (Gibson & Low 1998; Isavnin 2016;
Verbeke et al. 2019; Asvestari et al. 2021, 2022) to investigate the evolution
of the global FR structure of CMEs from Sun to Earth and their interactions
with the ambient solar wind. However, we need the understanding and
capabilities to model the smaller-scale features of CMEs and their sheath
regions. One important aspect is the transmission of Alfvénic fluctuations
from the surrounding ambient solar wind into the CME and the role it plays in
forming the sheath.

In this study, by using numerical simulations, we aim to enhance our
understanding of the formation of sheath structures by demonstrating the
effect of Alfvénic solar wind fluctuations on the large-scale structures of
the CME and to analyse the transmission of these fluctuations into the sheath
region. We find the CME shock speed influenced by the frequency of solar wind
fluctuations, with the CME sheath exhibiting non-radial flows, along with both
sunward and anti-sunward Alfvénic fluctuations. These results are obtained by
performing 2.5D MHD simulations of the solar corona assuming a radial solar
magnetic field, with the FR modelled using the Grad-Shafranov equation.

In Section 2 we introduce the MHD equations and associated boundary
conditions, the mechanism for Alfvén wave injection, and the CME model used
for the simulations. The influence of solar wind fluctuations on the CME and
their transmission to the sheath is discussed in Section 3. Section 4 presents
a statistical comparison of the shock location and sheath extent for varying
solar wind and CME parameters, including a case with no solar wind
fluctuations. The conclusions are summarised in Section 5.

##  2 Methodology

To perform our study we developed a 2.5D magnetohydrodynamic (MHD) simulation
from the low corona at $1.03$ solar radii ($R_{\odot}$) to
$30\leavevmode\nobreak\ R_{\odot}$. The simulation domain is 2D in space; the
velocity and electromagnetic field vectors have three components. The solar
wind is modelled assuming a global radial unipolar (outward) solar magnetic
field, which can be considered realistic for a limited region of the Sun, such
as a coronal hole. The MHD equations and the relevant physical processes of
gravity and ad hoc coronal heating are described by the following equations:

$\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0,$
(1) $\displaystyle\frac{\partial(\rho\mathbf{v})}{\partial
t}+\nabla\cdot[\rho\mathbf{v}\mathbf{v}+(P+\frac{B^{2}}{2\mu_{0}})\mathbf{I}-\frac{\mathbf{B}\mathbf{B}}{\mu_{0}}]=-\frac{GM_{\odot}\rho}{r^{2}}\hat{\mathbf{r}},$
(2) $\displaystyle\frac{\partial\mathcal{E}}{\partial
t}+\nabla\cdot[(\mathcal{E}+P-\frac{B^{2}}{2\mu_{0}})\mathbf{v}+\frac{1}{\mu_{0}}\mathbf{B}\times(\mathbf{v}\times\mathbf{B})]=-\frac{GM_{\odot}\rho
v_{r}}{r^{2}}+S,$ (3) $\displaystyle\nabla\cdot\mathbf{B}=0,$ (4)
$\displaystyle\frac{\partial\mathbf{B}}{\partial
t}-\nabla\times(\mathbf{v}\times\mathbf{B})=0,$ (5)

where

$\displaystyle\mathcal{E}=\frac{1}{2}\rho
v^{2}+\frac{P}{\gamma-1}+\frac{B^{2}}{2\mu_{0}},$ (6) $\displaystyle
S=S_{0}\mathrm{exp}\left(-\frac{r}{L}\right).$ (7)

Here the quantities $\rho$, $\mathbf{v}$, $\mathbf{B}$, $\mathcal{E}$, and $P$
correspond to the mass density, bulk plasma velocity, magnetic field, total
energy density, and thermal pressure. Equations 1-5 correspond to the mass
continuity, momentum, energy continuity, and induction equations,
respectively, and are subsequently referred to as such. The solar wind plasma
evolves by solving these MHD equations for an adiabatic polytropic index of
$\gamma=5/3$. Thus, to obtain a steady-state solar wind that approximates a
Parker-like outflow, we incorporate an additional energy source term in
Equation 7 (Pomoell et al. 2015; Mikić et al. 2018; Sishtla et al. 2022) with
$S_{0}=0.5\times 10^{-6}$ Wm-3 and $L=0.4\leavevmode\nobreak\ R_{\odot}$.

The numerical method used in this work to solve the MHD equations was employed
in previous studies of the solar corona (Pomoell & Vainio 2012). The method
utilises a strong stability preserving (SSP) Runge-Kutta method to advance the
semi-discretised equations in time, and employs the Harten–Lax–van Leer (HLL)
approximate Riemann solver supplied by piece-wise linear slope-limited
interface states. The equations are solved in spherical coordinates and the
magnetic field is ensured to be divergence free to the floating point accuracy
by utilising the constrained transport method (Kissmann & Pomoell 2012).

The MHD equations were integrated forward in time for a 2D meridional plane
with a radial extent of $r=r_{0}=1.03\leavevmode\nobreak\ R_{\odot}$ to
$r=30\leavevmode\nobreak\ R_{\odot}$, and an co-latitudinal extent of
$\theta=10^{\circ}$ to $\theta=170^{\circ}$. The domain is therefore symmetric
in the out-of-plane longitudinal $\phi$ direction. The solar magnetic field
was initialised to be radially outward with an associated vector potential
$\mathbf{A}=-B_{0}r_{0}(r_{0}/r)\cot{\theta}\leavevmode\nobreak\
\hat{\mathbf{\phi}}$, where $B_{0}=5$ G, and the magnetic field in the
simulation was then specified using $\mathbf{B=\nabla\times A}$. The
simulation grid was defined by $500$ cells logarithmically spaced in the
radial direction, and $128$ equidistant cells in the latitudinal direction.
Appendix A validates this choice of the radial grid resolution by verifying
the results presented in the following sections for a significantly higher
resolution.

At the inner radial boundary, representing the coronal base, we specified a
constant mass density and temperature along the boundary with
$\rho_{0}=8.5\times 10^{-13}\leavevmode\nobreak\ \mathrm{kg}$ and
$T_{0}=1.2\times 10^{6}\leavevmode\nobreak\ \mathrm{K}$. At the outer radial
and the latitudinal boundaries, we linearly extrapolated all the dynamical
quantities to enforce an outflow boundary condition.

###  2.1 Introducing Alfvénic perturbations

Figure 1: Coronal quasi-steady state. Panel (a) shows a snapshot of the plasma
temperature upon the injection of a $3\leavevmode\nobreak\ $mHz linearly
polarised Alfvén wave, with an annotation describing the viewing angle along
$105^{\circ}$. In panels (b) and (c) are shown the out-of-plane $v_{\phi}$
velocity and $B_{\phi}$ magnetic field components along the viewing angle. The
variations in the density $\rho$ from the quasi-steady values prior to the
injection of the Alfvén wave are presented in panel (d).

After achieving a steady-state solar wind by integrating Equations 1\- 7 in
time, we introduced Alfvénic fluctuations. The Alfvén waves were introduced at
the coronal base by utilising a time-dependent boundary condition for the
Elsässer variables, defined by

$\displaystyle\mathbf{z}^{\pm}_{\perp}=\mathbf{v}_{\perp}\pm\frac{\mathbf{B}_{\perp}}{\sqrt{\mu_{0}\rho}}.$
(8)

We continuously injected the monochromatic and linearly polarized Alfvénic
fluctuations in the out-of-plane $\phi$ direction by specifying the anti-
sunward (outgoing) Elsässer variable as
$\delta\mathbf{z}^{-}=\mathrm{Z_{0}}\sin{\left(2\pi
f_{0}t\right)}\leavevmode\nobreak\ \hat{\mathbf{\phi}}$ at the lower boundary
with $\mathrm{Z_{0}}=32\sqrt{2}\leavevmode\nobreak\ \mathrm{kms^{-1}}$ being
the amplitude and $f_{0}$ the frequency of the wave.

In Figure 1 we present the quasi-steady solar wind after the injection of a
$3\leavevmode\nobreak\ \mathrm{mHz}$ Alfvén wave. In general, the solar wind
response to the injected fluctuations depends on the polarization of the waves
(Goldstein 1978; Hollweg 1971). The propagation of the linearly polarised
injected Alfvén wave causes a fluctuating magnetic field strength which
results in the steepening of the Alfvén waves themselves (Cohen & Kulsrud
1974), in addition to generating density fluctuations due to the ponderomotive
force (Nakariakov et al. 1997). Due to this, we observed an increase in
temperature from 1.2 MK at the lower boundary to 1.4 MK near 3 $R_{\odot}$,
before decreasing again (see Figure 1(a)). The generation of density
fluctuations is a second-order non-linear effect, and is absent in
incompressible MHD. In this simulation the density fluctuations are absent as
the chosen grid resolution causes the Alfvén waves to be damped due to
numerical diffusion before the density fluctuations can be generated. This
damping ensures that we have only a pure monochromatic Alfvén wave in the
simulation that has not yet experienced any reflections from large-scale
density gradients in the solar wind (Verdini & Velli 2007; Van Ballegooijen et
al. 2011), and confines the waves to be present below $\approx
10\leavevmode\nobreak\ \mathrm{R_{\odot}}$. Thus, in this study we confined
our analysis to the wind below $\approx 10\leavevmode\nobreak\
\mathrm{R_{\odot}}$.

The Alfvénicity, steepening, and absence of density fluctuations in the
simulation are illustrated by considering the radial propagation of the
injected waves along a viewing angle (annotated in Figure 1(a)). In Figure 1,
panels (b) and (c), we present the out-of-plane velocity $v_{\phi}$ and
magnetic field $B_{\phi}$ components along this viewing angle. Upon comparing
the two panels, we observe an anti-correlation between $v_{\phi}$ and
$B_{\phi}$, which confirms both the Alfvénicity and anti-sunward direction of
the injected wave. Furthermore, to verify the lack of accompanying density
perturbations, we plot in Figure 1(d) the fluctuating component of the mass
density $\Delta\rho/\rho=(\rho-\rho(t=0))/\rho(t=0),$ where $\rho(t=0)$ is the
mass density in the coronal volume prior to the Alfvén wave injection. The
panel shows a large-scale variation in the density, but the absence of any
smaller-scale fluctuations.

###  2.2 Introducing a coronal mass ejection

In this study, we do not model the initiation and subsequent eruption of the
CME, but instead directly instantiate an erupting plasma structure mimicking
an eruptive CME. We superimpose an appropriate plasma structure on the quasi-
steady solar wind containing the Alfvénic fluctuations to achieve this.

The magnetic field of the CME is modelled as a force-free FR using the
Soloviev solution of the Grad-Shafranov (GS) equation (Solov’ev 1968). The
solutions to the GS equation represent axisymmetric MHD equilibria of
magnetized plasmas without flows such that the equilibrium condition

$\displaystyle\mathbf{J}\times\mathbf{B}=\nabla P$ (9)

is satisfied where $\mathbf{J}$ is the current density given by
$\mathbf{J}=\nabla\times\mathbf{B}/\mu_{0}$, and $P$ is the thermal pressure
of the plasma. Once the magnetic structure of the CME is modelled using
Equation 9 under the assumption of zero-beta ($P=0$) conditions we then
populate it with plasma to model a high-density ejecta. The density inside the
structure is specified as

$\displaystyle\rho_{\mathrm{cme}}=\frac{\rho_{\mathrm{cme,0}}}{2}\left[1-\cos\left(\pi\frac{d_{\mathrm{cme}}-d}{d_{\mathrm{cme}}}\right)\right],$
(10)

where $d$ is the distance from the centre of the structure and
$d_{\mathrm{cme}}$ is the radial extent, and $\rho_{\mathrm{cme,0}}$ is the
density specified at the centre. This formulation of $\rho_{\mathrm{cme}}$
ensures a continuous transition from the high-density $\rho_{\mathrm{cme,0}}$
CME core to the background density at the edge of the structure. We also
initialise the plasma with a constant temperature of $0.5\times
10^{6}\leavevmode\nobreak\ \mathrm{K}$, and an ejection velocity
$\mathbf{v_{\mathrm{ej}}}$ along the radial direction inside the CME. The
constructed CME (Equations 9 and 10) is then superimposed on the quasi-steady
solar wind including the Alfvénic fluctuations described in Section 2.1. We
note that due to the ad hoc specification of the thermal pressure inside the
CME and superposition of the structure on the quasi-steady wind the plasma in
and immediately surrounding the CME is not in equilibrium causing the FR to
expand and propagate.

Figure 2: CME insertion into the solar wind. The black curves show magnetic
field lines with the black arrows indicating the direction and relative
strength. The figure is annotated with the directions of the $\nabla P$ and
$\mathbf{J\times B}$ forces that comprise the Grad-Shafranov equilibrium
condition. An initial ejection velocity $\mathbf{v}_{\mathrm{ej}}$ is given to
the CME along the radial direction.

In Figure 2 we present a schematic showing the magnetic field configuration
and dynamic contributions acting on the CME at the onset. The poloidal field
of the FRs we used for this study is oriented in the anti-clockwise direction,
as seen by the direction of magnetic field vectors around the FR. This causes
them to deflect when reconnecting with the radially outward magnetic field
lines. The ejection velocity is also directed in the radial direction.

The plasma signatures encountered by a virtual spacecraft upon traversing such
an FR are shown in Figure 3. The dashed vertical lines demarcate the upstream,
CME sheath, and FR regions. The spacecraft is placed at 5 $R_{\odot}$, and a
viewing angle of 105 degrees with the time axis referenced from the time of
CME injection in the simulation. The CME is modelled using an initial speed
$v_{\mathrm{ej}}=500\leavevmode\nobreak\ $km s-1, peak density of $2\rho_{0}$
(where $\rho_{0}$ is the constant mass density at the coronal base at
$r=r_{0}$), and $B_{\phi}\approx 12$ G. Prior to encountering the CME, the
virtual spacecraft measures the pristine upstream solar wind conditions as
seen in Figures 3(a)-(d). We observe anti-sunward Alfvénic fluctuations by the
anti-correlated variations in $B_{\phi}$ and $v_{\phi}$. The first CME-related
signature registered is the shock at $t\approx 15$ min. The shock is followed
by the CME sheath. In this sheath region we observe larger non-radial flows
(compared to the upstream fluctuations) as non-zero values for $v_{\theta}$
and $v_{\phi}$ (Figure 3(b)). The CME sheath is also characterised by an
enhanced density and temperature (Figures 3(c), (d)) due to the shock
transition and plasma piling ahead of the CME. Finally, the spacecraft
encounters the FR. It features a smooth variation in $B_{\theta}$ (Figure
3(a)) indicating rotation of the field as it crosses the magnetic island
initialised in Equation 9. The FR region also has a relatively high density
(Figure 3(c)).

Figure 3: CME encounter with a virtual spacecraft located at 5 $R_{\odot}$ and
a viewing angle of $105^{\circ}$. The vertical lines differentiate the
upstream, sheath, and flux rope encountered by the spacecraft.

##  3 Results

In this section we describe the interaction of the Alfvénic fluctuations in
the quasi-steady solar wind (Section 2.1) with a CME modelled as in Section
2.2 with $v_{\mathrm{ej}}=500\leavevmode\nobreak\ $km s-1, peak density of
$2\rho_{0}$, and $B_{\phi}\approx 12$ G. The CME is deflected in the -X
direction as it reconnects with the anti-sunward-directed radial magnetic
field line due to the chosen poloidal field direction of the FR. In Figure
4(a)–(c) we show the density compression ratio computed as $\rho(t)/\rho(t=0)$
(Pomoell et al. 2015), the plasma beta
$\beta=p_{\mathrm{thermal}}/p_{\mathrm{magnetic}}$, and the out-of-plane
(longitudinal) velocity component $v_{\phi}$ at simulation time $t=$ 10.8 min.

The initial velocity of the FR ($\mathbf{v_{\mathrm{ej}}}$) and the out-of-
equilibrium $\mathbf{J\times B}-\nabla P$ force allows the plasma of the CME
to expand at a rate much higher than the ambient solar wind velocity. This
results in the FR driving a fast mode shock. In an ideal MHD system, the
maximum density compression ratio at a shock front is
$\frac{\gamma+1}{\gamma-1}$ (e.g. Koskinen 2011), which in our case, with
$\gamma=5/3$, gives a theoretical maximum compression of $4$. At the shock
front, located approximately at $2.5$ $R_{\odot}$, there is an observed
density compression jump from $1$ in the upstream region to $\approx 2$ inside
the sheath at the flank of the CME, and to $\approx 3$ near the head-on region
of the CME. The FR is driving a shock as a result of the large difference
between the CME ejection velocity and upstream solar wind velocity. The FR
trails behind the leading shock front and is identified by the closed field
lines forming the magnetic island. The sheath is the region between the shock
and FR. CME sheath regions are often characterised by non-radial flows and
build-up of density in a pile-up compression region (PUC) (Das et al. 2011).
In our simulation the presence of non-radial flows is due to the draping of
the flow around the magnetic island in the sheath (Siscoe & Odstrcil 2008).
This draping causes the formation of an oblique shock, which in turn causes
large-scale flows to be generated to maintain the non-radial continuities in
the Rankine-Hugoniot jump conditions. Additionally, the compression of plasma
in the sheath region causes the formation of a PUC. Figure 4(a) is annotated
with markings denoting the PUC, the sheath region, and the location of the
reconnection site causing the CME to deflect. The reconnection at the CME
flank reduces magnetic flux at this location, while the field still drapes
around the FR in the opposite flank. This drives a strong magnetic field
gradient that deflects the CME.

Figure 4: Snapshots of CME propagation. The figure presents snapshots of the
simulation as the CME is propagating in the low corona at $t=10.8$ min. In
panel (a) the colour intensity denotes the density compression compared to the
quasi-steady solar wind, with annotations indicating the PUC, sheath, and
reconnection site. The plot in panel (b) shows the plasma beta, and panels (c)
and (d) present the out-of-plane velocity component $v_{\phi}$ and the co-
latitudinal (meridional) component $v_{\theta}$.

In Figure 4(b) we plot the plasma beta in the simulation to investigate
whether plasma dynamics are dominated by the magnetic field (low $\beta$) or
gas dynamics (high $\beta$). We see that the whole steady-state solar wind
upstream of the shock has $\beta\ll 1$, which indicates a frozen-in plasma
condition is strongly met. We observe a region of high $\beta$ inside the
sheath as we view the CME head-on and around the reconnection site. A
comparison with Figure 4(a) shows that this high-$\beta$ region occurs when we
encounter a region of enhanced density inside the sheath. The FR is isolated
from the surrounding sheath region and maintains a low $\beta$.

Finally, in Figures 4(c) and 4(d) we present the $v_{\phi}$ and $v_{\theta}$
components in the simulation, respectively. At the CME flanks, we see that the
solar wind perturbations in $v_{\phi}$ are modified by the shock. The radially
directed wave vectors upstream of the shock are modified downstream to reflect
the non-radial topology of the shock front. However, in the regions of
enhanced $\beta$ from Figure 4(b), there are significant flows in the
$\pm\phi$ directions, as shown by the large $v_{\phi}$ magnitudes. These are
the non-radial flows that are generated as a result of the structure of the FR
that causes the solar magnetic field to drape around it, as well as the strong
guide field of the FR affecting the flow of the surrounding plasma. The
spatial extent of the non-radial flows, as depicted by the dark blue region in
Figure 4(c), indicates a similarity in size to the wavelength of Alfvén waves
at the flanks of the CME. The $v_{\theta}$ component does not have any
perturbations upstream as expected. However, downstream of the shock, we see
large flows in $\theta$ as the FR sweeps away the surrounding plasma as it
propagates.

###  3.1 CME modified solar wind fluctuations

The cut through the CME flanks in Figure 4(c) shows that the frequency of the
upstream solar wind fluctuations decreases downstream of the shock. Alfvénic
fluctuations such as these are characterised by a correlation in the velocity
and magnetic field (Belcher et al. 1969) and can be identified by the
accompanying Elsässer variables (Equation 8). Fluctuations can be identified
by subtracting the mean plasma flow speed from the accompanying Elsässer
variable. The simulation snapshots of the anti-sunward-propagating Elsässer
variable $z_{\phi}^{-}$ at various times is shown in Figure 5. This figure is
annotated with the viewing angle of $160^{\circ}$ corresponding to the flank
of the CME.

Figure 5: Snapshots of Elsässer variables. The figure presents the anti-
sunward-propagating Elsässer variable
$z_{\phi}^{-}=v_{\phi}-B_{\phi}/\sqrt{\mu_{0}\rho}$ during the CME evolution
shown at various times. The figures are annotated with a viewing angle of
$160^{\circ}$ corresponding to the CME flank.

The panels (a) and (b) present the anti-sunward Elsässer variable 6.25 and
8.75 minutes after the onset of the eruptive event, respectively. The
significant negative value of $z_{\phi}^{-}$ in the figure is due to the
positive $B_{\phi}$ inside the FR. The large positive $B_{\phi}$ field
compresses the plasma ahead of it, causing the large negative valued
$z_{\phi}^{-}$. This positive $B_{\phi}$ along with the anti-clockwise
direction of the poloidal field around the FR (Figure 2) denotes a positive
(right-handed) chirality for the FR. At the flanks, the initially expanding
CME amplifies the imposed fluctuations on the flanks as it ‘drags’ the solar
wind at speeds higher than the ambient Alfvén velocity prior to shock
formation (Figures 5(a)-(d)). After the formation of a shock along the
$160^{\circ}$ viewing angle in panel (e), these CME-modified anti-sunward
fluctuations are also present in the downstream region.

####  3.1.1 Shock transmitted solar wind fluctuations

Figure 6: Snapshot of the Alfvén velocity, defined as
$v_{a}=\mathbf{v_{a}}\cdot\hat{\mathbf{b}}$, at $t=37.5$ min. The figure is
annotated to show the CME flank at the viewing angle $160^{\circ}$.

The presence of a shock modifies the upstream anti-sunward solar wind
fluctuations as they are transmitted (propagating anti-sunward) and reflected
(propagating sunward) downstream of the shock (e.g. Vainio & Schlickeiser
1998, 1999). If a medium is stationary, a wave propagates conserving its
frequency. In the shock frame, the fluid structure is quasi-stationary on the
timescale it takes for the wave to be transported through the shock, so Alfvén
waves conserve their frequency in the shock frame. Another boundary condition
at the shock for the wave vector comes from the conservation of the tangential
wave length. Thus, for a transmitted, outward-propagating Alfvén wave,

$\displaystyle k_{1,n}(u_{1,n}-v_{a1,n})$ $\displaystyle=$ $\displaystyle
k_{2,n}(u_{2,n}-v_{a2,n}),$ (11) $\displaystyle k_{1,t}$ $\displaystyle=$
$\displaystyle k_{2,t},$ (12)

where $\mathbf{u}$ is the fluid velocity in the shock frame, $\mathbf{k}$ is
the wave vector, $\mathbf{v}_{a}$ is the Alfvén velocity, the subscripts 1 and
2 denote the upstream and downstream regions, and the subscipts $n$ and $t$
denote the normal and tangential components of the vector quantities in
relation to the shock surface normal. As the normal component of the magnetic
field is conserved at the shock, $v_{a2,n}=v_{a1,n}/\sqrt{X}$, where
$X=\rho_{2}/\rho_{1}=u_{1,n}/u_{2,n}$ is the compression ratio of the shock.
Thus, the downstream wave number

$k_{2,n}=k_{1,n}X\frac{M_{A}-1}{M_{A}-\sqrt{X}},$ (13)

where $M_{A}=u_{1,n}/v_{a1,n}$ is the Alfvénic Mach number, showing that the
wavelength in the shock normal direction is compressed by a factor exceeding
the gas compression ratio of the shock. For a low-Mach-number ($M_{A}\lesssim
2$) quasi-parallel fast-mode shock propagating in a low-$\beta$ plasma, the
compression ratio is approximately $X\lesssim M_{A}^{2}$ (Vainio &
Schlickeiser 1999), implying that the wave compression can be very
significant. We note that at the limit of a switch-on shock ($X=M_{A}^{2}$),
wave compression becomes infinite. For a reflected wave (i.e. the case where
the downstream wave is propagating towards the Sun), the wave compression is
less significant,

$k_{2,n}=k_{1,n}X\frac{M_{A}-1}{M_{A}+\sqrt{X}},$ (14)

in particular for a low-Mach-number shock. Thus, we expect the upstream Alfvén
wave to significantly decrease in wavelength as it propagates downstream of
the CME shock. Therefore, the expected composition of the downstream anti-
sunward solar wind fluctuations as a consequence of the shock transmission
consists of a long-wavelength component as the CME flank drags the waves that
were transmitted downstream through the early quasi-perpendicular phase of the
shock on a given field line ($k_{t}$ is conserved) and a short-wavelength
component due to the Alfvén wave transmission across the quasi-parallel part
of the shock. The waves transmitted in the early quasi-perpendicular stage
(Figure 5) have a longer wavelength as they experience a higher Alfvén
velocity downstream of the CME shock (Figure 6) causing the increase in
wavelength.

####  3.1.2 Fluctuations around the CME shock

In Figure 3, we observe the presence of a variety of radial (via the shock
propagation) and non-radial (in the sheath plasma) enhancements in the
velocity. In Figure 7 we attempt to exclude the radial flow by viewing the
Elsässer variables in the frame of reference of the shock. The CME shock is
detected by locating a jump in density compression that exceeds a factor of 2
and is always placed at the $x=0$ coordinate with the Elsässer variables shown
in the neighbourhood of 5 $R_{\odot}$. The positive x-axis values are upstream
of the shock, and the negative x-axis are downstream.

Figure 7: Evolution of the Elsässer variables at the CME flank. The spatio-
temporal evolution of the Elsässer variables for the non-radial directions in
the frame of reference of the shock ($x=0$) is presented. The quantities are
shown for a viewing angle of $160^{\circ}$, and an Alfvénic fluctuation
frequency of $3$ mHz. The x-axis denotes the shock neighbourhood in units of
$R_{\odot}$; positive values indicate the solar wind and negative values
indicate the region downstream of the shock.

In Figure 7 we present the sunward ($z_{\phi,\theta}^{+}$) and anti-sunward
Elsässer variables for a viewing angle of $160^{\circ}$ (CME flank) and the
Alfvén wave with frequency 3 mHz injected in the quiet solar wind. The x-axis
represents the distance along the given viewing angle, and the y-axis is the
simulation time. The magnitude of the Elsässer variables are described using
the colour intensity. In Figure 7(a) the upstream solar wind fluctuations can
be observed as they are incident onto the CME shock from $x>0$. The upstream
Alfvén waves in the simulation have significant amplitudes until $\approx
10\leavevmode\nobreak\ \mathrm{R_{\odot}}$ and therefore disappear beyond
$\mathrm{t}\approx 40\leavevmode\nobreak\ \mathrm{min}$. Further into the
downstream $x<0$ we see the long-wavelength component as the CME propagation
modifies the downstream waves. The white region between these two regions is
where the short-wavelength component of the shock-transmitted waves should be
present, as expected based on the analysis in Section 3.1.1. This white region
corresponds to a similar region in Figures 5(e)-(f) around the CME shock where
the upstream waves should be compressed. Additionally, we see the generation
of sunward fluctuations in Figure 7(b) due to the interaction of solar wind
fluctuations with the CME shock (Equation 14). In the $\theta$ direction, we
do not see any fluctuations upstream, as expected in Figure 4(d). However, in
the shock neighbourhood we can see the effect of non-radial flows due to it
being a non-radial shock, which was also seen in Figure 7(d).

Figure 8: Shock at the CME flank. Panel (a) is a simulation snapshot at
$t=25\leavevmode\nobreak\ $min of the anti-sunward Elsässer variable
$z_{\phi}^{-}$ with annotations describing the viewing angle along
$160^{\circ}$, the shock location, the approximate beginning of the CME
sheath, and the approximate shock velocity $v_{\mathrm{shock}}$. Panels (b)
and (c) are the density compression and $z_{\phi}^{-}$ along the viewing
angle, respectively. Panels (d) and (e) present the fluid velocity and Alfvén
speed along the direction of the background magnetic field.

The white region in Figure 7(a) is further investigated through Figure 8 where
the density compression, the anti-sunward Elsässer variable, the flow speed
($\mathbf{v\cdot b}$), and the Alfvén speed along the background field
($\mathbf{v}_{a}\cdot\mathbf{b}$) are presented at $t=25\leavevmode\nobreak\
$min. Panel (a) of the figure is an annotation of Figure 5(f) with the
location of the CME shock and the approximate beginning of the sheath, where
we start observing the long-wavelength fluctuations. The white region thus
corresponds to the location between these two markers. Panel (b) shows the
density compression utilised in identifying the shock, and panel (c) is the
anti-sunward Elsässer variable. The average shock velocity at the
$160^{\circ}$ viewing angle between $t=6.25\leavevmode\nobreak\ $min to
$t=31.25\leavevmode\nobreak\ $min is found to be $\approx
2078\leavevmode\nobreak\ $km s-1, as annotated in panel (a). The shock is
associated with a gas compression ratio of $\approx 2$ (panel (d)), with the
Alfvén speed increasing from upstream to downstream (panel (e)). Then,
downstream of the shock (through Equation 14) the wavelength of the upstream
wave would be compressed by about three times. The absence of the anticipated
compression of the upstream Alfvén wave in our simulation indicates that the
spatial grid does not adequately resolve this specific region. This causes the
transmitted waves to be of lower amplitude in this location, as observed in
panel (c), signifying numerical dissipation. Therefore, the downstream
fluctuations plotted in Figure 7 do not contain the additional shock-
compressed Alfvén waves. However, the restricted grid resolution for this
simulation is necessary to sustain a monochromatic Alfvén wave before the CME
injection by numerically damping the waves before their decay. In Appendix A
we present a modified simulation where the shock-compressed waves are
captured. The results in the Appendix show an adherence to the expected
composition of the downstream waves (see Section 3.1.2), with the downstream
waves becoming modified compared to Figure 7 only after $\mathrm{t}\approx
20\leavevmode\nobreak\ \mathrm{min,}$ due to possible wave steepening. Thus,
in practice the downstream solar wind fluctuations would contain long- and
short-wavelength components due to the CME passage and upstream waves
transmission, respectively, prior to the development of further non-linear
interactions.

Figure 9: Evolution of the Elsässer variables at the CME nose, similar to
Figure 7, but with a viewing angle of $105^{\circ}$. The x-axis similarly
denotes the shock neighbourhood in units of $R_{\odot}$ with the shock centred
at $x=0$.

In Figure 9, we show the solar wind fluctuations around the shock when viewing
the CME head-on instead of the flank. Panel (a) shows the upstream solar wind
fluctuations incident onto the shock. A similar white region corresponds to
the region where the incident waves would be compressed. However, in the far
downstream region $x<0$, we only observe large non-radial flows as the large
positive guide field $B_{\phi}$ of the FR affects the surrounding plasma to
generate a non-radial flow. These non-radial flows are additionally observed
in the sunward component (panel (b)). In the $\theta$ directions (panels (c)
and (d)) the large flows are generated due to the non-radial topology of the
CME shock.

Therefore, a primary difference between solar wind fluctuations downstream of
the CME shock for a head-on encounter (Figure 9) compared to a flank encounter
(Figure 7) is the absence of long-wavelength amplified fluctuations, which are
comparable in size to the non-radial flows. Through Figure 8(a) it is seen
that the CME shock is non-radial as the shock velocity is greater head-on (the
direction where the FR is expanding) than at the flank. This indicates that
the wavelengths of the shock-compressed upstream waves differ as the
compression depends on the Alfvén Mach number in the shock frame of reference.
Furthermore, as the shock expands faster than the ambient Alfvén speed, we
expect different characteristics of the fluctuations closer to the shock
(containing a mix of shock-transmitted and already present fluctuations) and
further downstream (with the CME-amplified fluctuations).

##  4 Formation of the CME Sheath

In Section 3 we discussed the dependence of the CME sheath fluctuations on the
upstream solar wind conditions and the shock properties. The interaction of
the solar wind fluctuations with the CME shock gave rise to both sunward and
anit-sunward Alfvénic fluctuations at the CME flank (Figures 7), along with
the compression of anti-sunward upstream waves. In addition, the CME sheath
contains non-radial flows due to the magnetic structure of the FR and the non-
radial CME shock (Figures 4(c)- 4(d)). The extent of the non-radial flows,
represented by the dark blue region in Figure 4(c) and Figure 5, suggests that
their spatial extent is comparable to the Alfvén waves at the CME flanks. This
limitation could hinder the presence of large-amplitude Alfvénic fluctuations
in the presence of similarly large non-radial flows. Thus, we now investigate
the influence of the Alfvén waves on the growth of the CME sheath region and
propagation of the CME shock. This allows us to understand the effect of
Alfvén waves on the shock properties and to infer the development of non-
radial flows as the CME propagates further in the solar wind.

Previous studies have shown significant variations of the sheath thickness
based on the physical properties of the CME, more precisely the properties of
the CME FR and the shock compression ratio (Russell & Mulligan 2002). Thus, we
investigate how the large-scale structure of the sheath depends on the density
and injection velocity of the FR driving it, and the frequency of the Alfvénic
fluctuations that are present in the solar wind. These different cases,
studied by varying a selection of the parameter values of the simulation set-
up, including the case studied in the previous sections (henceforth designated
as C1) are detailed in Table 1. We find that the large-scale structures of the
sheath, such as a PUC, high-speed flows, and magnetic field line draping, are
similar for all the cases considered in Table 1.

Case | CME Parameters | $f_{\mathrm{inj}}$ [mHz]
---|---|---
|  $\mathbf{v}_{\mathrm{ej}}$ [km s-1]
---
$\rho(t=0)/\rho_{0}$
C1 | 500 | 1 | 3
C2 | 500 | 2 | 3
C3 | 500 | 0.5 | 3
C4 | 1000 | 1 | 3
C5 | 500 | 1 | 5
C6 | 500 | 1 | 0
Table 1: Model parameters used in the different simulation runs. The ejection
velocity $\mathbf{v}_{\mathrm{ej}}$ is the initial velocity imparted to the
CME along $90^{\circ}$, the $\rho/\rho_{0}$ parameter is the density imparted
to the CME as a multiple of the low-coronal boundary density, and
$f_{\mathrm{inj}}$ denotes the frequency of the injected Alfvénic
perturbations. Figure 10: Formation of the CME sheath in the different
simulation runs. Shown are the evolution of the radial extent of the sheath
(a), the flux rope leading edge (b), and the location of the CME shock along a
viewing angle of $105^{\circ}$ (c) for the simulation runs detailed in Table
1. The individual runs at time $t=21$ min are visualised in (d)-(i); the
magnetic field lines are colour-coded according to the case number. In each
panel (d)-(i), the viewing angle, the flux rope, and shock locations are also
indicated.

To quantify the differences, for each model run the extent of the sheath,
location of the FR, and the shock location for a viewing angle of
$105^{\circ}$ (head-on encounter) are computed and presented in Figure
10(a)-(c) as a function of time. The computation of the shock’s location
relies on the density compression ratio, while the positioning of the FR is
determined by identifying the first closed magnetic field contour encountered
along the viewing angle directed towards the Sun. Subsequently, in Figures
10(d)-(i), we display the snapshot of the simulation for the various cases
(Table 1) at $t=21$ min from the event onset. These snapshots are overlaid
with markers displaying the viewing angle, shock location, and the FR leading
edge location.

Panels (b) and (c) in Figure 10 show that the lower the density of the FR, the
faster the FR and its leading shock propagate through the corona; this is seen
by comparing the high-density (C2; blue curve), nominal density (C1; black
curve), and low-density (C3; orange curve) cases. We note that for these three
cases the initial FR speed and the frequency of injected fluctuations were the
same ($500$ km s-1 and 3 mHz, respectively). From these cases, the low-density
FR (C3) that propagates fastest through the solar corona has the widest sheath
(panel a). This dependency of the propagation speed on density can be
understood by considering the deflection of case C3; as explained in Section
3, deflection is expected to result from magnetic reconnection at the FR
boundaries. The comparison between simulation snapshots in Figures 10(d)-(f)
show that the low-density case C3 deflects more than the higher-density cases
C1 and C2. This deflection causes our selected viewing angle to focus on the
flank of the CME for C3, while for C1 and C2 their higher inertia prevents
them from deflecting and they are probed head-on, as intended.

Next we explore the effect of FR injection velocity. The shock and the FR for
C4 (green curve) propagate faster through the corona than for C1, which has
the same density but slower speed (see Fig. 10, panels (b) and (c)). This
results in a much smaller sheath thickness for C4 than for C1 (Figure 10(a)).
Finally, when we increase the Alfvénic fluctuation frequency in C5, and
compare it to run C1, which otherwise has identical parameters, there is no
notable difference in the FR location (Figure 10(b)), but the shock propagates
faster (Figure 10(c)), causing the sheath extent to increase (Figure 10(a)).
Moreover, when not injecting any Alfvénic fluctuations, as is the case for run
C6, we still see no difference in the FR location compared to C1, but the
shock propagates more slowly and the sheath extent decreases.

The variations in the shock evolution (Figure 10(c)) for the different cases
can be understood through the steepening of the MHD waves. The initial
eruption of the CME onto a quasi-steady solar wind generates a fast wave
propagating ahead of the CME as it initially strongly expands in the
surrounding plasma. If we assume the wave driven by the FR to be a pressure
wave, then as this wave propagates it locally compresses the plasma and
increases the local sound speed. This would cause the next pressure wave pulse
generated by the outward-propagating FR to catch up to the preceding wave
modes, thus causing a shock to be generated by the steepening of large-
amplitude compressive disturbances. In the general case, the fast wave
generated by the FR eruption would be a MHD wave. The rate of steepening a
fast mode MHD wave, with no additional assumptions other than the
compressibility of the medium, was previously derived (e.g. Kantrowitz et al.
1966) to be given by

$\displaystyle\gamma_{s}=\omega\frac{\delta\rho}{\rho}\left[1+\frac{1}{2}\frac{(\gamma-1)v_{A}^{2}c_{s}^{2}\sin^{2}\theta+(v_{\mathrm{ph}}^{2}-c_{s}^{2})^{2}}{v_{A}^{2}c_{s}^{2}\sin^{2}\theta+(v_{\mathrm{ph}}^{2}+c_{s}^{2})^{2}}\right].$
(15)

Here $\omega$ is the wave frequency, $v_{A}$ the Alfvén speed, $c_{s}$ the
sound speed, $v_{ph}$ the phase speed of the wave, and $\theta$ the wave
normal angle relative to the magnetic field. The steepening rate depends
primarily on the compressibility of the medium ($\delta\rho/\rho$), where
$\rho$ corresponds to the undisturbed solar wind density, with minor
contributions from the term in brackets (Kennel et al. 1985; Tsurutani et al.
1987). Among the CME runs described in Table 1, the high-density FR (case C2)
corresponds to an increased $\delta\rho/\rho$, while for the low-density FR
(case C3) $\delta\rho/\rho$ is smaller than for C1. As a consequence, for C2
the wave steepens to a shock more quickly (at a lower starting height) than
for C1, while for C3 the shock forms later. The shock locations in Figure
10(c) grow linearly with time, which indicates that after the fast wave
steepens the shock propagates with a constant velocity in this region.
Therefore, for the case of C2 the fast wave steepens to a shock fastest from
the investigated cases and the shock thus forms at the lowest heights in the
corona (Figure 10(c)). For C3 in turn, the wave decelerates more slowly and
the shock forms at a greater height.

For run C4 the higher injection velocity does not have a direct influence on
the steepening rate (Equation 15), and the shock starts at approximately the
same height as for C1 in Figure 10(c). However, because the FR in C4
propagates much faster through the corona than for C1 (due to the higher
injection velocity), it drives the shock faster at the CME nose, and so the
difference in the shock location between C4 and C1 increases as the simulation
progresses. Finally, the cases C5 and C6 show that the frequency of the
upstream Alfvénic perturbations seem to affect the speed of the CME shock. We
note that this dependence of the shock speed on the Alfvén wave frequency is
independent of the grid resolution (Appendix A). In Section 2.1, we showed
that Alfvénic waves in the solar wind in our simulation could not steepen to
form shocks themselves. Only the interaction of the Alfvén waves with the
shock, and the initially propagating fast MHD wave prior to shock formation
may alter the shock speed. It should be noted that this effect of the Alfvén
wave frequency affecting the shock formation is shown for a quasi-parallel
shock in this simulation. In the case of a perpendicular shock, previous
studies (Lu et al. 2009) only indicated structural modifications at the shock
front without an influence on the propagation speed.

Thus, the propagation velocity of the CME shock depends initially on the
effect of the wave steepening followed by the FR driving it further. The FR
itself propagates based on the injection velocity and momentum contributing to
the resulting force imbalance at onset. This variation of how different
parameters affect the shock and FR location separately causes differences in
the CME sheath extent.

##  5 Conclusion

This study presents the interaction between small amplitude Alfvénic
fluctuations and a CME in the low corona using 2.5D time-dependent MHD
simulations. The fluctuations in the quasi-steady solar wind are linearly
polarised and monochromatic in frequency. They are injected using time-
dependent boundary conditions in the low corona.

In Section 2.1 we described the linear evolution of the injected Alfvén waves
without decaying into compressive and reflected wave modes. In this scenario
we found that the CME sheath would consist of short-wavelength waves that are
compressed by the shock and long-wavelength waves transmitted in the initial
quasi-perpendicular phase of the CME expansion, which were modified in
wavelength by the CME shock passage. The Alfvén waves downstream of the CME
shock were inhibited close to the FR due to non-radial flows. While this
result was obtained for a 2D simulation, we can extrapolate this argument into
a higher dimension. In a 3D case we would observe non-radial flows in $\phi$
at the CME flanks as well (in the same manner as we do for $\theta$). Thus, we
might expect the CME sheath fluctuations to consist of short-wavelength
components based on the non-radial flows present in each direction. Due to the
importance of the CME sheath structure in influencing the fluctuations present
in this region, we investigated the formation of the sheath in Section 4. We
found the CME-driven shock to be formed due to wave steepening, with an
additional constraint on the frequency of the fluctuations present in the
system. At the same time the FR evolution is unaffected by the frequency of
the fluctuations. In the discussion presented in this study, we do not address
the Alfvén waves generated by the magnetic recconection (Cranmer 2018; Lynch
et al. 2014) inside the CME sheath. We cannot capture these additional waves
in this simulation as they require a much finer simulation grid. The
properties of such reconnection-driven Alfvén waves depend on the rate of
reconnection, plasma $\beta$, and magnetic field strength (Kigure et al.
2010). A complete discussion of the impact of these waves on the observed
properties of fluctuations inside CME sheaths would require further study. The
results presented in this study are thus in the context of the shock
transmission of already-present solar wind fluctuations. Therefore, the
applicability of these results is valid close to the CME shock when compared
with spacecraft observations.

A primary result of this study is the transmission of the upstream Alfvén
waves based on the upstream solar wind conditions in the frame of reference of
the CME shock (Section 3.1.1). This transmission process naturally generates
sunward-propagating Alfvén waves, with the compression of the upstream anti-
sunward propagating waves varying in the latitudinal direction ($\theta$) due
to the varying shock speeds. This indicates that Alfvénic fluctuations have
only anti-sunward components upstream, and both sunward and anti-sunward
components downstream due to their interaction with the CME shock. This
behaviour has been observed across CME shocks (Good et al. 2020).
Additionally, the properties of the downstream Alfvénic fluctuations depend on
their relative distance to the CME shock; locations closer to the shock
contain more compressed upstream waves. This might indicate varying spectral
slopes in the near-shock, mid-sheath, and near-FR regions of the CME sheath
(Kilpua et al. 2020). In Section 4 we observed that the Alfvén wave frequency
affects the shock velocity. This requires further study, as previous studies
investigating this interaction for perpendicular shocks found no appreciable
differences in shock speeds. Thus, the result presented in this study might be
a feature of the quasi-parallel CME shock.

###### Acknowledgements.

The work has been supported by the Finnish Centre of Excellence in Research on
Sustainable Space (FORESAIL). This is a project under the Academy of Finland,
and this research has been supported by the European Research Council (SolMAG;
grant no. 724391) as well as Academy of Finland project SWATCH (343581). The
authors are also grateful to the anonymous referee for their constructive
input during the peer review process.

## References

  * Asvestari et al. (2021) Asvestari, E., Pomoell, J., Kilpua, E., et al. 2021, Astronomy & Astrophysics, 652, A27
  * Asvestari et al. (2022) Asvestari, E., Rindlisbacher, T., Pomoell, J., & Kilpua, E. K. 2022, The Astrophysical Journal, 926, 87
  * Bale et al. (2005) Bale, S., Kellogg, P., Mozer, F., Horbury, T., & Reme, H. 2005, Physical Review Letters, 94, 215002
  * Belcher & Davis Jr (1971) Belcher, J. & Davis Jr, L. 1971, Journal of Geophysical Research, 76, 3534
  * Belcher et al. (1969) Belcher, J., Davis Jr, L., & Smith, E. 1969, Journal of Geophysical Research, 74, 2302
  * Chen (2016) Chen, C. 2016, Journal of Plasma Physics, 82
  * Cohen & Kulsrud (1974) Cohen, R. H. & Kulsrud, R. M. 1974, The Physics of Fluids, 17, 2215
  * Coleman Jr (1968) Coleman Jr, P. J. 1968, The Astrophysical Journal, 153, 371
  * Cranmer & Van Ballegooijen (2005) Cranmer, S. & Van Ballegooijen, A. 2005, The Astrophysical Journal Supplement Series, 156, 265
  * Cranmer (2018) Cranmer, S. R. 2018, The Astrophysical Journal, 862, 6
  * D’Amicis & Bruno (2015) D’Amicis, R. & Bruno, R. 2015, ApJ, 805, 84
  * Das et al. (2011) Das, I., Opher, M., Evans, R., Loesch, C., & Gombosi, T. I. 2011, The Astrophysical Journal, 729, 112
  * Gershman et al. (2019) Gershman, D. J., Connerney, J. E., Kotsiaros, S., et al. 2019, Geophysical Research Letters, 46, 7157
  * Gibson & Low (2000) Gibson, S. & Low, B. 2000, Journal of Geophysical Research: Space Physics, 105, 18187
  * Gibson & Low (1998) Gibson, S. E. & Low, B. 1998, The Astrophysical Journal, 493, 460
  * Goldstein (1978) Goldstein, M. L. 1978, The Astrophysical Journal, 219, 700
  * González et al. (2021) González, C., Tenerani, A., Matteini, L., Hellinger, P., & Velli, M. 2021, The Astrophysical Journal Letters, 914, L36
  * Good et al. (2020) Good, S., Kilpua, E., Ala-Lahti, M., et al. 2020, The Astrophysical Journal Letters, 900, L32
  * Good et al. (2022) Good, S. W., Hatakka, L. M., Ala-Lahti, M., et al. 2022, MNRAS, 514, 2425
  * Hollweg (1971) Hollweg, J. V. 1971, Journal of Geophysical Research, 76, 5155
  * Isavnin (2016) Isavnin, A. 2016, The Astrophysical Journal, 833, 267
  * Kalliokoski et al. (2022) Kalliokoski, M. M., Henderson, M. G., Morley, S. K., et al. 2022, Authorea Preprints
  * Kalliokoski et al. (2020) Kalliokoski, M. M., Kilpua, E. K., Osmane, A., et al. 2020in , Copernicus GmbH, 683–701
  * Kantrowitz et al. (1966) Kantrowitz, A., Petschek, H., & Kunkel, W. 1966, McGraw-Hill Book Company, New York, 147
  * Kato et al. (2016) Kato, Y., Steiner, O., Hansteen, V., et al. 2016, The Astrophysical Journal, 827, 7
  * Kennel et al. (1985) Kennel, C. F., Edmiston, J. P., & Hada, T. 1985, Washington DC American Geophysical Union Geophysical Monograph Series, 34, 1
  * Kigure et al. (2010) Kigure, H., Takahashi, K., Shibata, K., Yokoyama, T., & Nozawa, S. 2010, Publications of the Astronomical Society of Japan, 62, 993
  * Kilpua et al. (2017a) Kilpua, E., Balogh, A., Von Steiger, R., & Liu, Y. 2017a, Space Science Reviews, 212, 1271
  * Kilpua et al. (2019) Kilpua, E., Fontaine, D., Moissard, C., et al. 2019, Space Weather, 17, 1257
  * Kilpua et al. (2021) Kilpua, E., Good, S., Ala-Lahti, M., et al. 2021, Frontiers in Astronomy and Space Sciences, 7, 610278
  * Kilpua et al. (2013) Kilpua, E., Hietala, H., Koskinen, H., Fontaine, D., & Turc, L. 2013in , Copernicus Publications Göttingen, Germany, 1559–1567
  * Kilpua et al. (2015) Kilpua, E., Hietala, H., Turner, D., et al. 2015, Geophysical Research Letters, 42, 3076
  * Kilpua et al. (2017b) Kilpua, E., Koskinen, H. E., & Pulkkinen, T. I. 2017b, Living Reviews in Solar Physics, 14, 1
  * Kilpua et al. (2020) Kilpua, E. K., Fontaine, D., Good, S. W., et al. 2020in , Copernicus GmbH, 999–1017
  * Kilpua et al. (2021) Kilpua, E. K. J., Good, S. W., Dresing, N., et al. 2021, A&A, 656, A8
  * Kissmann & Pomoell (2012) Kissmann, R. & Pomoell, J. 2012, SIAM Journal on Scientific Computing, 34, A763
  * Koskinen (2011) Koskinen, H. 2011, Physics of space storms: From the solar surface to the Earth (Springer Science & Business Media)
  * Lu et al. (2009) Lu, Q., Hu, Q., & Zank, G. P. 2009, The Astrophysical Journal, 706, 687
  * Lynch et al. (2014) Lynch, B., Edmondson, J., & Li, Y. 2014, Coronal Magnetometry, 421
  * Matthaeus et al. (1984) Matthaeus, W., Ambrosiano, J., & Goldstein, M. L. 1984, Physical review letters, 53, 1449
  * Mikić et al. (2018) Mikić, Z., Downs, C., Linker, J. A., et al. 2018, Nature Astronomy, 2, 913
  * Moissard et al. (2019) Moissard, C., Fontaine, D., & Savoini, P. 2019, Journal of Geophysical Research: Space Physics, 124, 8208
  * Nakariakov et al. (1997) Nakariakov, V., Roberts, B., & Murawski, K. 1997, Solar Physics, 175, 93
  * Perez & Chandran (2013) Perez, J. C. & Chandran, B. D. 2013, The Astrophysical Journal, 776, 124
  * Pomoell et al. (2015) Pomoell, J., Aran, A., Jacobs, C., et al. 2015, Journal of Space Weather and Space Climate, 5, A12
  * Pomoell et al. (2015) Pomoell, J., Aran, A., Jacobs, C., et al. 2015, Journal of Space Weather and Space Climate, 5, A12
  * Pomoell & Vainio (2012) Pomoell, J. & Vainio, R. 2012, The Astrophysical Journal, 745, 151
  * Ruohotie et al. (2022) Ruohotie, J., Kilpua, E. K. J., Good, S. W., & Ala-Lahti, M. 2022, Frontiers in Astronomy and Space Sciences, 9, 943247
  * Russell & Mulligan (2002) Russell, C. & Mulligan, T. 2002, Planetary and Space Science, 50, 527, solar System Magnetosheaths-A Special Issue in Honor of John Spreiter
  * Schekochihin et al. (2009) Schekochihin, A., Cowley, S., Dorland, W., et al. 2009, The Astrophysical Journal Supplement Series, 182, 310
  * Siscoe & Odstrcil (2008) Siscoe, G. & Odstrcil, D. 2008, Journal of Geophysical Research: Space Physics, 113
  * Sishtla et al. (2022) Sishtla, C. P., Pomoell, J., Kilpua, E., et al. 2022, Astronomy & Astrophysics, 661, A58
  * Soljento et al. (2023) Soljento, J. E., Good, S. W., Osmane, A., & Kilpua, E. K. 2023, The Astrophysical Journal Letters, 946, L19
  * Solov’ev (1968) Solov’ev, L. 1968, Sov. Phys. JETP, 26, 400
  * Tomczyk et al. (2007) Tomczyk, S., McIntosh, S., Keil, S., et al. 2007, Science, 317, 1192
  * Tsurutani et al. (1987) Tsurutani, B. T., Thorne, R. M., Smith, E. J., Gosling, J., & Matsumoto, H. 1987, Journal of Geophysical Research: Space Physics, 92, 11074
  * Tu & Marsch (1995) Tu, C.-Y. & Marsch, E. 1995, Space Science Reviews, 73, 1
  * Vainio & Schlickeiser (1998) Vainio, R. & Schlickeiser, R. 1998, Astronomy and Astrophysics, 331, 793
  * Vainio & Schlickeiser (1999) Vainio, R. & Schlickeiser, R. 1999, Astronomy and Astrophysics, v. 343, p. 303-311 (1999), 343, 303
  * Van Ballegooijen et al. (2011) Van Ballegooijen, A., Asgari-Targhi, M., Cranmer, S., & DeLuca, E. 2011, The Astrophysical Journal, 736, 3
  * Verbeke et al. (2019) Verbeke, C., Pomoell, J., & Poedts, S. 2019, A&A, 627, A111
  * Verdini & Velli (2007) Verdini, A. & Velli, M. 2007, The Astrophysical Journal, 662, 669
  * Zank & Matthaeus (1992) Zank, G. & Matthaeus, W. 1992, Journal of plasma physics, 48, 85

##  Appendix A Shock compressed Alfvén waves

In this Appendix we present a modified simulation of the nominal run (case C1
in Table 1) discussed in Section 3 where we improve the resolution between
$1.1\leavevmode\nobreak\ R_{\odot}$ and $10\leavevmode\nobreak\ R_{\odot}$ to
capture the compression of upstream Alfvén waves due to the shock that are not
resolved in Section 3.1.1. Here we detail the methodology for introducing the
modified resolution, and report the expected propagation of Alfvén waves based
on the discussion presented in Section 3. Therefore, the results presented in
this Appendix validate the approach presented in this manuscript where we
utilise a reduced simulation resolution, while still obtaining the proper
physics.

###  A.1 Modifying the grid resolution

As described in Section 2, the simulation is performed in three steps:

  1. 1.

generating a steady-state solar wind solution without fluctuations;

  2. 2.

generating a quasi-steady state wind by injecting Alfvén waves;

  3. 3.

introducing the CME.

Figure 11: Shock at the CME flank for the modified resolution simulation run.
This figure is similar to Figure 8, but with a modified grid (higher
resolution) between $1.1\leavevmode\nobreak\ R_{\odot}$ and
$10\leavevmode\nobreak\ R_{\odot}$. Panel (a) shows a simulation snapshot at
$t=25\leavevmode\nobreak\ $min of the anti-sunward Elsässer variable
$\mathbf{z_{\phi}^{-}}$ with annotations describing the viewing angle along
$160^{\circ}$, the shock location, the approximate beginning of the CME
sheath, and the approximate shock velocity $v_{\mathrm{shock}}$. Panels (b)
and (c) show the density compression and $\mathbf{z_{\phi}^{-}}$ along the
viewing angle, respectively. Panels (d) and (e) present the fluid velocity and
Alfvén speed along the direction of the background magnetic field.

A reduced resolution with $500$ grid cells logarithmically spaced in the
radial direction results in numerical damping of the Alfvén waves in the
corona, and inhibits the generation of density fluctuations and alternate
Alfvén wave modes (Section 2.1). Thus, after obtaining a quasi-steady wind
with a reduced resolution, we modify the simulation grid to have $4000$
equally spaced cells between $1.1\leavevmode\nobreak\ R_{\odot}$ and
$10\leavevmode\nobreak\ R_{\odot}$ solar radii, while preserving the original
grid below $1.1\leavevmode\nobreak\ R_{\odot}$ and beyond
$10\leavevmode\nobreak\ R_{\odot}$. This higher resolution in the region of
interest where the Alfvén waves are present in the simulation enables us to
capture the compression at the CME shock. The choice of $4000$ cells was made
after determining that the solar wind behaviour is unchanged for $3000$ and
$3500$ grid cells as well.

Figure 11 illustrates the effect of the modified grid on the Alfvén wave
transmission by presenting a figure similar to Figure 8. In panels (a) and (c)
we observe that Alfvén waves, seen as crests and troughs of the $z^{-}_{\phi}$
Elsässer variable, are of a higher amplitude than in Figure 8, due to a lower
level of numerical damping. However, the CME shock and the start of the longer
wavelength CME sheath region is seen to occur at the same location as in the
lower-resolution simulation, as annotated in the figure. The primary
difference between Figure 11 and Figure 8 is the shock compressed region in
panel (c) between the vertical annotations. In this region we observe the
upstream waves compressed in frequency (as described in Section 3.1.1), with
the modified grid capturing one wavelength of this compressed Alfvén wave.
However, after this compressed region the Alfvén waves observed further
downstream are also of longer wavelengths, as in the lower-resolution runs, as
the Alfvén speed increases again. To verify whether the grid resolution
captures the entire compressed wave, we plot the flow speed
($\mathbf{v\cdot}\hat{\mathbf{b}}$), and the Alfvén speed along the background
field ($\mathbf{v}_{a}\cdot\hat{\mathbf{b}}$) in Figure 11(d), (e). Assuming
the shock normal to be exactly in the direction of the upstream magnetic field
(in the radial direction), and by estimating the shock speed between
$t=6.25\leavevmode\nobreak\ $min and $t=31.25\leavevmode\nobreak\ $min as
$\approx 2087\leavevmode\nobreak\ \mathrm{km\leavevmode\nobreak\ s^{-1}}$, we
obtain the Alfvén Mach number in the shock frame to be $\approx 2.45$ and
$\approx 1.78$ in the upstream and downstream, respectively. Additionally,
taking the gas compression ratio to be $\approx 2$ (panel (d)), we can
estimate the downstream wavelength (Equation 13) to be three times smaller
than the upstream wavelength, which can be resolved by the higher-resolution
grid. It should be noted that the average shock velocity with the modified
grid ($v_{\mathrm{shock}}\approx 2087\leavevmode\nobreak\
\mathrm{km\leavevmode\nobreak\ s^{-1}}$) is marginally higher than for the
original grid ($v_{\mathrm{shock}}\approx 2078\leavevmode\nobreak\
\mathrm{km\leavevmode\nobreak\ s^{-1}}$), possibly due to the modified grid
better resolving the interactions of the wave with the shock.

Finally, the effect of the Alfvén waves on the propagation of the CME is
discussed in Section 4. The FR propagates based on its force imbalance, and
does not present a significant dependence on the Alfvén wave frequency. In
contrast, the shock velocity increases with the wave frequency. To verify
whether such a result is dependant on the grid resolution, we performed a
similar analysis as in Section 4 to check the nominal case C1 against the
simulation with the modified resolution as in the Appendix. The results showed
that the Alfvén wave has no further effect on the CME shock velocity with
increased resolution.

###  A.2 Alfvén wave transmission across the shock

The transmission of the upstream Alfvén wave occurs based on the gas
compression ratio across the shock and the upstream Alfvén Mach number in the
shock frame. To illustrate the compressed wave further, Figures 12 and 13 are
similar to Figure 7, but use the modified high-resolution grid, and are
presented for viewing angles of $160^{\circ}$ and $105^{\circ}$.

Figure 12: Evolution of the Elsässer variables at the CME flank for the
modified simulation grid. Similar to Figure 7, but using a modified grid
(higher resolution) between $1.1\leavevmode\nobreak\ R_{\odot}$ and
$10\leavevmode\nobreak\ R_{\odot}$. The x-axis similarly denotes the shock
neighbourhood in units of $R_{\odot}$; the shock is centred at $x=0$.

In Figure 12 (comparable to Figure 7) we see the upstream waves in panel (a)
incident to the shock with the far downstream region ($x<0$) showing the long-
wavelength Alfvén waves. The downstream region also shows an enhancement of
the Alfvén wave amplitudes beyond $\mathrm{t}=20\leavevmode\nobreak\
\mathrm{min}$. We now also observe the compressed waves near the shock, which
are amplified, but quickly dissipate as the shock propagates further. This
dissipation and subsequent enhancement of the downstream waves seems to
indicate steepening of the compressed waves. The Elsässer variables in
$\theta$ remain as presented in Section 3.1.2.

Figure 13: Evolution of the Elsässer variables at the CME nose for the
modified simulation grid, similar to Figure 12, but with a viewing angle of
$105^{\circ}$. The x-axis similarly denotes the shock neighbourhood in units
of $R_{\odot}$; the shock is centred at $x=0$.

For the viewing angle of $105^{\circ}$ (CME head-on) in Figure 13 we see a
similar compression of the upstream waves and subsequent dissipation as the
shock progresses further. The far downstream region shows the expected large
flow due to the presence of the FR magnetic field. Panels (b), (c), and (d)
show behaviour similar to that in Figure 9.

Therefore, the upstream Alfvén waves are compressed by the CME shock and
further interact non-linearly with each other to dissipate and steepen the
downstream waves. The high-resolution grid captures the expected behaviour of
the waves, as described in Section 3.1.1, with an additional effect based on
the wave dissipation as the CME propagates beyond $10\leavevmode\nobreak\
\mathrm{R_{\odot}}$. The resolution does not appear to have any effect on the
CME dynamics.