# Flux erosion of magnetic clouds by reconnection with the Sun’s open flux

Gindraft=false Geophysical Research Letters

Center of Excellence in Space Sciences India, Indian Institute of Science
Education and Research Kolkata, Mohanpur 741246, West Bengal, India Department
of Physical Sciences, Indian Institute of Science Education and Research
Kolkata, Mohanpur 741246, West Bengal, India

Dibyendu<EMAIL_ADDRESS>
Flux of magnetic clouds is eroded due to magnetic reconnection with the solar
open flux, impacting their geoeffectiveness in some cases.

A linear relationship is observed between eroded and open flux with larger
average flux erosion during the stronger cycle 23 compared to 24.

We establish a solar cycle link to the evolution of magnetic cloud flux during
their interplanetary propagation.

## Abstract

Magnetic clouds (MCs) are flux-rope magnetic structures forming a subset of
solar coronal mass ejections which have significant space weather impacts. The
geoeffectiveness of MCs depends on their properties which evolve during their
interplanetary passage. Based on an analysis of observations spanning two
solar cycles we establish that MCs interacting with the ambient solar wind
magnetic field (i.e., heliospheric open flux) lose a substantial amount of
their initial magnetic flux via magnetic reconnection, which in some cases,
reduce their geoeffectiveness. We find a linear correlation between the eroded
flux of MCs and solar open flux which is consistent with the scenario that MC
erosion is mediated via the local heliospheric magnetic field draping around
an MC during its interplanetary propagation. The solar open flux is governed
by the sunspot cycle. This work therefore uncovers a hitherto unknown pathway
for solar cycle modulation of the properties of MCs.

## Plain Language Summary

Vast eruptions of magnetized plasma from the Sun known as coronal mass
ejections (CMEs) are the main contributors to severe space weather conditions.
The magnetic properties of CMEs – which determine their geoeffectiveness –
evolve during their interplanetary journey. Therefore it is important to
understand the physical processes that govern their evolution. Based on in
situ satellite observations, and modeling here we provide evidence that the
magnetic flux of CME associated magnetic clouds erode over their
interplanetary journey due to reconnection with the ambient heliospheric open
flux. We also demonstrate that this flux erosion impacts the geoeffectiveness
of magnetic clouds in some cases. This mechanism establishes a novel
connection between the solar cycle variation of the heliospheric open flux and
space weather in the vicinity of solar system planets.

##  1 Introduction

Coronal mass ejections (CMEs) are magnetised plasma structures expelled
sporadically from the Sun and travel toward the heliosphere as interplanetary
coronal mass ejections (ICMEs). If an ICME has a smoothly rotating strong
magnetic field and a low proton temperature [Burlaga . (1981), Klein Burlaga
(1982)] it is categorised as a magnetic cloud (MC). The twisted flux ropes of
MCs [Goldstein (1983), Burlaga (1988), Lepping . (1990)] carry a significant
amount of solar magnetic flux and helicity [Pal . (2017)]. While propagating
through the interplanetary medium, the presence of a relative motion between
an MC and the ambient plasma would result in a draping of the ambient
interplanetary magnetic field (IMF) over the MC [Gosling McComas (1987)]. Such
draping has been directly observed (e.g., crooker1985magnetic) as well as
modeled (e.g., alksne1970magnetic). If the magnetic field orientation of the
MC is opposite to that of the draped IMF, the process of magnetic reconnection
(henceforth reconnection) is initiated [McComas . (1988)]. Due to this
reconnection a part of the MC’s magnetic flux is eroded away and an asymmetry
is generated in its azimuthal magnetic flux that accumulates along the
spacecraft trajectory. This fact has been established by observations [Dasso .
(2006), Dasso . (2007), Möstl . (2008), Ruffenach . (2012), Ruffenach .
(2015)] as well as by using global MHD simulations [Schmidt Cargill (2003),
Taubenschuss . (2010)]. 2014JGRA..119…26L showed that erosion of an MC’s
magnetic flux may reduce its geoeffectiveness by a significant amount. Figure
1 shows an idealized schematic diagram of the draping of a purely radial
ambient IMF about a fast moving MC and the MC’s cross-section in a plane
perpendicular to the ecliptic plane. The imbalance in the MC’s accumulated
azimuthal flux arises due to the reconnection at the MC’s front.

An approximate understanding of the erosion of the magnetic flux of MCs due to
reconnection with the IMF can be obtained with recourse to the Sweet-Parker
model [Parker (1957), Sweet (1958)]. In the diffusion region (a small-scale
region surrounding the reconnection site) 2007PhPl…14j2114C showed that the
reconnection rate scales as $E_{rec}\sim V_{1}B_{1}\sim V_{2}B_{2}$, where $V$
and $B$ are the plasma flow speed and magnetic field intensity, respectively,
and the subscripts ‘1’ and ‘2’ refer to either side of the inflow region. For
steady state reconnection $E_{rec}$ is uniform at the reconnection region,
where $E_{rec}$ defines the rate of magnetic flux that is transferred from the
inflow to the dissipation region [Nakamura . (2018)]. This transferred
magnetic flux determines the amount of flux erosion ($F_{er}$) and one expects
that $F_{er}$ would scale linearly with the background IMF field strength
[Cassak Shay (2007)].

The local Alfvén speed ($V_{A}$) in the plasma near the reconnection site
governs the reconnection rate [Parker (1973), Birn . (2001), Cassak Shay
(2007)]. As $V_{A}$ decreases with the increasing heliospheric distance, the
reconnection rate is expected to be higher closer to the Sun and in the inner
heliosphere (e.g. 2008JGRA..113.0B08L). 2007SoPh..244..115D found an extended
coherent magnetic region just behind an MC and interpreted that this region is
formed due to the reconnection process which started $\approx 54$ hrs before
the MC was observed. 2014JGRA..119…26L suggested that about 47% to 67% of the
MC erosion is expected to occur within the orbit of Mercury ($\approx 0.39$ AU
from the Sun). 2015JGRA..120…43R found no correlation between the eroded
amount of magnetic flux and the ambient solar wind properties observed in situ
at 1 AU. The lack of correlation suggests that most of the reconnection is
expected to occur during the MC’s passage through the inner heliosphere and
not at 1 AU.

Large scale open magnetic field originating from the solar corona extends out
into the heliosphere and shapes the IMF. Utilizing the observed solar
photospheric magnetic field information as boundary condition the heliospheric
magnetic field can be reconstructed following the simplest assumption that the
solar corona is current-free [Schatten . (1969)]. This approach is known as
Potential Field Source Surface extrapolation (PFSS). In the outer radial
boundary of the PFSS model the field lines are constrained to be radial which
is consistent with the observations of the coronal hole foot-points [Wang .
(1996)]. The open magnetic flux ($\phi_{open}$) derived using the Sun’s open
magnetic field lines can be used to estimate the radial IMF intensity at any
heliocentric distance from the Sun as the $Ulysses$ spacecraft measurements
show a latitudinal and longitudinal uniformity of the IMF’s radial component
[EJ. Smith Balogh (2008), Balogh . (1995), Erdős Balogh (2014)]. Therefore,
the radial magnetic field intensity ($|B_{r}|$) can be expressed as
$|B_{r}|=|\phi_{open}|/4\pi r^{2}$ at a heliocentric distance $r$
($r>>R_{\odot}$). If $\theta_{HG}$ and $\phi_{HG}$ are the heliographic
latitude and longitude, respectively, then the open magnetic flux can be
derived by integrating $|B_{r}(R_{ss},\theta_{HG},\phi_{HG})|$ over the
surface of a sphere concentric to the Sun with radius $R_{ss}$, known as
source surface. Using $\phi_{open}$, several studies have estimated $B_{r}$ at
$r=1$ AU and compared it with the in situ near-Earth radial IMF intensity. The
studies found a good correspondence between the estimated and observed values
[Wang Sheeley (1988), Wang Sheeley (1995), Wang . (2000), AR. Yeates . (2010),
Wang Sheeley Jr (2015)].

The present study aims to establish a connection between the MC’s eroded
magnetic flux and the IMF intensity. For achieving this we explore the
relationship between annual averages of MC’s eroded flux and the $\phi_{open}$
over two solar cycles. Also, we present a statistical analysis to investigate
the geoeffectiveness of eroded MCs. We describe the data and their sources in
section 2 and present our analysis methodology in section 3. Results are
presented in section 4 followed by discussion and conclusions in section 5 and
section 6, respectively.

##  2 Instrumentation and Overview of Data

We use 64s average level 2 (verified) data of merged interplanetary magnetic
field and solar wind parameters
(http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA_MAG-SWEPAM.html) from the
Advanced Composition Explorer (ACE) [Stone . (1998)] spacecraft’s Solar wind
Electron, Proton and Alpha Monitor (SWEPAM) [McComas . (1998)] and Magnetic
Field Experiment (MAG) [CW. Smith . (1998)] instruments. The selected data is
in Geocentric Solar Ecliptic (GSE) coordinate system and available since the
year 1998. In GSE coordinate system $\hat{x}_{GSE}$ points toward the Sun,
$\hat{z}_{GSE}$ points towards the north pole (of the ecliptic) and
$\hat{y}_{GSE}$ completes the right-handed system. To obtain the geomagnetic
response of MC we use the geomagnetic activity index ($Dst$) available at the
World Data Center, Kyoto (http://wdc.kugi.kyoto-u.ac.jp/dstdir/index.html) in
a temporal resolution of 1 hour.

We obtain the radial magnetic field data from synoptic magnetograms
(http://hmi.stanford.edu/data/synoptic.html) collected by the Michelson
Doppler Imager (MDI) [Scherrer . (1995)] onboard the Solar and Heliospheric
Observatory (SOHO) [Domingo . (1995)] and the Helioseismic and Magnetic Imager
(HMI) [Scherrer . (2012)] onboard the Solar Dynamics Observatory (SDO)
[Pesnell . (2012)]. The data from MDI and HMI is collected for a period of
1998-April 2010 and May 2010-2018, respectively. We utilize the radial
magnetic field data to extrapolate the coronal magnetic field till the source
surface at $r=R_{ss}=2.5R_{\odot}$ using a global PFSS extrapolation model.

###  2.1 MC events and criteria for selection

In general, MCs are identified following the criteria defined by
1981JGR….86.8893B which are (1) throughout an MC interval the magnetic field
intensity enhances (compared to the ambient solar wind), (2) smooth rotation
in both $B_{y,GSE}$ and $B_{z,GSE}$ components exists, (3) the proton
temperature falls below the expected solar wind temperature [Lopez Freeman
(1986)], and (4) the plasma $\beta$ parameter is less than 1. For solar cycle
23 (1998-2007) we consider 37 MCs from a list of 109 MCs selected by
2015JGRA..120…43R. The study categorized the MCs as Q1 (both the front and
rear boundaries of MCs are well determined), Q2 (any one of the two boundaries
is identified without ambiguity) and Q3 (both the boundaries are difficult to
identify). Of them, we select only Q1 and Q2 type MCs for our study as the
fitting of flux ropes to the MCs is impacted by incorrect boundary selection.
The events during 2008-2014 are obtained from the MC list published by
2015JGRA..120..9221G and the events of 2015-2018 are selected from the
Richardson2010 ICME catalogue
(http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm) after manually
identifying each of their boundaries. Thus, we select a total number of 35
events during solar cycle 24. Note that, each of the events selected here is
consistent with 1981JGR….86.8893B’s definition of MCs. For none of the events,
the spacecraft crosses the MC flux ropes parallel to the legs. This is because
those cases lead an azimuthal flux asymmetry which may be misinterpreted as a
result of erosion [Ruffenach . (2015)]. Also, there exist uncertainties in
fitting parameters [e.g. 2007AnGeo..25.2453M,2003JGRA..108.1356L] while those
MCs are fit with the cylindrical flux rope model. 2013A&A…556A..50J introduced
a location angle ($\lambda$) measured from the plane
($\hat{y}_{GSE}$,$\hat{z}_{GSE}$) towards the MC axis. It is a proxy for the
spacecraft crossing distance from the MC nose. The $\lambda$ evolves
monotonically along the MC flux rope with $\lambda=-90^{\circ}$ in one leg,
$\lambda=0^{\circ}$ at its nose, and $\lambda=90^{\circ}$ in the other leg.
2015JGRA..120…43R estimated that errors in the azimuthal flux imbalance is
less than 10% if $\lambda$ holds a value between $\pm 45^{\circ}$. Thus we
study only those MCs which cross the spacecraft trajectory sufficiently close
to their apex by selecting $\lambda<|45^{\circ}|$.

##  3 Analysis methodology

Several case studies [Dasso . (2006), Dasso . (2007)] show the presence of
unbalanced azimuthal magnetic flux in MCs although MCs maintain a classical
characteristic. Since the outer part of an MC is mainly affected by
reconnection and the azimuthal flux is much greater than the axial flux in
there [Ruffenach . (2015)], the asymmetry is mainly observed in the MC’s
azimuthal magnetic flux. To estimate this asymmetry we employ the “direct
method” developed by 2006A&A…455..349D.

###  3.1 Measurement of MC’s eroded magnetic flux

The direct method uses the principle of magnetic flux conservation
($\nabla\cdot\mathbf{B}=0$) across the plane formed by the spacecraft
trajectory and MC axis to determine and analyse the MC’s accumulated azimuthal
magnetic flux. The MC axis orientation is defined by a latitude angle $\theta$
(the angle between MC’s axis and the ecliptic plane) varying from
$-90^{\circ}$ to $90^{\circ}$ and a longitude angle $\phi$ (the angle between
the projection of MC’s axis on the ecliptic plane and the Earth-Sun direction)
varying from $0^{\circ}$ to $360^{\circ}$. Using $\theta$ and $\phi$, we
rotate the observed MC from GSE to the local MC coordinate defined by
2006A&A…455..349D and derive $\mathbf{B}_{cloud}$ and $\mathbf{V}_{cloud}$
from $\mathbf{B}_{GSE}$ and $\mathbf{V}_{GSE}$. To obtain $\theta$ and $\phi$
we use a constant-$\alpha$ force-free flux rope model [Burlaga (1988),
Marubashi (1986)] to least-square fit the observed MCs. The model follows the
equation $\nabla\times\mathbf{B}=\alpha\mathbf{B}$, where $\alpha$ is a
constant and allows self-similar expansion of an MC. The Lundquist’s solution
to the equation [Lepping . (1990)] provides the modeled magnetic field vectors
in the cylindrical coordinate system. After performing a series of iterations
through proper adjustment of model parameters based on minimizing the
difference between modeled and observed magnetic field vectors, a final set of
best-fit parameters including $\theta$ and $\phi$ are ascertained (e.g,
2007AnGeo..25.2453M, 2012JGRA..117.1101M).

If $\theta$, $\phi$ and the time of MC front ($t_{front}$) and rear boundaries
($t_{rear}$) are known, the accumulated azimuthal magnetic flux per unit
length ($\frac{\phi_{y}(t_{1},t_{2})}{L}$) can be measured using

$\frac{\phi_{y}(t_{1},t_{2})}{L}=\int_{t_{1}}^{t_{2}}B_{y,cloud}(t)V_{x,cloud}dt,$
(1)

where $L$ is the total length of the MC axis till 1 AU, $B_{y,cloud}$ and
$V_{x,cloud}$ are the MC’s magnetic field and velocity components computed in
the local cloud coordinate system, respectively. The times $t_{1}$ and $t_{2}$
represent generic times defining the boundaries of the integral. The
normalized perpendicular distance between spacecraft trajectory and MC axis is
called the impact parameter ($Y_{0}$). The accumulated azimuthal flux is more
precisely estimated when $Y_{0}$ is low. In Figure 2 we plot the time
evolution of $B_{y,cloud}$ (blue curve) along with $\phi_{y}/L$ (red curve) of
two MCs during their passage through the spacecraft. Figure 2a shows an MC
that is eroded at the front edge. The parameter $\phi_{y}/L$ is calculated
considering $t_{1}=t_{front}$ and $t_{2}=t_{rear}$. Due to erosion its
accumulated azimuthal flux shows an excess at its rear. The total azimuthal
flux per unit length ($\phi_{az}$) that was originally present in that MC
(before erosion) is estimated by the unsigned value of
$\phi_{y}(t_{center},t_{rear})/L$. Here $t_{center}$ (shown in dashed-dotted
black vertical lines in Figures 2a and 2b) is the time when $B_{y,cloud}$
changes its sign and results in a maximum value of unsigned $\phi_{y}/L$. The
time $t_{center}$ is the time when closest approach of the spacecraft to the
MC axis is achieved. The eroded azimuthal flux (per unit length) of the MC is
represented by the absolute value of $\phi_{y}/L$ at $t_{rear}$. The second MC
(Figure 2b) is eroded at its rear and shows a flux imbalance at its front.
Here $\phi_{y}/L$ is calculated considering $t_{1}=t_{rear}$ and
$t_{2}=t_{front}$. For this case $\phi_{az}$ is estimated by the unsigned
value of $\phi_{y}(t_{center},t_{front})/L$. The eroded flux of the MC is
defined by the absolute value of $\phi_{y}/L$ at $t_{front}$. We next define
$\phi_{erod}$ as the eroded flux normalized to $\phi_{az}$. The dashed-dotted
green vertical lines show the estimated rear (Figure 2a) and front (Figure 2b)
boundaries of the remaining flux-rope when it is observed in situ.

###  3.2 Estimation of the Sun’s open flux

An estimate of the Sun’s open flux is traditionally made based on potential
field extrapolation of the Sun’s coronal field (satisfying
$\nabla\times\mathbf{B}=0$, Schatten1969). Following van_Ballegooijen_2000 and
using a module developed by anthony_yeates_2018_1472183 we extrapolate the
coronal magnetic field utilizing solar surface synoptic maps from MDI and HMI
as the lower boundary condition. The unsigned open flux ($|\phi_{open}|$) is
estimated at $R_{ss}=2.5R_{\odot}$. We obtain a strong positive correlation
(with linear correlation coefficient, $r_{p}$= 0.89 at 99.99% confidence
level) between the yearly averaged IMF intensity observed in situ at 1 AU and
$|\phi_{open}|$. Therefore, $|\phi_{open}|$ can be used as a proxy of the IMF
intensity.

##  4 Analysis and Results

By applying the direct method to all the selected MCs we derive $\phi_{az}$,
$\phi_{erod}$, and the position of the flux imbalance (front or back) for each
of the MCs. In Figure 3 (a) we demonstrate the effect of MC’s erosion on its
geoeffectiveness by plotting MC’s remaining azimuthal flux ($\phi_{az}-$
eroded flux) versus its geomagnetic response (quantified by $Dst_{m}$– the
minimum value of $Dst$ during an MC’s passage). The correlation between the
remaining flux and $Dst_{m}$ ($r_{p}=-0.7$ at 99.99% confidence) is slightly
lower than the correlation between total flux and $Dst_{m}$ ($r_{p}=-0.76$ at
99.99% confidence) if all events are considered. However, it is to be noted
that the impact of MC’s azimuthal flux erosion on its geomagnetic response is
expected only if $B_{z,GSE}$ has a negative value at the reconnection site and
the MC’s azimuthal field component corresponds to $B_{z,GSE}$ (i.e.,
$\theta,\lambda\sim 0^{\circ}$) near the Earth (i.e., at 1 AU). Our sample has
9 MCs (depicted by red dots in Figure 3 (a)) which have negative $B_{Z,GSE}$
at their reconnection sites and $|\theta|,|\lambda|\leq 30^{\circ}$ (i.e.,
close to the above conditions, while still providing reasonable sample size).
For these events the correlation between the remaining flux of MCs and
$Dst_{m}$ is higher ($r_{p}=-0.95$ at 99.99% confidence) compared to that
between $\phi_{az}$ and $Dst_{m}$ ($r_{p}=-0.88$ at 99.99% confidence). This
reveals the impact of flux erosion on the geoeffectiveness of MCs.

We notice that out of 72, 40 MCs show the flux imbalance at rear with an
average value of $\phi_{erod}$ $\sim 0.26$ and 32 MCs show flux imbalance at
front with an average $\phi_{erod}$ $\sim 0.23$. The average eroded flux
irrespective of the erosion position during solar cycle 23 and 24 are found to
be 0.276 and 0.22, respectively. Our result matches well with the value of
normalized average eroded flux found by 2015JGRA..120…43R for the events
during solar cycle 23. MCs showing flux imbalance at the front or rear
boundaries are named as $MC_{F}$ and $MC_{R}$, respectively. In Table 1, we
show the number of total MCs (column 2), $MC_{R}$ (column 3), $MC_{F}$ (column
4), annual averages of $\phi_{erod}$ (column 5), and unsigned $\phi_{open}$
(column 6) with their standard deviations corresponding to each year mentioned
in column 1.

###  4.1 Comparison between the Sun’s unsigned open flux and MC flux erosion

The erosion of magnetic flux in an MC depends on the rate of reconnection
between the magnetic field of the MC and IMF. As it is stated before that the
reconnection rate scales with the IMF intensity, a correlation is expected
between $\phi_{erod}$ and the prevailing IMF strength whose proxy is the
unsigned $\phi_{open}$. We define the yearly average of the normalized eroded
azimuthal flux ($\bar{\phi}_{erod}$) by averaging over $\phi_{erod}$ with the
total number of events ($N_{MC}$) in each year. In Figure 3b, we plot
$\bar{\phi}_{erod}$ and the yearly average of unsigned $\phi_{open}$ (
$\bar{|\phi}_{open}|$) in solid red and dashed black curves, respectively,
along with error bars over the period encompassing solar cycle 23 and 24. The
error bars are calculated by estimating, $\frac{\sigma}{\sqrt{N_{event}}}$,
where $\sigma$ is the standard deviation and $N_{event}$ represents number of
MC events for the eroded flux and number of PFSS extrapolations for the
calculation of the open flux in each year. A correlation study is performed
between $\bar{\phi}_{erod}$ and $\bar{|\phi}_{open}|$. We obtain a Pearson
correlation coefficient ($r_{p}$) of $0.56$ and a Spearman’s rank correlation
coefficient ($r_{s}$) of $0.44$, at 99% and 95% confidence levels,
respectively. To confirm our result we compute $r_{p}$ including the
parameters’ error bar information and find the value of $r_{p}$ as $0.51\pm
0.03$ at 98% confidence level. In Figure 3c, we show a scatter plot between
$\bar{\phi}_{erod}$ and $\bar{|\phi}_{open}|$. The observed correspondence
between the eroded flux and the Sun’s open flux indicates an underlying linear
relationship between the reconnection rate and the IMF intensity. To relate
these two parameters, we perform a least-squares fit to $\bar{|\phi}_{open}|$
versus $\bar{\phi}_{erod}$ which gives the following equation

$\bar{\phi}_{erod}=0.06\bar{|\phi}_{open}|+0.06,$ (2)

where $\bar{|\phi}_{open}|$ is in units of $10^{22}Mx$. The solid blue line
over plotted on the scatter plot represents the least-squares fitting.

##  5 Discussion

To establish a correspondence between MC’s eroded flux and the IMF intensity,
we study the dependency of the MC’s eroded flux on the Sun’s open flux over
two solar cycles. We find that about $44\%$ of the total MCs studied here
shows an imbalance at their rear boundaries. A numerical simulation of
reconnection between an MC and solar wind performed by 2003JGRA..108.1023S
shows an enhancement in reconnection with the increasing relative speed of the
MC compared to the solar wind. If an MC is followed and compressed by a fast
solar wind, reconnection may occur at the MC’s rear boundary. Such high speed
solar wind primarily originates from polar coronal holes and may extend
towards low latitudes [Fenrich Luhmann (1998)]. Also, MCs can be compressed by
co-rotating interaction regions whose formation is related to the presence of
coronal holes [Rouillard . (2010)].

In our study we obtain the orientation of an MC axis by fitting the observed
MC using a force free cylindrical flux rope model. The model is impacted by
several factors, such as, incorrect boundary selection [Lepping . (2003)], non
circular MC cross section [Savani . (20112), Savani . (20111)] and high impact
parameter values [Démoulin . (2013), Riley . (2004)]. 2015JGRA..120…43R showed
that the flux rope fitting method has a tendency of deriving a consistently
lower value of eroded flux because the model is based on axisymmetric
geometry. Also, a high impact parameter lowers the total azimuthal magnetic
flux that causes an overestimation of the normalized eroded flux.

The imbalance in MC azimuthal flux accumulated along the spacecraft path may
occur due to various other reasons. When a high-speed CME propagates from Sun
to Earth it undergoes strong deceleration [Jones . (2007)]. If a cool, dense
filament material is located at the base of the CME, the momentum of the
filament results in its forward movement through the decelerating CME. Thus,
the protruding filament drives flow leading to a sideways transport of the
CME’s poloidal field. This can generate an azimuthal flux imbalance in the CME
flux rope by the time the CME reaches the Earth. 2014JGRA..119.5449M explained
such flux imbalances through simulation of a three-part structure CME observed
on 20 January 2005. In such cases, the eroded magnetic flux amount does not
depend on the rate of reconnection between the MC and the IMF. This may also
contribute to a lower correlation between the amount of eroded MC flux and
$\phi_{open}$. Nonetheless, we find the correlation to be significant and the
underlying relationship between $\phi_{erod}$ and $\phi_{open}$ to be linear.

##  6 Conclusions

We establish a relationship between the azimuthal magnetic flux erosion of MCs
and the Sun’s unsigned open flux that regulates the IMF intensity. Utilizing
the direct method we investigate 72 MCs spanning solar cycles 23 and 24 to
estimate the magnetic flux imbalance and erosion. We find that on average
$28\%$ and $22\%$ of the total azimuthal magnetic flux of MCs of solar cycle
23 and 24, respectively, are eroded during their propagation through the inner
heliosphere. The reconnection causing this erosion may occur on either side of
the MC boundaries and peel off an almost similar amount of magnetic flux. We
compare the annual averages of solar open flux to the average fraction of the
eroded magnetic flux of MCs over the past 21 years (from 1998 to 2018) and
find a significant positive correlation with an underlying linear
relationship. Since the solar open flux is strongly correlated with the IMF
intensity, this is suggestive of the latter’s role in MC flux erosion.

We note that the solar open flux is governed by the emergence and
redistribution of active region magnetic fields on the Sun’s surface due to
surface flux transport processes – a crucial component of the solar dynamo
mechanism [Bhowmik Nandy (2018)]. Given that the solar wind dispersed open
flux determines the ambient heliospheric magnetic field (IMF), the latter’s
solar cycle modulation provides a novel pathway via which the large-scale
solar cycle can govern flux erosion and thus possibly the geoeffectiveness of
interplanetary magnetic clouds.

###### Acknowledgements.

The Center of Excellence in Space Sciences India (CESSI) is funded by the
Ministry of Human Resource Development, Government of India under the Frontier
Areas of Science and Technology scheme. S.D. acknowledges funding from the
DST-INSPIRE program of the Government of India. The authors are grateful to
Benoit Lavraud for insightful discussions and suggestions. We thank Katsuhide
Marubashi for making available the constant-$\alpha$ force free cylindrical
flux rope model. We thank an anonymous referee for useful comments. The
authors acknowledge the use of data from the SDO/HMI, SOHO/MDI, ACE
instruments, and the World Data Center for Geomagnetism, Kyoto.

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Table 1: Total number of MCs, $MC_{R}$, $MC_{F}$ in each year, annual average of normalised eroded flux ($\bar{\phi}_{erod}$), and unsigned $\phi_{open}$ ($\bar{\phi}_{open}$) along with their standard deviations over the two solar cycles (1998-2018).  Year | $N_{MC}$ | $MC_{R}$ | $MC_{F}$ | $\bar{\phi}_{erod}$ | $\bar{\phi}_{open}$
---|---|---|---|---|---
| $\\#$ | $\\#$ | $\\#$ |  |  $(\times 10^{22}$Mx)
1998 | 5 | 2 | 3 | 0.24$\pm$0.13 | 2.18$\pm$0.21
1999 | 3 | 2 | 1 | 0.28$\pm$0.06 | 4.12$\pm$0.45
2000 | 5 | 4 | 1 | 0.47$\pm$0.11 | 5.11$\pm$0.25
2001 | 4 | 2 | 2 | 0.29$\pm$0.13 | 4.43$\pm$0.22
2002 | 3 | 3 | 0 | 0.42$\pm$0.14 | 7.08$\pm$0.28
2003 | 2 | 0 | 2 | 0.16$\pm$0.05 | 6.24$\pm$0.29
2004 | 5 | 4 | 1 | 0.17$\pm$0.10 | 3.17$\pm$0.21
2005 | 3 | 1 | 2 | 0.24$\pm$0.06 | 2.76$\pm$0.11
2006 | 3 | 3 | 0 | 0.24$\pm$0.09 | 1.76$\pm$0.16
2007 | 2 | 0 | 2 | 0.16$\pm$0.06 | 1.06$\pm$0.05
2008 | 2 | 2 | 0 | 0.12$\pm$0.12 | 1.01$\pm$0.08
2009 | 4 | 1 | 3 | 0.19$\pm$0.05 | 0.61$\pm$0.04
2010 | 3 | 1 | 2 | 0.11$\pm$0.08 | 1.17$\pm$0.06
2011 | 1 | 1 | 0 | 0.22$\pm$0.00 | 1.60$\pm$0.10
2012 | 9 | 4 | 5 | 0.20$\pm$0.05 | 1.74$\pm$0.09
2013 | 10 | 6 | 4 | 0.28$\pm$0.08 | 2.15$\pm$0.17
2014 | 3 | 0 | 3 | 0.31$\pm$0.17 | 3.06$\pm$0.41
2015 | 1 | 1 | 0 | 0.15$\pm$0.00 | 3.05$\pm$0.16
2016 | 2 | 2 | 0 | 0.11$\pm$0.10 | 2.20$\pm$0.10
2017a | 0 | 0 | 0 | – | 1.23$\pm$0.03
2018 | 2 | 1 | 1 | 0.27$\pm$0.23 | 0.92$\pm$0.05
aNone of the observed MCs follow our selection criteria. |  |  |  |

Figure 1: An idealized schematic of the ambient IMF draping about an MC
propagating radially outward from the Sun towards the Earth (left), the MC’s
magnetic structure in a plane perpendicular to the ecliptic plane (right-top)
and the expected variation in the MC’s accumulated azimuthal flux due to its
reconnection with the draped ambient IMF (right-bottom)

.

Figure 2: Plots depicting $B_{y,cloud}$ (blue) and $\phi_{y}/L$ (red)
variations of two MCs during their passage past the spacecraft. The black and
green vertical dashed-dotted lines represent $t_{center}$ and the zero
crossing of $\phi_{y}/L$ curves, respectively. (a) For an event where the flux
erosion is at the MC-front. Here $\phi_{az}=3.02\times 10^{21}$ Mx/AU and
eroded flux = $6.98\times 10^{20}$ Mx/AU. (b) For an event where the flux
erosion is at MC-rear. Here $\phi_{az}=6.06\times 10^{20}$ Mx/AU and eroded
flux = $2.04\times 10^{20}$ Mx/AU. Figure 3: (a) Scatter plot between the
remaining flux of MCs and their geoeffectivness as quantified by the minimum
$Dst$ index ($Dst_{m}$). Least-squares fit considering all events (red and
black dots) and the correlation are shown in black. For a subset of events
(red dots only) as described in the text, the corresponding least-squares fit
and correlation are depicted in red. (b) $\bar{\phi}_{erod}$ and
$\bar{|\phi}_{open}|$ plotted against each year for solar cycle 23 and 24 in
continuous red and dashed black curves, respectively. Vertical lines in
corresponding colors represent the error bars for $\bar{\phi}_{erod}$ and
$\bar{|\phi}_{open}|$. (c) Scatter plot of $\bar{|\phi}_{open}|$ versus
$\bar{\phi}_{erod}$. The over plotted blue solid line shows the least-squares
fit to the data points. The correlation is mentioned in blue.