# Why “solar tsunamis” rarely leave their imprints in the chromosphere

Ruisheng Zheng Institute of Space Sciences, Shandong University, Weihai
264209, China  Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao 266237, China  CAS Key Laboratory of Solar Activity,
National Astronomical Observatories, Chinese Academy of Sciences, Beijing
100101, China  Yihan Liu Institute of Space Sciences, Shandong University,
Weihai 264209, China  Wenlong Liu  Institute of Space Sciences, Shandong
University, Weihai 264209, China  Bing Wang  Institute of Space Sciences,
Shandong University, Weihai 264209, China  Zhenyong Hou School of Earth and
Space Sciences, Peking University, Beijing 100871, China  Shiwei Feng
Institute of Space Sciences, Shandong University, Weihai 264209, China
Xiangliang Kong Institute of Space Sciences, Shandong University, Weihai
264209, China  Institute of Frontier and Interdisciplinary Science, Shandong
University, Qingdao 266237, China  Zhenghua Huang Institute of Space Sciences,
Shandong University, Weihai 264209, China  Hongqiang Song Institute of Space
Sciences, Shandong University, Weihai 264209, China  Institute of Frontier and
Interdisciplinary Science, Shandong University, Qingdao 266237, China  Hui
Tian CAS Key Laboratory of Solar Activity, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100101, China  School of
Earth and Space Sciences, Peking University, Beijing 100871, China  State Key
Laboratory of Space Weather, National Space Science Center, Chinese Academy of
Sciences, Beijing 100190, China  Pengfei Chen School of Astronomy and Space
Science, Nanjing University, Nanjing 210023, China  Key Laboratory of Modern
Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210023, China  Robertus Erdélyi Solar Physics & Space Plasma Research
Center (SP2RC), School of Mathematics and Statistics, University of Sheffield,
Hicks Building, Hounsfield Road, S3 7RH, United Kingdom  Yao Chen Institute of
Space Sciences, Shandong University, Weihai 264209, China  Institute of
Frontier and Interdisciplinary Science, Shandong University, Qingdao 266237,
China

###### Abstract

Solar coronal waves frequently appear as bright disturbances that propagate
globally from the eruption center in the solar atmosphere, just like the
tsunamis in the ocean on Earth. Theoretically, coronal waves can sweep over
the underlying chromosphere and leave an imprint in the form of Moreton wave,
due to the enhanced pressure beneath their coronal wavefront. Despite the
frequent observations of coronal waves, their counterparts in the chromosphere
are rarely detected. Why the chromosphere rarely bears the imprints of solar
tsunamis remained a mystery since their discovery three decades ago. To
resolve this question, all coronal waves and associated Moreton waves in the
last decade have been initially surveyed, though the detection of Moreton
waves could be hampered by utilising the low-quality H$\alpha$ data from
Global Oscillations Network Group. Here, we present 8 cases (including 5 in
Appendix) of the coexistence of coronal and Moreton waves in inclined
eruptions where it is argued that the extreme inclination is key to providing
an answer to address the question. For all these events, the lowest part of
the coronal wavefront near the solar surface appears very bright, and the
simultaneous disturbances in the solar transition region and the chromosphere
predominantly occur beneath the bright segment. Therefore, evidenced by
observations, we propose a scenario for the excitation mechanism of the
coronal-Moreton waves in highly inclined eruptions, in which the lowest part
of a coronal wave can effectively disturb the chromosphere even for a weak
(e.g., B-class) solar flare.

##  1 Introduction

The solar atmosphere is composed of magnetized plasma and hosts a variety of
waves that are generally believed to account for many fundamental processes in
solar physics (De Pontieu et al., 2007; Jess et al., 2009). One of the most
spectacular waves are the solar coronal waves often detected as circular
disturbances traveling outward from their eruption source (Moses et al., 1997;
Thompson et al., 1998), similar to tsunamis on the Earth. Coronal waves are
often associated with large-scale coronal mass ejections (CMEs), flares, or
even small-scale jets. They have been interpreted differently, from e.g. the
true fast-mode magnetohydrodynamic (MHD) waves (Wu et al., 2001; Ofman &
Thompson, 2002), pseudo waves (Attrill et al., 2007; Delannée et al., 2008),
or hybrid waves (Chen & Wu, 2011; Liu & Ofman, 2014; Warmuth, 2015). Fast-mode
coronal waves, by means of solar magneto-seismology, can yield estimates of
the coronal magnetic field strengths that are hard to measure directly
(Nakariakov et al., 1999; Ballai, 2007), and may also accelerate solar
energetic particles that influence space weather (Carley et al., 2013).

Before their observational discovery, the existence of coronal waves was
already conjectured by the detection of the chromospheric Moreton waves, first
detected in the 1960s (Moreton, 1960; Moreton & Ramsey, 1960; Meyer, 1968). A
typical Moreton wave forms not far ($\sim$100 Mm) from an eruption center, and
appears as an arc-shaped chromospheric disturbance. They propagate with a high
speed ($\sim$1000 km s-1) for minutes before becoming irregular and diffusive.
Moreton waves are interpreted as the ground tracks of coronal fast-mode MHD
wavefronts sweeping over the chromosphere (Uchida, 1968, 1974). In theory, the
coronal wavefront leads to local enhanced pressure that compresses the
chromospheric plasma, and therefore these Moreton-wave fronts appear in
emission in the line core and blue wing of H$\alpha$, and in absorption in the
red wing (Vršnak et al., 2002; Vršnak & Cliver, 2008). In addition, Moreton
waves are frequently associated with solar type II radio bursts, which
indicates that their coronal counterparts have evolved into shocks (Mann,
1995).

With the high-quality observations from the Ahead (-A) and Behind (-B)
spacecraft of Solar Terrestrial Relations Observatory (Kaiser et al., 2008)
(STEREO) and the Solar Dynamics Observatory (Pesnell et al., 2012) (SDO),
hundreds of coronal waves have been found and studied. However, only tens of
Moreton waves have been observed. Since chromospheric Moreton waves are the
counterparts of coronal waves, some fundamental questions rise naturally: Why
did most coronal waves fail to ignite a detectable chromospheric disturbance?
Why was no circular-shaped Moreton wave observed for all the coronal waves
with circular-shaped fronts? Moreover, why are all previous Moreton waves only
associated with strong M- and X-class flares? On the other hand, if a coronal
wave can compress the chromosphere, the transition region (TR) as the
interface between them should be compressed first, but the wave signatures in
TR are also rarely reported. Hence, although it is generally believed that
Moreton waves are the chromospheric counterpart of coronal waves, the bridge
between two phenomena still remains elusive.

Combining observations from the dual perspectives by SDO and STEREO, we
initially surveyed all coronal waves observed in the last decade from 2010 to
2021 (https://www.lmsal.com/nitta/movies/AIA_Waves/oindex.html), and the
associated Moreton waves were checked by the line core of H$\alpha$ from
Global Oscillations Network Group (GONG). Note that most of the detected
Moreton waves are only distinguished in animations and very faint in still
images, due to the low-quality data of GONG. Though the low-quality data could
decrease the detection number of total Moreton waves, the continuous
observations can be related to all the coronal waves. Because the same
condition of low-quality GONG data are used to detect the related Moreton
waves for all coronal waves that are studied here, the data factor can be
neglected when we analyse the relationship between coronal waves and Moreton
waves. The fraction of coronal waves associated Moreton waves is in progress
in the upcoming statistics work.

In this Letter, we picked three cases of distinct chromospheric wave
signatures and coronal waves associated with inclined eruptions that are
identified by the observations in dual perspectives from the sample in the
quick view, to first propose a scenario of the plausible excitation mechanism
for Moreton waves. In this scenario, Moreton waves are easily ignited by the
enhanced pressure behind coronal waves when the eruptions are highly
“inclined” and thus the nose parts of coronal waves are close to the solar
surface. The strong reliance to highly inclined eruptions can reasonably
explain the rarity of detectable Moreton waves.

##  2 Results

Three coronal wave events, presented and analyzed here in details, are
respectively related to an X6.9-class flare on 2011 August 9 (E1), an
M1.4-class flare on 2021 May 22 (E2), and a B9.5-class flare on 2011 July 3
(E3). Each event is accompanied by a CME and a type II radio burst
(Supplementary Fig. S6 and Fig. S7). The inclined eruptions and coronal waves
are shown clearly in dual perspectives from Atmospheric Imaging Assembly (AIA)
(Lemen et al., 2012) on-board SDO and from Extreme Ultraviolet Imager (EUVI)
(Howard et al., 2008) on STEREO-A (Fig. 1 and Fig. 1 animation). In the full-
disk composite images in AIA 304, 171, and 193 Å (left panels), all the
eruptions move highly nonradially (the sectors), and the inclined angles with
respect to the solar radial direction are in the range of $\sim$63–76∘
estimated by the reconstruction of the eruption cores or lowest part of
coronal wavefronts (Supplementary Fig. S8 and Fig. S9). Following the inclined
eruptions, the coronal waves also propagate unilaterally. In the local
difference images of AIA 193 Å and EUVI-A 195 Å (middle and right columns in
Fig. 1), the wavefronts are arc-like (white arrows), and their lowest parts
close to the solar surface became very bright (red arrows).

Interestingly, different from the rest of the wavefront, the bright segments
in AIA 193 Å had evident associated signatures in AIA 171, 304 Å and H$\alpha$
(Fig. 2 and Fig. 2 animation). In AIA 171, the wavefronts are replaced by
coronal dimmings, which can be interpreted by the heating of local plasma (Liu
& Ofman, 2014). The wave signatures in AIA 304 Å are evident, and the
H$\alpha$ responses are faint. The signatures in He II 304 Å and H$\alpha$ can
respectively represent the TR and the chromosphere counterparts of coronal
waves, and both have arc-shapes similar to the bright segments of 193 Å wave,
indicating an intimate relation between three waves in different layers. They
have similar speeds in the range of $\sim$380–580 km s-1 according to the
time-distance plots (Supplementary Fig. S10). Assuming a fast magnetosonic
speed of $\sim$200–300 km s-1 at a low height for the bright segments (Wu et
al., 2001), the coronal waves may manifest as shocks with a fast-magnetosonic
Mach number of $\sim$2 (Warmuth et al., 2001), in line with the occurrence of
the radio type II bursts in Supplementary Fig. S7. Note that the disturbance
in E3 is the first example of a Moreton wave associated with a weak B-class
flare, though the wave signal in H$\alpha$ is nearly invisible in the time-
distance plot (Supplementary Fig. S10).

For the coronal wavefronts (cyan boxes in Fig. 2), we analyse the evolution of
intensity and emission at different passbands (Fig. 3). In the upper row, the
curves represent the changes of the normalized intensities in AIA 193 (green),
171 (black), 304 Å (red) , and GONG H$\alpha$ (purple). After the wave arrival
(pink solid lines), the intensity of AIA 171 Å begins to decrease suddenly,
and the intensities in AIA 193 and 304 Å increase abruptly. Though the rise
onset of the intensity in GONG H$\alpha$ is uncertain due to the poor data
quality, each of them achieves a distinct peak as those in AIA 193 and 304 Å
(dashed lines). The peaks in AIA 193, 304 Å, and GONG H$\alpha$ are almost
simultaneous for E1 and E2. For E3, the peak time in AIA 304 Å is 36 seconds
later than that in AIA 193 Å, but 30 seconds earlier than that in GONG
H$\alpha$ (panel (c)).

The bottom row of Fig. 3 shows the average evolution of the differential
emission measure (DEM) in the region of bright segments (cyan boxes in Fig. 2)
obtained by employing the sparse inversion code (Cheung et al., 2015; Su et
al., 2018). After the wave passage (pink solid lines), the DEM at
$\log(T/K)\approx 6.2$ increases obviously, and the DEM at $\log(T/K)\approx
6.0$ decreases simultaneously (white and pink arrows). It indicates that the
local plasma of wavefronts is heated from $\log(T/K)\approx 6.0$ to
$\log(T/K)\approx 6.2$, consistent with the bright wavefront in AIA 193 Å and
the dimmings in AIA 171 Å. Intriguingly, the DEM at $\log(T/K)\approx 5.7$
also increases simultaneously with that at $\log(T/K)\approx 6.2$ (red
arrows), though the increase in E3 is weak. Moreover, the continuous increase
of DEM at $\log(T/K)\approx 5.7$ seemed to be evolved from that at
$\log(T/K)\approx 5.9$, which possibly implies their compression relationship.
It indicates that the DEM increase at $\log(T/K)\approx 5.7$ likely represents
the emission enhancement for the plasma at the bottom of the corona and/or the
upper TR, due to the downward expansion and compression of the bright segment
of the coronal wave that is nearly horizontal and close to the solar surface.

##  3 Discussion and Conclusions

Vršnak et al. (Vršnak et al., 2016) conjectured that maybe an
asymmetric/nonradial eruption is an additional requirement for the generation
of the Moreton wave in a weak eruption, when they simulated the Moreton waves
associated with impulsive eruptions. In the few cases of coronal and Moreton
waves observed by SDO and ground-based observatories, all the related
eruptions are non-radial (Asai et al., 2012; Shen & Liu, 2012; Francile et
al., 2016; Cabezas et al., 2019; Long et al., 2019; Wang et al., 2020), and
the inclined eruptions could be the result of filaments erupting into weaker
magnetic field regions (Panasenco et al., 2013). For instance, the inclined
eruption and associated Moreton wave on 2014 March 29 is showed in
Supplementary Fig. S11 and Fig. S11 animation), which has been studied by
other researchers (Francile et al., 2016; Cabezas et al., 2019; Long et al.,
2019).

The AIA 304 Å channel is mainly dominated by the two He II 303.8 Å lines with
a temperature of $\log(T/K)\approx 4.7$ for coronal holes, quiet Sun regions,
active regions, and flare plasma. However, in QS off-limb the channel can also
make an important contribution by Si XI 303.33 Å line at the coronal
temperature of $\log(T/K)\approx 6.2$ (O’Dwyer et al., 2010). Some previous
observations have shown that wave signatures in AIA 304 Å channel could be
dominated in a coronal line of Si XI 303.33 Å (Long et al., 2010), or the He
II 303.8 Å lines in the transition region (Long et al., 2011). In this Letter,
it is very clear that the wavefronts in 304 Å were narrow, similar to
wavefronts in H$\alpha$, and were different with wavefronts in 193 Å. Like
Moreton waves, the wavefronts in 304 Å were narrow, and closely related to the
bright parts of coronal wavefronts (red and white arrows in Fig. 1 and the
dashed boxes in Fig. 2, similar to the case in Hou et al. (2022). In addition,
the wavefronts in 304 Å can be shown in the disk in the perspective of SDO or
STEREO. Hence, the morphological difference in the disk between the wavefronts
in 304 and 193 Å suggested that the wavefronts in 304 Å in this Letter were
much likely dominated by the He II line.

To further uncover the relationship between the coronal and Moreton waves, we
check four more coronal waves that are associated with either inclined limb
eruptions or radial eruptions. In the inclined limb eruptions (Supplementary
Fig. S12 and Fig. S12 animation), the bright segments of coronal waves and
disturbances in TR are prominent (blue arrows), though the chromospheric wave
signatures are hard to identify due to the poor H$\alpha$ data quality.
Although the radial eruptions are associated with strong X- or M-class flares
(Supplementary Fig. S13 and Fig. S13 animation), no wave signature is found in
He II 304 Å and H$\alpha$. This is then consistent with the fact that the bulk
of coronal waves is obvious in AIA 193 and 211 Å, while there is no signature
in AIA 304 Å. According to Uchida’s model (Uchida, 1968, 1974), the TR should
be compressed first if the coronal wave can disturb the chromosphere.
Therefore, the waves in He II 304 Å for the above cases likely represent the
response in TR, and can be regarded as the missing link between coronal and
Moreton waves. Furthermore, for the above cases of inclined eruptions, all
coronal waves successfully disturb the TR and chromosphere, including the one
associated with the weak B-class flare. This further implies that the inclined
eruption is the key to generating chromospheric Moreton waves.

For coronal waves in the inclined eruptions presented above, they display a
bright segment of wavefronts near the solar surface, distinct from other parts
(Fig. 1), and the disturbances in the TR and chromosphere simultaneously occur
just under the brightest segment (Fig. 2). Comparing with the evolution in
different regions of wavefronts, it is likely that the average DEM evolution
in the region of brightest segment reveals the emission enhancement in the
coronal base and the upper TR behind the coronal shocks/waves (Fig. 3),
consistent with the scenario of the Moreton wave (Vršnak et al., 2002). Hence,
the bright segments of coronal wavefronts and the related continuous increase
of DEM indicate that the TR and chromosphere can be effectively disturbed by
the lowest part of the inclined coronal waves, and provide further evidence of
the role of inclined eruptions in driving the Moreton waves.

Based on the above observational results, we propose a plausible excitation
mechanism for Moreton waves in Fig. 4. The coronal wave (the solid-dotted pink
curve) consists of a shocked nose (solid part) and weak flanks (dotted parts).
For the coronal wave in an inclined eruptions (panel (a)), the lowest part of
the wavefront is close to the nose part of the shock wave, hence it is
relatively strong at the onset (at $t_{0}$). The expanding lowest part exerts
a strong compression towards the solar surface resulting in the bright segment
(the red patch). The compression ignites a pressure jump that will cause a
density/temperature increase not only in the TR but also in the chromosphere,
in the form of wave signatures in the TR and chromosphere (the distortions and
the projected arcs). When the nose part of the coronal wavefront lifts higher
later (at $t_{1}$), the lowest part, being further away from the nose part of
the coronal wave, becomes so weak that it hardly compresses the TR beneath,
and consequently the wave signatures in the TR and chromosphere vanish. On the
contrary, in the radial eruption scenario (panel (b)), the nose of coronal
wave initially forms at a typical height of 50–100 Mm (Patsourakos &
Vourlidas, 2012) a few minutes after the early rising phase of the eruption
core. The lowest rim of the coronal wavefront is far from the nose part of the
shock (above a height of $h$) and becomes too weak to imprint the TR and the
chromosphere.

In this scenario, Moreton waves are easily generated by coronal waves that are
associated with inclined eruptions. However, would all coronal waves with
inclined eruptions leave imprints in the chromosphere? We show two more
Moreton waves associated with inclined eruptions (Supplementary Fig. S14 and
Fig. S14 animation). The Moreton wave associated with an X1.8 flare on 2011
September 7 was hardly distinguishable, and was only confirmed by the
oscillation of a low-lying filament and nearby loops (the pink and black
arrows in top panels in Fig. S14). The Moreton wave (blue arrows) associated
with an X1.0 flare on 2021 October 28 was also weak, and the westward erupting
prominence (yellow arrows) had an inclination angle of $\sim$30∘ calculated
with reconstructed points of the erupting prominence in dual perspectives
(bottom panels in Fig. S14). Hence, the scenario can be complimented by the
factors of the scale of magnetic structures beneath the inclined coronal
wavefronts and the range of inclination angles. Assuming that there exists an
underlying large-scale coronal loop system that would be intact during the
downward compression of the coronal wavefronts (Fig. 5a), the large-scale loop
system possibly invokes the MHD wave refractions from regions of high-Alvén
speed (Wang, 2000). The refraction from the ambient large-scale loop systems
indeed contributed to the lack of Moreton waves for some coronal waves in
inclined eruptions. On the other hand, if the inclination angle of the
eruption is small to the radial direction, it should become difficult for the
coronal wave to disturb the TR and the chromosphere significantly, and the
Moreton wave signature will be faint or may even disappear (Fig. 5b).

Therefore, the scenario introduced above suggests that the reliance on the
highly inclined eruptions is the key to resolving the rarity puzzle of Moreton
waves. Hence, the comprehensive relationship between the coronal waves and
Moreton waves is likely established by the highly inclined eruptions, with the
additional factors (the inclined angle and the magnetic structures beneath
downward coronal wavefronts) for the scenario. We also imply that the wave in
He II 304 Å may be the counterparts of Moreton waves when the Moreton waves
are undetectable in low-quality H$\alpha$ data. According to this scenario, a
coronal wave related to an impulsive X-class flare need a moderately inclined
eruption, and a coronal wave associated with a weak B-class flare can ignite a
detectable Moreton wave when the eruption and the associated coronal wavefront
are extremely “inclined”. The high-quality chromospheric data from the Chinese
H$\alpha$ Solar Explorer (Li et al., 2022), launched in October of 2021, will
provide more cases of Moreton waves to check the scenario. Further simulations
are demanded to verify the scenario proposed in this Letter.

SDO is a mission of NASA’s Living With a Star Program. We gratefully
acknowledge the usage of data from the SDO, STEREO, and from the ground-based
GONG project. This work is supported by grants of NSFC 11790303, 12073016,
11825301, 11973031, and 12127901. R.E. is also grateful to the Science and
Technology Facilities Council (STFC, grant No. ST/M000826/1) for the support
received.

Figure 1: Coronal waves in dual perspectives. Left column: The eruptions are
displayed in the full-disk composite images of AIA 304 (red), 171 (green), and
193 Å (blue). The sectors indicate the directions of eruptions and associated
coronal waves, and the white boxes represent the field of view (FOV) of middle
panels. Middle and right columns: Coronal waves (white arrows) and bright
segments (red arrows) are shown in difference images in AIA 193 Å and EUVI-A
195 Å. The red dashed boxes indicate the FOV of panels in Fig. 2. (An
animation of middle and right columns is available online.)

Figure 2: The wave signatures in different atmospheric layers. The cyan boxes
in difference images outline the bright segment of coronal waves in AIA 193 Å
and their responses in AIA 171, 304 Å and GONG H$\alpha$. The cyan boxes
indicate the regions for the DEM evolution in Fig. 3. (An animation of
associated with this figure is available online.)

Figure 3: The evolution for the intensity and emission of plasma. Upper row:
The light curves in the region of bright segments (cyan boxes in Fig. 2) in
AIA 193, 171, 304 Å and GONG H$\alpha$. Bottom row: The average DEM evolutions
in the region of bright segments (cyan boxes in Fig. 2). The pink solid lines
indicate the arrival times of coronal waves, and the colored dashed lines
represent the peak times in the associated passbands, and the colored arrows
show the DEM changes at different temperatures.

Figure 4: The scenario of exciting mechanism for Moreton waves. a): For an
inclined eruption with a core (orange), the lowest part of the associated
coronal wave (the solid-dotted pink front) becomes very bright (the red patch)
at the onset (t0), and compresses the TR (the purple layer) and chromosphere
(the blue layer) below. The projected purple and blue arcs represent the
disturbances in the TR and chromosphere. The nose part of the coronal
wavefront reaches a height of $h$ from the coronal base after the propagation
in a few minutes (t1). b): For a radial eruption, the nose of the coronal wave
forms at an initial height of $h$. Note that the coronal wave degenerates from
a shock wave near the nose part (solid line) to an ordinary wave further away
(dotted line).

Figure 5: The unsuccessful situations for Moreton waves in inclined eruptions.
a): The coronal wave encounters the low-lying coronal loops. b): The
inclination angle for the eruption is large. The coronal wave (the solid-
dotted pink curve) consists of a strong nose (solid) and weak flanks (dotted).

##  Appendix A Appendix information

Data The analysis of coronal waves is primarily based on the extreme
ultraviolet (EUV) observations in dual perspectives from AIA 304, 171, 193 Å
and EUVI 304, 171, and 195 Å (Fig. 1, Fig. 2, and associated animations). Each
AIA image has a cadence of 12 second and a pixel size of 0.6′′. EUVI images
have a pixel size of 1.58′′, and their cadences are 2.5 minutes, 5 minutes or
10 minutes. Moreover, to study the evolution of Moreton waves, we use the
H$\alpha$ filtergrams from the Global Oscillation Network Group (GONG) of the
National Solar Observatory. The H$\alpha$ images have a cadence of 1 minute
and a pixel size of $\sim$1′′ (Fig. 2).

The evolution of the CMEs in the high corona (white arrows) is shown by Large
Angle and Spectrometric Coronagraph (LASCO) (Brueckner et al., 1995) onboard
the Solar and Heliospheric Observatory (SOHO) spacecraft, and the related
flares are recorded by the Geostationary Operational Environment Satellite
(GOES) in the form of integrated full-disk soft X-ray emission from the Sun
(Supplementary Fig. S6). In addition, the associated radio bursts (white
arrows) was detected in the frequency range of 25-500 MHz by the metric
spectrometers from Hiraso RAdio Spectrograph (HiRAS) and YAMAGAWA of National
Institute of Information and Communications Technology (NICT) (Kondo et al.,
1995; Iwai et al., 2017), and Learmonth (LEAR) of the radio Solar Telescope
Network (RSTN) (Kennewell & Steward, 2003) (Supplementary Fig. S7).

Methods To identify well the wavefronts and CMEs, the image is subtracted by
one previous image, and the times of two images are attached (Fig. 1, Fig. 2,
Supplementary Fig. S6, Fig. S8, Fig. S11, Fig. S12, and associated
animations). The wave signatures in different layers are analysed by the time-
slice approach, in which a time-distance plot was constructed with a stack of
slices along a selected sector for a set of difference images. The wave speeds
and errors (blue lines) are obtained by fitting with the linear (linfit.pro)
function in the SolarSoftWare (SSW), assuming a measurement uncertainty of 4
pixels ($\sim 1.74$ Mm) for the selected points (Supplementary Fig. S10).

Benefiting with the observations in dual perspectives (upper panels in
Supplementary Fig. S9), we derive the real locations of the erupting cores
(the core material of erupting filaments/prominences and jets) and wavefronts
by the reconstruction procedure of scc_measure.pro in SSW. To estimate the
eruption inclination, we reconstruct some selected points from the erupting
core of E2 both that simultaneously appear in AIA 171 Å and EUVI-A 171 Å. For
E1 and E3, we choose the bright segment instead of invisible erupting cores
(Supplementary Fig. S8), assuming that the inclination of the erupting core is
approximately equal to that of the bright segment (the orange oval core and
the red segment in Fig. 4). Finally, we superimpose the reconstructed points
on one AIA 304 Å image in which the eruption center is rotated to the limb
(bottom panels in Supplementary Fig. S9).

We also investigated the emission evolution in the fronts of coronal waves
with the differential emission measure (DEM) method by employing the sparse
inversion code (Cheung et al., 2015; Su et al., 2018). The emission at a range
of temperature is obtained by a set of AIA images in six channels, i.e., 131 Å
(Fe XXI, $\sim$10 MK), 94 Å (Fe XVIII, $\sim$6.4 MK), 335 Å (Fe XVI, $\sim$2.5
MK), 211 Å (Fe XIV, $\sim$2.0 MK), 193 Å (Fe XII, $\sim$1.6 MK), and 171 Å (Fe
IX, $\sim$0.6 MK). The DEM evolution plots are constructed with the average
value in the selected regions (cyan boxes in Fig. 2) for a series of EM maps
(bottom panels in Fig. 3).

Figure S6: The associated CMEs and flares. Upper row: The related CMEs (white
arrows) are shown in difference images of LACSO C2. Bottom row: The related
flares are indicated by GOES X-rays 1–8 Å.

Figure S7: The associated radio bursts. The radio dynamic spectrums for three
cases are obtained from HiRAS, YAMAGAWA, and LEAR, respectively, and the white
arrows indicate the onsets of radio bursts.

Figure S8: The selected points for the reconstructions. The colored points in
dual perspectives from AIA and EUVI-A represent the positions for the
reconstructions in Supplementary Fig. S9.

Figure S9: The inclination angles of the eruptions. Upper row: The positions
of the Sun, the Earth, and the STEREO-A. Bottom row: The selected points in
the limb of rotated AIA 304 Å images to show the the inclination angles that
are $\sim 73^{\circ}$, $\sim 63^{\circ}$, and $\sim 76^{\circ}$, respectively.
The axis of $z$ points to the Earth, and the axe of $x$ and $y$ define the
coordinate plane of AIA images.

Figure S10: Wave velocities in different layers. The velocities in AIA 193,
171, 304 Å and GONG H$\alpha$ are calculated along the sectors in Fig. 1 for
three cases (left, middle, right columns), and the inclined lines indicate the
measured speeds. Note that the chromospheric wave signature for E3 is hardly
distinguished in the last panel.

Figure S11: The Moreton wave with an X1.0 flare on 2014 March 29. The inclined
erupting structure (the yellow arrow) is shown in AIA 94 AA. In the difference
images in dual perspectives, the blue arrows show the bright segments of
coronal waves (white arrows) in AIA 193 Å and EUVI 195 Å and the simultaneous
wave signatures in AIA 304 Å and GONG H$\alpha$.

Figure S12: Two cases of inclined limb eruptions. Two eruptions are separately
accompanied with a C6.2 flare on 2011 August 11 and an M3.5 flare on 2011
February 24. In the difference images in dual perspectives, the blue arrows
show the bright segments of coronal waves in AIA 193 Å and EUVI 195 Å and the
simultaneous wave signatures in He II 304 Å. (An animation of this figure is
available online.)

Figure S13: Two cases of radial eruptions. Two eruptions are separately
accompanied with an X4.9 flare on 2014 February 25 and an M1.7 flare on 2021
November 2. In the difference images in dual perspectives, the coronal waves
are obvious in AIA 193 Å and EUVI 195 Å, but there is no wave signature in He
II 304 Å and H$\alpha$. (An animation of this figure is available online.)

Figure S14: Two more cases of inclined eruptions. Two eruptions are separately
accompanied with an X1.8 flare on 2011 September 7 and an X1.0 flare on 2021
October 28. In the difference images in dual perspectives, the blue arrows
show the bright segments of coronal waves (white arrows) in AIA 193 Å and EUVI
195 Å and the simultaneous wave signatures in AIA 304 Å and GONG H$\alpha$.
The waves are closely associated with inclined erupting filaments (yellow
arrows), and are affected by the ambient coronal loops (the pink arrow) and
the underlying filament (black arrows). (An animation of this figure is
available online.)

##

## References

  * Asai et al. (2012) Asai, A., Ishii, T. T., Isobe, H., et al. 2012, ApJ, 745, L18, doi: 10.1088/2041-8205/745/2/L18
  * Attrill et al. (2007) Attrill, G. D. R., Harra, L. K., van Driel-Gesztelyi, L., Démoulin, P., & Wülser, J. P. 2007, Astronomische Nachrichten, 328, 760, doi: 10.1002/asna.200710794
  * Ballai (2007) Ballai, I. 2007, Sol. Phys., 246, 177, doi: 10.1007/s11207-007-0415-3
  * Brueckner et al. (1995) Brueckner, G. E., Howard, R. A., Koomen, M. J., et al. 1995, Sol. Phys., 162, 357, doi: 10.1007/BF00733434
  * Cabezas et al. (2019) Cabezas, D. P., Asai, A., Ichimoto, K., et al. 2019, ApJ, 883, 32, doi: 10.3847/1538-4357/ab3a35
  * Carley et al. (2013) Carley, E. P., Long, D. M., Byrne, J. P., et al. 2013, Nature Physics, 9, 811, doi: 10.1038/nphys2767
  * Chen & Wu (2011) Chen, P. F., & Wu, Y. 2011, ApJ, 732, L20, doi: 10.1088/2041-8205/732/2/L20
  * Cheung et al. (2015) Cheung, M. C. M., Boerner, P., Schrijver, C. J., et al. 2015, ApJ, 807, 143, doi: 10.1088/0004-637X/807/2/143
  * De Pontieu et al. (2007) De Pontieu, B., McIntosh, S. W., Carlsson, M., et al. 2007, Science, 318, 1574, doi: 10.1126/science.1151747
  * Delannée et al. (2008) Delannée, C., Török, T., Aulanier, G., & Hochedez, J. F. 2008, Sol. Phys., 247, 123, doi: 10.1007/s11207-007-9085-4
  * Francile et al. (2016) Francile, C., López, F. M., Cremades, H., et al. 2016, Sol. Phys., 291, 3217, doi: 10.1007/s11207-016-0978-y
  * Hou et al. (2022) Hou, Z., Tian, H., Wang, J.-S., et al. 2022, ApJ, 928, 98, doi: 10.3847/1538-4357/ac590d
  * Howard et al. (2008) Howard, R. A., Moses, J. D., Vourlidas, A., et al. 2008, Space Sci. Rev., 136, 67, doi: 10.1007/s11214-008-9341-4
  * Iwai et al. (2017) Iwai, K., Kubo, Y., Ishibashi, H., et al. 2017, Earth, Planets and Space, 69, 95, doi: 10.1186/s40623-017-0681-8
  * Jess et al. (2009) Jess, D. B., Mathioudakis, M., Erdélyi, R., et al. 2009, Science, 323, 1582, doi: 10.1126/science.1168680
  * Kaiser et al. (2008) Kaiser, M. L., Kucera, T. A., Davila, J. M., et al. 2008, Space Sci. Rev., 136, 5, doi: 10.1007/s11214-007-9277-0
  * Kennewell & Steward (2003) Kennewell, J., & Steward, G. 2003, Solar Radio Spectrograph [SRS] Data Viewer [srsdisplay] (Sydney: IPS Radio and Space Serv.)
  * Kondo et al. (1995) Kondo, T., Isobe, T., Igi, S., Watari, S., & Tokimura, M. 1995, J. Commun. Res. Lab, 42, 111
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17, doi: 10.1007/s11207-011-9776-8
  * Li et al. (2022) Li, C., Fang, C., Li, Z., et al. 2022, Science China Physics, Mechanics, and Astronomy, 65, 289602, doi: 10.1007/s11433-022-1893-3
  * Liu & Ofman (2014) Liu, W., & Ofman, L. 2014, Sol. Phys., 289, 3233, doi: 10.1007/s11207-014-0528-4
  * Long et al. (2010) Long, D., Deluca, E. E., & Gallagher, P. T. 2010, in AGU Fall Meeting Abstracts, Vol. 2010, SH23A–1825
  * Long et al. (2011) Long, D. M., DeLuca, E. E., & Gallagher, P. T. 2011, ApJ, 741, L21, doi: 10.1088/2041-8205/741/1/L21
  * Long et al. (2019) Long, D. M., Jenkins, J., & Valori, G. 2019, ApJ, 882, 90, doi: 10.3847/1538-4357/ab338d
  * Mann (1995) Mann, G. 1995, Journal of Plasma Physics, 53, 109, doi: 10.1017/S0022377800018043
  * Meyer (1968) Meyer, F. 1968, in Structure and Development of Solar Active Regions, ed. K. O. Kiepenheuer, Vol. 35, 485
  * Moreton (1960) Moreton, G. E. 1960, AJ, 65, 494, doi: 10.1086/108346
  * Moreton & Ramsey (1960) Moreton, G. E., & Ramsey, H. E. 1960, PASP, 72, 357, doi: 10.1086/127549
  * Moses et al. (1997) Moses, D., Clette, F., Delaboudinière, J. P., et al. 1997, Sol. Phys., 175, 571, doi: 10.1023/A:1004902913117
  * Nakariakov et al. (1999) Nakariakov, V. M., Ofman, L., Deluca, E. E., Roberts, B., & Davila, J. M. 1999, Science, 285, 862, doi: 10.1126/science.285.5429.862
  * O’Dwyer et al. (2010) O’Dwyer, B., Del Zanna, G., Mason, H. E., Weber, M. A., & Tripathi, D. 2010, A&A, 521, A21, doi: 10.1051/0004-6361/201014872
  * Ofman & Thompson (2002) Ofman, L., & Thompson, B. J. 2002, ApJ, 574, 440, doi: 10.1086/340924
  * Panasenco et al. (2013) Panasenco, O., Martin, S. F., Velli, M., & Vourlidas, A. 2013, Sol. Phys., 287, 391, doi: 10.1007/s11207-012-0194-3
  * Patsourakos & Vourlidas (2012) Patsourakos, S., & Vourlidas, A. 2012, Sol. Phys., 281, 187, doi: 10.1007/s11207-012-9988-6
  * Pesnell et al. (2012) Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3, doi: 10.1007/s11207-011-9841-3
  * Shen & Liu (2012) Shen, Y., & Liu, Y. 2012, ApJ, 752, L23, doi: 10.1088/2041-8205/752/2/L23
  * Su et al. (2018) Su, Y., Veronig, A. M., Hannah, I. G., et al. 2018, ApJ, 856, L17, doi: 10.3847/2041-8213/aab436
  * Thompson et al. (1998) Thompson, B. J., Plunkett, S. P., Gurman, J. B., et al. 1998, Geophys. Res. Lett., 25, 2465, doi: 10.1029/98GL50429
  * Uchida (1968) Uchida, Y. 1968, Sol. Phys., 4, 30, doi: 10.1007/BF00146996
  * Uchida (1974) —. 1974, Sol. Phys., 39, 431, doi: 10.1007/BF00162436
  * Vršnak & Cliver (2008) Vršnak, B., & Cliver, E. W. 2008, Sol. Phys., 253, 215, doi: 10.1007/s11207-008-9241-5
  * Vršnak et al. (2016) Vršnak, B., Žic, T., Lulić, S., Temmer, M., & Veronig, A. M. 2016, Sol. Phys., 291, 89, doi: 10.1007/s11207-015-0822-9
  * Vršnak et al. (2002) Vršnak, B., Warmuth, A., Brajša, R., & Hanslmeier, A. 2002, A&A, 394, 299, doi: 10.1051/0004-6361:20021121
  * Wang et al. (2020) Wang, J., Yan, X., Kong, D., et al. 2020, ApJ, 894, 30, doi: 10.3847/1538-4357/ab8565
  * Wang (2000) Wang, Y. M. 2000, ApJ, 543, L89, doi: 10.1086/318178
  * Warmuth (2015) Warmuth, A. 2015, Living Reviews in Solar Physics, 12, 3, doi: 10.1007/lrsp-2015-3
  * Warmuth et al. (2001) Warmuth, A., Vršnak, B., Aurass, H., & Hanslmeier, A. 2001, ApJ, 560, L105, doi: 10.1086/324055
  * Wu et al. (2001) Wu, S. T., Zheng, H., Wang, S., et al. 2001, J. Geophys. Res., 106, 25089, doi: 10.1029/2000JA000447