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A&A
Volume 675, July 2023
Article Number A129
Number of page(s) 6
Section The Sun and the Heliosphere
DOI https://doi.org/10.1051/0004-6361/202346378
Published online 10 July 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

During the ESA/NASA space mission Solar Heliospheric Observatory (SOHO), a new wave phenomenon initially referred to as a coronal transitent wave (Moses et al. 1997; Thompson et al. 1998) was discovered using the Extreme Ultraviolet Imaging Telescope (EIT; Delaboudiniere et al. 1995). These waves have subsequently become known as EIT waves, extreme-ultraviolet (EUV) waves, or more generally large-scale coronal propagating fronts (LCPFs). They appear as a bright rim (sometimes nearly circular) in EUV images, which propagates as a large-scale coronal disturbance over a significant fraction of the solar surface.

Immediately after their discovery, EUV waves were linked to Moreton waves (Moreton & Ramsey 1960), which are observed in the chromosphere. Moreton waves are often accompanied by solar type II radio bursts (Kai 1970), which are signatures of coronal shock waves (Wild et al. 1959). Because of this association, Moreton waves and the shock waves related to type II radio bursts were considered to be generated by the same phenomenon, namely a solar eruption (Uchida 1968). In contrast to Hα Moreton waves, EUV waves are visible in EUV spectral lines, for example at Fe XII at 195 Å, which is dominated by emission of coronal plasma at temperatures of ≈1.5 × 106 K. The velocities of Moreton waves seemed to be generally much higher than those of the EUV waves, which are found to have velocities in the range of 200−400 km s−1 (Klassen et al. 2000; Thompson & Myers 2009).

Apparent discrepancies such as this have led to a prolonged discussion on the physical nature of EUV waves. In the ‘classical’ scenario, they are considered as large-scale fast magnetosonic waves and/or shocks (see e.g., Mann et al. 1999; Klassen et al. 2000; Warmuth et al. 2004b; Grechnev et al. 2008; Temmer et al. 2009; Veronig et al. 2010). According to this interpretation, a flare or a coronal mass ejection (CME) excites a fast magnetosonic wave in the corona, which is observed as an EUV wave that travels laterally along the solar surface. It may steepen into a shock wave and generate type II radio emission. This scenario explains the relationship between EUV waves and solar type II radio bursts as reported by Klassen et al. (2000). Accepting this model, EUV waves may be used to conduct coronal seismology. As an example, a magnetic field strength of ≈3 G was deduced for the quiet corona (Mann et al. 1999; Warmuth & Mann 2005). The Alfvén-Mach number of the shocks associated with type II radio bursts were found to be well above 1.6 (Mann et al. 1999).

An alternative interpretation of EUV waves is that they are the result of a magnetic restructuring of the corona in the framework of an expanding CME or an erupting flux rope (see e.g., Delannée & Aulanier 1999; Delannée 2000, 2008; Chen et al. 2002, 2005; Zhukov & Auchére 2004; Attrill et al. 2007; Wills-Davey et al. 2007; Dai et al. 2010) In this model, the disturbances are not real waves in a physical sense, but are propagating density and temperature variations induced by the expanding CME.

The apparent speed discrepancy between Moreton and EIT waves could be resolved by showing that both phenomena are consistent with a single, decelerating disturbance (Warmuth et al. 2001). Due to its low cadence (12–15 min), EIT recorded the EUV waves only when their speed had already decreased significantly. This was corroborated by observations in soft X-rays (Warmuth et al. 2005) and Helium I (Vršnak et al. 2002). Subsequently, the observational capabilities in EUV were increased by the Extreme Ultra Violet Imagers (EUVI) (Howard et al. 2008) on board the twin spacecraft Solar-Terrestrial Relations Observatory (STEREO) (Kaiser et al. 2008). The EUVI was designed with a much better cadence in comparison to EIT, of namely ∼2.5 min. STEREO confers several advantages: it has a large field of view, high sensitivity, and can obtain simultaneous observations from two well-separated vantage points in space. Using these advantages of the STEREO spacecraft, EUV waves have been investigated by means of the EUVI data as reported in several papers (Long et al. 2008; Veronig et al. 2008; Gopalswamy et al. 2009; Patsourakos & Vourlidas 2009; Patsourakos et al. 2009; Attrill et al. 2009; Cohen et al. 2009; Kienreich et al. 2009; Ma et al. 2009; Zhukov et al. 2009; Dai et al. 2010; Podladchikova et al. 2010; Grechnev et al. 2011).

Finally, an additional step forward in terms of observational capabilities in EUV has been provided by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012) spacecraft. This instrument provides superior temporal cadence (12 s) as well as multi-temperature coverage due to the availability of six EUV channels. With AIA, it was possible to show that LCPFs propagate with an almost constant initial velocity of ≈1000 km s−1 from the chromosphere up to the corona (Chen & Wu 2011; Shen & Liu 2012a,b; Nitta et al. 2013; Shen et al. 2014).

Beyond individual case studies, a thorough physical understanding of coronal EUV waves requires the analysis of statistically significant samples. Warmuth & Mann (2011) analysed the kinematics of a sample of 176 EUV waves recorded by EIT and EUVI, and their findings revealed three different classes:

  1. waves with initial velocities of > 320 km s−1 showing pronounced deceleration during their evolution;

  2. waves propagating with a nearly constant speed of 170−320 km s−1;

  3. disturbances with low speeds (< 130 km s−1) and showing a rather erratic motion;

(see also Fig. 8 in Warmuth & Mann 2011). Class 1 events are interpreted as large-amplitude magnetohydrodynamic (MHD) waves or shocks. Their velocity decreases during their evolution. Class 2 waves are considered to be linear waves travelling with the local fast magnetosonic speed, because they travel with a nearly constant velocity. Some authors (Shen & Liu 2012a,b; Zhou et al. 2022a,b) reported on typical wave properties, such as interference and reflection effects during the interaction of EUV waves with coronal structures. These observations support the wave nature of EUV waves. Finally, class 3 events are interpreted as disturbances caused by magnetic reconfigurations (Warmuth & Mann 2011) and are not considered to be waves in a physical sense. Thus, class 1 and 2 waves represent freely propagating waves in the corona.

These results have been corroborated by statistical studies of coronal waves using both EUVI (Muhr et al. 2014) and AIA observations (Long et al. 2017a). In particular, the correlation between deceleration and speed in fast events is now firmly established. Together with other observations that have shown typical wave properties such as reflection (e.g., Long et al. 2008; Veronig et al. 2008), refraction (e.g., Shen et al. 2013), and transmission (e.g., Olmedo et al. 2012) at coronal structures, as well as results from numerical simulations (e.g., Downs et al. 2021), this has led to a growing consensus that at least a significant fraction of large-scale coronal waves are indeed fast-mode MHD waves (cf. Long et al. 2017b). High-cadence and stereoscopic EUV observations (e.g., Patsourakos & Vourlidas 2009; Kienreich et al. 2009; Patsourakos et al. 2010; Ma et al. 2011) have revealed that the waves are launched by erupting flux ropes. EUV waves can also be excited by coronal jets (Shen et al. 2018b) and loop expansions caused by external disturbances (Shen et al. 2017, 2018a). Initially, the aforementioned drivers cause the disturbance, while later the perturbation becomes decoupled from the driver and continues as a freely propagating wave. For more detailed descriptions of the EUV waves and associated phenomena, we refer to the reviews by Vršnak & Cliver (2008), Wills-Davey & Attrill (2009), Gallagher & Long (2011), Patsourakos & Vourlidas (2012), Warmuth (2015).

We now focus on the kinematics of class 1 EUV waves, which show a characteristic decrease in propagation speed during their evolution. As the EUV waves travel over one hemisphere, they travel mainly outside active regions, that is, they move through regions of the quiet Sun, where the magnetic field is predominantly radially directed. For this reason, the fast magnetosonic velocity should be regarded as almost constant in this region. Therefore, EUV waves start with a velocity greater than the local fast magnetosonic speed, but their velocities decrease during their further propagation. This property can be explained in terms of non-linear waves. The velocity of a non-linear wave is typically dependent on its amplitude (Landau & Lifschitz 1987). The EUV wave is initially excited in a small region in the corona; during its evolution, it is distributed over an increasing region, as in a spherical, cylindrical, or circular wave. This distribution leads to a decrease in the amplitude of the wave. As its velocity depends on its amplitude, the velocity becomes smaller during the evolution of the wave in the corona. This behaviour is only seen in the EUV waves of class 1 (see Fig. 8 in Warmuth & Mann 2011).

The aim of this paper is to describe this process in a quantitative manner. In the paper by Warmuth & Mann (2011), Fig. 8 reveals for class 1 EUV waves that the higher initial velocity is related with a higher deceleration. This relationship has yet to be explained theoretically. As an observational basis, we have chosen a sample of seven EUV waves from the large sample treated by Warmuth & Mann (2011). These waves belong to class 1 and were recorded by the EUVI instruments (Howard et al. 2008) on board the twin STEREO spacecraft. Because of the cadence of ∼2.5 min, the resulting data are appropriate for studying the kinematical properties of freely propagating large-amplitude EUV waves (see Sect. 2). The relationship between the initial velocity and the deceleration during the first 300 s is derived for each event of this sample and is compared with the theoretical results in Sect. 4. The corona is a magnetised plasma. Therefore, the EUV waves must be described in terms of non-linear MHD waves. A brief description of simple MHD waves is presented in Sect. 3. In Sect. 4, the theoretically obtained results regarding the kinematics of simple MHD waves are compared with the observations of EUV waves given in Sect. 2. The results of the paper are summarised in the last paragraph of Sect. 4.

2. Data analysis

Seven EUV waves of class 1 (observed with EUVI) were chosen from the sample studied by Warmuth & Mann (2011). They are observed at 171 Å and 195 Å. The high cadence of these instruments allows us to study the spatio-temporal evolution of these waves. The observational data of these events are summarised in Table 1. The event starts at the time ts. The bright rim of the EUV wave is approximated by a cycle, allowing us to determine the centre of the cycle (see Warmuth et al. 2001, 2004a for a detailed description of this method). Then, ds gives the distance of the bright rim from the centre of this cycle at the starting time ts. The event disappears at the time te at a distance de from the centre of the cycle. The EUV wave starts with a velocity vs and its final speed is ve. As vs > ve according to Table 1, the EUV wave is decelerated.

Table 1.

Parameters of the class 1 EUV waves studied in this paper.

Each event provides a series of distances di from the centre of the approximated cycle at subsequent times ti. This series can be approximated using a power-law approach (Warmuth et al. 2001, 2004a) according to

d ( t ) = d 0 · ( t − t 0 ∗ ) δ . $$ \begin{aligned} d(t) = d_{0} \cdot (t-t_{0}^{*})^{\delta }. \end{aligned} $$(1)

The quantities d0, t 0 ∗ $ t_{0}^{*} $, and δ are determined for each event from the measurements and are presented in Table 2. The initial velocities v0 of each event are calculated as follows:

v ( t ) = d d t · d ( t ) = d 0 · δ · ( t − t 0 ∗ ) δ − 1 , $$ \begin{aligned} v(t) = \frac{\mathrm{d}}{\mathrm{d}t} \cdot d(t) = d_{0} \cdot \delta \cdot (t-t_{0}^{*})^{\delta -1} ,\end{aligned} $$(2)

Table 2.

Parameters of the power-law approximation for the individual class 1 EUV waves studied in this paper.

as v0 = v(t = 0). The values resulting from this method are given for each individual event in Table 2. The initial deceleration is found by

a 300 = v ( t = 300 s ) − v ( t = 0 ) 300 s , $$ \begin{aligned} a_{300} = \frac{v(t=300\,\mathrm{s}) - v(t=0)}{300\,\mathrm{s}} ,\end{aligned} $$(3)

which gives the velocity decrease after the first 300 s (5 min) of the event. The values of a300 are presented in Table 2 for each event. The pairs of values (a300; v0) for each event (see Table 2) are inserted as bold dots in Fig. 3.

Averaging over all events, one obtains d0 = 5443 Mm, t 0 ∗ = − 217 $ t_{0}^{*} = -217 $ s, and δ = 0.662 as typical values for EUV waves (see bottom line in Table 2). Figure 1 shows the temporal behaviour of the local distance d(t) from the centre of the cycle, the local velocity v(t), and the local deceleration a(t) of a typical EUV wave according to Eqs. (1)–(3). The deceleration is calculated as

a ( t ) = d 2 d ( t ) d t 2 = d 0 · δ · ( δ − 1 ) · ( t − t 0 ∗ ) δ − 2 . $$ \begin{aligned} a(t) = \frac{\mathrm{d}^{2} d(t)}{\mathrm{d}t^{2}} = d_{0} \cdot \delta \cdot (\delta -1) \cdot (t-t_{0}^{*})^{\delta -2} .\end{aligned} $$(4)

thumbnail Fig. 1.

Temporal behaviour of the distance d from the origin of the EUV wave, and the velocity v(t) and deceleration a(t) of a typical EUV wave with d0 = 5443, t 0 ∗ = − 217 $ t_{0}^{*} = -217 $ s, and δ = 0.662 according to Eqs. (1), (2), and (4).

Figure 1 shows that a typical EUV wave starts at a distance of 192 Mm from the centre of the corresponding cycle with a velocity of 585 km s−1. The final velocity is 275 km s−1 after 30 min (1800 s). In the initial state, the wave is rapidly decelerated. For instance, the velocity decreases from 585 km s−1 to 436 km s−1 during the first 300 s leading to a deceleration of 497 m s−2. These values are inserted as a cross in Fig. 3.

3. Simple MHD waves

The large-scale EUV waves of class 1 according to the classification by Warmuth & Mann (2011) are considered as the manifestation of non-linear, fast magnetosonic waves. Therefore, non-linear MHD wave theory must be employed. This is done in terms of so-called simple waves in Riemann’s sense (Landau & Lifschitz 1987). The approach of simple waves was performed in MHD by Mann (1995). According to this approach, all varying quantities of the wave depend only on the spatial (x) and temporal (t) coordinates. However, there are relations between all varying quantities without any explicit dependence on x or t. In MHD, these varying quantities are the particle number density N, the streaming velocity v with its components vx, vy, and vz, and the magnetic field b with its components bx, by, and bz. The ambient magnetic field is put in the x − z plane and takes an angle ϑ to the x-axis. The relationships between these varying quantities, namely

0 = d v x − V N 2 · d N , $$ \begin{aligned}&0 = \mathrm{d}v_{x} - \frac{V}{N^{2}} \cdot \mathrm{d}N ,\end{aligned} $$(5)

0 = b y d b y + b z d b z + ( c s 0 2 v A 0 2 · N γ − 1 − V 2 ) · d N , $$ \begin{aligned}&0 = b_{y}\mathrm{d}b_{y} + b_{z}\mathrm{d}b_{z} + \left(\frac{c_{\rm s0}^{2}}{v_{\rm A0}^{2}} \cdot N^{\gamma -1} - V^{2} \right) \cdot \mathrm{d}N ,\end{aligned} $$(6)

0 = ( V 2 − b x 2 N ) N ˙ d b y − V 2 · b y · d N , $$ \begin{aligned}&0 = \left(V^{2} - \frac{b_{x}^{2}}{N} \right) \dot{N}\mathrm{d}b_{y} - V^{2} \cdot b_{y} \cdot \mathrm{d}N ,\end{aligned} $$(7)

0 = ( V 2 − b x 2 N ) N ˙ d b z − V 2 · b z · d N , $$ \begin{aligned}&0 = \left(V^{2} - \frac{b_{x}^{2}}{N} \right) \dot{N}\mathrm{d}b_{z} - V^{2} \cdot b_{z} \cdot \mathrm{d}N ,\end{aligned} $$(8)

are then derived from the ideal MHD equations (a detailed derivation of the Eqs. (5)–(8) is found in the paper by Mann 1995). Here, the particle number density N, the velocities, that is, the propagation velocity of the wave V, the components of the streaming velocity vx, vy, and vz, and the components of the magnetic field bx, by, and bz are normalised to the undisturbed particle number density N0, the Alfvén velocity v A , 0 = B 0 / ( 4 π μ ∼ m p N ) 1 / 2 $ v_{\mathrm{A,0}} = B_{0}/(4\pi \tilde \mu m_{\mathrm{p}}N)^{1/2} $ (mp, proton mass), and ambient magnetic field B0. The sound speed cs, 0 is given by c s , 0 = ( γ k B T 0 / μ ∼ m p $ c_{\mathrm{s,0}} = (\gamma k_{\mathrm{B}}T_{0}/\tilde \mu m_{\mathrm{p}} $), where γ is the ratio of specific heats, kB is the Boltzmann’s constant, T0 is the undisturbed temperature, and μ ∼ = 0.6 $ \tilde \mu = 0.6 $ is the mean molecular weight (Priest 1982), respectively. Consequently, one finds bx = cos ϑ. It should be emphasised that the propagation velocity V does not explicitly depend on the spatial and temporal coordinates, but might be a function of N, vx, by, and bz. Equations (5)–(8) represent a homogeneous system of equations; its non-trivial solutions are

V = v A , x $$ \begin{aligned} V = v_{A,x} \end{aligned} $$(9)

and

V = ( v A 2 + c s 2 ) 2 · [ 1 ± 1 − 4 v A , x 2 c s 2 ( v A 2 + c s 2 ) 2 ] , $$ \begin{aligned} V = \sqrt{\frac{(v_{\rm A}^{2}+c_{\rm s}^{2})}{2} \cdot \left[1 \pm \sqrt{1 - \frac{4v_{\mathrm{A},x}^{2}c_{\rm s}^{2}}{(v_{\rm A}^{2}+c_{\rm s}^{2})^{2}}} \right] } ,\end{aligned} $$(10)

with v A , x 2 = b x 2 / N $ v_{\mathrm{A},x}^{2} = b_{x}^{2}/N $, v A , y 2 = b y 2 / N $ v_{\mathrm{A},y}^{2} = b_{y}^{2}/N $, v A , z 2 = b z 2 / N $ v_{\mathrm{A},z}^{2} = b_{z}^{2}/N $, v A 2 = v A , x 2 + v A , y 2 + v z 2 = ( b x 2 + b y 2 + b z 2 ) / N = b 2 / N $ v_{\mathrm{A}}^{2} = v_{\mathrm{A},x}^{2}+v_{\mathrm{A},y}^{2}+v_{z}^{2} = (b_{x}^{2}+b_{y}^{2}+b_{z}^{2})/N = b^{2}/N $, and c s 2 = ( c s 0 2 / v A 0 2 ) N γ − 1 $ c_{\mathrm{s}}^{2} = (c_{\mathrm{s0}}^{2}/v_{\mathrm{A0}}^{2})N^{\gamma -1} $. These equations represent the non-linear generalisation of the intermediate (or Alfvén) (see Eq. (9)), fast (see Eq. (10), plus sign), and slow (see Eq. (10), minus sign) modes in MHD. Because the wave is travelling along the x-axis, there are no disturbances of the y- and z-component of the streaming velocity, that is vy = 0 and vz = 0. Equations (5) and (6) result in a system of non-linear differential equations:

d v x d N = V N $$ \begin{aligned}&\frac{\mathrm{d}v_{x}}{\mathrm{d}N} = \frac{V}{N} \end{aligned} $$(11)

d b d N = V 2 − c s 2 b . $$ \begin{aligned}&\frac{\mathrm{d}b}{\mathrm{d}N} = \frac{V^{2} - c_{\rm s}^{2}}{b} .\end{aligned} $$(12)

The initial conditions are b(N = 1) = 1 and vx(N = 1) = 0.

The EUV waves are travelling outside active regions, where the magnetic field is mostly radial directed, that is, the wave propagates nearly perpendicular to the ambient magnetic field. Thus, the angle θ = 90° has to be chosen, leading to vA, x = 0. Equation (10) then provides V 2 = v A 2 + c s 2 $ V^{2} = v_{\mathrm{A}}^{2}+c_{\mathrm{s}}^{2} $, as the plus sign has to be chosen for the fast magnetosonic mode. Equation (12) then reduces to

d b d N = v A 2 b = b N , $$ \begin{aligned} \frac{\mathrm{d}b}{\mathrm{d}N} = \frac{v_{\rm A}^{2}}{b} = \frac{b}{N} ,\end{aligned} $$(13)

leading to b(N) = N with the initial conditions. Finally, Eq. (10) can be written as

V = N + c s , 0 2 v A , 0 2 · N ( γ − 1 ) . $$ \begin{aligned} V = \sqrt{N + \frac{c_{\rm s,0}^{2}}{v_{\rm A,0}^{2}} \cdot N^{(\gamma -1)}} .\end{aligned} $$(14)

Equations (11) and (14) are numerically solved for vA, 0 = 209 km s−1 and cs, 0 = 179 km s−1 (see Sect. 4 for the choice of these parameters). In the case of simple waves (Landau & Lifschitz 1987), the top of the wave pulse propagates with the velocity W = V + vx. The dependence of W on N is presented in Fig. 2 showing that W is a monotonically increasing function of N.

thumbnail Fig. 2.

Dependence of W(N) on N for vA, 0 = 209 km s−1 and cs, 0 = 179 km s−1 as a solution of Eqs. (11) and (14).

4. Discussion

As described in Sect. 1, the EUV waves of class 1 have a basic property whereby their velocities decrease during their evolution; a high initial velocity is related with a strong deceleration during the initial phase of evolution (see Fig. 8 in Warmuth & Mann 2011). Such a property is typical for non-linear waves, where the propagation velocity of the wave is dependent on its amplitude. An EUV wave starts at a distance d0 from the origin with the velocity vs and propagates as a circular (or cylindrical) or spherical wave over the solar surface. During this evolution, its amplitude, and therefore also its velocity, become smaller, leading to a deceleration of the wave. This is demonstrated for a EUV wave with typical parameters in Fig. 1. At the end of its appearance, it becomes a linear wave travelling with the linear wave speed.

The EUV waves of class 1 travel outside active regions, where the magnetic field of the quiet Sun is mainly radially directed. Therefore, these EUV waves can be regarded as a manifestation of non-linear, fast magnetosonic MHD waves, which propagate nearly perpendicular to the ambient magnetic field. Their final speeds can therefore be identified by the fast magnetosonic speed v fms , 0 = ( v A , 0 2 + c s , 0 2 ) 1 / 2 $ v_{\mathrm{fms,0}} = (v_{\mathrm{A,0}}^{2}+c_{\mathrm{s,0}}^{2})^{1/2} $. The final velocity of a typical EUV wave is 275 km s−1. This value is taken for vfms, 0. It is assumed that the EUV waves travel at a height of ≈5 Mm above the photosphere along the solar surface. According to the one-fold Newkirk (1961) density model,

N e = N 0 · 10 4.32 R ⊙ / r , $$ \begin{aligned} N_{\rm e} = N_{0} \cdot 10^{4.32\,R_{\odot }/r} ,\end{aligned} $$(15)

with N0 = 4.2 × 104 cm−3 (R⊙, radius of the Sun), an electron number density of Ne = 8.18 × 108 cm−3 is expected there. The one-fold Newkirk model describes the radial behaviour of the electron number density very well above quiet equatorial regions (Koutchmy 1994). It corresponds to a barometeric height model with a temperature of 1.4 MK. A sound speed of cs, 0 = 179 km s−1 is obtained for such a temperature. Then, an Alfvén speed of 209 km s−1 is found by v A , 0 = ( v fms , 0 2 − c s , 0 2 ) 1 / 2 $ v_{\mathrm{A,0}} = (v_{\mathrm{fms,0}}^{2}-c_{\mathrm{s,0}}^{2})^{1/2} $. Assuming that the coronal plasma consists of electrons, protons, and double ionized helium, the full particle number density N is related to the electron number density Ne by N = 1.92Ne (Mann et al. 1999), a magnetic field of 3.0 G is found for the quiet Sun at a height of 5 Mm above the photosphere by means of B = v A , 0 · ( 4 π μ ∼ m p 1.92 N e ) 1 / 2 $ B = v_{\mathrm{A,0}} \cdot (4\pi \tilde \mu m_{\mathrm{p}}1.92N_{\mathrm{e}})^{1/2} $ (with μ ∼ = 0.6 $ \tilde \mu = 0.6 $ as the mean molecular weight Priest 1982). This is a typical value for the magnetic field of the quiet Sun at the bottom of the corona (Mann et al. 1999; Klassen et al. 2000).

Waves of the fast magnetosonic mode are accompanied with a density enhancement. As discussed in Sect. 3, the particle number density and the velocity of the wave are strongly related with each other in the case of simple waves. Therefore, the evolution of the density has to be studied in order to investigate the evolution of the wave. The evolution of the density is described in an iterative manner: At time ti, the top of the wave is located at di. There, it has a particle number density Nmax, i. It therefore propagates with the velocity W(Nmax, i)vA, 0. After the time step Δt, the top of the wave travels up to di + 1 = di + W(Nmax, i)vA, 0 ⋅ Δt. During this time step, the density at the top is diminished to Nmax, i + 1 according to

N max , i + 1 = 1 + ( N max , i − 1 ) [ 1 + W ( N max , i ) v A Δ t d i ] α , $$ \begin{aligned} N_{\mathrm{max},i+1} = 1 + \frac{(N_{\mathrm{max},i}-1)}{\left[1 + \frac{W(N_{\mathrm{max},i})v_{\rm A}\Delta t}{d_{i}} \right]^{\alpha }} ,\end{aligned} $$(16)

with α = 1 and 2 for a circular (or cylindrical) and spherical wave, respectively. This iteration is numerically treated with the choice of vA, 0 = 209 km s−1, di = 0 = 192 Mm, and Δt = 100 s. These parameters are typical for EUV waves, as seen in Fig. 1. The procedure is performed in the following way: Initially, a value of Nmax, i = 0 is chosen and the velocity W(Nmax, i = 0) is calculated according to the numerical results leading to Fig. 2. Then, the value of Nmax, i = 3 is iteratively determined according to Eq. (16) and, subsequently, W(Nmax, i = 3) is found, which is the velocity after 300 s of the evolution. Then, the deceleration a300, which the wave experiences within the initial period of 300 s, is found by

a 300 = ( W max , i = 3 − W max , i = 0 ) v A , 0 300 s . $$ \begin{aligned} a_{300} = \frac{(W_{\mathrm{max},i=3}-W_{\mathrm{max},i=0})v_{A,0}}{300\,\mathrm{s}} .\end{aligned} $$(17)

This procedure is done for Nmax, i = 0 = 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, and 6.0. In this way, a relationship between the initial velocity v0 = W(Nmax, i = 0)vA, 0 and a300 can be derived for the case of a circular (or cylindrical) (α = 1) and a spherical (α = 2) wave. The results are drawn in Fig. 3.

thumbnail Fig. 3.

Relationship between the initial velocity v0 and the initial deceleration a300 as obtained by the procedure described in Sect. 4 for α = 1 (dashed line) and α = 2 (full line). The pairs of values (a300, v0) derived from the sample of seven EUV waves (see Table 2) are inserted as bold dots. The cross denotes the pair (a300, v0) obtained for the representative EUV wave described in Sect. 2 (see Fig. 1).

The approach of simple MHD waves can explain that a non-linear wave with a high initial velocity experiences a high initial deceleration. Thus, a large-amplitude EUV wave is decelerated during its evolution due to non-linear effects. In order to compare the theoretical results with observations of EUV waves, the initial velocities and decelerations measured for a sample of seven EUV waves are inserted in Fig. 3. This comparison shows that the evolution of a spherical (α = 2) simple MHD wave agrees well with the observations.

5. Summary

A characteristic feature of initially fast (i.e., > 320 km s−1) large-scale EUV waves propagating in the solar corona is their pronounced deceleration, where the magnitude of deceleration correlates with the speed. The measured kinematics of such waves (Warmuth & Mann 2011) are compared with the results of a non-linear MHD wave model describing the evolution of a so-called simple MHD wave (Mann 1995). We find that a spherical large-amplitude magnetosonic wave reproduces the observed kinematics quite well. This provides further support for the interpretation of coronal EUV waves as fast-mode MHD waves.

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All Tables

Table 1.

Parameters of the class 1 EUV waves studied in this paper.

In the text
Table 2.

Parameters of the power-law approximation for the individual class 1 EUV waves studied in this paper.

In the text

All Figures

thumbnail Fig. 1.

Temporal behaviour of the distance d from the origin of the EUV wave, and the velocity v(t) and deceleration a(t) of a typical EUV wave with d0 = 5443, t 0 ∗ = − 217 $ t_{0}^{*} = -217 $ s, and δ = 0.662 according to Eqs. (1), (2), and (4).
In the text
thumbnail Fig. 2.

Dependence of W(N) on N for vA, 0 = 209 km s−1 and cs, 0 = 179 km s−1 as a solution of Eqs. (11) and (14).
In the text
thumbnail Fig. 3.

Relationship between the initial velocity v0 and the initial deceleration a300 as obtained by the procedure described in Sect. 4 for α = 1 (dashed line) and α = 2 (full line). The pairs of values (a300, v0) derived from the sample of seven EUV waves (see Table 2) are inserted as bold dots. The cross denotes the pair (a300, v0) obtained for the representative EUV wave described in Sect. 2 (see Fig. 1).
In the text

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