Numerical Simulation of Coronal Waves Interacting with Coronal Holes. III. Dependence on Initial Amplitude of the Incoming Wave

We performed 2.5D simulations of fast-mode MHD waves of different initial amplitudes interacting with low density regions by using our newly developed MHD code. This code is based on the so-called Total Variation Diminishing Lax-Friedrichs (TVDLF) method, which is a fully explicit scheme and was first described by Tóth & Odstrčil (1996). We numerically solve the standard MHD equations (see Equations (1)–(5)). By using the TVDLF-method, we achieve second-order accuracy in space and time. This second-order temporal and spatial accuracy is attained by using the Hancock predictor method, which was first described by van Leer (1984). As a limiter function, we apply the so-called Woodward limiter, which guarantees that the method behaves well near discontinuities and that no spurious oscillations are generated (for a detailed description, see van Leer 1977; Tóth & Odstrčil 1996). We use transmissive boundary conditions at the right and left boundaries of the computational box, which is equal to 1.0 both in the x- and y-directions. We perform the simulations using a resolution of 500 × 300.

The following set of equations with standard notations for the variables describes the two-dimensional MHD model we use for our simulation.

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\rm{\nabla }}\cdot (\rho {\boldsymbol{v}})=0,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial (\rho {\boldsymbol{v}})}{\partial t}+{\rm{\nabla }}\cdot (\rho {\boldsymbol{vv}})-{\boldsymbol{J}}\times {\boldsymbol{B}}+{\rm{\nabla }}p=0,\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}-{\rm{\nabla }}\times ({\boldsymbol{v}}\times {\boldsymbol{B}})=0,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial e}{\partial t}+{\rm{\nabla }}\cdot [(e+p){\boldsymbol{v}}]=0,\end{eqnarray} \tag{ 4 }$$

where the plasma energy e is given by

$$\begin{eqnarray}&&e=\displaystyle \frac{p}{\gamma -1}+\displaystyle \frac{\rho | {\boldsymbol{v}}{| }^{2}}{2}+\displaystyle \frac{| {\boldsymbol{B}}{| }^{2}}{2}\end{eqnarray} \tag{ 5 }$$

and γ = 5/3 denotes the adiabatic index.

The initial setup of our simulation consists of four different cases for the initial density amplitude, ρ_IA, of the incoming wave, starting from a density of ρ_IA = 1.3 and increasing by a stepsize of 0.2 to a value of ρ_IA = 1.9. The detailed initial conditions for all parameters are as follows:

$$\begin{eqnarray}\rho (x)=\left\{\begin{array}{cc}{\rm{\Delta }}\rho \cdot {\cos }^{2}(\pi \tfrac{x-{x}_{0}}{{\rm{\Delta }}x})+{\rho }_{0} & 0.05\leqslant x\leqslant 0.15\\ 0.3 & 0.4\leqslant x\leqslant 0.6\\ 1.0 & \mathrm{else}\end{array}\right.\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\bigtriangleup \rho =0.3\vee 0.5\vee 0.7\vee 0.9\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}{v}_{x}(x)=\left\{\begin{array}{lc}2\cdot \sqrt{\tfrac{\rho (x)}{{\rho }_{0}}}-2.0 & 0.05\leqslant x\leqslant 0.15\\ 0 & \mathrm{else}\end{array}\right.\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}{B}_{z}(x)=\left\{\begin{array}{cc}\rho (x) & 0.05\leqslant x\leqslant 0.15\\ 1.0 & \mathrm{else}\end{array}\right.\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{B}_{x}={B}_{y}=0,0\leqslant x\leqslant 1\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{v}_{y}={v}_{z}=0,0\leqslant x\leqslant 1,\end{eqnarray} \tag{ 11 }$$

where ρ₀ = 1.0, x₀ = 0.1, $\bigtriangleup x=0.1$.

In Figure 1, one can see a vertical cut through the 2D initial conditions (shown in Figure 2) for density, ρ, plasma flow velocity in the x-direction, v_x, and z-component of the magnetic field, B_z. Figure 1(a) shows an overlay of four different vertical cuts of the 2D density distribution at y = 0.3 (ρ_IA = 1.3), y = 0.5 (ρ_IA = 1.5), y = 0.7 (ρ_IA = 1.7), and y = 0.9 (ρ_IA = 1.9). Moreover, one can see the density drop from ρ = 1.0 to ρ = 0.3 in the range 0.04 ≤ x ≤ 0.6, which represents the CH in our simulation. The background density is equal to 1.0 everywhere. In the range of 0.05 ≤ x ≤ 0.15, the plasma flow velocity, v_x, and magnetic field component in the z-direction, B_z, are functions of ρ (see Figures 1(b) and (c)).

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In Figures 3(a)–(c), one can see how the primary waves with different initial amplitudes are moving in the positive x-direction toward the left CH boundary. We observe a decrease of the amplitudes and at the same time a steepening of the primary wave and a shock formation in all four cases (see Figures 3(b)–(c)). Moreover, we find that the larger the initial amplitude, the faster the primary wave moves toward the CH and the faster the evolution of the shock front. Such a behavior is consistent with the MHD wave theory (Vršnak & Lulić 2000).

In Figure 3(d), we can see how the wave for the case of ρ_IA = 1.9 (blue) starts traversing through the CH. Figures 3(e) and (f) show how the other three waves propagate through the CH. We find that the larger the initial amplitude, the larger the amplitude value of the traversing wave.

In Figure 3(d), we also start observing a first reflection for the case ρ_IA = 1.9 (blue) at x ≈ 0.38. This first reflective feature is located at the left side of the density depletion in all four cases of different initial amplitude. While this first reflection moves in the negative x-direction, its peak value decreases until it reaches a value where it is difficult to distinguish from the background density (see Figures 3(d)–(f) and Figures 4(a)–(f)). We find that the larger the initial amplitude, the larger the amplitude of the first reflection.

There is not only a reflection at the left CH boundary that is moving in the negative x-direction, but also a reflection inside the CH. Due to the fact that the amplitudes inside the CH are small compared to the surrounding area, especially in the cases of the reflected waves inside the CH, we zoom in on the region 0.04 ≤ x ≤ 0.6 in Figures 5, 6 and 7. Figure 5 shows how the first traversing waves propagate through the CH within t = 0.19729 and t = 0.24824. We find that the larger the initial amplitude, the larger the amplitude of the traversing wave, i.e., the largest amplitude can be observed for the case ρ_IA = 1.9 (blue), whereas the smallest amplitude can be found for the case of ρ_IA = 1.34 (magenta). Furthermore, one can observe that the larger the initial amplitude, the faster the first traversing wave. When this first traversing wave reaches the right CH boundary, one part of the wave leaves the CH while another part gets reflected at the inner CH boundary. This reflected wave (hereafter named the second traversing wave) then moves in the negative x-direction inside the CH (see Figure 6). At the time when this second traversing wave reaches the left CH boundary, again one part of the wave gets reflected at the CH boundary inside the CH and propagates in the positive x-direction (third traversing wave; shown in Figure 7). Another part of the second traversing wave leaves the CH and causes another transmitted wave, which can be seen as an additional bump inside the first transmitted wave (see Figure 8). This second transmitted wave moves together with the first transmitted wave in the positive x-direction until the end of the simulation run at t = 0.5. This can only be seen in the case of ρ_IA = 1.9 (blue) in Figure 8 at x ≈ 0.725.

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Due to the different phase speeds of the primary waves and the waves inside the CH, the traversing waves leave the CH at different times. Therefore, the larger the initial density amplitude, the earlier we observe the transmitted wave propagating outside the CH in the positive x-direction. The first transmitted wave that can be observed is the one with an initial amplitude of ρ_IA = 1.9 (see Figure 4(a)). In Figures 4(a)–(f), one can see that the larger the initial amplitude, ρ_IA, the larger the amplitude of the transmitted wave.

Furthermore, we find that the primary wave is capable of pushing the left CH boundary in the positive x-direction. We observe that the larger the initial amplitude, the stronger the CH boundary is being pushed to the right.

In Figure 3(d), we observe a first stationary feature at the left CH boundary for the case ρ_IA = 1.9 (blue), which appears as a stationary peak at x ≈ 0.4. This peak occurs in all four cases of different initial amplitude (see Figure 3(e) (red peak) and 3(f) (green peak) at x ≈ 0.4). For the cases ρ_IA = 1.9 (blue) and ρ_IA = 1.7 (red), the peaks can clearly be seen. In order to see and compare all the peak values as well as their lifetimes in all four cases, we zoom in on the area 0.3 ≤ x ≤ 0.45 in Figure 9. In this figure, one can see that the larger the initial amplitude, the larger the peak value of the first stationary feature. This stationary peak can be observed first in the case of ρ_IA = 1.9 (blue) at x ≈ 0.41, followed by the peaks in the cases ρ_IA = 1.7 (red), ρ_IA = 1.5 (green), and ρ_IA = 1.3 (magenta). The peak values decrease in time in all four cases and move slightly in the positive x-direction.

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In Figures 4(c)–(e), we find a second stationary feature at the CH boundary. It appears first at x ≈ 0.45 in Figure 4(c) for the case ρ_IA = 1.9 (blue), followed by the cases ρ_IA = 1.7 (red peak at x ≈ 0.44 in Figure 4(d)) and ρ_IA = 1.5 (green peak at x ≈ 0.425 in Figure 4(e)). In order to see the peak also in the case ρ_IA = 1.3 (magenta) and to compare the other peak values and their lifetimes, we zoom in the area 0.35 ≤ x ≤ 0.47 between t = 0.34092 and t = 0.43416 in Figure 10. In this figure one can see that, as in the case of the first stationary feature, the larger the initial amplitude, the larger the peak value of the second stationary feature. In contrary to the first stationary feature, the second stationary feature is slightly shifted in the negative x-direction. The peak values decrease to a value slightly above 1.1 and remain observable while the second reflection moves onward in the negative x-direction.

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In Figure 3(d), we observe how a density depletion in the case ρ_IA = 1.9 (blue) starts evolving at x ≈ 0.4. It is located at the left side of the first stationary feature and moves in the negative x-direction. We find that the larger the initial amplitude, the smaller the minimum value of the density depletion. In Figure 11, we zoom in on the area of the density depletion to study in detail its minimum values and the time of its appearance. One can see that the density depletion occurs first in the case ρ_IA = 1.9 (blue), followed by the cases ρ_IA = 1.7 (red), ρ_IA = 1.5 (green), and ρ_IA = 1.3 (magenta). While moving in the negative x-direction, the minimum value of the density depletion decreases.

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Figure 12 shows the 2D temporal evolution of the density distribution for nine different time steps, starting at t = 0 and ending at the end of the simulation run at t = 0.49981. In Figure 12(a), one can see that the initial amplitude increases linearly from ρ = 1.3 to ρ = 1.9, whereas the CH density has a fixed value of ρ_CH = 0.3 for all different cases of initial amplitude. In Figure 12(b), we can observe that the waves with larger initial amplitude move faster toward the left CH boundary than the ones with smaller amplitudes. Figure 12(c) shows the evolution of a first stationary feature at the left CH boundary for the waves with large initial amplitude. At the same time, the primary waves with smaller initial amplitude are still moving toward the left CH boundary. In Figure 12(d), we find that the waves with larger initial amplitude have already left the CH and propagate as transmitted waves in the positive x-direction (magenta peak), while the waves with small initial amplitude still traverse through the CH. Figures 12(e)–(i) show the evolution and propagation of the density depletion and the reflected wave at the left side of the CH as well as the propagation of the transmitted waves at the right side of the CH in all cases of different initial amplitude, ρ_IA.

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Figure 13 shows the temporal evolution of the density, ρ, plasma flow velocity, v_x, position of the wave crest, Pos_A, phase speed, v_w, and magnetic field component in the z-direction, B_z, for the primary waves in every different case of initial amplitude, ρ_IA. In Figure 13(a), we observe that the amplitude of the density remains approximately constant at their initial values until the time when the shock is formed and the density amplitude of the primary wave starts decreasing (see Vršnak & Lulić 2000), i.e., at t ≈ 0.03 (blue), t ≈ 0.04 (red), and t ≈ 0.055 (green). For the case of ρ_IA = 1.3 (magenta), a decrease of the amplitude of the primary can hardly be observed, as expected for low-amplitude wave (Warmuth 2015). One can see that the larger the initial amplitude, ρ_IA, the stronger the decrease of the primary wave's amplitude, which is consistent with observations (Warmuth & Mann 2011; Muhr et al. 2014; Warmuth 2015). The amplitudes decrease to values of ρ ≈ 1.6 (blue), ρ ≈ 1.5 (red), and ρ ≈ 1.4 (green) until the primary wave starts entering the CH. Due to the fact that the waves with larger initial amplitude enter the CH earlier than those with small initial amplitude, we can see in Figure 13(a) that the tracking of the parameters of the faster waves stops at an earlier time than the one for the slower waves. A similar behavior to the one of the density, ρ, can be observed for the plasma flow velocity, v_x, in Figure 13(b) and the magnetic field component, B_z, in Figure 13(e). Here, the amplitudes decrease from v_x = 0.75, B_z = 1.9 (for ρ_IA = 1.9, blue), v_x = 0.6, B_z = 1.7 (for ρ_IA = 1.7, red), v_x = 0.45, B_z = 1.5 (for ρ_IA = 1.5, green), and v_x = 0.27, B_z = 1.3 (for ρ_IA = 1.3, magenta) to v_x = 0.55, B_z = 1.6 (for ρ_IA = 1.9, blue), v_x = 0.46, B_z = 1.5 (for ρ_IA = 1.7, red), v_x = 0.36, B_z = 1.4 (for ρ_IA = 1.5, green), and v_x = 0.25, B_z = 1.25 (for ρ_IA = 1.3, magenta). Figure 13(c) shows how the primary waves propagate in the positive x-direction. In all four cases of different initial amplitude, ρ_IA, the phase speed decreases slighty (consistent with observations; see Warmuth et al. 2004 and Warmuth 2015) until the waves enter the CH at different times, i.e., the values for the phase speed start at v_w ≈ 2.2 (for ρ_IA = 1.9, blue), v_w ≈ 1.9 (for ρ_IA = 1.7, red), v_w ≈ 1.7 (for ρ_IA = 1.5, green), and v_w ≈ 1.4 (for ρ_IA = 1.3, magenta) and decrease to v_w ≈ 1.5 (for ρ_IA = 1.9, blue), v_w ≈ 1.39 (for ρ_IA = 1.7, red), v_w ≈ 1.2 (for ρ_IA = 1.5, green), and v_w ≈ 1.13 (for ρ_IA = 1.3, magenta).

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The kinematics of the first traversing wave are analyzed in Figure 14. In Figure 14(a), we find that the traversing waves propagate with a low amplitude through the CH. Moreover, one can see that the larger the initial amplitude, ρ_IA, the larger the amplitude inside the CH. When the wave enters the CH, it takes a short time until the amplitude achieves its highest value from where it starts decreasing slightly while propagating through the CH, i.e., ρ ≈ 0.42 (for ρ_IA = 1.9, blue), ρ ≈ 0.4 (for ρ_IA = 1.7, red), ρ ≈ 0.38 (for ρ_IA = 1.5, green), and ρ ≈ 0.36 (for ρ_IA = 1.3, magenta) decrease to ρ ≈ 0.41 (for ρ_IA = 1.9, blue), ρ ≈ 0.39 (for ρ_IA = 1.7, red), ρ ≈ 0.37 (for ρ_IA = 1.5, green), and ρ ≈ 0.357 (for ρ_IA = 1.3, magenta) until the traversing waves leave the CH and gets partly reflected at the right CH boundary at the same time. For the plasma flow velocity, v_w, and the magnetic field component in the z-direction, B_z, we find a similar decreasing behavior (see Figures 14(b) and (e)). In Figure 14(d), we observe that the larger the initial amplitude, ρ_IA, the faster the traversing wave propagates through the CH. In detail, this means that the amplitudes for the phase speed, v_w, start at v_w ≈ 2.38 (for ρ_IA = 1.9, blue), v_w ≈ 2.35 (for ρ_IA = 1.7, red), v_w ≈ 2.33 (for ρ_IA = 1.5, green), and v_w ≈ 2.3 (for ρ_IA = 1.3, magenta), and decrease to v_w ≈ 2.3 (for ρ_IA = 1.9, blue), v_w ≈ 2.24 (for ρ_IA = 1.7, red), v_w ≈ 2.14 (for ρ_IA = 1.5, green), and v_w ≈ 2.05 (for ρ_IA = 1.3, magenta). Figure 14(c) shows how the traversing waves propagate in the positive x-direction in all four cases of different initial amplitude, ρ_IA.

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Figure 15 describes the temporal evolution of the parameters of the transmitted waves. In Figure 15(a), we observe that the larger the initial amplitude, ρ_IA, the larger the amplitude of the transmitted wave. More specifically, the density values decrease from ρ ≈ 1.48 (for ρ_IA = 1.9, blue), ρ ≈ 1.4 (for ρ_IA = 1.7, red), ρ ≈ 1.3 (for ρ_IA = 1.5, green), and ρ ≈ 1.22 (for ρ_IA = 1.3, magenta) to ρ ≈ 1.35 (for ρ_IA = 1.9, blue), ρ ≈ 1.31 (for ρ_IA = 1.7, red), ρ ≈ 1.25 (for ρ_IA = 1.5, green), and ρ ≈ 1.18 (for ρ_IA = 1.3, magenta) at the end of the simulation run at t = 0.5. Furthermore, we observe that the larger the initial amplitude, ρ_IA, the earlier the transmitted wave appears at the right CH boundary and starts propagating in the positive x-direction. Analogous to the density, ρ, the plasma flow velocity, v_x, and magnetic field in the z-direction, B_z, also decrease with time (see Figures 15(b) and (e)). Figure 15(c) shows how the transmitted wave propagates in the positive x-direction in every case of different initial amplitude, ρ_IA. Figure 15(d) describes how the phase speed of the different transmitted waves decreases with time. We find that the larger the initial amplitude, ρ_IA, the faster the transmitted waves propagate in the positive x-direction, i.e., the values for the phase speed, v_w decrease from v_w ≈ 1.39 (for ρ_IA = 1.9, blue), v_w ≈ 1.27 (for ρ_IA = 1.7, red), v_w ≈ 1.2 (for ρ_IA = 1.5, green), and v_w ≈ 1.18 (for ρ_IA = 1.3, magenta) at the time when the transmitted wave starts appearing at the right CH boundary, to v_w ≈ 1.27 (for ρ_IA = 1.9, blue), v_w ≈ 1.27 (for ρ_IA = 1.7, red), v_w ≈ 1.23 (for ρ_IA = 1.5, green), and v_w ≈ 1.05 (for ρ_IA = 1.3, magenta) at the end of the simulation run. The fluctuations at the beginning of Figures 15(b) and (e) result from the resolution in the tracking algorithm for v_x and B_z.

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In Figure 16, we analyze the evolution of the first stationary feature for all different cases of initial density amplitude, ρ_IA. This feature appears first for the case ρ_IA = 1.9 (blue) at t ≈ 0.2, followed by the stationary features in the cases ρ_IA = 1.7 (red), ρ_IA = 1.5 (green), and ρ_IA = 1.3 (magenta). In Figure 16(a), one can see that the larger the initial density amplitude, ρ_IA, the larger the peak value of the first stationary feature, i.e., we observe the largest peak for the case ρ_IA = 1.9 (blue) and the smallest peak value for the case ρ_IA = 1.3 (magenta). The density values decrease from ρ ≈ 1.3 (for ρ_IA = 1.9, blue), ρ ≈ 1.24 (for ρ_IA = 1.7, red), ρ ≈ 1.18 (for ρ_IA = 1.5, green), and ρ ≈ 1.11 (for ρ_IA = 1.3, magenta) to ρ ≈ 1.08 (for ρ_IA = 1.9, blue), ρ ≈ 1.05 (for ρ_IA = 1.7, red), ρ ≈ 1.04 (for ρ_IA = 1.5, green), and ρ ≈ 1.02 (for ρ_IA = 1.3, magenta). Analogous to the temporal evolution of the density, ρ, the amplitudes for the plasma flow velocity, v_x, and the magnetic field component in the z-direction, B_z, decrease with time (see Figures 16(b) and (e)). Figure 16(c) shows that the first stationary feature moves slightly in the positive x-direction. In Figure 16(d), we find that the larger the initial density amplitude, ρ_IA, the faster the first stationary feature gets shifted in the positive x-direction.

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The kinematics of the second stationary feature are presented in Figure 17. The density plot in Figure 17(a) shows a correlation between the initial density amplitude, ρ_IA, and the peak values of the second stationary feature. Similar to the first stationary feature, we observe that the larger the initial amplitude, ρ_IA, the larger the peak values of the density of the second stationary feature (see Figure 17(a)), i.e., ρ ≈ 1.15 (for t ≈ 0.35 and ρ_IA = 1.9), ρ ≈ 1.09 (for t ≈ 0.39 and ρ_IA = 1.7), ρ ≈ 1.08 (for t ≈ 0.42 and ρ_IA = 1.5), and ρ ≈ 1.06 (for t ≈ 0.45 and ρ_IA = 1.3). The amplitudes of B_z are similar to the ones of the density (see Figure 17(e)). Due to the fact that the shift of this feature in the negative x-direction is extremely small (see Figure 17(c)), the plasma flow velocity, v_x, and the phase speed, v_w, are close to zero (see Figures 17(b) and (d)).

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In Figure 18, we present the temporal evolution of the density depletion for all cases of different initial amplitude, ρ_IA. In Figure 18(a), we observe that this feature occurs first for the case ρ_IA = 1.9 (blue) at t ≈ 0.25, followed by the cases ρ_IA = 1.7 (red), ρ_IA = 1.5 (green), and ρ_IA = 1.3 (magenta). We see that the larger the initial amplitude, ρ_IA, the smaller the minimum value of the density depletion, i.e., the final minimum values are ρ ≈ 0.83 (for ρ_IA = 1.9, blue), ρ ≈ 0.86 (for ρ_IA = 1.7, red), ρ ≈ 0.9 (for ρ_IA = 1.5, green), and ρ ≈ 0.93 (for ρ_IA = 1.3, magenta). An analogous behavior to the density evolution can be observed for the plasma flow velocity, v_x, and the magnetic field component in the z-direction, B_z (see Figures 18(b) and (e)). In Figure 18(c), one can see how the density depletion is moving in the negative x-direction. Figure 18(d) shows that the larger the initial density amplitude, ρ_IA, the larger the phase speed at which the depletion is moving in the negative x-direction.

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We assume two extreme cases for the phase speed. For the secondary waves, the largest phase speed can be achieved if we assume a large initial amplitude, ρ_IA, (corresponding to ρ_IA = 1.9) and a small CH density, ρ_CH (corresponding to ρ_CH = 0.1). The smallest phase speed, on the other hand, is a result of combining a small initial amplitude, ρ_IA, (corresponding to ρ_IA = 1.3) with a large CH density, ρ_CH (corresponding ρ_CH = 0.5). In Figure 19, one can see how the density amplitudes for the two extreme cases evolve with time. Figure 20 shows a comparison of the simulated extreme cases with observations of Kienreich et al. (2013) and Olmedo et al. (2012).

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In the upper panel of Figure 20 where we compare our simulation results to the observations in Kienreich et al. (2013) one can see how the primary waves propagate with different phase speeds toward the CH (blue shaded area). The stars (black) represent the primary wave of the observations and the simulation at the same time; we assume that the observational phase speed coincides, first, with the simulation phase speed of extreme case No. 1, and, second, with the simulation phase speed of extreme case No. 2. When the primary wave enters the CH, we observe that the simulated phase speed increases (blue solid lines). Subsequently, the phase speed decreases, when the waves leave the CH and propagate further as transmitted waves (green solid lines). Moreover, we can see that the phase speed of the observed reflected wave (red stars) is smaller than the extreme values of the phase speed of the simulated reflected waves (red solid lines). In order to get the percentage of the phase speed drop in the observations, we compared the mean velocity of the primary and the reflected wave in Kienreich et al. (2013). In our simulations, the mean phase speed of the reflected waves is smaller by approximately 39% relative to the incoming speed in extreme case No. 1 and 28% in extreme case No. 2, whereas in the observations of Kienreich et al. (2013) we find an average difference of about 48% (magenta stars in the upper panel). This deviation of the phase speed can be explained by the constraints we have in the simulation. First, we do not consider refraction in our study, i.e., every primary wave approches the CH boundary exactly perpendicularly. Second, we perform 2.5D simulations, i.e., the wave is not able to escape in the vertical direction when interacting with the CH. Hence, the smaller drop in the phase speed of the simulated reflected waves, compared to the observed ones, is consistent with what we expect due to our simulation constraints.

In the lower panel of Figure 20, we see the same simulated extreme cases for the phase speed as in the upper panel, but this time, compared to the observations of Olmedo et al. (2012). Again, we assume that the initial phase speed in the observations phase speed coincides, first, with the simulation phase speed of extreme case No. 1, and, second, with the simulation phase speed of extreme case No. 2. The phase speed of the reflected waves in these observations decreases by approximately 58% (magenta stars in the lower panel), which is again a larger drop than in the simulations. We find that the phase speed of the observed traversing wave (blue stars) is smaller than the simulated ones; the same is true for the observed phase speed of the transmitted wave (green stars). In the simulations, the phase speed of the traversing waves is about 130% (extreme case No. 1) and 24% (extreme case No. 2) larger than the phase speed of the primary waves. The observational phase speed changes were obtained by comparing the mean velocity of the primary wave with the mean velocity of reflected, traversing, and transmitted waves in Olmedo et al. (2012). Here the phase speed increases only by about 3% (blue stars) when the primary wave enters the CH. In the observations, the phase speed of the transmitted wave is smaller by approximately 64% (green stars) relative to the incoming speed, whereas in the simulations the phase speed of the transmitted wave is smaller by about 26% relative to the incoming speed in extreme case No. 1 and 19% in extreme case No. 2 (green solid line). Again, this deviation can be explained by the constraints we apply in our simulation. We assume a homogeneous magnetic field, which does not reflect the complex magnetic field structure inside an actual CH. Therefore, the faster phase speed of the traversing and transmitted waves in the simulation is compatible with observations.

Similar to the comparison of extreme values for the secondary waves, we assume two different extreme cases for the first stationary feature. In order to achieve the largest peak value for this feature, we combine a large initial density amplitude, ρ_IA = 1.9, with a large CH density, ρ_CH = 0.5. A small peak value for the first stationary feature can be achieved by combining a small initial density amplitude, ρ_IA = 1.3, with a small CH density, ρ_CH = 0.1. In Figure 21, one can see the temporal evolution of these two extreme cases and how the first stationary feature reaches its smallest and its largest values. Figure 22 shows how the peak values of the density for both extreme cases evolve with time. We find that the lifetime of the first stationary feature is between t ≈ 0.08 and t ≈ 0.16 in our simulations, which corresponds to approximately 10 and 20 minutes in realtime if we assume an Alfvén speed of approximately 300 km s⁻¹ and a CH width of about 300 Mm. This value shows good agreement with observations, where authors report about a lifetime of stationary brightenings of about 15 minutes (Delannée & Aulanier 1999). It is also observed that a bright front can lie at the same location even for several hours (Delannée 2000).

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