AN INTERPRETATION OF FLARE-INDUCED AND DECAYLESS CORONAL-LOOP OSCILLATIONS AS INTERFERENCE PATTERNS

Since we are only considering magnetic forces, fast MHD waves lack motion parallel to the field lines and the transverse motion is irrotational. The fluid velocity, u, can be decomposed into two related components,

$$\begin{eqnarray} &&{{{\boldsymbol {u}}}} = v {{{\boldsymbol {\hat{y}}}}} + w {{{\boldsymbol {\hat{\eta }}}}}, \end{eqnarray} \tag{ 2.2 }$$

$$\begin{eqnarray} &&\frac{\partial {v}}{\partial {\eta }} = \frac{\partial {w}}{\partial {y}}, \end{eqnarray} \tag{ 2.3 }$$

where v is the fluid velocity in the binormal direction and w is the component in the normal direction. Further, driven fast waves satisfy the following simple equation,

$$\begin{equation} \left[ \frac{\partial ^{2}{ }}{\partial {t}^{2}} - V_{\rm A}^2 \left(\frac{\partial ^{2}{ }}{\partial {y}^{2}} + \frac{\partial ^{2}{ }}{\partial {s}^{2}} + \frac{\partial ^{2}{ }}{\partial {\eta }^{2}} \right) \right] {{{\boldsymbol {u}}}} = {{{\boldsymbol {S}}}}({{{\boldsymbol {x}}}},t), \end{equation} \tag{ 2.4 }$$

where ${{{\boldsymbol {S}}}}({{{\boldsymbol {x}}}},t)$ is a wave driver, the Alfvén speed V_A is given by $V_{\rm A}^2 = B^2/4\pi \rho$, and ρ is the mass density. In observational contexts, motions in the direction of the principal normal, w, are often referred to as "vertical" oscillations, whereas velocities v in the binormal direction, along the axis of the arcade, are called "horizontal" oscillations. We will explore horizontal oscillations (w = 0) by supposing that the driver only acts in the binormal direction and lacks variation across the arcade's sheet, i.e., the driver is independent of the coordinate η. With these assumptions, only two-dimensional wave modes are excited and they obey the following equation:

$$\begin{equation} \left[ \frac{\partial ^{2}{ }}{\partial {t}^{2}} - V_{\rm A}^2 \left(\frac{\partial ^{2}{ }}{\partial {y}^{2}} + \frac{\partial ^{2}{ }}{\partial {s}^{2}} \right) \right] v = S(s,y,t). \end{equation} \tag{ 2.5 }$$

We make the arcade into a waveguide by imposing boundary conditions at the footpoints. Specifically, we apply the line tieing condition at the photosphere (i.e., v = 0 at s = 0 and s = L). The resonant modes of this waveguide form a discrete spectrum in the tangential s-direction and a continuous spectrum in the transverse y-direction down the axis of the waveguide,

$$\begin{equation} v_n(s,y,t; \kappa) = U_n(s) \, e^{i\kappa y} \, e^{-i \omega _{n}(\kappa) t}, \end{equation} \tag{ 2.6 }$$

$$\begin{equation} U_n(s) \equiv \left(\frac{2}{L}\right)^{1/2} \, \sin (\lambda _n s), \end{equation} \tag{ 2.7 }$$

for positive mode orders n = 1, 2, 3, 4, ⋅⋅⋅. The allowed parallel wavenumbers λ_n and the eigenfrequency ω_n(κ) are given by

$$\begin{equation} \lambda _n = \frac{n \pi }{L}, \end{equation} \tag{ 2.8 }$$

$$\begin{equation} \omega _{n}^2(\kappa) = \left(\lambda _n^2 + \kappa ^2\right) V_{\rm A}^2. \end{equation} \tag{ 2.9 }$$

The wave is a standing wave in the direction parallel to the magnetic field, with discrete wavenumbers λ_n, and a propagating wave in the y direction with continuous wavenumber κ. The temporal frequency ω_n(κ) depends on both wavenumbers and is illustrated in Figure 2. The parallel eigenfunctions, U_n(s), have been normalized such that they form an orthonormal set.

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Our general strategy for solving the driven Equation (2.5) is as follows: we will Fourier transform the equation in the invariant y-direction, decompose the source and solution into the eigenfunctions of the waveguide, solve for the amplitude of each mode in spectral space, and then return to configuration space by inverting the transform. After Fourier transforming the driven wave equation and projecting onto the eigenmodes of the waveguide, we obtain

$$\begin{equation} \left[ \frac{\partial ^{2}{ }}{\partial {t}^{2}} + \omega _{n}^2(\kappa)\right] \hat{v}_{n}(\kappa,t) = \hat{S}_{n}(\kappa,t), \end{equation} \tag{ 2.10 }$$

where

$$\begin{equation} \hat{S}_{n}(\kappa,t) = \int _{-\infty }^\infty dy \int _0^L ds \; S(s,y,t) \, U_{n}(s) \, e^{-i\kappa y}, \end{equation} \tag{ 2.11 }$$

$$\begin{equation} \hat{v}_{n}(\kappa,t) = \int _{-\infty }^\infty dy \int _0^L ds \; v(s,y,t) \, U_{n}(s) \, e^{-i\kappa y}. \end{equation} \tag{ 2.12 }$$

In configuration space, the solution is obtained by inverting the transform and summing over eigenmodes,

$$\begin{equation} v(s,y,t) = \frac{1}{2\pi } \int _{-\infty }^\infty dy \sum _{n=1}^\infty \, \hat{v}_n(\kappa,t) \, U_{n}(s) \, e^{i\kappa y}. \end{equation} \tag{ 2.13 }$$

A similar equation holds for the reconstruction of the source from its spectral decomposition,

$$\begin{equation} S(s,y,t) = \frac{1}{2\pi } \int _{-\infty }^\infty dy \sum _{n=1}^\infty \, \hat{S}_n(\kappa,t) \, U_{n}(s) \, e^{i\kappa y}. \end{equation} \tag{ 2.14 }$$

The background velocity resulting from the broad-band component of the source can be expressed as a superposition of waveguide modes. The amplitude and phase of each mode can be obtained by taking the temporal Fourier transform of Equation (2.10), solving for the velocity in spectral space, and inverting the temporal transform through contour integration (see Appendix A),

$$\begin{eqnarray} v_{\rm bg}(s,y,t) &=& \frac{1}{2\pi } \sum _{n=1}^\infty \int _{-\infty }^\infty d\kappa \; A_n(\kappa) \,U_n(s) \, \nonumber\\ && \times \sin\left[\kappa y - \omega _n(\kappa) t + \theta _n(\kappa)\right]. \end{eqnarray} \tag{ 3.17 }$$

In this equation, A_n(κ) is the mode amplitude of the mode and θ_n(κ) is the phase of the source function in spectral space, evaluated at the mode frequencies,

$$\begin{eqnarray} \hat{S}^{\rm (bg)}_{n}(\kappa,\omega) &=& \int _{-\infty }^\infty dt \int _{-\infty }^\infty dy \int _0^L ds \; S_{\rm bg}(s,y,t) \nonumber\\ & & \times U_{n}(s) \, e^{-i(\kappa y-\omega t)}, \end{eqnarray} \tag{ 3.18 }$$

$$\begin{eqnarray} \hat{S}^{\rm (bg)}_{n}(\kappa,\omega _n) &=& \left|\hat{S}^{\rm (bg)}_{n}(\kappa,\omega _n)\right| e^{i\theta _n(\kappa)}, \end{eqnarray} \tag{ 3.19 }$$

$$\begin{eqnarray} A_n(\kappa) &\equiv & \frac{\left|\hat{S}^{\rm (bg)}_{n}(\kappa,\omega _n)\right|}{\omega _n(\kappa)}. \end{eqnarray} \tag{ 3.20 }$$

The amplitude of the mode A_n(κ) depends on the modulus of the source function evaluated at the mode frequency. The factor of frequency appearing in the denominator of the mode amplitude causes a white source to excite modes such that they all have equal energy. Since the energy in each mode E_n is proportional to both the square of the amplitude A_n(κ) and the square of the frequency ω_n(κ), an amplitude that is inversely proportional to its frequency corresponds to an equipartition of kinetic energy. Even if the source function possesses wavenumber dependence, we still expect the waves with the lowest frequency to dominate the background signal as long as the source is not a rapidly increasing function of wavenumber.

The phase of the response depends on the phase θ_n of the source function and the variation of this phase over all wavenumbers κ comprising the signal. Since the signal has contribution from a range of frequencies around a dominant frequency, we expect the interference between the different frequencies to cause beat patterns that will slowly and randomly rotate the apparent phase of the oscillation in time. Therefore, at a given position along the waveguide, we should expect the resonant background to be dominated by the gravest mode and produce a signal with the following form,

$$\begin{equation} v_{\rm bg} \approx A_{\rm bg} \, \sin (\pi s/L) \, \sin [(\pi V_{\rm A}/L) t + \phi (t)], \end{equation} \tag{ 3.21 }$$

where the phase ϕ(t) slowly changes with time ($|\dot{\phi }_n| \ll \pi V_{\rm A}/L$) due to the continual excitation by the source.

We will see that this expectation holds true by exploring in more detail two types of background sources. In Figure 3(a) we show the source strength (red curve) for a source with a Gaussian dependence on wavenumber,

$$\begin{equation} \left|\hat{S}^{\rm (bg)}_{n}(\kappa,\omega _n)\right| = \tilde{S} \, \exp \left(-\frac{\kappa ^2}{2\Delta ^2}\right) \delta _{n1}. \end{equation} \tag{ 3.22 }$$

In this equation, $\tilde{S}$ is an arbitrary constant and Δ is the spectral width of the Gaussian. For simplicity we have assumed that the source only excites the fundamental mode n = 1. We model a stochastic source by imposing that the phase of the source θ_n(κ) is a random function with a uniform distribution between 0 and 2π. Figure 3(a) also shows the mode amplitude A_n(κ) as the black curve. This type of source generates an amplitude spectrum that is sharply peaked at zero wavenumber with little contribution from the wings. Thus we expect that the corresponding time-series (shown in Figure 3(b)) should be dominated by the frequency ω₁ = πV_A/L, with a phase that slowly wanders. This is indeed the case. This time-series was constructed by numerically evaluating the inverse spatial transform in Equation (3.17).

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For the second source we choose a white source whose strength (by definition) is a constant function of wavenumber. The phase of the source function is once again chosen to be random. Figure 4 presents the amplitude spectrum and the resulting time series. Since the white source contains a broader range of wavenumbers, it generates a time-series with richer frequency response. In particular, we can clearly see from Figure 4(b) that the time series possesses high-frequency jitter. The time-series generated by both sources are highly correlated with very similar low-frequency behavior. This is because the same realization of random phases was used to construct both sources. The equivalency of the set of phases also manifests in the temporal power spectra of the two time series (see Figure 5). The fine structure in the two power spectra is similar because this structure arises from wave interference, which of course is determined by the relative phases (which were chosen to be identical).

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The waves generated by the impulsive source are of course determined by the Green's function. Therefore, consider a single point source of unit amplitude that occurs at time t' and at location (s, y) = (s', y'). We perform a detailed derivation of the Green's function in Appendix B. The general procedure is to Fourier transform the wave equation in the axial direction y, decompose into waveguide modes, and solve for the temporal behavior. The solution is then reconstructed by summing over mode orders and inverting the spatial Fourier transform. After some manipulation of Equation (B9) this inverse transform is expressed as

$$\begin{eqnarray} G(s,s^\prime, y-y^\prime, t-t^\prime) &=& \frac{H(t-t^\prime)}{2\pi } \sum _{n=1}^\infty \int _{-\infty }^\infty d\kappa \; U_n(s) \, U_n(s^\prime) \nonumber\\ && \times \frac{\sin [\omega _n(\kappa) \,(t-t^\prime)]}{\omega _n(\kappa)} \; e^{i\kappa (y-y^\prime)}, \nonumber\\ \end{eqnarray} \tag{ 3.23 }$$

where H is the Heaviside step function. The inverse transform has an analytic solution (Weast et al. 1989) involving zero-order Bessel functions of the first kind, J₀,

$$\begin{equation} G(s,s^\prime, y-y^\prime, t-t^\prime) = \frac{H(\tau)}{2V_{\rm A}} \sum _{n=1}^\infty U_n(s) U_n(s^\prime) \, J_0\left(\lambda _n V_{\rm A}T\right). \end{equation} \tag{ 3.24 }$$

We have written the solution compactly by defining the following delayed times:

$$\begin{eqnarray} \tau &\equiv & (t-t^\prime) - \frac{|y-y^\prime|}{V_{\rm A}}, \end{eqnarray} \tag{ 3.25 }$$

$$\begin{eqnarray} &&\nonumber \\ T &\equiv & \sqrt{(t-t^\prime)^2 - \frac{(y-y^\prime)^2}{V_{\rm A}^2}}. \end{eqnarray} \tag{ 3.26 }$$

As expected, the Green's function is nonzero only after the excitation occurs and after the first wave fronts arrive at the observation point (s, y), i.e., when τ > 0. Further, since both the source and the observation point are within a waveguide, there are many paths that waves can take from the point source at y' to the observation point at y, each reflecting a different number of times from the walls of waveguide (in this case the footpoints). The superposition of waves traveling along all these paths generates the oscillation pattern seen at any given point. This superposition is a combination of waves with different wavenumbers κ and hence directions of initial launch from the source. This summation is represented by the integral in the inverse Fourier transform in Equation (3.23).

The response of the waveguide to the impulsive source appearing in Equation (3.15), S_imp(s)δ(t − t')δ(y − y') is of course the integral of the product of the Green's function and the source S_imp(s') over the point source's location s',

$$\begin{eqnarray} v_{\rm imp}(s,y,t) &=& \frac{H(\tau)}{2V_{\rm A}} \sum _{n=1}^\infty {\cal A}_n \, U_n(s) \, J_0 (\lambda _n V_{\rm A}T) \; \end{eqnarray} \tag{ 3.27 }$$

$$\begin{eqnarray} {\cal A}_n &\equiv & \int _0^L S_{\rm imp}(s^\prime) U_n(s^\prime)\, ds^\prime. \end{eqnarray} \tag{ 3.28 }$$

In Figures 6–8 we show this signal superimposed on the background oscillations. In all cases, the impulsive source occurs at time t = 0 and the waves are observed at the apex of the loop s = L/2. Further, for simplicity we assume that only the gravest mode n = 1 is excited to significant amplitude (i.e., $|{\cal A}_n| \ll |{\cal A}_1|$ for n ≠ 1). Figures 6 and 8 correspond to an impulsive event that occurs only a short distance away from the observation point, Δy = y − y' = 0.1 L, while for Figure 7 the distance between the source and observation point is a one-hundred times larger, Δy = 10 L. Not only is there increased delay between the event and reception of the first signal for the more distant source, but the fringe pattern is compressed near the time of first arrival. Thus, more distance sources generate signals with a wider range of apparent frequencies.

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The decayless oscillations seen by Nisticò et al. (2013) and Anfinogentov et al. (2013) appear to be reproduced with fidelity by considering the effect of a stochastic source that operates throughout the waveguide and continues for long durations. Such a source produces a profusion of waveguide modes with uncorrelated phases. For each longitudinal order n, these modes form a continuous spectrum in the transverse wavenumber κ and therefore frequency ω. Each spectrum has a low-frequency cut-off below which no modes exist (see Figure 5). This cut-off corresponds to modes that propagate parallel to the field lines and hence do not travel up and down the waveguide. As such, these modes are those that are most analogous to those that would be obtained in a one-dimensional model where one assumes that thin bundles of field lines are individually resonant.

For a source with wavenumber dependence that is sufficiently flat near κ = 0, i.e., near the cut-off, we expect that most of the relevant modes saturate such that they have equal energy. Therefore, since the mode energy is proportional to the square of the velocity amplitude and to the square of the frequency, the mode amplitude should be inversely proportional to the mode frequency. This means that the dominant frequency in the spectrum should be the low-frequency cut-off. Thus, in observations we should expect to primarily see the fundamental (n = 1) waveguide mode that propagates nearly parallel to the field lines (κ = 0). This is, of course, exactly what is observed for the large-amplitude flare-induced waves (Aschwanden et al. 1999; Nakariakov et al. 1999), but has yet to be verified for the low-amplitude decayless oscillations.

We further point out that while the spectrum of oscillations is dominated by the mode with the lowest frequency, the wavefield also contains higher frequency components. The relative importance of the high-frequency waves depends on the spectral content of the source. White spectra produce noticeable high-frequency jitter (see Figure 4) whereas a more narrow-band source has a smoother response (see Figure 3). The signal with high-frequency jitter is quite reminiscent of the decayless oscillations presented by Nisticò et al. (2013). Furthermore, the beating and slow modulation of the phase caused by interference between different nearby frequency components is also seen in these observations.

A point source located within the waveguide generates a circular wavefront that initially expands isotropically in two-dimensions across the arcade's magnetic sheet. This isotropic expansion stops, when the wavefront impacts the photosphere and reflection occurs. These reflections then begin to interfere with other portions of the wave front and after many reflections the interference pattern can become rather complicated. Observations made some distance down the waveguide from the point source will see an oscillatory fringe pattern produced by this interference. Thus, the oscillation signal that is seen does not arise from a resonance occurring on the field line where the observation is made. Instead, the oscillation is an interference pattern of many waves as they propagate past the observation point. The initial pulse arises from the segment of the wave front that propagated straight down the waveguide without reflection (i.e., waves with κ ≫ λ_n). At later times, the signal is the interference of segments of the initial wave front that have taken different paths down the waveguide, all with the same path length. As time passes the waves that arrive have undergone more and more reflections and therefore have smaller and smaller wavenumber κ. Asymptotically, for very long times all waves contributing to the signal have κ ≪ λ and thus nearly identical frequencies of ω_n = λ_nV_A. Thus, the signal stabilizes to the same frequency that one would obtain for a one-dimensional cavity.

One important consequence is that the signal at the observation point decays, but it does not do so because of physical damping. We have not included any dissipation mechanisms in our model. Because of this the decay does not have the exponential fall off with time as one would expect from physical damping. Instead, as indicated by the asymptotic form of the J₀ Bessel function, the signal decreases with time like a power law 1/t^1/2. This decay rate (and the fringe pattern itself) is a direct consequence of the shape of the waveguide and the distance from source to observation point. The shape of the waveguide determines the possible paths and therefore the interference. The distance between source and observation point is important because the excited waves have differing phase speeds parallel to the axis of the waveguide,

$$\begin{equation} \frac{\omega }{\kappa } = \left(1 + \frac{\lambda _n^2}{\kappa ^2}\right)^{1/2} V_{\rm A}. \end{equation} \tag{ 4.29 }$$

Therefore, the waves disperse as they travel down the waveguide and the wave packet elongates and changes shape. Thus, the resulting fringe pattern depends on how far the waves have traveled from the source. This effect manifests as the delayed time $T=\sqrt{\vphantom{A^A}\smash{{{\Delta t^2 - \Delta y^2/V_{\rm A}^2}}}}$ that appears in the argument of the Bessel function.

Finally, we comment that not all observations of flare-driven kink waves have a sudden onset followed by rapid day. Some appear to grow initially and only afterward begin to decay (see Nisticò et al. 2013; Wang et al. 2012). Such cases are likely the result of a source with a duration that is comparable to or longer than the period of the waves that are excited. The resulting signal would be the temporal convolution of the source with the Green's function. So, the fringe pattern that would be observed would not only depend on the shape of the waveguide and the distance from the source, but also the duration and temporal variation of the source itself.

The interpretation of coronal-loop oscillations that we suggest here involves the resonances of a two-dimensional arcade instead of a one-dimensional loop. Therefore, this new picture complicates how mode frequencies might be extracted from an observed time series as the time series has a richer high-frequency spectrum of waves that propagate obliquely to the field. Fortunately, the dominant frequencies correspond to the same type of wave that one would derive from a one-dimensional model. Thus, these frequencies can still be used in a seismic analysis as others have previously envisioned.

While none of our figures have included the signal from higher-frequency overtones (n > 1), such modes will certainly be excited. Their exact amplitude depends on the distribution of the driver along the field lines, but in all cases the amplitudes of overtones likely decrease with mode order as high-frequency modes tend to have lower amplitudes even for modes with equal energy. We wish to point out that if one is attempting to measure overtone frequencies from flare-induced oscillations, one must be careful. The response of the fundamental mode of the waveguide to a point source is polychromatic. A wavelet analysis would suggest that the frequency of the oscillation slowly decreases from onset until an asymptotic value is achieved. A distant source in particular may start oscillating with a rather high frequency compared to its eventual asymptotic value (see Figure 7). It is this asymptotic value that corresponds to the one-dimensional resonant frequencies. Of course in many observations a loop oscillation may only be visible for several cycles and the asymptotic regime may never be reached before the flare-induced signal falls below the background oscillations. Due to the polychromatic nature of flare-induced oscillations, the low-amplitude decayless oscillations may be a better frequency diagnostic as the frequency content of the signal is largely steady with time. This property might allow significant averaging of Fourier (or wavelet) power spectra such that the low-frequency cut-offs that should be present for each mode order become visible and measurable.

Finally, we emphasize that that the decay of the flare-induced signal may have nothing to do with physical damping. In our model the decay is a wave interference effect and the resulting fringe pattern is sensitive to the shape of the waveguide. An arcade comprised of loops with a wide variety of lengths should generate a very different fringe pattern and concomitant decay rate than the rectangular waveguide employed here. Further, spatial variation of the Alfvén speed within the waveguide will change the raypaths that combine to form a fringe. Thus with further analysis, the decay rate might prove to be a useful diagnostic of the wavespeed when utilized in tandem with the frequencies.