Fast Sausage Oscillations in Coronal Loops with Fine Structures

Fast sausage modes (FSMs) in flare loops have long been invoked to account for rapid quasi-periodic pulsations (QPPs) with periods of order seconds in flare lightcurves. However, most theories of FSMs in solar coronal cylinders assume a perfectly axisymmetric equilibrium, an idealized configuration apparently far from reality. In particular, it remains to examine whether FSMs exist in coronal cylinders with fine structures. Working in the framework of ideal magnetohydrodynamics (MHD), we numerically follow the response to an axisymmetric perturbation of a coronal cylinder for which a considerable number of randomly distributed fine structures are superposed on an axisymmetric background. The parameters for the background component are largely motivated by the recent IRIS identification of a candidate FSM in Fe XXI 1354 \AA~observations. We find that the composite cylinder rapidly settles to an oscillatory behavior largely compatible with a canonical trapped FSM. This happens despite that kink-like motions develop in the fine structures. We further synthesize the Fe XXI 1354 \AA~emissions, finding that the transverse Alfv\'en time characterizes the periodicities in the intensity, Doppler shift, and Doppler width signals. Distinct from the case without fine structuring, a non-vanishing Doppler shift is seen even at the apex. We conclude that density-enhanced equilibria need not be strictly axisymmetric to host FSM-like motions in general, and FSMs remain a candidate interpretation for rapid QPPs in solar flares.


INTRODUCTION
Rapid quasi-periodic pulsations (QPPs) with periods ranging from seconds to a couple of tens of seconds are frequently seen in solar flare light curves measured in various passbands (see reviews by Nakariakov & Melnikov 2009;Li et al. 2020;Zimovets et al. 2021). A possible interpretation for these rapid QPPs is associated with fast sausage modes (FSMs) supported in flare loops. The FSMs manifest axisymmetric oscillating property under a classical assumption that the magnetic waveguides are considered as straight axisymmetric cylinders (Edwin & Roberts 1983;Roberts et al. 1984). Divided by cutoff wavenumbers, the FSMs possess two regimes: wave energy is well confined in the trapped regime but is continuously lost to the surroundings when the leaky regime arises. In both regimes, the periods of the FSMs are found to be determined by the transverse Alfvén transit time, which typically evaluates to seconds in the corona. This makes the FSMs a potential candidate to account for the rapid QPPs in solar flares.
A departure from the perfectly axisymmetric straight equilibrium is more reasonable in the structured solar atmosphere. For simplicity, we insist on straight equilibria throughout. A straight cylinder with an elliptical cross-section is a natural consideration that breaks the axisymmetry (e.g., Ruderman 2003; Morton & Ruderman 2011;Guo et al. 2020). Actually, many pores and sunspots are measured to have elliptical cross-sections (e.g., Keys et al. 2018;Aldhafeeri et al. 2021), and FSMs have been proved to be supported in elliptical cylinders both theoretically (Erdélyi & Morton 2009;Aldhafeeri et al. 2021) and observationally (e.g., Keys et al. 2018). Equilibria with more realistic irregular cross-sections have been discussed in Aldhafeeri (2021). Furthermore, recent observations by the Interface Region Imaging Spectrograph (IRIS) have revealed that FSMs are supported by fine-structured flare loops (Tian et al. 2016, hereafter T16). As typical magnetic structures in the solar corona, coronal loops are generally not monolithic in realistic measurements, but consist of fine sub-structures instead (e.g., Aschwanden & Peter 2017;Reale 2014, for a review). The investigations associated with waves and oscillations in loops with fine structures have attracted substantial attention, such as eigenmode analysis in two parallel loops (Luna et al. 2008;Van Doorsselaere et al. 2008;Robertson et al. 2010;Gijsen & Van Doorsselaere 2014) and studies of transverse oscillations in more complex multistranded loop systems with application of the T-matrix theory (Luna et al. 2010(Luna et al. , 2019, which was first introduced to the solar context to study p-mode in sunspots by Bogdan & Fox (1991) and Keppens et al. (1994). In addition, transverse waves or oscillations have been examined from the initial value problem perspective in multistranded loops Magyar & Van Doorsselaere 2016;. However, these studies focus on transverse waves or oscillations. Regarding FSMs, a forward step has been made by considering concentric shells as the radial inhomogeneities of straight magnetic waveguides (Pascoe et al. 2007;Chen et al. 2015).
To our knowledge, there is no study on FSMs in a non-monolithic loop with fine structures so far and the existence of sausage modes in such kind of equilibrium remains unknown. We thus perform a three-dimensional MHD simulation involving FSMs in a composite loop with randomly distributed fine structures inside. This paper is organized as follows. Section 2 details the loop model we considered, including the equilibrium and numerical setup. In Section 3 we present the results and forward modelling. Section 4 summarizes the results, ending with some discussion.

Equilibrium Setup
We consider a loop model as a monolithic cylinder with fine structures randomly spread inside.
Note that this setup is our first attempt towards the non-axisymmetric equilibrium, it would be better to retain the monolithic background for reference. The parameters of the loop are uniform along the vertical direction. In the transverse direction, the density profile is given by where ρ mono (x, y) and ρ FS (x, y) represent density profiles of the monolithic background and the fine structures, respectively. They are prescribed by where with and We consider an electron-proton plasma throughout. The mass density ρ is then connected to the electron number density N through ρ = N m p with m p being the proton mass. We specify the internal loop density ρ i and external loop density ρ e in Equation (2) such that the corresponding N is 5.0 × 10 10 cm −3 and 0.8 × 10 9 cm −3 , respectively. The loop length is fixed at L = 45Mm, and the nominal loop radius is R = 5Mm. The radius of each fine structure is R FS = 0.8Mm. The steepnesses of density profiles of the monolithic background and the fine structures are determined by a parameter of α = 5. [x j , y j ] in Equation (5) represents the position of the center of each fine structure, which is supposed to be random in the monolithic region (|x, y| ≤ R). N FS represents the number of fine structures, we take N FS = 20 in practical runs. Figure 1 shows the initial snapshot of density distribution at z = L/2. In addition, the distribution of temperature follows the same profile of the density of the monolithic background. We take the temperature inside the loop as T i = 10MK and external temperature as T e = 2MK. Furthermore, to maintain magnetostatic pressure balance, the magnetic field has a variation from B i = 50G in the internal region of the monolithic cylinder to B e = 77.3G in the external medium. The resulting internal (external) Alfvén speed is v Ai = 496km s −1 (v Ae = 5965km s −1 ). The physical and geometrical parameters of the monolithic component follow rather closely from the IRIS measurements of the composite flare loop supporting a candidate FSM in T16.

Numerical Setup
Trying to excite FSMs in the composite loop, we adopt an axisymmetric initial velocity perturbation, which is similar to the one used in Chen et al. (2016), where v 0 = 10km s −1 is the amplitude of the initial velocity. σ r = 5.0Mm characterizes the extent of the perturbation in the radial direction. See Figure 1 for the initial velocity field (black arrows) at z = L/2. Note that our initial velocity perturbation is not intended to represent a realistic exciter.
Using this initial perturbation is computationally simple and can readily compare with the results in monolithic loops.
To solve the three-dimensional ideal MHD equations, we use the PLUTO code (Mignone et al. 2007). A piecewise parabolic scheme is employed for spatial reconstruction. The numerical fluxes are computed by the HLLD Riemann solver, and the second-order Runge-Kutta algorithm is used for time marching. A hyperbolic divergence cleaning method is adopted to maintain the divergence-free

Oscillations in the Composite Loop
Inspired by the IRIS observation in T16, we excite axial fundamental sausage modes in the composite loop, meaning that the axial wavenumber is π/L. Recall that the ratio between the loop length and radius is L/R = 9, and the steepness of the transverse structuring of the monolithic loop is α = 5, we can readily find that the normalized cutoff wavenumber in the monolithic background without fine structures is k c R = 0.32, by solving pertinent eigenvalue problem in monolithic loops.
The cutoff wavenumber is smaller than the axial wavenumber, meaning that sausage modes in the monolithic loop would be in the trapped regime. For a reference, we can also estimate the period of the FSM being 16.4s in the monolithic loop.
Now we examine the oscillations in the composite loop after the initial velocity perturbation. . This is a typical signature of standing Alfvén waves (see Guo et al. 2020).
The Alfvén waves can be clearly observed near the fine structures at the loop boundary at y = 0, hinting that these Alfvén waves originate from the oscillations of these fine structures.
The dynamics of fine structures can be clearly revealed by examining the z-component of vorticity (Ω z ) at different instants, as shown in Figure 3. The vorticity is zero at the initial state, showing no vortex of the initial perturbation. We find that the internal regions of fine structures possess an opposite oscillating direction against their boundary, indicating kink-like motions that are rapidly excited by the initial perturbation. Similar properties can also be found in the velocity field of kink modes (e.g., Goossens et al. 2014;Guo et al. 2020). Resonantly converted Alfvén waves are characterized by blue and red ripples around each fine structure (see also the associated animation of Figure 3). Before proceeding, we examine the temporal evolution of v y in Figure 4 resonances can be seen in all cases unless the strands are located at loop center. In addition, a lower resolution run with a set of 420 × 420 × 32 grid cells has been conducted. We find no significant difference in the oscillation curves from those in Figure 2. Even though fewer ripples around the strands resulting from phase mixing are seen, the above analysis is not influenced in the lower resolution run.
In fact, sausage modes characterized by coherent breathing motions have also been studied in the lower atmosphere, despite the irregular shape of the examined structures and the fine structuring therein. In the context of p-mode interaction with sunspots, sausage modes are found in a bundle of symmetric tubes, characterized by breathing motions of the tubes (Keppens et al. 1994). Unlike the FSMs in coronal loops, the sausage motions appear in tightly packed sunspot fibrils and are demonstrated as surface modes.

Observable Properties
Inspired by the fundamental FSMs measured in flare loops in T16, we forward-modelled the numerical results by using the FoMo code (Van Doorsselaere et al. 2016) to obtain their observational signals. We perform the synthetic process by using the Fe XXI 1354Å spectral line, which has been considered by Shi et al. (2019) in their forward modeling effort associated with the above-mentioned IRIS measurements. In our case, however, the coronal loop with substructures therein is a more realistic consideration. The possible influence of substructures on the detection of FSMs in flare loops is worth examining. We note further that the coronal approximation, inherent to FoMo, applies to  can be seen at the loop apex, illustrating a significant difference from the equilibrium without fine structuring. In general, magnetic equilibria need not be perfectly axisymmetric to support FSMs, which remain a candidate interpretation for rapid QPPs in solar flares.
The short periodic signals that appear in the non-axisymmetric substructured loop relax the strict definition of FSMs in coronal loops. In the corona, complex density distribution such as substructures in flare loops is still an observational obstacle for unambiguous identifications and measurements of FSMs. Our current results thus bring more confidence to the identification of FSMs in such kind of complex loop structure and the interpretation of rapid QPPs.
In reality, flare loops are more complex and dynamic than the current model. Chromospheric evaporation has been reported in the impulsive phase in T16. Simulations by Ruan et al. (2019) demonstrated that turbulent interactions may be induced by the chromospheric evaporation in a magnetic arcade, leading to the fast mode associated QPPs in soft X-ray. So both observations and simulations indicate a close relation between the evaporation and QPPs. However, this process cannot be evaluated in our current model. Although the curvature and many realistic processes have not been included, we stress again that the current cylinder aims to achieve a conceptional understanding of the existence of FSMs in the non-axisymmetric structure, rather than a full picture of a flare event. Realistic information has been considered in recent models (e.g., Cheung et al. 2019;Ruan et al. 2020), where X-ray lightcurves that are used for quantifying QPPs are obtained.
Another realistic information in flare loops is gravitational stratification. Including the stratification would induce a density reduction from loop top to bottom. The density scale height, which can be roughly estimated by k B T i /µm p g with k B the Boltzmann constant and µ the mean molecular weight, would be 500 Mm. This value is much larger than the loop height (L/π ∼ 14Mm) when deformed to a semi-circle. So the density would change only 3% from the loop top to bottom in the current model. Once the stratification is considered, the Rayleigh-Taylor dynamics that are believed to be important in prominences (e.g., Keppens et al. 2015;Terradas et al. 2015) is worth noticing at loop apex. In the current model, the largest growth rate of the Rayleigh-Taylor instability can be roughly obtained by γ = gπ/R ∼ 0.01s −1 (Goedbloed et al. 2019, and references therein). This means that the growth time of the Rayleigh-Taylor instability is much larger than the oscillation period.
The origins of QPPs are attributed to various mechanisms. Apart from the aforementioned chromospheric evaporation, repetitive magnetic reconnection can account for long-period QPPs (e.g., Ofman & Sui 2006). MHD oscillation is also a possible candidate to account for these QPPs, as demonstrated in T16. Using the FSM to interpret the the phase-difference between the intensity and Doppler shift seems more reasonable, and this interpretation is supported by forward models (Antolin & Van Doorsselaere 2013).
One caveat of the present model might be the composite loop seems artificial. As a first step towards the non-axisymmetry, a monolithic loop with substructures is worth considering since the well-known results in monolithic loops (e.g., Guo et al. 2016;Shi et al. 2019) can be a reference for the new findings. Besides, it would not be easy to unambiguously exclude the possibility of substructures interspersing a monolithic loop. Fine structures can also be induced by transverse oscillations in a monolithic loop (e.g., Antolin et al. 2014). Although the pre-existing fine structures are different from the transverse wave-induced ones, the similar strand-like structures manifested in forward models indicate that the two scenarios are not easy to distinguish. In addition, fine structures play a role to break the axisymmetry of the monolithic background. Although this can be readily achieved by considering a single strand inside the monolithic loop, a considerable number of strands are needed (e.g., Peter et al. 2013).
The authors acknowledge the funding from the National Natural Science Foundation of China (41974200,11761141002,41904150).

A. OSCILLATION PROFILES IN A CYLINDRICAL COORDINATE
The radially and azimuthally averaged evolutions in velocity and magnetic field along the loop boundary at z = L/2 are shown in Figure 6. The quantity q are calculated by where q represents the radial (azimuthal) velocity v r (v φ ) and the radial (azimuthal) magnetic field B r (B φ ). l is the length along the loop edge at z = L/2. A similar oscillation property as in the main text can be seen in both v r and B r profiles. A trapped FSM with a period of about 15s shows up.
However, the Alfvén signal in v φ and B φ evolutions can hardly be seen since the Alfvén waves appear locally near the fine strands and are naturally hidden by the averaged calculation.

B. ALFVÉN RESONANCE EXAMINATION
The density variation in the substructures inside the loop leads to local Alfvén resonance. Figure 7 shows Another LOS angle of π/4 with respect to the x-axis is examined in Figure 8. The oscillation properties are similar to that in the main text with an LOS along the y-axis. A similar periodicity as in Figure 5 can be seen here. However, the distribution of the intensity in the x-direction is different from the one in the main text. We see the Doppler velocity has a slight drop in Figure 8. The line width maps are similar but slightly larger in Figure 8. All the differences presented here reveal the non-axisymmetry of our loop.
Note that the resolution in Figure 5 and Figure 8 remains the numerical one, which is much higher than that of the real instruments. So the fine structures in Figure 5 and Figure 8 can hardly be observed in reality. Although the strands in our loop are not so thin and can be measured by many instruments that have a resolution of about 1 arcsec, to recognize more structures due to the overlap of different strands, for instance, may need a much higher resolution. A similar discussion about the potential of future instruments can be found in .        A spectral line at z = L/2 with an LOS of π/4 with respect to the x-axis.