Author: Valery Nakariakov, Dmitrii Kolotkov and Sihui Zhong at the University of Warwick.
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Introduction
The launching of the TRACE spacecraft ushered in a new era of high-resolution observations, boosting the confident detection of MHD waves and their use for probing unknown physical parameters in the coronal plasma, known as MHD seismology. In particular, this technique is crucial for the estimation of the magnetic field in corona where the Zeeman effect could not be resolved.
Solar coronal loops are known to host several types of waves and oscillations, among which kink oscillations are the most intensively applied in seismology [1]. The kink oscillation period is determined by the time-varying displacement extracted from time-distance maps. Then, the period is used to derive the kink and Aflvén speeds, the magnetic field strength, scale height of loop density, and transverse density structure etc., by assuming that the observed period is interpreted as the period of a standing kink wave.
Kink oscillations appear in two regimes; decaying and decayless. The former is triggered by impulsive energy releases, while the latter appears without a visible driver, and displays ubiquitously in coronal loops. Decayless kink oscillations are a promising tool for routine diagnostics of the quiescent corona. In addition, its undamped nature arouses interest in how the energy is continuously input into the loop system to balance the damping by e.g. resonant absorption. So far, decayless kink oscillations have been interpreted as self-oscillations sustained by a quasi-steady flow moving across the loop footpoints [2], or random motions of loop footpoints [3]. Alternatively, it is described as an apparent periodic brightness because of the Kelvin-Helmholtz instability and resonant absorption.
Typical dynamic processes of the corona are coloured noises manifesting as power-law-like Fourier power spectra of observables. In the context of MHD seismology, it is of interest to understand whether the noise affects the quality of seismological inversions. The self-oscillatory model of decayless kink oscillations could include noises associated with random footpoint and coronal motions which sustain the oscillations. Therefore, it is necessary to know to what extent the kink oscillation period is influenced by the existence of the noise.
In this work, we combine the randomly driven and self-sustained oscillation models of decayless kink oscillations of coronal loops to demonstrate that the oscillation period is almost insensitive to the noisy dynamics around the oscillating loop and its footpoints.
Results
We construct a zero-dimensional model describing the decayless kink oscillations based on a randomly driven Rayleigh oscillator with random coefficients. The model accounts for negative friction responsible for the interaction of the loop with external non-periodic flows, and random motions of footpoints. Both processes could compensate the damping. The governing equation includes multiplicative and additive noises which represent chromospheric and coronal noisy processes. The noises are independent of each other.
In the absence of noise, the governing equation reduces to the standard Rayleigh equation, which has an asymptotically stationary (decayless) oscillatory solution with constant period and amplitude.
In the presence of essentially broadband red noise, the oscillatory patterns are found to be almost monochromatic. For both multiplicative and additive noises of the amplitudes up to 120% of the corresponding model coefficients, the oscillation period is found to depart from the noise-free value by less than two percent (Figure 1).
In the noisy cases, the oscillation amplitude experiences fluctuations around the mean value. Random fluctuations of the coefficients associated with the external flows are found to result in a symmetric modulation of the oscillation pattern, while random footpoint motions lead to antisymmetric modulation. When both multiplicative and additive noises are present, the amplitude modulation is a superposition of the symmetric and antisymmetric modulations, see Figure 2. Such modulations are also qualitatively consistent with the observed behaviour, see Figure 3.
The results obtained justify the seismological diagnostics based on the kink oscillation periods, e.g., the construction of Alfvén speed and magnetic field maps of pre-flaring coronal active regions [4].
Conclusions
In terms of a zero-dimensional model based on a randomly driven Rayleigh oscillator with random coefficients, we investigate the effect of noise on decayless kink oscillations of solar coronal loops. It was found that the period of kink oscillations is practically independent of noises, justifying the coronal seismology techniques based on kink oscillation periods. Moreover, in the presence of noise, kink oscillatory signals experience long-durational amplitude modulations which are consistent with observational findings.
This work has been published in MNRAS. The full version could be found here.