Model fitting of kink waves in the solar atmosphere: Gaussian damping and time-dependence

Abstract

Aims. Observations of the solar atmosphere have shown that magnetohydrodynamic waves are ubiquitous throughout. Improvements in instrumentation and the techniques used for measurement of the waves now enables subtleties of competing theoretical models to be compared with the observed waves behaviour. Some studies have already begun to undertake this process. However, the techniques employed for model comparison have generally been unsuitable and can lead to erroneous conclusions about the best model. The aim here is to introduce some robust statistical techniques for model comparison to the solar waves community, drawing on the experiences from other areas of astrophysics. In the process, we also aim to investigate the physics of coronal loop oscillations.Methods. The methodology exploits least-squares fitting to compare models to observational data. We demonstrate that the residuals between the model and observations contain significant information about the ability for the model to describe the observations, and show how they can be assessed using various statistical tests. In particular we discuss the Kolmogorov-Smirnoff one and two sample tests, as well as the runs test. We also highlight the importance of including any observational trend line in the model-fitting process.Results. To demonstrate the methodology, an observation of an oscillating coronal loop undergoing standing kink motion is used. The model comparison techniques provide evidence that a Gaussian damping profile provides a better description of the observed wave attenuation than the often used exponential profile. This supports previous analysis from Pascoe et al. (2016). Further, we use the model comparison to provide evidence of time-dependent wave properties of a kink oscillation, attributing the behaviour to the thermodynamic evolution of the local plasma.

Coronal loop seismology using damping of standing kink oscillations by mode coupling

“…Numerical simulations performed by Ruderman & Terradas (2013) also demonstrated the Gaussian and exponential regimes. Observational evidence for the Gaussian damping regime was found in TRACE data by De Moortel et al (2002) and , and in SDO data by Pascoe et al (2016c), and supported by the statistical analysis of Morton & Mooroogen (2016). Pascoe et al (2016b) (which we will refer to as Paper I hereafter) produced the first seismological inversions of the transverse density profile using the general damping envelope.…”

“…Also, where data allows, we generally want to model the evolution of the background trend before the oscillation begins. -The period of oscillation is considered to be time-dependent (e.g., the linear variations modelled in Nisticò et al 2013;White et al 2013;Morton & Mooroogen 2016). In this paper we use a polynomial (3rd order) to allow increases and/or decreases during the oscillation and to test the correlation with the background trend, which may represent changes in loop length (when the loop has an appropriate orientation).…”

“…In this way our model differs from that of Morton & Mooroogen (2016) who consider the Gaussian damping regime with a time-dependent period of oscillation but a constant damping time characterised by the constant a 4 in their Eq. (5).…”

“…Changes in the period of oscillation of kink modes have been investigated using wavelet analysis by De Moortel et al (2002) and (these two studies also include the shape of the damping profile as a fitted parameter). White et al (2013) and Morton & Mooroogen (2016) fit loop oscillations using a period of oscillation that varies linearly in time (and all were found to increase). Nisticò et al (2013) related the (linear) increase of the decayless kink oscillation period to the observed expansion of the loop.…”

“…We note there is no theoretical justification for this choice, or the order of the polynomial, and no physical interpretation associated with polynomial coefficients. For example, Aschwanden et al (2002) contains examples of polynomial trends with order 1-6, use a 3rd order polynomial for the background, and Morton & Mooroogen (2016) compare results using 3rd and 4th order polynomials. Figure 4 demonstrates examples of fitting oscillation data with a model using a background trend given by a polynomial function (4th order with constant coefficients; top panels) or our spline procedure described above (bottom panels).…”