In this paper, we provide clear direct evidence of multiple concurrent higher-order magnetohydrodynamic (MHD) modes in circular and elliptical sunspots by applying both proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) techniques on solar observational data. These techniques are well documented and validated in the areas of fluid mechanics, hydraulics, and granular flows but are relatively new to the field of solar physics. While POD identifies modes based on orthogonality in space and provides a clear ranking of modes in terms of their contribution to the variance of the signal, DMD resolves modes that are orthogonal in time. The clear presence of the fundamental slow sausage and kink body modes, as well as higher-order slow sausage and kink body modes, have been identified using POD and DMD analysis of the chromospheric Hα line at 6562.808 Å for both the circular and elliptical sunspots. Additionally, for the various slow body modes, evidence for the presence of the fast surface kink mode was found in the circular sunspot. All of the MHD mode patterns were cross-correlated with their theoretically predicted counterparts, and we demonstrated that ellipticity cannot be neglected when interpreting MHD wave modes. The higher-order MHD wave modes are even more sensitive to irregularities in umbral cross-sectional shapes; hence, this must be taken into account for more accurate modeling of the modes in sunspots and pores.

Although theoretically predicted, the simultaneous excitation of several resonant modes in sunspots has not been observed. Like any harmonic oscillator, a solar magnetic flux tube can support a variety of resonances, which constitute the natural response of the system to external forcing. Apart from a few single low order eigenmodes in small scale magnetic structures, several simultaneous resonant modes were not found in extremely large sunspots. Here we report the detection of the largest-scale coherent oscillations observed in a sunspot, with a spectrum significantly different from the Sun’s global acoustic oscillations, incorporating a superposition of many resonant wave modes. Magnetohydrodynamic numerical modeling agrees with the observations. Our findings not only demonstrate the possible excitation of coherent oscillations over spatial scales as large as 30–40 Mm in extreme magnetic flux regions in the solar atmosphere, but also paves the way for their diagnostic applications in other astrophysical contexts.

The purpose of this paper is to study the behavior of magnetohydrodynamic (MHD) wave modes that propagate in compressible magnetic flux tubes with an elliptical cross section embedded in a magnetic environment. The dispersion relation that describes the behavior of MHD wave modes permitted in an elliptical magnetic flux tube is solved numerically. Distortion of the spatial structure of the purely real eigenmodes from the well-known circular flux tube model has been considered. It has been studied under both photospheric and coronal conditions. It has been shown that (i) solutions in the form of even Mathieu functions are more sensitive to the value of eccentricity than solutions with the form of odd Mathieu functions; (ii) if the ellipticity of the cross section of the magnetic flux tube increases, a sausage mode (m = 0) cannot be easily identified; (iii) even solutions that correspond to the fluting mode (m = 3) can be misinterpreted as a kink mode (m = 1) due to their similarities. In contrast to the fluting modes that are polarized along the major axis and strongly depend on the ellipticity of the magnetic flux tube, the kink and sausage surface modes are practically unaffected by ellipticity. Several examples of the spatial structure of the eigenmodes permitted in the pores and sunspots have been visualized. The solutions obtained in the approximation of cylindrical symmetry are in agreement with previous studies.

In this study, an advanced computational artificial neural network (ANN) procedure is designed using the novel characteristics of the Levenberg–Marquardt backpropagation (LBMBP), i.e., ANN-LBMBP, for solving the waste plastic management in the ocean system that plays an important role in the economy of any country. The nonlinear mathematical form of the waste plastic management in the ocean system is categorized into three groups: waste plastic material W( χ), marine debris M( χ), and reprocess or recycle R( χ). The learning based on the stochastic ANN-LBMBP procedures for solving mathematical waste plastic management in the ocean is used to authenticate the sample statics, testing, certification, and training. Three different statistics for the model are considered as training 70%, while for both validation and testing are 15%. To observe the performances of the mathematical model, a reference dataset using the Adams method is designed. To reduce the mean square error (MSE) values, the numerical performances through the ANN-LBMBP procedures are obtained. The accuracy of the designed ANN-LBMBP procedures is observed using the absolute error. The capability, precision, steadfastness, and aptitude of the ANN-LBMBP procedures are accomplished based on the multiple topographies of the correlation and MSE.