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Effect of Variable Background on an Oscillating Hot Coronal Loop

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Abstract

We investigate the effect of a variable, i.e. time-dependent, background on the standing acoustic (i.e. longitudinal) modes generated in a hot coronal loop. A theoretical model of 1D geometry describing the coronal loop is applied. The background temperature is allowed to change as a function of time and undergoes an exponential decay with characteristic cooling times typical for coronal loops. The magnetic field is assumed to be uniform. Thermal conduction is assumed to be the dominant mechanism for damping hot coronal oscillations in the presence of a physically unspecified thermodynamic source that maintains the initial equilibrium. The influence of the rapidly cooling background plasma on the behaviour of standing acoustic (longitudinal) waves is investigated analytically. The temporally evolving dispersion relation and wave amplitude are derived by using the Wenzel–Kramers–Brillouin theory. An analytic solution for the time-dependent amplitude that describes the influence of thermal conduction on the standing longitudinal (acoustic) wave is obtained by exploiting the properties of Sturm–Liouville problems. Next, numerical evaluations further illustrate the behaviour of the standing acoustic waves in a system with a variable, time-dependent background. The results are applied to a number of detected loop oscillations. We find a remarkable agreement between the theoretical predictions and the observations. Despite the emergence of the cooling background plasma in the medium, thermal conduction is found to cause a strong damping for the slow standing magneto–acoustic waves in hot coronal loops in general. In addition to this, the increase in the value of thermal conductivity leads to a strong decay in the amplitude of the longitudinal standing slow MHD waves.

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Acknowledgements

The authors would like to thank M.S. Ruderman and R.J. Morton for useful discussions. R.E. acknowledges M. Kéray for patient encouragement. The authors are also grateful to NSF, Hungary (OTKA, Ref. No. K83133), Science and Technology Facilities Council (STFC), UK, and Ministry of Higher Education, Oman for the financial support.

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Correspondence to K. S. Al-Ghafri.

Appendix

Appendix

Eliminating all terms that include \(\tilde{\sigma}\), Equation (45) becomes

$$ \frac{\partial^2Q_1}{\partial z^2}+\frac{\omega^2}{c_\mathrm{s}^2}Q_1=\frac{\mathrm {i}}{\omega c_\mathrm{s}^2} \biggl[ \biggl(\frac{7}{2}\omega^2+3\omega\frac{\mathrm {d}\omega}{ \mathrm {d}t_1} \biggr)Q_0+3\omega^2\frac{\partial Q_0}{\partial t_1}+c_\mathrm{s}^2 \frac{\partial^3Q_0}{\partial t_1\partial z^2}+\frac{5}{2}c_\mathrm{s}^2 \frac{\partial^2Q_0}{\partial z^2} \biggr]. $$
(53)

Since the second term on the right-hand side of Equation (53) can be written as

$$ 3\omega\frac{\mathrm {d}\omega}{ \mathrm {d}t_1}Q_0=(2+1)\omega\frac{\mathrm {d}\omega}{ \mathrm {d}t_1}Q_0= \biggl(-\omega^2+\omega\frac{\mathrm {d}\omega}{ \mathrm {d}t_1}\biggr)Q_0, \quad \mathrm{where}\ \frac{\mathrm {d}\omega}{ \mathrm {d}t_1}=-\frac{\omega}{2}, $$
(54)

and the fourth term, after using Equation (42), can have the form

$$ c_\mathrm{s}^2\frac{\partial^3Q_0}{\partial t_1\partial z^2}=-\omega^2 \frac {\partial Q_0}{\partial t_1}, $$
(55)

substituting Equations (54) and (55), Equation (53) is given as

$$ \frac{\partial^2Q_1}{\partial z^2}+\frac{\omega^2}{c_\mathrm{s}^2}Q_1=\frac{\mathrm {i}}{\omega c_\mathrm{s}^2} \biggl[ \biggl(\frac{5}{2}\omega^2+\omega\frac{\mathrm {d}\omega}{ \mathrm {d}t_1} \biggr)Q_0+2\omega^2\frac{\partial Q_0}{\partial t_1}+ \frac {5}{2}c_\mathrm{s}^2\frac{\partial^2Q_0}{\partial z^2} \biggr]. $$
(56)

Now, the first term and the last term on the right-hand side of Equation (56) can be combined to give

$$ \frac{5}{2}\omega^2Q_0+\frac{5}{2}c_\mathrm{s}^2 \frac{\partial^2Q_0}{\partial z^2}=\frac{5}{2}c_\mathrm{s}^2 \biggl[ \frac{\omega^2}{c_\mathrm{s}^2}Q_0+\frac{\partial^2Q_0}{\partial z^2} \biggr]=0. $$
(57)

Eventually, by employing Equation (57), Equation (56) will reduce to

$$ \frac{\partial^2Q_1}{\partial z^2}+\frac{\omega^2}{c_\mathrm{s}^2}Q_1=\frac{\mathrm {i}}{\omega c_\mathrm{s}^2} \biggl[ \omega\frac{\mathrm {d}\omega}{ \mathrm {d}t_1}Q_0+2\omega^2\frac {\partial Q_0}{\partial t_1} \biggr], $$
(58)

which is exactly Equation (36) after taking ω as a common factor from the bracket. Similarly, we can recover Equation (38) from Equation (47).

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Al-Ghafri, K.S., Erdélyi, R. Effect of Variable Background on an Oscillating Hot Coronal Loop. Sol Phys 283, 413–428 (2013). https://doi.org/10.1007/s11207-013-0225-8

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  • DOI: https://doi.org/10.1007/s11207-013-0225-8

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