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This chapter aims to apply the results of earlier chapters to solar observations, considering both historical cases and recently obtained ground- or space-based observations of the Sun’s atmosphere. Coronal loops, prominences and sunspots are used to illustrate the various theoretical results. Attention to historical contributions is also part of the treatment. The founding of coronal seismology is explored and some results are applied to coronal loops. Results for resonant absorption theory are illustrated. Prominences are also explored from the viewpoint of oscillation theory, illustrating some results of prominence seismology. Finally, sunspots are discussed in the context of slow mode propagation.
The effect of gravity is investigated in this chapter and the importance of the Klein-Gordon equation is demonstrated. The Klein-Gordon equation is solved for impulsive initial conditions and the phenomenon of an oscillating wake demonstrated. Cutoff frequency is determined. Waves in a stratified incompressible medium with a horizontal magnetic field are examined, leading to the Rayleigh-Taylor dispersion relation. The compressible case is related to the topic of magnetic helioseismology. Waves in a vertical magnetic field are also discussed. For this case, the slow mode dispersion relation is obtained and exhibits a cutoff frequency.
Connection formulas for a magnetic flux tube that describe the approximate behaviour of the perturbations across thin layers where dissipative processes (here electrical conductivity) act are derived for the Alfven singularity. The tube may be twisted or untwisted. In an appropriate limit these formulas reduce to jumps across a narrow region. Such jumps are described in terms of introduced functions $F$ and $G$ and their related functions. Jump relations are used to derive approximate dispersion relations, leading to the determination of resonant absorption decay rates. Decay rates are determined for two specific density profiles, the linear one and the sinusoidal profile. Jump conditions pertaining to the slow mode are also discussed. The equivalent jump relations holding for Cartesian geometry are obtained and illustrated for a single magnetic interface, obtaining decay rates.
The modes of oscillation of a magnetic flux tube are explored, working from the fundamental differential equations obtained in Chapter 3. Sausage modes and kink modes (as in a magnetic slab) are investigated and their dispersion relations understood. Fluting modes also occur. Dispersion relations and diagrams, each similar to those arising in a slab, are derived and displayed, for both photospheric and coronal conditions. Leaky waves are explored. Resonant absorption in a flux tube is examined, with the decay rate obtained for a $\beta = 0$ tube. Two profiles of density across a thin layer on the boundary of the tube are explored, the linear profile and the sinusoidal profile, with decay rates obtained for both.
The differential equations in Cartesian geometry are solved for the magnetoacoustic waves in a magnetic slab. The case of a field-free environment is also investigated as is the $\beta = 0$ plasma. Sausage and kink waves arise and their properties are described. The notion of surface waves and body waves is introduced. Dispersion diagrams are displayed under two sets of conditions, the photospheric medium and the coronal medium. Impulsive waves are examined. Also, waves in smoothly varying profiles are explored, especially the Epstein profile. Cutoff frequencies are obtained for a range of profiles.
Surface waves are introduced, and the surface wave dispersion relation derived. Some general properties of this relation are investigated. Surface waves in certain special cases, including when one interface is field-free or when both sides of the interface are $\beta = 0$ plasmas are discussed in detail.
The effect of damping by magnetic diffusivity and viscosity is examined for an Alfven wave in a non-uniform atmosphere, demonstrating the rapidity of damping when phase mixing operates. A cubic law of damping tends to apply, though this may apply only after a transition stage or time. Damping when phase mixing is absent and when it is operative is illustrated for coronal conditions. The various approximations used in the derivation of such results are examined. Damping by a slow wave under the influence of viscosity and thermal conductivity is explored at length. Results are illustrated for coronal conditions. Both temporal and spatial behaviours are investigated.
The thin tube theory for a kink wave in a stratified flux tube is determined and explored in the case when the tube is unstratified. Perturbations are also considered for this case. Using a multiple scales approach, the wave equation is derived for the kink mode of a thin magnetic flux tube in an unstratified atmosphere, demonstrating the importance of the kink speed. The theory is illustrated for standing waves in a uniform loop and also extended to structured loops with non-uniform density along the structure. Two density profiles are considered in detail. Period ratios for standing waves under coronal conditions are explored. The role of a non-uniform magnetic field is explored, and leads to a wave equation with non-uniform kink speed. Dispersive corrections in a uniform tube are examined and compared with earlier results. Gravity effects are also examined.
The thin tube theory for a sausage wave is developed from first principles and shown to lead to the Klein-Gordon equation. The equations that hold when gravity is negligible are explored and the dispersion relation obtained and compared with earlier results. The effects of stratification are explored in detail, with contributing terms to the cutoff frequency explored for various cases ranging from a rigid and straight tube to a diverging elastic tube of the shape expected for a thin flux tube. The cutoff frequency is illustrated for a range of conditions likely to arise in the solar atmosphere. The role of cutoff for coronal loops is explored.
Nonlinear aspects of wave propagation are investigated. Special attention is given to magnetic slabs and tubes, deriving the Benjamin-Ono equation for the slow mode in a slab and the Leibovich-Roberts equation for the slow mode in a tube. Soliton solutions are obtained and illustrated under various solar conditions. The role of Whitham’s equation is explored. Dissipative effects are also added, and shown to lead to the Benjamin-Ono-Burgers equation. Approximate solutions are given and illustrated for solar conditions. The roles of viscous and thermal damping of weakly nonlinear slow waves (sound waves) are also explored, and the effect of gravity is examined. Both standing waves and propagating waves are looked at. Finally, the nonlinear kink mode is presented.
The twisted flux tube is explored. Its dispersion relation is obtained for an incompressible plasma and examined in the special case of a thin tube. The case of a compressible medium is also discussed for small twist. The special case of a twisted annular region is also explored.
Here the fundamental problem of MHD waves in a uniform medium is discussed in detail, principally from the viewpoint of partial differential equations. The tube speed is introduced. Dispersion relations are obtained and their properties determined, as well as the properties of the perturbations. Two special cases are also discussed: the incompressible medium, and the $\beta = 0$ plasma.
Wave propagation in a non-uniform medium is formulated and the basic governing partial differential equations are derived. Two geometries are considered: the Cartesian system and the cylindrical polar system. The fundamental ordinary differential equations governing wave propagation are obtained. Singularities in the system are introduced. The idea of phase mixing is introduced. Again, the special cases of the incompressible medium and a $\beta = 0$ plasma are formulated
This chapter sets the scene for the discussion, presenting the MHD equations and their basic properties before turning to a discussion of the basic ideas of wave propagation. A variety of plasmas are also briefly reviewed with most attention devoted to the solar atmosphere and its observed features. Coronal loops and sunspots are given some attention.
The process of linearization of equations is described. Also, the two fundamental speeds that arise, the sound speed and Alfven speed, are defined and evaluated for illustrative purposes. The concepts of phase speed and group velocity are introduced.