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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.
This volume presents a full mathematical exposition of the growing field of coronal seismology which will prove invaluable for graduate students and researchers alike. Roberts' detailed and original research draws upon the principles of fluid mechanics and electromagnetism, as well as observations from the TRACE and SDO spacecraft and key results in solar wave theory. The unique challenges posed by the extreme conditions of the Sun's atmosphere, which often frustrate attempts to develop a comprehensive theory, are tackled with rigour and precision; complex models of sunspots, coronal loops and prominences are presented, based on a magnetohydrodynamic (MHD) view of the solar atmosphere, and making use of Faraday's concept of magnetic flux tubes to analyse oscillatory phenomena. The rapid rate of progress in coronal seismology makes this essential reading for those hoping to gain a deeper understanding of the field.
Basics concepts of the tides are discussed: the tidal movement of sea level, tidal currents, the tide as a wave phenomenon. A qualitative explanation of tidal generation is given. The connection between tidal dissipation and changes in the length of day and lunar recession is explained. An example of a tide-gauge record serves to illustrate the main semidiurnal signal, the spring-neap cycle, and the diurnal inequality. The chapter is concluded with a discussion on the scope of the book and an overview of the contents of the chapters, followed by a further reading section.
Based on the overview from the previous chapter, the main tidal constituents are understood intuitively when their frequencies are derived. This involves the three species of long-period, diurnal, and semidiurnal tides. The corrections needed for the lunar nodal cycle are discussed. The effect of the main constituents on the tidal signal are illustrated (modulation due to elliptic orbit, diurnal inequality, spring-neap, and related cycles). The principle of the method of harmonic analysis is explained. For the tidal signal as a whole, the notion of the tidal period is discussed, including its variability and long-term mean, as well as the presence of circa-tidal clocks in marine organisms.
In this chapter, expressions are derived for the tide-generating force and the associated tide-generating potential. The Moon and Sun act as the tide-generating bodies. The declination is introduced followed by an alternative expression for the tide-generating potential in terms of terrestrial coordinates, which serves as a starting point for Chapter 4. The Moon and Sun act as tide-generating bodies; their combined effect is qualitatively shown to result in a spring-neap cycle.
The propagation of waves at tidal frequencies is studied analytically for simple configurations involving a wall, a channel, semi-enclosed basins, or a continental slope with adjacent shelf sea. The equations of motions are presented, and are simplified using the linear and hydrostatic approximations. The fundamental wave types (Poincaré and Kelvin waves) are derived. The appearance of amphidromic points is explained. A detailed analysis is provided of the solution of the Taylor problem in the case of perfect reflection. The parameter space is explored for modified Kelvin waves in the presence of a shelf sea. As a special case, the double Kelvin wave is obtained.
This chapter provides a systematic qualitative overview of the periodicities involved in the motions of the Earth and Moon that are relevant for tides. Key features are the ellipticity of the orbits and the declination. The celestial origin of the different years (sidereal, tropical, anomalistic) is explained, and the same for months (sidereal, tropical, anomalistic, synodic) and days (sidereal, solar, lunar). The long-period variations (lunar apsidal precession and lunar nodal cycle) are also explained. The implications of the solar tide-generating force on the Earth–Moon system are outlined (evection and variation). The chapter ends with a convenient list of all the relevant periods.
This chapter focuses on tides in coastal seas and basins, where nonlinear and frictional effects are generally important. The depth-averaged shallow-water constituents are derived (Appendix B). The origin of shallow-water constituents is explained. A simple example is analyzed of tidal flow over a bank to explain the principles behind tide-induced residual circulation. Implications for chaotic stirring are discussed. Co-oscillation and resonance in tidal basins are analyzed for simple configurations, including the effects of frictional and radiation damping. The Helmholtz oscillator is explained.Finally, the focus shifts from depth-averaged currents to the vertical structure (Ekman dynamics, tidal straining, strain-induced periodic stratification in estuaries). The decomposition of tidal currents in phasors (rotary components) is elucidated.