It has become clear that the closed-field regions of the solar atmosphere are not static (as was once thought) but that many types of steady and unsteady flows and other dynamical, processes such as flares are continually occurring, in them. This thesis investigates some theoretical aspects of these dynamical phenomena. Steady, one-dimensional flow along a coronal loop is investigated first of all. Such a flow may be driven by a pressure difference between the foot points, and a wide range of shocked and unshocked flows are found. The presence of steady flows removes the symmetry present in most static loop models, and these models are shown to form only one class of a much wider range of dynamic solutions to the equations of motion. Thermal non-equilibrium in hot coronal loops occurs if the pressure in a loop becomes too big. The non-linear evolution of this non-equilibrium state is followed, and the loop is found to cool from of order 10[super]6 K to below 10[super]5 K in a few hours. An upflow is driven, and non-equilibrium is suggested as a means of formation of either cool loop cores or prominences. Thermal non-equilibrium is also discussed as a possible mechanism for the simple-loop flare. It is suggested that a cool equilibrium at a temperature of a few times 10[super]4 K can flare to over. 10[super]7 K if the mechanical heating in the cool loop becomes too large. The evolution is followed and the loop is found to flare to over 10[super]7 K in approximately 5 minutes. Magnetohydrodynamic shock waves have long been regarded as a potentially efficient heating mechanism. Their behaviour is re-examined here, and it is found that certain types of shock can release very large amounts of energy. These results are then applied to the heating of "post"-flare loops, for which temperatures of 10[super]7 K at the loop summit may be obtained. Finally, some solutions of the magnetostatic equation are discussed, and it is pointed out that if the gas pressure is too big then magnetostatic equilibrium will break down. It is suggested that the subsequent evolution may give rise to a surge or other mass ejection.

Ever since the first recorded observation of a solar flare in September 1859, it has been a key question - for physics as a whole and for astrophsics in particular - to ask what mechanism lies behind the sudden, violent release of energy from the sun. It has become increasingly apparent that the complex structure of the solar magnetic field lies at the heart of the answer. The process of magnetic reconnection has, over the years, become the accepted explanation by which magnetic energy can be released on both large and small scales in astrophysical and laboratory plasmas. The results of reconnection can be seen, for instance, in star formation, solar flares and the earth's aurorae; indeed the 1859 flare was followed by exceptional auroral activity. The mechanism of magnetic reconnection was first postulated by Giovanelli (1947) as a way of releasing the magnetic energy stored in the Sun. He, and later Dungey (1953), realised that the behaviour of the plasma in the vicinity of a magnetic neutral or null point, where the field disappears, is quite different from other regions of space. In this thesis the nature of magnetic neutral points and their role in the process of reconnection is investigated. Firstly, a general classification of magnetic neutral points is presented. The chapter includes equilibrium and steady-state solutions for two-dimensional magnetic neutral points. The differences in the field behaviour close to each type of neutral point are explained and criteria for the existence of steady-state solutions and equilibria involving pressure balance are presented. In the last section, a self-similar solution for a collapsed X-point is explored. The X-point necessarily becomes cusp-like in nature if shearing is applied in the ignorable direction. Two reconnection models are considered. The first is an extension of the Priest-Lee model (1990). It incorporates large pressure gradients in the inflow corresponding to the Forbes-Priest Almost-Uniform Model. The investigation includes both analytical and numerical solutions and a study of the separatrix jet. In the numerical study, current spikes are found at the end of the current sheets and a much increased reconnection rate is found analytically in the extreme flux file-up limit. The second reconnection model presented is also based on the Priest-Lee configuration. A uniform field is imposed on the basic structure producing a cusp-point with a non-zero field strength as the neutral point is approached from above. This results in the removal of the singularity in the flow above the separatrix. A non-singular solution is found analytically for a double-cusp. A much larger reconnection rate is found and a numerical solution is presented.

A technique has been developed for studying surface waves on liquids. The measuring device employed Fourier analyzes the surface wave being studied. This property of the technique has been verified by three independent tests. The method developed has been applied to the study of the Rayleigh-Taylor instability of fluid surfaces. The results of this study are in good agreement with theory. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

This study investigates the propagation of a surge wave in a power canal following load rejection or reduction. Dimensionless relationships are derived to predict (a) the initial wave height, (b) the variation of the wave height along the canal and (c) the maximum stage of water depth at the downstream end for straight prismatic canals of rectangular, triangular and trapezoidal cross-sections. The effects of various parameters, such as velocity and depth of initial flow, frictional coefficients, bed slope, cross-section of the canal, distance of wave propagation and initial wave height of the surge are studied. A computer program is developed for the calculations required. It is found that, as a positive surge propagates along the canal, the wave height decreases linearly with distance for a short canal, according to an exponential function for a long canal. An approximate logarithmic relationship is also found between the variation of wave height of a positive surge and canal cross-sectional parameters. The variation of water depth at the downstream end of the canal is not linear with respect to time. An almost linear relationship between the maximum water depth at the downstream end of the canal and the length of the canal is noted. The dimensionless relationships derived in this study may be used to establish design criteria for crest elevations of the banks and walls of power canals to avoid overtopping. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

Faraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. We investigate two-dimensional Faraday waves in a long rectangular container, both theoretically and experimentally. Hysteresis occurs when both finite amplitude solutions and the flat surface solution are available. We derive a nonlinear model of a standing wave, extending the Lagrangian method of Miles (1976). The model is used to investigate hysteresis. It is found necessary to retain cubic damping, cubic forcing and the fifth-order conservative term in order to achieve agreement with experiments. The fifth-order conservative term was omitted from all previous studies of Faraday waves. Stable limit cycles are found to arise from this single-mode equation. We examine the structure of this new solution in detail, both analytically and numerically. We describe local bifurcations using a multiple time scales analysis and global bifurcations using Melnikov's method. The coefficients of linear and cubic damping are derived for a standing wave in a rectangular container by considering energy dissipation in the main body of the fluid (due to potential flow and streaming) and in boundary layers at the sidewalls and at the surface. Surface contamination, due to the presence of a thin viscoelastic surface film, creates a boundary layer at the surface which causes enhanced dissipation comparable to, or greater than, that caused by the boundary layers at the walls of the container. Three-mode interaction equations are used to model intermittency and complex modulations which are found to arise from a sideband instability mechanism similar to that of Eckhaus (1963) and Benjamin & Feir (1967). The role of cubic and fifth-order nonlinear terms on this instability mechanism is examined. Theoretical results are found to compare quite favourably with experimental data.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains xii, 156 p. : ill. (some col.) + MPEG video files. Includes MPEG video files. Includes abstract. Includes bibliographical references.

Several significant wave propagation problems in the fields of nonlinear optics and hydrodynamics are studied in this thesis. In optics, the physical model considered is the two-core optical fiber (TCF), which is an essential component of lightwave technology. In hydrodynamics, the motion of a wave packet on the free surface of water of finite depth allowing modulations from two mutually perpendicular and horizontal directions, governed by the famous Davey-Stewartson (DS) equations, is taken into account. The main contributions of this thesis are: In optics, the effects of the intermodal dispersion (IMD) and the birefringence induced effects, both of which always exist in the TCFs, have been ignored in the previous studies of the modulation instability (MI) of continuous waves (CWs) in the TCFs. In this thesis, a detailed analysis of these effects on the MI spectra has been done. It is found that IMD does not seriously affect the MI spectra of the symmetric/antisymmetric CW states, but can significantly modify the MI spectra of the asymmetric CW states. In exploring the birefringence induced effects, a particular class of asymmetric CW states, which admits analytical solutions and has no counterpart in the single-core fibers, is focused on. It is found that the MI spectra of a birefringent TCF in the normal dispersion regime can be distinctively different from those of a zero-birefringence TCF especially for the circular-birefringence TCF. All the findings of MI analysis can be well verified by the wave propagation dynamics. Another contribution of this thesis is that we find the dramatic pulse distortion and even pulse splitting phenomenon due to IMD in TCFs, which is unexpected in many situations, can be effectively suppressed and even avoided by Kerr nonlinearity, which has never been reported in the literatures in the studies of TCFs. In hydrodynamics, DS equations describe the evolution of weakly nonlinear, weakly dispersive wavepackets with slow spanwise dependences on a fluid of finite depth. Generally, DS equations are divided into two types e, i.e. DSI and DSII equations, depending on the specific fluid configurations (fluid depth, wavelength of the water wave, surface tension etc). Due to the importance of DS equations, many exact solutions have been derived by different nonlinear wave methods over the years in the literature. In this thesis, two new exact doubly periodic wave patterns of DS equations are derived by the use of properties of the theta functions, or equivalently, the Jacobi elliptic functions, and the corresponding solitary waves are also deduced in the long wave limits. The new feature of the two wave patterns found is that they can be applied to both DSI and DSII systems at the same time. / published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

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