الفهرس 
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Abstract
This thesis is devoted to exploring the theory of extended MHD in the framework of Hamiltonian mechanics. The merit of extended MHD lies in its capability of describing of the scale hierarchy of plasma systems, i.e., it can explain the underlying small-scale physics both in the ion skin depth and electron skin depth regimes. from the perspective of Hamiltonian mechanics, the multi-scale effect on three phenomena in plasma physics has been elucidated by deriving fully-nonlinear exact solutions. This thesis consists of six chapters. In Chapter 1, a general introduction about magnetohydrodynamics (MHD) and its role in studying plasma physics is introduced. In Chapter 2: we have presented a brief review of non-canonical Hamiltonian mechanics and some of its features. In Chapter 3, the derivation of the extended MHD model starting from the two-fluid model and its Hamiltonian structure has been presented. A basic theorem on Lie algebras has been unearthed for proving the Jacobi’s identity. This Lie algebra has been used to generate the non-canonical Poisson bracket for the Hall, inertial and Hall MHD systems. Boundary conditions and Casimir invariants have also been investigated. In Chapter 4, we have started by constructing the linear theory of the extended MHD model and studying several limits of the dispersion relation. Based on the results obtained in Chapter 3, we have investigated the nonlinear Alfvén wave in extended MHD. The Casimir invariants of the system served as the root for studying these nonlinear structures. Via constructing equilibrium solutions (so-called Beltrami equilibrium) on Casimir leaves, we have derived nonlinear wave solutions. The dispersion relation is exactly that of the linear theory, while the wave amplitude may be arbitrarily large. In Chapter 5, a complete nonlinear theory for helicon and TG waves has been constructed. We have started with the linear theory for helicon waves. Subsequently, the exact solutions of the double curl Beltrami equation have been derived in the cylindrical geometry. To construct the nonlinear theory we have followed the same approach used in Chapter 4. Detailed discussions of the dispersion relations for different boundary conditions, energy deposition and the partition of the energy waves have been presented. In Chapter 6, extended MHD has been used to derive the kinetic and magnetic spectra by resorting to a Kolmogorov-like hypothesis based on the constant cascading rates of the energy and Casimir invariants of this model. The magnetic and kinetic spectra have been derived in the ideal, Hall, and electron inertia regimes. The resultant spectra have been compared against the observational evidence, and have been shown to be in good agreement. In Chapter 7, the main conclusions arising from the work accomplished in this thesis have been presented.