Recent developments in the study of Majorana fermions through braid theory have shown that there exists a set of interchanges that allow for the realization of true quantum computation. Alongside these developments there have been studies of topological superconductivity which show the existence of states that exhibit non-Abelian exchange statistics. Motivated by these developments we study the differences between Abelian and non-Abelian topological phase in the vortex state through the Bogoliubov de-Gennes (BdG) formalism. Due to our interests in low-energy states we first implement computationally efficient algorithms for calculating the mean fields and computing eigenpairs in an arbitrary energy window. We have shown that these algorithms adequately reproduce results obtained from a variety of other techniques and show that these algorithms retain spatial inhomogeneity information. Our results show topological superconductivity and vortex states can coexist; providing a means to realize zero-energy bound states, the number of which corresponds to the topological phase. With the use of our methods we present results contrasting the differences between Abelian and non-Abelian topological phase. Our calculations show that an increase in Zeeman field affects numerous parameters within topological superconductors. It causes the order parameter to become more sensitive to temperature variations in addition to a reduced rate of recovery to the bulk value from a vortex core. The increased field suppresses spin-up local density of states (LDOS) in close proximity to the vortex core for low-energy states. Further, it narrows the spectral gap at the lattice centre. Both energy spectrum and LDOS calculations confirm that trivial topological phase have no zero-energy bound states, Abelian phases have an even number, while non-Abelian phases have an odd number.

Many known extra-solar giant planets lie close to their host stars. Around 60 have their semi-major axes smaller than 0.05 AU. In contrast to planets further out, the vast majority of these close-in planets have low eccentricity orbits. This suggests that their orbits have been circularized likely due to tidal dissipation inside the planets. These exoplanets share with our own Jupiter at least one trait in common: when they are subject to periodic tidal forcing, they behave like a lossy spring, with a tidal ``quality factor'', Q, of order 10^5. This parameter is the ratio between the energy in the tide and the energy dissipated per period. To explain this, a possible solution is resonantly forced internal oscillation. If the frequency of the tidal forcing happens to land on that of an internal eigenmode, this mode can be resonantly excited to a very large amplitude. The damping of such a mode inside the planet may explain the observed Q value. The only normal modes that fall in the frequency range of the tidal forcing (~ few days) are inertial modes, modes restored by the Coriolis force. We present a new numerical technique to solve for inertial modes in a convective, rotating sphere. This technique combines the use of an ellipsoidal coordinate system with a pseudo-spectral method to solve the partial differential equation that governs the inertial oscillations. We show that, this technique produces highly accurate solutions when the density profile is smooth. In particular, the lines of nodes are roughly parallel to the ellipsoidal coordinate axes. In particular, using these accurate solutions, we estimate the resultant tidal dissipation for giant planets, and find that turbulent dissipation of inertial modes in planets with smooth density profiles do not give rise to dissipation as strong as the one observed. We also study inertial modes in density profiles that exhibit discontinuities, as some recent models of Jupiter show. We found that, in this case, our method could not produce convergent solutions for the inertial modes. Additionally, we propose a way to observe inertial modes inside Saturn indirectly, by observing waves in its rings that may be excited by inertial modes inside Saturn.

Many known extra-solar giant planets lie close to their host stars. Around 60 have their semi-major axes smaller than 0.05 AU. In contrast to planets further out, the vast majority of these close-in planets have low eccentricity orbits. This suggests that their orbits have been circularized likely due to tidal dissipation inside the planets. These exoplanets share with our own Jupiter at least one trait in common: when they are subject to periodic tidal forcing, they behave like a lossy spring, with a tidal ``quality factor'', Q, of order 10^5. This parameter is the ratio between the energy in the tide and the energy dissipated per period. To explain this, a possible solution is resonantly forced internal oscillation. If the frequency of the tidal forcing happens to land on that of an internal eigenmode, this mode can be resonantly excited to a very large amplitude. The damping of such a mode inside the planet may explain the observed Q value. The only normal modes that fall in the frequency range of the tidal forcing (~ few days) are inertial modes, modes restored by the Coriolis force. We present a new numerical technique to solve for inertial modes in a convective, rotating sphere. This technique combines the use of an ellipsoidal coordinate system with a pseudo-spectral method to solve the partial differential equation that governs the inertial oscillations. We show that, this technique produces highly accurate solutions when the density profile is smooth. In particular, the lines of nodes are roughly parallel to the ellipsoidal coordinate axes. In particular, using these accurate solutions, we estimate the resultant tidal dissipation for giant planets, and find that turbulent dissipation of inertial modes in planets with smooth density profiles do not give rise to dissipation as strong as the one observed. We also study inertial modes in density profiles that exhibit discontinuities, as some recent models of Jupiter show. We found that, in this case, our method could not produce convergent solutions for the inertial modes. Additionally, we propose a way to observe inertial modes inside Saturn indirectly, by observing waves in its rings that may be excited by inertial modes inside Saturn.

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