Nonlinear dynamics of warped astrophysical discs

The classical perspective of astrophysical discs models them as being flat, circular and coplanar. However, a wealth of observational evidence has highlighted the ubiquity of warped geometries in a variety of contexts. This has motivated theoretical efforts to explain the origins of such distortions and understand their ensuing dynamical evolution. Despite marked progress, a complete theory remains elusive, particularly in the observationally relevant regime of large warp amplitudes wherein nonlinear effects and hydrodynamic instabilities significantly modify the warp evolution -- both of which we aim to elucidate in this work. In the first part of this thesis, we construct a novel analytical model which proffers a flexible hydrodynamic framework for studying nonlinear warps. Crucially, this model allows us to investigate inviscid, Keplerian discs for which resonances between the orbital and epicyclic frequencies demand a separate treatment compared with previous theoretical frameworks. We solve the ideal, compressible fluid equations for a non-selfgravitating elliptical cylinder within a local shearing sheet. By restricting attention to flow fields which are a linear function of the coordinates, we capture the lowest order global motions (tilting, shearing and breathing modes) and reduce the dynamics to a set of coupled ordinary differential equations. The connection between tilting tori and warped discs is demonstrated, before the linear modes are confirmed in numerical grid based simulations. In the second part of this thesis we exploit this model to uncover two distinct nonlinear regimes as the warp amplitude is increased. Initially, we find a smooth modulation theory that describes warp evolution in terms of the averaged Lagrangian of the oscillatory vertical motions of the disc. Upon the warp amplitude exceeding a critical value, which scales as the square root of the aspect ratio of our ring, the disc enters into a bouncing regime with extreme vertical compressions twice per orbit. We develop an impulsive theory that predicts special retrograde and prograde precessing warped solutions. Such solutions emphasize the essential activation of nonlinear vertical oscillations within the disc and may have important implications for energy and warp dissipation. In the third part of this thesis, we investigate the growth, saturation and feedback of a parametric instability, which feeds off the free-energy associated with the warped geometry. Using a Lagrangian-Godunov code, we perform simulations of our tilting ring and identify several locally growing modes, as predicted by a three-mode coupling analysis of inertial waves. After finding decent agreement with the theoretical growth rates, we proceed to understand the saturation mechanism as a wave breaking process which suppresses the growth of shorter wavelength couplings first, whilst allowing the longest mode to dominate the final quasi-steady, wave-like turbulence. The associated Reynolds stresses can be effectively modelled using a time-dependent, anisotropic viscous alpha model which closely captures the amplitude and phase evolution of the warp. Finally, in the fourth part of this thesis we consider a simplified model for warps in young protostellar discs, where small dust grains are coupled to the gas. By using a two-fluid approach we find the internal shearing flows associated with the warp and then perform a stability analysis in the terminal velocity approximation. This reveals the suppression of parametric growth by the dusty damping of inertial waves.