Numerical and experimental investigations of instabilities with astrophysical implications

Over the past several decades, researchers have explored a family of mathematical analogies that relate inaccessible astrophysical phenomena to systems that can be manipulated in laboratory settings. Now, a variety of these phenomena have been successfully scrutinised in a laboratory setting, allowing for a deeper investigation of these analogies. This thesis presents results from a series of studies of instabilities with astrophysical implications, undertaken in the context of analogue gravity. From the perspective of analogue gravity, vortices in quantum fluids provide space-times with discrete and topological features. It has long been known that a multiply charged quantum vortex will decay into a cluster of singly charged vortices. Recently, it was pointed out that this instability is similar to that of the rotational superradiance of black holes. This relationship is interpreted further, and through numerical observations, a new phenomenon is encountered in the late stages of the decay. Finally, an upper bound on the orbital frequency of a vortex pair is found, and related to the sound wave responsible for the decay of a doubly charged vortex. On the other hand, the relaxation of compact clusters of quantum vortices involves a complex interplay between vortices and waves, with energy released as sound radiation. By applying techniques from gravitational physics, the study uncovers the emergence of circular sound trajectories, a large-scale feature that enables a straightforward prediction of radiated sound. This phenomenon, called sound-rings, is closely related to the ringdown process of black holes. Furthermore, the linear scaling of the sound-rings with the net charge of the cluster allows them to be located well outside the vortex core. In a completely different system, we investigate cosmological preheating via the parametric instability appearing from applying a vertical oscillation to two-fluid interfaces. Using methods adapted from field theories, a non-linear model is presented and compared with experimental results and numerical simulations. The appearance of secondary instabilities created by the nonlinear contributions to the primary instability is observed and is well predicted by the model. Experimental results suggest that the analogy with cosmological preheating persists in the nonlinear regime. Then, in preparation for the next generation of analogue gravity experiments, a new technique of digital holography for the measurement of deformations of fluid interfaces is introduced. Due to partial reflections of the optical beam, coherent light impinging on a fluid interface returns as a multitude of rays. By applying, or exploiting, a tilt of the bed on which the fluid rests, multiple off-axis holograms are formed and can independently be used to measure the interface. The method is realisable with only a few basic optical components and provides a versatile scheme for high-precision measurement of fluid interfaces.