| Summary: | It is well known that instabilities in rotational flows, such as those found on planets or in the solar tachocline, lead to the formation of long-lived zonal jets. Pioneered by the work of Rhines, after whom the fundamental length scale of these jets is named, much work has been put into simulating these formations for various situations. These models are often motivated by applications such as the cloud bands of Jupiter, or the geophysical stratospheric polar night jet. The exploration of a driven flow under rotational effects provides a fascinating subject for investigation. Many aspects of fluid behaviour can be observed; from the interaction of mean flows with small-scale turbulence, to the effects of wave-like motion and the transport of potential vorticity (PV). The gradient of PV produces anisotropic behaviour and an inverse energy cascade forming zonal jets with properties governed by the nonlinearity of the system. Starting on the basis of a simple two-dimensional beta-plane system (incompressible Navier-Stokes) under the effects of a body force, we implement a shearing box coordinate system in order to study the competing effects of shear and rotation. We use this in combination with spectral methods to numerically simulate the flow. Following the work of Moffatt, we use the flux of a passive scalar field to calculate and compare the effective diffusivity of the system over a range of the parameter space. In particular, we investigate the effect shear has of disrupting the formation of beta-plane jets, and the resulting modification to transport. We use quasi-linear analysis to further explore these systems. In doing so, we establish important mechanisms bought about by key parameters. We extend the scope of our investigation to include general mean flows. We show relationships between the mean flow and its feedback on the fluid, particularly regarding the perpetuation of zonal jets. We give important modifications to the flow bought about by frictional forces such as viscosity, and show the inherently complicated effect beta has on the mean flow feedback. We make an extension to the above work by looking at the corresponding magnetohydrodynamic system, investigating the effect of adding a magnetic field to a sheared/rotating flow. We find that the magnetic field disrupts beta-plane jets, creating a resonance-like peak in transport, suppressing it when the field strength is increased. We discuss the three predominant quantities governing the feedback for an MHD flow analytically; the Reynolds stress, Lorentz force and magnetic flux. We find that the magnetic flux allows for interactions between the vorticity gradient and magnetic field which potentially allow for zonal features in the mean field; we observe these in our numerical simulations.
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