From the atmospheres of giant planets to terrestrial oceans, self-organisation of turbulence into zonal jets is a ubiquitous feature of geophysical fluid dynamics. The relationship between jets and turbulence is rich, with many unanswered mathematical questions. The aim of this thesis is to extend the existing literature and understanding of jets using numerical and theoretical approaches.
It’s well known the number of jets that form in fluid dynamical models can spontaneously change. Such transitions can occur with increasing rarity as the model parameters vary. The first interesting problem in this thesis studies the use of a ‘rare event algorithm’ to calculate transition probabilities when it is too expensive to do so by direct numerical simulation. We verify the effectiveness of the algorithm in this context and successfully adapt the algorithm for application to a deterministic two-layer (baroclinic) model.
The second problem focuses on the stochastically forced single-layer (barotropic) model, with the motivating question ‘Can eddy momentum fluxes be parameterised?’. If so, the mean jet profile can be solved directly. The key idea is if the turbulent scale is small compared to the background flow, eddy motions only
‘see’ local properties of the flow. For the problem to be mathematically tractable we adopt a second order statistical truncation of the model (known variously as CE2 or SSST). Within this framework, we simplify an existing expression for eddy momentum flux in terms of the local background shear. The new expression can be readily computed and is shown to agree with the statistical model when the background flow is stable and steady. The expression is tested as a method of parameterising small scale turbulence in various scenarios with great success. An important discovery of the work is the role that emergent secondary (barotropic) instabilities play in the mean momentum balance for jets.