Chen Wang

This paper concerns the behaviour of reduced models for turbulence–mean-flow interactions, in particular for the modelling of zonostrophic instability, which accounts for the spontaneous generation of zonal flows observed in many planetary and geophysical fluid systems. In modelling, an idealised sea of random waves or turbulent eddies is usually driven by a stochastic body force. However, the stochasticity of the zonal mean flow is often neglected, following an ergodic assumption that the zonal mean is equal to the ensemble mean. In this paper, we examine the stochasticity of the zonal mean flow and its consequences for zonostrophic instability. From large numbers of realisations of an idealised fluid system, we show that the zonal mean flow typically exhibits considerable stochasticity. The actual assumption needed for the theory of zonostrophic instability is a weaker assumption, that the waves and mean flows are statistically decorrelated, and we examine its validity. From the evolution equation for mean flows in stochastic waves, we derive an analytical expression for the weak correlation between waves and mean flow. Results indicate that this correlation becomes stronger and has an impact on growth rates as the instability becomes prominent. We further show that neglecting the correlation will result in an under-estimation of the growth rate, and we derive an improved dispersion relation based on this correlation. Theoretical predictions are verified by numerical simulations.

The nonlinear dynamics of equatorial shallow-water waves is examined both theoretically and numerically. While unidirectional Kelvin waves are non-dispersive and can evolve into shocks, they also excite Poincaré waves which are dispersive. This dispersion significantly impacts the dynamics of the shock. A reduced model has been developed to describe the behaviours of the Kelvin wave and the first Poincaré wave, where the dispersive term appears in a form similar to that in the Ostrovsky equation. The findings indicate that a sharp spike forms in the profile of the first Poincaré wave, resulting in an overshoot of the shock. Theoretical predictions are validated through numerical simulations of the primitive equations.

Zonal flows are mean flows in the east–west direction, which are ubiquitous on planets, and can be formed through ‘zonostrophic instability’: within turbulence or random waves, a weak large-scale zonal flow can grow exponentially to become prominent. In this paper, we study the statistical behaviour of the zonostrophic instability and the effect of magnetic fields. We use a stochastic white noise forcing to drive random waves, and study the growth of a mean flow in this random system. The dispersion relation for the growth rate of the expectation of the mean flow is derived, and properties of the instability are discussed. In the limits of weak and strong magnetic diffusivity, the dispersion relation reduces to manageable expressions, which provide clear insights into the effect of the magnetic field and scaling laws for the threshold of instability. The magnetic field mainly plays a stabilising role and thus impedes the formation of the zonal flow, but under certain conditions it can also have destabilising effects. Numerical simulation of the stochastic flow is performed to confirm the theory. Results indicate that the magnetic field can significantly increase the randomness of the zonal flow. It is found that the zonal flow of an individual realisation may behave very differently from the expectation. For weak magnetic diffusivity and moderate magnetic field strengths, this leads to considerable variation of the outcome, that is whether zonostrophic instability takes place or not in individual realisations.

This paper studies the instability of two-dimensional magnetohydrodynamic systems on a sphere using analytical methods. The underlying flow consists of a zonal differential rotation and a toroidal magnetic field is present. Semicircle rules that prescribe the possible domain of the wave velocity in the complex plane for general flow and field profiles are derived. The paper then sets out an analytical study of the ‘clamshell instability’, which features field lines on the two hemispheres tilting in opposite directions (Cally, Sol. Phys., vol. 199, 2001, pp. 231–249). An asymptotic solution for the instability problem is derived for the limit of weak shear of the zonal flow, via the method of matched asymptotic expansions. It is shown that when the zonal flow is solid body rotation, there exists a neutral mode that tilts the magnetic field lines, referred to as the ‘tilting mode’. A weak shear of the zonal flow excites the critical layer of the tilting mode, which reverses the tilting direction to form the clamshell pattern and induces the instability. The asymptotic solution provides insights into properties of the instability for a range of flow and field profiles. A remarkable feature is that the magnetic field affects the instability only through its local behaviour in the critical layer.

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow, in the critical layer, is found to play a complementary role in the instability.

Baroclinic critical levels arise as singularities in the inviscid linear theory of waves propagating through a stratified, horizontally directed and sheared flow. For a steady wave forcing, disturbances grow secularly over the critical layers surrounding these levels, generating a jet-like defect in the mean flow. We use a matched asymptotic expansion to furnish a reduced model of the nonlinear dynamics of such defects. By solving the linear initial-value problem for small perturbations to the defect, we establish that secondary instabilities appear at later times. Because the defect is time dependent, conventional normal-mode analysis is quantitatively inaccurate, but does successfully predict the occurrence of the secondary instability. The instability has a singular character in that disturbances with the shortest horizontal wavelength grow most vigorously at late times, unless dissipation is included. The instability can be suppressed by weak viscosity; by itself, thermal dissipation delays, but does not arrest instability. Numerical computations with the dissipative reduced model demonstrate that the secondary instability saturates as the defect rolls up into a coherent vortical structure. This structure excites a new wave propagating at a different phase speed, thereby forcing a new set of baroclinic critical levels. The implications for self-replication are discussed.

In this paper, we study the forcing of baroclinic critical levels, which arise in stratified fluids with horizontal shear flow along the surfaces where the phase speed of a wave relative to the mean flow matches a natural internal wave speed. Linear theory predicts the baroclinic critical-layer dynamics is similar to that of a classical critical layer, characterized by the secular growth of flow perturbations over a region of decreasing width. By using matched asymptotic expansions, we construct a nonlinear baroclinic critical layer theory to study how the flow perturbations evolve once they enter the nonlinear regime. A key feature of the theory is that, because the location of the baroclinic critical layer is determined by the streamwise wavenumber, the nonlinear dynamics filters out harmonics and the modification to the mean flow controls the evolution. At late times, we show that the vorticity begins to focus into yet smaller regions whose width decreases exponentially with time, and that the addition of dissipative effects can arrest this focussing to create a drifting coherent structure. Jet-like defects in the mean horizontal velocity are the main outcome of the critical-layer dynamics.

In this paper, we study the forcing of baroclinic critical levels, which arise in stratified fluids with horizontal shear flow along the surfaces where the phase speed of a wave relative to the mean flow matches a natural internal wavespeed. Linear theory predicts the baroclinic critical layer dynamics is similar to that of a classical critical layer, characterized by the secular growth of flow perturbations over a region of decreasing width. By using matched asymptotic expansions, we construct a nonlinear baroclinic critical layer theory to study how the flow perturbation evolves once they enter the nonlinear regime. A key feature of the theory is that, because the location of the baroclinic critical layer is determined by the streamwise wavenumber, the nonlinear dynamics filters out harmonics and the modification to the mean flow controls the evolution. At late times, we show that the vorticity begins to focus into yet smaller regions whose width decreases exponentially with time, and that the addition of dissipative effects can arrest this focussing to create a drifting coherent structure. Jet-like defects in the mean horizontal velocity are the main outcome of the critical-layer dynamics.