Dynamics of Coherent Structures in Geophysical and Astrophysical Turbulence

Abstract

Geophysical turbulence frequently self-organizes into coherent structures, ranging from roll-streak structures (RSS) in planetary boundary layers to the planetary scale banded winds of the gas giants. In astrophysical environments such as the solar convection zone and accretion disks, these velocity fields are frequently coupled with large-scale magnetic fields. Despite their ubiquity, the fundamental mechanism by which these structures form and are sustained remains poorly understood. Remarkably, these structures often arise where the underlying base flow is hydrodynamically stable, rendering traditional modal theory inapplicable. Understanding their formation requires a theory that explicitly accounts for the interaction between an evolving coherent state and a background of turbulent fluctuations. This thesis utilizes the Statistical State Dynamics (SSD) formulation of the equations of motion which incorporates and isolates this interaction to provide an analytical understanding of how such structures arise and equilibrate at finite amplitude through meanstate/ background-turbulence interactions. Previous applications of SSD to laboratory wall-bounded shear flows, which are mostly linearly stable, have successfully elucidated the mechanism of RSS formation. In these cases, while the Navier- Stokes (NS) equations remain stable, their SSD reformulation reveals an instability. This ’Reynoldsstress torque instability’ occurs when a perturbation RSS mode systematically organizes the Reynolds stresses of an unstructured background turbulent shear flow. These organized stresses drive roll structures optimally oriented to sustain the companion streak by the lift-up mechanism, thereby destabilizing the RSS. In the first part of this thesis, this approach is extended to Langmuir turbulence by applying the SSD formulation to the Craik–Leibovich (CL) equations, which incorporate interaction of the surface ‐ wave-induced Stokes drift with the underlying Eulerian shear flow in the shear flow dynamics. While the classical CL2 instability that results from this interaction succeeds in predicting the formation of Langmuir circulations, in a turbulent shear flow Langmuir dynamics includes also the mean-state/turbulence interactions. Using a compact SSD formulation that resolves both the CL2 and Reynolds-stress torque mechanisms, this work demonstrates that these two processes act synergistically to drive RSS formation. In the second part of this thesis, RSS dynamics in the turbulent Ekman layer is examined. As in the case of Langmuir structure formation, inflectional instability provides a classical modal route to RSS formation, while SSD-based analysis reveals that Ekman layer roll structure formation involves synergistic interaction between the inflectional and Reynolds-stress torque instability mechanisms. In fully nonlinear statistical equilibrium, the turbulent Ekman roll regime maintains a background vorticity gradient in which inflectional instability plays only a minor role, suggesting that SSD-mediated interactions dominate the maintenance of the RSS at finite amplitude. In the fourth part of this thesis the dynamics the shallow-water MHD (SWMHD) equations on an equatorial β-plane are formulated in SSD form to investigate the dynamics of coupled zonal jet– toroidal field structures (ZJTFS) such as characterize the solar 22-year sunspot cycle. SSD analysis had previously explained the formation of an equatorial velocity zonal jet (ZJ) in a nonconducting fluid. In this work a large-scale dynamo instability that drives a transition from purely zonal jet (ZJ) equilibria to ZJTFS was found to form as an instability as the fluid conductivity increased. Moreover, it was found that the ZJTFS can assume both fixed-point and time-dependent states. The study distinguishes between several large-scale dynamo mechanisms across different parameter regimes. In one regime, the traditional ω-mechanism is sustained by fluctuation–fluctuation advection and tilting/ stretching that maintain a large-scale poloidal field. However, at parameter values representative of the 22-year solar cycle, the dynamo action is found to arise directly from fluctuation–fluctuation advection and tilting/stretching processes.