The submesoscales of the ocean range from 0.1 km to 10 km with time-scales on the order of hours to days. In this range, inertial, rotational, and stratification effects are all important. Submesoscale fronts with large horizontal density gradients are common in the upper ocean. These fronts are associated with enhanced vertical transport and are hotspots for biological activity. Dynamics excited here dictate the exchange rate of important biogeochemical tracers such as heat or CO₂ between the atmosphere and ocean interior. Submesoscale fronts in particular are susceptible to symmetric instability (SI) — a form of stratified inertial instability which can occur when the potential vorticity is of the opposite sign to the Coriolis parameter. The growing SI modes eventually break down through a secondary shear instability, leading to three-dimensional turbulence and vertically mixing the geostrophic momentum. Once out of thermal wind balance, the front undergoes inertial oscillations which can drive further small-scale turbulence, the details of which strongly depend on the ratio of the horizontal buoyancy gradient to the Coriolis frequency.

Here, we consider the idealised problem of a front with uniform horizontal buoyancy gradient in thermal wind balance and bounded by flat no-stress horizontal surfaces. We study the evolution to equi- libration of this unstable front using a linear stability analysis and three-dimensional nonlinear numerical simulations. We find drastically different behaviour emerging at late times depending on the relative strength of the front. While weak fronts develop frontlets and excite subinertial oscillations, stronger fronts produce bore-like gravity currents that propagate along the horizontal boundaries. Although the instantaneous turbulent dissipation rate is much larger in strong fronts, the turbulence is intermittent and peaks during periods of destratification. We describe the details of these energy pathways as the front evolves towards the final adjusted state in terms of the dimensionless front strength.

The shearing box approximation to the compressible unsteady Euler equations in a shearing, rotating, and stratified flow is investigated. A fundamental review is presented, detailing the development of a 2D finite volume method in a cartesian shearing box representing a circular ac- cretion disc. An overall 2nd order method is constructed using the HLLC approximate Riemann solver, a dimensionally split WAF flux-limited reconstruction scheme, and a symplectic source term integrator to conserve energy. The developed algorithm is validated against a comprehen- sive test suite modelled after that of Toro (1999) at each 1D, 2D, and source term construction step. The solver is finally employed to explore and further validate against a number of popular problems from accretion disc literature, looking both in the plane of the disc (r-φ) as well as at azimuthally invariant solutions in the r-z plane.

The shearing box approximation to the compressible Euler equations modelling shearing, rotating, and stratified flows in a local frame is presented in a generalised form for eccentric discs. A semi-analytic solution describing the nonlinear laminar vertical resonant solutions to this model is found for the special case of an isothermal disc. The overall 2nd order finite volume method constructed and validated in Part I of this series is modified to solve for the oscillatory geometric source terms arising from the eccentric orbit. This modified algorithm based on the well-balanced MUSCL TVD scheme using the HLLC approximate Riemann solver is employed to solve for the 1D horizontally invariant laminar solutions first identified in Ogilvie (2001). The finite volume implementation is consequently validated against the known semi-analytic isothermal solutions for a range of disc eccentricities.

Internal waves have recently been shown to be a crucial component of energy transport and mixing in the oceans. A number of mechanisms are known to facilitate the direct cascade of energy from the large scales of internal waves down to small-scale mixing. The parametric subharmonic instability (PSI) is perhaps the most studied mechanism recently, but is only well established for infinite plane waves in the inviscid limit as well as with limited viscosity (Koudella & Staquet 2006). Naturally occurring internal waves are of course finite width, and often of complex-valued envelope profile (Thomas & Stevenson 2006). It has further been shown experimentally (Bourget et al. 2013), and in limiting cases theoretically (Karimi & Akylas 2014), that finite beam width effects are significant for the ability of PSI to extract and transfer energy to smaller scales. Yet we show here that there is little agreement to the extent of this effect.

Thus we present the results of a series of numerical studies investigating the width ef- fects on the threshold amplitude for PSI, and compare these to two independent analyses (Bourget et al. 2014; Karimi & Akylas 2014). Finally, we present an investigation of the effects of mean flow and circulation on PSI and explain the influence on the evolution of the subharmonic daughter frequencies using a time-frequency analysis.

• Geophysical and Geological Flows
• Stratified Flows
• Internal Waves: Parametric Instability

Late protostellar accretion disks are often idealized as thin, Keplerian, and laminar in nature; however, many disk instabilities are not insensitive to the initial turbulence spec- trum. One such mode of turbulence driving in protostellar disks is by anisotropic core accretion. We build on the results of Gammie (2001), Rice et al. (2005), and Steiman- Cameron et al. (2013), now conducting global core collapse simulations with free ac- cretion. We investigate the effects of heavy anisotropic accretion on fragmentation and the proliferation of gravitational instabilities into a gravito-turbulent state. We use the adaptive mesh refinement (AMR) code, Orion, to perform high-resolution simulations of solar mass star-forming molecular cloud cores located in massive star-forming regions. We include self-gravity, use a baroclinic equation of state, and represent regions exceeding the maximum grid resolution with sink particles, accurately simulating Bondi accretion. The turbulence, fragmentation, and laminarization of the ensuing late protostellar and early protoplanetary disk is studied during periods of high mass-infall rates. We also outline the development of a global baroclinic cooling prescription for these core collapse simulations forming T Tauri protostellar systems. Our model self-consistently treats the viscously heated disk equilibrium temperature and cooling time using global disk proper- ties. These results will be used to initialize a culminating study of baroclinic instabilities in protoplanetary disks.

Keywords

• Accretion Disks
• Solar System: Formation
• Stars: Formation
• Rotating Turbulence