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Fronts with large horizontal density gradients and O(1) Rossby numbers are common in the upper ocean. Such fronts develop through a variety of mechanisms including by mesoscale eddy strains, coastal upwelling, or the input of freshwater via river discharge. These fronts may be unstable to symmetric instability — a form of convective-inertial instability which occurs when the potential vorticity is of opposite sign to the Coriolis param- eter. Symmetric instability is characterised by growing slantwise convection cells aligned with isopycnals and which encourage vertical transport of important biogeochemical tracers in addition to geostrophic momentum. We previously found that this momentum transport destabilises the balanced thermal wind and can prompt geostrophic adjustment which often leaves remnant inertial oscillations (Wienkers et al., 2021a,b).

Here we consider the equilibration of an initially balanced but symmetrically unstable front and explore the parameter space of front strength and Rossby number using theory & nonlinear numerical simulations. While fronts with Ro > 2.6 collapse to a self-similar profile dependent only on the deformation radius, we find that for small enough Ro ∼ 1, self-similar frontlets form during equilibration. These frontlets increase the energy of the equilibrated state and can interact with the near-surface currents to further enhance mixing. We finally propose a scaling model for the incurred diabatic mixing and to describe the ultimate state of the front compared to the classical (i.e. inviscid, adiabatic, & 0-PV) adjustment theory of Ou (1984).

Fronts with large lateral density gradients in geostrophic and hydrostatic balance are common in the upper ocean. Such strong fronts develop through a variety of mechanisms including frontogenesis forced by mesoscale eddies, coastal upwelling, and the input of freshwater via river discharge. These fronts may be unstable to symmetric instability (SI) — a form of stratified inertial instability which occurs when the potential vorticity is of opposite sign to the Coriolis parameter. SI encourages vertical transport of biogeochemical tracers as well as of geostrophic momentum. We previously found that this momentum transport destabilises the balanced thermal wind which further enhances small-scale turbulent mixing and often leaves remnant inertial oscillations.

Here, we consider the equilibration of an initially balanced front of finite width and which is bounded by flat no-stress horizontal surfaces. We examine how the adjustment depends on the aspect ratio and strength of the front using nonlinear numerical simulations, and develop a model to predict the resultant mixing and the energy in the final equilibrated state. While fronts with Ro > 2.6 collapse to a self-similar profile dependent only on the deformation radius, we find that for small enough Ro ≲ 1, frontlets form as the front equilibrates. These frontlets increase the kinetic and potential energy of the equilibrated state and interact with the near-surface currents if the front exhibits inertial oscillations.

The Texas-Louisiana Shelf in the Northern Gulf of Mexico is home to the second largest human-caused dead zone in the world. Here, the nutrient-laden, stratified waters of the Mississippi River plume condition hypoxia in a bottom layer during the summer. The plume also generates strong fronts, features of the circulation that are known pathways for the exchange of water between the ocean surface and the deep. Using a combination of high-resolution observations and numerical simulations, we show that the vertical exchange at these fronts can be quite rapid and can lead to the oxygenation of bottom water when the fronts are forced by the summer land-sea breeze. These winds generate strong inertial oscillations, which set up a diurnally-pulsing vertical circulation at the fronts that draws bottom waters up to the surface mixed layer. The simulations suggest that during these “breaths” the rate of oxygenation of the bottom waters is comparable to deoxygenation by the respiration of organic matter over the shelf and hence could play an important role in the evolution of the region’s dead zone.

Conference

American Physical Society: Division of Fluid Dynamics, Chicago, IL / Virtual

Publication year: 2020

Technical Report

Publication year: 2020

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 CO_{2} 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.

Here, we consider the problem of an initially balanced front with a horizontal buoyancy gradient of finite width and bounded by flat no-stress horizontal surfaces. We study the evolution to equilibration of this symmetrically-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 strength and width of the front. While weak fronts develop frontlets and excite subinertial oscillations, stronger fronts produce bore-like gravity currents that propagate along the top and bottom boundaries. Although the instantaneous turbulent dissipation rate can be much larger in these strong fronts, the turbulence is intermittent and peaks during periods of weak stratification. We explain the energetics as the front evolves towards the final adjusted state in terms of the dimensionless front strength and aspect ratio.

Conference

American Meteorological Society: Atmospheric and Oceanic Fluid Dynamics, Portland, ME

Publication year: 2019

Here, we consider an idealised problem with a front of finite width bounded by flat no-stress horizontal surfaces. The front is initially in thermal wind balance but there is no initial vertical stratification and the flow is unstable to SI. We study the evolution of the unstable front using a linear stability analysis and nonlinear numerical simulations. We find that the aspect ratio of the front and the ratio of the maximum horizontal buoyancy gradient to the Coriolis frequency are important parameters and influence the evolution of the front. Interesting behaviour emerges, particularly for fronts with large Rossby numbers. For example, fronts with relatively large horizontal density gradients develop bore-like gravity currents that propagate along the top and bottom boundaries. We then describe the final adjusted state in terms of the dimensionless parameters of the system.

Submesoscale fronts with large lateral buoyancy gradients and O(1) Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts 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. SI has linear eigenmodes which are capable of transporting buoyancy and geostrophic momentum. The unstable SI modes eventually break down through a secondary shear instability, leading to three-dimensional turbulence which further modifies the geostrophic momentum. An initially balanced front that is unstable to SI will evolve due to the momentum and buoyancy transport associated with SI and 3D turbulence.

Here, we consider an idealised problem with a front of finite width bounded by flat no-stress horizontal surfaces. The front is initially in thermal wind balance but there is no initial vertical stratification and the flow is unstable to SI. We study the evolution of the unstable front using a linear stability analysis and nonlinear numerical simulations. We find that the aspect ratio of the front and the ratio of the maximum horizontal buoyancy gradient to the Coriolis frequency are important parameters and influence the evolution of the front. Interesting behaviour emerges, particularly for fronts with large Rossby numbers. For example, fronts with relatively large horizontal density gradients develop bore-like gravity currents that propagate along the top and bottom boundaries. We then describe the energetics evolving towards the final adjusted state in terms of the dimensionless parameters of the system and understand these results in the context of the primary and secondary linear instability.

Conference

American Physical Society: Division of Fluid Dynamics, Seattle, WA

Publication year: 2019

Submesoscale fronts with large lateral buoyancy gradients and O(1) Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts 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. Growing SI modes eventually break down through a secondary shear instability, leading to 3D 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.

Here, we consider the idealised problem of a balanced front with uniform horizontal buoyancy gradient and bounded by flat no-stress horizontal surfaces. We study the equilibration of this unstable front using a linear stability analysis and 3D numerical simulations. We find drastically different behaviour emerging at late times. While weak fronts develop frontlets and excite subinertial oscillations, stronger fronts produce bore-like gravity currents. We describe the details of these energy pathways as the front evolves toward the final adjusted state in terms of the dimensionless front strength.

Journal Paper

Monthly Notices of the Royal Astronomical Society, Volume 477, Issue 4, July 2018, Pages 4838–4855

Publication year: 2018

Non-linear evolution of the parametric instability of inertial waves inherent to eccentric discs is studied by way of a new local numerical model. Mode coupling of tidal deformation with the disc eccentricity is known to produce exponentially growing eccentricities at certain mean-motion resonances. However, the details of an efficient saturation mechanism balancing this growth still are not fully understood. This paper develops a local numerical model for an eccentric quasi-axisymmetric shearing box which generalizes the often-used Cartesian shearing box model. The numerical method is an overall second-order well-balanced finite volume method which maintains the stratified and oscillatory steady-state solution by construction. This implementation is employed to study the non-linear outcome of the parametric instability in eccentric discs with vertical structure. Stratification is found to constrain the perturbation energy near the mid-plane and localize the effective region of inertial wave breaking that sources turbulence. A saturated marginally sonic turbulent state results from the non-linear breaking of inertial waves and is subsequently unstable to large-scale axisymmetric zonal flow structures. This resulting limit-cycle behaviour reduces access to the eccentric energy source and prevents substantial transport of angular momentum radially through the disc. Still, the saturation of this parametric instability of inertial waves is shown to damp eccentricity on a time-scale of a thousand orbital periods. It may thus be a promising mechanism for intermittently regaining balance with the exponential growth of eccentricity from the eccentric Lindblad resonances and may also help explain the occurrence of ‘bursty’ dynamics such as the superhump phenomenon.

- Accretion discs;
- Hydrodynamics;
- Instabilities;
- Turbulence;
- Waves

Technical Report

Publication year: 2017

Technical Report

Publication year: 2017

The nonlinear evolution of the parametric instability of inertial waves inherent to eccentric discs is studied by way of a new local numerical model. Mode coupling of tidal deformation with the disc eccentricity is known to produce exponentially growing eccentricities at mean-motion resonances, most prominently via the 3:1 eccentric Lindblad resonance in circumstellar binaries. However, the details of an efficient saturation mechanism for this growth are still not fully understood. Linear theory for the parametric instability of inertial waves in eccentric discs has previously been studied by Barker and Ogilvie (2014), the nonlinear evolution of which dictates the transport rates and may help saturate this unmitigated eccentricity growth. This thesis develops a generalised numerical model for an eccentric quasi-axisymmetric shearing box based on the theoretical model developed by Ogilvie and Barker (2014) which itself generalises the often-used local cartesian shearing box model. The numerical method is an overall 2nd order well-balanced finite volume method which maintains the stratified and oscillatory steady state solution to machine precision.

This implementation is employed to study the nonlinear outcome of this parametric instability in eccentric discs with vertical structure. Stratification is found to constrain the perturbation energy near the midplane, and localises the effective region of the secondary instability. A saturated marginally sonic turbulent state is then maintained by the breaking of inertial waves cascading energy into smaller scales. The resulting turbulence is quite inefficient at transporting angular momentum through the disc, as might be expected from a restricted axisymmetric instability. Still, the saturation of this parametric instability of inertial waves is shown to damp the eccentricity on the timescale of 50 to 100 orbital periods, and so is a promising mechanism for balancing the exponential growth of eccentricity from the eccentric Lindblad resonances.

*via* geometric modification of component parameters. Theoretical and experimental results for 3D fluidic capacitors demonstrated that transitioning from planar to non-planar diaphragm architectures improved component performance. Flow rectification experiments for 3D printed fluidic diodes revealed a diodicity of 80.6 ± 1.8. Geometry-based gain enhancement for 3D printed fluidic transistors yielded pressure gain of 3.01 ± 0.78. Consistent with additional additive manufacturing methodologies, the use of digitally-transferrable 3D models of fluidic components combined with commercially-available 3D printers could extend the fluidic routing capabilities presented here to researchers in fields beyond the core engineering community.

Technical Report

Publication year: 2015

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

Conference

American Physical Society: Division of Fluid Dynamics, Pittsburgh, PA

Publication year: 2013

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.

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