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The evolution & equilibration of an isolated symmetrically-unstable front

Technical Report
Wienkers, Aaron F., Thomas L., & Taylor, J. R.
Publication year: 2020

Abstract

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.

Equilibration of symmetric instability and inertial oscillations at an idealised submesoscale front

Conference
Wienkers, Aaron F., Thomas L., & Taylor, J. R.
American Physical Society: Division of Fluid Dynamics, Seattle, WA, 2019
Publication year: 2019

Abstract

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.

Nonlinear hydrodynamic instability and turbulence in eccentric astrophysical discs with vertical structure

Journal Paper
Wienkers, Aaron F. & Ogilvie, G. I.
Monthly Notices of the Royal Astronomical Society, Volume 477, Issue 4, July 2018, Pages 4838–4855
Publication year: 2018

Abstract

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.

Keywords

  • Accretion discs;
  • Hydrodynamics;
  • Instabilities;
  • Turbulence;
  • Waves

The effect of rapid rotation & stratification on homogeneous turbulent shear flows via linear kinematic simulations

Technical Report
Wienkers, Aaron F.
Publication year: 2017

Spatio-temporal stability of the stratified Ekman boundary layer

Technical Report
Wienkers, Aaron F. & Lele, S. K.
Publication year: 2017

On the secondary instability of Rayleigh-Bénard convection

Technical Report
Wienkers, Aaron F.
Publication year: 2017

Nonlinear oscillations and zonal flows in turbulent eccentric astrophysical discs with vertical structure

Conference
Wienkers, Aaron F. & Ogilvie, G. I.
American Physical Society: Division of Fluid Dynamics, Denver, CO, 2017
Publication year: 2017

Abstract

The classical theory of astrophysical discs (including Saturn’s rings, protoplanetary systems, and high-energy accretion discs around black holes) assumes circular orbital motion around a central mass. However, certain systems are known to contain eccentric forcing, necessitating a generaliza- tion to the shearing box model to include the oscillatory local geometry associated with this eccentricity. The hydrodynamic equations in this model are non-standard because of the use of time-dependent, non-orthogonal coordinates, and are known to lead to hydrodynamic instability involving the growth of internal waves. Here we present the results of the first ever local nonlinear simulations in an eccentric shearing box representing an elliptic disc with vertical structure. The nonlinear sat- uration of this parametric instability inherent to eccentric discs generates further self-regulating azimuthal zonal flows, and results in stable limit cycle behavior. We explore this energy pathway from the global eccentric mode into turbulence and finally the zonal flows, and discuss the viability of this instability to balance the eccentricity growth in systems exhibiting mean-motion orbital resonances such as the eccentric Lindblad resonance.

On the shearing box approximation for the compressible Euler equations

Technical Report
Wienkers, Aaron F.
MPhil Project, University of Cambridge
Publication year: 2016

Abstract

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.

On the fragmentation of liquid jets by neutron irradiation

Technical Report
Wienkers, Aaron F., Sharp, D. H., Cheng, B., & Urzay, J.
Publication year: 2016

Nonlinear hydrodynamic instability and turbulence in eccentric astrophysical discs

Thesis
Wienkers, Aaron F.
MPhil Thesis, University of Cambridge
Publication year: 2016

Abstract

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.

Computation of horizontally invariant nonlinear solutions in an eccentric shearing box

Technical Report
Wienkers, Aaron F.
MPhil Project, University of Cambridge
Publication year: 2016

Abstract

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.

3D printed microfluidic circuitry via multijet-based additive manufacturing

Journal Paper
Sochol, R. D., Wienkers, Aaron F., & Lin, L. et al.
Lab on a Chip, Volume 16, 2016, Pages 668-678
Publication year: 2016

Abstract

The miniaturization of integrated fluidic processors affords extensive benefits for chemical and biological fields, yet traditional, monolithic methods of microfabrication present numerous obstacles for the scaling of fluidic operators. Recently, researchers have investigated the use of additive manufacturing or “three-dimensional (3D) printing” technologies – predominantly stereolithography – as a promising alternative for the construction of submillimeter-scale fluidic components. One challenge, however, is that current stereolithography methods lack the ability to simultaneously print sacrificial support materials, which limits the geometric versatility of such approaches. In this work, we investigate the use of multijet modelling (alternatively, polyjet printing) – a layer-by-layer, multi-material inkjetting process – for 3D printing geometrically complex, yet functionally advantageous fluidic components comprised of both static and dynamic physical elements. We examine a fundamental class of 3D printed microfluidic operators, including fluidic capacitors, fluidic diodes, and fluidic transistors. In addition, we evaluate the potential to advance on-chip automation of integrated fluidic systems 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.

A brief review of localisation effects in finite-size parametric subharmonic instability

Technical Report
Wienkers, Aaron F., Brouzet, C., & Dauxois, T.
Publication year: 2015

Abstract

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.

 

Keywords

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

The role of interactions between waves and baroclinic critical layers in zombie vortex self-replication

Conference
Jiang, C. H., Wienkers, Aaron F., & Marcus, P. S. et al.
American Physical Society: Division of Fluid Dynamics, Pittsburgh, PA, 2013
Publication year: 2013

Abstract

Inertio-gravity waves are triggered from various types of perturbations in numerical simulations of rotating, vertically-stratified and horizontal-shearing flows (Marcus et al. 2013 PRL). The interactions of these waves and baroclinic critical layers can create large vortices when the shear is sufficiently strong. An important feature of these flows is that an instability at one critical layer can excite an instability at its neighboring critical layers and spawn new generations of waves and vortices. Because the self- replication of these vortices in simulations of “dead zones” in protoplanetary disks reminds us of zombies multiplying by infecting each other, we call them “zombie vortices.” However, not all interactions between waves and critical layers produce zombie vortices. The manner in which one “infected” critical layer infects its neigh- bor is not clear. The interaction of waves and critical layers are sensitive to the local Brunt-Vaisala frequency and to the wavelengths of the waves. Here we discuss how the interactions and formation of vortices depend upon the Brunt-Vaisala frequency (including its change in value as a function of vertical position) and our progress in understanding how the instability passes from a critical layer to its neighbor.

Development of an efficient baroclinic cooling prescription for global core collapse simulations into protoplanetary disks

Technical Report
Wienkers, Aaron F., Cunningham, A. J., Klein, R. I., Marcus, P. S., & McKee, C. F.
Lawrence Livermore National Laboratory, ICRL, 2013
Publication year: 2013

Abstract

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