The instability and breakdown of tall columnar vortices in a quasi-geostrophic fluid

1996 ◽  
Vol 328 ◽  
pp. 129-160 ◽  
Author(s):  
David G. Dritschel ◽  
Manuel De La Torre JuáRez
Keyword(s):  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Lower Surface ◽  
Buoyancy Frequency ◽  
Coriolis Parameter ◽  
Nonlinear Development ◽  
Aspect Ratios ◽  
Geostrophic Turbulence ◽  
Boussinesq Fluid

We examine the linear stability of elliptical columns of uniform potential vorticity subject to two-dimensional (horizontal) straining within a rapidly rotating, stratified (quasi-geostrophic) fluid. We find that horizontal straining can promote the exponential growth of three-dimensional disturbances when the vortex height-to-width aspect ratio exceeds, qualitatively, three times the ratio of the Coriolis parameter to the buoyancy frequency. This instability is not related to the usual baroclinic instability which operates on shallow vortex columns whose potential vorticity changes sign with height. The nonlinear development of these instabilities is investigated numerically using a high-resolution contour surgery algorithm. Simulations are conducted for both a Boussinesq (ocean-like) fluid and a compressible (atmospheric-like) fluid having exponentially decreasing density with height. The simulations reveal a generic nonlinear development that results in a semi-ellipsoidal baroclinic vortex dome at the lower surface and, in the case of a Boussinesq fluid, another such dome at the upper surface.The related problem of two interacting vortex columns is also examined. A generic three-dimensional instability and nonlinear development occurs no matter how great the distance between the vortex columns, provided that they are sufficiently tall.Our results may bear upon the observed structure of many atmospheric and oceanic vortices, whose height-to-width aspect ratios are consistent with our findings. Remarkably, even strongly ageostrophic vortices, such as tropical cyclones, fit the pattern. Our results furthermore re-open questions about the long-time nature of freely decaying quasi-geostrophic turbulence, for which recent simulations indicate a progressive two-dimensionalization by vortex alignment, while earlier simulations have indicated long-lived baroclinic vortices, not unlike what we find here.

scholarly journals Properties of Steady Geostrophic Turbulence with Isopycnal Outcropping

2012 ◽  
Vol 42 (1) ◽  
pp. 18-38 ◽  
Author(s):  
G. Roullet ◽  
J. C. McWilliams ◽  
X. Capet ◽  
M. J. Molemaker
Keyword(s):  
High Resolution ◽  
Large Scale ◽  
Mechanical Energy ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Primary Source ◽  
Eddy Diffusivity ◽  
Energy Cycle ◽  
Geostrophic Turbulence ◽  
Pv Gradient

Abstract High-resolution simulations of β-channel, zonal-jet, baroclinic turbulence with a three-dimensional quasigeostrophic (QG) model including surface potential vorticity (PV) are analyzed with emphasis on the competing role of interior and surface PV (associated with isopycnal outcropping). Two distinct regimes are considered: a Phillips case, where the PV gradient changes sign twice in the interior, and a Charney case, where the PV gradient changes sign in the interior and at the surface. The Phillips case is typical of the simplified turbulence test beds that have been widely used to investigate the effect of ocean eddies on ocean tracer distribution and fluxes. The Charney case shares many similarities with recent high-resolution primitive equation simulations. The main difference between the two regimes is indeed an energization of submesoscale turbulence near the surface. The energy cycle is analyzed in the (k, z) plane, where k is the horizontal wavenumber. In the two regimes, the large-scale buoyancy forcing is the primary source of mechanical energy. It sustains an energy cycle in which baroclinic instability converts more available potential energy (APE) to kinetic energy (KE) than the APE directly injected by the forcing. This is due to a conversion of KE to APE at the scale of arrest. All the KE is dissipated at the bottom at large scales, in the limit of infinite resolution and despite the submesoscales energizing in the Charney case. The eddy PV flux is largest at the scale of arrest in both cases. The eddy diffusivity is very smooth but highly nonuniform. The eddy-induced circulation acts to flatten the mean isopycnals in both cases.

scholarly journals Surface signature of Mediterranean water eddies in the Northeastern Atlantic: effect of the upper ocean stratification

Ocean Science ◽  
2012 ◽  
Vol 8 (6) ◽  
pp. 931-943 ◽  
Author(s):  
I. Bashmachnikov ◽  
X. Carton
Keyword(s):  
Potential Vorticity ◽  
Initial Period ◽  
Relative Vorticity ◽  
Sea Surface ◽  
Upper Ocean ◽  
Buoyancy Frequency ◽  
Coriolis Parameter ◽  
Ocean Stratification ◽  
Surface Signature ◽  
Mediterranean Water

Abstract. Meddies, intra-thermocline eddies of Mediterranean water, can often be detected at the sea surface as positive sea-level anomalies. Here we study the surface signature of several meddies tracked with RAFOS floats and AVISO altimetry. While pushing its way through the water column, a meddy raises isopycnals above. As a consequence of potential vorticity conservation, negative relative vorticity is generated in the upper layer. During the initial period of meddy acceleration after meddy formation or after a stagnation stage, a cyclonic signal is also generated at the sea-surface, but mostly the anticyclonic surface signal follows the meddy. Based on geostrophy and potential vorticity balance, we present theoretical estimates of the intensity of the surface signature. It appears to be proportional to the meddy core radius and to the Coriolis parameter, and inversely proportional to the core depth and buoyancy frequency. This indicates that surface signature of a meddy may be strongly reduced by the upper ocean stratification. Using climatic distribution of the stratification intensity, we claim that the southernmost limit for detection in altimetry of small meddies (with radii on the order of 10–15 km) should lie in the subtropics (35–45° N), while large meddies (with radii of 25–30 km) could be detected as far south as the northern tropics (25–35° N). Those results agree with observations.

scholarly journals Head-on collisions between two quasi-geostrophic hetons in a continuously stratified fluid

2015 ◽  
Vol 779 ◽  
pp. 144-180 ◽  
Author(s):  
Jean N. Reinaud ◽  
Xavier Carton
Keyword(s):  
Parameter Space ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Point Vortex ◽  
Point Vortices ◽  
Symmetric Pair ◽  
Horizontal Shear ◽  
Azimuthal Mode ◽  
Aspect Ratios ◽  
Vortex Merger

We examine the interactions between two three-dimensional quasi-geostrophic hetons. The hetons are initially translating towards one another. We address the effect of the vertical distance between the two poles (vortices) constituting each heton on the interaction. We also examine the influence of the horizontal separation between the poles within each heton. In this investigation, the two hetons are facing each other. Two configurations are possible depending on the respective locations of the like-signed poles of the hetons. When they lie at the same depth, we refer to the configuration as symmetric; the antisymmetric configuration corresponds to opposite-signed poles at the same depth. The first step in the investigation uses point vortices to represent the poles of the hetons. This approach allows us to rapidly browse the parameter space and to estimate the possible heton trajectories. For a symmetric pair, the hetons either reverse their trajectory or recombine and escape perpendicularly depending of their horizontal and vertical offsets. On the other hand, antisymmetric hetons recombine and escape perpendicularly as same-depth dipoles. In a second part, we focus on finite core hetons (with finite volume poles). These hetons can deform and may be sensitive to horizontal-shear-induced deformations, or to baroclinic instability. These destabilisations depend on the vertical and horizontal offsets between the various poles, as well as on their width-to-height aspect ratios. They can modify the volume of the poles via vortex merger, breaking and/or shearing out; they compete with the advective evolution observed for singular (point) vortices. Importantly, hetons can break down or reconfigure before they can drift away as expected from a point vortex approach. Thus, a large variety of behaviours is observed in the parameter space. Finally, we briefly illustrate the behaviour of tall hetons which can be unstable to an azimuthal mode $l=1$ when many vertical modes of deformation are present on the heton.

scholarly journals The universal aspect ratio of vortices in rotating stratified flows: experiments and observations

2012 ◽  
Vol 706 ◽  
pp. 34-45 ◽  
Author(s):  
Oriane Aubert ◽  
Michael Le Bars ◽  
Patrice Le Gal ◽  
Philip S. Marcus
Keyword(s):  
Aspect Ratio ◽  
Salt Water ◽  
Boussinesq Equations ◽  
Companion Paper ◽  
Buoyancy Frequency ◽  
Coriolis Parameter ◽  
Aspect Ratios ◽  
Horizontal Length ◽  
The Law ◽  
The Relationship

AbstractWe validate a new law for the aspect ratio $\ensuremath{\alpha} = H/ L$ of vortices in a rotating, stratified flow, where $H$ and $L$ are the vertical half-height and horizontal length scale of the vortices. The aspect ratio depends not only on the Coriolis parameter $f$ and buoyancy (or Brunt–Väisälä) frequency $\bar {N} $ of the background flow, but also on the buoyancy frequency ${N}_{c} $ within the vortex and on the Rossby number $\mathit{Ro}$ of the vortex, such that $\ensuremath{\alpha} = f \mathop{ [\mathit{Ro}(1+ \mathit{Ro})/ ({ N}_{c}^{2} \ensuremath{-} {\bar {N} }^{2} )] }\nolimits ^{1/ 2} $. This law for $\ensuremath{\alpha} $ is obeyed precisely by the exact equilibrium solution of the inviscid Boussinesq equations that we show to be a useful model of our laboratory vortices. The law is valid for both cyclones and anticyclones. Our anticyclones are generated by injecting fluid into a rotating tank filled with linearly stratified salt water. In one set of experiments, the vortices viscously decay while obeying our law for $\ensuremath{\alpha} $, which decreases over time. In a second set of experiments, the vortices are sustained by a slow continuous injection. They evolve more slowly and have larger $\vert \mathit{Ro}\vert $ while still obeying our law for $\ensuremath{\alpha} $. The law for $\ensuremath{\alpha} $ is not only validated by our experiments, but is also shown to be consistent with observations of the aspect ratios of Atlantic meddies and Jupiter’s Great Red Spot and Oval BA. The relationship for $\ensuremath{\alpha} $ is derived and examined numerically in a companion paper by Hassanzadeh, Marcus & Le Gal (J. Fluid Mech., vol. 706, 2012, pp. 46–57).

scholarly journals The universal aspect ratio of vortices in rotating stratified flows: theory and simulation

2012 ◽  
Vol 706 ◽  
pp. 46-57 ◽  
Author(s):  
Pedram Hassanzadeh ◽  
Philip S. Marcus ◽  
Patrice Le Gal
Keyword(s):  
Aspect Ratio ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Stratified Flows ◽  
Coriolis Parameter ◽  
Aspect Ratios ◽  
Ideal Gases ◽  
Horizontal Length ◽  
Background Density ◽  
Rotating Stratified Flows

AbstractWe derive a relationship for the vortex aspect ratio $\ensuremath{\alpha} $ (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt–Väisälä frequencies within the vortex ${N}_{c} $ and in the background fluid outside the vortex $\bar {N} $, the Coriolis parameter $f$ and the Rossby number $\mathit{Ro}$ of the vortex: ${\ensuremath{\alpha} }^{2} = \mathit{Ro}(1+ \mathit{Ro}){f}^{2} / ({ N}_{c}^{2} \ensuremath{-} {\bar {N} }^{2} )$. This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for $\ensuremath{\alpha} $ has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have ${ N}_{c}^{2} \gt {\bar {N} }^{2} $; weak anticyclones (with $\vert \mathit{Ro}\vert \lt 1$) must have ${ N}_{c}^{2} \lt {\bar {N} }^{2} $; and strong anticyclones must have ${ N}_{c}^{2} \gt {\bar {N} }^{2} $. We verify our relation for $\ensuremath{\alpha} $ with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localized region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically computed vortices validate our relationship for $\ensuremath{\alpha} $, and generally they differ significantly from the values obtained from the much-cited conjecture that $\ensuremath{\alpha} = f/ \bar {N} $ in quasi-geostrophic vortices.

The Emergence of Multiple Robust Zonal Jets from Freely Evolving, Three-Dimensional Stratified Geostrophic Turbulence with Applications to Jupiter

2008 ◽  
Vol 65 (12) ◽  
pp. 3947-3962 ◽  
Author(s):  
Kunio M. Sayanagi ◽  
Adam P. Showman ◽  
Timothy E. Dowling
Keyword(s):  
Three Dimensional ◽  
Vortical Flow ◽  
Spectral Space ◽  
Small Scale ◽  
Width Ratio ◽  
Coriolis Parameter ◽  
Zonal Jets ◽  
Turbulence Velocity ◽  
Geostrophic Turbulence ◽  
Flow Transitions

Abstract Three-dimensional numerical simulations of freely evolving stratified geostrophic turbulence on the β plane are presented as a simplified model of zonal jet formation on Jupiter. This study samples the parameter space that covers the low, middle, and high latitudes of Jupiter by varying the central latitude of the β plane. The results show that robust zonal jets can emerge from initial small-scale random turbulence through the upscale redistribution of the kinetic energy in the spectral space. The resulting flow’s sensitivities to the flow’s deformation radius LD and the two-dimensional Rhines length Lβ = U/β (U is the characteristic turbulence velocity and β is the meridional gradient of the planetary vorticity) are tested, revealing that whether the outcome of the upscale energy transfer becomes dominated by jets or vortices depends on the relative values of LD and Lβ. The values of Lβ and LD are varied by tuning the β-plane parameters, and it is found that the flow transitions from a jet-dominated regime in Lβ ≲ LD to a vortical flow in Lβ ≳ LD. A height-to-width ratio equal to f /N, the Coriolis parameter divided by the Brunt–Väisälä frequency, has previously been established for stable vortices, and this paper shows that this aspect ratio also applies to the zonal jets that emerge in these simulations.

scholarly journals Layering and turbulence surrounding an anticyclonic oceanic vortex: in situ observations and quasi-geostrophic numerical simulations

2013 ◽  
Vol 731 ◽  
pp. 418-442 ◽  
Author(s):  
Bach Lien Hua ◽  
Claire Ménesguen ◽  
Sylvie Le Gentil ◽  
Richard Schopp ◽  
Bruno Marsset ◽  
...  
Keyword(s):  
Potential Energy ◽  
Numerical Simulations ◽  
Potential Vorticity ◽  
Scaling Law ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Critical Levels ◽  
Vertical Derivative ◽  
Vertical Scales

AbstractEvidence of persistent layering, with a vertical stacking of sharp variations in temperature, has been presented recently at the vertical and lateral periphery of energetic oceanic vortices through seismic imaging of the water column. The stacking has vertical scales ranging from a few metres up to 100 m and a lateral spatial coherence of several tens of kilometres comparable with the vortex horizontal size. Inside this layering, in situ data display a $[{ k}_{h}^{- 5/ 3} { k}_{h}^{- 2} ] $ scaling law of horizontal scales for two different quantities, temperature and a proxy for its vertical derivative, but for two different ranges of wavelengths, between 5 and 50 km for temperature and between 500 m and 5 km for its vertical gradient. In this study, we explore the dynamics underlying the layering formation mechanism, through the slow dynamics captured by quasi-geostrophic equations. Three-dimensional high-resolution numerical simulations of the destabilization of a lens-shaped vortex confirm that the vertical stacking of sharp jumps in density at its periphery is the three-dimensional analogue of the preferential wind-up of potential vorticity near a critical radius, a phenomenon which has been documented for barotropic vortices. For a small-Burger (flat) lens vortex, baroclinic instability ensures a sustained growth rate of sharp jumps in temperature near the critical levels of the leading unstable modes. Such results can be obtained for a background stratification which is due to temperature only and does not require the existence of salt anomalies. Aloft and beneath the vortex core, numerical simulations well reproduce the $[{ k}_{h}^{- 5/ 3} { k}_{h}^{- 2} ] $ scaling law of horizontal scales for the vertical derivative of temperature that is observed in situ inside the layering, whatever the background stratification. Such a result stems from the tracer-like behaviour of the vortex stretching component and previous studies have shown that spectra of tracer fields can be steeper than $- 1$, namely in $- 5/ 3$ or $- 2$, if the advection field is very compact spatially, with a $- 5/ 3$ slope corresponding to a spiral advection of the tracer. Such a scaling law could thus be of geometric origin. As for the kinetic and potential energy, the ${ k}_{h}^{- 5/ 3} $ scaling law can be reproduced numerically and is enhanced when the background stratification profile is strongly variable, involving sharp jumps in potential vorticity such as those observed in situ. This raises the possibility of another plausible mechanism leading to a $- 5/ 3$ scaling law, namely surface-quasi-geostrophic (SQG)-like dynamics, although our set-up is more complex than the idealized SQG framework. Energy and enstrophy fluxes have been diagnosed in the numerical quasi-geostrophic simulations. The results emphasize a strong production of energy in the oceanic submesoscales range and a kinetic and potential energy flux from mesoscale to submesoscales range near the critical levels. Such horizontal submesoscale production, which is correlated to the accumulation of thin vertical scales inside the layering, thus has a significant slow dynamical component, well-captured by quasi-geostrophy.

On the stratified Taylor column

1973 ◽  
Vol 58 (3) ◽  
pp. 517-537 ◽  
Author(s):  
Nelson G. Hogg
Keyword(s):  
Three Dimensional ◽  
Flow Speed ◽  
Bottom Current ◽  
Buoyancy Frequency ◽  
Coriolis Parameter ◽  
Height Scale ◽  
Topographic Parameter ◽  
Upstream Flow ◽  
Baroclinic Current ◽  
Taylor Column

We analyse the effects of small, circularly symmetric topography on the slow flow of an inviscid, incompressible, diffusionless, horizontally uniform, baroclinic current and show that the vertical influence depends primarily on three parameters: a stratification measureS(the square of the ratio of buoyancy frequency times height scale to Coriolis parameter times length scale), a topographic parameter β (ratio of scaled topographic height multiplied by scaled bottom current to Rossby number ε) and the scaled upstream shearu′0(z) (the dimensional upstream shear divided by the ratio of the r.m.s. upstream flow speed to height scale).Investigating a linear stratification model we find that the topographic effect is depth independent ifS[lsim ] ε and a Taylor column, as indicated by the appearance of closed streamlines above the bump, exists when β > 2. Moderate stratification (S∼ 1) causes the flow to be fully three-dimensional and the Taylor column to be a conical vortex whose height depends on βSandu′0). The results are compared with Davies's (1971, 1972) experiments.Our results tend to support the Taylor column theory of Jupiter's Great Red Spot but effects due to variations in the Coriolis parameter with latitude have been (unjustifiably) ne glected. Using typical values for the earths oceans we find that Taylor columns of significant height could be found there. Some pertinent observations from the ocean are discussed.

scholarly journals Length Scale of the Finite-Amplitude Meanders of Shelfbreak Fronts

2015 ◽  
Vol 45 (10) ◽  
pp. 2598-2620 ◽  
Author(s):  
Weifeng G. Zhang ◽  
Glen G. Gawarkiewicz
Keyword(s):  
Numerical Models ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Buoyancy Frequency ◽  
Length Scale ◽  
Initial Growth ◽  
Coriolis Parameter ◽  
Frontal Instability ◽  
Symmetric Instability ◽  
Empirical Constants

AbstractThrough combining analytical arguments and numerical models, this study investigates the finite-amplitude meanders of shelfbreak fronts characterized by sloping isopycnals outcropping at both the surface and the shelfbreak bottom. The objective is to provide a formula for the meander length scale that can explain observed frontal length scale variability and also be verified with observations. Considering the frontal instability to be a mixture of barotropic and baroclinic instability, the derived along-shelf meander length scale formula is [b1/(1 + a1S1/2)]NH/f, where N is the buoyancy frequency; H is the depth of the front; f is the Coriolis parameter; S is the Burger number measuring the ratio of energy conversion associated with barotropic and baroclinic instability; and a1 and b1 are empirical constants. Initial growth rate of the frontal instability is formulated as [b2(1 + a1S1/2)/(1 + a2αS1/2)]NH/L, where α is the bottom slope at the foot of the front, and a2 and b2 are empirical constants. The formulas are verified using numerical sensitivity simulations, and fitting of the simulated and formulated results gives a1 = 2.69, b1 = 14.65, a2 = 5.1 × 103, and b2 = 6.2 × 10−2. The numerical simulations also show development of fast-growing frontal symmetric instability when the minimum initial potential vorticity is negative. Although frontal symmetric instability leads to faster development of barotropic and baroclinic instability at later times, it does not significantly influence the meander length scale. The derived meander length scale provides a framework for future studies of the influences of external forces on shelfbreak frontal circulation and cross-frontal exchange.

scholarly journals Three Dimensional Simulation of Rayleigh-Be´nard Convection for Rapid Microscale Polymerase Chain Reaction

2009 ◽  
Author(s):  
Radha Muddu ◽  
Victor M. Ugaz ◽  
Yassin A. Hassan
Keyword(s):  
Polymerase Chain Reaction ◽  
Three Dimensional ◽  
Lower Surface ◽  
High Concentration ◽  
Chain Reaction ◽  
Aspect Ratios ◽  
Temperature Distributions ◽  
Temperature Regimes ◽  
Specific Order ◽  
Polymerase Chain

A variety of biochemical reactions involve cycling reagents through different temperature regimes in a specific order as dictated by chemical kinetics. One such reaction is the polymerase chain reaction (PCR), a fundamental process used to selectively amplify specific regions of interest within a larger DNA strand to very high concentration levels. Natural convection has recently been explored as a novel way to perform PCR by providing a passive approach to circulate reactants through the correct temperature zones. Optimal reactor design requires the spatial velocity and temperature distributions to be precisely controlled to ensure that the reactants sequentially occupy the correct temperature zones for a sufficient period of time. Rayleigh-Be´nard convection in vertical cylindrical reaction chambers maintained at fixed upper and lower surface temperatures provides a convenient model system to study these phenomena. In this work, we present results of three dimensional numerical simulations of the flow and temperature distributions in cylindrical reactors of different aspect ratios (height (h) / diameter (d)). These results helped identify different flow profiles that are possible in each of the geometries and led to an optimized redesign of the Rayleigh-Be´nard PCR convection system for rapid DNA replication.

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