Non-geostrophic instabilities of an equilibrium baroclinic state

2013 ◽  
Vol 734 ◽  
pp. 535-566 ◽  
Author(s):  
Alexandre B. Pieri ◽  
F. S. Godeferd ◽  
C. Cambon ◽  
A. Salhi
Keyword(s):  
Numerical Simulations ◽  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Direct Numerical Simulations ◽  
Vertical Shear ◽  
Dimensionless Numbers ◽  
Stability Bound ◽  
Simultaneous Existence ◽  
The Stability ◽  
Symmetric Instability

AbstractWe consider non-geostrophic homogeneous baroclinic turbulence without solid boundaries, and we focus on its energetics and dynamics. The homogeneous turbulent flow is therefore submitted to both uniform vertical shear $S$ and stable vertical stratification, parametrized by the Brunt–Väisälä frequency $N$, and placed in a rotating frame with Coriolis frequency $f$. Direct numerical simulations show that the threshold of baroclinic instability growth depends mostly on two dimensionless numbers, the gradient Richardson number $\mathit{Ri}= {N}^{2} / {S}^{2} $ and the Rossby number $\mathit{Ro}= S/ f$, whereas linear theory predicts a threshold that depends only on $\mathit{Ri}$. At high Rossby numbers the nonlinear limit is found to be $\mathit{Ri}= 0. 2$, while in the limit of low $\mathit{Ro}$ the linear stability bound $\mathit{Ri}= 1$ is recovered. We also express the stability results in terms of background potential vorticity, which is an important quantity in baroclinic flows. We show that the linear symmetric instability occurs from the presence of negative background potential vorticity. The possibility of simultaneous existence of symmetric and baroclinic instabilities is also investigated. The dominance of symmetric instability over baroclinic instability for $\mathit{Ri}\ll 1$ is confirmed by our direct numerical simulations, and we provide an improved understanding of the dynamics of the flow by exploring the details of energy transfers for moderate Richardson numbers.

scholarly journals Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation

2016 ◽  
Vol 806 ◽  
pp. 165-204 ◽  
Author(s):  
Corentin Herbert ◽  
Raffaele Marino ◽  
Duane Rosenberg ◽  
Annick Pouquet
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Direct Numerical Simulations ◽  
Vertical Shear ◽  
Energy Spectra ◽  
Stratified Turbulence ◽  
Viscous Effects ◽  
Inverse Cascade ◽  
Main Effects ◽  
The Waves

We study the partition of energy between waves and vortices in stratified turbulence, with or without rotation, for a variety of parameters, focusing on the behaviour of the waves and vortices in the inverse cascade of energy towards the large scales. To this end, we use direct numerical simulations in a cubic box at a Reynolds number $Re\approx 1000$, with the ratio between the Brunt–Väisälä frequency $N$ and the inertial frequency $f$ varying from $1/4$ to 20, together with a purely stratified run. The Froude number, measuring the strength of the stratification, varies within the range $0.02\leqslant Fr\leqslant 0.32$. We find that the inverse cascade is dominated by the slow quasi-geostrophic modes. Their energy spectra and fluxes exhibit characteristics of an inverse cascade, even though their energy is not conserved. Surprisingly, the slow vortices still dominate when the ratio $N/f$ increases, also in the stratified case, although less and less so. However, when $N/f$ increases, the inverse cascade of the slow modes becomes weaker and weaker, and it vanishes in the purely stratified case. We discuss how the disappearance of the inverse cascade of energy with increasing $N/f$ can be interpreted in terms of the waves and vortices, and identify the main effects that can explain this transition based on both inviscid invariants arguments and viscous effects due to vertical shear.

scholarly journals Double-Diffusive Recipes. Part II: Layer-Merging Events

2014 ◽  
Vol 44 (5) ◽  
pp. 1285-1305 ◽  
Author(s):  
T. Radko ◽  
J. D. Flanagan ◽  
S. Stellmach ◽  
M.-L. Timmermans
Keyword(s):  
Numerical Simulations ◽  
Double Diffusion ◽  
Power Laws ◽  
Direct Numerical Simulations ◽  
Qualitative Description ◽  
The Arctic ◽  
Specific Power ◽  
Evolutionary Patterns ◽  
Salt Fingering ◽  
The Stability

Abstract This study explores the dynamics of thermohaline staircases: well-defined stepped structures in temperature and salinity profiles, commonly observed in regions of active double diffusion. The evolution of staircases in time is frequently characterized by spontaneous layer-merging events. These phenomena, the authors argue, are essential in regulating the equilibrium layer thickness in fully developed staircases. The pattern and mechanics of merging events are explained using a combination of analytical considerations, direct numerical simulations, and data analysis. The theoretical merger model is based on the stability analysis for a series of identical steps and pertains to both forms of double diffusion: diffusive convection and salt fingering. The conceptual significance of the proposed model lies in its ability to describe merging events without assuming from the outset specific power laws for the vertical transport of heat and salt—the approach adopted by earlier merging models. The analysis of direct numerical simulations indicates that merging models based on the four-thirds flux laws offer adequate qualitative description of the evolutionary patterns but are less accurate than models that do not rely on such laws. Specific examples considered in this paper include the evolution of layers in the diffusive staircase in the Beaufort Gyre of the Arctic Ocean.

scholarly journals Stability of an isolated pancake vortex in continuously stratified-rotating fluids

2016 ◽  
Vol 801 ◽  
pp. 508-553 ◽  
Author(s):  
Eunok Yim ◽  
Paul Billant ◽  
Claire Ménesguen
Keyword(s):  
Angular Velocity ◽  
Baroclinic Instability ◽  
Base Flow ◽  
Shear Instability ◽  
Vertical Shear ◽  
Rossby Number ◽  
Centrifugal Instability ◽  
Pancake Vortex ◽  
To Come ◽  
The Stability

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

Hypersonic boundary layer transition on a concave wall: stationary Görtler vortices

2019 ◽  
Vol 865 ◽  
pp. 1-40 ◽  
Author(s):  
X. Chen ◽  
G. L. Huang ◽  
C. B. Lee
Keyword(s):  
Boundary Layer ◽  
Numerical Simulations ◽  
Stability Theory ◽  
Direct Numerical Simulations ◽  
Boundary Layer Transition ◽  
Hypersonic Boundary Layer ◽  
Mode Instability ◽  
Görtler Vortices ◽  
Gortler Vortices ◽  
The Stability

This study investigates the stability and transition of Görtler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. In the simulations, Görtler vortices are separately excited by wall blowing and suction with spanwise wavelengths of 3, 6 and 9 mm. In addition to primary streaks with the same wavelength as the blowing and suction, secondary streaks with half the wavelength also emerge in the 6 and 9 mm cases. The streaks develop into mushroom structures before breaking down. The breakdown processes of the three cases are dominated by a sinuous-mode instability, a varicose-mode instability and a combination of the two, respectively. Both fundamental and subharmonic instabilities are relevant in all cases. Multiple modes are identified in the secondary-instability stage, some of which originate from the primary instabilities (first and second Mack modes). We demonstrate that the first Mack mode can be destabilized to either a varicose-mode or sinuous-mode streak instability depending on its frequency and wavelength, whereas the second Mack mode undergoes a stabilizing stage before turning into a varicose mode in the 6 and 9 mm cases. An energy analysis reveals the stabilizing and destabilizing mechanisms of the primary instabilities under the influence of Görtler vortices, highlighting the role played by the spanwise production based on the spanwise gradient of the streamwise velocity in both varicose and sinuous modes. The effects introduced by the secondary streaks are examined by filtering the secondary streaks in two new simulations with nominally identical conditions to those of the 6 and 9 mm cases. Remarkably, the secondary streaks can destabilize the Görtler vortices, therefore advancing the transition. The stability theory results are in good agreement with those from direct numerical simulations.

scholarly journals Bottom-trapped currents as statistical equilibrium states above topographic anomalies

2012 ◽  
Vol 699 ◽  
pp. 500-510 ◽  
Author(s):  
A. Venaille
Keyword(s):  
Velocity Field ◽  
Statistical Mechanics ◽  
Numerical Simulations ◽  
Potential Vorticity ◽  
Linear Relation ◽  
Direct Numerical Simulations ◽  
The Self ◽  
Self Organization ◽  
Probable Outcome ◽  
Geostrophic Turbulence

AbstractOceanic geostrophic turbulence is mostly forced at the surface, yet strong bottom-trapped flows are commonly observed along topographic anomalies. Here we consider the case of a freely evolving, initially surface-intensified velocity field above a topographic bump, and show that the self-organization into a bottom-trapped current can result from its turbulent dynamics. Using equilibrium statistical mechanics, we explain this phenomenon as the most probable outcome of turbulent stirring. We compute explicitly a class of solutions characterized by a linear relation between potential vorticity and streamfunction, and predict when the bottom intensification is expected. Using direct numerical simulations, we provide an illustration of this phenomenon that agrees qualitatively with theory, although the ergodicity hypothesis is not strictly fulfilled.

Effects of Topography on Baroclinic Instability

2013 ◽  
Vol 43 (4) ◽  
pp. 790-804 ◽  
Author(s):  
Changheng Chen ◽  
Igor Kamenkovich
Keyword(s):  
Numerical Model ◽  
Numerical Simulations ◽  
Potential Vorticity ◽  
Growth Rates ◽  
Vertical Structure ◽  
Bottom Topography ◽  
Baroclinic Instability ◽  
Critical Value ◽  
Zonal Flows ◽  
Topographic Slope

Abstract The importance of bottom topography in the linear baroclinic instability of zonal flows on the β plane is examined by using analytical calculations and a quasigeostrophic eddy-resolving numerical model. The particular focus is on the effects of a zonal topographic slope, compared with the effects of a meridional slope. A zonal slope always destabilizes background zonal flows that are otherwise stable in the absence of topography regardless of the slope magnitude, whereas the meridional slopes stabilize/destabilize zonal flows only through changing the lower-level background potential vorticity gradient beyond a known critical value. Growth rates, phase speeds, and vertical structure of the growing solutions strongly depend on the slope magnitude. In the numerical simulations configured with an isolated meridional ridge, unstable modes develop on both sides of the ridge and propagate eastward of the ridge, in agreement with analytical results.

scholarly journals Quasigeostrophic Turbulence with Explicit Surface Dynamics: Application to the Atmospheric Energy Spectrum

2009 ◽  
Vol 66 (2) ◽  
pp. 450-467 ◽  
Author(s):  
Ross Tulloch ◽  
K. Shafer Smith
Keyword(s):  
Surface Temperature ◽  
Potential Vorticity ◽  
Baroclinic Instability ◽  
Temperature Gradients ◽  
Vertical Shear ◽  
Model Driven ◽  
Function Solution ◽  
Mean Temperature ◽  
Small Scales ◽  
Quasigeostrophic Model

Abstract The horizontal wavenumber spectra of wind and temperature near the tropopause have a steep −3 slope at synoptic scales and a shallower −5/3 slope at mesoscales, with a transition between the two regimes at a wavelength of about 450 km. Here it is demonstrated that a quasigeostrophic model driven by baroclinic instability exhibits such a transition near its upper boundary (analogous to the tropopause) when surface temperature advection at that boundary is properly resolved and forced. To accurately represent surface advection at the upper and lower boundaries, the vertical structure of the model streamfunction is decomposed into four parts, representing the interior flow with the first two neutral modes, and each surface with its Green’s function solution, resulting in a system with four prognostic equations. Mean temperature gradients are applied at each surface, and a mean potential vorticity gradient consisting both of β and vertical shear is applied in the interior. The system exhibits three fundamental types of baroclinic instability: interactions between the upper and lower surfaces (Eady type), interactions between one surface and the interior (Charney type), and interactions between the barotropic and baroclinic interior modes (Phillips type). The turbulent steady states that result from each of these instabilities are distinct, and those of the former two types yield shallow kinetic energy spectra at small scales along those boundaries where mean temperature gradients are present. When both mean interior and surface gradients are present, the surface spectrum reflects a superposition of the interior-dominated −3 slope cascade at large scales, and the surface-dominated −5/3 slope cascade at small scales. The transition wavenumber depends linearly on the ratio of the interior potential vorticity gradient to the surface temperature gradient, and scales with the inverse of the deformation scale when β = 0.

scholarly journals The Generation of Turbulence below Midlevel Cloud Bases: The Effect of Cooling due to Sublimation of Snow

2013 ◽  
Vol 52 (4) ◽  
pp. 819-833 ◽  
Author(s):  
Atsushi Kudo
Keyword(s):  
Numerical Simulations ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Vertical Motion ◽  
Vertical Shear ◽  
Cloud Base ◽  
Base Temperature ◽  
Sequence Of Events ◽  
Frontal Zones ◽  
The Stability

AbstractIn the author’s experience as a forecaster, commercial aircraft sometimes report turbulence beneath midlevel clouds that extend above upper frontal zones. Turbulence caused by Kelvin–Helmholtz instability occurs in upper frontal zones with strong vertical shear of horizontal winds. However, the turbulence seems to occur not only in the cloud bases (where upper frontal zones are) but also below the cloud bases where the vertical shear is not strong. Because those clouds are usually accompanied by precipitation that does not reach the ground, cooling by evaporation or sublimation seems to contribute to the generation of turbulence. In this paper, the mechanisms generating turbulence below midlevel cloud bases are examined by using observations and high-resolution three-dimensional numerical simulations with idealized initial conditions. The numerical simulations showed that the following sequence of events led to turbulence. Falling snow sublimated below cloud bases and cooled the air, which created absolute instability. This generated Rayleigh–Bénard convection cells. The vertical motion caused turbulence. The horizontal scale of the convection was about 800–1000 m, and the variations of vertical wind velocity were up to about 7 m s−1. The cloud base was accompanied by a virga-like distribution of snow. Sensitivity experiments showed the appropriate conditions to cause the turbulence: 1) the cloud-base temperature was between about 0° and −15°C, 2) the relative humidity in subcloud layers was sufficiently low, and 3) the stability in subcloud layers was weak. The results of the numerical simulations agreed well with the observations.

Baroclinic stability under non-hydrostatic conditions

1971 ◽  
Vol 45 (4) ◽  
pp. 659-671 ◽  
Author(s):  
Peter H. Stone
Keyword(s):  
Richardson Number ◽  
Laboratory Experiments ◽  
Baroclinic Instability ◽  
Hydrostatic Equilibrium ◽  
Rossby Number ◽  
Helmholtz Instability ◽  
Rotating System ◽  
The Stability ◽  
Two Parameters ◽  
Symmetric Instability

Eady's model for the stability of a thermal wind in an inviscid, stratified, rotating system is modified to allow for deviations from hydrostatic equilibrium. The stability properties of the flow are uniquely determined by two parameters: the Richardson number Ri, and the ratio of the aspect ratio to the Rossby number δ. The latter parameter may be taken as a measure of the deviations from hydrostatic equilibrium (δ = 0 in Eady's model). It is found that such deviations decrease the growth rates of all three kinds of instability which can occur in this problem: ‘geostrophic’ baroclinic instability, symmetric instability, and Kelvin–Helmholtz instability. The unstable wavelengths for ‘geostrophic’ and Kelvin–Helmholtz instability are increased for finite values of δ, while the unstable wavelengths for symmetric instability are unaffected. The ‘non-hydrostatic’ effects (δ ≠ 0) are significant for symmetric and Kelvin–Helmholtz instability when δ [gsim ] 1, but not for ‘geostrophic’ instability unless δ [Gt ] 1. Consequently, the first two types of instability tend to be suppressed relative to ‘geostrophic’ instability by ‘non-hydrostatic’ conditions. Figure 3 summarizes the different instability régimes that can occur. In laboratory experiments symmetric instability can be studied best when δ [lsim ] 1, while Kelvin–Helmholtz instability can be studied best when δ [Lt ] 1.

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.

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