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.

The viscous nonlinear symmetric baroclinic instability of a zonal shear flow

1975 ◽  
Vol 68 (4) ◽  
pp. 757-768 ◽  
Author(s):  
I. C. Walton
Keyword(s):  
Constant Velocity ◽  
Richardson Number ◽  
Baroclinic Instability ◽  
Lower Boundary ◽  
Length Scale ◽  
Ekman Number ◽  
Zonal Current ◽  
Horizontal Length ◽  
The Stability ◽  
Zonal Momentum

The stability of a baroclinic zonal current to symmetric perturbations on a meridionally unboundedf-plane is considered. The lower boundary is at rest but the upper one moves with a constant velocity in keeping with the velocity of the zonal current. Following Stone (1966) a horizontal length scaleO(Ro) is taken, whereRois the Rossby number, with the Richardson numberRi=O(1). Instability sets in when the wavelength isO(E1/3), whereEis the Ekman number based on the distance between the rigid horizontal boundaries, which corresponds to Stone's inviscid value zero, and to McIntyre's (1970) value infinity on a length scaleO(E½).A nonlinear analysis about the point of onset of instability yields the result that for the monotonic mode zonal momentum is convected polewards. The possible implications of this result for the dynamics of Jupiter's atmosphere are discussed.

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.

On the stability of elliptical vortex solutions of the shallow-water equations

1987 ◽  
Vol 183 ◽  
pp. 343-363 ◽  
Author(s):  
P. Ripa
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Normal Modes ◽  
Finite Amplitude ◽  
Rossby Number ◽  
Coriolis Parameter ◽  
Vortex Solutions ◽  
The Stability ◽  
The One ◽  
Two Parameters

The one-layer reduced gravity (or ‘shallow water’) equations in the f-plane have solutions such that the active layer is horizontally bounded by an ellipse that rotates steadily. In a frame where the height contours are stationary, fluid particles move along similar ellipses with the same revolution period. Both motions (translation along an elliptical path and precession of that orbit) are anticyclonic and their frequencies are not independent; a Rossby number (R0) based on the combination of both of them is bounded by unity. These solutions may be taken, with some optimism, as a model of ocean warm eddies; their stability is studied here for all values of R0 and of the ellipse eccentricity (these two parameters determine uniquely the properties of the solution).Sufficient stability conditions are derived from the integrals of motion; f-plane flows that satisfy them must be either axisymmetric or parallel. For the model vortex, the circular case simply corresponds to a solid-body rotation, and is found to be stable to finite-amplitude perturbations for all values of R0. This includes R0 > ½, which implies an anticyclonic absolute vorticity.The stability of the truly elliptical cases are studied in the normal modes sense. The height perturbation is an n-order polynomial of the horizontal coordinates; the cases for 0 ≤ n ≤ 6 are analysed, for all possible values of the Rossby number and of the eccentricity. All eddies are stable to perturbations with n ≤ 2. (A property of the shallow-water equations, probably related to the last result, is that a general finite-amplitude n-order field is an exact nonlinear solution for n ≤ 2.) Many vortices - noticeably the more eccentric ones - are unstable to perturbations with n ≥ 3; growth rates are O(R02f) where f is the Coriolis parameter.

Note on the paper of Kochar & Jain on Howard's semicircle theorem

1984 ◽  
Vol 140 ◽  
pp. 1-10 ◽  
Author(s):  
Yu. N. Makov ◽  
Yu. A. Stepanyants
Keyword(s):  
Maximum Rate ◽  
Richardson Number ◽  
Laboratory Experiments ◽  
Wave Number ◽  
Dimensionless Wave Number ◽  
Parallel Flows ◽  
Complex Wave ◽  
Stratified Shear Flow ◽  
The Stability ◽  
Theoretical Results

A further refinement of the Howard-Kochar-Jain theorem is given which allows the estimation of the range of complex wave velocity for growing perturbations in a stratified shear flow. According to the results obtained, the boundary of this region depends both on the minimum Richardson number and on the wavenumber of the perturbations. The effect of external boundaries on the stability of parallel flows is defined. An estimate of the maximum rate of growth versus dimensionless wave-number is found. The theoretical results are compared with numerical computations and laboratory experiments of other authors.

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 Three-dimensional baroclinic instability of a Hadley cell for small Richardson number

1983 ◽  
Vol 137 ◽  
pp. 423-445 ◽  
Author(s):  
Basil N. Antar ◽  
William W. Fowlis
Keyword(s):  
Stability Analysis ◽  
Richardson Number ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Maximum Growth ◽  
Hadley Cell ◽  
Navier Stokes ◽  
Zonal Structure ◽  
The Stability ◽  
Energy Equations

A three-dimensional linear stability analysis of a baroclinic flow for Richardson number Ri of order unity is presented. The model considered is a thin, horizontal, rotating fluid layer which is subjected to horizontal and vertical temperature gradients. The basic state is a Hadley cell which is a solution of the Navier–Stokes and energy equations and contains both Ekman and thermal boundary layers adjacent to the rigid boundaries; it is given in closed form. The stability analysis is also based on the Navier–Stokes and energy equations; and perturbations possessing zonal, meridional and vertical structures were considered. Numerical methods were developed for the solution of the stability problem, which results in an ordinary differential eigenvalue problem. The objectives of this work were to extend the previous theoretical work on three-dimensional baroclinic instability for small Ri to a more realistic model involving the Prandtl number σ and the Ekman number E, and to finite growth rates and a wider range of the zonal wavenumber. The study covers ranges of 0.135 [les ] Ri [les ] 1.1, 0.2 [les ] σ [les ] 5.0, and 2 × 10−4 [les ] E [les ] 2 σ 10−3. For the cases computed for E = 10−3 and σ ≠ 1, we found that conventional baroclinic instability dominates for Ri > 0.825 and symmetric baroclinic instability dominates for Ri < 0.675. However, for E [ges ] 5 × 10−4 and σ = 1 in the range 0.3 [les ] Ri [les ] 0.8, conventional baroclinic instability always dominates. Further, we found in general that the symmetric modes of maximum growth are not purely symmetric but have weak zonal structure. This means that the wavefronts are inclined at a small angle to the zonal direction. The results also show that as E decreases the zonal structure of the symmetric modes of maximum growth rate also decreases. We found that when zonal structure is permitted the critical Richardson number for marginal stability is increased, but by only a small amount above the value for pure symmetric instability. Because these modes do not substantially alter the results for pure symmetric baroclinic instability and because their zonal structure is weak, it is unlikely that they represent a new type of instability.

scholarly journals Inhibition or Facilitation? Contrasted Inter-Specific Interactions in Sphagnum under Laboratory and Field Conditions

Plants ◽  
2020 ◽  
Vol 9 (11) ◽  
pp. 1554
Author(s):  
Chao Liu ◽  
Zhao-Jun Bu ◽  
Azim Mallik ◽  
Yong-Da Chen ◽  
Xue-Feng Hu ◽  
...  
Keyword(s):  
Field Experiment ◽  
Species Interactions ◽  
Laboratory Experiments ◽  
Resource Competition ◽  
Specific Interactions ◽  
Interaction Type ◽  
Negative Effects ◽  
Experiment Data ◽  
The Stability ◽  
Positive Effect

In a natural environment, plants usually interact with their neighbors predominantly through resource competition, allelopathy, and facilitation. The occurrence of the positive effect of allelopathy between peat mosses (Sphagnum L.) is rare, but it has been observed in a field experiment. It is unclear whether the stability of the water table level in peat induces positive vs. negative effects of allelopathy and how that is related to phenolic allelochemical production in Sphagnum. Based on field experiment data, we established a laboratory experiment with three neighborhood treatments to measure inter-specific interactions between Sphagnum angustifolium (Russ.) C. Jens and Sphagnum magellanicum Brid. We found that the two species were strongly suppressed by the allelopathic effects of each other. S. magellanicum allelopathically facilitated S. angustifolium in the field but inhibited it in the laboratory, and relative allelopathy intensity appeared to be positively related to the content of released phenolics. We conclude that the interaction type and intensity between plants are dependent on environmental conditions. The concentration of phenolics alone may not explain the type and relative intensity of allelopathy. Carefully designed combined field and laboratory experiments are necessary to reveal the mechanism of species interactions in natural communities.

scholarly journals Persistent Liquidity Effects and Long-Run Money Demand

2014 ◽  
Vol 6 (2) ◽  
pp. 71-107 ◽  
Author(s):  
Fernando Alvarez ◽  
Francesco Lippi
Keyword(s):  
Interest Rates ◽  
Money Demand ◽  
Asset Markets ◽  
Propagation Mechanism ◽  
Monetary Model ◽  
Long Run ◽  
Short Run ◽  
The Stability ◽  
Two Parameters

We present a monetary model with segmented asset markets that implies a persistent fall in interest rates after a once-and-for-all increase in liquidity. The gradual propagation mechanism produced by our model is novel in the literature. We provide an analytical characterization of this mechanism, showing that the magnitude of the liquidity effect on impact, and its persistence, depend on the ratio of two parameters: the long-run interest rate elasticity of money demand and the intertemporal substitution elasticity. The model simultaneously explains the short-run “instability” of money demand estimates as well as the stability of long-run interest-elastic money demand. (JEL E13, E31, E41, E43, E52, E62)

On the Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems

1959 ◽  
Vol 26 (3) ◽  
pp. 377-385
Author(s):  
R. M. Rosenberg ◽  
C. P. Atkinson
Keyword(s):  
Free Vibrations ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Natural Modes ◽  
Theoretical Predictions ◽  
Stability Chart ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

Abstract The natural modes of free vibrations of a symmetrical two-degree-of-freedom system are analyzed theoretically and experimentally. This system has two natural modes, one in-phase and the other out-of-phase. In contradistinction to the comparable single-degree-of-freedom system where the free vibrations are always orbitally stable, the natural modes of the symmetrical two-degree-of-freedom system are frequently unstable. The stability properties depend on two parameters and are easily deduced from a stability chart. For sufficiently small amplitudes both modes are, in general, stable. When the coupling spring is linear, both modes are always stable at all amplitudes. For other conditions, either mode may become unstable at certain amplitudes. In particular, if there is a single value of frequency and amplitude at which the system can vibrate in either mode, the out-of-phase mode experiences a change of stability. The experimental investigation has generally confirmed the theoretical predictions.

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