Variation in the linear stability property of resistive ballooning mode including the parametric dependence of its growth rate in L-mode edge

2018 ◽  
Vol 25 (9) ◽  
pp. 092506 ◽  
Author(s):  
J. Y. Kim ◽  
H. S. Han
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Stability Property ◽  
Parametric Dependence ◽  
Ballooning Mode

scholarly journals Influence of Bottom Topography on Vortex Stability

2019 ◽  
Vol 49 (12) ◽  
pp. 3199-3219 ◽  
Author(s):  
Bowen Zhao ◽  
Emma Chieusse-Gérard ◽  
Glenn Flierl
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Bottom Topography ◽  
Relative Vorticity ◽  
Subsequent Evolution ◽  
Vortex Stability ◽  
Number Of Layers ◽  
One Step ◽  
Baroclinic Vortices ◽  
Dynamics Simulations

AbstractThe effects of topography on the linear stability of both barotropic vortices and two-layer, baroclinic vortices are examined by considering cylindrical topography and vortices with stepwise relative vorticity profiles in the quasigeostrophic approximation. Four vortex configurations are considered, classified by the number of relative vorticity steps in the horizontal and the number of layers in the vertical: barotropic one-step vortex (Rankine vortex), barotropic two-step vortex, and their two-layer, baroclinic counterparts with the vorticity steps in the upper layer. In the barotropic calculation, the vortex is destabilized by topography having an oppositely signed potential vorticity jump while stabilized by topography of same-signed jump, that is, anticyclones are destabilized by seamounts while stabilized by depressions. Further, topography of appropriate sign and magnitude can excite a mode-1 instability for a two-step vortex, especially relevant for topographic encounters of an otherwise stable vortex. The baroclinic calculation is in general consistent with the barotropic calculation except that the growth rate weakens and, for a two-step vortex, becomes less sensitive to topography (sign and magnitude) as baroclinicity increases. The smaller growth rate for a baroclinic vortex is consistent with previous findings that vortices with sufficient baroclinic structure could cross the topography relatively easily. Nonlinear contour dynamics simulations are conducted to confirm the linear stability analysis and to describe the subsequent evolution.

Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis

2013 ◽  
Vol 721 ◽  
pp. 268-294 ◽  
Author(s):  
L. Talon ◽  
N. Goyal ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Stokes Equations ◽  
Variable Density ◽  
Local Concentration ◽  
Miscible Displacements ◽  
Instability Modes ◽  
Finger Tip

AbstractA computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

scholarly journals Decomposition of the temporal growth rate in linear instability of compressible gas flows

2015 ◽  
Vol 778 ◽  
pp. 120-132 ◽  
Author(s):  
Mario Weder ◽  
Michael Gloor ◽  
Leonhard Kleiser
Keyword(s):  
Growth Rate ◽  
Energy Balance ◽  
Linear Stability ◽  
Couette Flow ◽  
Stability Theory ◽  
Compressible Flows ◽  
Linear Instability ◽  
Linear Stability Theory ◽  
Gas Flows ◽  
Temporal Growth Rate

We present a decomposition of the temporal growth rate ${\it\omega}_{i}$ which characterises the evolution of wave-like disturbances in linear stability theory for compressible flows. The decomposition is based on the disturbance energy balance by Chu (Acta Mech., vol. 1 (3), 1965, pp. 215–234) and provides terms for production, dissipation and flux of energy as components of ${\it\omega}_{i}$. The inclusion of flux terms makes our formulation applicable to unconfined flows and flows with permeable or vibrating boundaries. The decomposition sheds light on the fundamental mechanisms determining temporal growth or decay of disturbances. The additional insights gained by the proposed approach are demonstrated by an investigation of two model flows, namely compressible Couette flow and a plane compressible jet.

Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations

2002 ◽  
Vol 451 ◽  
pp. 261-282 ◽  
Author(s):  
F. GRAF ◽  
E. MEIBURG ◽  
C. HÄRTEL
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Linear Stability ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interface Thickness ◽  
Gap Width ◽  
Stability Results ◽  
Hele Shaw Cell

We consider the situation of a heavier fluid placed above a lighter one in a vertically arranged Hele-Shaw cell. The two fluids are miscible in all proportions. For this configuration, experiments and nonlinear simulations recently reported by Fernandez et al. (2002) indicate the existence of a low-Rayleigh-number (Ra) ‘Hele-Shaw’ instability mode, along with a high-Ra ‘gap’ mode whose dominant wavelength is on the order of five times the gap width. These findings are in disagreement with linear stability results based on the gap-averaged Hele-Shaw approach, which predict much smaller wavelengths. Similar observations have been made for immiscible flows as well (Maxworthy 1989).In order to resolve the above discrepancy, we perform a linear stability analysis based on the full three-dimensional Stokes equations. A generalized eigenvalue problem is formulated, whose numerical solution yields both the growth rate and the two-dimensional eigenfunctions in the cross-gap plane as functions of the spanwise wavenumber, an ‘interface’ thickness parameter, and Ra. For large Ra, the dispersion relations confirm that the optimally amplified wavelength is about five times the gap width, with the exact value depending on the interface thickness. The corresponding growth rate is in very good agreement with the experimental data as well. The eigenfunctions indicate that the predominant fluid motion occurs within the plane of the Hele-Shaw cell. However, for large Ra purely two-dimensional modes are also amplified, for which there is no motion in the spanwise direction. Scaling laws are provided for the dependence of the maximum growth rate, the corresponding wavenumber, and the cutoff wavenumber on Ra and the interface thickness. Furthermore, the present results are compared both with experimental data, as well as with linear stability results obtained from the Hele-Shaw equations and a modified Brinkman equation.

Linear Stability Analysis in a Compressible Axisymmetric Swirling Jet

2014 ◽  
Vol 574 ◽  
pp. 15-20
Author(s):  
Zhi Wei Guo ◽  
Si Min Shen ◽  
Wei Min Feng ◽  
Bo Fu Wang
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Previous Analysis ◽  
Optimal Growth ◽  
Flow Dynamics ◽  
Swirl Number ◽  
Swirling Jet ◽  
Optimal Growth Rate ◽  
The Stability

Temporal linear stability of a compressible axisymmetric swirling jet is investigated. The present work extends a previous analysis to include the effects of swirl number on the stability of flow dynamics. Results obtained show that the optimal growth rate of disturbance for azimuthal wavenumber n = -1 is larger than that for n = -2 while the corresponding frequencies for both n increases as axial wavenumber increases. As swirl number q increases, the optimal growth rate of disturbance also increases. What is more, there is an optimal swirl number for small axial wavenumbers, which is different from the situation for medium and large axial wavenumbers.

Sediment-laden fresh water above salt water: linear stability analysis

2011 ◽  
Vol 691 ◽  
pp. 279-314 ◽  
Author(s):  
P. Burns ◽  
E. Meiburg
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
Fresh Water ◽  
Settling Velocity ◽  
Interface Thickness ◽  
Particle Settling ◽  
Salinity Field ◽  
Unstable Layer ◽  
Stability Results ◽  
Double Diffusive

AbstractWhen a layer of particle-laden fresh water is placed above clear, saline water, both Rayleigh–Taylor and double diffusive fingering instabilities may arise. For quasi-steady base profiles, we obtain linear stability results for such situations by means of a rational spectral approximation method with adaptively chosen grid points, which is able to resolve multiple steep gradients in the base state density profile. In the absence of salinity and for a step-like concentration profile, the dominant parameter is the ratio of the particle settling velocity to the viscous velocity scale. As long as this ratio is small, particle settling has a negligible influence on the instability growth. However, when the particles settle more rapidly than the instability grows, the growth rate decreases inversely proportional to the settling velocity. This damping effect is a result of the smearing of the vorticity field, which in turn is caused by the deposition of vorticity onto the fluid elements passing through the interface between clear and particle-laden fluid. In the presence of a stably stratified salinity field, this picture changes dramatically. An important new parameter is the ratio of the particle settling velocity to the diffusive spreading velocity of the salinity, or alternatively the ratio of the unstable layer thickness to the diffusive interface thickness of the salinity profile. As long as this quantity does not exceed unity, the instability of the system and the most amplified wavenumber are primarily determined by double diffusive effects. In contrast to situations without salinity, particle settling can have a destabilizing effect and significantly increase the growth rate. Scaling laws obtained from the linear stability results are seen to be largely consistent with earlier experimental observations and theoretical arguments put forward by other authors. For unstable layer thicknesses much larger than the salinity interface thickness, the particle and salinity interfaces become increasingly decoupled, and the dominant instability mode becomes Rayleigh–Taylor-like, centred at the lower boundary of the particle-laden flow region.

On the Surface Stability of a Spherical Void Embedded in a Stressed Matrix

2005 ◽  
Vol 74 (1) ◽  
pp. 8-12 ◽  
Author(s):  
Jérôme Colin
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Applied Stress ◽  
Spherical Harmonics ◽  
Linear Stability Analysis ◽  
Spherical Cavity ◽  
Spherical Void ◽  
Surface Stability ◽  
Infinitesimal Perturbation

The linear stability analysis of the shape of a spherical cavity embedded in an infinite-size matrix under stress has been performed when infinitesimal perturbation from sphericity of the rod is assumed to appear by surface diffusion. Developing the perturbation on a basis of complete spherical harmonics, the growth rate of each harmonic Ylm(θ,φ) has been determined and the conditions for the development of the different fluctuations have been discussed as a function of the applied stress and the order l of the perturbation.

Linear stability analysis of two-way coupled particle-laden density current

2017 ◽  
Vol 95 (3) ◽  
pp. 291-296 ◽  
Author(s):  
Pouriya Amini ◽  
Ehsan Khavasi ◽  
Navid Asadizanjani
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Base Flow ◽  
Interfacial Instability ◽  
Linear Stability Theory ◽  
Flow Density ◽  
Density Current ◽  
Density Currents

Stability of two-way coupled particle-laden density current is studied with the aim of linear stability analysis. Interfacial instability can be found in density currents, which effects entrainment and the rate of effective mixing. In this paper, we investigate the density current interfacial instability using linear stability theory, considering the particles attendance. The ultimate goal is to extract the governing equation for current with particles and study the effect of different parameters on stability of such currents. Base flow has velocity and density profiles of tangent hyperbolic type. Main current and particles are studied in two separate phases. It is found that current will be more stable as M0 (M0 = S∗N∗/ρ∗ where ρ∗ is the non-dimensional flow density, S∗ is the Stokes’ drag coefficient, and N∗ is the particles’ number density) grows, this is a result of number of particles and their radius, and also viscosity effects. The current is more stable as the growth rate increases. As the Richardson number in M0 rises, the growth rate value decreases. As the slope of the river bed increases, the current is less stable.

scholarly journals The Growth and Saturation of Submesoscale Instabilities in the Presence of a Barotropic Jet

2018 ◽  
Vol 48 (11) ◽  
pp. 2779-2797 ◽  
Author(s):  
Megan A. Stamper ◽  
John R. Taylor ◽  
Baylor Fox-Kemper
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Initial Conditions ◽  
Boussinesq Equations ◽  
Small Region ◽  
Computational Domain ◽  
Mean Flow ◽  
The Mean

AbstractMotivated by recent observations of submesoscales in the Southern Ocean, we use nonlinear numerical simulations and a linear stability analysis to examine the influence of a barotropic jet on submesoscale instabilities at an isolated front. Simulations of the nonhydrostatic Boussinesq equations with a strong barotropic jet (approximately matching the observed conditions) show that submesoscale disturbances and strong vertical velocities are confined to a small region near the initial frontal location. In contrast, without a barotropic jet, submesoscale eddies propagate to the edges of the computational domain and smear the mean frontal structure. Several intermediate jet strengths are also considered. A linear stability analysis reveals that the barotropic jet has a modest influence on the growth rate of linear disturbances to the initial conditions, with at most a ~20% reduction in the growth rate of the most unstable mode. On the other hand, a basic state formed by averaging the flow at the end of the simulation with a strong barotropic jet is linearly stable, suggesting that nonlinear processes modify the mean flow and stabilize the front.

scholarly journals Scales of Linear Baroclinic Instability and Macroturbulence in Dry Atmospheres

2009 ◽  
Vol 66 (6) ◽  
pp. 1821-1833 ◽  
Author(s):  
Timothy M. Merlis ◽  
Tapio Schneider
Keyword(s):  
Growth Rate ◽  
Linear Stability ◽  
General Circulation ◽  
Baroclinic Instability ◽  
Circulation Model ◽  
Length Scale ◽  
Stability Analyses ◽  
Scaling Relationship ◽  
Wide Range ◽  
Nonlinear Simulations

Abstract Linear stability analyses are performed on a wide range of mean flows simulated with a dry idealized general circulation model. The zonal length scale of the linearly most unstable waves is similar to the Rossby radius. It is also similar to the energy-containing zonal length scale in statistically steady states of corresponding nonlinear simulations. The meridional length scale of the linearly most unstable waves is generally smaller than the energy-containing meridional length scale in the corresponding nonlinear simulations. The growth rate of the most unstable waves increases with increasing Eady growth rate, but the scaling relationship is not linear in general. The available potential energy and barotropic and baroclinic kinetic energies of the linearly most unstable waves scale linearly with each other, with similar partitionings among the energy forms as in the corresponding nonlinear simulations. These results show that the mean flows in the nonlinear simulations are baroclinically unstable, yet there is no substantial inverse cascade of barotropic eddy kinetic energy from the baroclinic generation scale to larger scales, even in strongly unstable flows. Some aspects of the nonlinear simulations, such as partitionings among eddy energies, can be understood on the basis of linear stability analyses; for other aspects, such as the structure of heat and momentum fluxes, nonlinear modifications of the waves are important.

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