Non-normal stability analysis of a shear current under surface gravity waves

The stability of a horizontal shear current under surface gravity waves is investigated on the basis of the Rayleigh equation. As the differential operator is non-normal, a standard modal analysis is not effective in capturing the transient growth of a perturbation. The representation of the stream function by a suitable basis of bi-orthogonal eigenfunctions allows one to determine the maximum growth rate of a perturbation. It turns out that, in the considered range of parameters, such a growth rate can be two orders of magnitude larger than the maximum eigenvalue obtained by standard modal analysis.

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The effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000012467 ◽

1998 ◽

Vol 40 (2) ◽

pp. 190-206

Author(s):

Sudebi Bhattacharyya ◽

K. P. Das

Keyword(s):

Growth Rate ◽

Evolution Equation ◽

Gravity Waves ◽

Fourth Order ◽

Deep Water ◽

Water Surface ◽

Spectral Width ◽

Surface Gravity ◽

Surface Gravity Waves ◽

The Stability

AbstractThe effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline is studied. A previously derived fourth order nonlinear evolution equation is used to find a spectral transport equation for a narrow band of surface gravity wave trains. This equation is used to study the stability of an initially homogeneous Lorentz shape of spectrum to small long wave-length perturbations for a range of spectral widths. The growth rate of the instability is found to decrease with the increase of spectral widths. It is found that the fourth order term in the evolution equation produces a decrease in the growth rate of the instability. There is stability if the spectral width exceeds a certain critical value. For a vanishing bandwidth the deterministic growth rate of the instability is recovered. Graphs have been plotted showing the variations of the growth rate of the instability against the wavenumber of the perturbation for some different values of spectral width, thermocline depth, angle of perturbation and wave steepness.

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Instabilities of a Baroclinic, Double Diffusive Frontal Zone

Journal of Physical Oceanography ◽

10.1175/2007jpo3770.1 ◽

2008 ◽

Vol 38 (4) ◽

pp. 840-861 ◽

Cited By ~ 10

Author(s):

W. D. Smyth

Keyword(s):

Growth Rate ◽

Frontal Zone ◽

Baroclinic Instability ◽

Flux Ratio ◽

Maximum Growth ◽

Maximum Growth Rate ◽

Variable Diffusivity ◽

The Stability ◽

Parameter Values ◽

Double Diffusive

Abstract The linear theory of double diffusive interleaving is extended to take account of baroclinic effects. This study goes beyond previous studies by including the possibility of modes with nonzero tilt in the alongfront direction, which allows for advection by the baroclinic frontal flow. This requires that the stability equations be solved numerically. The main example is based on observations of interleaving on the lower flank of Meddy Sharon, but a range of parameter values is covered, leading to conclusions that are relevant in a variety of oceanic regimes. The frontal zone is treated as infinitely wide with uniform gradients of temperature, salinity, and alongfront velocity. The stationary, vertically symmetric interleaving mode is shown to have maximum growth rate when its alongfront wavenumber is zero, providing validation for previous studies in which this property was assumed. Besides this, there exist two additional modes of instability: the ageostrophic Eady mode of baroclinic instability and a mode not previously identified. The new mode is oblique (i.e., it tilts in the alongfront direction), vertically asymmetric, and propagating. It is strongly dependent on boundary conditions, and its relevance in the ocean interior is uncertain as a result. Effects of variable diffusivity and buoyancy flux ratio are also considered.

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Stability of the solar wind against the whistler mode

Journal of Plasma Physics ◽

10.1017/s0022377800021528 ◽

1978 ◽

Vol 20 (2) ◽

pp. 225-230 ◽

Cited By ~ 5

Author(s):

P. Revathy

Keyword(s):

Growth Rate ◽

Solar Wind ◽

Relative Velocity ◽

Maximum Growth ◽

Maximum Growth Rate ◽

The Sun ◽

Thermal Velocity ◽

Whistler Mode ◽

Minimum Value ◽

The Stability

The stability of the solar wind against the whistler mode is analyzed. It is shown that the solar wind can become unstable owing to this mode after a distance of 100 R from the sun. The minimum value of the relative velocity (Ur) between ∝-particles and protons for the excitation of this instability is the proton thermal velocity αi. The maximum growth rate at 200 R occurs for the values of the parameters kρi = 0·3,(T┤/T∥)i = 0·3 and Ur/αi = 0·5.

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Kelvin–Helmholtz instability under horizontal rotation and magnetic fields

Journal of Plasma Physics ◽

10.1017/s0022377897006314 ◽

1998 ◽

Vol 59 (2) ◽

pp. 193-209

Author(s):

L. A. DÁVALOS-OROZCO

Keyword(s):

Magnetic Field ◽

Growth Rate ◽

Fluid Layer ◽

Maximum Growth ◽

Maximum Growth Rate ◽

Density Stratification ◽

Velocity Shear ◽

Sufficient Condition ◽

Horizontal Shear ◽

Helmholtz Instability

The author's previous work on the Rayleigh–Taylor instability is extended to the Kelvin–Helmholtz instability, and the maximum growth rate of a perturbation and an estimate of its upper bound is obtained for an infinite fluid layer under horizontal rotation where the density, horizontal velocity (shear) and magnetic field are continuously stratified in the direction of gravity. Conclusions are drawn about the possibility of stability for some directions of propagation of the perturbation, even in the case of unstably stratified density. It is also shown that the new terms that appear owing to the interaction of the horizontal shear flow, horizontal rotation and stratified magnetic field increase the range of values that contribute to the estimate of the maximum growth rate compared with previous work. Furthermore, a generalization of the sufficient condition for stability under horizontal rotation alone obtained by Johnson is calculated in the presence of density stratification. A new method is also given to obtain a sufficient condition for stability when a magnetic field is present in addition to rotation and density stratification.

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On the inviscid stability of bi-layer axisymmetric coatings

Journal of Fluid Mechanics ◽

10.1017/s0022112008001560 ◽

2008 ◽

Vol 605 ◽

pp. 389-400

Author(s):

P. A. BLYTHE ◽

P. G. SIMPKINS

Keyword(s):

Growth Rate ◽

Double Layer ◽

Composite Coatings ◽

Density Ratio ◽

Maximum Growth ◽

Reynolds Numbers ◽

Maximum Growth Rate ◽

Length Scale ◽

Layer Composite ◽

The Stability

This paper is concerned with the stability of fibre coatings at large Reynolds numbers. Both single- and double-layer coatings are considered; no restriction is placed on the coating thicknesses. Calculations for the maximum growth rate, together with the corresponding length scale of the instability, are presented. Rescaling with respect to the maximum growth rate generates universal dispersion relations over the unstable wavenumber range. For double-layer composite coatings, modifications are required when the density ratio becomes large.

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Water-bag stability theory for planar bounded plasmas with counter-streaming injection. Part 1. The neutralized diode

Journal of Plasma Physics ◽

10.1017/s0022377800025599 ◽

1975 ◽

Vol 14 (1) ◽

pp. 143-152 ◽

Cited By ~ 2

Author(s):

K. M. Hu ◽

E. H. Klevans

Keyword(s):

Growth Rate ◽

Lower Order ◽

Stability Theory ◽

Electron Distribution ◽

Electron Beams ◽

Maximum Growth ◽

Maximum Growth Rate ◽

Higher Order Modes ◽

The Stability ◽

System Approaches

The stability of a bounded, homogeneous, neutralized plasma with counter- streaming electron beams is analysed. A water-bag model is used to describe the electron distribution in velocity space, so that finite beam temperature and a background plasma are included in the theory. For boundary conditions, the absorber– source wall (the diode boundary) and the reflecting wall are considered. For the former, growth-rate calculations indicate that the instability is a combination of charge bunching (counter-streaming) and diode circuit effect. As the diode length increases, the growth rate of all modes in the system approaches the maximum growth rate. For the reflecting wall, as the length increases, the maximum growth rate transfers to higher and higher order modes with shorter wavelength, while the growth rate of the lower-order modes goes to zero.

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