Growth Rates Of Unstable Waves Research Articles

Baroclinic Instability of Nonzonal Flows and Bottom Slope Effects on the Propagation of the Most Unstable Wave

AbstractA linear, two-layer potential vorticity (PV) equation model on the β plane is employed to study the baroclinic instability of nonzonal basic currents flowing over a uniform bottom slope. Criteria for the instability, phase speed, and growth rate of unstable waves are all given as functions of the basic shear velocity, β, and bottom slope. The study suggests two kinds of long wave cutoff, one induced by the slope and the other by β; the first one exists in all directions, while the second requires at least a slight deviation of the wave vector from the meridians. Subtle differences between configurations of the PV gradient lead to completely different characteristics of unstable perturbations, such as propagation and scale. In the case of a positive slope (the bottom slope in the same direction as the isopycnal tilt), the fastest-growing wave is capable of propagating across the basic flow streamlines. By contrast, in the case of a negative slope (the bottom slope opposed to the isopycnal tilt), the most unstable wave always propagates along the streamlines. In addition, the spatial scale of the most unstable mode can be heavily reduced by a negative slope.

Current-Driven Low-Frequency Electrostatic Waves in a Collisional Strongly Coupled Magnetized Dusty Plasma

Current-driven low-frequency electrostatic waves are studied in a collisional strongly coupled magnetized dusty plasma. The theoretical model for the longitudinal and transverse modes is developed in the regime $\omega \tau _{m} \ll 1$ for the collisional strongly coupled magnetized dusty plasma in the presence of drifting currents. It is observed that the dispersion properties of low-frequency electrostatic waves are modified under the effect of magnetic field-aligned drifting ion currents in a strongly coupled collisional magnetized dusty plasma. The frequency rate and growth rate of longitudinal and transverse modes increase with normalized wave number. Moreover, the growth rate of unstable waves is destabilized with normalized drift velocity of ions for both longitudinal and transverse modes. The critical drift velocity of ions decreases with normalized wave number but increases with increase in the strong correlation parameter $\Gamma $ . The effect of strong coupling parameter on the frequency and growth rate of low-frequency electrostatic modes in the presence of current drift and collisions is also investigated.

Toward the Selection of Control Parameters in Models of the Purposeful Weather and Climate Modification

A unified methodology for planning and implementation of weather and climate modification (geoengineering) can be apparently developed based on the ideas and methods of geophysical cybernetics in which the climate system and its processes represent a control object and the role of the controlling subsystem is given to the appropriate social structures and, in particular, to operators having the necessary resources. This problem can be solved based on the sensitivity theory of dynamical systems. In this paper, as an example, we study the impact of the main parameters that control the development of baroclinic instability in the atmosphere on the growth rate of unstable waves. Analytical expressions for the absolute and relative sensitivity coefficients are obtained, which allows estimating the model response to the control parameters and, therefore, drawing a conclusion about the hypothetical ability to control the large-scale wave dynamics in the atmosphere and ocean. Selecting the baroclinic instability as a subject of this study is due to the significant role of this physical mechanism in the formation of the general circulation of the atmosphere and ocean, and, consequently, the Earth's climate.

Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack

The direct numerical simulation of boundary layer transition over a 5° half-cone-angle blunt cone is performed. The free-stream Mach number is 6 and the angle of attack is 1°. Random wall blow-and-suction perturbations are used to trigger the transition. Different from the authors’ previous work [Li et al., AIAA J. 46, 2899 (2008)], the whole boundary layer flow over the cone is simulated (while in the author’s previous work, only two 45° regions around the leeward and the windward sections are simulated). The transition location on the cone surface is determined through the rapid increase in skin fraction coefficient (Cf). The transition line on the cone surface shows a nonmonotonic curve and the transition is delayed in the range of 20°≤θ≤30° (θ=0° is the leeward section). The mechanism of the delayed transition is studied by using joint frequency spectrum analysis and linear stability theory (LST). It is shown that the growth rates of unstable waves of the second mode are suppressed in the range of 20°≤θ≤30°, which leads to the delayed transition location. Very low frequency waves (VLFWs) are found in the time series recorded just before the transition location, and the periodic times of VLFWs are about one order larger than those of ordinary Mack second mode waves. Band-pass filter is used to analyze the low frequency waves, and they are deemed as the effect of large scale nonlinear perturbations triggered by LST waves when they are strong enough.

Relativistic warm plasma waveguide under the effects of plasma electrons and HF electrical field on Buneman instability

Abstract The stabilization effect of a strong HF (pump) electrical field and plasma electrons on a two-stream (Buneman) instability in a plane relativistic warm plasma waveguide is investigated; using the separation method to solve the two-fluid plasma model we separate the problem into two parts. The “temporal” (dynamical) part enables us to determine the frequencies and growth rates of unstable waves; this part within the redefinition of natural frequencies coincides with the system describing HF suppression of Buneman instability in uniform unbounded plasma. Natural frequencies of oscillations and spatial distribution of the amplitude of the self-consistent electrical field are determined from the solution of a boundary-value problem (“space part”) taking into account specific spatial distribution of plasma density. Plasma electrons are considered to have a relativistic thermal velocity. It is shown that the growth rate of instability in relativistic warm plasma is reduced compared to non-relativistic (cold or warm) plasma and relativistic cold plasma. In addition, it is found that the plasma electrons have no effect on the solution of the space part of the problem.

Interaction between an eddy and a zonal jet: Part II. Two-and-a-half-layer model

In a two-and-a-half-layer quasi-geostrophic model, a process study is conducted on the interaction between a vortex and a zonal jet, both with constant potential vorticity. The vortex is a stable anticyclone, initially located north of the eastward jet. The potential vorticity of the jet is allowed to have various vertical structures, while the vortex is concentrated in only one layer. The flow parameters are set to values characteristic of the Azores region. First, the jet is stable. Weak vortices steadily drift north of the jet without crossing it while strong vortices can cross the jet and tear off a cyclone with which they pair as a heton (baroclinic dipole). This heton often breaks later in the shear exerted by the jet; the two vortices finally drift apart. When crossed by deep anticyclones, the jet develops meanders with 375 km wavelength. These results exhibit a noticeable similarity with the one-and-a-half-layer case studied in Part I. Secondly, the jet is allowed to be linearly unstable. In the absence of the vortex, it develops meanders with 175 km wavelength and 25-day e-folding time on the β-plane. For various vertical structures of the jet, baroclinic instability is shown to barely affect jet–vortex interaction if the linear growth rate of unstable waves is smaller than 1/(14 days). Further simulations with a linearly unstable, nonlinearly equilibrated jet evidence its strong temporal variability when crossed by a deep vortex on the β-plane. In particular, long waves can dominate the spectrum for a few months after jet crossing by the vortex. Again in this process, the deep vortex couples with a surface cyclone and both drift southwestward.

Stabilization effect of a strong HF electrical field on beam–plasma interaction in a relativistic plasma waveguide

The influence effect of a strong HF electrical field on the excitation of surface waves by an electron beam under the development of instability of low-density electron beam passing through plane relativistic plasma is investigated. Starting from the two fluid plasma model we separate the problem into two parts. The “temporal” (dynamical) part enables us to find the frequencies and growth rates of unstable waves. This part within the redefinition of natural (eigen) frequencies coincides with the system describing HF suppression of the Buneman instability in a uniform unbounded plasma. Natural frequencies of oscillations and spatial distribution of the amplitude of the self-consistent electrical field are obtained by solving a boundary-value problem (“spatial” part) considering a specific spatial distribution of plasma density. Plasma electrons are considered to have a relativistic velocity. It is shown that a HF electric field has no essential influence on dispersion characteristics of unstable surface waves excited in a relativistic plasma waveguide by a low-density electron beam. The region of instability only slightly narrowing and the growth rate decreases by a small parameter and this result has been reduced compared to nonrelativistic plasma. Also, it is found that the plasma electrons have not affected the solution of the space part of the problem.

Linear stability of a gas boundary layer flowing past a thin liquid film over a flat plate

The flow of a gas stream past a flat plate under the influence of rainfall is investigated. As raindrops sediment on the flat plate, they coalesce to form a water film that flows under the action of shear from the surrounding gas stream. In the limit of (a) large Reynolds number, Re, in the gas phase, (b) small rainfall rate, r˙, compared to the free-stream velocity, U∞, and (c) small film thickness compared to the thickness of the boundary layer that surrounds it, a similarity solution is obtained that predicts growth of the liquid film like x3/4; x denotes dimensionless distance from the leading edge. The flow in the gas stream closely resembles the Blasius solution, whereas viscous dissipation dominates inside the film. Local linear stability analysis is performed, assuming nearly parallel base flow in the two streams, and operating in the triple-deck regime. Two distinct families of eigenvalues are identified, one corresponding to the well-known Tollmien–Schlichting (TS) waves that originate in the gas stream, and the other corresponding to an interfacial instability. It is shown that, for the air–water system, the TS waves are convectively unstable whereas the interfacial waves exhibit a pocket of absolute instability, at the streamwise location of the applied disturbance. Moreover, it is found that as the inverse Weber number (We−1) increases, indicating the increasing effect of surface tension compared to inertia, the pocket of absolute instability is translated towards larger distances from the leading edge and the growth rate of unstable waves decreases, until a critical value is reached, We−1 ≈ We−1c, beyond which the family of interfacial waves becomes convectively unstable. Increasing the inverse Froude number (Fr−1), indicating the increasing effect of gravity compared to inertia, results in the pocket of absolute instability shrinking until a critical value is reached, Fr−1 ≈ Fr−1c, beyond which the family of interfacial waves becomes convectively unstable. As We−1 and Fr−1 are further increased, interfacial waves are eventually stabilized, as expected. In this context, increasing the rainfall rate or the free-stream velocity results in extending the region of absolute instability over most of the airfoil surface. Owing to this behaviour it is conjectured that a global mode that interacts with the boundary layer may arise at the interface and, eventually, lead to three-dimensional waves (rivulets), or, under extreme conditions, even premature separation.

Advanced Sub-Model for Airblast Atomizers

A three-dimensional viscous stability analysis has been carried out to model airblast atomization. The model considers an annular liquid sheet downstream of an airblast atomizer and incorporates essential features, such as three-dimensional disturbances, liquid viscosity, innerand outerair swirl, and e nite e lm thickness. Effects of axial velocity, air swirl, and liquid viscosity on the growth rates of various disturbance modes have been examined in detail.Ithasbeenfoundthatincreasingtherelativeaxialvelocitybetweentheliquid andthegasphasessignie cantly improves the fuel atomization. The inner and outer air moving together enhances the instability of the liquid sheet more signie cantly than only the inner or outer air. When air swirl is absent, theaxisymmetricmode dominates the breakup process of the liquid sheet. Liquid viscosity is found to have a twofold effect: reduce the growth rates of unstable waves and shift the dispersion diagram toward long waves. Air swirl not only promotes the instability of theliquidsheet,butalso switchesthedominantmodefrom theaxisymmetricmodetoahelicalmode.Acombination of the inner and outer air swirl improves airblast atomization more signie cantly than a single air swirl.

The effect of air swirl profile on the instability of a viscous liquid jet

A temporal linear stability analysis has been carried out to predict the instability of a viscous liquid jet surrounded by a swirling air stream with three-dimensional disturbances. The effects of flow conditions and fluid properties on the instability of the liquid jet are investigated via a parametric study by varying axial Weber number axial velocity ratio of the gas to liquid phase, swirl Weber numbers, density ratio and the Ohnesorge number. It is observed that the relative axial velocity between the liquid and gas phases promotes the interfacial instability. As the axial Weber number increases, the growth rates of unstable waves, the most unstable wavenumber and the unstable range of wavenumbers increase. Meanwhile, the increasing importance of helical modes compared to the axisymmetric mode switches the breakup regime from the Rayleigh regime to the first wind-induced regime and on to the second wind-induced regime. The predicted range of wavenumbers in which the first helical mode has higher growth rates than the axisymmetric mode agrees very well with experimental data. Results show that the destabilizing effects of the density ratio and the axial Weber number are nearly the same. Liquid viscosity inhibits the disintegration process of the liquid jet by reducing the growth rate of disturbances and by shifting the most unstable wavenumber to a lower value. Moreover, it damps higher helical modes more significantly than the axisymmetric mode. Air swirl has a stabilizing effect on the liquid jet. As air swirl strength increases, the growth rates of helical modes are reduced more significantly than that of the axisymmetric mode. The air swirl profile is found to have a significant effect on the instability of the liquid jet. The global, as well as local, effects of the swirl profile are examined in detail.

Nonparallel instabilities of a supersonic mixing layer

Abstract Contrary to the parallel flow approach most often studied, the growth rates of unstable waves of the supersonic mixing layer are analyzed with the consideration of flow nonparallelism by employing the method of multiple scales. Flow parameters including Reynolds number, Mach number, velocity ratios and temperature ratios influencing on the instability‐wave solutions are investigated. The mean flow is determined by solving the compressible boundary layer equations and applying the Sutherland law relating the viscosity to the temperature. In this work, two dimensional compressible stability equations are solved to secure the spatial growth rate of instability waves of the flow.

Ion-acoustic wave instability driven by drifting electrons in a generalized Lorentzian distribution

A generalized Lorentzian (kappa) particle distribution function is useful for modelling plasma distributions with a high-energy tail that typically occur in space. The modified plasma dispersion function is employed to study the instability of ion-acoustic waves driven by electron drift in a hot isotropic unmagnetized plasma modelled by a kappa distribution. The real and imaginary parts of the wave frequency ω0 + ιγ are obtained as functions of the normalized wavenumber kλD, where λD is the electron Debye length. Marginal stability conditions for instability are obtained for different ion-to-electron temperature ratios. The results for a kappa distribution are compared with the classical results for a Maxwellian. In all cases studied the ion-acoustic waves are strongly damped at short wavelengths, kλD ≫ 1, but they can be destabilized at long wavelengths. The instability for both the kappa and Maxwellian distributions can be quenched by increasing the ion-electron temperature ratio Ti/Te. However, both the marginally unstable electron drift velocities and the growth rates of unstable waves can differ significantly between a generalized Lorentzian and a Maxwellian plasma; these differences are also influenced by the value of Ti/Te.

Baroclinic Instability of Flows Along Sloping Boundaries

In the two-layer quasi-geostrophic model with boundaries sloping perpendicular to the basic flow, the ratios of the slopes of the bottom and the top to that of the interface between the fluid layers in the basic state are important parameters in the expression of the growth rate of unstable waves. When Eady's (1949) model is extended to include sloping bottom and top boundaries, the growth rates of unstable waves depend on the ratios of the slopes of the bottom and the top to that of the isentropes of the basic state. For the Eady model with sloping bottom, an important parameter characterizing the instability is the ratio between the vertical and horizontal heat transports by the wave divided by the slope of the isentropes of the basic state. An interpretation of these ratios and their relations clarifies the stabilization of the system for large slopes, the variation of the wavelength of the most unstable wave with the bottom slope, and the destabilization of some short waves for negative bottom slopes. It is found that the most unstable wave of the system has zero vertical energy flux convergence at the sloping bottom.

The effect of the Helmholtz term on a two-layer model baroclinic atmosphere

The effects of the inclusion of a Helmholtz term in a quasi-geostrophic model, formulated as part of a lower boundary condition, are investigated using a two-layer model. The Helmholtz term has two main effects, both of which appear desirable in a quasi-geostrophic model. As well as reducing the phase speed of all waves, and thereby controlling the otherwise large retrograde motion of ultra-long Rossby waves, it reduces growth rates of unstable waves. The Helmholtz term reduces the westward slope with height of unstable waves, more so for shorter wave lengths. The Helmholtz term is shown to be important for all quasi-geostrophic models. However, its use with an enhanced factor of about 10, although desirable for “best” forecasts, has little physical basis and merely serves to offset shortcomings of the model. DOI: 10.1111/j.2153-3490.1978.tb00866.x

The effect of the Helmholtz term on a two-layer model baroclinic atmosphere

The effects of the inclusion of a Helmholtz term in a quasi-geostrophic model, formulated as part of a lower boundary condition, are investigated using a two-layer model. The Helmholtz term has two main effects, both of which appear desirable in a quasi-geostrophic model. As well as reducing the phase speed of all waves, and thereby controlling the otherwise large retrograde motion of ultra-long Rossby waves, it reduces growth rates of unstable waves. The Helmholtz term reduces the westward slope with height of unstable waves, more so for shorter wave lengths. The Helmholtz term is shown to be important for all quasi-geostrophic models. However, its use with an enhanced factor of about 10, although desirable for “best” forecasts, has little physical basis and merely serves to offset shortcomings of the model. DOI: 10.1111/j.2153-3490.1978.tb00866.x

Resonant interactions of shelf waves with wind-generated effects

Abstract A periodic wind produces an oscillating response when it blows across an ocean overlying a continental shelf. In this paper, the response of a frictionless, horizontally non-divergent ocean to a wind system which travels along the shelf is considered. It is shown that the periodic response produced by such wind systems may have a resonant non-linear interaction with shelf waves whose frequency is close to half that of the wind. The theory may have applications to east-west continental shelves (for example, the south coast of Australia) when strong weather systems move eastward along the shelf. A prerequisite for the onset of resonance is that the travelling wind system may be characterised as a progressive wave whose frequency and wavenumber are close to twice those of any free shelf wave. The growth rate of unstable waves is exponential in time or distance with exponent proportional to the strength of the non-linear interaction. The bandwidth of the unstable wavenumbers and the bandwidth of wavenumbers and frequencies characterising possible resonant wind systems are also proportional to the strength of the interaction.

Growing transverse waves in a plasma in a magnetic field

Instabilities in uniform collisionless plasmas may be relevant to the under-standing of several astrophysical and geophysical problems and to the confinement of plasmas in the laboratory. This paper presents a practicable method for ascertaining the instability of such plasmas in a uniform steady magnetic field with respect to disturbances propagating along the direction of that field. The method is applied to two examples of plasmas having anisotropic pressure, that is, having the pressure p ‖ along the steady magnetic field not equal to the pressure p ⊥ perpendicular to the field. The examples given are those of an electron plasma (with fixed neutralizing background) and of a plasma with two species having equal masses and opposite but equally large charges. The latter example is treated because it is mathematically simpler than an electron-ion plasma, and is referred to as an “electron-positron” plasma. In both examples when p ⊥ > p ‖ unstable electromagnetic type waves occur for all values of the dc magnetic field strength. When p ⊥ > p ‖ a new family of unstable hydromagnetic waves is found in the “electron-positron” plasma. Steady waves with frequency Ω(p ⊥ − p ‖) p ⊥ , where Ω is the electron gyrofrequency, are found in both examples for p ⊥ > p ‖. When p ‖ > p ⊥, classical results for the “hose” instability are recovered and are extended outside the guiding center limit in the case of the plasma with two oppositely charged species. The electron gas with fixed neutralizing background is found to be stable for p ‖ > p ⊥. Most of the results are found to be applicable to real (electron-ion) plasmas. Growth rates of unstable waves are treated in some detail.

Baroclinic instability in two layer systems

The stability properties of the two layer baroclinic model with arbitrary horizontal shear are discussed. Necessary conditions for instability are derived and show that the potential vorticity gradient must be both positive and negative. Bounds are placed on the propagation speed and growth rate of unstable waves by showing that the complex phase speed must lie in a certain semi-circle in the complex velocity plane. Sufficient conditions for instability are also discussed for a certain clam of flows.