Unstable Linear Modes Research Articles

Balance regimes for the stability of a jet in an f-plane shallow water system

To study a limit of validity of balanced models, stability of a zonal jet is investigated both linearly and nonlinearly in an f-plane shallow water system for a wide range of parameter. It is shown that quasi-geostrophic approximation gives not only a good estimation of maximum growth rate for high Rossby number, Ro, but also is valid even in the nonlinear phase of instability for high Ro as long as Froude number, Fr, is low. While the maximum growth rate of unstable modes is well estimated by the quasi-geostrophic approximation, dominant balance is different between high and low Ro. In the low Ro regime ( Ro < 5 ), geostrophic balance is dominant in the perturbation field, the ratio ∥ φ ∥ / ∥ ψ ∥ of the amplitudes of divergent flow to that of rotational flow is proportional to Fr 2 / Ro , where φ and ψ are velocity potential and streamfunction, respectively. On the other hand, in the high Ro regime ( Ro > 5 ), cyclostrophic balance with basic shear is dominant, ∥ φ ∥ / ∥ ψ ∥ ∝ Fr 2 . Considering that the barotropic instability is caused by the resonance of neutral Rossby wave modes, we can explain the difference of the ratio in each regime. Using the ratio ∥ φ ∥ / ∥ ψ ∥ being small, different approximation of the linear shallow water equations for each regime is deduced. Properties of the linear unstable modes are explained with these approximations.

Magnetic field induced by elliptical instability in a rotating spheroid

The tidal or the elliptical instability of the rotating fluid flows is generated by the resonant interaction of the inertial waves. In a slightly elliptically deformed rotating sphere, the most unstable linear mode is called the spin-over mode, and is a solid body rotation versus an axis aligned with the maximum strain direction. In the non-viscous case, this instability corresponds to the median moment of the inertial instability of the solid rotating bodies. This analogy is furthermore illustrated by an elliptical top experiment, which shows the expected inviscid heteroclinic behaviour. In geophysics, the elliptical instability may appear in the molten liquid cores of the rotating planets, which are slightly deformed by the tidal gravitational effects of the close bodies. It may then participate in the general outer core dynamics and possibly the geodynamo process. In this context, Kerswell and Malkus (Kerswell, R.R. and Malkus, W.V.R., Tidal instability as the source for Io's magnetic signature. Geophys. Res. Lett., 1998, 25, 603–606) showed that the puzzling magnetic field of the Jovian satellite Io may indeed be induced by the elliptically unstable motions of its liquid core that deflect the Jupiter's magnetic field. Our magnetohydrodynamics (MHD) experiment is a toy-experiment of this geophysical situation and demonstrates for the first time the possibility of an induction of a magnetic field by the flow motions due to the elliptical instability. A full analytical calculation of the magnetic dipole induced by the spin-over is presented. Finally, exponential growths of this induced magnetic field in a slightly deformed rotating sphere filled with galinstan liquid metal are measured for different rotating rates. Their growth rates compare well with the theoretical predictions in the limit of a vanishing Lorentz force.

Experimental characterization of the instability of the vortex ring. Part I: Linear phase

The results of experiments performed to study the linear phase of the instability of vortex rings are presented. The experiments were performed in water. The vortex rings are generated by pushing water through the cylindrical nozzle of a pipe submerged in an aquarium. The experiments were made with the help of planar laser induced fluorescence as well as 2D2C and 2D3C particle image velocimetry. They show the straining field causing the instability, and for the first time experimentally the growth of a band of linear unstable modes. They also confirm previous studies concerning the shape of the instability and theories predicting the number of waves and the bandwidth of unstable modes. However, the measurement of the growth rate shows the influence of viscous damping, and consequently, the limit of the theories based on the hypothesis of an ideal fluid.

Collisionless reconnection of magnetic field lines in dusty plasmas

Stability of a thin current sheet in an inhomogeneous dusty plasma is investigated. A new mechanism for the collisionless reconnection of the magnetic field is found. It is associated with the linear instability of a surface shear-Alfvén mode which exists at the boundary between two plasma regions with different orientations of the sheared magnetic field. This new mode is possible only in dusty plasmas in which the electron and ion densities are inhomogeneous and different from each other. The dispersion properties of the surface mode are found to be strongly dependent on the spatial profile of the current sheet. The unstable linear mode is found in the case of a current sheet with steep edges. It is identified as the bending- (or kink-) mode, whose electrostatic and vector potentials, φ and Az, are symmetric and antisymmetric functions, respectively. In such a plasma configuration, the surface tearing mode (φ-antisymmetric, Az-symmetric) cannot propagate as an eigenmode. It arises only as a forced mode through the resonant coupling with the bending mode and the local convective cell. The increment of the bending/tearing instability is estimated from the second-order contribution of the resonance.

The paradox of the scale-free discs

Scale-free discs have no preferred length or time-scale. The question has been raised whether such discs have a continuum of unstable linear modes or perhaps no unstable modes at all. We resolve this paradox by analysing the particular case of a gaseous, isentropic disc with a completely flat rotation curve (the Mestel disc) exactly. The heart of the matter is this: what are the correct boundary conditions to impose at the origin or central cusp? We argue that the linear stability problem is ill-posed and that similar ambiguities may afflict general disc models with power-law central cusps. From any finite radius, waves reach the origin after finite time but with logarithmically divergent phase. Instabilities exist, but their pattern speeds depend upon an undetermined phase with which waves are reflected from the origin. For any definite choice of this phase, there is an infinite but discrete set of growing modes. The ratio of growth rate to pattern speed is independent of the central phase. This ratio is derived in closed form for non-self-gravitating normal modes and is shown to agree with approximate results obtained from the shearing sheet in the short-wavelength limit. This provides the first exact, analytically solved stability analysis for a differentially rotating disc. For self-gravitating normal modes, the ratio of growth rate to pattern is found numerically by solving recurrence relations in Mellin-transform space.

On the stability of stationary nongyrotropic distribution functions: Coupling and purely growing waves

Unperturbed particle distributions exhibiting gyrophase bunching can linearly couple the eigenmodes of parallel propagation and destabilize otherwise stable magnetoplasmas. Previous research on the parallel stability of nongyrotropic populations has either considered homogeneous but time‐varying distributions (TNG) associated with closed phase spaces (solutions of the homogeneous Vlasov equation) or homogeneous, stationary distributions (SNG) arising from open phase spaces (solutions of the Vlasov, or transport equation with source and sink terms). Destabilization of otherwise stable media in the TNG case was only reported when the unperturbed gyrophase bunching generated a finite perpendicular current that coupled the electrostatic and electromagnetic parallel eigenmodes at frequencies shifted by multiples of the cyclotron frequency of the TNG species. Here we demonstrate that introduction of SNG in a particle population of an otherwise stable magnetoplasma can (1) bring about unstable linear mode coupling among the parallel eigenmodes for finite perpendicular currents (as was the case for TNG distributions) albeit now without the frequency shifts, and (2) stimulate nonoscillatory purely growing waves (zero real frequency within a finite wavenumber band) in nongyrotropic environments that only couple parallel electromagnetic waves (zero unperturbed perpendicular current), behavior not encountered in TNG media.

Steady states and oscillatory instability of swirling flow in a cylinder with rotating top and bottom

In this study we present a numerical investigation of steady states, onset of oscillatory instability, and slightly supercritical oscillatory states of an axisymmetric swirling flow of a Newtonian incompressible fluid in a cylinder, with independently rotating top and bottom. The first part of the study is devoted to the influence of co- and counter-rotation of the bottom on the steady vortex breakdown, which takes place in the well-known problem of flow in a cylinder with a rotating top. It is shown that weak counter-rotation of the bottom may suppress the vortex breakdown. Stronger counter-rotation may induce a stable steady vortex breakdown at relatively large Reynolds numbers for which a vortex breakdown does not appear in the case of the stationary bottom. Weak corotation may promote the vortex breakdown at lower Reynolds numbers than in the cylinder with the stationary bottom. Stronger corotation leads to the detachment of the recirculation zone from the axis and the formation of an additional vortex ring. The second part of the study is devoted to the investigation of the onset of oscillatory instability of steady flows. It is shown that the oscillatory instability sets in due to a Hopf bifurcation. The critical Reynolds number and the critical frequency of oscillations were calculated as a function of the rotation ratio (ξ=Ωbottom/Ωtop) for a fixed value of the aspect ratio γ (height/radius) of the cylinder γ=1.5. The stability analysis showed that there are several most unstable linear modes of the perturbation that become successively dominant with a continuous change of ξ. It is shown that the oscillatory instability may lead to an appearance and coexistence of more than one oscillating separation vortex bubble.

Nonlinear shear instabilities of alongshore currents on plane beaches

Evidence for the existence in the nearshore surf zone of energetic alongshore propagating waves with periods O(100 s) and wavelengths O(100 m) was found from observations by Oltman-Shay et al. (1989). These oscillations have wavelengths that are much too short to be surface gravity waves at the observed frequencies. The existence and properties of the wave-like motions were found to be related to the presence, strength and direction of an alongshore current in the surf zone. Based on a linear stability analysis of a mean alongshore current with offshore scale O(100 m), Bowen &amp; Holman (1989) described these fluctuations as unstable waves associated with a shear instability. Good agreement of wavelengths and wave speeds from observations and from predictions based on the most unstable linear mode was obtained by Dodd et al. (1992). The nonlinear dynamics of finite-amplitude shear instabilities of alongshore currents in the surf zone are studied here utilizing numerical experiments involving finite-difference solutions to the shallow water equations for idealized forced dissipative initial-value problems. Plane beach (i.e. constant slope) geometry is used with periodic boundary conditions in the alongshore direction. Forcing effects from obliquely incident breaking surface waves are approximated by an across-shore-varying steady force in the alongshore momentum equation. Dissipative effects are modelled by linear bottom friction. The solutions depend on the dimen-sionless parameter Q, which is the ratio of an advective to a frictional time scale. The steady frictionally balanced, forced, alongshore current is linearly unstable for Q less than a critical value Qc. The response of the fluid is studied for different values of ΔQ = Qc - Q. In a set of experiments with the alongshore scale of the domain equal to the wavelength 2π/k0 of the most unstable linear mode, disturbances that propagate alongshore in the direction of the forced current with propagation velocities similar to the linear instability values are found for positive ΔQ. The disturbances equilibrate with constant amplitude for small ΔQ and with time-varying amplitudes for larger ΔQ. For increasing values of ΔQ the behaviour of this fluid system, as represented in a phase plane with area-averaged perturbation kinetic energy and area-averaged energy conversion as coordinates, is similar to that found in low-dimensional nonlinear dynamical systems including the existence of non-trivial steady solutions, bifurcation to a limit cycle, period-doubling bifurcations, and irregular chaotic oscillations. In experiments with the alongshore scale of the domain substantially larger than the wavelength of the most unstable linear mode, different behaviour is found. For small positive ΔQ, propagating disturbances grow at wavelength 2π/k0. If ΔQ is small enough, these waves equilibrate with constant or spatially varying amplitudes. For larger ΔQ, unstable waves of length 2π/k0 grow initially, but subsequently evolve into longer-wavelength nonlinear propagating steady or unsteady wave-like disturbances with behaviour dependent on ΔQ. The eventual development of large-scale nonlinear propagating disturbances appears to be a robust feature of the flow response over plane beach geometry for moderate, positive values of ΔQ and indicates the possible existence in the nearshore surf zone of propagating finite-amplitude shear waves with properties not directly related to results of linear theory.

Nonlinear Pulsation Driven by Strange Modes in Hydrodynamic Models

Abstract The characteristics of the nonlinear pulsation of hydrodynamic models, which have only strange modes as pulsational unstable modes, are compared with those by conventional modes. It is shown that nonlinear pulsation with the strange modes has considerable amplitudes only at the hydrogen ionization zone and in the atmosphere. This is sharp in contrast to that by conventional modes, and is identical with the characteristics of the unstable linear strange modes. We thus conclude that the unstable strange modes may be responsible for the nonlinear pulsation. We also discuss the physical origin of the instability of the strange modes. Since the modes are pulsationally unstable without any luminosity perturbation, other physical mechanisms must be involved for the instability along with the conventional kappa and gamma mechanisms, which are familiar in Cepheid variables. We suggest that the density inversion in the H ionization region, including the adjacent He I ionization zone, may be the origin. Finally, we suggest that the different pulsation driving zones may cause the feature of the spatial amplitude variations.

Density and magnetic field fluctuations observed by ISEE 1-2 in the quiet magnetosheath

Abstract. We analyse the fluctuations of the electron density and of the magnetic field in the Earth's magnetosheath to identify the waves observed below the proton gyrofrequency. We consider two quiet magnetosheath crossings i.e. 2 days characterized by small-amplitude waves, for which the solar wind dynamic pressure was low. On 2 August 1978 the spacecraft were in the outer magnetosheath. We compare the properties of the observed narrow-band waves with those of the unstable linear wave modes calculated for an homogeneous plasma with Maxwellian electron and bi-Maxwellian (anisotropic) proton and alpha particle distributions. The Alfvén ion cyclotron (AIC) mode appears to be dominant in the data, but there are also density fluctuations nearly in phase with the magnetic fluctuations parallel to the magnetic field. Such a phase relation can be explained neither by the presence of a proton or helium AIC mode nor by the presence of a fast mode in a bi-Maxwellian plasma. We invoke the presence of the helium cut-off mode which is marginally stable in a bi-Maxwellian plasma with α particles: the observed phase relation could be due to a hybrid mode (proton AIC+helium cut-off ) generated by a non-Maxwellian or a non-gyrotropic part of the ion distribution functions in the upstream magnetosheath. On 2 September 1981 the properties of the fluctuations observed in the middle of the magnetosheath can be explained by pure AIC waves generated by protons which have reached a bi-Maxwellian equilibrium. For a given wave mode, the phase difference between B\\Vert and the density is sensitive to the shape of the ion and electron distribution functions: it can be a diagnosis tool for natural and simulated plasmas.

Nonlinear Motion of a Shallow Water Barotropic Vortex

Nonlinear motions of a shallow water barotropic vortex on a β plane differ substantially from the analogous linear motions. The nonlinear model described here, in which wavenumber 1–3 asymmetries interact with each other and the mean vortex, predicts that an initially completely cyclonic vortex will accelerate toward the NNW, reaching a speed of 2.5 m s−1 at 48 h. During the rest of the 240-h calculation, the speed varies by <0.5 m s−1 as the direction turns from NNW to NW. The vortex accelerations are in phase with temporal changes of vortex-relative angular momentum (LR). The turning of the track coincides with a transition of the wavenumber 1 asymmetry from a single dipole to a double dipole. The latter structure appears to be another orthogonal solution of the second-order radial structure equation for a neutral linear normal mode. The corresponding linear model, in which β forces only wavenumber 1, shows only the single dipole structure and straight NNW accelerating motion that reaches a speed of 9 m s−1 at 240 h. The slower motion in the nonlinear model stems from wave-induced changes in the axisymmetric vortex and vacillation between the orthogonal modal structures. A nonlinear calculation with zero initial LR on a β plane follows a curving path dictated by a barotropically unstable linear mode for the first 144 h. Subsequently, the double dipole structure for that mode appears as the track turns toward the NW and the speed accelerates from 1 to 2 m s−1. A spatially uniform geostrophic environmental current on an f plane causes vortex motion by advection and by propagation. The potential vorticity (PV) gradient due to the current acts much as β does. Although the PV gradient is typically 0.1 of that due to β, the induced propagation toward high potential vorticity is ½–¼ of that on a β plane because super-position of the vortex on the geopotential gradient amplifies the PV gradient's effect. In a quiescent environment on an f plane, initial asymmetries that project onto the normal modes induce long-lasting motion that retains about half its speed to 240 h. If the initial speed is ≥2 m s−1, vacillation between orthogonal modal structures may cause dramatic turns and accelerations of the vortex track.

Momentum Flux, Flow Symmetry, and the Nonlinear Barotropic Governor

Abstract A number of idealized life-cycle simulations of baroclinically unstable waves are systematically analyzed to study the effects of eddy momentum flux and of zonal mean horizontal shear on the finite-amplitude evolution of the waves. Twenty-level quasigeostrophic and primitive equation models with channel geometry are numerically integrated with the most unstable linear normal mode as an initial condition. The flows are inviscid except for weak second-order horizontal diffusion. It is found that the finite-amplitude baroclinic waves are sensitively influenced by the vertically integrated eddy momentum flux of the normal mode via the large barotropic shear it spins up in the mean flow. This “barotropic governor” mechanism prevents the eddy from attaining all the available potential energy stored in the domain, leading to irreversible barotropic decay. Only in the purely baroclinic, f-plane, quasigeostrophic problem, where the vertically integrated eddy momentum flux identically vanishes due to symme...

Linear instability of a mixed‐layer front

The linear instabilities of a geostrophically balanced frontal jet are computed numerically in an effort to understand the origin of intense small‐scale cold‐core features observed in the North Atlantic subtropical convergence zones during the Frontal Air‐Sea Interaction Experiment (FASINEX). A simple analytic continuously‐stratified basic state is used to represent the observed frontal structure. The most unstable linear modes have e‐folding timescales of 2–3 days, alongfront wavelengths near 70 km, and significant asymmetry across the front. The results provide indirect evidence that baroclinic instability of the frontal jet is a possible source of the cold‐core features observed during FASINEX.

Self-consistent drift wave turbulence, paradigm for transport

As a paradigm for electron-based anomalous transport involving several fluctuating quantities, collisional drift wave turbulence is studied numerically in a sheared slab configuration. Nonlinear mode structure changes allow self-sustaining of the turbulence, even though no unstable linear modes exist. The responsible physics violates all the assumptions behind mixing-length, or turbulence-as-diffusion models, and so the turbulence is necessarily not of that type. The key ingredients are shared by all important transport candidates. Further, this single model displays several important properties known from tokamak transport-isotope mass trends, up-gradient transport ('inward pinch'), and ExB shear effects-which have lacking or dubious explanations within the existing framework. It can therefore serve as a useful basis for building new, and more realistic, scenarios for inhomogeneous turbulent transport.

The Role of Diffusive Effects On Potential Vorticity In Fronts

AbstractTheoretical studies of two‐dimensional fronts have shown that frictional processes at the time of occlusion and/or diabatic effects imply low‐level, positive potential vorticity (PV) anomalies. It is shown here that significant amounts of PV can be produced in an evolving baroclinic wave (taken to be the two‐dimensional semigeostrophic Eady wave) purely owing to the presence of a no‐slip, insulating lower boundary. A simple analytic expression is given for the time evolution of the mass‐weighted volume‐average PV, (PV), up to the point of frontal collapse. For the most unstable linear normal mode it is found, at the time of the formation of the frontal discontinuity, that the PV has increased by approximately a third of its basic‐state value.The bulk formulation for surface fluxes of heat and momentum is used to illustrate the general role of boundary‐layer processes in modifying PV locally. Turbulent fluxes which decrease linearly from maximum values at the surface to zero at the top of the planetary boundary layer (PBL) are used to recover a simple analytical form for the PV generation tendency. the magnitudes of the PV anomalies produced by flows typical of mid‐latitude cyclones and associated fronts are highlighted by vertically integrating the tendency equation over the PBL. It is shown that the generation of PV in the PBL depends on the vertical differences of velocity and potential temperature over the depth of the PBL; and on the vertical vorticity and horizontal potential‐temperature gradient at the top of the PBL. For example, there is a local source of PV if there is cooler air at the top of the boundary layer to the right of the surface wind vector, i.e. if the thermal wind has a component opposing the wind at anemometer level. the production of PV anomalies by surface processes, using this formulation, is independent of the occlusion process.Purely horizontal mixing of heat in the vicinity of a potential‐temperature anomaly is examined to predict the magnitude of local PV anomalies, consistent with the internal ‘eddy’ viscosity often used in numerical models.Numerical integrations of a two‐dimensional Eady wave with a detailed surface‐layer description and stability‐dependent eddy viscosity coefficients also illustrate that significant growth in the PV occurs before the warm sector is lifted off the surface. There is a direct correspondence between this growth and the decrease of the horizontally‐averaged surface potential temperature, as predicted from the no‐slip, insulating boundary theory. Also, the local PV anomalies predicted by the bulk PBL representation are shown to be in close agreement with those produced in the numerical simulation, which uses a more comprehensive scheme.

Dynamics of the Coastal Transition Zone jet: 2. Nonlinear finite amplitude behavior

The finite amplitude nonlinear behavior of the coastal transition zone (CTZ) jet, assumed to be governed by quasi‐geostrophic dynamics, is studied utilizing numerical experiments in an idealized geometry. Finite difference solutions to initial value problems are obtained for a stratified, six‐layer fluid in a periodic, flat bottom, ƒ plane channel. The initial flow field in all experiments includes a uniform along‐channel jet with horizontal and vertical structure obtained from CTZ hydrographic and current measurements made in May 1987. The maximum velocity in the jet is about 0.5 m s−1, the vertical shear is such that the velocities below 500 m depth are small, and the jet width is about 60 km. The Rossby radius for the first baroclinic mode is 24.6 km. An analysis of the linear stability of the jet flow field in part 1 shows that the jet is linearly unstable to disturbances with along‐jet wavelengths greater than 50 km. The largest growth rate is found for a wavelength around 260 km. The objectives here are to find the characteristics of the finite amplitude nonlinear jet instabilities and to examine local dynamical balances for signatures of instability processes that would help with the physical interpretation of results from limited‐area CTZ data assimilation models. Experiments are run with periodic channels of different length L(x), where the initial flow includes the basic jet and a small perturbation in the form of the most unstable linear mode for a wavelength equal L(x). In the basic experiment (BC) L(x) = 250 km. One experiment (LC) is run in a long channel L(x) = 1280 km with initial disturbances forced by application of a weak wind stress curl for three days. For times less than about 70 days, experiments BC and LC show the growth and development of finite amplitude meanders in the CTZ jet with spatial variations similar to those observed in spring 1987. During this early time period, the amplitudes of the meanders grow at rates of 1–4 km d−1, increasing with meander amplitude, and the meanders propagate in the direction of the jet upper layer flow at phase velocities of 5–3 km d−1, decreasing with meander amplitude. Initially, at small amplitudes the instability involves comparable contributions from barotropic and baroclinic processes in agreement with linear theory, but for large amplitude meanders the baroclinic energy conversion processes dominate. The vertical velocity field develops a characteristic spatial structure in relation to the jet meanders as do terms in the kinetic energy balance corresponding to the time rate of change of kinetic energy following fluid particles and to the rate of conversion from potential energy. The spatial structure of the latter field may provide a useful indication for jet baroclinic instability processes in limited‐area models. At later times in experiment BC (days 90–110), the meander growth is limited by a pinching‐off process that results in the formation of detached eddies on both sides of the jet. This process is characterized by a large relative increase of the kinetic energy in the lower layers and in the depth‐independent component of the flow. Experiments with L(x) = 180 and 150 km give qualitatively different behavior than that found in experiments BC and LC. The meander growth is bounded, and the jet flow field exhibits time‐dependent variations in the volume integrated kinetic and potential energies. For large time the flow may asymptotically approach a nearly steady state or states with irregular or nearly periodic oscillations in the integrated kinetic and potential energies.

An experimental investigation of resonant heat transfer enhancement in grooved channels

Experiments on hydrodynamic resonance and its effect on heat transfer in laminar flows through ducts with periodically spaced transverse grooves cut into one wall are presented. Oscillatory perturbation of the flow rate at the frequency of the most unstable linear modes results in a resonant response with associated mixing and significant heat transfer enhancement. Measurements confirm previous calculations demonstrating that a 20% flow rate oscillation at the optimal frequency more than doubles the convective heat transfer coefficient. The experimental frequency band for significant enhancement is, however, broader than the numerically predicted one.

Nonlinear behavior of weakly unstable linear modes in plasmas with aligned equilibrium mass flow

The nonlinear evolution of weakly unstable linear magnetohydrodynamic waves in plasma configurations characterized by equilibrium mass flow aligned parallel to the magnetic field is investigated. The analysis is based on field representations expressed as expansions in Fourier harmonics and computation of modifications to the linear theory (first harmonic) arising from the nonlinear coupling between the zeroth and second harmonics. It is found that Landau’s equation governs the nonlinear evolution of the system. A correction to the linearized theory appears at third order in the mode amplitude and is proportional to the linear growth rate. Furthermore, the conditions under which the nonlinearities provide a saturation mechanism to limit the growth of the linear instabilities is explored. Finally, this saturation phenomenon for a specific configuration consisting of a Z pinch with a small axial magnetic field is illustrated.

The gravitational instability of collisionless particles in a cosmological self-similar shell

A flat Friedmann-Robertson-Walker (FRW) model has been used to study the nature of gravitationally unstable linear modes of an expanding, self-similar, spherical shell made of cold collisionless particles. Dimensionless growth rates of the fundamental radial modes were calculated as a function of the spherical eigennumber l, and compared with previous calculations of linear stability in the thin shell approximation. Two major differences were observed: first, the growth rates for small l were significantly smaller in this case, although it remains true that even l = 1 modes are gravitationally unstable. Second, the dimensionless growth rate grows with logarithmic dependence on l when l is very large. The latter result is partly due to the fact that the collisionless particles have no nonradial motions, and partly due to the long-range nature of gravitational interactions. It is found that the spherical shell is unstable on all scales. Consequently, a generic positive energy perturbation in a cold self-gravitating medium will not evolve into a spherical configuration. The effects of including a finite thermal velocity dispersion and the astrophysical implications of the results are discussed. 35 refs.

Instability and saturation of drift-convective modes in an inhomogeneous plasma

It is found that the inclusion of the electron inertia effect (parallel to an external magnetic field) can provide a linear coupling between the electrostatic drift and the convective modes in a non-uniform plasma. This coupling leads to new branches of rapidly growing modes, which are calculated in the kinetic as well as in the hydrodynamic regimes. To study the saturation of the linear unstable modes, we account for the mode coupling and derive a set of model nonlinear fluid equations. A perturbation technique is employed to obtain a nonlinear evolution equation. In the steady state, the latter yields the saturated electric potential. It is argued that the enhanced low-frequency fluctuations can cause anomalous particle transport in a magnetoplasma.