Marginal Stability Criterion Research Articles

Ion and electron beam effects on drift wave instability with different distribution functions - particle aspect analysis

Electrostatic drift waves are studied in the presence of ion and electron beams with various distribution functions. The method of particle aspect analysis has been adopted to evaluate the dispersion relation and growth rate of the wave. It is assumed that a low β (ratio of plasma pressure to magnetic pressure) plasma is composed of resonant and non-resonant particles. The resonant particles participate in the energy exchange with the wave while non-resonant particles support the oscillatory motion of the drift wave. Marginal stability criteria have been discussed. The effect of ion and electron beam velocities on the dispersion relation and the growth rate for various distribution indices has been discussed. An increase of growth rate with the distribution index is predicted. Dispersion relations and growth rate are calculated for space plasma parameters.

Stabilization of collisionless trapped particle modes in tokamaks

A quadratic form is derived to study the stability of collisionless trapped particle modes in tokamaks. Marginal stability criteria show that stability can be obtained by controlling the electron temperature profile. In a low aspect ratio equilibrium (such as the recently proposed Comet [Comments Plasma Phys. XII, 125 (1989)]), the stability requirement is ηe≂O(1) (ηe=∂ ln Te/∂ ln ne); in a standard large aspect ratio tokamak with a broad density profile, the requirement is LTe/R≂O(1) (L−1Te=−∂ln Te/∂r). Possible application of the second scenario to H‐mode plasmas is noted.

Kinetic instability of a gyrating ring distribution with application to satellite pickup in planetary magnetospheres

We assume that pickup ions released by a satellite in a rapidly rotating planetary magnetosphere can be modeled by a gyrating ring distribution with a finite isotropic thermal velocity spread. This distribution can provide a source of free plasma energy for the excitation of plasma waves, and we explore the kinetic instability of the distribution. Specifically, we obtain general analytical solutions for the (linear) growth rate of oblique plasma waves due to resonant particle-wave interactions. Advantages of the analytical approach are that in the limits of either quasi-longitudinal or quasi-transverse propagation the solutions reduce to simple algebraic expressions that may be readily used to compare contributions from any specific harmonic resonance. Further, the analytical solutions are especially powerful for treating the case of oblique wave propagation, the standard procedure for which has hitherto been tedious and potentially inaccurate numerical integration. We obtain a marginal stability criterion appropriate to the ring distribution, and we use the analytical solutions to calculate the marginal stability frequency as a function of propagation angle for first-order cyclotron resonance with whistler-mode waves. Ramifications of the results for pickup ions and electrons released by satellites in the rapidly rotating magnetospheres of the giant planets are briefly discussed.

Three-dimensional equilibrium and stability of ionospheric plasma clouds

The equilibrium and stability of three-dimensional (3-D) ionospheric plasma clouds is investigated. The equilibrium density profile is assumed to be a 3-D water bag with a circular cross section transverse to B, and an ellipsoidal cross section parallel to B. The equilibrium potential consists of two parts: a polarization potential driven by a uniform neutral wind, and an ambipolar potential driven by parallel electron pressure. The latter potential is shown to cause sheared azimuthal plasma flows perpendicular to B. A stability analysis of this equilibrium is carried out. It is found that the E×B instability, which causes the structuring of ionospheric plasma clouds, can be stabilized by the shear in the azimuthal flow. A marginal stability criterion is derived. On the basis of this stability criteria, striations of barium clouds in the ionospheric F region should be stable to further bifurcation, as is observed

Toroidal internal kink stability in tokamaks with ultra flat q profiles

Linear stability properties of the ideal internal kink mode in toroidal geometry are calculated for equilibria in which the q(r) profile is very flat and q ≈ 1 in the core region of the plasma. Marginal stability criteria and growth rates are calculated analytically in the large aspect ratio limit and compared with numerical results from the FAR code. The theory is developed for m = n = 1 modes and for higher m (= n) modes. The temperature perturbation due to adiabatic expansion in the linear phase of the 1/1 mode is calculated and compared with experimental data, showing that the predictions from ideal MHD theory cannot account for the observed temperature increase on JET. Comparison with collisionless theory suggests that the observed temperature increase can be accounted for by compression of the trapped particles.

Analytical solutions for the growth of oblique waves in a plasma with a field-aligned beam

The theory of kinetic instabilities in a plasma when the instabilities are driven by wave-particle resonance is addressed. Recent progress on the linear theory of resonant oblique wave growth in plasmas is extended to plasmas modeled by three standard forms of distribution, a bi-Maxwellian, a bi-Lorenzian, and a loss cone, each incorporating a field-aligned beam. The wave growth rates are shown to be functions of dimensionless integrals that can be expressed in terms of Bessel functions of argument equal to a normalized wave-normal variable. A marginal stability criterion is obtained, and accordingly we identify a threshold beam velocity for unstable growth of the waves. The simple analytical results derived can be readily applied to a wide variety of problems on oblique wave growth in space plasmas that hitherto were amenable only to extensive computer calculations.

A WKB type analysis of marginal stability in directional solidification

We propose a simple and physical approach to the marginal stability criterion of pattern selection in directionally solidifying systems. The technique involves a WKB type analysis of instability caused by perturbations on the steady state shape of the cell.

Cellular and dendritic growth in a binary alloy melt: A marginal stability approach

A simple model describing “constrained” growth of an “array” of cells or dendrites in a binary alloy melt, in the presence of an imposed positive temperature gradient in the liquid, has been presented. The dendrite or cell tip radius is calculated using the marginal stability criterion proposed by Langer and Müller-Krumbhaar. The results obtained are compared with the predictions deriving from two other approaches: the first approach adopts the “ad hoc” assumption of “minimum undercooling” at the cell or dendrite tips, the second approach is based on a closely similar stability criterion proposed by Trivedi. It has been shown that all three approaches predict tip radii, differing by no more than 30%. All three approaches yield the following simple relationship between the tip radius, rt, and the growth conditions: σc = A(1 − s) where σc = 2lcDL/Rr2t and s = DLGL/RΔT0 are dimensionless parameters, and A is a constant whose value depends on the criterion used to determine the tip radius. Recent experimental data in succinonitrile-acetone system have been shown to be in good agreement with these predictions. These comparisons also indicate that during “constrained” dendritic growth in an alloy, in the presence of a positive temperature gradient in the liquid, σc = 1/(2π2), exactly twice the value of σ∗ = 1/(4π2) = 0.0253 obtained by Langer and Müller-Krumbhaar for “free” dendritic growth in a supercooled pure melt. It is shown that the increase in the value of the tip stability parameter, σc, may be related to a transition in morphology from a cellular structure, with just a few side branches (for which s = 0.50), to a more “fully developed” dendritic structure (for which s → 0 or GL/R → 0).

Salt concentration profiles for marginal stability in solar ponds

In salt-stabilized solar ponds, the establishment and maintenance of salt gradients forms one of the significant factors controlling the economics of energy conversion. Hence, considerable attention is directed towards minimizing the expenditure. Recently, Elwell's analysis of the marginal stability criterion has highlighted the interesting possibility of stabilizing solar ponds with much smaller concentrations than the conventionally adopted saturation concentration at the pond bottom. A general treatment for the use of the stability criterion for evaluating concentration profiles has been given. The stabilities of different layers of pond fluids with marginal stability conditions have been discussed.

Theory of dendritic growth under rapid solidification conditions

A first order perturbation analysis is developed to study the stability of a moving interface and the results are applied to characterize the growth rate and tip radius of dendritic plates under rapid solidification conditions. A marginal stability criterion is developed which approaches the Langer and Müller-Krumbhaar stability result, VR 2 = constant, as the Peclet number goes to zero. Our results for the constrained growth also approach the Mullins-Sekerka absolute stability condition when the solute Peclet number becomes very large. The role of crystalline symmetry and interface constraints on the stability criterion is discussed.

Interdendritic Spacing: Part I. Experimental Studies

Directional solidification experiments have been carried out in a succinonitrile-5.5 mol pct acetone system to characterize dendrite tip radius and interdendrite spacings as functions of growth rate and temperature gradient in the liquid. A maximum in primary dendrite spacing as a function of growth rate is observed, and this maximum is found to occur at the dendrite-cellular transition velocity. A scaling law is shown to exist between the secondary dendrite arm spacing, λ2, near the tip and the dendrite tip radius, p, which is λ2/ρ = 2.2 ± 0.3. Experimental results on ρ have been found to agree with the theoretical model based on the marginal stability criterion.

Propagating Pattern Selection

Pattern selection is discussed in regard to a situation where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. The velocity of the propagating front and the wavelength of the pattern formed behind the front are determined by a marginal-stability criterion. The special system studied here has a Lyapunov functional, but the periodic state which propagates is not the one which minimizes the functional.

Linear theory of the E×B instability with an inhomogeneous electric field

A general linear theory of the E×B instability is developed which considers an ambient electric field that is at an arbitrary angle to the density gradient and allows the electric field component parallel to the density gradient to be inhomogeneous. A differential equation is derived which describes the mode structure of the unstable waves in the direction of the inhomogeneities. The theory (1) includes ion inertia effects, (2) allows for arbitrary density and electric field profiles, and (3) is valid in the long‐wavelength regime, i.e., kyL < 1, where L is the width of the boundary layer. The main results of the analysis are as follows. First, the inhomogeneous velocity flow caused by the inhomogeneous electric field can stabilize the instability. Second, short‐wavelength modes are preferentially stabilized over longer‐wavelength modes. Third, the stabilization mechanism is associated with the velocity shear due to an x‐dependent resonance [ω − kyVy(x)]−1, where Vy(x) = −cEx(x)/B, and not velocity shear terms explicitly proportional to ∂Vy/∂x or ∂²Vy/∂x². Fourth, the marginal stability criterion is weakly dependent on the magnitude of νin/ω. Applications of these results to ionospheric phenomena are discussed, viz., barium cloud striations and high‐latitude F region irregularities.

Marginal stability of impulsively initiated Couette flow and spin-decay

A viscous, incompressible fluid is contained within an infinitely long circular cylinder or between a pair of infinitely long concentric cylinders. In both cases, the entire system is in a state of rigid-body rotation. At time t = 0 the outer flow boundary is impulsively brought to rest, giving rise to a potentially unstable, unsteady swirl flow. The stability of these flows is examined by employing energy theory with a marginal stability criterion to obtain lower bounds on the onset times for instability. In some cases, the asymptotic steady-state flow will be stable, with any instability merely a transient, while in other cases, stability of the asymptotic state is not guaranteed.

Particle aspect analysis of electromagnetic ion cyclotron instability

The electromagnetic ion cyclotron instability, incorporating the details of charged particles trajectory, has been discussed. The wave is assumed to propagate along the ambient magnetic field in an anisotropic plasma. The plasma is considered to consist of resonant and non-resonant particles. Here it is assumed that resonant particles participate in energy exchange with the wave whereas the non-resonant particles support the oscillatory motion of the wave. Growth rate has been obtained by an energy conservation method for an arbitrary velocity distribution function. The effect of non-resonant particles on the resultant growth rate has also been discussed. The maximum growth rate has been evaluated for various values of β (ratio of kinetic pressure to magnetic pressure) and [Formula: see text]. The marginal stability criterion has also been derived. The time development of the ion cyclotron instability and the change in the resonant particles energy have been computed for magnetospheric plasma.

Ion acoustic stability analysis of the Earth's bow shock

Application of a marginal ion acoustic stability criterion to five thin bow shock observations provides new evidence in support of the criterion as the basis for determining the thickness of the magnetic gradients of subcritical, quasi‐perpendicular shock structures.

Study of drift-wave turbulence by microwave scattering in a toroidal plasma

A study of drift-wave turbulence by miowave scattering technique was carried out in a toroidal plasma confinement device, the FM-1 spherator. The principal results are: (a) Observation of the linear dispersion relation of drift waves in the high-magnetic-shear condition. The linear dispersion relation is followed up to the weak turbulence state (γ ≲ ω). (b) An experimental demonstration of the stabilization of the drift waves with an increase of the shear strength. In particular, the ion-mass dependence of the observed marginal stability criterion for drift waves (ω≈ω*) was in reasonably good agreement with theoretical expectations. Further, very-low-frequency (ω≪ω*) fluctuations were enhanced when the shear strength was lowered, (c) A detailed study of the behaviour of drift waves in a weak shear condition where a state of isotropic turbulence was observed. The turbulent density fluctuations showed a strong dependence on frequency, but no dependence on wave number (in sharp contrast to theoretical expectations). The possible dependence of the anomalous loss process on the observed drift-wave turbulence is also discussed.

The deepening of the wind-Mixed layer

Abstract A simple model is given that describes the response of the upper ocean to an imposed wind stress. The stress drives both mean and turbulent flow near the surface, which is taken to mix thoroughly a layer of depth h, and to erode the stably stratified fluid below. A marginal stability criterion based on a Froude number is used to close the problem, and it is suggested that the mean momentum has a strong role in the mixing process. The initial deepening is predicted to obey where u. is the friction velocity of the imposed stress, N the ambient buoyancy frequency, and t the time. After one-half inertial period the deepening is arrested by rotadeon at a depth h = 22/4 u.{(Nf)+ where f is the Coriolis frequency. The flow is then a “mixed Ekman” layer, with strong inertial oscillations superimposed on it. Three quarters of the mean energy of the deepening layer is found to be kinetic, and only one-quarter potential. Heating and cooling are included in the model, but stress dominates for time-scales of ...

Prevalent Instability of Nonthermal Plasmas

In thermally anisotropic plasmas or sheet pinches, the zero-order counterstreaming or directed streaming of like particles gives rise to instability modes that correspond essentially to the formation of local first-order pinches within the plasma. The marginal stability criteria for all such modes can be interpreted in terms of Bennett's pinch condition. The range of examples discussed includes Weibel's thermally anisotropic hot-electron plasma instability in null field, which is generalized for hot ions and finite magnetic field, and which is applied to interpenetrating neutral plasma streams, to a spherical electrostatic plasma-containment scheme, and to the Astron thermonuclear experiment. The familiar ``mirror instability'' is shown to be another example of a ``pinch''-type mode. Sheet pinches with or without longitudinal magnetic field are generally unstable against tearing into subsidiary linear pinches.