Threshold Condition For Instability Research Articles

Regulation of Solar Wind Electron Temperature Anisotropy by Collisions and Instabilities

Typical solar wind electrons are modeled as being composed of a dense but less energetic thermal “core” population plus a tenuous but energetic “halo” population with varying degrees of temperature anisotropies for both species. In this paper, we seek a fundamental explanation of how these solar wind core and halo electron temperature anisotropies are regulated by combined effects of collisions and instability excitations. The observed solar wind core/halo electron data in (β ∥, T ⊥/T ∥) phase space show that their respective occurrence distributions are confined within an area enclosed by outer boundaries. Here, T ⊥/T ∥ is the ratio of perpendicular and parallel temperatures and β ∥ is the ratio of parallel thermal energy to background magnetic field energy. While it is known that the boundary on the high-β ∥ side is constrained by the temperature anisotropy-driven plasma instability threshold conditions, the low-β ∥ boundary remains largely unexplained. The present paper provides a baseline explanation for the low-β ∥ boundary based upon the collisional relaxation process. By combining the instability and collisional dynamics it is shown that the observed distribution of the solar wind electrons in the (β ∥, T ⊥/T ∥) phase space is adequately explained, both for the “core” and “halo” components.

Linear Theory Analysis and One‐Dimensional Hybrid Simulations of High‐Frequency EMIC Waves in a Dipole Magnetic Field

AbstractNarrowband (Δf ≲ 0.1fcp), high‐frequency (0.9fcp ≲ f < fcp) electromagnetic ion cyclotron (EMIC) waves, or HFEMIC waves for short, are a relatively new type of EMIC waves, where fcp is the proton cyclotron frequency. Here, we investigate the instability threshold conditions and the nonlinear evolution of HFEMIC waves at parallel propagation. First, linear theory analysis is extended to a regime relevant to HFEMIC waves (parallel proton beta β‖p < 0.01). The instability threshold follows a similar anisotropy‐β‖p relation and requires a large value of anisotropy (T⊥p/T‖p ≳ 10) for wave growth. As a result of decreasing group velocity, the convective growth rate at a fixed threshold exhibits a similar inverse relation with β‖p. Heavy ions affect the instability only weakly, primarily through the introduction of stop bands. Second, we carry out one‐dimensional hybrid simulations in a parabolic background magnetic field, with initial parameters constrained by observation. Despite the narrow source region (within about ±3° latitude), HFEMIC waves can grow well above the thermal noise level due in large part to a small group velocity of HFEMIC waves. The saturation level is well within the range of observational amplitudes, and the quasilinear process primarily determines the wave evolution. Finally, we demonstrate that the present results compare favorably to the recent statistical results, thereby supporting anisotropic low‐energy protons as free energy source for HFEMIC waves.

Ion velocity shear‐induced drift wave and momentum transport in non‐Maxwellian magnetoplasma flows

AbstractWe have investigated the diamagnetic flow in a non‐uniform partially ionized plasma with non‐Maxwellian electron population to explain the dynamics of ion velocity shear‐induced low‐frequency drift mode and associated instabilities. The dispersion relations are found, and instability threshold conditions are pointed out along with ion‐parallel momentum transport due to drift motion with relative phase shift of fluctuating quantities in a non‐conservative system. The real frequencies and instability growth rates are studied numerically and illustrated for typical space and laboratory plasmas. This study should be useful in understanding some aspects of low‐frequency time‐delayed perturbations with sheared flow leading to drift instabilities and cross‐field parallel ion momentum transport in nonuniform magnetoplasmas containing a non‐Maxwellian electron population.

Effects of Electron Temperature Anisotropy on Proton-beam Instability in the Solar Wind

Abstract Solar wind observations often show that the drift velocity of a proton beam relative to a background proton decreases with the heliocentric distance. Proton-beam instability has been suggested to play an important role in the deceleration of the proton-beam velocity; the effects of electron temperature anisotropy on the proton-beam instability have not been examined. Based on a general kinetic dispersion relation solver for magnetized plasma (PDRK), we investigate the effects of electron temperature anisotropy on the oblique Alfvén/ion-cyclotron (A/IC) and parallel magnetosonic/whistler (M/W) instabilities driven by proton beams in the solar wind. The results show that the growth rates, real frequencies, and threshold conditions for both instabilities are sensitive to the electron temperature anisotropy T e⊥/T e∣∣ and the parallel electron beta β e∣∣. In the low-beta regime with , where is a critical plasma beta for which the threshold velocities of both instabilities are equal, the growth rate of the oblique A/IC instability is weakly dependent on the electron temperature anisotropy. In the high-beta regime with , the growth rate of the parallel M/W instability increases with decreasing T e⊥/T e∣∣. Moreover, the threshold velocities of both instabilities are shifted to lower values as T e⊥/T e∣∣ decreases, especially for the parallel M/W instability in the regime with β e∣∣ ≥ 1. The theoretical results for the threshold velocity together with the observed parallel electron beta and/or electron temperature anisotropy are used to explain the observed proton-beam drift velocity in the solar wind.

Solar Wind Temperature Isotropy.

Reliable models of the solar wind in the near-Earth space environment may constrain conditions close to the Sun. This is relevant to NASA's contemporary innerheliospheric mission Parker Solar Probe. Among the outstanding issues is how to explain the solar wind temperature isotropy. Perpendicular and parallel proton and electron temperatures near 1AU are theoretically predicted to be unequal, but insitu observations show quasi-isotropy sufficiently below the instability threshold condition. This has not been satisfactorily explained. The present Letter shows that the dynamical coupling of electrons and protons via collisional processes and instabilities may contribute toward the resolution of this problem.

On standing gravity wave-depression cavity collapse and jetting

Parametrically forced gravity waves can give rise to high-velocity surface jets via the wave-depression cavity implosion. The present results have been obtained in circular cylindrical containers of 10 and 15 cm in diameter (Bond number of order $10^{3}$) in the large fluid depth limit. First, the phase diagrams of instability threshold and wave breaking conditions are determined for the working fluid used, here water with 1 % detergent added. The collapse of the wave-depression cavity is found to be self-similar. The exponent $\unicode[STIX]{x1D6FC}$ of the variation of the cavity radius $r_{m}$ with time $\unicode[STIX]{x1D70F}$, in the form $r_{m}/R\propto \unicode[STIX]{x1D70F}^{\unicode[STIX]{x1D6FC}}$, is close to 0.5, indicative of inertial collapse, followed by a viscous cut-off of $\unicode[STIX]{x1D6FC}\approx 1$. This supports a Froude number scaling of the surface jet velocity caused by cavity collapse. The dimensionless jet velocity scales with the cavity depth that is shown to be proportional to the last stable wave amplitude. It can be expressed by a power law or in terms of finite time singularity related to a singular wave amplitude that sets the transition from a non-pinching to pinch-off cavity collapse scenario. In terms of forcing amplitude, cavity collapse and jetting are found to occur in bands of events of non-pinching and pinching of a bubble at the cavity base. At large forcing amplitudes, incomplete cavity collapse and splashing can occur and, at even larger forcing amplitudes, wave growth is again stable up to the singular wave amplitude. When the cavity is formed, an impulse model shows the importance of the singular cavity diameter that determines the strength of the impulse.

Electron temperature anisotropy regulation by whistler instability

AbstractThe solar wind electron temperature anisotropy is regulated by a number of physical processes, which include adiabatic expansion, electron Coulomb collisions, and microinstabilities. In the collisionless limit, the measured electron temperature anisotropy is constrained by the marginal threshold conditions for whistler (electromagnetic electron cyclotron or EMEC) and firehose instabilities, which are excited by excessive perpendicular and parallel temperature anisotropies, respectively. In the literature, these thresholds are expressed as inverse relationships between the electron temperature ratio and parallel beta, which are constructed on the basis of linear stability analysis and empirical fitting. In the present paper, macroscopic quasi‐linear kinetic theory of whistler (or EMEC) instability is employed in order to investigate the time development of the instability. One‐dimensional particle‐in‐cell (PIC) simulation is also carried out, and it is found that PIC simulation confirms the validity of the macroscopic quasi‐linear approach. It is also found that the saturation stage of the instability naturally corresponds to the threshold condition, thus confirming the inverse relationship. The present finding shows that the macroscopic quasi‐linear kinetic theory may be a valid theoretical tool for dynamical description of the solar wind.

Interplay of Electron and Proton Instabilities in Expanding Solar Wind

Abstract Protons and electrons observed in the solar wind possess temperature anisotropies for which upper and lower bounds appear to be partially regulated by marginal conditions associated with various kinetic plasma instabilities. Such features are most clearly seen when a collection of measurements is plotted as a two-dimensional histogram in phase space. While the partial outer boundaries of such data distribution may well be explained by various instability threshold conditions, an outstanding issue is that the majority of data points are actually located sufficiently away from the boundaries and reside in near isotropic conditions. This implies that certain processes are operative that counteract the adiabatic effect in the radially expanding solar wind, without which solar wind plasma will inexorably be forced to proceed toward the marginal firehose condition. A number of physical processes have been proposed in the literature to explain such a feature. The present paper suggests yet another mechanism. It considers dynamic electrons and protons in the quasilinear evolution of anisotropy-driven instabilities, which is in contrast to previous studies where either protons or electrons are assumed to be stationary when considering the dynamics of the other particle species. It is shown that the dynamical interplay between the two species during the quasilinear development of parallel electron firehose and proton–cyclotron instabilities leads to a counter-balancing effect, which prevents the uniform progression of the solar wind protons toward the marginal firehose state.

Macroscopic quasilinear theory of parallel electron firehose instability associated with solar wind electrons

A number of different microinstabilities are known to be responsible for regulating the upper bound of temperature anisotropies in solar wind protons, alpha particles, and electrons. In the present paper, quasilinear kinetic theory is employed to investigate the time variation in electron temperature anisotropies in response to the excitation of parallel electron firehose instability in homogeneous and non-collisional solar wind plasma under the condition of T∥e>T⊥e. By assuming the bi-Maxwellian form of velocity distribution functions, various velocity moments of the particle kinetic equation are taken in order to reduce the theory to macroscopic model in which the wave-particle interaction is incorporated, hence, the macroscopic quasilinear theory. The threshold condition for the parallel electron firehose instability, empirically constructed as a curve in (β∥e,T⊥e/T∥e) phase space, is implicit in the present macroscopic quasilinear calculation. Even though the present calculation excludes the oblique firehose instability, which is known to possess a higher growth rate, the basic methodology may be further extended to include such a mode. Among the findings is that the parallel electron firehose instability dynamically couples the electrons and protons, which implies that this instability may be important for overall solar wind dynamics. The present analysis shows that the macroscopic quasilinear approach may eventually be incorporated in global-kinetic models of the solar wind electrons and ions.

Role of inertial instability in the West African monsoon jump

AbstractThe West African monsoon jump is a sudden shift in the latitude of the West African precipitation maximum from the Guinean coast near 4°N into Sahel near 12°N in late June or early July. An examination of reanalyses and observations indicates that the Sahel rainy season develops smoothly and the monsoon jump occurs because of an abrupt decrease in Guinean coast rainfall. We show that this abrupt end of the coastal rainy season occurs when inertial instability develops over the region, 1 month later than it develops in the vicinity of the marine Atlantic Intertropical Convergence Zone. The reason for this delay is the presence of the African easterly jet, which places strong negative meridional zonal wind gradients over the coast to preserve the inertially stable environment. When the African easterly jet moves farther north due to the seasonal solar forcing, these gradients weaken and then reverse to satisfy the threshold condition for inertial instability; the rapid end of the Guinean coast rainy season follows. The northward movement and intensity of the African easterly jet are controlled by the seasonal development of strong meridional land surface temperature gradients and are independent of the formation of the Atlantic cold tongue. This explanation for the West African monsoon jump relates the phenomenon to the shape and location of the African continent, including the low‐latitude position of the Guinean coast and the large expanse of the continent to the north.

Nonlinear stable steady solutions to the Ostroumov problem

We investigate the linear and weakly nonlinear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov’s linear stability analysis yields instability threshold conditions that are characterized by zero wavenumber for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the supercritical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is attributed to the fact that the nonlinear interaction of the linear modes vanish identically. In this paper, we put forth exact and stable similarity type solutions to the Ostroumov problem. These solutions are characterized by the same linear threshold conditions as Ostroumov’s solutions. Moreover, we are able to extend the analysis to the supercritical regime through a small amplitude analysis to obtain steady two-dimensional solutions for a small range of Prandtl numbers. These solutions are found to be stable to general two-dimensional, time-dependent disturbances. Furthermore, when the analysis is extended to the case where the fluid layer thickness is also allowed to be infinite, we find that the infinite two-dimensional fluid region becomes linearly unstable through a Batchelor–Nitsche (BN) instability mechanism. Thus, the nonlinear solutions are obtained through a long wavelength expansion, and consequently our analysis provides the nonlinear development of the BN instability for the case of an unstable and uniform density profile. Finally, numerical solutions of the nonlinear problem are also presented which shed light on the flow patterns and temperature distributions for both the vertical channel and the infinitely extended stratified region problems.

Marginal instability threshold condition of the aperiodic ordinary mode in equal-mass plasmas

The purely growing ordinary (O) mode instability for counter-streaming bi-Maxwellian plasma particle distribution functions has recently received renewed attention due to its importance for the solar wind plasma. Here, the analytical marginal instability condition is derived for magnetized plasmas consisting of equal-mass charged particles, distributed in counter-streams with equal temperatures. The equal-mass composition assumption enormously facilitates the theoretical analysis due to the equality of the values of the electron and positron (positive and negative ion) plasma and gyrofrequencies. The existence of a new instability domain of the O-mode at small plasma beta values is confirmed, when the parallel counter-stream free energy exceeds the perpendicular bi-Maxwellian free energy.

Theoretical Study of Carbon Adsorption on Re Surfaces: Morphological Instability

We report results from new experiments on C/Re( $$11\bar{2}1$$ ) to identify threshold conditions for morphological instability of Re( $$11\bar{2}1$$ ). We have found that adsorption of carbon from 0.35 to 0.85 ML (0.3–6.0 L exposure of C2H2) at T ≥ 800 K leads to faceting of Re( $$11\bar{2}1$$ ) with formation of three-sided pyramids. Using density functional theory we have investigated binding sites and binding energies of C on planar and faceted Re surfaces as well as generated a surface phase diagram of C/Re to obtain an atomistic understanding of C-induced pyramidal faceting of Re( $$11\bar{2}1$$ ). The calculations reveal that at low to intermediate coverage, C atoms prefer binding at four-fold sites on the Re surfaces and formation of three-sided pyramids is thermodynamically favored. Using density functional theory and thermodynamic considerations as well as AES and LEED measurements we studied the structure of Re(11–21) surfaces in contact with C2H2. The experiments show that adsorption of carbon from 0.35 to 0.85 ML (0.3–6.0 L exposure of C2H2) at T ≥ 800 K leads to faceting of Re(11–21) with formation of three-sided pyramids. The calculations reveal that at low to intermediate coverage, C atoms prefer binding at four-fold sites on the Re surfaces and formation of three-sided pyramids is thermodynamically favored.

Ion temperature anisotropy effects on the dispersion relation and threshold conditions of a sheared current-driven electrostatic ion-acoustic instability with applications to the collisional high-latitude F-region

Plasma instabilities play a important role in producing small-scale irregularities in the ionosphere. In particular, current-driven electrostatic ion-acoustic (CDEIA) instabilities contribute to high-latitude F-region electrodynamics. Ion temperature anisotropies with enhanced perpendicular temperature often exist in the high-latitude F-region. In addition to temperature anisotropies, ion velocity shears are observed near auroral arc edges, sometimes coexisting with thermal ion upflow processes and field-aligned currents (FAC). We investigated whether ion temperature anisotropy lowers the threshold conditions required for the onset of sheared CDEIA instabilities. We generalised a dispersion relation to include ion thermal anisotropy, finite Larmor radius corrections and collisions. We derived new fluid-like analytical expressions for the threshold conditions required for instability that depend explicitly on ion temperature anisotropy. We studied how the instability threshold conditions vary as a function of the wave vector direction in both fluid and kinetic regimes. We found that, despite the dampening effect of collisions on ion-acoustic waves, ion temperature anisotropy lowers in some cases the threshold drift requirements for a large range of oblique wave vector angles. More importantly, realistic ion temperature anisotropies contribute to reducing the instability threshold velocity shears that are associated with small drift thresholds, for modes propagating almost perpendicularly to the geomagnetic field. Small shear thresholds that seem to be sustainable in the ionospheric F-region are obtained for low-frequency waves. Such instabilities could play a role in the direct generation of field-aligned irregularities in the collisional F-region that could be observed with the Super Dual Auroral Radar Network (SuperDARN) array of high-frequency radars. These modes would be very sensitive to the radar probing direction since they are restricted to very narrow angular intervals. The ion temperature anisotropy is an important parameter that needs to be considered in the studies of sheared and collisional CDEIA waves and instabilities in the high-latitude F-region.

On the marginal instability threshold condition of the aperiodic ordinary mode

The purely growing ordinary (O) mode instability has recently received renewed attention owing to its potential applicability to the solar wind plasma. Here, an analytical marginal instability condition is derived for counter-streaming bi-Maxwellian plasma particle distribution functions. The derived marginal instability condition as a function of the temperature anisotropy and plasma beta agrees remarkably well with the numerically determined instability condition. The existence of a new instability domain of the O-mode at small plasma beta values is confirmed with the leading A∝β∥−1-dependence, if the counter-stream parameter Pe exceeds a critical value. At small plasma beta values at large enough counter-stream parameter, the O-mode also operates for temperature anisotropies A = T⊥/T∥ > 1 even larger than unity, as the parallel counter-stream free energy exceeds the perpendicular bi-Maxwellian free energy.

Stability of a three-dimensional diffusive Lotka–Volterra system of type-K with delays

A three-dimensional diffusive Lotka–Volterra system of type-K with delays is investigated. We give a stability analysis in detail for all equilibria of the system and obtain some threshold conditions for linear instability and linear asymptotic stability of each equilibrium. We develop the analytical method for stability analysis of reaction–diffusion equations with multi-delays.

Experimental and theoretical studies on density wave instabilities in helically coiled tubes

This paper reports on the advancement in the study of thermal-hydraulic dynamic instabilities with reference to the helical-coiled tube geometry.A full-scale open-loop experimental facility simulating a helically coiled steam generator was built and operated at SIET labs in Piacenza (Italy). The facility comprises two helical tubes (1m coil diameter, 32m length, 8m height), connected via lower and upper headers. Nearly 100 flow instability threshold conditions were identified, in a test matrix of pressures (80bar, 40bar, 20bar), mass fluxes (600kg/m2s, 400kg/m2s, 200kg/m2s), inlet subcooling (from −30% up to ∼0), and inlet throttling (four different entrance resistance conditions). The long test section feature and the helical-coiled tube geometry render the present facility a quite unique test case in the outline of two-phase flow instability experimental studies. Parametric effects of the operating pressure, flow rate, inlet subcooling and inlet throttling on the threshold power are discussed. The period of oscillations is also discussed. Superimposition of Density Wave Oscillations (DWOs) with Ledinegg flow excursions is finally described.Theoretical modelling of DWO occurrence in helical pipes was addressed by means of a lumped parameter analytical model, which was exploited to highlight some peculiarities of DWO phenomena and respective stability boundary with respect to classical straight geometry. In the end, numerical simulation results with RELAP5/MOD3.3 code were compared.

Whistler instability in a semi-relativistic bi-Maxwellian plasma

Employing linearized Vlasov-Maxwell system, the Weibel instability embedded in an ambient magnetic field is discussed for a semi-relativistic bi-Maxwellian distribution hoping such a scenario occurs in some relativistic environments e.g., gamma-ray burst sources and relativistic jet sources, supernovae, and galactic cosmic rays where the perpendicular temperature is much dominated over the parallel . The dispersion relations are analyzed analytically along with the graphical representation and the estimates of the growth rate are presented along with the instability threshold condition in the limiting cases i.e., xi<=1 (resonant case) and xi>>1 (non-resonant case). It is observed that the relativistic effect suppresses the instability and also lowers the threshold for the instability to set in. The ambient magnetic field contribution to instability appears only in non-resonant case resulting in reduction of growth rate. However, the effect of the ambient magnetic field is diminished as we go from the weak relativistic regime to the highly relativistic one. We also note that the ambient magnetic field generates real oscillations and the reletivistic effect reduces these oscillations. Further for field free case i.e., B0=0, the growth rates for Weibel instabilities are also presented in a semi-relativistic bi- Maxwellian plasma for both the limiting cases.

Electromagnetic electron whistler-cyclotron instability in bi-Kappa distributed plasmas

Context. Recent studies of the electromagnetic electron whistler-cyclotron instability in anisotropic bi-Kappa distributed plasmas claim that the instability threshold conditions do not depend on the power index, κ e , of the electron distribution function, but that the maximum growth rate (γ m ) strongly depends on this parameter. But these two statements contradict each other because the instability threshold conditions are derived with respect to the threshold levels of the maximum growth rates (e.g.,γ m /Ω = 10-1 ,10-2 , etc.).Aims. This paper proposes to clarify this inconsistency, refining the analysis of the electron-whistler cyclotron instability. In anisotropic plasmas far from Maxwellian equilibrium, this instability represents one of the most plausible constraints for the electron temperature anisotropy T e, ⊥ > T e, ∥ , (where ∥ and ⊥ denote directions relative to the local stationary magnetic field). Methods. In the context of a suprathermal solar wind, where the electron populations are well fitted by the advanced Kappa distribution functions, these models are expected to provide a more realistic description for the critical stability conditions. The unstable solutions are derived exactly numerically, providing accurate physical correlations between the maximum growth rates and the threshold conditions.Results. Thresholds of the temperature anisotropy are derived for the full range of values of the plasma beta including both the solar wind and magnetospheric plasma conditions. The lowest thresholds, which are the most relevant for the marginal stability, are found to decrease with the increase in density of the suprathermal populations. This result is correlated with an opposite effect on the corresponding growth rates (at low anisotropies), because their maximum values are enhanced in the presence of suprathermal electrons. The new marginal thresholds calculated with a bi-Kappa model are expected to provide better predictions for the limits of the temperature anisotropy in the solar wind.

CONVECTION IN COLLOIDAL SUSPENSIONS OF SOLID PARTICLES: A COMPARATIVE STUDY BETWEEN THE HOMOGENEOUS MIXTURE AND THE PARTICULATE MEDIUM MODELS

The convection process in colloidal suspensions of solid particles is usually investigated by adopting either a homogeneous binary mixture (HM) or a particulate medium (PM) model. The present study aims only at comparing the instability threshold conditions for the onset of convection in the particle-dominated regime obtained through these two models. We consider a colloidal suspension of solid particles in a Rayleigh-Bénard set up. A PM model is adopted which accounts for the effects of thermophoresis, sedimentation, and Brownian diffusion. We assume that the suspension layer is subjected to a vertical temperature gradient that is not thermally destabilizing as in the case of negligibly small coefficient of thermal expansion. Thus, we consider the idealized situation wherein the thermally induced buoyancy is absent and the focus is solely on the particle-dominated convection regime. We show that the coupled effects of thermo-migration, sedimentation, and Brownian diffusion give rise to a weak convection regime that sets in on the long particle diffusion time scale. A small wave number expansion is undertaken to determine the stability threshold conditions. When contrasted with those obtained through the HM model, the latter are found to have the same functional dependence on parameters as those describing the onset of Soret-driven convection for the case of a positive and large separation ratio. However, we show that a PM model, unlike the HM model, yields stability threshold conditions that can be mapped to corresponding experimental parameters such as the size of the particles and of the fluid cell.