Marginal Stability Condition Research Articles

Gravitational stability and mass estimation of stellar disks

AbstractWe estimate masses of galaxy disks using the marginal gravitational stability criterion and compare them with the photometric disk‐mass evaluations. The comparison reveals that stellar disks of most spiral galaxies we have considered cannot be substantially overheated (at least within several radial scale lengths) and are therefore unlikely to have experienced a significant merging event in their history. However, for a substantial part of S0‐type galaxies, their stellar velocity dispersion is well in excess of the gravitational stability threshold suggesting a major merger event in the past. For four low‐surface‐brightness galaxies, we find the disk masses corresponding to the marginal stability condition to be significantly higher than one may expect from their brightness. Either their disks are dynamically overheated, or they contain a large amount of non‐luminous matter. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Implications of a heuristic model of auroral Farley Buneman waves and heating

The global implications, particularly with respect to altitude dependence, of the heuristic model of Farley Buneman waves put forward initially by Milikh and Dimant (2002) are studied. This model prescribes a relationship between the background convection electric field that excites the waves and the transverse electric fields of the waves that grow in response. It also prescribes the magnetic aspect angle of the waves, which is related to their ability to heat the auroral E region. The prescription is based on the condition of marginal stability. We reformulate the basic model, which is local, and embed it in the SAMI2 ionospheric model, which includes wave and Joule heating, heat transport, cooling, temperature‐dependent collisions, and related chemistry. Within the limits of its underlying assumptions, the combined model can be used to predict the phase‐speeds and magnetic‐aspect widths of Farley Buneman waves in the auroral zone and the heating they can cause, all as functions of altitude. Model predictions are compared with experimental results, and the efficacy of the model assessed. This modeling exercise highlights the importance of the thickness of the layer in which Farley Buneman waves exist, the strong variations in wave characteristics across the layer, and the consequences this has for coherent scatter radar measurements of the phenomenon.

Towards a complete parametrization of the ordinary-mode electromagnetic instability in counterstreaming plasmas. II. Ion effects

The linear marginal instability analysis of the ordinary perpendicular mode instability of drifting bi-Maxwellian plasma particle distributions with and without temperature anisotropy is extended by including the modifications of heavier ion species. For general values of the temperature anisotropy, the streaming velocity, and the parallel plasma beta, accurate marginal stability conditions are derived, which enable a better understanding of the interplay of counterstreaming and temperature anisotropy.

The effect of rotation on the Rayleigh-Bénard stability threshold

The standard method used to solve the Rayleigh-Bénard linear stability problem for a rotating fluid leads to a complex expression which can only be evaluated numerically. Here the problem is solved by a different method similar to that used in a recent paper on the non-rotating case [A. Prosperetti, “A simple analytic approximation to the Rayleigh-Bénard stability threshold,” Phys. Fluids 23, 124101 (2011)10.1063/1.3662466]. In principle the method leads to an exact result which is not simpler than the standard one. Its value lies in the fact that it is possible to obtain from it an approximate explicit analytic expression for the dependence of the Rayleigh number on the wave number of the perturbation and the rate of rotation at marginal stability conditions. Where the error can be compared with exact results in the literature, it is found not to exceed a few percent over a very broad Taylor number range. The relative simplicity of the approach permits us, among others, to account for the effects of a finite thermal conductivity of the plates, which have not been studied before.

Quasilinear theory of anisotropy‐beta relation for combined mirror and proton cyclotron instabilities

The solar wind and the Earth's magnetosheath are often characterized by proton temperature anisotropies that cannot be discussed by adiabatic fluid theory. An excessive perpendicular temperature anisotropy may result when the plasma undergoes compression. The proton temperature anisotropy, T⊥/T∥ > 1, leads to the proton cyclotron and mirror instabilities. Marginal stability conditions for these instabilities may be expressed as inverse correlations between T⊥/T∥ and parallel beta, β∥. In the literature, these correlations are constructed on the basis of linear theory, hybrid simulations, or observational fitting. The present paper makes use of quasilinear theory for the proton cyclotron and mirror instabilities. In such an approach the inverse correlation naturally emerges as the time‐asymptotic states of self‐consistent evolution. The inverse correlation thus constructed shows the predominance of proton cyclotron instability for lowβ∥ regime, while for high β∥ values, the mirror instability dictates the inverse correlation.

Current Fragmentation and Particle Acceleration in Solar Flares

Particle acceleration in solar flares remains an outstanding problem in plasma physics and space science. While the observed particle energies and timescales can perhaps be understood in terms of acceleration at a simple current sheet or turbulence site, the vast number of accelerated particles, and the fraction of flare energy in them, defies any simple explanation. The nature of energy storage and dissipation in the global coronal magnetic field is essential for understanding flare acceleration. Scenarios where the coronal field is stressed by complex photospheric motions lead to the formation of multiple current sheets, rather than the single monolithic current sheet proposed by some. The currents sheets in turn can fragment into multiple, smaller dissipation sites. MHD, kinetic and cellular automata models are used to demonstrate this feature. Particle acceleration in this environment thus involves interaction with many distributed accelerators. A series of examples demonstrate how acceleration works in such an environment. As required, acceleration is fast, and relativistic energies are readily attained. It is also shown that accelerated particles do indeed interact with multiple acceleration sites. Test particle models also demonstrate that a large number of particles can be accelerated, with a significant fraction of the flare energy associated with them. However, in the absence of feedback, and with limited numerical resolution, these results need to be viewed with caution. Particle in cell models can incorporate feedback and in one scenario suggest that acceleration can be limited by the energetic particles reaching the condition for firehose marginal stability. Contemporary issues such as footpoint particle acceleration are also discussed. It is also noted that the idea of a “standard flare model” is ill-conceived when the entire distribution of flare energies is considered.

Gravitational stability and dynamical overheating of galactic stellar disks

We use the marginal stability condition for galactic disks and the stellar velocity dispersion data published by different authors to place upper limits on the disk local surface density at two radial scalelengths $R=2h$. Extrapolating these estimates, we constrain the total mass of the disks and compare these estimates to those based on the photometry and color of stellar populations. The comparison reveals that the stellar disks of most of spiral galaxies in our sample cannot be substantially overheated and are therefore unlikely to have experienced a significant merging event in their history. The same conclusion applies to some, but not all of the S0 galaxies we consider. However, a substantial part of the early type galaxies do show the stellar velocity dispersion well in excess of the gravitational stability threshold suggesting a major merger event in the past. We find dynamically overheated disks among both seemingly isolated galaxies and those forming pairs. The ratio of the marginal stability disk mass estimate to the total galaxy mass within four radial scalelengths remains within a range of 0.4---0.8. We see no evidence for a noticeable running of this ratio with either the morphological type or color index.

Alfvén Ion-Cyclotron Instability in a Mirror Trap with Highly Anisotropic Plasma

The Alfvén ion cyclotron instability is studied for mirror-confined bi-Maxwellian highly anisotropic plasmas. In such plasmas the wave length of unstable modes is of the order of the plasma scale length. Another specific feature is that a typical ion can execute several bounce oscillations along the strongly non-uniform plasma during the time of the phase divergence between the wave and cyclotron rotation. Traditional approaches such as WKB method and local dispersion relation fail under these conditions.An integral equation for the modes is derived. The spatial distribution of the eigenmodes as well as the marginal stability conditions are found by numerical solution of this equation. The asymptotics of these results in the limit of infinitely large anisotropy are obtained analytically. It is found that the mirror-confined highly anisotropic plasma can be much more stable than it follows from the traditionally used scaling.

Linear instability of the horizontal throughflow in a plane porous layer saturated by a power-law fluid

The onset of the convective instability in the horizontal throughflow of a power-law fluid saturating a horizontal porous layer heated from below is studied. A linear stability analysis of the basic flow is carried out and the disturbance equations are solved analytically. The problem examined here is an extension of the classical Prats problem for Newtonian fluids. It is shown that the marginal stability condition, as well as the critical values of the wave number and of the Darcy–Rayleigh number, is affected by the value of the Péclet number associated with the basic flow, except for the special case of a Newtonian fluid. The limit of a vanishingly small Péclet number is considered leading to the special case of the Horton–Rogers–Lapwood (HRL) problem for a power-law fluid, i.e., the Prats problem with a vanishing basic throughflow. It is shown that the generalized HRL problem is always linearly stable for pseudoplastic fluids and always linearly unstable for dilatant fluids.

FORMING PLANETESIMALS BY GRAVITATIONAL INSTABILITY. II. HOW DUST SETTLES TO ITS MARGINALLY STABLE STATE

Dust at the midplane of a circumstellar disk can become gravitationally unstable and fragment into planetesimals if the local dust-to-gas density ratio mu is sufficiently high. We simulate how dust settles in passive disks and ask how high mu can become. We settle the dust using a 1D code and test for dynamical stability using a 3D shearing box code. This scheme allows us to explore the behavior of small particles having short but non-zero stopping times in gas: 0 < t_stop << the orbital period. The streaming instability is thereby filtered out. Dust settles until shearing instabilities in the edges of the dust layer threaten to overturn the entire layer. In this state of marginal stability, mu=2.9 for a disk whose bulk (height-integrated) metallicity is solar. For a disk whose bulk metallicity is 4x solar, mu reaches 26.4. These maximum values of mu, which depend on the background radial pressure gradient, are so large that gravitational instability of small particles is viable in disks whose bulk metallicities are just a few (<4) times solar. Earlier studies assumed that dust settles until the Richardson number Ri is spatially constant. Our simulations are free of this assumption but provide support for it within the dust layer's edges, with the proviso that Ri increases with bulk metallicity in the same way that we found in Paper I. Only modest enhancements in bulk metallicity are needed to spawn planetesimals directly from small particles.

Matrix norms, BPS bounds and marginal stability in $ \mathcal{N} = 8 $ supergravity

We study the conditions of marginal stability for two-center extremal black holes in N-extended supergravity in four dimensions, with particular emphasis on the N=8 case. This is achieved by exploiting triangle inequalities satisfied by matrix norms. Using different norms and relative bounds among them, we establish the existence of marginal stability and split attractor flows both for BPS and some non-BPS solutions. Our results are in agreement with previous analysis based on explicit construction of multi-center solutions.

Extended Oberbeck–Boussinesq approximation study of convective instabilities in a porous layer with horizontal flow and bottom heating

A study of the onset of convective instabilities in a porous layer with a horizontal basic flow is performed by including the effects of viscous dissipation and pressure work in the energy balance. Firstly, the so-called extended Oberbeck–Boussinesq approximation, i.e. the model based on the enthalpy formulation of the energy balance, is adopted. Then, the results for the marginal stability condition are compared with those obtained by the so-called Chandrasekhar approximation, i.e. the model based on the internal-energy formulation of the energy balance. It is shown that a marked discrepancy occurs between the two approaches, that becomes specially evident for high values of the Gebhart number. According to the extended Oberbeck–Boussinesq approximation, the effects of the viscous dissipation and of the pressure work result in a stabilization of the basic flow. On the contrary, the Chandrasekhar approximation predicts a destabilization of the basic flow induced by the viscous dissipation. The destabilization can be so intense that the onset of convective rolls may occur even in the absence of a boundary temperature difference, i.e. with a vanishing Darcy–Rayleigh number.

Linear analysis of incompressible Rayleigh-Taylor instability in solids

The study of the linear stage of the incompressible Rayleigh-Taylor instability in elastic-plastic solids is performed by considering thick plates under a constant acceleration that is also uniform except for a small sinusoidal ripple in the horizontal plane. The analysis is carried out by using an analytical model based on the Newton second law and it is complemented with extensive two-dimensional numerical simulations. The conditions for marginal stability that determine the instability threshold are derived. Besides, the boundary for the transition from the elastic to the plastic regime is obtained and it is demonstrated that such a transition is not a sufficient condition for instability. The model yields complete analytical solutions for the perturbation amplitude evolution and reveals the main physical process that governs the instability. The theory is in general agreement with the numerical simulations and provides useful quantitative results. Implications for high-energy-density-physics experiments are also discussed.

Limit on stably trapped particle fluxes in planetary magnetospheres

We reexamine the Kennel‐Petschek concept of self‐limitation of stably trapped particle fluxes in a planetary magnetosphere. In contrast to the original Kennel‐Petschek formulation, we carry out a fully relativistic analysis. In addition, we replace the wave reflection criterion in the Kennel‐Petschek theory by the condition that the limit on the stably trapped particle flux is attained in the steady state condition of marginal stability when electromagnetic waves generated at the magnetic equator acquire a specified gain over a given convective growth length. We derive relativistic formulae for the limiting electron integral and differential fluxes for a general planetary radiation belt at a given L shell. The formulae depend explicitly on the spectral index and pitch angle index of the assumed particle distribution and on the ratio of the electron gyrofrequency to the electron plasma frequency. We compare the theoretical limits on the trapped flux with observed energetic electron fluxes at Earth, Jupiter, and Uranus.

A Theory of Baroclinic Turbulence

Abstract Understanding the physical mechanism maintaining fluid turbulence remains a fundamental theoretical problem. The two-layer model is an analytically and computationally simple system in which the dynamics of turbulence can be conveniently studied; in this work, a maximally simplified model of the statistically steady turbulent state in this system is constructed to isolate and identify the essential mechanism of turbulence. In this minimally complex turbulence model the effects of nonlinearity are parameterized using an energetically consistent stochastic process that is white in both space and time, turbulent fluxes are obtained using a stochastic turbulence model (STM), and statistically steady turbulent states are identified using stochastic structural stability theory (SSST). These turbulent states are the fixed-point equilibria of the nonlinear SSST system. For parameter values typical of the midlatitude atmosphere, these equilibria predict the emergence of marginally stable eddy-driven baroclinic jets. The eddy variances and fluxes associated with these jets and the power-law scaling of eddy variances and fluxes are consistent with observations and simulations of baroclinic turbulence. This optimally simple model isolates the essential physics of baroclinic turbulence: maintenance of variance by transient perturbation growth, replenishment of the transiently growing subspace by nonlinear energetically conservative eddy–eddy scattering, and equilibration to a statistically steady state of marginal stability by a combination of nonlinear eddy-induced mean jet modification and eddy dissipation. These statistical equilibrium states provide a theory for the general circulation of baroclinically turbulent planetary atmospheres.

Deconstructing the D0-D6 system

We find the complete classical moduli space of two-centered supersymmetric solutions carrying D0 and D6 brane charge in the STU model delimited by walls of marginal stability of co-dimension one. U-duality guarantees our conclusions hold for any BPS state with negative quartic invariant. The analysis explicitly shows that the conditions of marginal stability, i.e. the integrability conditions, are generically insufficient to provide a regular supergravity solution in this model.

Effect of pressure work and viscous dissipation in the analysis of the Rayleigh–Bénard problem

The classical Rayleigh–Bénard problem in an infinitely wide horizontal fluid layer with isothermal boundaries heated from below is revisited. The effects of pressure work and viscous dissipation are taken into account in the energy balance. A linear analysis is performed in order to obtain the conditions of marginal stability and the critical values of the wave number a and of the Rayleigh number Ra for the onset of convective rolls. Mechanical boundary conditions are considered such that the boundaries are both rigid, or both stress-free, or the upper stress-free and the lower rigid. It is shown that the critical value of Ra may be significantly affected by the contribution of pressure work, mainly through the functional dependence on the Gebhart number and on a thermodynamic Rayleigh number. While the pressure work term affects the critical conditions determined through the linear analysis, the viscous dissipation term plays no role in this analysis being a higher order effect. A nonlinear analysis is performed showing that the superadiabatic Rayleigh number replaces Ra in the functional dependence of the excess Nusselt number. Finally, a reasoning is proposed to show how the results obtained may be used as a test on the most appropriate formulation of the Oberbeck–Boussinesq model.

The Horton–Rogers–Lapwood Problem Revisited: The Effect of Pressure Work

The problem of the onset of convective roll instabilities in a horizontal porous layer with isothermal boundaries at unequal temperatures, well known as the Horton–Rogers–Lapwood problem, is revisited including the effect of pressure work and viscous dissipation in the local energy balance. A linear stability analysis of rolls disturbances is performed. The analysis shows that, while the contribution of viscous dissipation is ineffective, the contribution of the pressure work may be important. The condition of marginal stability is investigated by adopting two solution procedures: method of weighted residuals and explicit Runge–Kutta method. The pressure work term in the energy balance yields an increase of the value of the Darcy–Rayleigh number at marginal stability. In other words, the effect of pressure work is a stabilizing one. Furthermore, while the critical value of the Darcy– Rayleigh number may be considerably affected by the pressure work contribution, the critical value of the wave number is affected only in rather extreme cases, i.e. for very high values of the Gebhart number. A nonlinear stability analysis is also performed pointing out that the joint effects of viscous dissipation and pressure work result in a reduction of the excess Nusselt number due to convection, when the Darcy–Rayleigh number is replaced by the superadiabatic Darcy–Rayleigh number.

New Composite Models of Partially Ionized Protoplanetary Disks

We study an accretion disk in which three different regions may coexist: MHD turbulent regions, dead zones and gravitationally unstable regions. Although the dead zones are stable, there is some transport due to the Reynolds stress associated with waves emitted from the turbulent layers. We model the transport in each of the different regions by its own $\alpha$ parameter, this being 10 to $10^{3}$ times smaller in dead zones than in active layers. In gravitationally unstable regions, $\alpha$ is determined by the fact that the disk self-adjusts to a state of marginal stability. We construct steady-state models of such disks. We find that for uniform mass flow, the disk has to be more massive, hotter and thicker at the radii where there is a dead zone. In disks in which the dead zone is very massive, gravitational instabilities are present. Whether such models are realistic or not depends on whether hydrodynamical fluctuations driven by the turbulent layers can penetrate all the way inside the dead zone. This may be more easily achieved when the ratio of the mass of the active layer to that of the dead zone is relatively large, which in our models corresponds to $\alpha$ in the dead zone being about 10% of $\alpha$ in the active layers. If the disk is at some stage of its evolution not in steady-state, then the surface density will evolve toward the steady-state solution. However, if $\alpha$ in the dead zone is much smaller than in the active zone, the timescale for the parts of the disk beyond a few AU to reach steady-state may become longer than the disk lifetime. Steady-state disks with dead zones are a more favorable environment for planet formation than standard disks, since the dead zone is typically 10 times more massive than a corresponding turbulent zone at the same location.

Instability of the salinity profile during the evaporation of saline groundwater

In this paper we investigate salt transport during the evaporation and upflow of saline groundwater. We describe a model in which a sharp evaporation–precipitation front separates regions of soil saturated with an air–vapour mixture and with saline water. We then consider two idealized problems. We first investigate equilibrium configurations of the freshwater system when the depth of the soil layer is finite, obtaining results for the location of the front and the upflow of water induced by the evaporation. We then develop a solution for a propagating front in a soil layer of infinite depth and investigate the gravitational stability of the salinity profile which develops below the front, obtaining marginal linear stability conditions in terms of a Rayleigh number and a dimensionless salt saturation parameter. Applying our findings to realistic parameter regimes, we predict that salt fingering is unlikely to occur in low-permeability soils, but is likely in high-permeability (sandy) soils under conditions of relatively low evaporative upflow.