On the stability of the solar wind

The stability of the solar wind is studied in the case of spherical symmetry and constant temperature. It is shown that the stability problem must be formulated as a mixed initial and boundary-value problem in which are prescribed the perturbation values of velocity and density at an initial time and additionally the velocity perturbation at the base of the corona for all times. The solution is constructed by linear superposition of normal solutions, which contain the time only in an exponential factor. The stability problem becomes a singular eigenvalue problem for the amplitudes of the velocity and pressure perturbations, since additionally to the boundary condition at the base of the corona one must add the condition that the amplitudes behave regularly at the critical point. It is proved that only stable eigenvalues exist.     

Radiative and Free Convective Effects of a MHD Flow Through a Porous Medium Between Infinite Parallel Plates with Time-dependent Suction

The problem of magnetohydrodynamic free-convection flow, with radiative heat transfer in porour media subject to time-dependent suction of an incompressible and optically transparent medium has been solved making fairly realistic assumption. For a small-time-dependent perturbation of the fluid velocity and temperatures, the nonlinear problem is tackled by asymptotic approximation, giving solutions for steady-flow on which a first-order transient component is superimposed. The effect of heat radiation and free convection on the flow of the fluid is demonstrated analytically and quantitatively. The flow field is seen to be affected mainly by radiation and convection parameters, in addition to the porosity and magnetic factors.     

Stability of thermally balanced gas flow

The linear analysis of hydrodynamic stability of the local thermal balance in a homogeneous moving gas is revisited to get information about the development of a spatially limited perturbation as seen at a fixed location. The consideration concerns both the evolution of the perturbed quantities inside a domain where the perturbation initially localizes and spreading the perturbation outside this domain. Inside the initial perturbation domain, the conditions for the exponential growth/decay are found to coincide with the well-known Field's criteria, ensuing the analysis of the normal modes. However, as soon as the modal isentropic stability criterion is satisfied the perturbation outside its initial domain asymptotically spreads out with a subsonic velocity not depending on the initial perturbation field. It enables the gas flow to carry the disturbances away and leads to an improved stability criterion for inhomogeneous thermally balanced flows where the modally unstable region appears to be spatially bounded. The spreading velocity, playing a key role in the new stability criterion, is calculated as a function of the same derivatives of the heating/cooling function as the modal instability criteria exploit.     

Evolution of initially localized perturbations in stratified ionized discs

A detailed solution of an initial value problem of a vertically localized initial perturbation in rotating magnetized vertically stratified disc is presented. The appropriate linearized magnetohydrodynamics equations are solved by employing the Wentzel–Kramers–Brillouin (WKB) approximation and the results are verified numerically. The eigenfrequencies as well as eigenfunctions are explicitly obtained. It is demonstrated that the initial perturbation remains confined within the disc. It is further shown that thin enough discs are stable but as their thickness grows increasing number of unstable modes participate in the solution of the initial value problem. However, it is demonstrated that due to the localization of the initial perturbation, the growth time of the instability is significantly longer than the calculated inverse growth rate of the individual unstable eigenfunctions.     

Compressible Fluid Model for the Seismic Waves Generated by A Sunquake

The solar convection zone is modeled as a horizontally stratified atmosphere with a constant gravitational field and an adiabatic temperature gradient (a neutrally stratified polytrope). At equilibrium, the gas pressure and density decreases to zero at the solar surface so that the solar surface is treated as a free surface which is bounded by vacuum. The evolution of small amplitude perturbations about the equilibrium state is described by the linearized Euler equations for an inviscid compressible fluid with an adiabatic equation of state. A sunquake is initiated at time zero by means of an initial perturbation with a Gaussian velocity profile and the exact solution of the initial value problem is obtained in terms of a Fourier integral. Comparisons between theory and observations indicate that this highly simplified model is able to predict the propagation of sunquake waves across the solar surface with an error of approximately 10% or 20%.     

Instabilities in a nonstationary model of self-gravitating disks. III. The phenomenon of lopsidedness and a comparison of perturbation modes

This is an examination of the gravitational instability of the major large-scale perturbation modes for a fixed value of the azimuthal wave number m = 1 in nonlinearly nonstationary disk models with isotropic and anisotropic velocity diagrams for the purpose of explaining the displacement of the nucleus away from the geometric center (lopsidedness) in spiral galaxies. Nonstationary analogs of the dispersion relations for these perturbation modes are obtained. Critical diagrams of the initial virial ratio are constructed from the rotation parameters for the models in each case. A comparative analysis is made of the instability growth rates for the major horizontal perturbation modes in terms of two models, and it is found that, on the average, the instability growth rate for the m = 1 mode with a radial wave number N = 3 almost always has a clear advantage relative to the other modes. An analysis of these results shows that if the initial total kinetic energy in an isotropic model is no more than 12.4% of the initial potential energy, then, regardless of the value of the rotation parameter Ω, an instability of the radial motions always occurs and causes the nucleus to shift away from the geometrical center. This instability is aperiodic when Ω = 0 and is oscillatory when Ω ≠ 0 . For the anisotropic model, this kind of structure involving the nucleus develops when the initial total kinetic energy in the model is no more than 30.6% of the initial potential energy.     

"Quasi Satellite Orbits" to observe a possible small moon of Pallas

The purpose of this paper is to make a numerical search for natural orbits that can be used for a spacecraft to study a possible small moon of Pallas. There are many speculations about the existence of a small companion around this large asteroid, so finding and classifying orbits around this possible celestial body is an interesting problem in astrodynamics and that can be used for a spacecraft to observe this body. It is assumed that this moon has a radius that can vary from 0.125 to 1 km and that is located 750 or 500 km away from the center of Pallas. The idea is to show the effects of this parameter in the orbits around this moon. It means that the moon is much smaller than Pallas, so Keplerian orbits are not possible around it. To solve this problem, it is possible to find some special orbits that are called "Quasi Satellite Orbits" (QSO). They are orbits dominated by the gravity of Pallas, but that use the smaller perturbation from the moon to keep the spacecraft close to it. The present work searches for orbits that make the spacecraft to remain at given limits in its distance from the moon, like in the range from 3 to 50 km, the values used as an example in the present paper. This value is used because it is a good range to observe the body without getting to close to it, so reducing the risks of collisions. Each trajectory can be identified by the initial conditions of the spacecraft with respect to the moon, which means its initial position and velocity. The dynamics considers the restricted three-body problem and the influence of the solar radiation pressure, because some spacecraft may have higher values for the area-to-mass ratio, which gives a non-negligible effect in the trajectory of the spacecraft.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid

We study the unsteady free-convection flow near a moving infinite flat plate in a totating medium by imposing a time-dependent perturbation on a constant plate temperature. The temperatures involved are assumed to be very large so that radiative heat transfer is significant, which renders the problem very nonlinear even on the assumption of a differential approximation for the radiative flux. When the perturbation is small, the transient flow is tackled by the Laplace transform technique. Complete first-order solutions are deduced for an impulsive motion.     

Absolute and convective instabilities in an inviscid compressible mixing layer

We consider the stability of a compressible shear flow separating two streams of different speeds and temperatures. The velocity and temperature profiles in this mixing layer are hyperbolic tangents. The normal mode analysis of the flow stability reduces to an eigenvalue problem for the pressure perturbation. We briefly describe the numerical method that we used to solve this problem. Then, we introduce the notions of the absolute and convective instabilities and examine the effects of Mach number, and the velocity and temperature ratios of each stream on the transition between convective and absolute instabilities. Finally, we discuss the implication of the results presented in this paper for the heliopause stability. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Recovery of the cosmological peculiar velocity from the density field in the weakly non-linear regime

Using third-order perturbation theory, we derive a relation between the divergence of the peculiar velocity and the density. Specifically, we compute the expectation value of the divergence given density. Our calculations assume Gaussian initial conditions and are valid for Gaussian filtering of the evolved density and velocity fields. The mean velocity divergence turns out to be a third-order polynomial in the density contrast. We test the power-spectrum dependence of the coefficients of the polynomial for scale-free and standard CDM spectra and find it rather weak. Over scales larger than about 5  h ⁻¹ Mpc, the scatter in the relation is small compared with that introduced by random errors in the observed density and velocity fields. The relation can be useful for recovering the peculiar velocity from the associated density field, and also for non-linear analyses of the anisotropies of structure in redshift surveys.     

Fragmentation of a Molecular Cloud Core versus Fragmentation of the Massive Protoplanetary Disk in the Main Accretion Phase

We present three-dimensional numerical simulations on binary formation through fragmentation. The simulations follow gravitational collapse of a molecular cloud core up to growth of the first core by accretion. At the initial stage, the gravity is only slightly dominant over the gas pressure. We made various models by changing initial velocity distribution (rotation speed, rotation law, and bar-mode perturbation). The cloud fragments whenever the cloud rotates sufficiently slowly to allow collapse but faster enough to form a disk before first-core formation. The latter condition is equivalent to Ω₀ t _ff ? 0.05, where Ω₀ and t _ff f denote the initial central angular velocity and the freefall time measured from the central density, and the condition is independent of the initial rotation law and bar-mode perturbation. Fragmentation is classified into six types. When the initial cloud rotates rigidly the cloud collapses to form a adiabatic disk supported by rotation. When the bar-mode perturbation is very minor, the disk deforms to a rotating bar, and the bar fragments. Otherwise, the adiabatic disk evolves into a central core surrounded by a circumstellar disk, and the the circumstellar disk fragments. When the initial cloud rotates differentially, the cloud deforms to a ring or bar in the isothermal collapse phase. The ring fragments into free or more cores, while the bar fragments into only two cores. In the latter case, the core merges due to low orbital angular momentum and new satellite cores form in the later stages.     

Interplanetary magnetic field structure and solar wind parameters as inferred from solar magnetic field observations and by using a numerical 2-D MHD model

An attempt is made to infer parameters of the solar corona and the solar wind by means of a numerical, self-consistent MHD simulation. Boundary conditions for the magnetic field are given from the observations of the large-scale magnetic field at the Sun. A two-region, planar (the ecliptic plane is assumed) model for the solar wind flow is considered. Region I of transonic flow is assumed to cover the distances from the solar surface up to 10R _S (R _S is the radius of the Sun). Region II of supersonic, super-Alfvénic flow extends between 10R _S and the Earth's orbit. Treatment for region I is that for a mixed initial-boundary value problem. The solution procedure is similar to that discussed by Endler (1971) and Steinolfson, Suess, and Wu (1982): a steady-state solution is sought as a relaxation to the dynamic equilibrium of an initial state. To obtain a solution to the initial value problem in region II with the initial distribution of dependent variables at 10R _S (deduced from the solution for region I), a numerical scheme similar to that used by Pizzo (1978, 1982) is applied. Solar rotation is taken into account for region II; hence, the interaction between fast and slow solar wind streams is self-consistently treated. As a test example for the proposed formulation and numerical technique, a solution for the problem similar to that discussed by Steinolfson, Suess, and Wu (1982) is obtained. To demonstrate the applicability of our scheme to experimental data, solar magnetic field observations at Stanford University for Carrington rotation 1682 are used to prescribe boundary conditions for the magnetic field at the solar surface. The steady-state solution appropriate for the given boundary conditions was obtained for region I and then traced to the Earth's orbit through region II. We compare the calculated and spacecraft-observed solar wind velocity, radial magnetic field, and number density and find that general trends during the solar rotation are reproduced fairly well although the magnitudes of the density in comparison are vastly different.     

The two-body problem with radiation pressure in a rotating reference frame

We study a perturbed Newtonian two-body problem, in which the perturbation is due to a force field of constant magnitude but rotating direction. By considering this system as a perturbation of the non-rotating case a Melnikov-type analysis allows us to show the existence of horseshoes in the level sets of the Hamiltonian and the subsequent sensitive dependence on initial conditions and non-integrability. We discuss the consequences of these results for a particular planar restricted three-body problem.Supported by a grant from the Royal Swedish Academy of Sciences and AFOSR NM 91-0329.     

Rayleigh-Taylor instability of a two-fluid layer under a general rotation field and a horizontal magnetic field

In this paper the Rayleigh-Taylor instability (RTI) of a two-fluid layer system under the simultaneous action of a general rotation field and a horizontal magnetic field is presented. An approximate and an exact solution of the eigenvalue equation are calculated. These solutions are important not only to understand more deeply the physical problem but also to determine the correct numerical solutions. Numerical calculations are done for an unstable density stratification in the cases of horizontal magnetic field parallel and perpendicular to the horizontal component of the angular velocity. For an adverse density stratification, it is shown that in comparison to previous works, the horizontal magnetic field creates new angular areas (of the angle of propagation of the perturbation) at which the perturbation is stable and propagates as traveling waves. It is also shown that the vertical component of the angular velocity has a destabilizing effect because it works to eliminate the stable angular areas.     

Unsteady flow of a relativistic radiating gas in a gravitational field

The unsteady flow of a relativistic radiating neutrino gas is studied in a gravitational field. The curved body is assumed to be a vertical flat plate on which is imposed a time-dependent perturbation on a basic flow. For small perturbations, the ill-posed problem is reduced to a well-posed one and analytical solutions are developed.     

Numerical Study for the Bursty Nature of Spontaneous Fast Reconnection

In this paper, spontaneous fast reconnection in a neutral current sheet, which is initially perturbed by a localized resistivity, is studied by the newly developed Space-Time Conservation Element and Solution Element (CESE) method. After the initial perturbation is switched off, an anomalous resistivity is allowed to occur if a threshold of the local electron-ion drift velocity is exceeded. For a given threshold value, the amount of the reconnected magnetic flux introduced by the initial perturbation is very crucial for the onset of the anomalous resistivity. The numerical results indicate that fast reconnection can develop self-consistently with slow shocks extending between the diffusion region and a large-scale plasmoid-like structure, which is pushed forward by the reconnection outflow. A Petschek-like configuration is then built up, but it can not be sustained as a quasi-steady state. In fact, during the reconnection evolution, the diffusion region undergoes an elongation process so that after the dynamic process is nonlinearly saturated secondary tearing is subject to occur at the center of the system. This leads to enhanced and time-dependent reconnection. The reconnection evolution is further studied in various physical situations, also confirming the bursty nature of the spontaneous fast reconnection mechanism.     

Rotational modes in a slowly and uniformly rotating star

A perturbation method is derived forr-modes in a slowly and uniformly rotating star. In contrast to previous studies, the perturbation of the gravitational potential is included in the perturbation method.On the assumption that the effects of the centrifugal force are taken into account in the equilibrium model up to the second order in the angular velocity, an eigenvalue problem of sixth-order in the radial coordinate is derived that allows one to determine the zeroth-order toroidal displacement field and the third-order term in the expansion of the eigenfrequency. Furthermore, another eigenvalue problem is derived that governs the first-order toroidal displacement field and the fourth-order term in the expansion of the eigenfrequency. This second eigenvalue problem is also of the sixth-order in the radial coordinate.It is shown that the third-order term in the expansion of the eigenfrequency is real, and that the fourth-order term is zero.     

Primeval Turbulence?

Some difficulties with the Primeval Turbulence picture for galaxy formation are considered. It is shown that the evolution of the matter turbulence one decoupled from the radiation depends very much on the assumed turbulence velocity. If the velocity is small the turbulence simply adds to the amplitude of the growing density perturbation mode, an effect almost indistinguishable from the equally ad hoc assumption of a somewhat larger initial density perturbation. If the turbulence velocity field were made large enough to provide the angular velocity of the Galaxy, or the nominal peculiar velocities of galaxies, the galaxies would have formed sooner than one would otherwise have speculated.Research supported in part by the National Science Foundation.     

Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass

The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used. The application of the averaging method to the perturbed two-body problem is possible but involves many technical difficulties which in the case of the two-body problem with variable mass are avoided by deriving new and more suitable equations for these perturbation problems. Application of the averaging method to these perturbation problems yields asymptotic approximations which are valid on a long time-scale. It is shown by comparison with results obtained earlier that in the case of the two-body problem with slow decrease of mass the averaging method cannot be applied if the initial conditions are nearly parabolic. In studying the two-body problem with quick decrease of mass it is shown that the new formulation of the perturbation problem can be used to obtain matched asymptotic approximations.     

Deformation of a line-element in the phase space at the triangular libration point

The flow in the projection of the phase space into the configuration space is presented in the neighborhood of a neutrally (or critically) stable equilibrium point in the restricted problem of three bodies. The projection is a line-element every point of which has zero initial velocity. After the elapse of various times the mapping (the rotations and elongations) of the line-element is described showing chaotic behavior.