Plane-parallel Atmosphere Research Articles

Nonlinear Evolution of the Magnetothermal Instability in Two Dimensions

In weakly magnetized, dilute plasmas in which thermal conduction along magnetic field lines is important, the usual convective stability criterion is modified. Instead of depending on entropy gradients, instability occurs for small wavenumbers when (∂P/∂z)(∂ ln T/∂z) > 0, which we refer to as the Balbus criterion. We refer to the convective instability that results in this regime as the magnetothermal instability (MTI). We use numerical MHD simulations that include anisotropic electron heat conduction to follow the growth and saturation of the MTI in two-dimensional, plane-parallel atmospheres that are unstable according to the Balbus criterion. The linear growth rates measured in the simulations agree with the weak-field dispersion relation. We investigate the effect of strong fields and isotropic conduction on the linear properties and nonlinear regime of the MTI. In the nonlinear regime, the instability saturates and convection decays away when the atmosphere becomes isothermal. Sustained convective turbulence can be driven if there is a fixed temperature difference between the top and bottom edges of the simulation domain, and if isotropic conduction is used to create convectively stable layers that prevent the formation of unresolved, thermal boundary layers. The largest component of the time-averaged heat flux is due to advective motions. These results have implications for a variety of astrophysical systems, such as the temperature profile of hot gas in galaxy clusters and the structure of radiatively inefficient accretion flows.

Effects of gravitational darkening on the determination of fundamental parameters in fast-rotating B-type stars

In this paper we develop a calculation code to account for the effects carried by fast rotation on the observed spectra of early-type stars. Stars are assumed to be in rigid rotation and the grid of plane-parallel model atmospheres used to represent the gravitational darkening are calculated by means of a non-LTE approach. Attention is paid on the relation between the apparent and parent non-rotating counterpart stellar fundamental parameters and apparent and true vsini parameters as a function of the rotation rate Omega/Omega_c, stellar mass and inclination angle. It is shown that omission of gravitational darkening in the analysis of chemical abundances of CNO elements can produce systematic overestimation or underestimation, depending on the lines used, rotational rate and inclination angle. The proximity of Be stars to the critical rotation is re-discussed by correcting not only the vsini of 130 Be stars, but also their effective temperature and gravity to account for stellar rotationally induced geometrical distortion and for the concomitant gravitational darkening effect. We concluded that the increase of the vsini estimate is accompanied by an even higher value of the stellar equatorial critical velocity, so that the most probable average rate of angular velocity of Be stars attains Omega/Omega_c ~ 0.88.

Analytic Green’s Function for Radiative Transfer in Plane-Parallel Atmospheres

Abstract Green’s function is a widely used approach for boundary value problems. In problems related to radiative transfer, Green’s function has been found to be useful in land, ocean, and atmosphere remote sensing. It is also a key element in higher order perturbation theory. This paper presents an explicit expression of the Green’s function, in terms of the source and radiation field variables, for a plane-parallel atmosphere with either vacuum boundaries or a reflecting [atmosphere–bidirectional reflectance distribution function (BRDF)] surface. A FORTRAN 95 code, Green’s function and discrete ordinate method (GDOM), has been developed to efficiently compute the Green’s function. This code also integrates with an implementation of the discrete ordinate method with several extensions and improvements. Computing complexity of the Green’s function algorithm is analyzed, and validation of the code is discussed.

A peculiar discontinuity in the intensity of light reflected by a plane-parallel atmosphere

A peculiar discontinuity in the intensity of light reflected by a plane-parallel atmosphere for horizontal directions of incidence and reflectance is uncovered. A formula is derived describing this discontinuity as a function of the albedo of single scattering and the phase (scattering) function, both at the top of the atmosphere, and the manner in which the directions of incidence and emergence become horizontal. This formula holds for atmospheres that are homogeneous or inhomogeneous, semi-infinite or finite with or without a reflecting surface underneath. Some illustrative examples and sample applications are also discussed.

Coupling of acoustic waves to clouds in the jovian troposphere

Seismology is the best tool for investigating the interior structure of stars and giant planets. This paper deals with a photometric study of jovian global oscillations. The propagation of acoustic waves in the jovian troposphere is revisited in order to estimate their effects on the planetary albedo. According to the standard model of the jovian cloud structure there are three major ice cloud layers (e.g., [Atreya et al., 1999. A comparison of the atmospheres of Jupiter and Saturn: Deep atmospheric composition, cloud structure, vertical mixing, and origin. Planet Space Sci. 47, 1243–1262]). We consider only the highest layers, composed of ammonia ice, in the region where acoustic waves are trapped in Jupiter's atmosphere. For a vertical wave propagating in a plane parallel atmosphere with an ammonia ice cloud layer, we calculate first the relative variations of the reflected solar flux due to the smooth oscillations at about the ppm level. We then determine the phase transitions induced by the seismic waves in the clouds. These phase changes, linked to ice particle growth, are limited by kinetics. A Mie model [Mishchenko et al., 2002. Scattering, Absorption, and Emission of Light by Small Particles. Cambridge Univ. Press, Cambridge, pp. 158–190] coupled with a simple radiation transfer model allows us to estimate that the albedo fluctuations of the cloud perturbed by a seismic wave reach relative variations of 70 ppm for a 3-mHz wave. This albedo fluctuation is amplified by a factor of ∼70 relative to the previously published estimates that exclude the effect of the wave on cloud properties. Our computed amplifications imply that jovian oscillations can be detected with very precise photometry, as proposed by the microsatellite JOVIS project, which is dedicated to photometric seismology [Mosser et al., 2004. JOVIS: A microsatellite dedicated to the seismic analysis of Jupiter. In: Combes, F., Barret, D., Contini, T., Meynadier, F., Pagani, L. (Eds.), SF2A-2004, Semaine de l'Astrophysique Francaise, Les Ulis. In: EdP-Sciences Conference Series, pp. 257–258].

Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere

Green's function is a widely used approach for boundary value problems. In problems related to radiative transfer, Green's function has been found to be useful in land, ocean and atmosphere remote sensing. It is also a key element in higher order perturbation theory. This paper presents an explicit expression of the Green's function, in terms of the source and radiation field variables, for a plane-parallel atmosphere with either vacuum boundaries or a reflecting (BRDF) surface. Full polarization state is considered but the algorithm has been developed in such way that it can be easily reduced to solve scalar radiative transfer problems, which makes it possible to implement a single set of code for computing both the scalar and the vector Green's function.

Self‐similar Evolution of Relativistic Shock Waves Emerging from Plane‐parallel Atmospheres

We study the evolution of the ultra-relativistic shock wave in a plane-parallel atmosphere adjacent to a vacuum and the subsequent breakout phenomenon. When the density distribution has a power law with the distance from the surface, there is a self-similar motion of the fluid before and after the shock emergence. The time evolution of the Lorentz factor of the shock front is assumed to follow a power law when the time is measured from the moment at which the shock front reaches the surface. The power index is found to be determined by the condition for the flow to extend through a critical point. The energy spectrum of the ejected matter as a result of the shock breakout is derived and its dependence on the strength of the explosion is also deduced. The results are compared with the self-similar solution for the same problem with non-relativistic treatment.

Is Arcturus a well-understood K giant?

We present near-IR interferometric measurements of the K1.5 giant Arcturus (alpha Bootis), obtained at the IOTA interferometer with the FLUOR instrument, in four narrow filters with central wavelengths ranging from 2.03 to 2.39 micron. These observations were expected to allow us to quantify the wavelength dependence of the diameter of a typical K giant. They are compared to predictions from both plane-parallel and spherical model atmospheres. Unexpectedly, neither can explain the observed visibilities. We show how these data suggest the presence of a companion, in accordance with the Hipparcos data on this star, and discuss this solution with respect to Arcturus' single star status.

The effect of aerosol shape in retrieving optical properties of cloud particles in the planetary atmospheres from the photopolarimetric data. Jupiter

The influence of the assumed shape of aerosols on the estimates of the refractive index and size of particles (based on the data of ground-based spectropolarimetric measurements) is investigated for Jupiter’s cloud layer. In the present analysis, we supposed the atmospheric particles to be chaotically oriented spheroids and cylinders with a gamma size distribution. Their single-scattering characteristics were calculated with the T-matrix method, and the intensity and degree of linear polarization of the radiation reflected by the center of the planetary disk were found by solving the vector radiative-transfer equation with consideration for multiple scattering in a plane-parallel atmosphere. We considered a spectral interval from 0.423 to 0.798 µm and phase angles α from 0° to 11°. It has been shown that, if we use the optical characteristics of aerosols found within the frames of a spherical model (Mishchenko, 1990a), the models with the nonspherical particles considered here cannot fit the observational data. The refractive index and the sizes of spheroidal and cylindrical particles were estimated from a comparison of the data of measurements and calculations, and the simplest models for the Jovian cloud layer structure have been considered. We have concluded that the optical parameters of cloud particles (specifically, the refractive index) cannot be reliably estimated only on the basis of measurements made in a narrow range of phase angles.

Radiative properties of magnetic elements

For the understanding of more complex numerical simulation results, a basic facular model is created consisting of a magnetic flux sheet embedded in a plane parallel atmosphere. The atmosphere within the flux sheet is similar to the external one but shifted in the downward direction to result in a Wilson depression of 150 km. It is shown that the horizontal spatial extension of the contrast enhancement produced by this model increases from center to limb from a few tenths to up to 1 �� as a consequence of enhanced radiation from the limbward surface outside (behind) the magnetic flux concentration. For a conceivable explanation of this radiative effect it is noted that a plasma parcel on the solar surface sideways of the flux sheet sees a more transparent sky in the direction towards the flux sheet compared to a direction away from it because of the rarefied atmosphere within the flux sheet. This facular model also produces a dark lane at the disk-center side (in front) of the flux sheet even though no flow is present. It it is due to the deep layers of the flux sheet that have a lower temperature gradient and are cooler than the surrounding atmosphere at equal geometrical depth. This implies that limb observations off er ag limpse of the cool bottom of magnetic elements. The center-to-limb variation of the size and the dark-lane effect derived from this basic model is recovered in a self-consistent, two-dimensional non-stationary simulation of a magnetic flux concentration. These findings are in excellent agreement with and provide an interpretation of recent observations of faculae at very high resolution by Lites et al. (2004, Sol. Phys., 221, 65).

Exact and unique solution of a transport equation in a semi-infinite medium by Laplace transform and Wiener-Hopf technique

The equation of radiative transfer in non-conservative case for diffuse reflection in a plane-parallel semi-infinite atmosphere with axial symmetry has been solved by Laplace transform and Wiener-Hopf technique. We have determined the emergent intensity in terms of Chandrasekhar's H-function and the intensity at any optical depth by inversion.

Transmission of radiation through an aerosol medium

The properties of radiation through an aerosol medium have been achieved. This has been done by employing Mie scattering theory to calculate the radiation transfer scattering parameters in the form of extinction, absorption and scattering efficiencies. The equation of radiative transfer for the heat flux through a plane parallel atmosphere of aerosol has been solved. The aerosol size distributions are found in practical systems. Average efficiencies over size distribution for spherical particles of complex refractive index are calculated. Therefore, the radiative properties of stratospheric aerosols have been done. The obtained results found to be in a good agreement with the previous work.

A solution of radiative transfer in isotropic plane-parallel atmosphere by integrating Milne equation

In this paper we solve the inversion problem of the radiative transfer process in the isotropic plane-parallel atmosphere by iterative integrations of the Milne integral equation. As a result, we obtain the scattering function in the form of a cubic polynomial in optical thickness. The author has already solved the same problem by iterative integrations of Chandrasekhar's integral equation. In the Milne integral equation, both the cosines of the viewing angles and the optical thickness are integral variables, while in Chandrasekhar's integral equation the cosines of the viewing angles are variables but the optical thickness is not. We derive several series of exponential-like functions as intermediate derivations. Their convergences are evaluated by the author's previous work in the solution of Chandrasekhar's integral equation. The truncated scattering function up to the third order in optical thickness thus obtained is identical to that obtained from Chandrasekhar's integral equation, though their apparent forms are different. Chandrasekhar pointed out that the solution of Chandrasekhar's integral equation does not have a uniqueness of solution. The Milne equation, in contrast, has been proven to have a unique solution. We discuss the uniqueness of the solution by these two methods.

Solution of the [formula omitted] radiative transfer equation in an inhomogeneous plane-parallel atmosphere by the decomposition method

In this work, we report a closed-form solution for the S N radiative transfer equation in an inhomogeneous plane-parallel atmosphere by the decomposition method. We present numerical simulation and comparison with available results in the literature.

On the Oxygen Abundance of HE 0107-5240

We have determined the oxygen abundance of HE 0107-5240 from UV-OH lines detected in VLT/UVES spectra. Using a plane-parallel LTE model atmosphere, we derive [O/Fe] = +2.3, and a similar analysis of CD -38°245 yields [O/Fe] = +0.9. We estimate systematic errors due to three-dimensional effects to be on the order of 0.3-0.4 dex. That is, our derived O abundances are likely overestimates: effects from thermal inhomogeneities due to convection may require that the abundances should be reduced by 0.3-0.4 dex or even more. Radial velocity data for HE 0107-5240 based on high-resolution spectra show that over a time span of 373 days, the radial velocity was constant at 44.5 km s-1, with a 1 σ scatter of the measurements of 0.5 km s-1. However, it cannot yet be ruled out that HE 0107-5240 is a very long period and/or low amplitude binary. These results provide new constraints on scenarios for the origin of the abundance pattern of HE 0107-5240. In particular, it seems unlikely that the large overabundances of CNO have been produced in a medium-mass asymptotic giant branch star that later evolved into a white dwarf. The oxygen abundance of HE 0107-5240 is significantly smaller than the prediction of Umeda & Nomoto from calculated yields of an ~25 M☉ Population III star exploding as a supernova of low explosion energy (Eexp = 3 × 1050 ergs) with mixing and fallback. The scenario of Limongi et al., involving two Population III supernovae, predicts an oxygen abundance of [O/Fe] = +4.1 for HE 0107-5240, in strong contradiction to the observed value. In conclusion, none of the above-mentioned scenarios, in their present realizations, can satisfactorily explain the abundance pattern of HE 0107-5240.

A fast multistream scattering-based Jacobian for microwave radiance assimilation

The full utilization of satellite-based passive microwave imagery for weather forecasting rests on the ability to assimilate radiances into numerical weather prediction (NWP) models for highly scattering and absorbing hydrometeor states. State vector updates need to be performed rapidly enough to maintain pace with the sensor data stream and require, in particular, rapid calculation of the tangent linear relationship (Jacobian) between the observed antenna temperatures and the NWP prognostic hydrometeor parameters. To facilitate the use of both spaceborne and airborne passive microwave data in numerical forecasting, we present a new rapid multiple-stream discrete-ordinate algorithm for calculating the Jacobian under arbitrary scattering and absorbing conditions. The algorithm is based on the layer-adding method for a plane-parallel atmosphere for which the number of operations required to compute the solution is proportional to the number of layers. A nontrivial aspect of the problem is the stable calculation of the reflectance and transmittance operators in highly scattering layers for which a diagonalization technique and analytical factorization of specific matrices are used to ensure stability. Scaling calculations suggest that the new algorithm will be suitable for use in real-time all-weather microwave radiance assimilation.

The effect of particle shape on microphysical properties of Jovian aerosols retrieved from ground-based spectropolarimetric observations

In this paper we study potential errors in the retrieved values of the cloud particle refractive index and size that may result from the neglect of nonsphericity in analyses of ground-based spectropolarimetric observations of Jupiter. Using T-matrix and vector radiative-transfer codes, we have carried out numerous computations in the spectral range from 0.423 to 0.798μm for a plane-parallel model atmosphere comprising polydisperse ensembles of randomly oriented oblate and prolate spheroids with sizes comparable to the wavelength. The particle models were specified by a gamma size distribution with varying values of the effective radius and effective variance and by a varying refractive index. Two models of the Jovian atmosphere were considered: (i) a homogeneous semi-infinite layer composed of gas and cloud particles; and (ii) a two-layered medium with a layer of pure gas on top of a semi-infinite homogeneous layer composed of gas and cloud particles. Good agreement between ground-based observational data of the brightness and the degree of linear polarization for the center of the Jovian disk and model computations was found for the following combinations of the real part of the refractive index mR, the effective radius reff, and the effective variance veff: mR=1.45, reff=0.35μm, veff=0.40 for oblate spheroids with an axial ratio of a/b=1.3; mR=1.50, reff=0.35μm, veff=0.30 for prolate spheroids with a/b=1/1.3; mR=1.52, reff=0.40μm, veff=0.35 for oblate spheroids with a/b=1.5, and mR=1.54, reff=0.90μm, veff=0.30 for prolate spheroids with a/b=1/1.5. These values (especially the real part of the refractive index) differ quite significantly from those retrieved previously for the model of spherical cloud particles and emphasize nonuniqueness of remote-sensing retrievals based on the ground-based data alone.

Eruption of a Buoyantly Emerging Magnetic Flux Rope

We present a three-dimensional numerical magnetohydrodynamic simulation designed to model the emergence of a magnetic flux rope passing from below the photosphere into the corona. For the initial state, we prescribe a plane-parallel atmosphere that comprises a polytropic convection zone, photosphere, transition region, and corona. Embedded in this system is an isolated horizontal magnetic flux rope located 10 photospheric pressure scale heights below the photosphere. The flux rope is uniformly twisted, with the plasma temperature inside the rope reduced to compensate for the magnetic pressure. Density is reduced in the middle of the rope, so that this section buoyantly rises. The early evolution proceeds with the middle of the rope rising to the photosphere and expanding into the corona. Just as it seems the system might approach equilibrium, the upper part of the flux rope begins to separate from the lower, mass-laden part. The separation occurs through stretching of the field, which forms a current sheet, where reconnection severs the field lines to form a new system of closed flux. This flux then erupts into the corona. Essential to the eruption process are shearing motions driven by the Lorentz force, which naturally occur as the rope expands in the pressure-stratified atmosphere. The shearing motions transport axial flux and energy to the expanding portion of the magnetic field, driving the eruption.

Exact solution of the standard transfer problem in a stellar atmosphere

We come back to the analytical solution of the standard transfer problem in a stellar atmosphere. It consists in solving the radiative transfer equation in a homogeneous and isothermal plane-parallel atmosphere, with light scattering taken as isotropic and monochromatic. The literature on the subject is reviewed and the existing solution in a finite slab is improved thanks to the introduction of non classical auxiliary functions. Eleven-figure tables of the solution are given for typical values of the input parameters currently met in stellar atmospheres.

Radiative transfer in one-dimensional inhomogeneous atmospheres

We use Ambartsumian's method of addition of layers to show that various problems, including the standard ones, of radiation transfer in a plane-parallel inhomogeneous atmosphere may be reduced to the solution of the Cauchy problems for linear differential equations. This allows avoiding the known difficulties arising in solving the boundary-value problems to which the classical approach leads. For the purpose of exposition, the paper deals with the simplest one-dimensional problem of multiple scattering for an atmosphere of finite optical thickness. The idea of the approach is that we start with determining the reflection and transmission coefficients of an atmosphere by solving the initial-value problem for a set of linear differential equations of the first order. After that the internal radiation field is found immediately without solving any new equation. The approach is applied to several classical problems of astrophysical interest. In particular, we evaluate the mean number of scatterings undergone by different types of photons. The transfer of radiation in an atmosphere with arbitrarily distributed internal sources is considered. Analytical solutions for these problems are obtained. Simple recursion formulas are derived to find the radiation intensity emitted by a multicomponent atmosphere. The problem of multiple scattering of radiation with partial redistribution over frequencies is discussed to demonstrate the generalization of the approach to the matrix case. The results of numerical calculations are given.