Radiative Transfer in Inhomogeneous Atmospheres. I

This series of papers is devoted to multiple scattering of light in plane parallel, inhomogeneous atmospheres. The approach proposed here is based on Ambartsumyan's method of adding layers. The main purpose is to show that one can avoid difficulties with solving various boundary value problems in the theory of radiative transfer, including some standard problems, by reducing them to initial value problems. In this paper the simplest one dimensional problem of diffuse reflection and transmission of radiation in inhomogeneous atmospheres with finite optical thicknesses is considered as an example. This approach essentially involves first determining the reflection and transmission coefficients of the atmosphere, which, as is known, are a solution of the Cauchy problem for a system of nonlinear differential equations. In particular, it is shown that this system can be replaced with a system of linear equations by introducing auxiliary functions P and S. After the reflectivity and transmissivity of the atmosphere are determined, the radiation field in it is found directly without solving any new equations. We note that this approach can be used to obtain the required intensities simultaneously for a family of atmospheres with different optical thicknesses. Two special cases of the functional dependence of the scattering coefficient on the optical thickness, for which the solutions of the corresponding equations can be expressed in terms of elementary functions, are examined in detail. Some numerical calculations are presented and interpreted physically to illustrate specific features of radiative transport in inhomogeneous atmospheres.     

The first three orders of scattering in vertically inhomogeneous scattering-absorbing media

In order to facilitate the computations of the intensities of radiation reflected and/or transmitted by plane-parallel, vertically inhomogenous, scattering-absorbing media, we carry out the optical thickness integrations of the Cauchy systems (normally referred to as Invariant Imbedding Equations) for reflection and transmission functions originating from the first three orders of scattering: the medium in question is represented by a stack of a certain number of homogeneous slabs, each of which is characterized by a constant single scattering albedo and a constant phase function together with the optical thickness.The results are a set of recurrence formulae involving only the angular intergrations, a convenient feature for numerical computations, and should prove useful particularly for finding approximate values of the high frequency Fourier coefficients of reflection and transmission functions of inhomogeneous media or efficiently initializing the solution for a thin layer to perform rigorous multiple scattering computations by means of other techniques such as the Doubling-Adding method.     

Scattering and transmission functions of radiation by finite atmospheres with reflecting surfaces

In the present paper, with the aid of invariance principles in connection with the scattering matrix, we get the exact solution of diffuse reflection and transmission problems by finite inhomogeneous, anisotropically scattering atmospheres bounded by reflecting sufaces. On making use of the reflection and transmission integral operators, we show how to obtain the non-linear integro-differential equations for these operators, which do not depend on the initial condition. Then, we have a system of the required integro-differential equations for the scattering and transmission functions. The obtained result is new, so far as we know. Finally, using the scattering matrix, we reduce the diffuse reflection and transmission problems for planetary atmospheres with reflecting surfaces to the standard diffuse reflection and transmission problems.Supported by the National Science Foundation under Grant No. GP29049 and the Atomic Energy Comminission, Division of Research under Contract No. AT(40-3)-133, Project 19.     

Reflection Properties of Gravito-MHD Waves in an Inhomogeneous Horizontal Magnetic Field

We derive the dispersion equation for gravito-magnetohydrodynamical (MHD) waves in an isothermal, gravitationally stratified plasma with a horizontal inhomogeneous magnetic field. Sound and Alfvén speeds are constant. Under these conditions, it is possible to derive analytically the equations for gravito-MHD waves. The high values of the viscous and magnetic Reynolds numbers in the solar atmosphere imply that the dissipative terms in the MHD equations are negligible, except in layers around the positions where the frequency of the MHD wave equals the local Alfvén or slow wave frequency. Outside these layers the MHD waves are accurately described by the equations of ideal MHD. We consider waves that propagate energy upward in the atmosphere. For the plane boundary, z=0, between two isothermal plasma regions with horizontal but different magnetic fields, we discuss the boundary conditions and derive the equations for the reflection and transmission coefficients. In the simpler case of a gravitationally stratified plasma without magnetic field, these coefficients describe the reflection and transmission properties of gravito-acoustic waves.     

Invariant imbedding and Chandrasekhar's planetary problem of radiative transfer

In connection with Chandrasekhar's planetary problem of radiative transfer the total scattering and the diffuse transmission functions have been discussed by several authors (cf. Chandrasekhar, 1950; van de Hulst, 1948; Sobolev, 1948; Bellman,et al., 1967; Kagiwada and Kalaba, 1971). With the aid of the Bellman-Krein formula for the resolvent kernel of the auxiliary equation governing the source function, we show how the invariant imbedding equations governing the diffuse scattering and transmission functions can readily be obtained. So far as we know, the Cauchy system of the functional equations for the scattering and transmission functions is new and is well-suited for the numerical computation.Supported by the National Science Foundation under Grant No. GP 29049, and by the Atomic Energy Commission under Grant No. AT (40-3)-113 Project 19.     

On combined operations method for transfer problems in homogeneous,cylindrical media

The problem of diffuse reflection by a homogeneous, isotropically scattering, infinite cylindrical medium has been considered. The relevant auxiliary equation has been formulated, the scattering function defined and the integro-differential equation for such function deduced. For a medium having cylindrical distribution of source in addition to the incident flux at the outer surface, the integro-differential equation for the emergent intensity has been established.     

Particle transfer in multiregions

Asymptotic calculations for reflection and transmission coefficients for particles incident on an inhomogeneous plane parallel medium are performed. The medium is assumed to consist of several different optically thick homogeneous layers. Functional relations between the reflection and transmission coefficients for the sublayers are obtained. The invariant embedding concepts are used to calculate the albedo for sublayers. Numerical calculations and comparisons are performed.     

Combined-Operations method for diffuse reflection by an isotropic,non-coherent scattering homogeneous sphere

Combined-Operations method has been utilised to solve the problem of diffuse reflection by a homogeneous, isotropic, non-coherent scattering spherical medium. The source function is considered to be frequency independent. The auxiliary equation has been formulated, the scattering function defined, and the integro-differential equation for this function deduced. A method for obtaining the emergent intensity and the internal source function for non-zero internal source distribution has been suggested for a given line profile.     

A new technique for exact and unique solution of transfer equations in finite media

In this paper we develop a new method, combined with Laplace transformation and Wiener-Hopf technique, to obtain unique solutions of transport equations in finite media. For this purpose we consider the simple transfer equation for diffuse reflection by a plane-parallel finite atmosphere scattering radiation with moderate anisotropy. It is transformed, by Laplace transformation, into two coupled linear integral equations which are then reduced to two uncoupled Fredholm integral equations admitting of unique solutions by the method of iteration for values of the breadth of the atmosphere greater than that specified, depending on the scattering process.     

Fluctuations in the heliospheric hydrogen distribution induced by generalized and time-dependent interstellar boundary conditions

It is well known that the neutral component of the local interstellar medium can effectively pass through the plasma interface ahead of the solar system and can penetrate deeply into the inner heliosphere. Here we present a newly-developed theoretical approach to describe the distribution function of LISM neutral hydrogen in the heliosphere, also taking into account time-dependent solar and interstellar boundary conditions. For this purpose we start from a Boltzmann-Vlasov equation, Fourier-transformed with respect to space and time coordinates, in connection with correspondingly transformed solar radiation forces and ionization rates, and then arrive at semi-analytic solutions for the transformed hydrogen velocity distribution function. As interstellar boundary conditions we allow for very general, non-Maxwellian and time-dependent distribution functions to account for the case that some LISM turbulence patterns or nonlinear wave-like shock structures pass over the solar system. We consider this theoretical approach to be an ideal instrument for the synoptic interpretation of huge data samples on interplanetary Ly- resonance glow intensities registered from different celestial directions over extended periods of time. In addition we feel that the theoretical approach presented here, when applied to interplanetary resonance glow data, may permit the detection of genuine fluctuations in the local interstellar medium.     

A method of computing the emergent radiation by the atmosphere in the region ranging from ultraviolet to infrared

A method of computing the diffuse reflection and transmission radiation by an inhomogeneous, plane-parallel planetary atmosphere with internal emission source is discussed by use of the adding method. If the atmosphere is simulated by a number of homogeneous sub-layers, the radiation diffusely reflected or transmitted by the atmosphere can be expressed in terms of the reflection and transmission matrices of the radiation of sub-layers. The diffusely transmitted radiation due to the internal emission source can be also easily computed in the same manner. These equations for the emergent radiation are in a quite general form and are applicable to radiative transfer in the atmosphere in the region from ultraviolet to infrared radiation. With this method, the tiresome treatment due to the polarity effect of radiation is overcome.     

Diffuse reflection and transmission of radiation by finite atmospheres with specular reflectors

It is supposed that solar rays are incident uniformly on the top of a plane-parallel, anisotropically scattering, and homogeneous atmosphere, whose bottom is bounded by a specular reflector. An initial-value solution for the scattering and transmission functions is obtained, starting with an auxiliary equation. Some results of numerical experiments for given optical parameters are presented in Figures.     

Nonlinear diffuse reflection and transmission of radiative energy by a layer of finite thickness

The major results for the linear problem of diffuse reflection and transmission of radiation by a layer of finite thickness are carried over to the nonlinear case by successive application of Ambartsumyan’s approach for a one dimensional anisotropic medium. Formulas are given for nonlinear addition of layers which can be used to construct recurrence calculation procedures for uniform, periodic, and arbitrary stratified media. A complete set of differential equations for invariant imbedding is derived with the aid of these formulas. These equations are used to obtain a system of total invariance equations, which, in turn, offer the possibility of reducing the nonlinear problem of diffuse reflection and transmission during irradiation of a layer from both sides to the simpler problem of illuminating this medium from only one side, with the thickness of the layer remaining only as a fixed parameter. Finally, it is shown that the results obtained for the single frequency case (two-level atom) remain valid in the polychromatic case (multilevel atom), which is important for interpreting astrophysical data.     

Evaluation of the internal radiation field in a composite planetary atmosphere with the lambert surface calculated by matrix method

If the atmosphere is simulated by a number of homogeneous sublayers, it was shown (Takashima, 1973a) that the intensity and polarization parameters emerging from any boundary of internal sublayer's field can be determined, provided that the diffuse reflection and transmission matrices of radiation of sublayers are known. Furthermore, if the surface (ground) is assumed to reflect light in accordance with the Lambert law, it is shown in this paper that the internal radiation field at boundary of any sublayer can be also computed in terms of the diffuse radiation matrices of sublayers rather than in terms of that of the entire atmosphere (Sekera, unpublished). The effect of polarization is included.     

Order-of-scattering and doubling method for the solution of transfer equations

A method to obtain scattering and transmission functions for computational purposes has been developed in this paper. This method can be easily extended to the scattering atmosphere involving internal energy and may be to the time-dependent case. Special cases such as the homogeneous process, the Helmholtz's principle of reciprocity and the self-adjoint are presented.     

A hybrid mode of diffuse and specular reflector for computation of the emergent radiation by the adding method

A method of computing the diffuse reflection and transmission radiation from an inhomogenous, plane-parallel planetary atmosphere bounded by the hybrid surface of a diffuse and specular reflector is discussed by using the addition method. If the atmosphere is simulated by a number of homogeneous sublayers, the radiation diffusely reflected and transmitted by the atmosphere can be expressed in terms of the diffuse reflection and transmission matrices of radiation of sublayers (Laciset al., 1974; Takashima, 1973, 1975). With this method (Takashima, 1975), the troublesome treatment due to the effect of polarity of radiation is overcome. Moreover, if the surface reflects radiation in accordance with the Lambert law as well as a quite arbitrary phase matrix (Takashima, 1974), the addition method can be easily extended. It is shown in this paper that the addition method is suitable for numerical computation even if the surface reflects light according to the hybrid mode of the diffuse and specular law (Uenoet al., 1974; Mukai, 1976).On leave from the Meterological Research Institute, Tokyo, Japan.     

Non-stationary one-dimensional probabilistic radiative transfer in a bounded medium

Monochromatic time-dependent diffusion of a photon in a homogeneous bounded medium is considered as a one-dimensional probabilistic process in a medium with absorbing boundaries. Simple and intuitive relations are shown to exist between Laplace transforms of time-dependent distributions of photons in infinite medium and inside a bounded medium, as well as for the corresponding emerging intensities. One obtains in particular an explicit and exact expression for the original time-dependent distribution inside a bounded one-dimensional medium.     

Magnetogasdynamic plane shock motion

In the present work the reflection of a plane shock wave is studied in order to achieve a high pressure and temperature state by a reflected shock wave. We consider the plane geometry and solve the one-dimensional, time-dependent system of hyperbolic equations by Rusanov's method.     

Investigation of numerical properties of Hovenier's Exit Function equation for multiple scattering of light

We show that Hovenier's Exit Function equation describing reflection and transmission by a plane-parallel layer can be obtained from the Invariant Imbedding equations. As an immediate extension we obtain a similar equation for an Exit Function defined in terms of reflection and transmission functions for successive orders of scattering. These equations allow the reflection and transmission functions of a homogeneous atmosphere of arbitrary optical thickness to be obtained from angle integrations of only one function.A technique based on successive iterations is developed to solve Hovenier's equation. The numerical behavior of this equation is then investigated employing a few representative (i.e., isotropic, Rayleigh, and Henyey-Greenstein) phase functions with the following conclusions. (i) As long as the deviation from isotropy is small (cos 0.15), the Exit Function equation can be numerically solved with an efficiency comparable to that of the standard Doubling technique, which is one of the fastest algorithms available. (ii) The reflection function generated from the Exit Function is usually more accurate than the corresponding transmission function, particularly in the case of large optical thickness. (iii) As the degree of anisotropy increases, so does the difficulty in obtaining the numerical solution for the Exit Function. The solution of the equation depends sensitively on the treatment of the numerical singularities which arise from the integrands and also on the initial approximation employed for the iteration. An improved scheme is required for numerically obtaining the Exit Function in order for this method to yield accurate reflection and transmission functions for strongly anisotropic scattering.     

Group-theoretical description of radiative transfer in one-dimensional media

Group theory is used to describe a procedure for adding inhomogeneous absorbing and scattering atmospheres in a one-dimensional approximation. The inhomogeneity originates in the variation of the scattering coefficient with depth. Group representations are derived for the composition of media in three different cases: inhomogeneous atmospheres in which the scattering coefficient varies continuously with depth, composite or multicomponent atmospheres, and the special case of homogeneous atmospheres. We extend an earlier proposal to solve problems in radiative transfer theory by first finding global characteristics of a medium (reflection and transmission coefficients) and then determining the internal radiation field for an entire family of media without solving any new equations. Semi-infinite atmospheres are examined separately. For some special depth dependences of the scattering coefficients it is possible to obtain simple analytic solutions expressed in terms of elementary functions. An algorithm for numerical solution of radiative transfer problems in inhomogeneous atmospheres is described.