Sterrewacht "Sonnenborgh", Servaas Bolwerk 13, Utrecht, The Netherlands
M. M. R. WILLIAMS
Department of Nuclear Engineering, Queen Mary College, University of
London, London, England
Preface: Garry Hunt
The Symposium on Interdisciplinary Applications of Transport Theory
was sponsored by the Science Research Council and organised by the
SRC Atlas Computer Laboratory, Chilton, Berkshire, as the Third
Atlas Symposium. The meetings were held at the Mathematical
Institute, Oxford on 1-4 September 1970 and a hundred scientists
participated in the programme which included ten reviews and twenty eight
contributed papers.
The main reason for organising such a
meeting developed from the observation that there are many physical
problems in which Transport Theory plays an important role;
astrophysics, meteorology and neutron transport are particular
examples. The physical conditions and quantities vary widely from one
discipline to another and these constraints have resulted in the
production of specialised methods of solution being used in each
research area. Furthermore, in contrast to the classical approach
favoured by many research workers, the advent of the high speed
digital computer has influenced some scientists to analyse their
problems from a computational viewpoint. In view of the growing
diversity of the problems in which Transport Theory plays a major
role and the rapid development of the specialised techniques which
are available to solve them it was proposed to hold an
interdisciplinary meeting. Hopefully this would generate an exchange
which would yield new insight into the analysis of each field of
application.
Interdisciplinary meetings of this type are not new. The
idea has been tested on varying scales at Boulder, Colorado (1965);
Ankara, Turkey (1965); Philadelphia (1966); and Blacksburg, Virginia
(1968) for a variety of disciplines. Encouraged by the apparent
success of these earlier meetings it was suggested that a meeting be
held in Oxford to cover transport problems in astrophysics,
meteorology and neutron transport; disciplines that had not
previously been represented at a single interdisciplinary meeting.
The programme for each discipline was arranged around state-of-the-art
review papers with additional contributed papers of both
mathematical and physical content. In all cases the interdisciplinary
nature of the meeting was stressed so that topics of common interest
to several applied fields generated many active discussions. This, in
itself, provides us with some indication of the success of this type
of meeting.
For the Success of this meeting, we are greatly indebted
to the Science Research Council for providing the necessary financial
support, to Dr Howlett for providing the mechanism that enabled this
meeting to be the subject of the Third Atlas Symposium and for
allowing the Administrative Division of the Laboratory to be used for
the organisation of the meeting in addition to their normal duties.
It is also a pleasure to thank Professor Penner for allowing the
proceedings of this meeting to appear as a special issue of JQSRT and
for the cooperation of the publishers, Pergamon Press.
I count it a remarkable privilege to be able to welcome to Oxford
such a distinguished gathering of astrophysicists: a term which I use
in a very general sense, to include everyone who is concerned with
the subject of this Symposium: Radiation Transport. I find it hardly
believable that I should, if only for a few moments, be on the same
platform as Professor Chandrasekhar. But at the same time I am only
too well aware that it would be sheer impertinence of me to say
anything to you about the subject which you will be discussing for the
rest of the week: though I did make some contact with some of the
mathematics when I reviewed Dr Busbridge's book for the Mathematical
Gazette - and what's more, actually read it.
What I want to do now,
very briefly, is to say just a little about these Atlas Laboratory
Symposia, of which you will see from the programme that this is the
third. The Atlas Computer Laboratory is one of the establishments of
the Science Research Council, which is in turn one of the bodies
through which the British Government supports civil science, in this
context meaning mainly academic science. The Laboratory contributes
to this support by providing computing services to research workers,
mainly in universities but also to some in Government-associated
laboratories. Thus as a body whose professional concern is computing
we are interested in all kinds of applications of computers, and as a
part of the Science Research Council we wish to contribute to the
development of science.
A few years ago some fortunate circumstances
led my senior colleagues and myself to the idea of arranging an
intensive discussion, lasting a week, on a well-defined subject,
Computational Problems of Abstract Algebra; attendance was to be by
invitation only and limited to about 100; we had the warmest
collaboration of the mathematicians in Oxford and were able to hold
the meeting here in this very fine Mathematical Institute, and the
whole thing was a great success, That was in 1967 and, encouraged by
this success, we decided that this meeting should be the first of a
series. The second, in 1968, was on Computational Problems in Number
Theory and was also a great success and here we are at the start of
the third, which is clearly going to be as intensive and as
successful as the first two.
Let me say a few thank-you's. Dr Hunt,
as will have been evident to you from the correspondence you will
have had, has done most of the organising and has been quite tireless
in this. The Laboratory's first contact with the subject of radiation
transport theory was made through Dr Ian Grant, who held one of our
Fellowship posts for five years and who now holds a full Fellowship
at Pembroke College here: I'm glad to have the opportunity to
acknowledge our very great debt to him. Once again I'm very grateful
to Professor Coulson and all the Institute staff for not just
allowing us to meet here but for making us so welcome - as I hope you
are realising, Oxford is a wonderful place for this kind of a
gathering. The Atlas Laboratory administrative staff, under the ever-
watchful eye of Mr C. L. Roberts, has done all the hard work and,
quite literally, nothing has been too much trouble to them. And
finally, I am most grateful to the Science Research Council for
allowing me to use some of the Laboratory's funds and resources to
support this meeting.
Conference Proceedings
Preface
Contents
List of contributors
Introductory remarks by Dr J. Howlett at start of symposium on 1 September 1970
Solution of non-LTE transfer problems, Eugene H. Avrett, Smithsonian Astrophysical Observatory, Cambridge, Mass.
In the outer layers of a stellar atmosphere, the radiation at line and continuum wavelengths strongly influences
the relative number of atoms in various states of excitation and ionization.
Conversely, the radiation at a given point often depends on the atomic populations throughout a large surrounding region.
Various procedures involving either differential equations or integral equations can be used to solve the problem.
The main topic of this review is the integral-equation approach, with emphasis on numerical methods developed in recent years
for the solution of physically realistic problems.
Line formation in pulsating variable stars, W. Kalkofen, C.A. Whitney, Smithsonian Astrophysical Observatory, Cambridge, Mass.
We describe an integral equation method that solves the equation of radiative transfer in a spectral line for an opacity that
depends on depth, frequency, and angle, and derive an expression for the line source function of a two-level atom in a gas
that is traversed by a plane, steady shock wave.
Population inversion in the outer layers of a radiating gas, John T. Jefferies, University of Hawaii
On the basis of solutions to the coupled radiative transfer equations for multiplet lines,
we present simple physical arguments on the expected distribution of population among closely spaced energy levels
in the ground, or a low-lying state, of an atom or molecules.
It is shown that uncoupling of the multiplet lines will occur in the outer layers of a radiating gas,
producing population ‘anomalies’ in the fine structure states associated with the upper and/or lower levels
of the multiplet transitions.
On the application of the generalized Newton-Raphson method in radiative transfer problems, A. Skumanich, B.A. Domenico, National Center for Atmospheric Research and Department of Physics and Astrophysics, University of Colorado, Boulder.
The simulataneous excitation and transfer of radiation in a plasma gives rise to non-linear operator equations
for the atomic occupation numbers. An iterative method for solving such equations, based on a generalized Newton-Raphson scheme,
is described and applied to the problem of the formation (under the condition of complete redistribution) of a
resonance emission line interacting with two associated subsidiary lines.
Radiative transfer in two-component stellar atmospheres, Eugene H. Avrett, Rudolf Loeser, Smithsonian Astrophysical Observatory, Cambridge, Mass.
We describe a method for solving the line-transfer equations for a two-component atmosphere consisting of an array of columns,
characterized by a particular variation of temperature with height, which are embedded in a medium in which the temperature is
a different function of height. Radiative interaction between the two regions is taken into account, and the opacity is allowed
to vary arbitrarily with depth. Some calculations that correspond to the solar Ca K-line are described.
The stellar atmospheres problem, L.H. Auer, Yale University Observatory
This paper discusses the construction of model stellar atmospheres placing particular stress on the mathematical methods used.
The basic equations and assumptions of the problem are presented. Techniques which reduce the transfer equation to a set of
linear equations are developed. It is shown how the constraint of radiative equilibrium may then be included directly
into the transfer equation, yielding much more powerful methods for constructing model atmospheres than previously available.
A modification of the Feautrier method of solving transfer problems is presented that allows advantage to be taken
of such simplifications as isotropic scattering and complete frequency redistribution.
The method has timing and storage requirements comparable to integral equation methods,
but retains the ease of formulation of differential methods.
Grey radiative transfer, G.C. Pomraning, Science Applications, Inc, La Jolla, California
The general problem of reducing a frequency-dependent radiative transfer problem to an equivalent grey problem is discussed.
In particular, we construct, from the integral transport equation, an asymptotic solution to be used in forming grey opacities.
This is closely related, but not identical, to the so-called B-N method widely used in neutron transport work.
This leads to generalizations, to non-zero temperature gradients, of the standard Planck and Rosseland mean opacities.
Some aspects of the non-LTE physics of the helium atom in hot stars, A.B. Underhill, A.G. Hearn, Sonneborgh Observatory, Utrecht and Observatoire de Nice
In the atmospheres of hot stars, the populations of the states of the helium atom are significantly
different from the LTE populations. This not only gives a change in the line source function with a consequent change
in the predicted line depths, but it also changes the scale of the optical depths in the lines, with a consequent change
in the deduced abundance of helium. Some exploratory numerical results are given to illustrate these effects,
which could explain the observed variations of the strengths of the He I and He II lines in O and B type stars.
Line and continuum problems in gaseous nebulae, D.E. Osterbrock, Washburn Observatory, University of Wisconsin
Gaseous nebulae are effectively transparent to nearly all the strong emission lines in the ordinarily observed spectral region,
and if the physical conditions are known the calculation of the expected spectrum simply becomes a
calculation of emission coefficients, without any intervention of radiative-transfer theory.
For this reason astronomers tend to think of stellar- or planetary-atmosphere problems when they think of applications
of radiative transfer theory. Actually, however, there are many important nebular radiative-transfer problems, particularly
those of the ultraviolet continuum and resonance lines, which in fact fix the physical conditions throughout the nebula.
Some of these problems are reviewed in the present paper.
Bellman's new approach to the numerical solution of Fredholm integral equations with positive kernels, Jürgen Gruschinske, Sueo Ueno, Institute of Astrophysics, Kyoto University
In this paper, using Bellman's new technique, we show how to solve numerically the integral equation of the source
function of the radiation field in a homogeneous, isotropically scattering spherical medium or hollow spherical shell
medium with no flux of radiation incident on the outer surface and with an isotropically emitting central point source.
Radiative transfer effects in Cepheid atmospheres, Cecil G. Davis, Los Alamos Scientific Laboratory
The transfer equation, without retardation and considering scattering as absorption, is transformed by the moments
expansion method for use in studying the dynamics of Cepheid atmospheres.
A review of computational techniques for analysing the transfer of radiation through a model cloudy atmosphere, G.E. Hunt, Atlas Computer Laboratory
We consider the problem of constructing a realistic model cloudy atmosphere.
Discussion is made of the computational difficulties associated with computing the single scattering phase diagram
which is a problem central to all models. A brief review is made of recent developments in computational techniques
for analysing radiative transfer problems. Further discussion is made of various computational economies to reduce
computer execution time and storage requirements, and an assessment is made of the penalties paid for implementing these
approximations.
A method for computing the transfer of solar radiation through clouds of hexagonal ice crystals, H. Jacobowitz, NOAA, Washington, D.C.
The angular patterns that result from the scattering of solar radiation by hexagonal ice prisms were computed
under the assumption that the sizes of the prisms were many times the wavelengths of the incident radiation.
Using the results of geometric ray optics, a sufficiently large number of equally spaced, parallel rays were mathematically
traced through the prisms (including internal reflections) in order to determine the contribution to the scattering
pattern that was due to refraction and reflection. The contribution due to forward diffraction was approximated by
the Kirchhoff formula.
Radiative heat transfer in water clouds by infrared radiation, Giichi Yamamoto, Masayuki Tanaka, Shoji Asano, Geophysical Institute, Tohoku University, Sendai, Japan
Radiative heat transfer in water clouds is studied by the method of discrete ordinates,
taking into account not only scattering, absorption and emission by cloud droplets but also absorption and emission by
water vapor in the cloud.
Effet de la diffusion sur l'échauffement radiatif dû au rayonnement solaire, Y. Fouquart, Laboratoire d'Optique Atmosphé rique, Lille, France
The effect of scattering by a haze on heating rate due to absorption of solar radiation by water vapor
is computed in the near infrared and the results are compared with those obtained by neglecting scattering.
General theory of radiative transfer across the random atmosphere-ocean interface, Rudolph W. Preisendorfer, Naval Postgraduate School, Monterey
The general problem of radiative transfer across a random air-water interface is formulated and several
important cases are solved. In particular the reflected sky radiance from a rough sea with given wave height and wave
slope statistics is determined by solving a novel class of stochastic integral equations.
On the solar radiation field in a polluted atmosphere, K.L. Coulson, Department of Agricultural Engineering, University of California, Davis
Measurements of the fields of intensity and polarization of radiation from the sunlit sky,
made in the heavily polluted atmosphere of Los Angeles, are compared with similar measurements obtained
in Davis under particularly clear atmospheric conditions, and with computed fields for selected atmospheric
models. Data for a thick stratus overcast are also included. The measurements are in six spectral intervals
from the ultraviolet to the near infrared, the intervals being defined by interference type optical filters
in combination with a photomultiplier tube having an S-20 response.
A direct method for the integration of the equation of radiative transfer in a turbid atmosphere, Günter Eschelbach, Institut für Meteorologie, Johannes Gutenberg-Universität, Mainz
In this paper we apply a numerical method on the equation of radiative transfer in a turbid atmosphere.
The solution is obtained by means of direct integration of the equation of radiative transfer without any circuitous
series development.
Some theoretical aspects of remote sounding in the earth's atmosphere, C.D. Rodgers, Clarendon Laboratory, Oxford
Methods of solving the inverse problem of atmospheric radiation are reviewed,
for the particular case of remote sounding of temperature and composition of the Earth's atmosphere.
Optimum methods are described for the linear problem, including generalised least squares, minimum variance and
maximum probability. Extension of these optimum methods to the general non-linear problem is indicated, and some
particular cases are described.
Radiative transfer: Terrestrial clouds, S. Twomey, Division of Radiophysics, CSIRO, Sydney
The terrestrial atmospheric physicist is interested in radiative transfer in clouds from two distinct points
of view: firstly, the profound and as yet imperfectly understood effects of clouds on the atmospheric radiation balance
and secondly the inference of cloud characteristics (size, drop size, dispersion, etc.) through measurements of the angular
or wavelength dependence of radiation scattered by clouds.
The work reported here relates to the second aspect>
Multiple scattering in planetary atmospheres, H.C. van de Hulst, Sterrewacht, Leiden
Numerical results of reflection and transmission on plane-parallel layers with strongly anisotropic
scattering are reviewed. A convenient method is to use successive scattering, doubling, and asymptotic fitting,
in three size ranges. The eigenvalues determining the convergence of series in the doubling method are shown and
eigenvalue problems in the other steps are described.
Formation of absorption spectra by diffuse reflection from a semi-infinite planetary atmosphere, Akira Uesugi, William M. Irvine, Yoshiyuki Kawata, Department of Physics and Astronomy, University of Massachusetts
The diffuse reflection from a semi-infinite, plane-parallel, homogeneous atmosphere for arbitrary single scattering
albedo may be computed if the reflection functions for n-times scattered radiation are known in the conservative case.
Rapid convergence is obtained by utilizing asymptotic expressions for the higher order scattering.
Tables of these nth order reflection functions are presented for isotropic scattering and for two phase functions
representative of clouds and hazes.
The necessary relations for computation of the shape and equivalent width of absorption lines formed in a scattering atmosphere
are also presented.
The doubling method applied to multiple scattering of polarized light, J.E. Hansen, J.W. Hovenier, Institute for Space Studies, Goddard Space Flight Center
The doubling method is used for computations of multiple scattering with full account of polarization.
Several checks on the accuracy of the method are indicated. Graphs are presented for the polarization of sunlight
reflected by the visible disk of Venus.
The results show that photons scattered two or more times still have a high degree of polarization which must be included in a
complete interpretation of observations.
The composition of planetary atmospheres, M.B. McElroy, Division of Engineering and Applied Physics, Harvard University
The present status of research on the composition of planetary atmospheres is reviewed.
It is shown that CO2 is the major constituent in the atmospheres of Venus and Mars, and that H2 is
almost certainly the dominant constituent for the outer planets Jupiter, Saturn, Uranus, and Neptune.
Ground based i.r. spectroscopy is the largest single source of information on atmospheric composition and much
of the paper is concerned with difficulties involved in the interpretation of i.r. spectra. The mechanisms for line
formation are examined carefully and it is concluded that multiple scattering by cloud materials must play a dominant
role for Jupiter, Saturn, and Venus. There is no evidence for visible cloud layers on either Uranus or Neptune.
Both planets, however, exhibit deep molecular atmospheres and Rayleigh scattering will affect the strength of absorption
features.
The searchlight problem with isotropic scattering, G.B. Rybicki, Joint Institute for Laboratory Astrophysics,† Boulder, Colorado
A solution to Chandrasekhar's searchlight problem with isotropic scattering is presented for semi-infinite
and finite geometries. The scattering and transmission functions for the Fourier components of intensity are given
by expressions analogous to those of the plane-parallel case.
Some analytical methods in neutron transport theory, Noel Corngold, Division of Engineering and Applied Science, California Institute of Technology
We discuss some distinctive features of neutron transport theory.
The linearized transport equation is similar to those met in theories of gas dynamics and radiative transfer.
We describe the analytical techniques that have been most successful in analyzing the neutron equation.
Multigroup transport equation with a separable kernel, P. Silvennoinen, P.F. Zweifel, Virginia Polytechnic
We consider an N-group, one-dimensional, time-independent transport equation with a separable scattering kernel.
Full and partial range completeness relations are proved using the methods described in the book of Vekua.
The procedures used are simple generalizations of a similar problem with two energy groups treated by the same authors elsewhere.
Fundamental eigenvalues of the linear transport equation, Janusz Mika, Institute of Nuclear Research, Swierk, Otwock, Poland
In neutronic applications of the linear transport theory, an important role is played by two parameters characterizing
a given reactor system. The fundamental time constant α, which is an eigenvalue of the transport operator with the largest
real part, and the constant γ which is the number with the largest absolute value such that if the fission neutron source
in the transport equation is multiplied by 1/γ then the equation has a solution.
It follows from the physical reasons that both α and γ should be real and the corresponding eigenfunctions non-negative.
This fact has also been confirmed theoretically in many important cases.
In this paper there will be considered an auxiliary eigenvalue problem for the neutron transport equation leading to a
convenient method of calculating the constants α and γ and the corresponding eigenfunctions.
Milne's problem for two-dimensional transport in a quarter space, P.S.K. Lam, A. Leonard, Nuclear Engineering Division, Stanford University
Double Fourier transformation of the one-speed transport equation is used to determine a family of solutions to the
homogeneous (Milne's) problem for the quarter space.
All contributions to the density asymptotically away from the corner are determined explicitly by considering two appropriate
half-space problems and by extensive use of contour integration in two complex variables.
An iterative scheme previously derived may be used to determine the correction near the corner.
Numerical methods applied to problems of neutron transport, J.R. Askew, Atomic Energy Establishment, Winfrith
Numerical methods applied to problems of neutron transport are reviewed, with emphasis upon those less widely
used in other fields. The treatment of the energy variable is discussed followed by collision probability, discrete
and Monte Carlo methods. An attempt is made to identify possible improvements to existing approximations.
Properties of new numerical approximations to the transport equation, K.D. Lathrop, B.G. Carlson, Los Alamos Scientific Laboratory
Two new numerical methods are formulated for the solution of the Boltzmann transport equation.
Both methods are designed to reduce distortion of angular flux distributions produced by conventional SN or discrete
ordinates formulations.
Numerical solution of the unsteady radiative transfer equation, B.C. Hankin, J.G. Hill, A.G.P. Warham, Mathematical Physics Department, A.W.R.E., Aldermaston
In problems involving the transport of radiation through matter, it is necessary to solve the radiative transfer
equation numerically. The scope for analytic solutions is restricted because of the complexity of the radiation-material
interaction problem; this is particularly severe if the material is compressible.
A study is made of various difference schemes with a view to obtaining a well-behaved solution.
A method for combining the lower and higher order discrete ordinate approximation to the transport equation, Hiroshi Takahashi, Department of Applied Science, Brookhaven National Laboratory
A method for combining lower order, Sn, and higher order, SN, discrete ordinate approximations that apply to the regions
in which the fluxes are weakly anisotropic and strongly anisotropic is studied.
Methods of solution of the linearized Boltzmann equation for rarefied gas dynamics, C. Cercignani, A.R.S., S.p.A., and Politecnico di Milano
A survey is made of various methods of solution of the linearized Boltzmann equation for rarefied gas dynamics.
Included are spectral problems, analytical methods, variational and numerical techniques as well as existence and uniqueness theory.
The propagation of sound waves and neutron waves in bounded plane geometry, J.S. Cassell, M.M.R. Williams, Nuclear Engineering Department, Queen Mary College, University of London
Integral equations are formulated for neutron wave propagation parallel to the faces of a slab and sound wave propagation
in a gas confined between parallel plates. The main physical differences arise from the different conditions satisfied by the
particle distribution functions at the boundaries of the systems,
i.e. neutrons can pass through a boundary unhindered whilst gas atoms are reflected from it.
By specializing to simple diffuse reflection for gas atoms and using very crude separable kernel scattering models,
it has been possible to obtain by the methods of generalized analytic functions approximate expressions for the discrete eigenvalues,
i.e. the propagating modes. These eigenvalues are shown to depend upon the frequency of the wave and the transverse dimensions
of the system. However, because of the boundary conditions, the results for neutrons and gas atoms show some interesting differences.
Rarefied gas models with velocity dependent collision frequencies, B. Nicolaenko, J.K. Thurber, J. Dorning, New York University, Purdue University, Brookhaven National Laboratory
Linearized model Boltzmann equations with realistic velocity dependent collision frequencies are investigated.
Results obtained on the resolvent of the full hard-sphere and Maxwell Boltzmann equations establish qualitatives similarity
in behavior between the classical Boltzmann equations of rarefied gases and their degenerate kernel approximations.
This similarity holds also for the analytic continuation into the continuous spectrum of the Boltzmann operators.
The free molecular velocity distributions of an optically excited slab, J.W. Cipolla, T.F. Morse, Max-Planck-Institut fur Stromungsforschung, Göttingen, Brown University
For steady state operation of some gas lasers and gas laser amplifiers, optimum conditions are frequently obtained at
pressures so low that the mean free path of the gas is comparable to some macroscopic dimension of the device.
The velocity distributions associated with such a gas may then experience a severe distortion
from their equilibrium values due to the presence of the walls and due to the sensitivity of the process of
absorption and emission on the velocity of the particles undergoing the radiative event.
Furthermore appreciable changes in the operating pressure can cause a gas laser to cease functioning.
Summary of symposium, Ivan Kuscer, Department of Physics, University of Ljubljana, Yugoslavia
Transport theory is used for the mathematical discipline dealing with the Boltzmann equation and with Boltzmann-like equations.
Transport theory can claim the respectable age of almost exactly one hundred years
so the Symposium certainly came at an appropriate time.
Of course not all applications of transport theory are that old. Radiative transfer started about two decades after kinetic theory,
and it grew to a mature science in the thirties and forties, much through the work of Chandrasekhar.
It was a great privilege to hear from Professor Chandrasekhar himself an account of that development,
and to get a feeling of the enthusiasm he had with his work.
It was especially illuminating to hear about the origin of some of the fundamental ideas and of the basic methods - the same ones
which still constitute the foundation of present day development.
Neutron transport theory as a specialized branch of kinetic theory is relatively young, but was forced to develop at a faster pace.
All three subjects were well represented at the Symposium, although radiative transfer dominated.
In spite of the common basis of all these subjects, and of the equivalence of some simple models, the more realistic and more
difficult applications differ widely.
One strong impression is the tremendous impact of the computer, as reflected in the general shift away from oversimplified models.
Today realistic problems, no matter how dirty, can be attacked by brute force. We see the machine producing numbers or diagrams,
which with sufficient good luck, and cleverness, can even be matched to physical reality.
Some references that might be of interest regarding the authors are:
