Download 2n-Stream Radiative Transfer here 

Kees de Lange October 2, 2022 

The understanding of Earth’s climate depends to a large extent on our knowledge of radiative transfer processes in the atmosphere. Short wavelength radiation in the visible range from the sun enters the atmosphere and finds its way to the surface to warm it. Long wavelength radiation in the infrared range is emitted from the surface to find its way to the universe and cools the planet. The energy balance between these two streams of radiation has a profound influence on the temperature and the conditions to sustain life as we know it on our planet. Atmospheric physics is essential in understanding the relevant processes.  

The Sun emits radiation in the visible wavelength range. The Sun’s spectrum can be approximated by blackbody radiation at a temperature of ~ 5800 K. This radiation warms the surface of the earth to a temperature of ~255 K if an albedo of 0.30 is assumed [1]. In a similar vein the conditions on the Moon, with a darker surface than Earth, lead to a daytime surface temperature at the equator of ~ 390 K. This temperature drops at the end of the lunar night to ~100 K [2]. Since the Moon has no atmosphere, slow surface heat conduction is the only mechanism to even out these temperatures. On Earth the presence of an atmosphere has important consequences. Lateral atmospheric currents even out temperature differences between Earth’s sun-lit and dark sides relatively quickly. Temperature differences between night and day on Earth are therefore much smaller than on the Moon. 

On Earth the balance between short wavelength radiation that warms the surface and long wavelength radiation that cools it is significantly affected by molecular gases in the atmosphere. The atmosphere consists mainly of the diatomic gases nitrogen (78.1 %) and oxygen (20.9 %) that do not possess an electric dipole moment. Hence, the only way in which these gases can interfere with the outgoing long wavelength radiation is via very weak quadrupole-induced absorption.  

The gases water (H2O) and carbon dioxide (CO2) in the atmosphere do interfere with the outgoing long wavelength radiation. Transfer of infrared radiation is inhibited by electric-dipole induced absorption of these gases, leading to the so-called greenhouse effect. As a result, the global mean surface temperature is ~288 K, approximately 30 K warmer than it would be without these gases. Of course, this greenhouse effect depends on the concentration of these greenhouse gases. Water on our planet is the main greenhouse gas which can occur in different states of aggregation (gas, clusters, liquid micro-droplets, micro-particles of ice), all with their typical infrared absorption spectrum, in concentrations that can vary enormously as a function of local temperature. CO2 occurs at present in a concentration of ~ 420 ppm and is fairly evenly distributed around the globe.    

In order to understand the climate of our planet, a thorough understanding of radiative transfer of radiation is required. However, in order to treat radiation transfer from the fundamental point of view of atomic, molecular and optical (AMO) physics one is well advised to attack the problem step by step. In this stepwise approach the role of scattering is crucial. As a first logical approach radiative transfer through an atmosphere without clouds should be considered. When that problem can be solved satisfactorily, the role of clouds can be considered next.  

In a previous ground-breaking article [3] Van Wijngaarden and Happer studied the problem of radiation transfer in the atmosphere in the absence of clouds, and hence in the absence of scattering, but in the presence of the five most abundant greenhouse gases water (H2O), carbon dioxide (CO2), ozone (O3), nitrous oxide (N2O) and methane (CH4). This study took satellite observations over a wide range of infrared frequencies as starting point. In the theoretical description the Schwarzschild Equation was solved and simulations of the experimental results were obtained for three regions on earth, viz. the Mediterranean, the Sahara, and Antarctica. The correspondence between experimental satellite data and the simulations was truly remarkable [3]. An assessment of this paper is also available [4]. A key result of this work lies in the saturation effects that occur when the concentration of greenhouse gases is increased.   

Clouds are the Achilles heel of climate science because complicated scattering processes take place in clouds. Clouds can consist of water molecules that as a function of temperature and pressure occur as a variety of aggregates, ranging from single molecules in the gas phase to different oligomers and molecular ensembles in liquid and solid phases. Clouds can also contain particulate matter of many origins. All these aggregates and particles absorb and scatter infrared radiation at their own typical wavelengths. Scattering of radiation is a complicated phenomenon that depends to a large degree on the wavelength of the incident radiation and on the dimensions of the scattering particles. Well-known elastic scattering processes are Rayleigh scattering, where the wavelength is much larger than the particle size, and Mie scattering were the scatterers have a diameter similar to or larger than the wavelength of the incident light. The physics of radiation transfer under atmospheric conditions should therefore be considered in great detail. 

Since radiative transfer in physics is described by often coupled integro-integral equations, solving these equations under all kinds of physical circumstances is a demanding exercise in mathematical physics at a very high level. The study of such complex equations is not new. An important mathematical-physical paper by G.C. Wick (in German) already dates from 1943 [5], the ground-breaking book “Radiative Transfer” by Chandrasekhar [6] was published in 1960 and is still a key reference. In these references atmospheric scattering is discussed employing sophisticated mathematics, but what is lacking is a mathematical framework that can be applied without great difficulty, not just to a single scattering problem, but to a range of different scattering issues. The paper of Van Wijngaarden and Happer aims to fill this gap that still exists after so many years.  

In their new paper Van Wijngaarden and Happer [7] direct their attention to radiation transfer in the atmosphere, but with clouds to scatter incoming and outgoing radiation. In this context the role of greenhouse gases is only secondary. The main purpose of this paper is to develop a flexible mathematical-physical framework to deal with all kinds of different scattering processes. Let us turn to its detailed contents now. 

The most effective direction for long wavelength radiation to leave the atmosphere is vertical. Hence the projection of any direction that makes an angle Θ with the vertical is proportional to cos Θ. By only introducing a cos Θ-dependence, it is implicitly assumed that the relevant streams of radiation possess axial symmetry. Radiative transfer in semi-transparent media involves absorption, emission and scattering. These processes are described with an equation of transfer for I(μ, τ, ϑ) where μ = cos Θ,  τ the optical depth which is a measure of the altitude above the surface, and ϑ the relative time (Eq. 4 of ref. [7]). This intensity I(μ, τ, ϑ)  can be thought of as a stream of monochromatic photons at optical depth τ, making various angles Θ with the vertical. In a mathematical sense this equation shows similarities with the Schrödinger equation of quantum mechanics. In this paper the techniques to solve the equation of transfer are borrowed from quantum mechanics, and the description is phrased in terms of slightly modified Dirac bra and ket vectors, using a notation where bra vectors are not simply Hermitian conjugates of ket vectors. Many radiative-transfer variables in 2n-space can be conveniently represented with non-Hermitian matrices. Hence, it is not always possible to express left and right eigenvectors as Hermitian-conjugate pairs. Because of the dependence of the intensity I on μ, a series expansion in terms of the complete orthogonal set of Legendre polynomials Pl(μ) [8] is introduced. This approach is reminiscent of the more familiar Fourier analysis where an expansion in terms of orthogonal sines and cosines is employed. In this way the power of matrix algebra can be unleashed to solve the equation of transfer.  

In the equation of transfer an important quantity is the phase function p(μ, μ’), the probability for elastic scattering of incident radiation with direction cosine μ’ to scattered radiation with direction cosine μ. Here a random orientation of the scattering particles is assumed, and inelastic scattering processes are neglected. If the single scattering albedo is less than 1, some of the radiation can be absorbed.  

Employing their new notation, the equation of transfer (Eq. 4) can now be written in vector form (Eq. 52). If we assume a time-independent atmosphere, and neglect scattering completely, it is pleasing to note that equation of transfer (Eq. 52) now simplifies to the more familiar Schwarzchild equation (Eq. 65) which describes the transfer of thermal radiation through a cloud-free atmosphere containing greenhouse gases [3].The more general Eq. 62 which describes a combination of absorption, emission and scattering is much harder to solve. 

In order to solve Eq. 62 for a completely general combination of absorption, emission and scattering, a 2n-stream method is used. In order to calculate integrals numerically, the Gauss-Legendre quadrature method [9] is employed. A 2n-stream, whose intensity is sampled at the 2n nodes of the Legendre polynomials P2n, allows for 2n parameters, the n independent nodes (occurring as pairs with opposite signs) of the Legendre functions and the corresponding weights. This means that polynomials of degree 2n-1 can be represented exactly [9]. These nodes and weights of the Legendre functions are tabulated in detail (ref [8], pages 916-917), and are thus readily available. 

The angular dependence of scattering processes is complicated. A well-known example is elastic Rayleigh scattering, where the incident wavelength is much larger than the particle size. The phase function for Rayleigh scattering is given by Eq. (132) and ref [10]. The angular dependence of Rayleigh scattering is not too different from isotropic. Another example is Mie scattering were the scattering particles have a diameter similar to or larger than the wavelength of the incident light. The angular dependence is generally more forward peaked, but depends on the wavelength of the incident light in relation to the particle size. 

A key result of the paper is Eq. (134) where a 2n-stream is constructed such that the phase function maximizes forward scattering. The proof is found in the Appendix, and is a real tour de force that employs Lagrange multipliers. In general, with the 2n-stream method one can engineer the phase functions that one wishes to use without great difficulty. Since the angular dependence of the phase function is an important issue for the many scattering processes that can occur in the atmosphere, this is an important novel aspect of the present theory.  

After the extensive development of the new formulation of the theory of radiation transfer in Earth’s atmosphere, the authors take a lot of trouble to apply the theory to many situations involving all kinds of clouds. Since in general changes in time are negligibly slow, the corresponding term in Eq. (52) can be replaced by Eq. (62) which is valid for a steady-state atmosphere. This equation represents an inhomogeneous differential equation that contains absorption, emission and scattering in all possible combinations.  

In order to get some feeling for the solutions of this equation, the authors first treat the simplified case of non-emissive clouds. These clouds are too cold to emit radiation at frequencies of interest. Under these conditions the right-hand side of Eq. (62) can be set to zero. This assumption leads to a homogeneous differential equation which is easier to solve. Assuming various types of scattering (Rayleigh, isotropic, maximum forward scattering according to Eq. (134)), many examples are discussed. In particular, it can be calculated what fraction of the incident radiation is transmitted through the cloud, and what percentage is absorbed and reflected. 

Of course the real challenge is in solving the inhomogeneous steady-state Eq. (62), with the right-hand side not equal to zero. This equation describes the general problem of clouds which absorb, emit and scatter incident radiation. A convenient way to solve this inhomogeneous differential equation is with the use of Green’s functions. The physical meaning of the Green’s function G(x0, x) (sometimes called an influence function) is that this formulation describes the effect that a source placed at position x0 has at position x [11]. The total Green’s function of the cloud is given in Eq. (274). 

The theory developed in the present paper only works for finite absorption and the single scattering albedo ῶ < 1. However, these methods fail for the somewhat academic case of conservative scattering, when ῶ = 1 and no energy is exchanged between the radiation and scatterers, In a follow-up paper the authors show that minor modifications to the fundamental 2n-scattering theory for ῶ < 1 make it suitable for ῶ = 1 [12].   

Within the new formalism all the required integrals can be computed numerically with MATLAB [13]. In addition, MATLAB allows matrix manipulations, and plotting of functions and data. The computations performed with this extremely useful mathematical toolbox only require limited coding and can be performed on a laptop. 

In summary, the authors have produced a remarkable, ground-breaking and most valuable study in radiation transfer in Earth’s atmosphere in the presence of clouds where absorption, emission, and scattering all play a role. Their admirable achievement in mathematical physics, based on advanced atomic, molecular and optical physics, is phrased in terms that are reminiscent of the language and notation familiar from modern quantum mechanics. The novel scientific framework created in this work offers numerous new possibilities for studying radiation transfer processes in the presence of scattering caused by a large variety of molecules and particles. 

As always, the proof of the pudding will be in the eating. With these novel theoretical techniques now in place there is a strong need for experimental results against which the methods developed by both authors can be tested. This work poses a strong challenge to experimental atmospheric scientists to produce detailed reliable information that can serve, together with the present theoretical treatment, to improve our much needed understanding of scattering processes in atmospheric clouds.   

About the author: 

Cornelis Andreas “Kees” de Lange 

Dr. C.A. de Lange is a Guest Professor in the Faculty of Science at Vrije Universiteit Amsterdam and was a Member of the Senate in The Netherlands from 7 June 2011 until 1 May 2015 as well as the Chairman of The Netherlands Organization for Pensions from 12 May 2009 until 10 January 2011. He has a PhD in Theoretical Chemistry from the University of Bristol (UK) in June 1969 with a Dissertation on Nuclear Magnetic Resonance in Oriented Molecules. His CV can be found here, publications list here, and website here. He is a member of the CO2 Coalition.  

References: 

[1] Murry L. Salby, Atmospheric Physics, Academic Press (1996). 

[2] J.-P. Williams, D.A. Paige, B.T. Greenhagen, E. Sefton-Nash,  The global surface temperatures of the Moon as measured by the Diviner Lunar Radiometer Experiment, Icarus, Volume 283, 300-325 (2017).   

[3] W.A. van Wijngaarden, W. Happer: https://co2coalition.org/wp-content/uploads/2022/03/Infrared-Forcing-by-Greenhouse-Gases-2019-Revised-3-7-2022.pdf   

[4] C.A. de Lange, Van Wijngaarden and Happer Radiative Transfer Paper for Five Greenhouse Gases Explained,  

  https://co2coalition.org/publications/van-wijngaarden-and-happer-radiative-transfer-paper-for-five-greenhouse-gases-explained/ 

[5] G.C. Wick, Über ebene Diffusionsprobleme, Z. Physik 121, 702–718 (1943). 

[6] S. Chandrasekhar, Radiative Transfer, Dover Publications (January 1, 1960). 

[7] W. A van Wijngaarden, W. Happer, 2n-Stream Radiative Transfer, http://arxiv.org/abs/2205.09713 

[8] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards (1964). 

[9] www.dam.brown.edu/people/alcyew/handouts/GLquad.pdf 

[10] https://en.wikipedia.org/wiki/Rayleigh_scattering 

[11] www.math.arizona.edu/~kglasner/math456/greens.pdf 

[12] W. A van Wijngaarden, W. Happer, 2n-Stream Conservative Scattering, https://arxiv.org/pdf/2207.03978 

[13] https://en.wikipedia.org/wiki/MATLAB