Abstract

Polarimetric observations of planets are providing increasing details of the three-dimensional (3D) atmospheric structure. The one-dimensional plane-parallel approximation model neglects horizontally polarized radiative transfer. Multidimensional polarized radiative transfer models, especially 3D models, are required to contain the horizontal polarization mechanism. Here, we propose a lattice Boltzmann (LB) model for multidimensional polarized radiative transfer, which enables a simple solution of the multidimensional vector radiative transfer equation (VRTE) by performing collision and streaming processes. Through the Chapman–Enskog analysis, we rigorously derive the multi-dimensional VRTE from the proposed LB model. 2D and 3D numerical tests demonstrate that the proposed LB model is effective and accurate for simulating multidimensional polarized radiative transfer. Furthermore, we apply the proposed LB model to investigate the effects of multiple scattering on radiation intensity and degree of polarization in a 3D case and find that multiple scattering enhances the radiation intensity but dampens the degree of polarization throughout almost the whole angular space in multidimensional polarized radiative transfer. This work is expected to provide a simple and effective mesoscopic tool for multidimensional polarized radiative transfer.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Polarization is the vector property of electromagnetic radiation, and the polarization state determines how light interacts with matter [13]. Direct light from solar-type stars can be regarded as unpolarized integrated over the whole disk [4], while the reflected light will usually be polarized through planetary atmospheric scattering. Different atmospheric structures and particle natures induce different polarization properties. This polarization property provides characteristic information about the planetary atmosphere. As a result, numerous polarimetric observations have been carried out to characterize planetary atmospheres, including planets in the solar system and exoplanets [513].

For numerous polarimetric observations, polarized radiative transfer simulations are required to interpret and analyze these measurements and thus constrain parameters of planetary atmospheres. One-dimensional (1D) polarized radiative transfer adopts the locally plane-parallel approximation, which assumes the atmosphere is horizontally homogeneous. Nevertheless, in recent years, many polarimetric measurements have observed inhomogeneous planetary atmospheres, especially horizontally inhomogeneous atmospheres, such as zones and belts [14,15], cyclonic spots [1619], and polar hazes [20,21]. 1D polarized radiative transfer models [22] neglect the horizontal radiation transport and cannot effectively take into account the relevant contributions due to atmospheric horizontal inhomogeneities. Therefore, multidimensional polarized radiative transfer models, especially three-dimensional (3D) models, are required. The International Polarized Radiative Transfer working group of the International Radiation Commission [23] initiated and completed an intercomparison project of multidimensional polarized radiative transfer models, including the vector Monte Carlo models (VMCM) and the spherical harmonics discrete ordinate model (SHDOM), and presented the results of several two-dimensional (2D) and 3D test cases.

Polarized radiative transfer simulations rely on solving the vector radiative transfer equation (VRTE), and the VRTE is a complicated equation consisting of four coupled equations for the Stokes vector. What is more, the numerical solution of the multidimensional VRTE is a challenging task. In this work, we propose a lattice Boltzmann (LB) model for multidimensional polarized radiative transfer, which enables a simple solution of the multidimensional VRTE. The lattice Boltzmann method (LBM) is a mesoscopic method owing to its kinetic origin. This kinetic origin gives the LBM prominent features of simple formulation and high parallel efficiency. Inspired by lattice gas automata, the LBM has developed into a popular approach of computational fluid dynamics [2426] and has been successfully applied to many other fields, such as phonon transport [27], reaction diffusion [28], convection diffusion [29], and wave propagation [30]. Due to the attractive features, more recently, the LBM was further extended to scalar radiative transfer [3139]. However, due to the complexity of the VRTE, the LBM has rarely been applied to polarized radiative transfer. Zhang et al. [40] developed a pioneering LB model for 1D polarized radiative transfer and demonstrated that the LBM is accurate, flexible, and effective to solve polarized radiative transfer in 1D participating media. So far, to the best of our knowledge, no LB model has been successfully established for multidimensional polarized radiative transfer.

The main aim of this work is to fill the gap of lacking a LB model for multidimensional polarized radiative transfer. In Section 2, we propose our LB model for multidimensional polarized radiative transfer and give the rigorous derivation from the vector LB equation to the multidimensional VRTE. In Section 3, two numerical cases are conducted to test the proposed LB model for multidimensional polarized radiative transfer. Further, as an application, the present LBM is extended to investigate the effects of multiple scattering on radiation intensity and degree of polarization in multidimensional polarized radiative transfer. In Section 4, we summarize the main conclusions.

2. PHYSICAL MODEL AND MATHEMATICAL FORMULATIONS

We first introduce the Stokes vector and VRTE to model multidimensional polarized radiative transfer in Section 2.A. Then the LB model for multidimensional polarized radiative transfer is presented in Section 2.B. In Section 2.C, we adopt the Chapman–Enskog (CE) analysis to derive the multidimensional VRTE from the vector LB equation. Section 2.D presents the boundary treatment of the proposed LB model for multidimensional polarized radiative transfer. Finally, Section 2.E provides the detailed solution procedures of our LBM for simulating multidimensional polarized radiative transfer.

A. Vector Radiative Transfer Equation

The monochromatic VRTE describing multidimensional polarized radiative transfer can be written as [41]

$$\frac{1}{c_l}\frac{{\partial \textbf{I}({\textbf{r},\boldsymbol{\Omega},t} )}}{\partial t} + \boldsymbol{\Omega} \cdot \nabla \textbf{I}({\textbf{r},\boldsymbol{\Omega},t} ) + {\boldsymbol{\beta} \textbf{I}}\big({\textbf{r},\boldsymbol{\Omega},t} \big) = \textbf{S}\big({\textbf{r},\boldsymbol{\Omega},t} \big),$$
where ${c_l}$ is the speed of light in the medium; $\textbf{I} = {({I,Q,U,V})^{T}}$ is the Stokes vector with $I$ being the total radiation intensity, $Q$ and $U$ being the linear polarization, and $V$ being the circular polarization; $\textbf{r}$ is the location coordinate, $t$ is the time, $\boldsymbol{\Omega}$ is the radiation direction, $\boldsymbol{\beta} = {\rm diag}({\beta ,\beta ,\beta ,\beta})$ is the extinction coefficient matrix, and $\textbf{S}$ is the source term given by
$$\textbf{S}\big({\textbf{r},\boldsymbol{\Omega},t} \big) = {k_{a}}{\textbf{I}_{b}}\big({\textbf{r},\boldsymbol{\Omega},t} \big) + \frac{{{k_{s}}}}{{4\pi}}\int_{4\pi}\! {\textbf{Z}\big({\textbf{r},{\boldsymbol{\Omega}^\prime},\boldsymbol{\Omega}} \big)\textbf{I}\big({\textbf{r},{\boldsymbol{\Omega}^\prime},t} \big)} {\rm d}{\boldsymbol{\Omega}^\prime},$$
where ${k_{a}}$ and ${k_{s}}$ are the absorption and scattering coefficients, respectively, ${\textbf{I}_{b}} = {({{I_{b}},0,0,0})^{T}}$ is the emission vector of the medium with ${I_{b}}$ being the blackbody intensity, ${\boldsymbol{\Omega}^\prime}$ is the radiation direction before the scattering event, the relations of radiation directions $\boldsymbol{\Omega}$ and ${\boldsymbol{\Omega}^\prime}$ are illustrated in Fig. 1, and $\textbf{Z}$ is the scattering phase matrix and can be obtained via rotating the Mueller matrix $\textbf{P}$ as
$$\textbf{Z}\big({\textbf{r},{\boldsymbol{\Omega}^\prime},\boldsymbol{\Omega}} \big) = \textbf{L}({\pi - \psi} )\textbf{P}\big({\textbf{r},{\boldsymbol{\Omega}^\prime} \to \boldsymbol{\Omega}} \big)\textbf{L}({- \psi ^\prime} ),$$
where $\psi ^\prime $ and $\psi$ denote the rotation angles between the scattering plane and the incident and scattering meridian planes shown in Fig. 1, respectively, the computation of rotation angles can follow [42], and the rotation matrix is defined as follows [43]:
$$\textbf{L}(\psi) = \left({\begin{array}{* {20}{c}}1&\;\;0&\;\;0&\;\;0\\0&\;\;{\cos 2\psi}&\;\;{\sin 2\psi}&\;\;0\\0&\;\;{- \sin 2\psi}&\;\;{\cos 2\psi}&\;\;0\\0&\;\;0&\;\;0&\;\;1\end{array}} \right).$$

The boundary condition for the VRTE is expressed as [42]

$$\begin{split}\textbf{I}\big({{\textbf{r}_w},\boldsymbol{\Omega},t} \big) &= \left[{{f_{Fr}}{\boldsymbol{\varepsilon}_{Fr}}\big({| {{\textbf{n}_w} \cdot \boldsymbol{\Omega}} |} \big) + {f_{La}}{\boldsymbol{\varepsilon}_{La}}} \right]{I_b}\big({{\textbf{r}_w},t} \big)\\&\quad + {\textbf{I}_c}\big({{\textbf{r}_w},t} \big)\delta \big({\boldsymbol{\Omega} - {\boldsymbol{\Omega}_c}} \big)\\&\quad +{f_{Fr}}{\textbf{R}_{Fr}}\big({| {{\textbf{n}_w} \cdot \boldsymbol{\Omega}} |} \big)\textbf{I}\big({{\textbf{r}_w},{\boldsymbol{\Omega}^{\prime \prime}},t} \big)\\&\quad +\frac{{{f_{La}}}}{\pi}\int_{{\textbf{n}_w} \cdot {\boldsymbol{\Omega} ^\prime} \gt 0} {\textbf{R}_{La}}\textbf{I}\big({{\textbf{r}_w},{\boldsymbol{\Omega} ^\prime},t} \big)\left| {{\textbf{n}_w} \cdot {\boldsymbol{\Omega} ^\prime}} \right|{\rm d}{\boldsymbol{\Omega} ^\prime} ,\end{split}$$
where ${f_{Fr}}$ and ${f_{La}} = 1 - {f_{Fr}}$ represent the fractions of Fresnel and Lambertian reflection, respectively; ${\boldsymbol{\varepsilon}_{Fr}}$ and ${\boldsymbol{\varepsilon}_{La}}$ denote the surface emissivity vectors of Fresnel and Lambertian emission, respectively; ${\textbf{R}_{Fr}}$ and ${\textbf{R}_{La}}$ denote the Fresnel and Lambertian reflection matrixes, respectively; ${\textbf{n}_w}$ denotes the unit outer normal vector of the boundary; ${\textbf{I}_c}$ is the Stokes vector of the collimated irradiation beam with the incident direction ${\boldsymbol{\Omega}_c}$; and ${\boldsymbol{\Omega^{\prime \prime}}}$ denotes the incident direction of the present reflected beam with direction $\boldsymbol{\Omega}$. Specific expressions of ${\boldsymbol{\varepsilon}_{Fr}}$, ${\boldsymbol{\varepsilon}_{La}}$, ${\textbf{R}_{Fr}}$, and ${\textbf{R}_{La}}$ can follow [42].
 figure: Fig. 1.

Fig. 1. Schematic of the rotation angles ($\psi ^\prime $ and $\psi$), the scattering plane ($\textit{OPP}^\prime$), the incident direction ${\boldsymbol{\Omega}^\prime}(\theta ^\prime ,\varphi ^\prime)$, the scattering direction $\boldsymbol{\Omega}(\theta ,\varphi)$, and the meridian planes ($OzP^\prime$ and $OzP$).

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The VRTE Eq. (1) is a complete equation involving the transient effect of light. Neglecting the transient effect of light or reaching the steady state, the complete VRTE Eq. (1) will degenerate to the common steady-state form:

$$\boldsymbol{\Omega} \cdot \nabla \textbf{I}\big({\textbf{r},\boldsymbol{\Omega}} \big) + {\boldsymbol{\beta} \textbf{I}}\big({\textbf{r},\boldsymbol{\Omega}} \big) = \textbf{S}\big({\textbf{r},\boldsymbol{\Omega}} \big).$$

The total polarization state is quantitatively described by the degree of polarization $P$, defined as

$$P = \frac{{\sqrt {{Q^2} + {U^2} + {V^2}}}}{I}.$$

B. LB Model for Multidimensional Polarized Radiative Transfer

In this section, a LB model for multidimensional polarized radiative transfer will be proposed. First, the dimensionless time ${t^*} = {c_l}t/L$ and dimensionless location coordinate ${\textbf{r}^*} = \textbf{r}/L$ are introduced into the VRTE Eq. (1), and then the temporally and spatially dimensionless VRTE is obtained:

$$\frac{{\partial \textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)}}{{\partial {t^*}}} + \boldsymbol{\Omega} \cdot {\nabla ^*}\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) = \textbf{F}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),$$
with
$$\textbf{F}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) = L\textbf{S}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) - L{\boldsymbol{\beta} \textbf{I}}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),$$
where ${\nabla ^*}=\frac{\delta}{{\delta {x^{{*}}}}}\textbf{i} + \frac{\delta}{{\delta {y^{{*}}}}}\textbf{j} + \frac{\delta}{{\delta {z^{{*}}}}}\textbf{k}$, ${x^{{*}}}$, ${y^{{*}}}$, and ${z^{{*}}}$ denote the dimensionless coordinate component in Cartesian coordinates; $L$ is the reference length and can take the characteristic length of the medium.

Instead of directly discretizing the VRTE, the LBM is applied to solve the VRTE through the LB equation and obtain the Stokes vector through the distribution function. In our previous works [38,39], LB models for the scalar radiative transfer equation (SRTE) were successfully developed. Here, we extend the idea of constructing the LB model of SRTE to the VRTE, and then we formulate the vector LB equation and the vector equilibrium distribution function for the multidimensional VRTE and Stokes vector, respectively. The vector LB equation for the temporally and spatially dimensionless VRTE Eq. (8) reads

$$\begin{split}&{\textbf{f}_i}\big({{\textbf{r}^*} + {\textbf{c}_i}\Delta {t^*},\boldsymbol{\Omega},{t^*} + \Delta {t^*}} \big) - {\textbf{f}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) = - \frac{1}{\tau}\big[{\textbf{f}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) \\&\quad- \textbf{f}_i^{\rm eq}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) \big] +\Delta {t^*}\left[\vphantom{\frac{{\Delta {t^*}}}{2}}{\textbf{G}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) + {\textbf{F}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) \right.\\&\quad+\left. \frac{{\Delta {t^*}}}{2}{\partial _{{t^*}}}{\textbf{F}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) \right],\end{split}$$
where $\tau$ is the dimensionless relaxation time; $\Delta {t^*}$ is the dimensionless time step; ${\textbf{c}_i}$ is the discrete lattice velocity; the subscript “$ i $” denotes the index of discrete lattice direction; ${\textbf{f}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}}) = {({{f_{{Ii}}},{f_{{Qi}}},{f_{{Ui}}},{f_{{Vi}}}})^{T}}$ is the vector distribution function with ${f_{{Ii}}}$, ${f_{{Qi}}}$, ${f_{{Ui}}}$, and ${f_{{Vi}}}$ corresponding to distribution functions of four Stokes parameters $I$, $Q$, $U$, and $V$, respectively; $\textbf{f}_i^{\rm eq}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}}) = {({f_{{Ii}}^{\rm eq},f_{{Qi}}^{\rm eq},f_{{Ui}}^{\rm eq},f_{{Vi}}^{\rm eq}})^{T}}$ is the vector equilibrium distribution function with $f_{{Ii}}^{\rm eq}$, $f_{{Qi}}^{\rm eq}$, $f_{{Ui}}^{\rm eq}$, and $f_{{Vi}}^{\rm eq}$ corresponding to equilibrium distribution functions of $I$, $Q$, $U$, and $V$, respectively; and ${\textbf{F}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$ and ${\textbf{G}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$ are the vector source term distribution function and vector auxiliary source term distribution function, respectively. It should be noted that the explicit difference scheme ${{[{{\textbf{F}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}}) - {\textbf{F}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*} - \Delta {t^*}})}]} /{\Delta {t^*}}}$ is used to compute ${\partial _{{t^*}}}{\textbf{F}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$.

The vector equilibrium distribution function for the Stokes vector reads

$$\begin{split}\textbf{f}_i^{\rm eq}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) &= {\varpi _i}\left[{1 + \frac{{{\textbf{c}_i} \cdot \boldsymbol{\Omega}}}{{c_s^2}}+\frac{{\big({{\boldsymbol{\Omega \Omega}} - c_s^2\textbf{E}} \big):\big({{\textbf{c}_i}{\textbf{c}_i} - c_s^2\textbf{E}} \big)}}{{2c_s^4}}} \right]\\&\quad\times\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),\end{split}$$
where ${\varpi _i}$ denotes the weight coefficient of corresponding lattice direction, ${c_s}$ is the lattice sound speed, and $\textbf{E}$ denotes the unit tensor.

The vector source term distribution function and vector auxiliary source term distribution function are respectively expressed as

$${\textbf{F}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) = {\varpi _i}\textbf{F}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big), $$
$${{\textbf{G}}_{i}}( {{\textbf{r}}^{*}},\boldsymbol{\Omega },{{t}^{*}})={{\varpi }_{i}}\frac{{{\textbf{c}}_{i}}\cdot ( {1}-1/2\tau)\boldsymbol{\Omega \textbf F}( {{\textbf{r}}^{*}},\boldsymbol{\Omega },{{t}^{*}})}{c_{s}^{2}}.$$

In this work, the commonly used D2Q9 and D3Q15 lattice models [2426] are applied to multiple dimensions, and the lattice structures are illustrated in Fig. 2.

 figure: Fig. 2.

Fig. 2. Schematic of D2Q9 and D3Q15 lattice structures.

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The lattice velocities, lattice sound speed, and weight coefficients of the D2Q9 model are expressed as

$${\textbf{c}_i} = {\textbf{e}_i}c = \left\{{\begin{array}{* {20}{l}}{({0,0} ),}&{i = 0}\\{\big({\cos({i - 1} )\frac{\pi}{2}, \sin({i - 1} )\frac{\pi}{2}} \big)c,}&{i = 1 - 4}\\{\sqrt 2 \big({\cos({2i - 1} )\frac{\pi}{4}, \sin\big({2i - 1} \big)\frac{\pi}{4}} \big)c,}&{i = 5 - 8}\end{array}} \right.,$$
$${c_s} = \frac{c}{{\sqrt 3}},\quad {\varpi _i} = \left\{\begin{array}{* {20}{l}}4/9,&i = 0\\1/9,&i = 1 - 4\\1/36,&i = 5 - 8\end{array} \right..$$

The lattice velocities, lattice sound speed, and weight coefficients of the D3Q15 model are expressed as

$${\textbf{c}_i} = {\textbf{e}_i}c = \left\{\!\begin{array}{* {20}{l}}({0,0,0} ),&i = 0\\({\pm 1,0,0} )c,({0, \pm 1,0} )c,({0,0, \pm 1} )c,&i = 1 - 6\\({\pm 1, \pm 1, \pm 1} )c,&i = 7 - 14\end{array} \right.\!,$$
$${c_s} = \frac{c}{{\sqrt 3}},\quad {\varpi _i} = \left\{\begin{array}{* {20}{l}}2/9,&i = 0\\1/9,&i = 1 - 6\\1/72,&i = 7 - 14\end{array} \right..$$

Here, $c$ is the lattice velocity defined as $c=\Delta {{x}^{*}}/\Delta {{t}^{*}}$, in which $\Delta {x^ *}$ is the dimensionless lattice size. To ensure the numerical stability, the dimensionless time step is set to $\Delta {t^*} = CFL\Delta {x^ *}$, where $CFL$ is the Courant–Friedrichs–Lewy coefficient, which satisfies $CFL \le 1$ [44].

C. From Vector LB Equation to Multidimensional VRTE

In this section, the CE analysis [29,45] is conducted to link the multidimensional VRTE with the vector LB equation. In the above LB model, the Stokes vector can be calculated by

$$\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)=\sum\limits_i {{\textbf{f}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)} .$$

$\textbf{f}_i^{\rm eq}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$, ${\textbf{F}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$, and ${\textbf{G}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$ satisfy the following relations:

$$\sum\limits_i {\textbf{f}_i^{\rm eq}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)} = \textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),$$
$$\sum\limits_i {{\textbf{c}_i}\textbf{f}_i^{\rm eq}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)} = {\boldsymbol{\Omega}\textbf{I}}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),$$
$$\sum\limits_i {{\textbf{c}_i}{\textbf{c}_i}\textbf{f}_i^{\rm eq}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)} = {\boldsymbol{\Omega \Omega }\textbf{I}}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),$$
$$\begin{split}&\sum\limits_i {{\textbf{G}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)} = 0,\quad \sum\limits_i {{\textbf{c}_i}{\textbf{G}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)}\\&\quad = \left({{1} - \frac{1}{{2\tau}}} \right){\boldsymbol{\Omega} \textbf{F}}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),\end{split}$$
$$\sum\limits_i {{\textbf{F}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)} = \textbf{F}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big),\quad \sum\limits_i {{\textbf{c}_i}{\textbf{F}_i}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big)} = 0.$$

Here, for the sake of simplicity, the notations ${\textbf{f}_i}$, $\textbf{f}_i^{\rm eq}$, ${\textbf{F}_i}$, ${\textbf{G}_i}$, and $\textbf{F}$ are used in place of ${\textbf{f}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$, $\textbf{f}_i^{\rm eq}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$, ${\textbf{F}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$, ${\textbf{G}_i}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$, and $\textbf{F}({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}})$, respectively. First, the CE expansion with respect to time and space is applied:

$${\textbf{f}_i} = \textbf{f}_i^{\rm eq} + \varepsilon \textbf{f}_i^{(1)} + {\varepsilon ^2}\textbf{f}_i^{(2)},$$
$${\textbf{F}_i} = \varepsilon \textbf{F}_i^{(1)} + {\varepsilon ^2}\textbf{F}_i^{(2)},$$
$${\textbf{G}_i} = \varepsilon \textbf{G}_i^{(1)} + {\varepsilon ^2}\textbf{G}_i^{(2)},$$
$${\partial _{{t^*}}} = \varepsilon {\partial _{t_1^*}} + {\varepsilon ^2}{\partial _{t_2^*}},\quad {\nabla ^*}=\varepsilon \nabla _1^*,$$
$$\textbf{F} = \varepsilon {\textbf{F}^{(1)}} + {\varepsilon ^2}{\textbf{F}^{(2)}},$$
where $\varepsilon$ is a small expansion parameter and satisfies $\varepsilon \ll 1$.

By using the Taylor expansion on Eq. (9), we obtain

$$\begin{split}{D_i}{\textbf{f}_i} &+ \frac{{\Delta {t^*}}}{2}D_i^2{\textbf{f}_i} + \cdots = - \frac{1}{{\tau \Delta {t^*}}}\big({{\textbf{f}_i} - \textbf{f}_i^{\rm eq}} \big) \\&+ {\textbf{G}_i} + {\textbf{F}_i} + \frac{{\Delta {t^*}}}{2}{\partial _{{t^*}}}{\textbf{F}_i},\end{split}$$
where ${D_i} = {\partial _{{t^*}}} + {\textbf{c}_i} \cdot {\nabla ^*} = \varepsilon {D_{i1}} + {\varepsilon ^2}{\partial _{t_2^*}}$, and ${D_{i1}} = {\partial _{t_1^*}} + {\textbf{c}_i} \cdot \nabla _1^*$.

Substituting Eqs. (17a)–(17d) into Eq. (18) and comparing different orders of $\varepsilon$, we obtain

$${\varepsilon ^1}{{:}}{D_{i1}}\textbf{f}_i^{\rm eq} = - \frac{1}{{\tau \Delta {t^*}}}\textbf{f}_i^{(1)} + \textbf{G}_i^{(1)} + \textbf{F}_i^{(1)},$$
$$\begin{split}{\varepsilon ^2}:{\partial _{t_2^*}}\textbf{f}_i^{\rm eq} &+ {D_{i1}}\textbf{f}_i^{(1)} + \frac{{\Delta {t^*}}}{2}D_{i1}^2\textbf{f}_i^{\rm eq} = - \frac{1}{{\tau \Delta {t^*}}}\textbf{f}_i^{(2)} + \textbf{G}_i^{(2)}\\&+ \textbf{F}_i^{(2)} + \frac{{\Delta {t^*}}}{2}{\partial _{t_1^*}}\textbf{F}_i^{(1)}.\end{split}$$

Applying Eq. (19a) to Eq. (19b), Eq. (19b) can be rewritten as

$$\begin{split}&{\partial _{t_2^*}}\textbf{f}_i^{\rm eq} + \left({1 - \frac{1}{{2\tau}}} \right){D_{i1}}\textbf{f}_i^{(1)} + \frac{{\Delta {t^*}}}{2}{D_{i1}}\left[{\textbf{G}_i^{(1)} + \textbf{F}_i^{(1)}} \right]\\&\quad= - \frac{1}{{\tau \Delta {t^*}}}\textbf{f}_i^{(2)} + \textbf{G}_i^{(2)} + \textbf{F}_i^{(2)} + \frac{{\Delta {t^*}}}{2}{\partial _{t_1^*}}\textbf{F}_i^{(1)}.\end{split}$$

With the help of Eqs. (16), summing Eqs. (19a) and (20) over $ i $, we obtain

$${\partial _{t_1^*}}\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) + \boldsymbol{\Omega} \cdot \nabla _1^*\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) = {\textbf{F}^{(1)}},$$
$$\begin{split}{\partial _{t_2^*}}\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) &+ \left({1 - \frac{1}{{2\tau}}} \right)\sum\limits_i {\big({{\textbf{c}_i} \cdot \nabla _1^*} \big)\textbf{f}_i^{(1)}} \\&+ \frac{{\Delta {t^*}}}{2}\nabla _1^* \cdot \left({1 - \frac{1}{{2\tau}}} \right)\boldsymbol{\Omega}{\textbf{F}^{(1)}} = {\textbf{F}^{(2)}}.\end{split}$$

Substituting the expression of $\textbf{f}_i^{(1)}$ in Eq. (19a) into Eq. (21b), Eq. (21b) can eventually be written as

$${\partial _{t_2^*}}\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) = {\textbf{F}^{(2)}}.$$

Taking $\varepsilon \times$ Eq. (21a) $+{\varepsilon ^2} \times$ Eq. (22), we can obtain

$${\partial _{{t^*}}}\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) + \boldsymbol{\Omega} \cdot {\nabla ^*}\textbf{I}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big) = \textbf{F}\big({{\textbf{r}^*},\boldsymbol{\Omega},{t^*}} \big).$$

Ultimately, the multidimensional VRTE is rigorously derived from the vector LB equation through CE analysis.

D. Boundary Treatment

The boundary treatment of our LB model for multidimensional polarized radiative transfer is presented in this section. Equation (5) is applied to obtain the Stokes vector at the upstream boundary of the radiative transfer direction, and at the downstream boundary the Stokes vector is obtained by [39]

$$\textbf{I}\big({\textbf{r}_b^*,\boldsymbol{\Omega},{t^*}} \big) = 2\textbf{I}\big({\textbf{r}_a^*,\boldsymbol{\Omega},{t^*}} \big) - \textbf{I}\big({\textbf{r}_{na}^*,\boldsymbol{\Omega},{t^*}} \big),$$
where $\textbf{r}_b^*$ represents the boundary lattice node, $\textbf{r}_a^*$ represents the node neighboring the boundary lattice node, and $\textbf{r}_{na}^*$ represents the next neighboring node of $\textbf{r}_a^*$, which are depicted in Fig. 3.
 figure: Fig. 3.

Fig. 3. Schematic of the boundary treatment with $\textbf{r}_b^*$ denoting the boundary lattice node; $\textbf{r}_a^*$ and $\textbf{r}_{na}^*$ denote the neighboring node of $\textbf{r}_b^*$ and the next neighboring node of $\textbf{r}_a^*$, respectively.

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 figure: Fig. 4.

Fig. 4. Schematic of physical models for polarized radiative transfer in multidimensional media: (a) 2D case; (b) 3D case.

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In addition, the vector distribution function at the boundary is unknown after performing collision and streaming processes in Eq. (9). In this work, the nonequilibrium extrapolation scheme [46] is employed to obtain the vector distribution function at the boundary, which is expressed as

$${\textbf{f}_i}\big({\textbf{r}_b^*,\boldsymbol{\Omega},{t^*}} \big) = \textbf{f}_i^{\rm eq}\big({\textbf{r}_b^*,\boldsymbol{\Omega},{t^*}} \big) + \left[{{\textbf{f}_i}\big({\textbf{r}_a^*,\boldsymbol{\Omega},{t^*}} \big) - \textbf{f}_i^{\rm eq}\big({\textbf{r}_a^*,\boldsymbol{\Omega},{t^*}} \big)} \right].$$

E. Solution Procedures

Here, the detailed solution procedures of our LBM for simulating multidimensional polarized radiative transfer are as follows:

  • (1) Discretize the angular and spatial domains with appropriate discrete directions and lattices, respectively. Initialize all physical quantities.
  • (2) Calculate the Stokes vector distributions in each discrete direction by the following LBM program:
    • (a) Perform the vector LB Eq. (9) to get the vector distribution function distributions for next time.
    • (b) Calculate the Stokes vector distributions by Eq. (15).
    • (c) Implement the boundary treatment and update the Stokes vector and vector distribution function distributions at the boundaries.
    • (d) Loop the steps (a)–(c) until achieving the steady state. The following steady-state convergence criterion is adopted:
      $$\min \frac{{\sum\nolimits_j {\left| {{\textbf{S}_k}\big({\textbf{r}_j^*,{t^*}} \big) - {\textbf{S}_k}\big({\textbf{r}_j^*,{t^*} - 10\Delta {t^*}} \big)} \right|}}}{{\sum\nolimits_j {{\textbf{S}_k}\big({\textbf{r}_j^*,{t^*}} \big)}}} \lt \xi ,$$
      where the integration of the absolute value of the Stokes vector $\textbf{S}({\textbf{r}_j^*}) = \int_{4\pi} {| {\textbf{I}({\textbf{r}_j^*,\boldsymbol{\Omega}})} |{\rm d}\boldsymbol{\Omega}}$ is taken; $k$ denotes the index of four Stokes parameters $I$, $Q$, $U$, and $V$; $j$ denotes the index of the lattice nodes; and $\varepsilon$ is the convergence criterion. We note that in this work $\xi$ is set as ${10^{- 6}}$.

In general, polarized radiative transfer does not take the transient effect into account and usually presents steady-state results. We also noted that the steady-state polarized radiative transfer is considered in this work. Since the LBM is a transient evolutionary algorithm, it is necessary to execute the time convergence process to obtain steady-state results.

3. RESULTS AND DISCUSSION

We first present a 2D case and a 3D case to test the proposed LB model for multidimensional polarized radiative transfer in Section 3.A. Next, we will apply the present LBM to a new three-dimensional case in Section 3.B and investigate the effects of multiple scattering on radiation intensity and degree of polarization in multidimensional polarized radiative transfer. In this work, the dimensionless relaxation time $\tau$ is taken as 1, and the CFL coefficient is taken as 0.5.

A. Numerical Validations

In order to test the proposed LB model for multidimensional polarized radiative transfer, two cases are taken into account in this section: a 2D case shown in Fig. 4(a) and a 3D case shown in Fig. 4(b). In both cases, Rayleigh scattering is considered, and the Rayleigh scattering matrix is expressed as [41,47]

$$\textbf{P}(\Theta) = \frac{{3}}{{4}}\left({\begin{array}{* {20}{c}}{{{1 + }}{{\cos}^2}\Theta}&\;\;{- {{\sin}^2}\Theta}&\;\;0&\;\;0\\{- {{\sin}^2}\Theta}&\;\;{{{1 +}}{{\cos}^2}\Theta}&\;\;0&\;\;0\\0&\;\;0&\;\;{2\cos \Theta}&\;\;0\\0&\;\;0&\;\;0&\;\;{2\cos \Theta}\end{array}} \right),$$
where $\Theta$ is the scattering angle shown in Fig. 1.

1. 2D Test

As the first test, we consider polarized radiative transfer in a 2D medium filled with Rayleigh scattering atmosphere. As shown in Fig. 4(a), a collimated beam with polarization state vector ${({1,0.5,0.5,0.5})^{T}}$ is incident on the left boundary of the medium. The incident direction of the beam is zenith angle ${\theta _0} = 90^\circ$, azimuth angle ${\varphi _0} = 90^\circ$. The size of the medium is ${L_y} \times {L_z} = 30 \times 30\;{\rm m}^2$. The single scattering albedo of the medium is 0.99, and the optical thickness is 1. The bottom boundary is a Fresnel surface (specular reflection) with a refractive index 1.33, and the refractive index of the medium above the bottom boundary is 1. The other three boundaries are transparent.

The spatial domain is discretized into $20 \times 20$ lattices after an independence verification. The angular domain is divided into ${N_\theta} \times {N_\varphi} = 20 \times 40$ directions based on the piecewise constant approximation (PCA) scheme [48], in which ${N_\theta}$ and ${N_\varphi}$ represent the number of directions in the zenith and azimuth angular spaces, respectively. Figure 5 shows the dimensionless radiative fluxes of four Stokes parameters along the top and right boundaries obtained by the present LBM. The radiative fluxes of four Stokes parameters along the right boundary are obviously larger than those along the top boundary, due to the existence of collimated radiation at the right boundary. The results obtained by the VMCM [49] are taken as benchmark results also presented in Fig. 5 for comparison. The results obtained by the present LBM are observed to be in good agreement with the benchmark results, which demonstrates that the proposed LB model can effectively and accurately simulate 2D polarized radiative transfer.

 figure: Fig. 5.

Fig. 5. Dimensionless radiative fluxes of four Stokes parameters along different boundaries in the 2D test: (a) top boundary; (b) right boundary.

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 figure: Fig. 6.

Fig. 6. Dimensionless radiative fluxes of four Stokes parameters and the Stokes vector distributions at different locations in the 3D test: (a) the dimensionless radiative fluxes of four Stokes parameters along the top wall centerline $ l_{1} $, (b) the dimensionless radiative fluxes of four Stokes parameters along the right wall centerline $ l_{2} $, (c) the Stokes vector distributions of the top wall center point A for the viewing azimuth angle $\varphi = 90^\circ$, (d) the Stokes vector distributions of the right wall center point B for the viewing azimuth angle $\varphi = 90^\circ$.

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2. 3D Test

The second test we consider is polarized radiative transfer in a 3D cubic medium filled with Rayleigh scattering atmosphere. As shown in Fig. 4(b), a collimated beam with the direction of zenith angle ${\theta _0} = 90^\circ$, azimuth angle ${\varphi _0} = 90^\circ$ is incident on the left wall. The polarization state vector of the beam is also ${({1,0.5,0.5,0.5})^{T}}$. The size of the medium is ${L_x} \times {L_y} \times {L_z} = 1 \times 1 \times 1\;{\rm m}^3$. The single scattering albedo of the medium is 0.99, and the extinction coefficient is $1\;{{\rm m}^{- 1}}$. Similar to the two-dimensional case, the bottom wall of the medium is also a Fresnel surface with a refractive index 1.33, and the other five walls are transparent.

 figure: Fig. 7.

Fig. 7. Contours of radiation intensity and degree of polarization in the hemisphere space at point A for different scattering albedos of 1, 0.7, 0.4, and 0.1 with the bottom surface albedo $\rho = 1$.

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The spatial domain is discretized into $16 \times 16 \times 16$ lattices after an independence verification, and the angular domain is divided into ${N_\theta} \times {N_\varphi} = 40 \times 60$ directions based on the PCA scheme [48]. Figures 6(a) and 6(b) show the dimensionless radiative fluxes of four Stokes parameters along the wall centerline $ l_{1} $ and $ l_{2} $ (see Fig. 4) obtained by the present LBM. It is found that the radiative fluxes along the line $ l_{1} $ and $ l_{2} $ have similar distributions as those along the top and right boundaries in the above 2D case, because 2D polarized radiative transfer can be regarded as a 3D simplification where the size in one direction is assumed to be infinite. The results obtained by the discontinuous finite element method (DFEM) [50] are taken as benchmark results also presented in Figs. 6(a) and 6(b). We can see that the LBM results agree very well with the benchmark results. Furthermore, to validate the accuracy of Stokes vector distributions obtained by the present LBM in the angular domain, we choose the center points A and B of the walls (see Fig. 4) as the viewing points. For the viewing azimuth angle $\varphi = 90^\circ$, the Stokes vector distributions of points A and B along the zenith angle are shown in Figs. 6(c) and 6(d), respectively. For point A, the Stokes parameters exist in the range of zenith angle 0° to 90°, and for point B, the Stokes parameters exist in the range of zenith angle 0° to 180°. The LBM and DFEM results of Stokes vector distributions in Figs. 6(c) and 6(d) exhibit good agreement. The excellent agreement indicates that the proposed LB model is effective and accurate for simulating 3D polarized radiative transfer.

Overall, our proposed LB model has been proven to be able to deal with multidimensional polarized radiative transfer accurately. For the computational efficiency of our proposed LB model compared with other traditional models, we would like note that among these models, the vector Monte Carlo models consume the most computational time due to statistical properties. Based on our previous work [39], the computational efficiency of the LB model is roughly equivalent to those of the vector discrete ordinate models, with little difference in order of magnitude, and they are more efficient than those of vector Monte Carlo models. Compared with other traditional methods for multidimensional polarized radiative transfer, the prominent feature of present LBM lies in its mesoscopic nature based on the vector evolution equation and vector distribution function. The present LBM is a very simple algorithm for simulating multidimensional polarized radiative transfer by performing collision and streaming processes, avoiding not only complex matrix operations but also the extensive ray-tracing work of statistical methods.

B. 3D Application

After having shown that the proposed LB model performs well in multidimensional tests, we next apply the present LBM to investigate a new 3D case. Here, we also consider a 3D cubic medium filled with Rayleigh scattering atmosphere where thermal emission is not taken into account. The schematic of the physical model is also shown in Fig. 4(b). The size of the medium is the same as in Section 3.A.2. The extinction coefficient of the atmosphere is $1\;{{\rm m}^{- 1}}$. The bottom wall of the medium is set as a Lambertian surface (diffuse reflection), and the other five walls remain transparent. An unpolarized collimated beam with the same direction as in Section 3.A.2 is incident on the left wall. The Stokes vector of the unpolarized incident light is ${({\pi ,0,0,0})^{T}}$. Here, the unpolarized collimated beam can be regarded as natural light from solar-type stars integrated over the whole disk. We will focus on the effects of multiple scattering on radiation intensity and degree of polarization in multidimensional polarized radiative transfer, aiming to improve the understanding of the relationship between the polarized property of light and atmospheric scattering.

The discretization of the spatial and angular domains is the same as in Section 3.A.2. First, we treat the bottom wall as a perfectly white Lambertian surface, i.e., with the surface albedo $\rho = 1$, to discuss the effects of different scattering albedos on radiation intensity and degree of polarization. Taking the reflected light at the center point A of the top wall as a representative, we plot in Fig. 7 the contours of radiation intensity and degree of polarization in the hemisphere space ($\theta = [{0^\circ ,90^\circ}]$, $\varphi = [{0^\circ ,360^\circ}]$) at point A for different scattering albedos of 1, 0.7, 0.4, and 0.1, respectively. It is obvious that the distribution patterns of radiation intensity and degree of polarization are basically the same throughout the hemispheric space for different scattering albedos. The reflected light has a larger degree of polarization in the region showing weak radiation intensity, which indicates that in this region the linear horizontal and vertical polarization $Q$ and linear $\pm45^\circ$ $U$ have a larger share with respect to the radiation intensity. It should be noted that the circular polarization $V$ in the Rayleigh scattering atmosphere under unpolarized incidence is 0. We can see from the color scale that as the scattering albedo increases, the radiation intensity throughout the hemispheric space increases. This is obvious because, with the increase of scattering albedo, the scattering effect is enhanced and the absorption effect is diminished. However, as the scattering albedo increases, the degree of polarization of the reflected light decreases in almost the whole hemispheric space, except for the region where the degree of polarization is very small. We should note that the minimum values of degree of polarization in the whole hemispheric space increase slightly with increasing scattering albedo, but they are all close to zero and distributed in very individual directions. Overall, the increasing scattering albedo dampens the degree of polarization of the reflected light in almost the whole hemispheric space. Figure 8(a) shows the radiation intensity and degree of polarization at points A and B for the viewing direction $(\theta ,\varphi) = (42.75^\circ ,30^\circ)$ versus the scattering albedo. It is seen that for the given viewing direction, the radiation intensity of both points A and B is proportional to the scattering albedo, while the corresponding degree of polarization is inversely proportional to the scattering albedo. Figure 8(b) shows the maximum radiation intensity and peak polarization of the reflected light at points A and B in the whole hemispheric space versus the scattering albedo. Clearly, in the whole hemispheric space, the maximum radiation intensity is proportional to the scattering albedo, while the peak polarization is inversely proportional to the scattering albedo.

 figure: Fig. 8.

Fig. 8. Radiation intensity and degree of polarization of the reflected light at points A and B versus the scattering albedo: (a) the radiation intensity and degree of polarization at points A and B for the viewing direction $(\theta ,\varphi) = (42.75^\circ ,30^\circ)$ versus the scattering albedo; (b) the maximum radiation intensity and peak polarization at points A and B in the whole hemispheric space versus the scattering albedo.

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 figure: Fig. 9.

Fig. 9. Contours of radiation intensity and degree of polarization in the hemisphere space at point A for different bottom surface albedos of 1, 0.6, 0.3, and 0 with the scattering albedo $\omega = 1$.

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Further, we take the atmospheric scattering albedo as 1 (pure scattering, no absorption), to discuss the effects of different surface albedos of the bottom wall on radiation intensity and degree of polarization. We plot in Fig. 9 the contours of radiation intensity and degree of polarization in the hemisphere space at point A for different surface albedos of 1 (perfectly white surface), 0.6, 0.3, and 0 (black surface), respectively. We can see from the color scale that with increasing surface albedo, the radiation intensity throughout the hemispheric space increases because increasing the surface albedo enhances the scattering effect within the entire atmosphere. However, as the surface albedo increases, the degree of polarization of the reflected light also decreases in almost the whole hemispheric space, except for the region with very small degree of polarization. Also, the minimum values of degree of polarization in the whole hemispheric space increase slightly with increasing surface albedo; likewise, they are all close to zero and distributed in very individual directions. Overall, the increasing surface albedo also dampens the degree of polarization of the reflected light in almost the whole hemispheric space. Figure 10(a) shows the radiation intensity and degree of polarization at points A and B for the viewing direction $(\theta ,\varphi) = (42.75^\circ ,30^\circ)$ versus the surface albedo. It is observed that for the given viewing direction, the radiation intensity of both points A and B is proportional to the surface albedo, while the corresponding degree of polarization is inversely proportional to the surface albedo. Figure 10(b) shows the maximum radiation intensity and peak polarization of the reflected light at points A and B in the whole hemispheric space versus the surface albedo. Obviously, in the whole hemispheric space, likewise, the maximum radiation intensity is proportional to the surface albedo, while the peak polarization is inversely proportional to the surface albedo.

 figure: Fig. 10.

Fig. 10. Radiation intensity and degree of polarization of the reflected light at points A and B versus the surface albedo: (a) the radiation intensity and degree of polarization at points A and B for the viewing direction $(\theta ,\varphi) = (42.75^\circ ,30^\circ)$ versus the surface albedo; (b) the maximum radiation intensity and peak polarization at points A and B in the whole hemispheric space versus the surface albedo.

Download Full Size | PPT Slide | PDF

Through discussing the effects of scattering albedo and surface albedo on radiation intensity and degree of polarization, we conclude that in multidimensional polarized radiative transfer, multiple scattering enhances the radiation intensity but dampens the degree of polarization throughout almost the whole angular space. Scattering strongly increases the proportion of multiply scattered photons with randomized polarization directions. If the scattering is very weak, then the reflected light essentially consists only of photons that underwent one Rayleigh scattering and thus exhibit a high polarization state.

4. CONCLUSIONS

In this work, a LB model is established for multidimensional polarized radiative transfer. The vector LB equation and the vector equilibrium distribution function are formulated for the multidimensional VRTE and Stokes vector, respectively. The multidimensional VRTE is rigorously derived from the vector LB equation through CE analysis. The Stokes vector can be calculated from the vector LB equation through simple collision and streaming processes.

A 2D case and a 3D case are conducted to test the proposed LB model for multidimensional polarized radiative transfer. The results obtained by the proposed LB model show good agreement with the benchmark results, which demonstrate that the proposed LB model is effective and accurate for simulating multidimensional polarized radiative transfer. Thanks to its mesoscopic nature, the present LBM is a very simple algorithm for simulating multidimensional polarized radiative transfer by performing collision and streaming processes, avoiding not only complex matrix operations but also the extensive ray-tracing work of statistical methods. Furthermore, as an application, we use the present LBM to investigate the effects of scattering albedo and surface albedo on radiation intensity and degree of polarization in a 3D case. We find that in multidimensional polarized radiative transfer, multiple scattering enhances the radiation intensity but dampens the degree of polarization throughout almost the whole angular space. Finally, extending our LB model to simulate multidimensional polarized radiative transfer in atmospheres with complex structures will be the future work.

Funding

National Natural Science Foundation of China (51876004); Academic Excellence Foundation of BUAA for PhD Students.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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References

  • View by:

  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  2. D. S. Kliger, J. M. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy (Academic, 1990).
  3. X. Zhang, S. Yang, W. Yue, Q. Xu, C. Tian, X. Zhang, E. Plum, S. Zhang, J. Han, and W. Zhang, “Direct polarization measurement using a multiplexed Pancharatnam–Berry metahologram,” Optica 6, 1190–1198 (2019).
    [Crossref]
  4. J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
    [Crossref]
  5. J. E. Hansen and J. W. Hovenier, “Interpretation of the polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974).
    [Crossref]
  6. M. G. Tomasko and L. R. Doose, “Polarimetry and photometry of Saturn from pioneer 11: observations and constraints on the distribution and properties of cloud and aerosol particles,” Icarus 58, 1–34 (1984).
    [Crossref]
  7. P. H. Smith and M. G. Tomasko, “Photometry and polarimetry of Jupiter at large phase angles,” Icarus 58, 35–73 (1984).
    [Crossref]
  8. L. A. Sromovsky, P. M. Fry, K. H. Baines, S. S. Limaye, G. S. Orton, and T. E. Dowling, “Coordinated 1996 HST and IRTF imaging of Neptune and Triton I. Observations, navigation, and differential deconvolution,” Icarus 149, 416–434 (2001).
    [Crossref]
  9. S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “First detection of polarized scattered light from an exoplanetary atmosphere,” Astrophys. J. 673, L83–L86 (2008).
    [Crossref]
  10. S. Sengupta, “Cloudy atmosphere of the extrasolar planet HD 189733b: a possible explanation of the detected B-band polarization,” Astrophys. J. 683, L195–L198 (2008).
    [Crossref]
  11. S. J. Wiktorowicz, L. A. Nofi, D. Jontof-Hutter, P. Kopparla, G. P. Laughlin, N. Hermis, Y. L. Yung, and M. R. Swain, “A ground-based albedo upper limit for HD 189733b from polarimetry,” Astrophys. J. 813, 48–58 (2015).
    [Crossref]
  12. K. Bott, J. Bailey, L. Kedziora-Chudczer, D. V. Cotton, P. W. Lucas, J. P. Marshall, and J. H. Hough, “The polarization of HD 189733,” Mon. Not. R. Astron. Soc. Lett. 459, L109–L113 (2016).
    [Crossref]
  13. K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
    [Crossref]
  14. A. P. Ingersoll, P. J. Gierasch, D. Banfield, and A. R. Vasavada, “Moist convection as an energy source for the large-scale motions in Jupiter’s atmosphere,” Nature 403, 630–632 (2000).
    [Crossref]
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  18. H. B. Hammel, L. A. Sromovsky, P. M. Fry, K. Rages, M. Showalter, I. de Pater, M. A. van Dam, R. P. LeBeau, and X. Deng, “The dark spot in the atmosphere of Uranus in 2006: discovery, description, and dynamical simulations,” Icarus 201, 257–271 (2009).
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  46. Z. L. Guo, C. G. Zheng, and B. C. Shi, “Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method,” Chin. Phys. 11, 366–374 (2002).
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2021 (2)

P. Lallemand, L. S. Luo, M. Krafczyk, and W. A. Yong, “The lattice Boltzmann method for nearly incompressible flows,” J. Comput. Phys. 431, 109713 (2021).
[Crossref]

X. Liu, Y. Huang, C. H. Wang, and K. Zhu, “A multiple-relaxation-time lattice Boltzmann model for radiative transfer equation,” J. Comput. Phys. 429, 110007 (2021).
[Crossref]

2020 (2)

X. Liu, Y. Huang, C. H. Wang, and K. Zhu, “Solving steady and transient radiative transfer problems with strong inhomogeneity via a lattice Boltzmann method,” Int. J. Heat Mass Transfer 155, 119714 (2020).
[Crossref]

Z. Chai and B. Shi, “Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: modeling, analysis, and elements,” Phys. Rev. E 102, 023306 (2020).
[Crossref]

2019 (1)

2018 (3)

K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project—three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transf. 209, 19–44 (2018).
[Crossref]

C. H. Wang, L. Qu, Y. Zhang, and H. L. Yi, “Three-dimensional polarized radiative transfer simulation using discontinuous finite element method,” J. Quant. Spectrosc. Radiat. Transf. 208, 108–124 (2018).
[Crossref]

2017 (1)

Y. Wang, L. Yan, and Y. Ma, “Lattice Boltzmann solution of the transient Boltzmann transport equation in radiative and neutron transport,” Phys. Rev. E 95, 063313 (2017).
[Crossref]

2016 (5)

Y. Zhang, H. Yi, and H. Tan, “Lattice Boltzmann method for one-dimensional vector radiative transfer,” Opt. Express 24, 2027–2046 (2016).
[Crossref]

Y. Guo and M. Wang, “Lattice Boltzmann modeling of phonon transport,” J. Comput. Phys. 315, 1–15 (2016).
[Crossref]

A. Mink, G. Thäter, H. Nirschl, and M. J. Krause, “A 3D lattice Boltzmann method for light simulation in participating media,” J. Comput. Sci. 17, 431–437 (2016).
[Crossref]

H. L. Yi, F. J. Yao, and H. P. Tan, “Lattice Boltzmann model for a steady radiative transfer equation,” Phys. Rev. E 94, 023312 (2016).
[Crossref]

K. Bott, J. Bailey, L. Kedziora-Chudczer, D. V. Cotton, P. W. Lucas, J. P. Marshall, and J. H. Hough, “The polarization of HD 189733,” Mon. Not. R. Astron. Soc. Lett. 459, L109–L113 (2016).
[Crossref]

2015 (4)

L. A. Sromovsky, I. de Pater, P. M. Fry, H. B. Hammel, and P. Marcus, “High S/N Keck and Gemini AO imaging of Uranus during 2012-2014: new cloud patterns, increasing activity, and improved wind measurements,” Icarus 258, 192–223 (2015).
[Crossref]

S. J. Wiktorowicz, L. A. Nofi, D. Jontof-Hutter, P. Kopparla, G. P. Laughlin, N. Hermis, Y. L. Yung, and M. R. Swain, “A ground-based albedo upper limit for HD 189733b from polarimetry,” Astrophys. J. 813, 48–58 (2015).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

J. M. Zhao, J. Y. Tan, and L. H. Liu, “Monte Carlo method for polarized radiative transfer in gradient-index media,” J. Quant. Spectrosc. Radiat. Transf. 152, 114–126 (2015).
[Crossref]

2013 (1)

2012 (2)

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E 86, 016706 (2012).
[Crossref]

L. A. Sromovsky, H. B. Hammel, I. de Pater, P. M. Fry, K. A. Rages, M. R. Showalter, W. J. Merline, P. Tamblyn, C. Neyman, J. L. Margot, J. Fang, F. Colas, J. L. Dauvergne, J. M. Gómez-Forrellad, R. Hueso, A. Sánchez-Lavega, and T. Stallard, “Episodic bright and dark spots on Uranus,” Icarus 220, 6–22 (2012).
[Crossref]

2011 (1)

Y. Ma, S. Dong, and H. Tan, “Lattice Boltzmann method for one-dimensional radiation transfer,” Phys. Rev. E 84, 016704 (2011).
[Crossref]

2010 (3)

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer B 57, 126–146 (2010).
[Crossref]

C. K. Aidun and J. R. Clausen, “Lattice Boltzmann method for complex flows,” Annu. Rev. Fluid Mech. 42, 439–472 (2010).
[Crossref]

J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111, 433–446 (2010).
[Crossref]

2009 (2)

H. B. Hammel, L. A. Sromovsky, P. M. Fry, K. Rages, M. Showalter, I. de Pater, M. A. van Dam, R. P. LeBeau, and X. Deng, “The dark spot in the atmosphere of Uranus in 2006: discovery, description, and dynamical simulations,” Icarus 201, 257–271 (2009).
[Crossref]

B. Shi and Z. Guo, “Lattice Boltzmann model for nonlinear convection-diffusion equations,” Phys. Rev. E 79, 016701 (2009).
[Crossref]

2008 (4)

N. Barrado-Izagirre, A. Sánchez-Lavega, S. Pérez-Hoyos, and R. Hueso, “Jupiter’s polar clouds and waves from Cassini and HST images: 1993-2006,” Icarus 194, 173–185 (2008).
[Crossref]

L. N. Fletcher, P. G. J. Irwin, G. S. Orton, N. A. Teanby, R. K. Achterberg, G. L. Bjoraker, P. L. Read, A. A. Simon-Miller, C. Howett, R. De Kok, N. Bowles, S. B. Calcutt, B. Hesman, and F. M. Flasar, “Temperature and composition of Saturn’s polar hot spots and hexagon,” Science 319, 79–81 (2008).
[Crossref]

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “First detection of polarized scattered light from an exoplanetary atmosphere,” Astrophys. J. 673, L83–L86 (2008).
[Crossref]

S. Sengupta, “Cloudy atmosphere of the extrasolar planet HD 189733b: a possible explanation of the detected B-band polarization,” Astrophys. J. 683, L195–L198 (2008).
[Crossref]

2006 (1)

L. H. Liu, L. Zhang, and H. P. Tan, “Finite element method for radiation heat transfer in multi-dimensional graded index medium,” J. Quant. Spectrosc. Radiat. Transf. 97, 436–445 (2006).
[Crossref]

2002 (1)

Z. L. Guo, C. G. Zheng, and B. C. Shi, “Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method,” Chin. Phys. 11, 366–374 (2002).
[Crossref]

2001 (2)

H. H. Tynes, G. W. Kattawar, E. P. Zege, I. L. Katsev, A. S. Prikhach, and L. I. Chaikovskaya, “Monte Carlo and multicomponent approximation methods for vector radiative transfer by use of effective Mueller matrix calculations,” Appl. Opt. 40, 400–412 (2001).
[Crossref]

L. A. Sromovsky, P. M. Fry, K. H. Baines, S. S. Limaye, G. S. Orton, and T. E. Dowling, “Coordinated 1996 HST and IRTF imaging of Neptune and Triton I. Observations, navigation, and differential deconvolution,” Icarus 149, 416–434 (2001).
[Crossref]

2000 (3)

A. P. Ingersoll, P. J. Gierasch, D. Banfield, and A. R. Vasavada, “Moist convection as an energy source for the large-scale motions in Jupiter’s atmosphere,” Nature 403, 630–632 (2000).
[Crossref]

R. Blaak and P. M. A. Sloot, “Lattice dependence of reaction-diffusion in lattice Boltzmann modeling,” Comput. Phys. Commun. 129, 256–266 (2000).
[Crossref]

Y. Guangwu, “A lattice Boltzmann equation for waves,” J. Comput. Phys. 161, 61–69 (2000).
[Crossref]

1998 (1)

S. Chen and G. D. Doolen, “Lattice Boltzmann method for fluid flows,” Annu. Rev. Fluid Mech. 30, 329–364 (1998).
[Crossref]

1995 (1)

H. B. Hammel, G. W. Lockwood, J. R. Mills, and C. D. Barnet, “Hubble space telescope imaging of Neptune’s cloud structure in 1994,” Science 268, 1740–1742 (1995).
[Crossref]

1991 (1)

R. A. West and P. H. Smith, “Evidence for aggregate particles in the atmospheres of Titan and Jupiter,” Icarus 90, 330–333 (1991).
[Crossref]

1987 (1)

J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the Sun measured at a sensitivity of parts in ten million,” Nature 326, 270–273 (1987).
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1984 (2)

M. G. Tomasko and L. R. Doose, “Polarimetry and photometry of Saturn from pioneer 11: observations and constraints on the distribution and properties of cloud and aerosol particles,” Icarus 58, 1–34 (1984).
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P. H. Smith and M. G. Tomasko, “Photometry and polarimetry of Jupiter at large phase angles,” Icarus 58, 35–73 (1984).
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1974 (2)

J. E. Hansen and J. W. Hovenier, “Interpretation of the polarization of Venus,” J. Atmos. Sci. 31, 1137–1160 (1974).
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J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
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Achterberg, R. K.

L. N. Fletcher, P. G. J. Irwin, G. S. Orton, N. A. Teanby, R. K. Achterberg, G. L. Bjoraker, P. L. Read, A. A. Simon-Miller, C. Howett, R. De Kok, N. Bowles, S. B. Calcutt, B. Hesman, and F. M. Flasar, “Temperature and composition of Saturn’s polar hot spots and hexagon,” Science 319, 79–81 (2008).
[Crossref]

Aidun, C. K.

C. K. Aidun and J. R. Clausen, “Lattice Boltzmann method for complex flows,” Annu. Rev. Fluid Mech. 42, 439–472 (2010).
[Crossref]

Asinari, P.

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer B 57, 126–146 (2010).
[Crossref]

Bailey, J.

K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
[Crossref]

K. Bott, J. Bailey, L. Kedziora-Chudczer, D. V. Cotton, P. W. Lucas, J. P. Marshall, and J. H. Hough, “The polarization of HD 189733,” Mon. Not. R. Astron. Soc. Lett. 459, L109–L113 (2016).
[Crossref]

Baines, K. H.

L. A. Sromovsky, P. M. Fry, K. H. Baines, S. S. Limaye, G. S. Orton, and T. E. Dowling, “Coordinated 1996 HST and IRTF imaging of Neptune and Triton I. Observations, navigation, and differential deconvolution,” Icarus 149, 416–434 (2001).
[Crossref]

Banfield, D.

A. P. Ingersoll, P. J. Gierasch, D. Banfield, and A. R. Vasavada, “Moist convection as an energy source for the large-scale motions in Jupiter’s atmosphere,” Nature 403, 630–632 (2000).
[Crossref]

Barlakas, V.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project—three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transf. 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

Barnet, C. D.

H. B. Hammel, G. W. Lockwood, J. R. Mills, and C. D. Barnet, “Hubble space telescope imaging of Neptune’s cloud structure in 1994,” Science 268, 1740–1742 (1995).
[Crossref]

Barrado-Izagirre, N.

N. Barrado-Izagirre, A. Sánchez-Lavega, S. Pérez-Hoyos, and R. Hueso, “Jupiter’s polar clouds and waves from Cassini and HST images: 1993-2006,” Icarus 194, 173–185 (2008).
[Crossref]

Berdyugin, A. V.

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “First detection of polarized scattered light from an exoplanetary atmosphere,” Astrophys. J. 673, L83–L86 (2008).
[Crossref]

Berdyugina, S. V.

S. V. Berdyugina, A. V. Berdyugin, D. M. Fluri, and V. Piirola, “First detection of polarized scattered light from an exoplanetary atmosphere,” Astrophys. J. 673, L83–L86 (2008).
[Crossref]

Bindra, H.

H. Bindra and D. V. Patil, “Radiative or neutron transport modeling using a lattice Boltzmann equation framework,” Phys. Rev. E 86, 016706 (2012).
[Crossref]

Bjoraker, G. L.

L. N. Fletcher, P. G. J. Irwin, G. S. Orton, N. A. Teanby, R. K. Achterberg, G. L. Bjoraker, P. L. Read, A. A. Simon-Miller, C. Howett, R. De Kok, N. Bowles, S. B. Calcutt, B. Hesman, and F. M. Flasar, “Temperature and composition of Saturn’s polar hot spots and hexagon,” Science 319, 79–81 (2008).
[Crossref]

Blaak, R.

R. Blaak and P. M. A. Sloot, “Lattice dependence of reaction-diffusion in lattice Boltzmann modeling,” Comput. Phys. Commun. 129, 256–266 (2000).
[Crossref]

Borchiellini, R.

P. Asinari, S. C. Mishra, and R. Borchiellini, “A lattice Boltzmann formulation for the analysis of radiative heat transfer problems in a participating medium,” Numer. Heat Transfer B 57, 126–146 (2010).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Bott, K.

K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
[Crossref]

K. Bott, J. Bailey, L. Kedziora-Chudczer, D. V. Cotton, P. W. Lucas, J. P. Marshall, and J. H. Hough, “The polarization of HD 189733,” Mon. Not. R. Astron. Soc. Lett. 459, L109–L113 (2016).
[Crossref]

Bowles, N.

L. N. Fletcher, P. G. J. Irwin, G. S. Orton, N. A. Teanby, R. K. Achterberg, G. L. Bjoraker, P. L. Read, A. A. Simon-Miller, C. Howett, R. De Kok, N. Bowles, S. B. Calcutt, B. Hesman, and F. M. Flasar, “Temperature and composition of Saturn’s polar hot spots and hexagon,” Science 319, 79–81 (2008).
[Crossref]

Calcutt, S. B.

L. N. Fletcher, P. G. J. Irwin, G. S. Orton, N. A. Teanby, R. K. Achterberg, G. L. Bjoraker, P. L. Read, A. A. Simon-Miller, C. Howett, R. De Kok, N. Bowles, S. B. Calcutt, B. Hesman, and F. M. Flasar, “Temperature and composition of Saturn’s polar hot spots and hexagon,” Science 319, 79–81 (2008).
[Crossref]

Chai, Z.

Z. Chai and B. Shi, “Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: modeling, analysis, and elements,” Phys. Rev. E 102, 023306 (2020).
[Crossref]

Chaikovskaya, L. I.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford University, 1960).

Chen, S.

S. Chen and G. D. Doolen, “Lattice Boltzmann method for fluid flows,” Annu. Rev. Fluid Mech. 30, 329–364 (1998).
[Crossref]

Clausen, J. R.

C. K. Aidun and J. R. Clausen, “Lattice Boltzmann method for complex flows,” Annu. Rev. Fluid Mech. 42, 439–472 (2010).
[Crossref]

Colas, F.

L. A. Sromovsky, H. B. Hammel, I. de Pater, P. M. Fry, K. A. Rages, M. R. Showalter, W. J. Merline, P. Tamblyn, C. Neyman, J. L. Margot, J. Fang, F. Colas, J. L. Dauvergne, J. M. Gómez-Forrellad, R. Hueso, A. Sánchez-Lavega, and T. Stallard, “Episodic bright and dark spots on Uranus,” Icarus 220, 6–22 (2012).
[Crossref]

Cornet, C.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project—three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transf. 209, 19–44 (2018).
[Crossref]

C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
[Crossref]

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K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
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K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
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C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project—three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transf. 209, 19–44 (2018).
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S. J. Wiktorowicz, L. A. Nofi, D. Jontof-Hutter, P. Kopparla, G. P. Laughlin, N. Hermis, Y. L. Yung, and M. R. Swain, “A ground-based albedo upper limit for HD 189733b from polarimetry,” Astrophys. J. 813, 48–58 (2015).
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H. B. Hammel, L. A. Sromovsky, P. M. Fry, K. Rages, M. Showalter, I. de Pater, M. A. van Dam, R. P. LeBeau, and X. Deng, “The dark spot in the atmosphere of Uranus in 2006: discovery, description, and dynamical simulations,” Icarus 201, 257–271 (2009).
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J. M. Zhao, L. H. Liu, P. F. Hsu, and J. Y. Tan, “Spectral element method for vector radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 111, 433–446 (2010).
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K. Bott, J. Bailey, L. Kedziora-Chudczer, D. V. Cotton, P. W. Lucas, J. P. Marshall, and J. H. Hough, “The polarization of HD 189733,” Mon. Not. R. Astron. Soc. Lett. 459, L109–L113 (2016).
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P. Lallemand, L. S. Luo, M. Krafczyk, and W. A. Yong, “The lattice Boltzmann method for nearly incompressible flows,” J. Comput. Phys. 431, 109713 (2021).
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C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project—three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transf. 209, 19–44 (2018).
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C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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L. A. Sromovsky, I. de Pater, P. M. Fry, H. B. Hammel, and P. Marcus, “High S/N Keck and Gemini AO imaging of Uranus during 2012-2014: new cloud patterns, increasing activity, and improved wind measurements,” Icarus 258, 192–223 (2015).
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L. A. Sromovsky, H. B. Hammel, I. de Pater, P. M. Fry, K. A. Rages, M. R. Showalter, W. J. Merline, P. Tamblyn, C. Neyman, J. L. Margot, J. Fang, F. Colas, J. L. Dauvergne, J. M. Gómez-Forrellad, R. Hueso, A. Sánchez-Lavega, and T. Stallard, “Episodic bright and dark spots on Uranus,” Icarus 220, 6–22 (2012).
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K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
[Crossref]

K. Bott, J. Bailey, L. Kedziora-Chudczer, D. V. Cotton, P. W. Lucas, J. P. Marshall, and J. H. Hough, “The polarization of HD 189733,” Mon. Not. R. Astron. Soc. Lett. 459, L109–L113 (2016).
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Mayer, B.

C. Emde, V. Barlakas, C. Cornet, F. Evans, Z. Wang, L. C. Labonotte, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project—three-dimensional test cases (phase B),” J. Quant. Spectrosc. Radiat. Transf. 209, 19–44 (2018).
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C. Emde, V. Barlakas, C. Cornet, F. Evans, S. Korkin, Y. Ota, L. C. Labonnote, A. Lyapustin, A. Macke, B. Mayer, and M. Wendisch, “IPRT polarized radiative transfer model intercomparison project - phase A,” J. Quant. Spectrosc. Radiat. Transf. 164, 8–36 (2015).
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K. Bott, J. Bailey, D. V. Cotton, L. Kedziora-Chudczer, J. P. Marshall, and V. S. Meadows, “The polarization of the planet-hosting WASP-18 system,” Astrophys. J. 156, 293–308 (2018).
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J. Comput. Sci. (1)

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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)

Fig. 1.
Fig. 1. Schematic of the rotation angles ($\psi ^\prime $ and $\psi$), the scattering plane ($\textit{OPP}^\prime$), the incident direction ${\boldsymbol{\Omega}^\prime}(\theta ^\prime ,\varphi ^\prime)$, the scattering direction $\boldsymbol{\Omega}(\theta ,\varphi)$, and the meridian planes ($OzP^\prime$ and $OzP$).
Fig. 2.
Fig. 2. Schematic of D2Q9 and D3Q15 lattice structures.
Fig. 3.
Fig. 3. Schematic of the boundary treatment with $\textbf{r}_b^*$ denoting the boundary lattice node; $\textbf{r}_a^*$ and $\textbf{r}_{na}^*$ denote the neighboring node of $\textbf{r}_b^*$ and the next neighboring node of $\textbf{r}_a^*$, respectively.
Fig. 4.
Fig. 4. Schematic of physical models for polarized radiative transfer in multidimensional media: (a) 2D case; (b) 3D case.
Fig. 5.
Fig. 5. Dimensionless radiative fluxes of four Stokes parameters along different boundaries in the 2D test: (a) top boundary; (b) right boundary.
Fig. 6.
Fig. 6. Dimensionless radiative fluxes of four Stokes parameters and the Stokes vector distributions at different locations in the 3D test: (a) the dimensionless radiative fluxes of four Stokes parameters along the top wall centerline $ l_{1} $, (b) the dimensionless radiative fluxes of four Stokes parameters along the right wall centerline $ l_{2} $, (c) the Stokes vector distributions of the top wall center point A for the viewing azimuth angle $\varphi = 90^\circ$, (d) the Stokes vector distributions of the right wall center point B for the viewing azimuth angle $\varphi = 90^\circ$.
Fig. 7.
Fig. 7. Contours of radiation intensity and degree of polarization in the hemisphere space at point A for different scattering albedos of 1, 0.7, 0.4, and 0.1 with the bottom surface albedo $\rho = 1$.
Fig. 8.
Fig. 8. Radiation intensity and degree of polarization of the reflected light at points A and B versus the scattering albedo: (a) the radiation intensity and degree of polarization at points A and B for the viewing direction $(\theta ,\varphi) = (42.75^\circ ,30^\circ)$ versus the scattering albedo; (b) the maximum radiation intensity and peak polarization at points A and B in the whole hemispheric space versus the scattering albedo.
Fig. 9.
Fig. 9. Contours of radiation intensity and degree of polarization in the hemisphere space at point A for different bottom surface albedos of 1, 0.6, 0.3, and 0 with the scattering albedo $\omega = 1$.
Fig. 10.
Fig. 10. Radiation intensity and degree of polarization of the reflected light at points A and B versus the surface albedo: (a) the radiation intensity and degree of polarization at points A and B for the viewing direction $(\theta ,\varphi) = (42.75^\circ ,30^\circ)$ versus the surface albedo; (b) the maximum radiation intensity and peak polarization at points A and B in the whole hemispheric space versus the surface albedo.

Equations (40)

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1 c l I ( r , Ω , t ) t + Ω I ( r , Ω , t ) + β I ( r , Ω , t ) = S ( r , Ω , t ) ,
S ( r , Ω , t ) = k a I b ( r , Ω , t ) + k s 4 π 4 π Z ( r , Ω , Ω ) I ( r , Ω , t ) d Ω ,
Z ( r , Ω , Ω ) = L ( π ψ ) P ( r , Ω Ω ) L ( ψ ) ,
L ( ψ ) = ( 1 0 0 0 0 cos 2 ψ sin 2 ψ 0 0 sin 2 ψ cos 2 ψ 0 0 0 0 1 ) .
I ( r w , Ω , t ) = [ f F r ε F r ( | n w Ω | ) + f L a ε L a ] I b ( r w , t ) + I c ( r w , t ) δ ( Ω Ω c ) + f F r R F r ( | n w Ω | ) I ( r w , Ω , t ) + f L a π n w Ω > 0 R L a I ( r w , Ω , t ) | n w Ω | d Ω ,
Ω I ( r , Ω ) + β I ( r , Ω ) = S ( r , Ω ) .
P = Q 2 + U 2 + V 2 I .
I ( r , Ω , t ) t + Ω I ( r , Ω , t ) = F ( r , Ω , t ) ,
F ( r , Ω , t ) = L S ( r , Ω , t ) L β I ( r , Ω , t ) ,
f i ( r + c i Δ t , Ω , t + Δ t ) f i ( r , Ω , t ) = 1 τ [ f i ( r , Ω , t ) f i e q ( r , Ω , t ) ] + Δ t [ Δ t 2 G i ( r , Ω , t ) + F i ( r , Ω , t ) + Δ t 2 t F i ( r , Ω , t ) ] ,
f i e q ( r , Ω , t ) = ϖ i [ 1 + c i Ω c s 2 + ( Ω Ω c s 2 E ) : ( c i c i c s 2 E ) 2 c s 4 ] × I ( r , Ω , t ) ,
F i ( r , Ω , t ) = ϖ i F ( r , Ω , t ) ,
G i ( r , Ω , t ) = ϖ i c i ( 1 1 / 2 τ ) Ω F ( r , Ω , t ) c s 2 .
c i = e i c = { ( 0 , 0 ) , i = 0 ( cos ( i 1 ) π 2 , sin ( i 1 ) π 2 ) c , i = 1 4 2 ( cos ( 2 i 1 ) π 4 , sin ( 2 i 1 ) π 4 ) c , i = 5 8 ,
c s = c 3 , ϖ i = { 4 / 9 , i = 0 1 / 9 , i = 1 4 1 / 36 , i = 5 8 .
c i = e i c = { ( 0 , 0 , 0 ) , i = 0 ( ± 1 , 0 , 0 ) c , ( 0 , ± 1 , 0 ) c , ( 0 , 0 , ± 1 ) c , i = 1 6 ( ± 1 , ± 1 , ± 1 ) c , i = 7 14 ,
c s = c 3 , ϖ i = { 2 / 9 , i = 0 1 / 9 , i = 1 6 1 / 72 , i = 7 14 .
I ( r , Ω , t ) = i f i ( r , Ω , t ) .
i f i e q ( r , Ω , t ) = I ( r , Ω , t ) ,
i c i f i e q ( r , Ω , t ) = Ω I ( r , Ω , t ) ,
i c i c i f i e q ( r , Ω , t ) = Ω Ω I ( r , Ω , t ) ,
i G i ( r , Ω , t ) = 0 , i c i G i ( r , Ω , t ) = ( 1 1 2 τ ) Ω F ( r , Ω , t ) ,
i F i ( r , Ω , t ) = F ( r , Ω , t ) , i c i F i ( r , Ω , t ) = 0.
f i = f i e q + ε f i ( 1 ) + ε 2 f i ( 2 ) ,
F i = ε F i ( 1 ) + ε 2 F i ( 2 ) ,
G i = ε G i ( 1 ) + ε 2 G i ( 2 ) ,
t = ε t 1 + ε 2 t 2 , = ε 1 ,
F = ε F ( 1 ) + ε 2 F ( 2 ) ,
D i f i + Δ t 2 D i 2 f i + = 1 τ Δ t ( f i f i e q ) + G i + F i + Δ t 2 t F i ,
ε 1 : D i 1 f i e q = 1 τ Δ t f i ( 1 ) + G i ( 1 ) + F i ( 1 ) ,
ε 2 : t 2 f i e q + D i 1 f i ( 1 ) + Δ t 2 D i 1 2 f i e q = 1 τ Δ t f i ( 2 ) + G i ( 2 ) + F i ( 2 ) + Δ t 2 t 1 F i ( 1 ) .
t 2 f i e q + ( 1 1 2 τ ) D i 1 f i ( 1 ) + Δ t 2 D i 1 [ G i ( 1 ) + F i ( 1 ) ] = 1 τ Δ t f i ( 2 ) + G i ( 2 ) + F i ( 2 ) + Δ t 2 t 1 F i ( 1 ) .
t 1 I ( r , Ω , t ) + Ω 1 I ( r , Ω , t ) = F ( 1 ) ,
t 2 I ( r , Ω , t ) + ( 1 1 2 τ ) i ( c i 1 ) f i ( 1 ) + Δ t 2 1 ( 1 1 2 τ ) Ω F ( 1 ) = F ( 2 ) .
t 2 I ( r , Ω , t ) = F ( 2 ) .
t I ( r , Ω , t ) + Ω I ( r , Ω , t ) = F ( r , Ω , t ) .
I ( r b , Ω , t ) = 2 I ( r a , Ω , t ) I ( r n a , Ω , t ) ,
f i ( r b , Ω , t ) = f i e q ( r b , Ω , t ) + [ f i ( r a , Ω , t ) f i e q ( r a , Ω , t ) ] .
min j | S k ( r j , t ) S k ( r j , t 10 Δ t ) | j S k ( r j , t ) < ξ ,
P ( Θ ) = 3 4 ( 1 + cos 2 Θ sin 2 Θ 0 0 sin 2 Θ 1 + cos 2 Θ 0 0 0 0 2 cos Θ 0 0 0 0 2 cos Θ ) ,

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