EAGER: Develop Robust Light-Scattering Computational Capability Based on the Method of Separation of Variables in Spheroidal Coordinates for Small-to-Large Spheroids

尘埃气溶胶通过部分吸收和反射大气和地表发出的入射阳光和热能来影响全球气候。尘埃颗粒的光学性质对于减少目前对尘埃气溶胶在气候系统中的作用的认识中的不确定性至关重要，因此对预测未来气候很重要。尘埃颗粒的光学性质也是从空间和地面遥感观测中推断尘埃气溶胶特征的基础。尘埃颗粒几乎都是非球形的。与球形模型相比，球形粒子形状模型在计算非球形粒子的光学性质方面代表着一种量子飞跃，已被广泛证明。目前，小颗粒到大颗粒的光学性质只能针对球体进行计算。迫切需要一种精确而稳健的计算能力来计算球形粒子的光学性质。利用计算数学的进步，电磁散射理论的进步，以及现代计算机技术和计算机编码技术，该项目旨在开发一个新的程序来计算从小到大颗粒尺寸范围内的球形颗粒的光学性质。由于许多细菌、微藻、海洋粒子和星际尘埃粒子的形状接近球形，该项目的成果还将在气候科学(特别是气候系统中的辐射能量收支)、遥感、工业、生物光学、海洋光学、天体物理、行星科学等大气科学以外的领域获得广泛应用。由于该项目侧重于一个尚未解决的重大跨学科问题，而且面临重大挑战，特别是从计算电磁学和数学的角度来看，该项目是探索性的，但具有潜在的变革性，即“高风险-高回报”。除了其科学价值外，该项目还包含一个教育部分，以培训上述跨学科领域的职业早期研究人员。该项目旨在解决球体在球面坐标下的光散射问题。虽然在球面坐标下用分离变量法求解电磁波方程已有探索，但以前发展的模型仅适用于相对于入射波长较小且实际应用不多的粒子。以前的工作遇到的主要挑战是球谐函数的数值不稳定性。这个项目将寻求通过使用先进的算法来实现球谐函数的数值稳定性，例如用Wigner-d函数来表示球函数。计算球面函数的关键是求出相应球面方程的特征值。径向椭球方程和角椭球方程是Sturm-Liouville类型的。本征值的计算采用不变量嵌入法，数值稳定性好，精度高。因此，即使在极端参数的情况下，球函数也应该是准确的。这个项目的首要目标是开发一种数值稳定的能力，以准确计算超出其他现有计算能力的当前适用的颗粒尺寸和长宽比范围的椭球体的光学性质，如离散偶极子近似方法(DDA)、有限差分时间域(FDTD)方法、扩展边界条件方法(EBCM)和不变嵌入T矩阵方法(IITM)。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Dust aerosols affect global climate by partially absorbing and reflecting incoming sunlight and heat energy emitted by the atmosphere and the surface. The optical properties of dust particles are critical to reducing uncertainties in the current knowledge of the role of dust aerosols in the climate system, and thus are important for predicting future climate. The dust particle optical properties are also fundamental for inferring dust aerosol characteristics from space-borne and ground-based remote sensing observations. Dust particles are almost exclusively nonspherical. It has been extensively demonstrated that the spheroidal particle shape model represents a quantum leap forward, compared to the spherical model, for computing the optical properties of nonspherical particles. At present, the optical properties of small-to-large particles can be computed only for spheres. There is a pressing need to have an exact and robust computational capability to compute the optical properties of spheroidal particles. Leveraging advances in computational mathematics, advances in electromagnetic scattering theories, and modern computer technologies and computer coding techniques, this project aims to develop a novel program to compute the optical properties of spheroidal particle in the small-to-large particle size range. Because many bacteria, microweeds, oceanic particles, and interstellar dust particles have approximately spheroidal shapes, the outcome of this project will also find extensive applications in climate science (particularly the radiative energy budget in the climate system), remote sensing, industry, bio-optics, oceanic optics, astrophysics, planetary sciences, and other fields beyond atmospheric sciences. Because this project focuses on a major unsolved interdisciplinary problem and because of significant challenges, particularly from the perspective of computational electromagnetics and mathematics, this project is exploratory but potentially transformative, i.e., “high risk – high payoff”. In addition to its scientific merit, this project contains an educational component to train an early-career researcher in the interdisciplinary area mentioned above. This project aims to solve light scattering by a spheroid in spheroidal coordinates. Although solving the electromagnetic wave equation via the method of separation of variables in spheroid coordinates has been explored, the previously developed models are applicable only to particles that are small with respect to the incident wavelength and have little practical use. The major challenge encountered by the previous effort is numerical instability of spheroidal harmonic functions. This project will seek to achieve numerical stability of spheroidal harmonic functions by using advanced algorithms, such as expressing spheroidal functions in terms of the Wigner-d function. The key to computing spheroidal functions is to find eigenvalues of corresponding spheroidal equations. The radial and angular spheroidal equations are of the Sturm-Liouville type. The eigenvalues will be calculated by the invariant-imbedding method, which is expected to be numerically stable and accurate. Thus, the spheroidal functions are expected to be accurate even with extreme parameters. The overarching goal of this project is to develop a numerically stable capability for accurately computing the optical properties of a spheroid beyond the currently applicable particle size and aspect ratio ranges of other existing computational capabilities, such as the discrete dipole approximation method (DDA), the finite-difference time domain (FDTD) method, the extended boundary condition method (EBCM), and the invariant imbedding T-matrix method (IITM).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.