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

沙尘气溶胶通过部分吸收和反射入射的太阳光以及大气和地表释放的热能来影响全球气候。沙尘粒子的光学特性对于减少目前关于沙尘气溶胶在气候系统中作用的知识中的不确定性至关重要，因此对于预测未来气候非常重要。尘埃粒子的光学特性也是从空间和地面遥感观测推断尘埃气溶胶特性的基础。尘埃粒子几乎都是非球形的。它已被广泛证明，球形粒子形状模型代表了一个量子飞跃相比，球形模型，用于计算非球形粒子的光学性质。目前，从小到大的粒子的光学性质只能计算球体。有一个迫切需要有一个准确的和强大的计算能力来计算球形粒子的光学性质。本项目旨在利用计算数学的发展、电磁散射理论的发展、现代计算机技术和计算机编码技术，开发一种新的程序来计算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、海洋颗粒和星际尘埃颗粒具有近似球形的形状，因此该项目的成果也将在气候科学（特别是气候系统中的辐射能量预算）、遥感、工业、生物光学、海洋光学、天体物理学、行星科学和大气科学以外的其他领域中得到广泛应用。由于该项目侧重于一个重大的未解决的跨学科问题，并且由于重大挑战，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，但可能具有变革性，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早期职业研究人员。本计画的目的是在球坐标系下求解光散射问题。虽然已经探索了通过在椭球坐标系中分离变量的方法来求解电磁波方程，但是先前开发的模型仅适用于相对于入射波长较小的粒子，并且几乎没有实际用途。以前的努力所遇到的主要挑战是数值不稳定性的球谐函数。本计画将利用先进的演算法，例如以Wigner-d函数来表示球函数，以达到球调和函数的数值稳定性。计算球函数的关键是求相应球方程的特征值。径向和角向椭球方程是Sturm-Liouville型的。本征值将通过不变量嵌入法计算，预计该方法在数值上稳定且准确。因此，即使具有极端参数，也期望球形函数是准确的。该项目的首要目标是开发一种数值稳定的能力，用于精确计算球体的光学特性，超出当前适用的其他现有计算能力的颗粒尺寸和纵横比范围，例如离散偶极子近似法（DDA），时域有限差分法（FDTD），扩展边界条件法（EBCM），该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.