An Approximate Solver for Riemann and Riemann‐like Ellipsoidal Configurations

Abstract

We introduce a new technique for constructing three-dimensional models of incompressible Riemann S-type ellipsoids and compressible triaxial configurations that share the same velocity field as that of Riemann S-type ellipsoids. Our incompressible models are exact steady-state configurations; our compressible models represent approximate steady-state equilibrium configurations. Models built from this method can be used to study a variety of relevant astrophysical and geophysical problems.

Generalized Heine’s identity for complex Fourier series of binomials

“…Fourier expansions for algebraic distance functions have a rich history, and this expansion makes its appearance in the theory of arbitrarily-shaped charge distributions in electrostatics ( [33], [41], [5], [35], [34], [40]), magnetostatics ( [39], [6]) quantum direct and exchange Coulomb interactions ( [11], [20], [16], [32], [4]), Newtonian gravity ( [17], [37], [26], [25], [19], [31], [36], [9], [28], [7], [38]), the Laplace coefficients for planetary disturbing function ( [14], [15]), and potential fluid flow around actuator discs ( [8], [24]), just to name a few direct physical applications. A precise Fourier e imφ analysis for these applications is extremely useful to fully describe the general non-axisymmetric nature of these problems.…”

“…Algebraic functions of the form (z − cos j) −m arise naturally in classical physics through the theory of fundamental solutions of Laplace's equation, and they represent powers of distance between two points in a Euclidean space. Their expansions in Fourier series have a rich history, appearing in the theory of arbitrarily shaped charge distributions in electrostatics (Barlow 2003;Popsueva et al 2007;Pustovitov 2008a,b;Verdú et al 2008), magnetostatics (Selvaggi et al 2008b;Beleggia et al 2009), quantum direct and exchange Coulomb interactions (Cohl et al 2001;Enriquez & Blum 2005;Gattobigio et al 2005;Poddar & Deb 2007;Bagheri & Ebrahimi 2008), Newtonian gravity (Fromang 2005;Huré 2005;Huré & Pierens 2005;Chan et al 2006;Ou 2006;Saha & Jog 2006;Boley & Durisen 2008;Mellon & Li 2008;Selvaggi et al 2008a;Even & Tohline 2009;Schachar et al 2009), the Laplace coefficients of the planetary disturbing function (D'Eliseo 1989(D'Eliseo , 2007, and potential fluid flow around actuator discs (Hough & Ordway 1965;Breslin & Andersen 1994), just to name a few physical applications. A precise Fourier analysis is extremely useful to fully describe the general non-axisymmetric nature of these problems.…”

Further Evidence for an Elliptical Instability in Rotating Fluid Bars and Ellipsoidal Stars

“…In a previous paper (Ou 2006, hereafter paper I), we introduced a new self-consistentfield technique that is capable of constructing three-dimensional (3D) models of incompressible Riemann S-type ellipsoids and compressible triaxial configurations that share the same velocity fields as those of Riemann S-type ellipsoids. These compressible triaxial configurations represent fairly good quasi-equilibrium states and can be used to examine the dynamical stability of Riemann S-type ellipsoids in the nonlinear regime.…”