Algorithms and codes for the Macdonald function: recent progress and comparisons

AbstractThe modified Bessel function Kiν(x), also known as the Macdonald function, finds application in the Kontorovich–Lebedev integral transform when x and ν are real and positive. In this paper, a comparison of three codes for computing this function is made. These codes differ in algorithmic approach, timing, and regions of validity. One of them can be tested independent of the other two through Wronskian checks, and therefore is used as a standard against which the others are compared

Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha equations

We examine the effects of turbulence on elliptic instability of rotating stratified incompressible flows, in the context of the Lagragian-averaged Euler-Boussinesq-alpha, or \laeba, model of turbulence. We find that the \laeba model alters the instability in a variety of ways for fixed Rossby number and Brunt-V\"ais\"al\"a frequency. First, it alters the location of the instability domains in the $(\gamma,\cos\theta)-$parameter plane, where $\theta$ is the angle of incidence the Kelvin wave makes with the axis of rotation and $\gamma$ is the eccentricity of the elliptic flow, as well as the size of the associated Lyapunov exponent. Second, the model shrinks the width of one instability band while simultaneously increasing another. Third, the model introduces bands of unstable eccentric flows when the Kelvin wave is two-dimensional. We introduce two similarity variables--one is a ratio of the Brunt-V\"ais\"al\"a frequency to the model parameter $\Upsilon_0 = 1+\alpha^2\beta^2$, and the other is the ratio of the adjusted inverse Rossby number to the same model parameter. Here, $\alpha$ is the turbulence correlation length, and $\beta$ is the Kelvin wave number. We show that by adjusting the Rossby number and Brunt-V\"ais\"al\"a frequency so that the similarity variables remain constant for a given value of $\Upsilon_0$, turbulence has little effect on elliptic instability for small eccentricities $(\gamma \ll 1)$. For moderate and large eccentricities, however, we see drastic changes of the unstable Arnold tongues due to the \laeba model.Comment: 23 pages (sigle spaced w/figure at the end), 9 figures--coarse quality, accepted by Phys. Fluid

Scattering and delay time for 1D asymmetric potentials: the step-linear and the step-exponential cases

We analyze the quantum-mechanical behavior of a system described by a one-dimensional asymmetric potential constituted by a step plus (i) a linear barrier or (ii) an exponential barrier. We solve the energy eigenvalue equation by means of the integral representation method, classifying the independent solutions as equivalence classes of homotopic paths in the complex plane. We discuss the structure of the bound states as function of the height U_0 of the step and we study the propagation of a sharp-peaked wave packet reflected by the barrier. For both the linear and the exponential barrier we provide an explicit formula for the delay time \tau(E) as a function of the peak energy E. We display the resonant behavior of \tau(E) at energies close to U_0. By analyzing the asymptotic behavior for large energies of the eigenfunctions of the continuous spectrum we also show that, as expected, \tau(E) approaches the classical value for E -> \infty, thus diverging for the step-linear case and vanishing for the step-exponential one.Comment: 14 pages, 10 figure

Craik-Criminale solutions and elliptic instability in nonlinear-reactive closure models for turbulence

The Craik–Criminale class of exact solutions is examined for a nonlinear-reactive fluids theory that includes a family of turbulence closure models. These may be formally regarded as either large eddy simulation or Reynolds-averaged Navier–Stokes models of turbulence. All of the turbulence closure models in the class under investigation preserve the existence of elliptic instability, although they shift its angle of critical stability as a function of the rotation rate Ω of the coordinate system, the wave number β of the Kelvin wave, and the model parameter α, the turbulence correlation length. Elliptic instability allows a comparison among the properties of these models. It is emphasized that the physical mechanism for this instability is not wave–wave interaction, but rather wave, mean-flow interaction as governed by the choice of a model’s nonlinearity

Quantum and Classical divide: the gravitational case

We study the transition between quantum and classical behavior of particles in a gravitational quantum well. We analyze how an increase in the particles mass turns the energy spectrum into a continuous one, from an experimental point of view. We also discuss the way these effects could be tested by conducting experiments with atoms and fullerene-type molecules.Comment: Revtex4, 5 pages, 2 figures; version to appear in Physics Letters

Tidal instability in a rotating and differentially heated ellipsoidal shell

The stability of a rotating flow in a triaxial ellipsoidal shell with an imposed temperature difference between inner and outer boundaries is studied numerically. We demonstrate that (i) a stable temperature field encourages the tidal instability, (ii) the tidal instability can grow on a convective flow, which confirms its relevance to geo- and astrophysical contexts and (iii) its growth rate decreases when the intensity of convection increases. Simple scaling laws characterizing the evolution of the heat flux based on a competition between viscous and thermal boundary layers are derived analytically and verified numerically. Our results confirm that thermal and tidal effects have to be simultaneously taken into account when studying geophysical and astrophysical flows

Computation of Asymptotic Expansions of Turning Point Problems via Cauchy's Integral Formula: Bessel Functions

Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy's integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville´Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.The authors acknowledge support from Ministerio de Economía y Competitividad, project MTM2015-67142-P (MINECO/FEDER, UE). A.G. and J.S. acknowledge support from Ministerio de Economía y Competitividad, project MTM2012-34787. A.G. acknowledges the Fulbright/MEC Program for support during her stay at SDSU. J.S. acknowledges the Salvador de Madariaga Program for support during his stay at SDSU

Recent software developments for special functions in the Santander–Amsterdam project

We give an overview of published algorithms by our group and of current activities and future plans. In particular, we give details on methods for computing special functions and discuss in detail two current lines of research. Firstly, we describe the recent developments for the computation of central and non-central ÷-square cumulative distributions (also called Marcum Q.functions), and we present a new quadrature method for computing them. Secondly, we describe the fourth-order methods for computing zeros of special functions recently developed, and we provide an explicit example for the computation of complex zeros of Bessel functions. We end with an overview of published software by our group for computing special functions