Laguerre Polynomial Basis Research Articles

Gyrokinetic moment-based simulations of the Dimits shift

We present a convergence study of the gyromoment (GM) approach, which is based on projecting the gyrokinetic distribution function onto a Hermite–Laguerre polynomial basis, focused on the cyclone base case (CBC) (Lin et al., Phys. Rev. Lett., vol. 83, no. 18, 1999, pp. 3645–3648) and Dimits shift (Dimits et al., Phys. Plasmas, vol. 7, no. 3, 2000, pp. 969–983) as benchmarks. We report that the GM approach converges more rapidly in capturing the nonlinear dynamics of the CBC than the continuum GENE code (Jenko et al., Phys. Plasmas, vol. 7, no. 5, 2000, pp. 1904–1910) when comparing the number of points representing the velocity space. Increasing the velocity dissipation improves the convergence properties of the GM approach, albeit yielding a slightly larger saturated heat flux. By varying the temperature equilibrium gradient, we show that the GM approach successfully reproduces the Dimits shift (Dimits et al., Phys. Plasmas, vol. 7, no. 3, 2000, pp. 969–983) and effectively captures its width, which is in contrast to the gyrofluid framework. In the collisional regime, the convergence properties of the GM approach improve and a good agreement with previous global particle-in-cell results on transport is obtained (Lin et al., Phys. Rev. Lett., vol. 83, no. 18, 1999, pp. 3645–3648). Finally, we report that the choice of collision model has a minimal impact both on the ion temperature gradient growth rate and on the nonlinear saturated heat flux, at tokamak-relevant collisionality.

An eigen-representation of the Navier–Stokes equations

In this paper we demonstrate that the incompressible Navier–Stokes equations can be formulated as a system of quadratic polynomial equations with a regular, tractable structure. This is first shown to be possible by using the combination matrix of Cheung and Zaki (2014) on the time-harmonic Navier–Stokes with periodic boundaries. We also show that the initial value problem can be rewritten in a similar fashion using the Laguerre polynomial basis and the appropriate version of the combination matrix. The solution to both these formulations, when using a finite number of terms in the series expansion, can be found through the null space of the Macaulay matrix and leads to an eigenvalue problem. We also provide two examples which illustrate the methods discussed in this work. The approach is demonstrated on a nonlinear, univariate model differential equation, and also on the two dimensional Taylor Green vortex problem.

Four-dimensional drift-kinetic model for scrape-off layer plasmas

A four-dimensional plasma model able to describe the scrape-off layer region of tokamak devices at arbitrary collisionality is derived in the drift-reduced limit. The basis of the model is provided by a drift-kinetic equation that retains the full nonlinear Coulomb collision operator and describes arbitrarily far from equilibrium distribution functions. By expanding the dependence of the distribution function over the perpendicular velocity in a Laguerre polynomial basis and integrating over the perpendicular velocity, a set of four-dimensional moment equations for the expansion coefficients of the distribution function is obtained. The Coulomb collision operator as well as Poisson's equation are evaluated explicitly in terms of perpendicular velocity moments of the distribution function.

Geometric Relation Distribution for Place Recognition

In this letter, we illustrate geometric relation distribution (GRD), a novel signature for place recognition and loop closure with landmark maps. GRD encodes geometric pairwise relations between landmark points into a continuous probability density function. The pairwise angles are represented by von Mises distribution whereas two alternative distributions, Erlang or biased Rayleigh, are proposed for distances. The GRD function is represented through its expansion into Fourier series and Laguerre polynomial basis. Such orthogonal basis representation enables efficient computation of the translation and rotation invariant metric used to compare signatures and find potential loop closure candidates. The effectiveness of the proposed method is assessed through experiments with standard datasets.

Optimal input design for continuous-time system identification: application to fractional systems

An optimal input design framework for continuous-time system identification using fractional models is presented. This framework extends some existing results in discrete-time systems. The problem is formulated in convex finite dimensional form. Input spectrum decomposition in Laguerre polynomial basis is applied and an LMI solution is proposed. This method permits to synthesize optimal input spectrum for LTI continuous-time systems including fractional ones.

TMD PDFs in the Laguerre polynomial basis

We suggest the modified matching procedure for TMD PDF to the integrated PDF aimed to increase the amount of perturbative information in the TMD PDF expression. The procedure consists in the selection and usage of the non-minimal operator basis, which restricts the expansion to desired general behavior. The implication of OPE allows to systematic account of the higher order corrections. In the case of TMD PDF we assume the Gaussian behavior, which suggests Laguerre polynomial basis as the best for the convergence of OPE. We present the leading and next-to-leading expression of TMD PDF in this basis. The obtained perturbative expression for the TMD PDF is valid in the wide region of $b_T$ (we estimate this region as $b_T\lesssim 2-3$ GeV$^{-1}$ depending on $x$).

A LAGUERRE POLYNOMIAL ORTHOGONALITY AND THE HYDROGEN ATOM

The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis, an analogous strange orthogonality is obtained for Meixner polynomials.

Ground-state dt micro fusion rate and sticking probability using perimetric coordinates.

The ground-state energy, fusion rate, and sticking probability are calculated using a Laguerre polynomial basis. The results are compared with previous calculations, and the advantages are pointed out.

A nonlocal analysis of drift and drift-cyclotron waves in cylindrical geometry

An analysis of electron drift waves and drift-cyclotron waves has been carried out in cylindrical geometry using a k-space integral equation. The model assumes a low-β electrostatic plasma in a straight magnetic field with a Gaussian density profile. Above the threshold for the drift-cyclotron instability, a complicated spectrum of modes is found, with the fundamental radial eigenmode not necessarily the most unstable. In solving the integral equation, improved computational efficiency is obtained by decomposing the eigenfunctions on an associated Laguerre polynomial basis, a method which works best for low-azimuthal (m<25) and low-radial (n<10) mode numbers.