Spectral radial basis functions for full sphere computations

Livermore, Philip W.; Jones, C. A.; Worland, Steven J.

doi:10.1016/j.jcp.2007.08.026

Cited by 39 publications

(58 citation statements)

References 23 publications

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“…The radial and horizontal rms velocities in both models differ from less than 0.1%. For a better numerical stability, a more stringent condition could be to impose Z, W → r l+1 as r → 0 (Bayliss et al 2007;Livermore et al 2007;Glatzmaier 2013).…”

Section: Resultsmentioning

confidence: 99%

Theoretical seismology in 3D: nonlinear simulations of internal gravity waves in solar-like stars

Alvan

Brun

Mathis

2014

A&A

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Context. Internal gravity waves (IGWs) are studied for their impact on the angular momentum transport in stellar radiation zones and the information they provide about the structure and dynamics of deep stellar interiors. We present the first 3D nonlinear numerical simulations of IGWs excitation and propagation in a solar-like star. Aims. The aim is to study the behavior of waves in a realistic 3D nonlinear time-dependent model of the Sun and to characterize their properties. Methods. We compare our results with theoretical and 1D predictions. It allows us to point out the complementarity between theory and simulation and to highlight the convenience, but also the limits, of the asymptotic and linear theories. Results. We show that a rich spectrum of IGWs is excited by the convection, representing about 0.4% of the total solar luminosity. We study the spatial and temporal properties of this spectrum, the effect of thermal damping, and nonlinear interactions between waves. We give quantitative results for the modes' frequencies, evolution with time and rotational splitting, and we discuss the amplitude of IGWs considering different regimes of parameters. Conclusions. This work points out the importance of high-performance simulation for its complementarity with observation and theory. It opens a large field of investigation concerning IGWs propagating nonlinearly in 3D spherical structures. The extension of this work to other types of stars, with different masses, structures, and rotation rates will lead to a deeper and more accurate comprehension of IGWs in stars.

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Section: Resultsmentioning

confidence: 99%

Theoretical seismology in 3D: nonlinear simulations of internal gravity waves in solar-like stars

Alvan

Brun

Mathis

2014

A&A

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“…Since the existing orthogonal polynomials, such as the Legendre and Chebyshev polynomials, represent the special cases of the Jacobi polynomial family that contains two free parameters (Courant & Hilbert 1953), it would be no surprise that the new geostrophic polynomial G 2k−1 (r sin q) is also connected with the Jacobi polynomial family by making a proper choice of the two free parameters together with a suitable transformation. Moreover, it should be noted that the geostrophic polynomial G 2k−1 (r sin q) is fundamentally different from the Worland polynomial W l n (r) (Livermore et al 2007), which has been used as a set of radial basis functions in full sphere computations.…”

Section: A New Legendre-type Polynomialmentioning

confidence: 99%

“…Equations (4.2) and (4.3) are then solved numerically by expanding the velocity potentials in terms of Legendre polynomials P l (cos q) and of radial functions that satisfy the no-slip boundary condition 6) where T n (x) denotes the standard Chebyshev function, and w ln and v ln are unknown coefficients to be determined. Note that the factor r l in the expansions (4.5) and (4.6) is required to remove the numerical singularity at the origin r = 0 in a fluid sphere (see Livermore et al 2007). On substitution of equations (4.5) and (4.6) into equations (4.2) and (4.3) and application of the standard numerical procedure, we are able to derive a system of linear algebraic equations for w ln and v ln that can be found by an iterative numerical procedure.…”

Section: Numerical Analysismentioning

confidence: 99%

A new Legendre-type polynomial and its application to geostrophic flow in rotating fluid spheres

Liao

Zhang

2010

Proc. R. Soc. A.

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In rapidly rotating spheres, the whole fluid column, extending from the southern to northern spherical boundary along the rotation axis, moves like a single fluid element, which is usually referred to as geostrophic flow. A new Legendre-type polynomial is discovered in undertaking the asymptotic analysis of geostrophic flow in spherical geometry. Three essential properties characterize the new polynomial: (i) it is a function of r and q but takes a single argument (r sin q), which is restricted by 0 ≤ r ≤ 1 and 0 ≤ q ≤ p, where (r, q, f) denote spherical polar coordinates with q = 0 at the rotation axis; (ii) it is odd and vanishes at the axis of rotation q = 0, and (iii) it is defined withinand orthogonal over-the full sphere. As an example of its application, we employ the new polynomial in the asymptotic analysis of forced geostrophic flows in rotating fluid spheres for small Ekman and Rossby numbers. Fully numerical analysis of the same problem is also carried out, showing satisfactory agreement between the asymptotic solution and the numerical solution.

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“…A list of possible radial functions can be found in [24], and in some recent works [25,26]. In [25], how much a expansion in even or odd degree Chebyshev polynomials violates the regularity conditions at the origin is quantified, and two basis of one-sided Jacobi polynomials (Verkley and Worland) are compared.…”

Section: Introductionmentioning

confidence: 99%

“…In [25], how much a expansion in even or odd degree Chebyshev polynomials violates the regularity conditions at the origin is quantified, and two basis of one-sided Jacobi polynomials (Verkley and Worland) are compared. The main conclusion is that unless there is a need of a faithful representation of the solution, preserving the regularity near the origin, Chebyshev methods would be the choice, in general, due to the availability of fast transforms.…”

Section: Introductionmentioning

confidence: 99%

Radial collocation methods for the onset of convection in rotating spheres

Sánchez

García

Net

2016

Journal of Computational Physics

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The viability of using collocation methods in radius and spherical harmonics in the angular variables to calculate convective flows in full spherical geometry is examined. As a test problem the stability of the conductive state of a self-gravitating fluid sphere subject to rotation and internal heating is considered. A study of the behavior of different radial meshes previously used by several authors in polar coordinates, including or not the origin, is first performed. The presence of spurious modes due to the treatment of the singularity at the origin, to the spherical harmonics truncation, and to the initialization of the eigenvalue solver is shown, and ways to eliminate them are presented. Finally, to show the usefulness of the method, the neutral stability curves at very high Taylor and moderate and small Prandtl numbers are calculated and shown.

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Spectral radial basis functions for full sphere computations

Cited by 39 publications

References 23 publications

Theoretical seismology in 3D: nonlinear simulations of internal gravity waves in solar-like stars

Theoretical seismology in 3D: nonlinear simulations of internal gravity waves in solar-like stars

A new Legendre-type polynomial and its application to geostrophic flow in rotating fluid spheres

Radial collocation methods for the onset of convection in rotating spheres

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