Radial collocation methods for the onset of convection in rotating spheres

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. Several radial collocation meshes have been tested to solve convective flows in full spherical geometry.The inclusion of the origin in the mesh together with a single regularity condition is enough to obtain good results for eigenvalue problems.Several sources of spurious modes have been detected, and methods to eliminate them are proposed.The onset of convection in an internally heated sphere has been studied, and asymptotic laws for high Taylor and moderate Prandtl numbers have been found.

[1]  Marta Net,et al.  Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers , 2016, Journal of Fluid Mechanics.

[2]  C. W. Clenshaw.  The numerical solution of linear differential equations in Chebyshev series , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Licai Deng,et al.  Numerical simulation of core convection by a multi-layer semi-implicit spherical spectral method , 2011, J. Comput. Phys..

[4]  P. Fischer,et al.  High-Order Methods for Incompressible Fluid Flow , 2002 .

[5]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[6]  Sigal Gottlieb,et al.  Spectral Methods , 2019, Numerical Methods for Diffusion Phenomena in Building Physics.

[7]  Marta Net,et al.  Antisymmetric polar modes of thermal convection in rotating spherical fluid shells at high taylor numbers. , 2008, Physical review letters.

[8]  P. Merilees,et al.  An Alternative Scheme for the Summation of a Series of Spherical Harmonics , 1973 .

[9]  Friedrich H. Busse,et al.  Thermal instabilities in rapidly rotating systems , 1970, Journal of Fluid Mechanics.

[10]  Steven A. Orszag,et al.  Fourier Series on Spheres , 1974 .

[11]  B. Fornberg.  Generation of finite difference formulas on arbitrarily spaced grids , 1988 .

[12]  Gene H. Golub,et al.  Matrix computations , 1983 .

[13]  D. Anderson,et al.  Algorithms for minimization without derivatives , 1974 .

[14]  Juan Sánchez,et al.  Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells , 2014, J. Comput. Phys..

[15]  Claudio Canuto,et al.  Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation) , 2007 .

[16]  C. Bernardi,et al.  Approximations spectrales de problèmes aux limites elliptiques , 2003 .

[17]  Bengt Fornberg,et al.  A Pseudospectral Approach for Polar and Spherical Geometries , 1995, SIAM J. Sci. Comput..

[18]  Weizhang Huang,et al.  Pole condition for singular problems: the pseudospectral approximation , 1993 .

[19]  Evangelos A. Coutsias,et al.  Pseudospectral Solution of the Two-Dimensional Navier-Stokes Equations in a Disk , 1999, SIAM J. Sci. Comput..

[20]  Sheehan Olver,et al.  A Fast and Well-Conditioned Spectral Method , 2012, SIAM Rev..

[21]  D. Funaro.  Polynomial Approximation of Differential Equations , 1992 .

[22]  T. Colonius,et al.  Numerical Treatment of Polar Coordinate Singularities , 2000 .

[23]  Heli Chen,et al.  A Direct Spectral Collocation Poisson Solver in Polar and Cylindrical Coordinates , 2000 .

[24]  F. Busse,et al.  An experimental and theoretical investigation of the onset of convection in rotating spherical shells , 1983, Journal of Fluid Mechanics.

[25]  Marta Net,et al.  A comparison of high-order time integrators for thermal convection in rotating spherical shells , 2010, J. Comput. Phys..

[26]  D. Sorensen,et al.  4. The Implicitly Restarted Arnoldi Method , 1998 .

[27]  Thomas Hagstrom,et al.  An efficient spectral method for ordinary differential equations with rational function coefficients , 1996, Math. Comput..

[28]  Marta Net,et al.  Numerical simulations of thermal convection in rotating spherical shells under laboratory conditions , 2014 .

[29]  John P. Boyd,et al.  The Choice of Spectral Functions on a Sphere for Boundary and Eigenvalue Problems: A Comparison of Chebyshev, Fourier and Associated Legendre Expansions , 1978 .

[30]  Jan S. Hesthaven,et al.  Spectral Methods for Time-Dependent Problems: Contents , 2007 .

[31]  Keke Zhang.  Spiralling columnar convection in rapidly rotating spherical fluid shells , 1992, Journal of Fluid Mechanics.

[32]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[33]  A. Tilgner,et al.  Spectral methods for the simulation of incompressible flows in spherical shells , 1999 .

[34]  T. Hagstrom,et al.  Multidomain, sparse, spectral-tau method for helically symmetric flow , 2014 .

[35]  Wilhelm Heinrichs,et al.  Spectral collocation schemes on the unit disc , 2004 .

[36]  P. Roberts.  On the thermal instability of a rotating-fluid sphere containing heat sources , 1968, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[37]  J. Sánchez,et al.  A comparison of high-order time integrators for the Boussinesq Navier-Stokes equations in rotating spherical shells , 2009 .

[38]  A. Soward,et al.  The onset of thermal convection in a rapidly rotating sphere , 2000, Journal of Fluid Mechanics.

[39]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

[40]  A. Soward,et al.  The onset of thermal convection in rotating spherical shells , 2004, Journal of Fluid Mechanics.

[41]  Philip W. Livermore,et al.  Galerkin orthogonal polynomials , 2010, J. Comput. Phys..

[42]  Joel Ferziger,et al.  Higher Order Methods for Incompressible Fluid Flow: by Deville, Fischer and Mund, Cambridge University Press, 499 pp. , 2003 .

[43]  Philip W. Livermore,et al.  Spectral radial basis functions for full sphere computations , 2007, J. Comput. Phys..

[44]  Fu Yu,et al.  Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier Series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions , 2011, J. Comput. Phys..

[45]  Marta Net,et al.  On the onset of low-Prandtl-number convection in rotating spherical shells: non-slip boundary conditions , 2008, Journal of Fluid Mechanics.

[46]  X. Liao,et al.  A new asymptotic method for the analysis of convection in a rapidly rotating sphere , 2004, Journal of Fluid Mechanics.

[47]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[48]  Andrew Jackson,et al.  An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere , 2010, J. Comput. Phys..

[49]  T. J. Rivlin.  An Introduction to the Approximation of Functions , 2003 .

[50]  Wilhelm Heinrichs,et al.  Some results about the pseudospectral approximation of one-dimensional fourth-order problems , 1990 .

[51]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[52]  Juan Sánchez Umbría,et al.  Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a double-Hopf bifurcation , 2013 .

[53]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[54]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[55]  Weiwei Sun,et al.  Convergence Analysis of Spectral Collocation Methods for a Singular Differential Equation , 2003, SIAM J. Numer. Anal..

[56]  K. Meerbergen,et al.  Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems , 1996 .

[57]  Sheehan Olver,et al.  The automatic solution of partial differential equations using a global spectral method , 2014, J. Comput. Phys..

[58]  R. Peyret.  Spectral Methods for Incompressible Viscous Flow , 2002 .

[59]  J. Sánchez,et al.  Continuation and stability of convective modulated rotating waves in spherical shells. , 2016, Physical review. E.