×

Continuation of double Hopf points in thermal convection of rotating fluid spheres. (English) Zbl 1465.76040

MSC:

76E06 Convection in hydrodynamic stability
76E07 Rotation in hydrodynamic stability
76E20 Stability and instability of geophysical and astrophysical flows

Software:

pde2path; AUTO; LOCA; AUTO-86
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] F. Al-Shamali, M. Heimpel, and J. Aurnou, Varying the spherical shell geometry in rotating thermal convection, Geophys. Astrophys. Fluid Dyn., 98 (2004), pp. 153-169.
[2] J. M. Aurnou, Planetary core dynamics and convective heat transfer scaling, Geophys. Astrophys. Fluid Dyn., 101 (2007), pp. 327-345.
[3] A. P. Bassom, A. M. Soward, and S. V. Starchenko, The onset of strongly localized thermal convection in rotating spherical shells, J.Fluid Mech., 689 (2011), pp. 376-416. · Zbl 1241.76184
[4] F. H. Busse, Thermal instabilities in rapidly rotating systems, J.Fluid Mech., 44 (1970), pp. 441-460. · Zbl 0224.76041
[5] F. H. Busse, Convective flows in rapidly rotating spheres and their dynamo action, Phys. Fluids, 14 (2002), pp. 1301-1313. · Zbl 1185.76070
[6] P. Cardin and P. Olson, Chaotic thermal convection in a rapidly rotating spherical shell: Consequences for flow in the outer core, Phys. Earth Planet. Inter., 82 (1994), pp. 235-259.
[7] C. R. Carrigan and F. H. Busse, An experimental and theoretical investigation of the onset of convection in rotating spherical shells, J.Fluid Mech., 126 (1983), pp. 287-305. · Zbl 0526.76055
[8] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, New York, 1961. · Zbl 0142.44103
[9] U. Christensen, Zonal flow driven by strongly supercritical convection in rotating spherical shells, J.Fluid Mech., 470 (2002), pp. 115-133. · Zbl 1075.76027
[10] U. Christensen and J. Aubert, Scaling properties of convection-driven dynamos in rotating spherical shells and application to planetary magnetic fields, Geophys. J. Int., 166 (2006), pp. 97-114.
[11] K. A. Cliffe, Numerical calculations of the primary-flow exchange process in the Taylor problem, J.Fluid Mech., 197 (1988), pp. 57-79. · Zbl 0656.76087
[12] W. O. Criminale, T. L. Jackson, and R. D. Joslin, Theory and Computation in Hydrodynamic Stability, 2nd ed., Cambridge Monogr. Mech., Cambridge University Press, Cambridge, UK, 2018. · Zbl 1098.76002
[13] H. A. Dijkstra, F. W. Wubs, A. K. Cliffe, E. Doedel, I. F. Dragomirescu, B. Eckhardt, A. Gelfgat, A. Hazel, V. Lucarini, A. Salinger, J. Sánchez, H. Schuttelaars, L. Tuckerman, and U. Thiele, Numerical bifurcation methods and their application to fluid dynamics: Analysis beyond simulation, Commun. Comput. Phys., 15 (2014), pp. 1-45. · Zbl 1373.76026
[14] N. Dinar and H. B. Keller, Computation of Taylor vortex flows using multigrid continuation methods, in Recent Advances in Computational Fluid Dynamics, C. C. Chao, S. A. Orszag, and W. Shyy, eds., Lecture Notes in Engrg. 43, Springer, 1989, pp. 191-262. · Zbl 0692.76036
[15] E. Doedel, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Tech. report, California Institute of Technology, Pasadena CA, 1986.
[16] E. Doedel and L. S. Tuckerman, eds., Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, IMA Vol. Math. Appl. 119, Springer-Verlag, 2000.
[17] E. Dormy, A. M. Soward, C. A. Jones, D. Jault, and P. Cardin, The onset of thermal convection in rotating spherical shells, J.Fluid Mech., 501 (2004), pp. 43-70. · Zbl 1058.76026
[18] P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, UK, 1981.
[19] B. Favier and E. Knobloch, Robust wall states in rapidly rotating Rayleigh-Bénard convection, J.Fluid Mech., 895 (2020), R1. · Zbl 1460.76719
[20] F. Garcia, L. Bonaventura, M. Net, and J. Sánchez, Exponential versus IMEX high-order time integrators for thermal convection in rotating spherical shells, J.Comput. Phys., 264 (2014), pp. 41-54. · Zbl 1349.76607
[21] F. Garcia, M. Net, B. García-Archilla, and J. Sánchez, A comparison of high-order time integrators for the Boussinesq Navier-Stokes equations in rotating spherical shells, J.Comput. Phys., 229 (2010), pp. 7997-8010. · Zbl 1207.80009
[22] F. Garcia, J. Sánchez, E. Dormy, and M. Net, Oscillatory convection in rotating spherical shells: Low Prandtl number and non-slip boundary conditions, SIAM J.Appl. Dyn. Syst., 14 (2015), pp. 1787-1807, https://doi.org/10.1137/15M100729X. · Zbl 1330.35457
[23] F. Garcia, J. Sánchez, and M. Net, Antisymmetric polar modes of thermal convection in rotating spherical fluid shells at high Taylor numbers, Phys. Rev. Lett., 101 (2008), 194501.
[24] F. Garcia, J. Sánchez, and M. Net, Numerical simulations of high-Rayleigh-number convection in rotating spherical shells under laboratory conditions, Phys. Earth Planet. Inter., 230 (2014), pp. 28-44.
[25] A. Gelfgat, Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, Comput. Methods Appl. Sci. 50, Springer, Cham, 2018. · Zbl 1398.76005
[26] N. Gillet, D. Brito, D. Jault, and H.-C. Nataf, Experimental and numerical studies of convection in a rapidly rotating spherical shell, J.Fluid Mech., 580 (2007), pp. 83-121. · Zbl 1113.76002
[27] M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, Springer, New York, 1988. · Zbl 0691.58003
[28] W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000, https://doi.org/10.1137/1.9780898719543. · Zbl 0935.37054
[29] A. Griewank and G. Reddien, The calculation of Hopf points by a direct method, IMA J. Numer. Anal., 3 (1983), pp. 295-303. · Zbl 0521.65070
[30] B. W. Hindman, N. A. Featherstone, and K. Julien, Morphological classification of the convective regimes in rotating stars, Astrophys. J., 898 (2020), 120.
[31] C. A. Jones, A. M. Soward, and A. I. Mussa, The onset of thermal convection in a rapidly rotating sphere, J.Fluid Mech., 405 (2000), pp. 157-179. · Zbl 0990.76021
[32] D. D. Joseph, Stability of Fluid Motions, Vol. I, Springer, 1976. · Zbl 0345.76022
[33] D. D. Joseph, Stability of Fluid Motions, Vol. II, Springer, 1976. · Zbl 0345.76022
[34] G. Kawahara, M. Uhlmann, and L. van Veen, The significance of simple invariant solutions in turbulent flows, Ann. Rev. Fluid Mech., 44 (2012), pp. 203-225. · Zbl 1352.76031
[35] H. B. Keller, Numerical solution of bifurcation and nonlinear eigenvalue problems, in Applications of Bifurcation Theory, P. H. Rabinowitz, ed., Academic Press, New York, 1977, pp. 359-384.
[36] I. Kitiashvili and A. G. Kosovichev, Application of data assimilation method for predicting solar cycles, Astrophys. J., 688 (2008), pp. L49-L52.
[37] D. Kong, K. Zhang, K. Lam, and A. P. Willis, Axially symmetric and latitudinally propagating nonlinear patterns in rotating spherical convection, Phys. Rev.E, 98 (2018), 031101.
[38] M. Kubíček, Algorithm 502: Dependence of solution of nonlinear systems on a parameter, ACM Trans. Math. Softw., 2 (1976), pp. 98-107. · Zbl 0317.65019
[39] R. B. Lehoucq and D. C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J.Matrix Anal. Appl., 17 (1996), pp. 789-821, https://doi.org/10.1137/S0895479895281484. · Zbl 0863.65016
[40] R. Meyer-Spasche and H. B. Keller, Computation of the axisymmetric flow between rotating cylinders, J.Comput. Phys., 35 (1980), pp. 100-109. · Zbl 0425.76023
[41] M. Net, F. Garcia, and J. Sánchez, On the onset of low-Prandtl-number convection in rotating spherical shells: Non-slip boundary conditions, J.Fluid Mech., 601 (2008), pp. 317-337. · Zbl 1151.76461
[42] M. Net, F. Garcia, and J. Sánchez, Numerical study of the onset of thermosolutal convection in rotating spherical shells, Phys. Fluids, 24 (2012), 064101.
[43] M. Net and J. Sánchez, Continuation of bifurcations of periodic orbits for large-scale systems, SIAM J.Appl. Dyn. Syst., 14 (2015), pp. 674-698, https://doi.org/10.1137/140981010. · Zbl 1370.37101
[44] P. H. Roberts, On the thermal instability of a rotating fluid sphere containing heat sources, Phil. Trans. R. Soc. Lond.A, 263 (1968), pp. 93-117. · Zbl 0207.26903
[45] Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, UK, 1992. · Zbl 0991.65039
[46] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, New York, 1996. · Zbl 1031.65047
[47] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J.Sci. Stat. Comput., 7 (1986), pp. 856-869, https://doi.org/10.1137/0907058. · Zbl 0599.65018
[48] A. G. Salinger, N. M. Bou-Rabee, R. P. Pawlowsky, E. D. Wilkes, E. A. Burroughs, R. B. Lehoucq, and L. A. Romero, LOCA 1.1. Library of Continuation Algorithms: Theory and Implementation Manual, Sandia National Laboratories, Albuquerque, NM, 2002.
[49] J. Sánchez, F. Garcia, and M. Net, Critical torsional modes of convection in rotating fluid spheres at high Taylor numbers, J.Fluid Mech., 791 (2016), R1. · Zbl 1382.76284
[50] J. Sánchez, F. Garcia, and M. Net, Radial collocation methods for the onset of convection in rotating spheres, J.Comput. Phys., 308 (2016), pp. 273-288. · Zbl 1351.76205
[51] J. Sánchez and M. Net, Numerical continuation methods for large-scale dissipative dynamical systems, Eur. Phys. J. Special Topics, 225 (2016), pp. 2465-2486.
[52] J. Sánchez Umbría and M. Net, Generation of bursting magnetic fields by nonperiodic torsional flows, Phys. Rev. E, 100 (2019), 053110.
[53] J. Sánchez Umbría and M. Net, Torsional solutions of convection in rotating fluid spheres, Phys. Rev. Fluids, 4 (2019), 013501.
[54] P. Schmid and D. Henningson, Stability and Transition in Shear Flows, Appl. Math. Sci. 142, Springer, New York, 2000. · Zbl 0966.76003
[55] R. Simitev and F. H. Busse, Prandtl-number dependence of convection-driven dynamos in rotating spherical fluid shells, J.Fluid Mech., 532 (2005), pp. 365-388. · Zbl 1073.76080
[56] A. Tilgner and F. Busse, Finite amplitude convection in rotating spherical fluid shells, J.Fluid Mech., 332 (1997), pp. 359-376. · Zbl 0887.76070
[57] H. Uecker, D. Wetzel, and J. Rademacher, pde2path-a MATLAB package for continuation and bifurcation in 2D elliptic systems, Numer. Math. Theory Methods Appl., 7 (2014), pp. 58-106. · Zbl 1313.65311
[58] J. I. Yano, Asymptotic theory of thermal convection in a rapidly rotating system, J.Fluid Mech., 243 (1992), pp. 103-131. · Zbl 0759.76076
[59] K. Zhang, Spiralling columnar convection in rapidly rotating spherical fluid shells, J.Fluid Mech., 236 (1992), pp. 535-556. · Zbl 0747.76055
[60] K. Zhang, K. Lam, and D. Kong, Asymptotic theory for torsional convection in rotating fluid spheres, J.Fluid Mech., 813 (2017), R2. · Zbl 1383.76168
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.