Modulated waves in rotating spherical convection

Patterns and Simulations

The understanding of thermal convection in fluid spheres and spherical shells is fundamental for improving the knowledge of many astrophysical and geophysical phenomena. Bifurcation diagrams and flow patterns of rotating (RW) and modulated rotating waves (MRW) up to the appearance of chaos are presented in terms of their physical properties. They are obtained from direct numerical simulations of Boussinesq thermal convection in rotating spherical shells with constrained azimuthal symmetry. The parameters are taken so close as possible to those of the Earth's outer core. The periodicity and quasiperiodicity of the solutions is quantified by means of an accurate analysis of their frequency spectrum, together with the Poincaré sections. In addition an heuristic five degree model, based on the symmetries of the solutions, that reproduces with significative agreement the sequence of bifurcations and the type of solutions of the simulations is derived and explored.
Author Institutional AffiliationUniversitat Politècnica de Catalunya and Ecole Normale Supérieure Paris
Author Email
Author Postal MailCampus Nord, Mòdul B4, Jordi Girona Salgado 1--3, 08034 Barcelona, Spain.
This media entry is based on the paper SIAM Journal on Applied Dynamical Systems 2015 14:4, 1787--1807.
Keywordsthermal convection, rotating spherical geometry, modulated waves, period doubling, reduced model

Documents to download

  • Bifurcations and Poincaré sections of the DNS(.gif, 60.79 KB) - 3121 download(s) The bifurcation diagrams for the convection problem of some time averaged physical properties (mean kinetic energy density, ratio between the total and the non-axisymmetric mean kinetic energy densities, and the maximum and minimum of the kinetic energy) are plotted versus the Rayleigh number on the first row of plots. For each azimuthal wave number m the bifurcations typically gives rise to RW (rotating Rossby wave, cyan), followed by MRW (magenta), which undergo a period doubling cascade leading to chaos. Stable/unstable motions are noted with solid/dashed lines. The period doubling bifurcations can be identified in the loops of the Poincaré sections shown in the second row of plots. Notice in the last section the band corresponding to the appearance of chaos.
  • Reduced model bifurcations and Poincaré sections(.gif, 62.15 KB) - 3097 download(s) The bifurcation diagrams of some time averaged physical properties (kinetic energy, ratio between the total and the non-axisymmetric kinetic energy, and the maximum and minimum of the kinetic energy) are plotted versus the control parameter on the first row of plots. In the second row the Poincaré sections are shown. Notice that either the bifurcation diagrams and the Poincaré sections of the reduced model resemble very much to those obtained simulating the full three-dimensional convection problem.
  • Patterns: Temperature deviation and kinetic energy(.gif, 15.56 MB) - 3077 download(s) Contour plots of the temperature perturbation from the basic conductive state (left group of three plots) and of the kinetic energy density (right group of plots). In each group of plots the first section corresponds to a projection on a sphere, the second on the equatorial plane, and the last on a meridional section. The simulation covers a period of modulation (the last snapshot corresponds to an azimuthal rotation of the first). The pattern is azimuthally rotating while modulating its amplitude. Notice that an abrupt increase of the zonal circulations occurs due the connection between the convective cells.
  • Patterns: Axial vorticity and azimuthal velocity(.gif, 16.49 MB) - 3127 download(s) Same as Figure 3 but for the axial vorticity at left, and the azimuthal velocity at right. The meridional sections of Figure 3 and 4 show clearly that the z-dependence of the flow is weak due to the Taylor-Proudman theorem, and that it is symmetric with respect to the equatorial plane. In addition, convection inside the tangent cylinder is nearly absent, but extends in almost the remaining part of the shell. The equatorial sections of the azimuthal velocity and the kinetic energy density display a double-layered pattern with spiraling cells. The spiral vortices shown in the equatorial section of the axial vorticity develop close to the shear region between the double-layered structure and reach their maxima close to the outer surface.
  • Patterns: Radial dependence at the equator(.gif, 1.86 MB) - 3120 download(s) Radial profiles of the mean zonal flow (left) and of the mean axial vorticity (right), on the equatorial plane during a period of modulation. Because of these physical properties are azimuthally averaged they become periodic. In these movies it can be clearly identified that the solution has doubled his period. The mean axial vorticity is usually used to quantify the location and radial length scale of the convective cells. Near the inner boundary the mean zonal flow is negative (retrograde), and positive (prograde) in the middle of the shell. The value of the axial vorticity becomes maximum near r=0.9, meaning that large scale cells are located close to the radial point where the mean zonal flow is nearly stagnant and with nearly zero modulus.

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