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Torsional flows are preferred at the onset of thermal convection in fluid spheres with stress-free and perfectly conducting boundary conditions, in a narrow region of the parameter space for Prandtl numbers Pr¿0.9 and ratios Pr/Ek=O(10)¿, with Ek being the Ekman number. In this case, the transport of heat to the exterior is supposed instantaneous. When the thermal conductivity of the internal fluid is large, and the external convective heat transfer or radiative emissivity is low, the heat transmission is less efficient, and the thermal energy retained in the interior increases, enhancing the onset of convection. This study is devoted to analyzing the combined influence of the thermal conductivity and external conditions (temperature and resistance to heat transport) on the onset of the torsional convection by taking a Robin boundary condition for the temperature at the surface of the sphere. It is shown, by means of the numerical computation of the curves of simultaneous transitions to torsional flows and Rossby waves, that when the heat flux through the boundary decreases, the region where the axisymmetric flows are preferred shrinks, but it never strangles to an empty set. It has been found that with adequate scalings the curves delimiting the transition to torsional flows, and those of the critical Rayleigh number, Rac¿, and the frequencies of the modes vs Ek become almost independent of the parameter of the Robin boundary condition., Postprint (author's final draft)

The aim of this work is to elucidate the type of transitions that take place when the periodic axisymmetric flows, which can set up at the onset of thermal convection in rotating fluid spheres, lose stability and to study the behavior of the new stable velocity fields until the flows become temporally chaotic. The computations for Prandtl numbers Pr=0.715, 0.1, and 0.01 show that when it decreases, the range of stability of these flows becomes narrower because the kinetic energy of the axisymmetric periodic solutions increases very fast, favoring their instability. From the stability analysis and direct three-dimensional simulations it is found that the transition to stable quasiperiodic flows through Neimark-Sacker bifurcations is supercritical when Pr=0.01. For Pr=0.1 two branches of stable periodic flows emerging from the conduction state have been found due to the proximity to a double Hopf bifurcation. However, only the branches bifurcating from the azimuthal rotating waves are stable at large Rayleigh numbers. Far from this bifurcation the stable flows keep the influence of the axisymmetric dynamics up to large Rayleigh numbers. For small Pr they behave as repeated transients of mixed dynamics, controlled by the azimuthal wave numbers m=0, m=1, and m=2., Postprint (author's final draft)

The linear stability of the periodic and axisymmetric solutions of the convection in rotating, internally heated, and self-gravitating fluid spheres is presented. The transition to quasiperiodic flows via Neimark–Sacker bifurcations of different azimuthal wave numbers, m, is studied using matrix-free continuation and Floquet theory. Several pairs of Ekman and Prandtl numbers are considered in the region where the first bifurcation from the conduction state gives rise to the axisymmetric solutions. It is shown that the azimuthal wave numbers m=1 and m=2 are preferred and that, for small Ekman and Prandtl numbers, the secondary bifurcations to different m accumulate close to the onset of convection. This study confirms some results previously found just by direct simulations. The methods presented can be applied to systems of parabolic partial differential equations with O(2) or SO(2) symmetry group, when a periodic orbit, invariant under the group, loses stability through a Neimark–Sacker bifurcation., Postprint (published version)

The thermal convection of rotating fluids in spherical geometry is a classical problem with application to many geophysical and astrophysical problems. The study of the transition to periodic solutions from the steady conduction state of a rotating and self-gravitating fluid sphere, heated uniformly from the inside, is discussed here. The continuation of double Hopf points is used to determine the region of the parameter space in which the first bifurcation is to solutions independent of the longitude (axisymmetric solutions). It is limited by three segments of curves separated by two triple Hopf points. This type of so-called torsional solutions was recently found, and it is shown here that they are the preferred solutions at the onset of convection for a wide range of fluids of Prandtl numbers, ${Pr}$, extending from ${Pr}=0$ to ${Pr}\approx 0.9$, which includes, for instance, liquid metals and gases. Although the corresponding interval of Ekman numbers, ${E}$, narrows when ${Pr}\rightarrow 0$, it is shown that there is always a small gap of parameters, relevant to geophysics and astrophysics, where the torsional solutions are preferred. The limits of the double Hopf curves when ${Pr\rightarrow 0}$ follow linear laws of the form ${E}=c(m_1,m_2){Pr}$, $c(m_1,m_2)$ being constant depending on the two azimuthal wavenumbers, $m_1$ and $m_2$, of the eigenfunctions that define the double Hopf problem., Postprint (author's final draft)

The aim of this study is to determine through numerical simulations the extent and robustness of the three-dimensional torsional dynamics of the thermal convection in rotating spherical fluids at very low Prandtl numbers. It is known that the kinetic energy of the periodic axisymmetric flows propagates latitudinally on the surface of the sphere. Here, it is shown that when the axisymmetry is broken at a secondary Hopf bifurcation, the flow starts to drift in the azimuthal direction giving rise to a quasiperiodic motion that propagates the energy in latitude and longitude. The double direction of propagation gives rise to a meandering path of the kinetic energy, which is still concentrated on the surface, but highly localized. Several new stable states of convection with different symmetries have been identified in a large range of Rayleigh numbers, all of them retaining the torsional motion of the basic velocity field. Particular attention is paid to their dependence on the Rayleigh number and on the values of the frequencies, of the mean zonal flow, and of the kinetic energy of the fluid., Postprint (author's final draft)

This paper deals with the analysis of numerically obtained spatio-temporal data for thermal convection in a two-dimensional circular annulus. These data are post-processed using a recent method called the spatio-temporal Koopman decomposition, which applies to the frequent case in which the underlying dynamics exhibit oscillatory (possibly growing or decaying) behavior in both time and one of the spatial directions (the azimuthal coordinate for the present problem). When this holds, the method decomposes the data into Fourier-like series in both the distinguished spatial direction and time. In the general case, the obtained series account for, not only the involved temporal frequencies and spatial wavenumbers, but also the spatial and temporal growth rates. In the simpler situation of attractors showing spatially periodic behavior, the spatial and temporal growth rates vanish (or are very small). In this case, the analysis of the wavenumber–frequency pairs that are present allows for uncovering the spatio-temporal structure of the flow in the circular annulus. The analysis focuses on periodic and quasi-periodic attractors, which exhibit spatio-temporal symmetries that are identified by the method. In addition, for quasi-periodic attractors, the method gives semi-analytic descriptions for the tori densely covered by particular trajectories. Although the paper concentrates on the thermal convection problem in an annulus, it will become clear that the method applies to other related dynamics as well., Postprint (author's final draft)

A numerical study of the nonlinear torsional solutions of convection in rotating, internally heated, self-gravitating fluid spheres is presented. Their dependence on the Rayleigh number has been found for two pairs of Ekman (E) and small Prandtl (Pr) numbers in the region of parameters where, according to Zhang et al. [J. Fluid Mech. 813, R2 (2017)], the linear stability of the conduction state predicts that they can be preferred at the onset of convection. The bifurcation to periodic torsional solutions is supercritical for sufficiently small Pr. They are not rotating waves, unlike the nonaxisymmetric case. Therefore they have been computed by using continuation methods for periodic orbits. Their stability with respect to axisymmetric perturbations and physical characteristics have been analyzed. It was found that the time- and space-averaged equatorially antisymmetric part of the kinetic energy of the stable orbits splits into equal poloidal and toroidal parts, while the symmetric part is much smaller. Direct numerical simulations for E=10-4 at higher Rayleigh numbers (Ra) show that this trend is also valid for the nonperiodic flows and that the mean values of the energies remain almost constant with Ra. However, the modulated oscillations bifurcated from the quasiperiodic torsional solutions reach a high amplitude, compared with that of the periodic, increasing slowly and decaying very fast. This repeated behavior is interpreted as trajectories near heteroclinic chains connecting unstable periodic solutions. The torsional flows give rise to a meridional propagation of the kinetic energy near the outer surface and an axial oscillation of the hot nucleus of the metallic fluid sphere., Postprint (published version)

A mechanism for the cyclic generation of bursts of magnetic fields by nonlinear torsional flows of complex time dependence but very simple spatial structure is described. These flows were obtained numerically as axisymmetric solutions of convection in internally heated rotating fluid spheres in the Boussinesq approximation. They behave as repeated transients, which start with nearly periodic oscillations of the velocity field of slowly increasing amplitude. This regime is followed by a chaotic fast increase and a final decrease of the amplitude of, at least, one order of magnitude. The magnetic field decays due to the magnetic diffusion during the regular oscillations, but it grows in the form of bursts during the intervals of irregular time dependence of the velocity. The magnetic field is strongly localized in spirals, with spatial- and temporal-dependent intensity., Postprint (published version)

The steady and oscillatory dynamics of binary fluids contained in slots heated by the side is studied by using continuation methods, and stability analysis. The bifurcation points on the branches of solutions are determined with precision by calculating their spectra for a large range of Rayleigh numbers. It will be seen that continuation and stability methods are a powerful tool to analyze the origin of the hydrodynamic instabilities leading to steady and time periodic flows, and their dynamics. The role played by the shear stresses of the steady field, and the solutal and thermal buoyancies, at the onset of the oscillations is studied by means of the energy equation of the perturbations. With the parameters used, it is found that the shear is always the main responsible for the instabilities, and that the work done by the two buoyancies can even help to stabilize the fluid. The results also show that binary mixtures of Prandtl number order one, like pure gases, present multiple stable periodic flows coexisting in the same range of parameters, since several unstable leading multipliers remain attached to the unit circle and go back into it. However, at lower Prandtl numbers only the first branch of periodic orbits bifurcating directly from the steady state is found to be stable, because some of the unstable multipliers of the other branches quickly increase their modulus and never re-enter the unit circle., Postprint (author's final draft)

The influence of the Prandtl number on the stable stationary and periodic flows of the fluids contained in laterally heated slots under realistic conditions is analyzed by using continuation methods. Namely, non-slip boundaries, insulated top and bottom horizontal limits and perfectly conducting lateral sides are considered. The branches of solutions are computed by decreasing the Prandtl number, for four Rayleigh numbers. The dynamical behavior depends strongly on both parameters. For a Rayleigh number, ${\rm Ra}=10^3$ the steady flow remains stable in the wide range of Prandtl numbers computed. At ${\rm Ra}=10^4$ and ${O(10^5)}$ the first bifurcations are of Hopf type giving rise to a type of oscillations that affect the bulk of the fluid, alternating from a general circulation to multi-vortex solutions, or the boundary layer, respectively. However, it is found that, in any case, the location of the shear determines the type of the oscillations. Moreover, at ${\rm Ra}=10^4$ the critical multipliers at the secondary bifurcations on the main branch of POs are real, giving rise to different kinds of very bounded stable periodic states of different symmetries and periods. At ${\rm Ra}$ of order $10^5$ the instability of the periodic orbits gives rise directly to quasi-periodic flows., Postprint (author's final draft)

This study elucidates the origin of the multiplicity of stable oscillatory flows detected by time integration in tall rectangular cavities heated from the side. By using continuation techniques for periodic orbits, it is shown that initially unstable branches, arising at Hopf bifurcations of the basic steady flow, become stable after crossing Neimark-Sacker points. There are no saddle-node or pitchfork bifurcations of periodic orbits, which could have been alternative mechanisms of stabilization. According to the symmetries of the system, the orbits are either fixed cycles, which retain at any time the center symmetry of the steady flow, or symmetric cycles involving a time shift in the global invariance of the orbit. The bifurcation points along the branches of periodic flows are determined. By using time integrations, with unstable periodic solutions as initial conditions, we determine which of the bifurcations at the limits of the intervals of stable periodic orbits are sub- or supercritical., Postprint (author's final draft)

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., Peer Reviewed, Postprint (published version)

A tutorial on continuation and bifurcation methods for the analysis of truncated dissipative partial differential equations is presented. It focuses on the computation of equilibria, periodic orbits, their loci of codimension-one bifurcations, and invariant tori. To make it more self-contained, it includes some definitions of basic concepts of dynamical systems, and some preliminaries on the general underlying techniques used to solve non-linear systems of equations by inexact Newton methods, and eigenvalue problems by means of subspace or Arnoldi iterations., Postprint (author's final draft)

A numerical study of the onset of convection in rotating internally heated self-gravitating fluid spheres is presented. The exploration of the stability of the conduction state versus the Taylor and Prandtl numbers supplies a detailed idea of the laws that fulfil the four types of solutions obtained at low Prandtl numbers. The main result found is that axisymmetric (torsional) modes of convection are preferred at high Taylor numbers in the zero-Prandtl-number limit. This instability appears at low Rayleigh numbers and gives rise to an oscillating single vortex of very high frequency., Postprint (author's final draft)

Modulated rotating waves (MRW), bifurcated from the thermal-Rossby waves that arise at the onset of convection of a fluid contained in a rotating spherical shell, and their stability, are studied. For this purpose, Newton-Krylov continuation techniques are applied. Nonslip boundary conditions, an Ekman number E=10-4, and a low Prandtl number fluid Pr=0.1 in a moderately thick shell of radius ratio ¿=0.35, differentially heated, are considered. The MRW are obtained as periodic orbits by rewriting the equations of motion in the rotating frame of reference where the rotating waves become steady states. Newton-Krylov continuation allows us to obtain unstable MRW that cannot be found by using only time integrations, and identify regions of multistability. For instance, unstable MRW without any azimuthal symmetry have been computed. It is shown how they become stable in a small Rayleigh-number interval, in which two branches of traveling waves are also stable. The study of the stability of the MRW helps to locate and classify the large sequence of bifurcations, which takes place in the range analyzed. In particular, tertiary Hopf bifurcations giving rise to three-frequency stable solutions are accurately determined., Postprint (published version)

A new efficient methodology for the continuation of the codimension-one bifurcations of periodic orbits, including pitchfork bifurcations present in reflection--symmetric systems, has been developed. It is based on the combination of Newton-Krylov techniques applied to extended systems, and the integration of systems of variational equations up to second order. The extended systems are adapted from those usually found in the literature for fixed points of maps. A deflation term is needed in some cases to remove the trivial +1 multiplier of the periodic orbits. It will be shown that to evaluate the action of the Jacobian it is only necessary to integrate systems of ODEs of dimension at most four times that of the original system. This minimizes the computational cost. Two main tools are required to implement the algorithms presented here, a generic continuation code, and a time integrator for the particular problem at hand including the required variational equations. The thermal convection of a mixture of two fluids in a two-dimensional rectangular box is used as test problem. It is known that the onset of convection is oscillatory below a certain negative value of one of the parameters (the separation ratio), giving rise to a rich dynamics. A non-trivial diagram of periodic orbits is first deployed, by varying the Rayleigh number. Some of the bifurcations found on the main branch of periodic orbits are followed by adding as second parameter the Prandtl number. Several codimension-two points are found, and a double-Hopf bifurcation is studied in more detail. Finally the boundaries of a resonance region, found on a branch of invariant tori, are also continued., Postprint (published version)

Postprint (published version)

A methodology to track bifurcations of periodic orbits in large-scale dissipative systems depending on two parameters is presented. It is based on the application of iterative Newton-Krylov techniques to extended systems. To evaluate the action of the Jacobian it is necessary to integrate variational equations up to second order. It is shown that this is possible by integrating systems of dimension at most four times that of the original equations. In order to check the robustness of the method, the thermal convection of a mixture of two fluids in a rectangular domain has been used as a test problem. Several curves of codimension-one bifurcations, and the boundaries of an Arnold's tongue of rotation number 1/8, have been computed., Peer Reviewed, Postprint (author’s final draft)

A five-degree model, which reproduces faithfully the sequence of bifurcations and the type of solutions found through numerical simulations of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells with fixed azimuthal symmetry, is derived. A low Prandtl number fluid of s=0. 1 subject to radial gravity, filling a shell of radius ratio ¿=0.35, differentially heated, and with non-slip boundary conditions, is considered. Periodic, quasi-periodic, and temporal chaotic flows are obtained for a moderately small Ekman number, E=10-4,andatsupercritical Rayleigh numbers of order Ra~O(2Rac). The solutions are classified by means of frequency analysis and Poincaré sections. Resonant phase locking on the quasi-periodic branches,as well as a sequence of period doubling bifurcations, are also detected., Peer Reviewed, Postprint (published version)

Background: In a previous study on a nocturnal, every-other-day online haemodiafiltration scheme, different removal patterns were observed for urea, creatinine, ß2-¿microglobulin, myoglobin and prolactin. The aim of this study was to evaluate the influence of dialysis duration and infusion flow (Qi) on the removal of different molecular weight (MW) solutes, and to quantify the effect of the different treatments on the kinetics of the solutes by using a classical two-compartment model. Methods: This prospective, in-center study was carried out in 10 patients on a nocturnal, every-other-day online post-dilution haemodiafiltration program. Each patient received four dialysis sessions with different conditions, two 4-h sessions (with infusion flows of 50 or 100 ml/min) and two 8-h sessions (with infusion flows of 50 or 100 ml/min). To analyze the solute kinetics, blood samples were obtained hourly during the dialysis treatments and in the first 3 h post-dialysis. Results: Removal patterns differed in the molecules studied, which were quantified by means of the two-compartment mathematical model. The main results show the impact of dialysis duration on the removal of low molecular weight molecules (urea and creatinine), while the impact of Qi is clearly shown for high molecular weight molecules (myoglobin and prolactin). For middle molecular weight solutes, such as ß2-microglobulin, both factors (duration and Qi) enhance the removal efficiency of the dialyzer. Conclusions: Our study evaluates experimentally and mathematically how treatment time and infusion flow affect the filtration of solutes of different MW during post-dilution haemodiafiltration. The results provided by the present study should help physicians to select and individualise the most appropriate schedules to deliver an optimum diffusive and convective dialysis dose for each patient., Peer Reviewed, Postprint (author's final draft)

Many geophysical and astrophysical phenomena such as magnetic fields generation, or the differential rotation observed in the atmospheres of the major planets are studied by means of numerical simulations of the Navier-Stokes equations in rotating spherical shells. Two different computational codes, spatially discretized using spherical harmonics in the angular variables, are presented. The first code, PARODY, solves the magneto-hydrodynamic anelastic convective equations with finite a difference discretization in the radial direction. This allows the parallelization on distributed memory computers to run massive numerical simulations of second order in time. It is mainly designed to perform direct numerical simulations. The second code, SPHO, solves the fully spectral Boussinesq convective equations, and its variationals, parallelized on shared memory architectures and it uses optimized linear algebra libraries. High-order time integration methods are implemented to allow the use of dynamical systems tools for the study of complex dynamics., Postprint (published version)

Many geophysical and astrophysical phenomena such as magnetic fields generation, or the differential rotation observed in the atmospheres of the major planets are studied by means of numerical simulations of the Navier-Stokes equations in rotating spherical shells. Two different computational codes, spatially discretized using spherical harmonics in the angular variables, are presented. The first code, PARODY, solves the magneto-hydrodynamic anelastic convective equations with finite a difference discretization in the radial direction. This allows the parallelization on distributed memory computers to run massive numerical simulations of second order in time. It is mainly designed to perform direct numerical simulations. The second code, SPHO, solves the fully spectral Boussinesq convective equations, and its variationals, parallelized on shared memory architectures and it uses optimized linear algebra libraries. High-order time integration methods are implemented to allow the use of dynamical systems tools for the study of complex dynamics., Postprint (published version)

An exhaustive study, based on numerical three-dimensional simulations, of the Boussinesq thermal convection of a fluid confined in a rotating spherical shell is presented. A moderately low Prandtl number fluid (ro = 0.1) bounded by differentially-heated solid spherical shells is mainly considered. Asymptotic power laws for the mean physical properties of the flows are obtained in the limit of low Rossby number and compared with laboratory experiments and with previous numerical results computed by taking either stress-free boundary conditions or quasi-geostrophic restrictions, and with geodynamo models. Finally, using parameters as close as possible to those of the Earth's outer core, some estimations of the characteristic time and length scales of convection are given. © 2014 Elsevier B.V., Postprint (published version)

The efficiency of implicit and semi-implicit time integration codes based on backward differentiation and extrapolation formulas for the solution of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells was studied in [5] at weakly supercritical Rayleigh numbers R, moderate (10−3) and low (10−4) Ekman numbers, E, and Prandtl number = 1. The results presented here extend the previous study and focus on the effect of and R by analyzing the efficiency of the methods for obtaining solutions at E = 10−4, = 0.1 and low and high supercritical R. In the first case (quasiperiodic solutions) the decrease of one order of magnitude does not change the results significantly. In the second case (spatio-temporal chaotic solutions) the differences in the behavior of the semi-implicit codes due to the different treatment of the Coriolis term disappear because the integration is dominated by the nonlinear terms. As in [5], high order methods, either with or without time step and order control, increase the efficiency of the time integrators and allow to obtain more accurate solutions., Postprint (author’s final draft)

We assess the accuracy and efficiency of several exponential time integration methods coupled to a spectral discretization of the three-dimensional Boussinesq thermal convection equations in rotating spherical shells. Exponential methods are compared to implicit–explicit (IMEX) multi-step methods already studied previously in [1]. The results of a wide range of numerical simulations highlight the superior accuracy of exponential methods for a given time step, especially when employed with large time steps and at low Ekman number. However, presently available implementations of exponential methods appear to be in general computationally more expensive than those of IMEX methods and further research is needed to reduce their computational cost per time step. A physically justified extrapolation argument suggests that some exponential methods could be the most efficient option for integrating flows near Earthʼs outer core conditions., Postprint (published version)

A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, Newton-Krylov, and Arnoldi methods. An application to the study of a double-Hopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E=10−4 and σ=0.1, respectively, radius ratios η∈[0.32,0.35], Rayleigh numbers R∈[1.8×105,6×105], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves., Postprint (published version)

Resolved

A parallelizable algorithm to compute invariant tori of high-dimensional dissipative systems, obtained upon discretization of PDEs is presented. The size of the set of equations to be solved is only a small multiple of the dimension of the original system. The sequential and parallel implementations are compared with a previous method (Sánchez et al. (2010)) [11], showing that important savings in wall-clock time can be achieved. In order to test it, a thermal convection problem of a binary mixture of fluids has been used. The new method can also be applied to problems with very low rotation numbers, for which the previous is not suitable. This is tested in two examples of two-dimensional maps., Postprint (published version)

The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known Rec computed without mixture. For positive and sufficiently large compositional gradients, Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored., Postprint (published version)

Tesi per compendi de publicacions, The study of thermal convection in rotating spherical geometry is fundamental to explain many geophysical and astrophysical phenomena such as the generation of the magnetic fields, or the differential rotation observed in the atmospheres of the major planets. The difficulties associated with the experimental studies enhance the importance of the three-dimensional numerical simulations, such as those presented in this dissertation. In order to obtain the evolution equations, the Boussinesq approximation is applied to the mass, momentum and energy conservation equations, which are rewritten in terms of toroidal and poloidal potentials. Together with the temperature field, they are expanded in spherical harmonics over the sphere, and in the radial direction a collocation method is used. Semi- implicit schemes, based in backward differentiation formulae (IMEX-BDF), implemented with order and time step control (VSVO), are used for time integration. Applying non-slip boundary conditions with internal heating and very low Prandtl numbers (ratio between the thermal diffusive and the viscous time scales), one of the first exhaustive analysis of the linear stability of the conductive state has been performed. In addition, the existence of preferred polar antisymmetric modes at the onset of convection for high rotation rates has been described. A study of the efficiency of different high order time integration schemes, either with fixed time-step or VSVO, has been carried out. In our own time evolution codes we apply the IMEX-BDF formulae with an explicit treatment of the nonlinear terms of the equations. The use of 'matrix-free' methods allows the implicit treatment of the Coriolis term, and makes the implementation of a step and order control easier. The results show that the use of high order methods, especially those with time-step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions. Finally, at low Prandtl numbe, L'estudi de la convecció tèrmica en geometria esfèrica en rotació es fonamental per explicar molts fenòmens geofísics i astrofísics, com la generació de camps magnètics, o la rotació diferencial observada en l'atmosfera dels planetes majors. Les dificultats associades amb els estudis experimentals afavoreixen que les simulacions numèriques tridimensionals, com les que es presenten en aquesta memòria, siguin una eina molt important en aquest camp. Per a l'obtenció de les equacions d'evolució, s'aplica l'aproximació de Boussinesq a les equacions de conservació de la massa, la quantitat de moviment i l'energia, i es reescriuen en funció dels potencials toroidal i poloidal. Els potencials i la temperatura es desenvolupen, sobre l'esfera, en harmònics esfèrics i en la variable radial s'usa col·locació. Per a la integració es fan servir esquemes semi-implícits, que en el nostre cas, estan basats en les fórmules de diferenciació regressiva (IMEX-BDF), que s'han implementat amb control d'ordre i pas (VSVO). En primer lloc, sota condicions de contorn d'adherència, calentament intern i nombre de Prandtl (quocient entre les escales de temps de difusió tèrmica i viscosa) molt baix s'ha realizat un dels primers anàlisis exhaustius de l'estabilitat lineal de l'estat conductiu, gràcies a la millora dels mètodes numèrics emprats. Així mateix, s'ha descrit per primera vegada l'existència de modes polars antisimètrics a l'inici de la convecció amb rotacions elevades. En segon lloc s'ha realizat un estudi de l'eficiència de diferents integradors temporals d'ordre alt, amb pas fix o VSVO. En els nostres propis codis temporals apliquem les fórmules IMEX-BDF amb un tractament explícit dels termes no lineals de les equacions. L'ús de mètodes 'matrix-free' fa rentable el tractament implícit del terme de Coriolis i facilita la implementació d'un control d'ordre i pas temporal adequat. Els resultats mostren que amb ordre elevat, amb o sense control de pas i ordre, s'incrementa l'eficiència d, Postprint (published version)

Azimuthal waves appearing in the thermal convection of a pure fluid contained in a spherical shell with both boundaries at different temperatures are studied. They are computed by using continuation methods as steady solutions in the reference system of the wave. There stability is also studied, and the secondary bifurcations to modulated waves are detected., Postprint (author's final draft)

The stability analysis of the basic conductive state of a rotating binary mixture of two fluids bounded by spherical shells is studied. The Boussinesq approximation of the mass conservation, Navier-Stokes, energy and concentration equations is used, and results for a moderate Ekman number E = 10-3 are presented for positive and negative external compositional gradients. Preliminary results are compared with those obtained for a pure fluid in the same range of parameters. They show an important influence of the presence of a mixture on the onset of the convection for solutal Rayleigh numbers Rc at least of the order of the thermal Rayleigh number Re. On the sphere the leading eigenvectors, which give the patterns of convection, have sectorial structure like those of a pure fluid, although slightly deformed due to the presence of two components., Postprint (author's final draft)

The efficiency of implicit and semi-implicit time integration codes based on bac- kward differentiation and extrapolation formulae for the solution of the three-dimensio- nal Boussinesq thermal convection equations in rotating spherical shells is studied. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The results show that high order methods, either with or without time step and order control, increase the efficiency of the time integrators, and allow to obtain more accurate solutions., Peer Reviewed, Postprint (published version)

A numerical study of several time integration methods for solving the threedimensional Boussinesq thermal convection equations in rotating spherical shells is presented. Implicit and semi-implicit time integration techniques based on backward differentiation and extrapolation formulae are considered. The use of Krylov techniques allows the implicit treatment of the Coriolis term with low storage requirements. The codes are validated with a known benchmark, and their efficiency is studied. The results show that the use of high order methods, especially those with time step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions., Postprint (author’s final draft)

The application of the multiple shooting method to the continuation of periodic orbits in large-scale dissipative systems is analyzed. A preconditioner for the linear systems which appear in the application of Newton’s method is presented. It is based on the knowledge of invariant subspaces of the Jacobians at nearby solutions. The possibility of speeding up the process by using parallelism is studied for the thermal convection of a binary mixture of fluids in a rectangular domain, with positive results., Postprint (author’s final draft)

Accurate numerical computations of the onset of thermal convection in wide rotating spherical shells are presented. Low-Prandtl-number (σ) fluids, and non-slip boundary conditions are considered. It is shown that at small Ekman numbers (E), and very low σ values, the well-known equatorially trapped patterns of convection are superseded by multicellular outer-equatorially-attached modes. As a result, the convection spreads to higher latitudes affecting the body of the fluid, and increasing the internal viscous dissipation. Then, from E<10−5, the critical Rayleigh number (Rc) fulfils a power-law dependence Rc ∼E −4/3, as happens for moderate and high Prandtl numbers. However, the critical precession frequency (|ωc|) and the critical azimuthal wavenumber (mc) increase discontinuously, jumping when there is a change of the radial and latitudinal structure of the preferred eigenfunction. In addition, the transition between spiralling columnar (SC), and outer-equatorially-attached (OEA) modes in the (σ, E)-space is studied. The evolution of the instability mechanisms with the parameters prevents multicellular modes being selected from σ 0.023. As a result, and in agreement with other authors, the spiralling columnar patterns of convection are already preferred at the Prandtl number of the liquid metals. It is also found that, out of the rapidly rotating limit, the prograde antisymmetric (with respect to the equator) modes of small mc can be preferred at the onset of the primary instability., Postprint (published version)

The onset of thermal convection in a rotating spherical shell of intermediate radius ratio ¼ 0:4 is studied numerically for Taylor numbers Ta 1011 and the Prandtl number of the liquid sodium ( ¼ 0:01). For the first time, it is shown that at very high Taylor numbers the first unstable mode can be antisymmetric with respect to the equator and confined inside a cylinder tangent to the inner sphere at the equator (polar mode). The exponent of the power law determined from the asymptotic dependence of the critical Rayleigh number for very high Ta is 0.57, lower than 2=3, given theoretically for the spiraling columnar modes, and than 0.63, found numerically for the outer equatorially attached modes, Peer Reviewed, Postprint (published version)

En esta tesis se estudia, mediante simulación numérica directa de las ecuaciones de Navier-Stokes, la dinámica de un fluido en el interior de un anillo cilíndrico en rotación con las tapas ligeramente inclinadas y sometido a un gradiente de temperatura y una gravedad en dirección radial. En el presente trabajo se ha introducido la curvatura de las paredes de forma natural mediante la resolución de las ecuaciones en coordenadas cilíndricas. Se han considerado condiciones de contorno de adherencia en las paredes laterales y esfuerzos cortantes nulos en las tapas del anillo. Si la rotación es suficientemente elevada es conocido que la dinámica se desarrolla en planos perpendiculares al eje de rotación, en forma de ondas que precesionan en el sentido de la rotación, llamadas ondas térmicas de Rossby. Se ha estudiado la dinámica en función de la rotación, la razón de radios del anillo, del número de Prandtl. El análisis de estabilidad lineal del estado conductivo indica que el inicio de la convección depende fuertemente de una característica del fluido como es el número de Prandtl. Si este parámetro es moderado o elevado las inestabilidades térmicas son dominantes y se han encontrado dos familias de modos dominantes. Se han obtenido modos en los que la convección se desarrolla principalmente cerca del pared interior y están inclinados en el sentido de la rotación (espirales); y modos prácticamente rectos, en los que la convección se desarrolla en toda la capa (normales). Para bajos números de Prandtl todos los modos dominantes son del segundo tipo. Mediante técnicas de continuación se han analizado las soluciones no lineales relacionadas con los dos tipos de modos existentes para el caso más interesante de los estudiados. Esto es cuando el número de Prandtl es 0.7. Las soluciones obtenidas se han comparado con las encontradas en el caso de capa plana. También se ha analizado la estabilidad lineal de estas soluciones mediante una técnica de Floquet. Se, Postprint (published version)

En la tesis doctoral Convección térmica en un anillo cilíndrico en rotación se estudian diferentes aspectos de la dinámica lineal y no lineal de un fluido Boussinesqconfinado en un anillo cilíndrico, en rotación alrededor de su eje de simetría y sometido a la acción de un campo gravitatorio y de un gradiente de temperaturadesestabilizador en la dirección radial. Su principal objetivo es analizar la influencia de la rotación, de la geometría cilíndrica y de las condiciones de contorno en la dinámica de un sistema físico que constituye un modelo simple de la convección libre en las capasecuatoriales de las atmósferas planetarias y estelares. En eltrabajo, que se enmarca en el campo de la Simulación Numérica en Dinámica de Fluidos, se utilizan técnicas espectrales para hacer una simulación numérica directade las ecuaciones de Navier-Stokes.El estudio se inicia analizando la estabilidad lineal del estado conductivo imponiendo condiciones de contorno stress-free en las superficies horizontales del anillo.Cuando la rotación es alta, la convección se inicia en forma de columnas de Taylor estacionarias, que son soluciones bidimensionales formadas por estructuras alineadas con el eje del anillo. Para rotaciones bajas, existen dos tipos diferentes de modos convectivos: rollosconvectivos bidimensionales estacionarios y celdastridimensionales con precisión. Este último tipo de solución no se puede encontrar si no se tiene en cuenta la curvatura de la capa. La dinámica cambia cuando seconsideran condiciones de contorno no-slip en las tapas. En el caso de rotaciones bajas, las condiciones de contorno fuerzan que los modos críticos seantridimensionales y se produce un aumento en el número de Rayleigh necesario para que se inicie la convección. A medida que la rotación aumenta, las solucionestridimensionales se alargan progresivamente hasta que se forman columnas de Taylor prácticamente bidimension., Postprint (published version)

An exhaustive study, based on numerical three-dimensional simulations, of the Boussinesq thermal convection of a fluid confined in a rotating spherical shell is presented. A moderately low Prandtl number fluid (ro = 0.1) bounded by differentially-heated solid spherical shells is mainly considered. Asymptotic power laws for the mean physical properties of the flows are obtained in the limit of low Rossby number and compared with laboratory experiments and with previous numerical results computed by taking either stress-free boundary conditions or quasi-geostrophic restrictions, and with geodynamo models. Finally, using parameters as close as possible to those of the Earth's outer core, some estimations of the characteristic time and length scales of convection are given. © 2014 Elsevier B.V., Postprint (published version)