Subcritical convective instability Part 2. Spherical shells (original) (raw)
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The onset of thermo-compositional convection in rotating spherical shells
Geophysical & Astrophysical Fluid Dynamics, 2019
Double-diffusive convection driven by both thermal and compositional buoyancy in a rotating spherical shell can exhibit a rather large number of behaviours often distinct from that of the single diffusive system. In order to understand how the differences in thermal and compositional molecular diffusivities determine the dynamics of thermo-compositional convection we investigate numerically the linear onset of convective instability in a double-diffusive setup. We construct an alternative equivalent formulation of the non-dimensional equations where the linearised double-diffusive problem is described by an effective Rayleigh number, Ra, measuring the amplitude of the combined buoyancy driving, and a second parameter, α, measuring the mixing of the thermal and compositional contributions. This formulation is useful in that it allows for the analysis of several limiting cases and reveals dynamical similarities in the parameters space which are not obvious otherwise. We analyse the structure of the critical curves in this Ra−α space, explaining asymptotic behaviours in α, transitions between inertial and diffusive regimes, and transitions between large scale (fast drift) and small scale (slow drift) convection. We perform this analysis for a variety of diffusivities, rotation rates and shell aspect ratios showing where and when new modes of convection take place.
GeoFlow: On symmetry-breaking bifurcations of heated spherical shell convection
Journal of Physics: Conference Series, 2008
Convective motion in a spherical shell under the influence of a central force has been investigated numerically as a part of the GeoFlow experiment which will run under microgravity conditions on the International Space Station (ISS). An approach combining numerical simulations with a spectral time-stepping code and path-following techniques allows the computation of both stable and unstable solution branches of stationary states. The stability of the solutions has been determined by computing the leading eigenvalues of the Jacobian matrix. Additionally, special attention is paid to the symmetry-breaking bifurcations which are controlled by subgroups of the full spherical symmetry group O(3), under which the governing equations and the primary motionless conductive state are invariant. Finally, the transition from the stationary to the time-dependent regime is described.
Journal of Fluid Mechanics, 2008
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 (O...
Experiments on convection in a rotating hemispherical shell: Transition to chaos
Geophysical Research Letters, 1993
Thermal convection in a self-gravitating rotating fluid shell is modeled using a hemispherical fluid shell that can be rotated about its axis of symmetry. In this apparatus, a tertiary convective state begins to exist at a Rayleigh number approximately equal to 2.1 times the critical value for the onset of convection (Rc•). This state is characterized by the coexistence of three waves. In this tertiary state noise is always present. At a slightly higher Rayleigh number, a strong interaction was observed to develop. Frequency locking takes place at 2.4 Rc•. Later, the flow exhibits chaotic behavior as shown by the broad band Fourier spectra of the temperature records. Planetary implications of these findings are discussed.
Global and Convective Stability of Horizontal Convection
The current study seeks to identify the mechanism responsible for the instability leading to unsteady flow in horizontal convective flow. A one-dimensional (1D) linear stability analysis algorithm is developed which linearises the momentum and energy equations under the Boussinesq approximation, and is applied to vertical velocity and temperature profiles extracted from computed two-dimensional flow solutions. It is demonstrated that a Rayleigh-Bénard type transverse roll instability within the thermal layer ultimately leads to formation of plumes and consequently gives rise to unsteady flow in horizontal convection.
Experiments on convection in rotating hemispherical shells - Transition to a quasi-periodic state
Geophysical Research Letters, 1992
A•bstract. Convection driven by thermal buoyancy temperature gradient between the spherical in the presence of the Coriolis force occurs in boundaries the centrifugal force generates the planetary atmospheres and interiors. In order to same buoyancy force as the perpendicular model convection subject to nearly spherically component of a gravity distribution that symmetric distributions of gravity and increases linearly with radius. While the temperature, a hemisphere has been constructed dependence on the rotation rate of the critical which can be rotated about its axis of symmetry. Rayleigh number for the onset of convection was
Journal of Applied Sciences and Environmental Management, 2010
The effect of radiation on the onset of Rayleigh-Benard convection is studied in the case of a radiating Newtonian fluid in a fluid-saturated horizontal porous layer heated from below. The radiative heat transfer is treated using the differential approximation for optically thin limiting case. The linear stability theory is employed to predict the onset of buoyancy-driven convective motion. It is seen that neither radiation on the static temperature nor on the disturbances can be neglected. In addition, radiation delayed the onset of instability, and higher radiation values led to greater stabilization of a gravity-driven flow in a fluid-saturated porous medium heated from below. @JASEM
Incompressible Viscous Fluid Flows in a Thin Spherical Shell
Journal of Mathematical Fluid Mechanics, 2007
Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier-Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier-Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.