Instabilities of rotating inclined buoyancy layers.

The boundary layer near a cooled inclined plate, which is immersed in a stably stratified fluid rotating about an axis parallel to the direction of gravity, is a model for katabatic flows at high latitudes. In this paper the base flow of such an inclined buoyancy layer is solved analytically for arbitrary Prandtl numbers. By applying linear stability analyses, five unstable modes are identified for both the fixed temperature and the isoflux boundary conditions, i.e., the stationary longitudinal roll (LR) mode, the oblique roll with low streamwise wave-number (OR-1) and high streamwise wave-number (OR-2) modes, and the Tolmien-Schlichting (TS) wave with low streamwise wave-number (TS-1) and high streamwise wave-number (TS-2) modes. It is indicated that the Coriolis effect induced by the rotation leads the critical modes to be three dimensional, and a larger tilt angle of the plate and stronger Coriolis effect cause both TS wave modes to be more unstable for both thermal boundary conditions. When the Coriolis effect is considered, the OR-1 and OR-2 modes are the most unstable mode at low and high tilt angles, respectively, but the TS-1 wave mode may be the most unstable one when the plate is nearly vertical. In addition, the spanwise phase velocities of the TS wave modes change directions as the tilt angle passes some threshold values for both thermal boundary conditions except for the TS-1 wave mode with a fixed temperature boundary condition, which propagates in the same spanwise direction for all explored tilt angles.

Convection in a rotating layer: a simple case of turbulence.

Convection in a layer heated from below and rotating about a vertical axis exhibits a unique phenomenon in fluid dynamics in that the small-amplitude motion is governed by random effects in both its spatial and its time dependence. A simple theoretical description of the phenomenon is compared with laboratory observations. A more detailed mathematical description appears to be feasible because of the weakly nonlinear nature of the problem.

Laboratory simulation of thermal convection in rotating planets and stars.

Because of dynamical constraints in a rotating system, the component of gravity perpendicular to the axis of rotation is the dominant driving force of convection in liquid planetary cores and in stars. Except for the sign, the centrifugal force closely resembles the perpendicular component of gravity. Convection processes in stars and planets can therefore be modeled in laboratory experiments by using the centrifugal force with a reversed temperature gradient.

Reorientation of a hexagonal pattern under broken symmetry: the hexagon flip.

An unexpected pattern transition has been found experimentally in the transformation from hexagons to stripes caused by an applied anisotropy effect. The particular system studied is the surface instability of a horizontal layer of magnetic liquid in a tilted magnetic field. Two orthogonal Helmholtz pairs of coils provide a vertical and a tangential magnetic field. Whereas the vertical component destabilizes the flat layer, the tangential one preserves its stability. The ensuing surface patterns comprise regular hexagons, anisotropic hexagons, and stripelike ridges. The phase diagram for the tilted field instability is measured using a radioscopic technique. The investigation reveals an interesting effect: the flip from one hexagonal pattern to another under an increasing tangential field component, which is explained in terms of amplitude equations as a saddle-node bifurcation.

Square patterns in rotating Rayleigh-Bénard convection.

The Küppers-Lortz instability occurs in rotating Rayleigh-Bénard convection and is a paradigmatic example of spatiotemporal chaos. Since the steady state of convection rolls is unstable to disturbance rolls oriented with an angle of about 60 degrees with respect to the given rolls in the prograde direction [G. Küppers and D. Lortz, J. Fluid Mech. 35, 609 (1969)], a spatiotemporally chaotic pattern is realized with patches of rolls continuously replaced by other patches in which the roll axis is switched by about 60 degrees. Surprisingly and contrary to this established scenario, Bajaj [Phys. Rev. Lett. 81 (1998)] observed experimentally square patterns in a cylindrical layer in the range of parameters where Küppers-Lortz instability was expected. In this paper we present square patterns which we have obtained in a numerical study by taking into account realistic boundary conditions. The Navier-Stokes and heat transport equations have been solved in the Oberbeck-Boussinesq approximation. The numerical method is pseudospectral and second order accurate in time. The rotation velocity of the square pattern increases linearly with the control parameter epsilon=Ra/R a(c) -1 , as in the experiment of Bajaj Furthermore, it was observed that this velocity decreases when the aspect ratio of the cylinder increases. These results indicate that the square pattern appears when the flow is laterally confined. The range of epsilon for which this pattern is stable tends to vanish for more extended layers.

Theoretical study of flow coupling mechanisms in two-layer Rayleigh-Bénard convection.

Rayleigh-Bénard convection in a system of two superimposed immiscible fluids, heated from below, is investigated theoretically. In a two-layer system, stationary convection is characterized by two distinct modes of flow coupling, namely, thermal coupling and viscous coupling. We derive two coupled amplitude equations in order to describe the nonlinear interaction of the viscous and the thermal coupling modes, whereby we restrict our analysis to the two-dimensional case. By analyzing the amplitude equations for varying fluid parameters, we make predictions concerning the stability of the involved coupling modes in the weakly nonlinear regime.

Bounds on the convective heat transport in a rotating layer.

Previous bounds on the convective heat transport in a horizontal layer heated from below and rotating about a vertical axis have been improved through the use of separate energy balances for the poloidal and toroidal components of the velocity field. Because the additional constraint imposed for the solution of the variational problem for the extremalizing vector field leads to Euler-Lagrange equations which can no longer be solved analytically, numerical methods must be employed. A Galerkin scheme is introduced and the variational problem is solved in the case when stress-free conditions are assumed at the upper and lower boundaries. Results are presented as a function of the Rayleigh number and the rotation parameter for the Prandtl numbers P=7, 0.7, 0.1, and 0.025.

Instability of a thin film flowing on a rotating horizontal or inclined plane.

In this paper the instability of a thin fluid film flowing under the effects of gravity, Coriolis, and centrifugal forces is investigated. It is supposed that the film flows far from the axis of rotation on a plane which may be horizontal or inclined with respect to the horizontal. In the former case, the flow is only driven by the centrifugal force while in the latter case, the flow is driven by the components of centrifugal force and gravity along the plane. This case may also be considered as the flow down a rotating cone but far from the apex. The stabilizing influence of rotation on the film flow increases with the rotation rate. Up to a certain critical rate of rotation, the film flowing down the rotating inclined plane (or cone) is more stable than the flow on the horizontal rotating plane while above this rate of rotation the situation is reversed. The instability above the critical rate is associated with a finite wave number in contrast to the vanishing wave number of the instability below the critical rate. The possibility of Ekman layer instabilities is also investigated. An equation describing the nonlinear evolution of surface waves is also obtained. Moreover, this equation is simplified for the case in which the amplitudes are very small. An equation including dissipation as well as dispersion is derived whose solutions may possess solitary waves, as in the case of similar equations considered in the literature. These solutions are likely to correspond to the solitary spiral waves observed in experiments.

Convection driven zonal flows and vortices in the major planets.

The dynamical properties of convection in rotating cylindrical annuli and spherical shells are reviewed. Simple theoretical models and experimental simulations of planetary convection through the use of the centrifugal force in the laboratory are emphasized. The model of columnar convection in a cylindrical annulus not only serves as a guide to the dynamical properties of convection in rotating sphere; it also is of interest as a basic physical system that exhibits several dynamical properties in their most simple form. The generation of zonal mean flows is discussed in some detail and examples of recent numerical computations are presented. The exploration of the parameter space for the annulus model is not yet complete and the theoretical exploration of convection in rotating spheres is still in the beginning phase. Quantitative comparisons with the observations of the dynamics of planetary atmospheres will have to await the consideration in the models of the effects of magnetic fields and the deviations from the Boussinesq approximation.

Geometry effects on Rayleigh-Bénard convection in rotating annular layers.

Rayleigh-Bénard convection is investigated in rotating annular cavities at a moderate dimensionless rotation rate Ω=60. The onset of convection is in the form of azimuthal traveling waves that set in at the sidewalls and at values of the Rayleigh number significantly below the value of the onset of convection in an infinitely extended layer. The present study addresses the effects of curvature and confinement on the onset of sidewall convection by using three-dimensional spectral solutions of the Oberbeck-Boussinesq equations. Such solutions demonstrate that the curvature of the outer boundary promotes the onset of the wall mode, while the opposite curvature of the inner boundary tends to delay the onset of the wall mode. An inner sidewall with a radius as low as one tenth of its height is sufficient, however, to support the onset of a sidewall mode. When radial confinement is increased the two independent traveling waves interact and eventually merge to form a nearly steady pattern of convection.

Nonlinear Rayleigh-Taylor instability of rotating inviscid fluids.

It is demonstrated theoretically that the nonlinear stage of the Rayleigh-Taylor instability can be retarded at arbitrary Atwood numbers in a rotating system with the axis of rotation normal to the acceleration of the interface between two uniform inviscid fluids. The Coriolis force provides an effective restoring force on the perturbed interface, and the uniform rotation will always decrease the nonlinear saturation amplitude of the interface at any disturbance wavelength.

Homologous onset of double layer convection.

The onset of convection in two superimposed fluid layers of the same height is considered. It is found that the neutral curve for R(a) for the onset Rayleigh number R in dependence on the wave number a is an invariant of a multidimensional parameter space of property ratios of the system even though the corresponding convection solutions may vary strongly with these property ratios. For each neutral curve R(a) two manifolds of solutions are found one of which can be understood on the basis of symmetry properties of the system, while the other does not exhibit simple symmetry features. In particular the neutral curves R(a) for various single Rayleigh-Bénard convection layers are shown to correspond to two two-dimensional manifolds of solutions. Analytical expressions for the latter are derived in the case of outer stress-free boundary conditions.

Drifting convection cells in rotating fluid layers heated from below.

It is shown that hexagonal convection cells in a rotating horizontal fluid layer heated from below will in general exhibit a drift in contrast to convection rolls except in the case of a vertical axis of rotation. The direction of the drift is prograde (retrograde) for cells with rising (descending) motion in the center of the convection cell. In addition a mean flow generated by convection is derived. An application to solar convection is discussed.

On thermal convection in slowly rotating systems.

The dynamical properties of convection patterns in a fluid layer heated from below and rotating slowly about a horizontal axis are reviewed. Applications to the equatorial regions of planetary and stellar atmospheres are emphasized. Attention is drawn to the wavelike drift of hexagonal convection cells in the azimuthal direction and to the mean flow generated by all convection patterns except for rolls aligned with the axis of rotation.

New patterns of centrifugally driven thermal convection.

An experimental study is described of convection driven by thermal buoyancy in the annular gap between two corotating coaxial cylinders, heated from the outside and cooled from the inside. Steady convection patterns of the hexaroll and of the knot type are observed in the case of high Prandtl number fluids, for which the Coriolis force is sufficiently small. Oblique rolls and phase turbulence in the form of irregular patterns of convection can also be observed in wide regions of the parameter space.

Large scale structures in Rayleigh-Bénard convection at high Rayleigh numbers.

Direct numerical simulations of Rayleigh-Bénard convection in a plane layer with periodic boundary conditions at Rayleigh numbers up to 10(7) show that flow structures can be objectively classified as large or small scale structures because of a gap in spatial spectra. The typical size of the large scale structures does not always vary monotonically as a function of the Rayleigh number but broadly increases with increasing Rayleigh number. A mean flow (whose average over horizontal planes differs from zero) is also excited but is weak in comparison with the large scale structures. The large scale circulation observed in experiments should therefore be a manifestation of the large scale structures identified here.

Convection-driven quadrupolar dynamos in rotating spherical shells.

It is found that for Taylor numbers of the order 10(8) quadrupolar dynamos aligned with the axis of rotation are preferred in comparison with dipolar dynamos. This preference holds for a range of Prandtl numbers P and magnetic Prandtl numbers P(m) in the neighborhood of unity. The main time-dependent feature of the quadrupolar dynamos are polward traveling waves.