F. H. Busse

Thermal instabilities in rapidly rotating systems

Thermal instabilities of a contained fluid are investigated for a fairly general class of problems in which the dynamics are dominated by the effects of rotation. In systems of constant depth in the direction of the axis of rotation the instability develops when the buoyancy forces suffice to overcome the stabilizing effects of thermal conduction and of viscous dissipation in the Ekman boundary layers. Owing to the Taylor–Proudman theorem, a slight gradient in depth exerts a strongly stabilizing influence. The theory is applied to describe the instability of the ‘lower symmetric regime’ in the rotating annulus experiments at high rotation rates. An example of geophysical relevance is the instability of a self-gravitating, internally heated, rotating fluid sphere. The results of the perturbation theory for this problem agree reasonably well with the results of an extension of the analysis by Roberts (1968).

On the stability of steady finite amplitude convection

The static state of a horizontal layer of fluid heated from below may become unstable. If the layer is infinitely large in horizontal extent, the Boussinesq equations admit many different steady solutions. A systematic method is presented here which yields the finite-amplitude steady solutions by means of successive approximations. It turns out that not every solution of the linear problem is an approximation to the non-linear problem, yet there are still an infinite number of finite amplitude solutions. A similar procedure has been applied to the stability problem for these steady finite amplitude solutions with the result that three-dimensional solutions are unstable but there is a class of two-dimensional flows which are stable. The problem has been treated for both rigid and free boundaries.

Transition to time-dependent convection

Steady solutions in the form of two-dimensional rolls are obtained for convection in a horizontal layer of fluid heated from below as a function of the Rayleigh and Prandtl numbers. Rigid boundaries of infinite heat conductivity are assumed. The stability of the two-dimensional rolls with respect to three-dimensional disturbances is analysed. It is found that convection rolls are unstable for Prandtl numbers less than about 5 with respect to an oscillatory instability investigated earlier by Busse (1972) for the case of free boundaries. Since the instability is caused by the momentum advection terms in the equations of motion the Rayleigh number for the onset of instability increases strongly with Prandtl number. Good agreement with various experimental observations is found.

Instabilities of convection rolls in a high Prandtl number fluid

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967 a ) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

Instabilities of convection rolls in a fluid of moderate Prandtl number

The instabilities of two-dimensional convection rolls in a horizontal fluid layer heated from below are investigated in the case when the Prandtl number is seven or lower. Two new mechanisms of instability are described theoretically as well as experimentally. The knot instability causes the transition to spoke-pattern convection at higher Rayleigh numbers while the skewed varicose instability accomplishes a change to larger horizontal wavelengths of the convection rolls. Both instabilities disappear in the limits of small and large Prandtl number. Although the experimental methods fail in realizing closely the infinitely conducting boundaries assumed in the theory, the observations agree in all qualitative aspects with the theoretical predictions.

A simple model of convection in the Jovian atmosphere

Abstract A theoretical model for the latitudinal structure of alternating zones and belts in the Jovian atmosphere is proposed. The explanation is based on the theory of convection in rapidly rotating spherical fluid shells heated from within. The Boussinesq approximation is used and effects of turbulences are taken into account in terms of a constant eddy viscosity. The main conclusion is that the boundary of the polar region corresponds to the latitude for which the distance from the axis is equal to the radius of Jupiters metallic hydrogen core.

The oscillatory instability of convection rolls in a low Prandtl number fluid

The instability of convection rolls in a fluid layer heated from below is investigated for stress-free boundaries in the limit of small Prandtl number. It is shown that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value. The instability corresponds to the generation of vertical vorticity, a mechanism which is likely to operate in the case of a variety of roll-like motions. In all aspects in which the theory can be related to experiments, reasonable agreement with the observations is found.

Generation of planetary magnetism by convection

Abstract Reliable measurements of the magnetic fields of Jupiter and Mercury have been obtained recently. Convection appears to be the most probable origin of Jovian and Hermaean magnetism as well as of geomagnetism. The similarity of the dynamo mechanism in the electrically conducting core of these planets offers opportunities for comparing different hypotheses and testing theoretical models. It is proposed in this paper that the realized magnetic field reaches a maximum amplitude in accordance with the dynamical constraints of nearly geostrophic motion and the condition for dynamo action.

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.

Patterns of convection in spherical shells

The problem of the pattern of motion realized in a convectively unstable system with spherical symmetry can be considered without reference to the physical details of the system. Since the solution of the linear problem is degenerate because of the spherical homogeneity, the nonlinear terms must be taken into account in order to remove the degeneracy. The solvability condition leads to the selection of patterns distinguished by their symmetries among spherical harmonics of even order. It is shown that the corresponding convective motions set in as subcritical finite-amplitude instabilities.