ALBERT

Description: Dissipative heating is produced by irreversible processes, such as viscous or ohmic heating, in a convecting fluid; its importance depends on the ratio d/HT of the depth of the convecting region to the temperature scale height. Integrating the entropy equation for steady flow yields an upper bound to the total rate of dissipative heating in a convecting layer. For liquids there is a regime in which the ratio of dissipative heating to the convected heat flux is approximately equal to c(d/HT), where the constant c is independent of the Rayleigh number. This result is confirmed by numerical experiments using the Boussinesq approximation, which is valid only if d/HT is small. For deep layers the dissipative heating rate may be much greater than the convected heat flux. If the earth's magnetic field is maintained by a convectively driven dynamo, ohmic losses are limited to 5% of the convected flux emerging from the core. In the earth's mantle viscous heating may be important locally beneath ridges and behind island arcs. © 1975, Cambridge University Press. All rights reserved.

Description: Plate tectonics provides a remarkably accurate kinematic description of the motion of the earth's crust but a fully dynamical theory requires an understanding of convection in the mantle. Thus the properties of plates and of the mantle must be related to a systematic study of convection. This paper reviews both the geophysical information and the fluid dynamics of convection in a Boussinesq fluid of infinite Prandtl number. Numerical experiments have been carried out on several simple two-dimensional models, in which convection is driven by imposed horizontal temperature gradients or else by heating either internally or from below. The results are presented and analysed in terms of simple physical models. Although the computations are highly idealized and omit variation of viscosity and other major features of mantle convection, they can be related to geophysical measurements. In particular, the external gravity field depends on changes in surface elevation; this suggests an observational means of investigating convection in the upper mantle. © 1974, Cambridge University Press. All rights reserved.

Description: Two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments. In a layer heated from below steady convection becomes unstable to oscillatory modes caused by the formation of hot or cold blobs in thermal boundary layers. Convection driven by internal heating shows a transition from steady motion through periodic oscillations to a chaotic regime, owing to the formation of cold blobs which plunge downwards and eventually split the roll. The interesting feature of this idealized problem is the interaction between constraints imposed by nonlinear dynamics and the obvious spatial structures associated with the sinking sheets and changes in the preferred cell size. These spatial structures modify the bifurcation patterns that are familiar from transitions to chaos in low-order systems. On the other hand, even large-amplitude disturbances are constrained to show periodic or quasi-periodic behaviour, and the. bifurcation sequences can be followed in considerable detail. There are examples of quasi-periodic behaviour followed by intermittency, of period-doubling cascades and of transitions from quasi-periodicity to chaos, associated with a preference for narrower rolls as the Rayleigh number is increased. © 1988, Cambridge University Press. All rights reserved.

Description: During the postcumulus stage of solidification in layered intrusions, fluid dynamic phenomena play an important role in developing the textural and chemical characteristics of the cumulate rocks. One mechanism of adcumulus growth involves crystallization at the top of the cumulate pile where crystals are in direct contact with the magma reservoir. Convection in the chamber can enable adcumulus growth to occur to form a completely solid contact between cumulate and magma. Another important process may involve compositional convection in which light differentiated melt released by intercumulus crystallization is continually replaced by denser melt from the overlying magma reservoir. This process favours adcumulus growth and can allow adcumulus growth within the pore space of the cumulate pile. Calculations indicate that this process could reduce residual porosities to a few percent in large layered intrusions, but could not form pure monomineralic rocks. Intercumulus melt may also be replaced by more primitive melt during episodes of magma chamber replenishment. Dense magma, emplaced over a cumulate pile containing lower density differentiated melt may sink several metres into the underlying pile in the form of fingers. Reactions between melt and matrix may lead to changes in mineral compositions, mineral textures and whole rock isotope compositions. Another important mechanism for forming adcumulate rocks is compaction, in which the imbalance of the hydrostatic and lithostatic pressures in the cumulate pile causes the crystalline matrix to deform and intercumulus melt to be expelled. For cumulate layers from 10 to 1000 metres in thickness, compaction can reduce porosities to very low values (〈 1%) and form monomineralic rocks. The characteristic time-scale for such compaction is theoretically short compared to the time required to solidify a large layered intrusion. During compaction changes of mineral compositions and texture may occur as moving melts interact with the surrounding matrix. Both compaction and compositional convection can be interrupted by solidification in the pore spaces. Compositional convection will only occur if the Rayleigh number is larger than 40, if the residual melt becomes lower in density, and the convective velocity exceeds the solidification velocity (measured by the rate of crystal accumulation in the chamber). Orthocumulates are thus more likely to form in rapidly cooled intrusions where residual melt is frozen into the pore spaces before it can be expelled by compaction or replaced by convection.