Joseph Werne

Rapidly rotating turbulent Rayleigh-Bénard convection

Turbulent Boussinesq convection under the influence of rapid rotation (i.e. with comparable characteristic rotation and convection timescales) is studied. The transition to turbulence proceeds through a relatively simple bifurcation sequence, starting with unstable convection rolls at moderate Rayleigh ( Ra ) and Taylor numbers ( Ta ) and culminating in a state dominated by coherent plume structures at high Ra and Ta . Like non-rotating turbulent convection, the rapidly rotating state exhibits a simple power-law dependence on Ra for all statistical properties of the flow. When the fluid layer is bounded by no-slip surfaces, the convective heat transport ( Nu − 1, where Nu is the Nusselt number) exhibits scaling with Ra 2/7 similar to non-rotating laboratory experiments. When the boundaries are stress free, the heat transport obeys ‘classical’ scaling ( Ra 1/3 ) for a limited range in Ra , then appears to undergo a transition to a different law at Ra ≈ 4 × 10 7 . Important dynamical differences between rotating and non-rotating convection are observed: aside from the (expected) differences in the boundary layers due to Ekman pumping effects, angular momentum conservation forces all plume structures created at flow-convergent sites of the heated and cooled boundaries to spin-up cyclonically; the resulting plume/cyclones undergo strong vortex-vortex interactions which dramatically alter the mean state of the flow and result in a finite background temperature gradient as Ra → ∞, holding Ra / Ta fixed.

Plumes in rotating convection. Part 1. Ensemble statistics and dynamical balances

Atmospheric and oceanic convection often occurs over areas occupied by many localized circulation elements known as plumes . The convective transports therefore may depend not only on the individual elements, but also on the interactions between plumes and the turbulent environment created by other plumes. However, many attempts to understand these plumes focus on individual isolated elements, and the behaviour of an ensemble is not understood. Geophysical convection may be influenced by rotation when the transit time of a convecting element is long compared to an inertial period (for example in deep oceanic convection). Much recent attention has been given to the effect of rotation on individual plumes, but the role of rotation in modifying the behaviour of an ensemble is not fully understood. Here we examine the behaviour of plumes within an ensemble, both with and without rotation, to identify the influence of rotation on ensemble plume dynamics. We identify the coherent structures (plumes) present in numerical solutions of turbulent Rayleigh–Benard convection, a canonical example of a turbulent plume ensemble. We use a conditional sampling compositing technique to extract the typical structure in both non-rotating and rotating solutions. The dynamical balances of these composite plumes are evaluated and compared with entraining plume models. We find many differences between non-rotating and rotating plumes in their transports of mass, buoyancy and momentum. As shown in previous studies, the expansion of the turbulent plume by entrainment of exterior fluid is suppressed by strong rotation. Our most significant new result is quantification of the continuous mixing between the plume and ambient fluid which occurs at high rotation without any net changes in plume volume. This mixing is generated by the plume–plume interactions and acts to reduce the buoyancy anomaly of the plume. By contrast, in the non-rotating case, no such loss of buoyancy by mixing occurs. As a result, the total buoyancy transport by upwardly moving plumes diminishes across the layer in the rotating case, while remaining approximately constant in the non-rotating case. At high values of rotation, the net vertical acceleration is considerably reduced compared to the non-rotating case due to loss of momentum through entrainment and mixing and a decelerating pressure gradient which partially balances the buoyancy-driven acceleration of plumes. As a result of the dilution of buoyancy, the pressure-gradient deceleration and the loss of momentum due to mixing with the environment in the rotating solutions, the conversion of potential energy to kinetic energy is significantly less than that of non-rotating plumes. The combination of efficient lateral mixing and slow vertical movement by the plumes accounts for the unstable mean temperature gradient that occurs in rotating Rayleigh–Benard convection, while the less penetrative convection found at low Rossby number is a consequence of the reduced kinetic energy transport. Within the ensemble of plumes identified by the conditional sampling algorithm, distributions of vertical velocity, buoyancy and vorticity mimic those of the volume as a whole. Plumes cover a small fraction of the total area, yet account for most of the vertical heat flux.

Generalized quasi-geostrophy for spatially anisotropic rotationally constrained flows

Closed reduced equations analogous to the quasi-geostrophic equations are derived in the extratropics for small Rossby numbers and vertical scales that are comparable to or much larger than horizontal scales. On these scales, significant vertical motions are permitted and found to couple to balanced geostrophic dynamics. In the equatorial regions, similar reduced equations are derived for meridional scales much larger than the vertical and zonal scales. These equations are derived by a systematic exploration of different aspect ratios, and Froude and buoyancy numbers, and offer advantages similar to the standard quasi-geostrophic equations for studies of smaller-scale processes and/or of the equatorial regions.

Penetrative convection in rapidly rotating flows: preliminary results from numerical simulation

Turbulent convection forced by a surface heat flux into a stably stratified region is a feature of both the atmospheric and oceanic planetary boundary layers. Of particular interest is the interface between the convective layer and the stable stratification, where the entrainment of fluid into the convective layer by penetrating plumes may lead to a reverse buoyancy flux, and an enhancement of the stable stratification. Whereas in the atmosphere the influence of rotation on this penetrative convection is negligible, oceanic convection may be subjected to lower Rossby numbers and hence greater rotational influence. To isolate the effects of rotation, we present three numerical solutions for turbulent penetrative convection, characterised by different rotation rates, with all other parameters being held constant. Our results indicate that at lower Rossby numbers the lateral scale of the plumes is reduced, whereas the vertical vorticity of the plumes is much enhanced. Vertical transports of buoyancy and kinetic energy across the convective layer are reduced, leading to less efficient penetration at the interface with the stratified layer, and hence less reverse buoyancy flux in this region.

Plume dynamics in quasi-2D turbulent convection

We have studied turbulent convection in a vertical thin (Hele-Shaw) cell at very high Rayleigh numbers (up to 7x10(4) times the value for convective onset) through experiment, simulation, and analysis. Experimentally, convection is driven by an imposed concentration gradient in an isothermal cell. Model equations treat the fields in two dimensions, with the reduced dimension exerting its influence through a linear wall friction. Linear stability analysis of these equations demonstrates that as the thickness of the cell tends to zero, the critical Rayleigh number and wave number for convective onset do not depend on the velocity conditions at the top and bottom boundaries (i.e., no-slip or stress-free). At finite cell thickness delta, however, solutions with different boundary conditions behave differently. We simulate the model equations numerically for both types of boundary conditions. Time sequences of the full concentration fields from experiment and simulation display a large number of solutal plumes that are born in thin concentration boundary layers, merge to form vertical channels, and sometimes split at their tips via a Rayleigh-Taylor instability. Power spectra of the concentration field reveal scaling regions with slopes that depend on the Rayleigh number. We examine the scaling of nondimensional heat flux (the Nusselt number, Nu) and rms vertical velocity (the Peclet number, Pe) with the Rayleigh number (Ra(*)) for the simulations. Both no-slip and stress-free solutions exhibit the scaling NuRa(*) approximately Pe(2) that we develop from simple arguments involving dynamics in the interior, away from cell boundaries. In addition, for stress-free solutions a second relation, Nu approximately nPe, is dictated by stagnation-point flows occurring at the horizontal boundaries; n is the number of plumes per unit length. No-slip solutions exhibit no such organization of the boundary flow and the results appear to agree with Priestleys prediction of Nu approximately Ra(1/3). (c) 1997 American Institute of Physics.

Vertical transport by convection plumes: modification by rotation

Abstract Convection, generated by destabilising buoyancy forcing, is a significant source of vertical mixing in the ocean and atmosphere. Ensembles of convective elements or plumes transport buoyancy and tracers across the convective layer. Here we examine numerical simulations of turbulent convection to identify the modifications to the vertical transports induced by strong rotation (typical of deep ocean convection). We extract the typical convection plumes from the numerical simulations using a conditional sampling compositing technique. The plume budgets of mass, heat and momentum are then compared with an entraining plume model. In the presence of strong rotation the vertical transports are significantly reduced by vigorous lateral mixing, which dilutes plume anomalies. This mixing is generated by the energetic cyclonic vortices associated with the individual plumes and the interaction between the many vortices in the plume ensemble.

Dynamics and scaling in quasi-two-dimensional turbulent convection

High Rayleigh number (4 x 104-3 × 107) quasi-two-dimensional convection is studied experimentally, numerically, and theoretically. The fluid is contained in an isothermal Hele-Shaw cell, which enforces the two dimensionality, and the convection is driven by an externally imposed vertical concentration gradient. The full two dimensional concentration field is imaged experimentally and is sampled at regular intervals spanning between five and fifty convective times (the time it takes a plume to cross the cell). The most prominent and dynamically significant features of the flow are solutal plumes which form within the concentration boundary layer. Tip splitting of plumes leading to smaller length scales and merging of plumes leading to larger length scales are observed. A model of Hele-Shaw convection, which includes a friction term due to the boundaries parallel to the velocity field, is simulated. The evolution of the pattern and dynamics of the concentration field, as well as time averaged spatial spectra, are obtained from the experiment and simulation. Results for exponents which describe how the Nusselt number and Reynolds number scale with Rayleigh number are obtained theoretically and compared with the values from simulation.

The Effect of Rotation on Convective Overshoot

In this note we discuss some preliminary results from our 3D numerical simulations of incompressible penetrative convection in the presence of rotation. Though these simulations pertain to a particular case of a deepening mixed layer with overshoot, we believe the results to have significance in the solar scenario. In particular, we conjecture that rotation has a constraining effect on convective overshoot.

A Reduced Description of Rapidly Rotating Turbulent Convection

The tendency towards two-dimensionality in rapidly rotating flows is described by the Taylor-Proudman theorem. In applications strict two-dimensionality is usually broken by the presence of boundary and/or thermal forcing, or by initial conditions. Both types of forcing are present in thermal convection in a rapidly rotating horizontal layer, described by the equations n n