Multiple solutions are common in fluid mechanics, from either the quadratic nature of the Euler or Navier-Stokes equations, or nonlinear terms associated with convection, free interfaces, or Non Newtonian effects. But problems where many solutions are steady and stable for identical boundary conditions are relatively rare. Many recent examples have been found in oceanic climate problems in which two states are both steady and completely stable. A number have been reproduced in the laboratory. If convection is driven by both heat and salinity differences, for instance, a convection cell is readily constructed that is driven principally by either temperature or salinity difference for the same forcing conditions. If convective motion is opposed by surface stress, an analog device can be made using pressure difference instead of stress, and air instead of heat, which possesses two stable, steady states. Multiple state effects are found in by a broad class of circulation and estuary models such as a coastal current of fresh water in salty rotating water and convection cooled from above when the water is salt stratified. The simplified equations for these types of examples will be discussed, and their probable occurrence in other areas of fluid mechanics, such as ventilation, combustion, and geophysics will be mentioned.
[Ch.02] Spin-up of a Rayleigh-Bénard cell.
Peter Vorobieff, Robert Ecke (Los Alamos National Laboratory)
Flow patterns forming during the spin-up of a cylindrical Rayleigh-Bénard convection cell about its vertical axis are studied via particle image velocimetry (PIV) and thermochromic liquid crystal (TLC) temperature field visualization. Initially the cell is at rest, and heat is transported from the lower to the upper boundary by thermal plumes. In the range of Rayleigh numbers R between 5\times10^7 and 5\times10^8 and for spin-up to dimensionless rotation rates Ømega up to 8\times10^4, we observe formation of ring-shaped regions of downwelling flow upon spin-up. The number of rings grows with the final rotation rate, while the characteristic radii of each ring structure depend on R and Ømega rather weakly. The rings are characterized by a drop in temperature and azimuthal velocity, the latter leading to shear instability and roll-up of Kelvin-Helmholz vortices eventually destroying the regular ring pattern and completing the transition to the stationary rotating convection flow regime.
[Ch.03] Turbulence in rotating Rayleigh-Bénard convection
Yuanming Liu, Peter Vorobieff, Robert Ecke (Los Alamos National Laboratory)
We report experimental results for turbulent rotating Rayleigh-Bénard convection in water with Prandtl number \sigma \approx 6. Heat transport measurements and flow visualization for Rayleigh number, 5\times 10^7\leq R\leq 5\times 10^8 and dimensionless rotation rates, 0 < Ømega < 8\times 10^4 are compared to obtain a consistent picture of turbulent boundary layer stability and the mechanisms for the enhancement of heat transport by rotation. The flow diagnostics consist of thermochromic-liquid-crystal visualization for the temperature field and PIV for the velocity field. A scanning PIV system allows us to measure the horizontal velocity field over a vertical distance of 3 cm below the upper cold boundary. The structure of the vortical plumes erupting from the boundary layer was determined to be cyclonic with Ekman suction at the core and Ekman pumping along a ring. No evidence for large scale circulation was found for strongly rotating convection.
\hfil Present Address: Jet Propulsion Lab, Pasadena, CA
[Ch.04] One-Dimensional Stochastic Simulation of Staircase Development in Turbulent Double-Diffusive Convection.
Alan R. Kerstein (Sandia National Laboratories, Livermore, CA)
The thermohaline staircase, observed in laboratory and oceanographic studies of thermally-driven convective turbulence in salt-stratified media, is simulated using a Monte Carlo model, One-Dimensional Turbulence (ODT). ODT is a fully resolved 1D simulation in which molecular transport (viscosity, species diffusion) is implemented deterministically, punctuated by rearrangement events (mappings) representing turbulent eddies. The stochastic event sequence is governed by a rate expression reflecting the shear kinetic energy and gravitational potential energy available to drive eddies. This formulation, previously applied to various shear-driven and buoyancy-driven flows, is found to predict (with no empirical input) the experimentally observed regimes of heat:salt flux-ratio dependence on the heat:salt buoyancy stability ratio across a double-diffusive interface. Other familiar features of staircase development and structure are reproduced.
[Ch.05] Influence of non-Boussinesq effects on patterns in salt-finger convection
Yuriko Renardy, Michael Renardy (Dept. of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123)
The influence of non-Boussinesq effects on the quadratic coefficient of the amplitude equations for patterns that are doubly periodic with respect to the hexagonal lattice is discussed. The vertical length scale is assumed to be long compared with the horizontal. Within this approximation, we consider a variety of non-Boussinesq effects including the dependence of the fluid properties on temperature and salinity and the nonlinearity of the basic temperature and salinity profiles. We include some numerical calculations for the latter case. Oceanic data show a pronounced nonlinearity of the basic temperature profile. Our calculations, however, show this effect to be small, and, moreover, the dominant contribution in this case is not covered by our asymptotic ``thin finger" analysis. The reasons for this are discussed. A nonlinear salinity profile, on the other hand, produces a much stronger effect, and our analysis seems to be applicable to this case. Funded by NSF-CTS9612308, ONR N00014-92-J-1664, NSF-DMS9622735.
[Ch.06] The coupling between penetrative convection and internal waves
M. Michaelian, T. Maxworthy, L. G. Redekopp (University of Southern California)
Past studies (Townsend 1963, Deardorff 1968) have indicated an important connection between penetrative convection and internal wave motion in an adjacent stratified layer. The rationale behind the present work is to shed new light on the physical mechanisms which govern the transfer of energy between convection and internal waves. Experiments were performed to quantify the existence of a long wave mode in the stratified layer and to illuminate its effect on convective motion. It is hypothesized that individual convective plumes transfer some of their energy to short wavelength internal waves, which can then interact non-linearly to generate long wave motion in the stratified layer. The long wave motion, in turn, is thought to modulate the convective motions, thus closing the feedback mechanism for the energy of the system. Results using a temperature stratified experimental apparatus have shown the existence of this long wave mode, and the effects of the stratification as well as the apparatus length, on wavenumber/frequency selection.
[Ch.07] Rotational effects on localized buoyancy driven convection
H. Nassef, T. Maxworthy (University of Southern California)
A circular, surface-mounted salt-water-source issues fluid into a homogeneous ambient fluid in order to simulate a natural local buoyancy source. DPIV images are taken to study the resulting flow with or without background rotation. Side view images show a stable two layer flow when no background rotation is present. Density data taken beneath the source show the significance of the buoyancy time scale defined by the water column height divided by a characteristic fall velocity. Once background rotation is added, a frontal instability immediately forms on the outflowing gravity current. Top view DPIV images show that this instability results in uniformly spaced vortices around the source. Each
vortex is shown to be a Taylor-like column moving under the influence of the induced velocity fields of its neighbors. When a flat bottom is used, the vortices are observed to wander about freely in the tank. However, when a sloping bottom is present, the vortices are trapped around the source. Power law dependencies of the length and vorticity scales of the instability vortices on the dimensionless parameters are presented.
[Ch.08] A Thermal Wind in Pancake Vortices in a Stably Stratified Flow
D. M. Kidani, P. S. Marcus (University of California, Berkeley)
In rapidly rotating flows, vortices form Taylor columns aligned with the vertical rotation axis \hatz. When the flow is thermally stratified the Taylor-Proudman theorem is modified so that \partial \vecv/\partial z is no longer zero but proportional to the horizontal temperature gradient \nabla_H T. This is the Thermal Wind Equation (TWE). With a proper \nabla_H T, pancake-shaped rather than columnar vortices form. Planetary vortices are pancake-like. An isothermal pancake vortex will not satisfy the TWE and creates a secondary flow similar to Ekman pumping. This flow creates a \nabla_H T in accord with the TWE. However, the vortex decays slowly, and for small Prandtl number, with the thermal rather than viscous, timescale: thermal conductivity acts to destroy \nabla_H T, but it is maintained by the secondary flow (which is forced by the imbalance in the TWE if \nabla_H T decays). Thus the thermal timescale regulates the secondary flow. Kinetic energy flows from the primary vortex into the secondary flow which transforms it to potential energy (as it advects heat against the stable stratification). The potential energy (stored as cold above hot fluid) is then thermally dissipated.
[Ch.09] Persistence of Mean Flows in Turbulent Non-Penetrative Thermal Convection
Ajay K. Prasad (Department of Mechanical Engineering, University of Delaware)
Numerical simulations have been performed of turbulent, non-penetrative thermal convection at Ra = 2 \times 10^7, in which the lower boundary condition is one of constant heat flux, and the upper boundary is insulated. While the bulk temperature of the fluid increases linearly with time, the spatially-averaged temperature profile exhibits a time-invariant behavior along the vertical coordinate. The numerical study has replicated earlier PIV measurements of the same flow, in a 76.2~mm thick layer of water at an aspect ratio of 6.6. Ongoing simulations are focused on the phenomenon of ``mean-flow'', i.e., a global flow whose length scale is on the order of the horizontal dimension of the flow domain, which has been reported in a number of experiments. We have artificially created such a mean-flow in our simulations by introducing a perturbation in the established flow; the perturbation comprises of an excess of heat flux, for a specified duration, along a narrow strip on the lower boundary, adjacent to one side wall. The magnitude and self-sustaining ability of the resulting mean-flow will be determined as a function of the excess heat-flux, its duration, and the container aspect ratio.
[Ch.10] Penetrative convection and the breakup of a magnetic layer
Nicholas Brummell, Thomas Clune, Steven Tobias, Juri Toomre (JILA, University of Colorado, Boulder)
One of the main reasons for studying penetrative convection is more accurately to mimic the base of the solar convection zone. The stable layers below the solar convection zone are widely touted as the seat of the solar dynamo and therefore believed to contain strong magnetic fields and. As a first attempt to understand the interaction of magnetic fields and turbulent pentrative convection, we present the results of some initial magnetoconvection simulations. A slab of magnetic field is placed below the convection zone in the stable region. When the field is weak and positioned within range of the penetrative plumes, magnetic energy is wrapped up by the intense vortex action of the coherent structures and carried into the convection as tubes. If the field is strong, the initial slab may be subject to magnetic buoyancy instabilities forming structures immediately. The subsequent interaction with convection is complicated: depending on the choice of parameters the strong field may dominate the convection, or the convection may shred the magnetic structures. Whether or not magnetic field survives amongst the turbulent convection or is expelled back into the lower stable layers will also provide insight into theoretical solar dynamo mechanisms. This transport of magnetic field by the turbulent convection may act so as to build up a strong layer of magnetic field in the solar overshoot region.