GFD-1 LAB #6 DIFFUSION, STIRRING, MIXING, CONVECTION
P.B. Rhines, E.G. Lindahl Feb. 16 2001

This is the essence of an 'advection-diffusion' problem: fluid circulations ventilate the boundaries by steepening the temperature gradients near them. You blow on your soup to cool it. Kinematic formulations of advection-diffusion are very useful in this context: for example given a prescribed 2D flow with streamfunction = xz (hyperbolic streamlines), the equation
dT/dt + u*grad T = K Laplacian T
has closed-form solutions for an initial temperature field To = sin(kx+mz). The 'stripes' of hot and cold are compressed along the outflow axis (the x-axis) and extended along it. It is easy to solve for particle trajectories in this pure-strain flow field, and you will find that pairs of particles separate along the outflow axis exponentially with respect to time. As the temperature stripes squeeze together eventually diffusion and advection come into balance and the wavelength reaches a steady asymptote. Using Fourier analysis this single sin-wave of temperature can be combined into a single stripe of, say, hot fluid, and one can see its diffusive heat transport accelerated as its width is compressed by the flow, until a steady advection/diffusion balance is achieved.

Below are 3D numerical simulations of convection from uniformly heated lower boundary/cooled upper boundary, from ENSEEIHT (Ecole Nationale Superieure d'Electrotechnique, d'Electronique, d'Informatique, d'hydraulique,et des Telecommunications)), France for Rayleigh numbers of 10^6 and 10^7.

We demonstrated during the lab that the curve Nu as a function of Ra would be a 1/3 power law if the actual heat flux were to be independent of layer thickness, h. This is motivated by the idea of thin boundary layers ejecting thermal plumes which pass freely through the nearly neutrally buoyant interior. In fact there is a large literature on this, some of it arguing for a transition to a 2/7 power of Ra, others suggesting transition to a 1/2 power-law: though recent experiments suggest the robustness of the 1/3 power law (Niemela et al, Nature (404) 1999, p837 explore 11 orders of magnitude of Ra, from 10^6 to 10^17, finding a single power law, Nu ~ Ra^0.31).

The Rayleigh numbers reached in geophysical flows are enormous. The diffusive boundary layers become vanishingly thin, and the complexity of the boundary becomes important. But for all such flows at large Ra, convection involves strong, discrete plumes as well as larger scale cells (and very often rolls aligned with the boundary layer shear). An isosurface of constant temperature from the above numerical experiments shows these plumes:

There are many important variants on the basic Rayleigh-Benard problem: Coriolis effects, topographic effects, mean horizontal flow with vertical shear (as in the atmosphere) which picks out roll-cells with axes along the flow. One interesting such varient is to convect into a fluid with stable stratification, as occurs daily in the upper ocean, and lower atmosphere. Turbulent plumes can be 'penetrative', that is poke their heads into the stable layers, exciting gravity waves and producing a small region of reversed vertical heat flux, as their kinetic energy does some mixing. This situtation is shown in the numerics just below.


A major change in convection occurs when the boundary temperature or heat flux varies horizontally. This can be gradual, say a linear change from heating to cooling along the lower boundary, or it can be abrupt, with a concentrated heat source. It can 'atmospheric' as above or 'oceanic' with the boundary temperature varying along the i>upper surface of the fluid.
In the images just below we have a 'cold polar' region to the right where fluid is made more dense (here by salt rather than cold). As time proceeds, we change colors of the dense, sinking fluid at regular intervals. You see an overturning circulation driven by the analog of heating and cooling at the same level. The sinking branch is very narrow and turbulent while the rest of the fluid rises almost everwhere. This is Henry Stommel's famous GFD result predicting the 'narrowness of sinking regions in the ocean'. Similarly, the problem turned upsidedown shows that tropical heating should lead to a rising Hadley cell branch which is quite narrow, with subsidence occurring much more widely.
The narrow plume entrains a great quantity of fluid, so that the overturning circulation is much greater than the flow in the upper boundary layer would suggest. This 'entrainment amplifier' is seen in many flows in GFD. The lower images show fluorescein dye streaks, initially vertical, indicating the profile of horizontal velocity.
When the boundary conditions are non-uniform in this way, the fluid becomes , rather than the unstable or neutral stratification of the Rayleigh-Benard problem. We build the stable layering of the ocean and atmosphere with buoyancy forcing that varies with latitude. This geography of buoyancy forcing ('thermography') is essential to understanding of the general circulation, whether it be the standing waves of the atmosphere, the storm tracks of winter, or the sinking and meridional overturning of the world ocean beneath. These images were produced in the UW GFD laboratory for a BBC-television program on radical climate change ('The Big Chill', 1999).


Diffusion in geophysical fluids is greatly enhanced by turbulence. Here a very fine fresh-water jet is injected into the upper right corner of a thin cell of saline water. The momentum of the jet would drive a circulation downward/rightward but in fact it goes the other way. The buoyancy flux overcomes the momentum flux, and the large volume of low salinity water formed, moves as a gravity current to the left, and deep fluid moves upward to feed the mixing zone. This is how an ocean estuary works, having a large overturning circuation driven by river buoyancy (and vastly greater flux than the river water inflow itself).
A close in view of a jet, below, injected downward into the fluid, sweeping more fluid into motion as it dilutes itself: the volume flux increases rapidly, as the concentration of any tracer carried by the jet decreases.


Stable stratification can orient convection to be 'tilted', 'slant-wise' or virtually horizontal. Here a stable salinity gradient is heated from the side (at the end-wall of the container). It is an example of 'double diffusion' where the greater diffusivity of heat means that warmed fluid parcels cannot rise indefinitely far up the wall...they lose their buoyancy. Layers form, each one a miniature meridional circulation, with fluid moving outward from the wall at the top of each layer, and back at the bottom. Double diffusion has two essential aspects. Firstly, in the interior, parcels of fluid lose their temperature anomalies more quickly than they lose their salinity anomalies. This leads to salt-finger convection and other plumes like 'smoke fingers' that fall out of a layer of warm smoke hanging in the air of a bar. Secondly, the differing boundary conditions for heat and freshwater (or salt) in the ocean lead to convective motions which are not dependent on diffusivity differences. This is a subtle aspect of climate dynamics, where fresh-water layers from the Arctic and subArctic float on top of the more saline ocean: upper-ocean temperature anomalies have smaller persistence (because of active feedback with the atmosphere) than upper-ocean salinity anomalies, and this leads to interesting and relevant circulations.





Shadowgraph view of the density field in the double-diffusive layers.