Why is fluid dynamics interesting and challenging?
The dynamics is just Newton's laws of motion but (a) needs an entirely different treatment from the familiar particle dynamics, and (b) needs two psychologically very different descriptive frameworks ('Eulerian' and 'Lagrangian'). The Lagrangian description is relevant to the transport of pollutants and other chemical constituents by fluid motion. It turns out that we also need both descriptions for a full understanding of the dynamics.
The exquisitely surprising character of fluid motion is demonstrated by a few examples, including the Plumb-McEwan experiment, a laboratory analogue of the celebrated quasi-biennial oscillation of the east-west winds in the equatorial stratosphere. The latter is a remarkable example of long-term predictability, and order-out-of-chaos, in the Earth's atmosphere.
Even irrotational flow can be interesting. What is it, and why is it peculiar? The answer can be clearly given with minimal mathematics. What would it be like to stir 'irrotational coffee'? A hands-on computer demonstration shows this. Irrotational flow is interesting not least because of its conspicuously unreal character in most situations. Birds and aircraft could not become airborne if the three-dimensional flow around them were irrotational. The demonstration conveys vividly, by implication, the crucial importance of vorticity -- a measure of the departure from irrotationality in real fluid motion including, for instance, motion resulting in pollutant dispersion.
ORDERS OF MAGNITUDE AND SCALING
The importance of identifying and estimating dimensionless parameters is illustrated. For example, low Mach number implies approximately incompressible behaviour. A live demonstration of spin-down illustrates the difference between the spin-down time and the characteristic diffusion time: a puzzle!
What is ordinary (Fickian) diffusion? The most basic answer -- pointing to the most far-reaching generalizations, including the phenomena, plural, called turbulent diffusion -- is that ordinary diffusion results from a random walk in small steps. It requires scale separation; 'small' means small in comparison with the other spatial scales of the problem. This is made conspicuous by considering the simplest numerical difference scheme for the diffusion equation. Turbulent diffusion can have much larger steps.
Why is the simple diffusion-time estimate wrong? A careful qualitative discussion is given to prepare for the quantitative laboratory experiment.
ADVECTION AND THE EXPONENTIALLY FAST SHRINKAGE OF SCALES
We zoom in on a material point and look at the relative motion in its neighbourhood. The random-straining model, one of the few robust ideas from turbulence and chaotic advection theory, is illustrated. This sets the single most serious, ubiquitous, and inescapable limitation on the numerical modelling of real fluid flows.
The elementary one-dimensional solution is given. A computer demonstration shows robustness -- any initial distribution quickly relaxes to the same fundamental state. This is the key to understanding another large range of basic phenomena including boundary layers, Kolmogorov turbulent microscales, and, quite generally, how molecules are brought into thermal and chemical contact, i.e. what 'mixing' is.
TWO-DIMENSIONAL VORTEX DYNAMICS
This too is a paradigm of central importance, pointing to far-reaching generalizations. The main reason is that an enormous range of high-Reynolds-number, low-Mach-number problems in fluid dynamics have more or less the same generic structure as two-dimensional vortex dynamics. That is, they are all governed entirely by the evolution either of the vorticity or, in the case of stratified systems like the oceans and atmosphere, by an equally fundamental quantity known as potential vorticity.
The evolution may be a purely advective evolution, or an advective-diffusive or other modified-advective evolution, depending on the physical processes acting. An 'invertibility principle' makes explicit the idea that once you know the potential vorticity distribution you can deduce everything else by 'inversion'.
The foregoing is central to an understanding, for instance, of atmospheric cyclones and anticyclones, and of the transport of chemicals by ocean eddies like the celebrated Atlantic 'Meddies' whereby water from the Mediterranean Sea is transported over long distances within the Atlantic Ocean.
SIMPLE VORTEX INTERACTIONS
Quick description of point-vortex interactions; experimental vortex-pair demonstration on the overhead projector. Computer demonstration to verify the basic vortex-interaction phenomena asserted in the quick description.
VORTEX MERGING AND VORTEX EROSION/STRIPPING
These are the basic two-dimensional vortex interactions outside the scope of the point-vortex model. Merging exemplifies, incidentally, the possibility of 'up-gradient transport' of vorticity and potential vorticity.
This is another basic and ubiquitously important mechanism, whose simplest illustration can again be found within two-dimensional vortex dynamics, carrying insight into more general cases. The restoring mechanism or 'quasi-elasticity' to which Rossby waves owe their existence can be most simply and clearly understood through the idea of potential-vorticity inversion. The restoring mechanism depends on a kind of 'sideways stratification' associated with sideways gradients of potential vorticity.
This quasi-elasticity is fundamental to very many large-scale atmospheric dynamical processes. In the context of the Antarctic ozone hole, for instance, it is a key factor, along with horizontal shear, in inhibiting the advective exchange of chemicals between the ozone hole and its surroundings. Rossby-wave elasticity is equally fundamental to understanding the mechanism of ordinary shear instability and its large-scale counterparts, such as the 'baroclinic instability' underlying certain weather developments and modes of eddy formation.
In the two-dimensional case this is the simplest paradigm for a large range of environmental fluid dynamical processes. Emphasis is on a qualitative description exposing the role of Rossby-wave elasticity in shear instabilities. Naturally-occurring examples are shown, including examples from satellite pictures, and the small-scale Kelvin-Helmholtz billows that any keen observer can see in certain cloud patterns.