Abstract:

A gravity current appears when fluid of one density, $\roc$, propagates into
another fluid of a different density, $\roa$, and the motion is mainly in
the horizontal direction.  A gravity current is formed when we open the door 
of a heated house and cold air from outside flows over the floor into the less
dense warm air inside.  A gravity current is formed when we pour honey on a
pancake and we let it spread out on its own. Gravity currents originate in 
many natural and industrial circumstances and are present in the atmosphere, 
lakes and oceans as winds, cold or warm streams or currents, volcanic clouds,
polluted discharges, etc. A gravity current which propagates inside a
stratified fluid (rather than along a boundary) is called ``intrusion.''

Simple qualitative consideration and observations indicate that the gravity
current is a very complex, multi-faced, and parameter-rich physical
manifestation. Nevertheless, the gravity current also turns out to be a
modeling-friendly phenomenon. Indeed, visualizations of the real flow field
reveal an extremely complicated three-dimensional motion, with an irregular
interface, billows, mixing, and instabilities. The accurate numerical
simulation of this flow from the full set of governing equations (the
Navier-Stokes system) requires weeks of number-crunching on powerful
computer arrays.  On the other hand, there are ``mathematical models'' based 
on a long line of assumptions such as hydrostatic pressure, sharp interface, 
Boussinesq system, thin layer, idealized release conditions. This simplified 
set of equations enables us to determine the behavior of the averaged variables
entirely from analytical considerations and/or numerical solutions that 
require insignificant CPU time.

The lecture gives a brief presentation of some typical models and solutions.
We see that: (a)  Qualitatively, the simplified theory is able to provide
the governing dimensionless parameters and the salient features of the
various flow regimes; and (b) Quantitatively, the simple models predict
velocities of propagation which agree with experiments and full Navier-Stokes 
simulations within a few percent, sometimes within the range of the 
experimental errors. The fact that such simple models give useful results merits 
attention: is the predictive power of the model a consequence of postulated 
mathematical simplicity, or is it rather a result of well-selected physical 
components?  Some tentative answers to this universally relevant question, 
as provided by the test-case of gravity currents and intrusions, are discussed.


References

[1] M. Ungarish.  ``An Introduction to Gravity Currents and Intrusions.''
Chapman & Hall/CRC press, Boca Raton London New York, 2009.