🚀 go-pugleaf

mroussel@alchemy.chem.utoronto.ca (Marc Roussel) writes:
: >	dX/dt = a(Y-X)
: >	dY/dt = bX-Y-XZ
: >	dZ/dt = XY - cZ
: >
: >				where a = 10
: >				      c = 8/3
: >				      b = control parameter (try 28)
: 
:      These are indeed the Lorenz equations (as they are now known) but
: these aren't the original weather simulation equations where Lorenz
: noticed sensitive dependence on initial conditions.  Lorenz actually
: started out with some specialized version of the Navier-Stokes equation
: (a PDE) and noticed sensitive dependence in these equations under the conditions
: of his weather simulation.  The Lorenz ODEs shown above are a truncation
: of the Navier-Stokes equation for the special case of (I think)
: Rayleigh-Benard convection under certain fairly restrictive conditions.

There is a physical system for which the Lorenz equations are a
"not incredibly bad" approximation (to use them for simple R-B convection is a 
"well that's really pretty cheezy" approximation).

It's called a thermosyphon, and it actually gets used for some engineering
applications but I don't know how really.

Think of a loop of pipe in an 'O' standing up vertically, like a bicycle
wheel, with fluid in the interior.

Now, keep the walls on the bottom half of the "O" hot, and the
walls on the top half cold. Given enough of a temperature difference,
you can start to get convective motion of the fluid as it tries to
transport heat from hot to cold.

In this circumstance, the "X" variable is proportional to fluid
velocity (in some complicated nondimensionalization), the Y and Z
variables are coefficients sine and cosine modes of the temperature.

"a" is the Prandtl number of the fluid (ratio of shear viscosity to
thermal diffusivity), "b" is the Rayleigh number (proportional to
temperature difference between hot and cold boundary conditions and
other geometrical factors), and "c" is always one.

: Lorenz cooked up these equations as a simple example of what he was
: seeing in his more sophisticated weather models.


: 				Marc R. Roussel
:                                 mroussel@alchemy.chem.utoronto.ca

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