soreff (01/08/83)
Does anyone out there know of an example of a physical system with an instability which truly requires a three dimensional representation or description? I don't mean a system where some vector quantities need components in all three directions, but rather one where for each direction, there is at least one physical function (temperature, electrostatic potential, magnitude of magnetic field strength) which varies along that direction in a way that is qualitatively important to the instability. Basically I am looking for systems which don't have an analog in any system with full translational symmetry in any direction, or with full rotational symmetry about any axis (which would also allow a two dimensional representation). A number of instabilities (those associated with convection, for instance) can produce three dimensional patterns (the hexagonal convection cells, in this instance), but all the ones that I know of have two dimensional analog, so the three-dimensionality of the system only affects the size of critical onset parameters and the like, but not the qualitative instability. I'd like this to be a discussion question, so please post answers to the net. -Jeffrey Soreff (hplabsb!soreff)
leichter (01/08/83)
I don't know of one off-hand, but a reasonable place to look is in magnetic fluids. The theory of "magneto-hydrodynamics" got written up in Scientific American about 6 months ago. The reason I suggest it is that the equations of motion for such a fluid are the Bernoulli equations with additional terms for the magnetic interaction. If convection produces a two-dimensional system, it might be possible to cook up a three-dimensional one by making the fluid magnetic and adding a magnetic field. (Sounds like it would be somewhat arti- ficial; but I think that may be because all of our "non-artificial" examples are physical systems which we can solve - and we produce them by simplification of real systems.) -- Jerry decvax!yale-comix!leichter