Introduction to Talk at SIAM DS07 Conference Snowbird Utah May 28th-June 1

The equations of motion of a fluid are obtained by averaging over the equations of motion of the large number of molecules that occupy even a small volume: we are not for the most part interested in the details of the motion of individual molecules. The equations of a fluid so obtained ( Euler or Navier-Stokes) are quite different from those of particle mechanics, being partial differential equations. Nevertheless the fundamental symmetries (translation and rotation invariance) of particle mechanics are preserved in this reformulation.

The conservation laws (energy, momentum,angular momentum) are preserved for ideal flow (Euler). In the next approximation, the effect of the transport of these conserved quantities to molecular scales are incorporated (viscosity). Even higher order corrections from molecular scales can be incorporated ( Chapman and Enskog) but are rarely needed.

A fluid is a notoriously unstable system, small perturbations in the initial data tend to grow exponentially fast. In spite of the instability, there are some large scale structures that emerge which are remarkably stable. The ocean currents and the jet stream are features that have persisted for centuries if not more. A hurricane can last weeks and extend over hundreds of kilometers. These large objects have a small number of degrees of freedom: the position of the center of a hurricane, its center of mass velocity and total angular momentum.

We suspect that there is a simpler theory ( ironically a system again of ordinary differential equations) Again, we are not interested in the small scale fluctuations (certainly less than a few kilometers in the case of the atmosphere or the ocean). Thus we would like to have an effective theory of `macroscopic’ fluid dynamics.

This is much like the passage from molecules to fluids; instead of averaging over the motion of molecules, we want to average over small scale fluid motion.

This effective theory of macroscopic fluids should still have the same symmetries as the Euler or Navier-Stokes equations. It should have loss terms analogous to viscosity that describe the loss of energy to small scale fluid motion. Because the demarkation between large and small scales is somewhat arbitrary, the theory should admit an additional invariance under scale transformations provided that the parameters of the macrofluid are varied accordingly.

All this is reminiscent of the Landau-Ginzburg theory of magnetism. A magnet is made of a very large of molecules, each with a spin and hence a small magnetic moment. We can average these spins over a region of size much larger than the molecular separation , to get a magnetization . Once such a magnetization develops it becomes energetically favorable for even more spins to align. The energy is roughly for some positive constant and mean magnetization . Thus over distances of order , the magnet can be described by a single vector

,

its `order parameter’. If we are to consider distances much larger even than , the magnetization can vary slowly. The energy is given by Landau-Ginzaburg theory by

The constants depend on the scale over which we have chosen to average: this dependence can be calculated using renormalization group invariance, which is a sophisticated form of scale invariance that includes the effects of fluctuations. In this way we avoid the dependence of the theory on the artificial parameter .

The nature of the basic degrees of freedom changed when we passed from the molecular to the fluid description: from the position and momentum vectors of molecules ( points on a manifold) to the density and velocity fields of the fluid, ( functions and vector fields on a differentiable manifold). In the same way the degrees of freedom of macroscopic fluid mechanics are not necessarily functions on a manifold. We will argue that, in order to incorporate the symmetries of the fluid dynamics, these are in effect functions on a non-commutative manifold. More precisely, they are elements of an associative but not commutative algebra.

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