Macroscopic Fluid Mechanics 2

Part 1
To understand the origin of this non-commutativity, let us again consider the example of a hurricane. It is an extended object, whose radius is of the order of 100 km. It wouldn’t make sense to have two such objects within a 100 km of each other: the two hurricanes will interact strongly with each other and combine into a single one. (This phenomenon of a reverse cascade’ can been seen clearly in some simulations.) Thus there is a limit to the resolution of the co-ordinates of a hurricane, given by the area of a hurricane. This is reminiscent of the uncertainty principle of quantum mechanics, except that there it is the co-ordinates of phase space that is fuzzy: the analogue of the area is Plank’s constant.

Remarkably, this analogy is quite precise (See for example this paper.)
). The motion of tracer particles advected by a two dimensional incompressible fluid ( an approximation to the atmosphere) is hamiltonian motion where the two components of position are canonically conjugate to each other. Thus it makes sense to apply the ideas of quantum mechanics to this purely classical situation. The analogue of quantization is simply the averaging over an area $a$ of space. We will see that such averages are matrices. The dimension $N={A\over a}$ of the matrix is the number of cells, the total area of the fluid $A$ divided by the area $a$ of each cell. Given a function $f$ , there is a unique $N\times N$ matrix $\check{f}$ which is its average. Given an $N\times N$ matrix $K$ there is a (not unique!) smooth function $\hat{K}$ with it as average. By combining these two operations we can define a non-commutative product of functions on the sphere:
$f\star g=\widehat{\check{f}\check{g}$

This Moyal’ product contains within it both the pointwise product and the Poisson bracket of function:
$f\star g=fg+{1\over N}\{f,g\}+{\rm O}(N^{-2}).$

This will allow us to recover the original fluid dynamics from macrofluid equations in the limit as $N\to \infty$.

Ordinarily we think of classical mechanics as an approximation to quantum mechanics. Here, the non-commutative mechanics is the approximation to the exact commutative theory.

What effect will the neglected `micro’ degrees of freedom of the fluid have on its large scale motion? As in the passage from molecular to fluid mechanics, there will be a flow of energy from large scale to small scales which will show up as dissipation. This dissipation will be larger than that of the original fluid as it describes the loss of energy not only to molecular motion but also small scale fluid motion. Conversely, there is a flow of information from small scales to the large: a small eddy affects the overall motion of the hurricane. Since we have no knowledge any more of the state of the small eddy, we will be forced to treat this as a random
forcing of the hurricane. In an equilibrium situation, the energy pumped into the system by such random effects and large scale effects such as condensation of vapor will be balanced by the outflow through dissipation: the fluctuation-dissipation theorem. However, the fluid need not be in this equilibrium state always: departures from it are probably of interest as well.

Thus we start from ordinary differential equations for the positions of molecules and pass first to partial differential equations of the fluid motion. Then we average over small scale fluid motion to get an effective theory of macroscopic degrees of freedom of large vortices. This will be an ordinary differential equation for a matrix $\check{\omega}$ representing the averaged vorticity. In the final stage we may allow a continuous variation of this $\check{\omega}$ to get partial differential equations for matrix-valued functions. This matrix-valued Navier-Stokes equation is our sugession for the equation of motion for macroscopic objects in two dimensional fluid mechanics.

See also the Talk at SIAM DS07 Conference Snowbird Utah May 28th-June 1

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