Research in Engineering and Aviation

A path integral for turbulence in incompressible fluids


Author(s): Thacker, W.D.

Journal: Journal of Mathematical Physics, Vol. 38, pp. 300-320 (1997). DOI: 10.1063/1.532190


In this paper a classical path integral is formulated for incompressible fluids that evolve according to the Navier–Stokes equation. The path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields. We construct a set of ISp(2) charges associated with the geometry of gvol and its Poisson structure, and a pair of supersymmetry charges connected with the Hamiltonian. These charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero. When the effect of dissipation is included, the charges associated with the Poisson structure and the Hamiltonian are no longer conserved. Charges that generate Kolmogorov scaling and Galilean transformations are also constructed. The classical path integral is formulated in terms of vorticity as well.