A new approach to strong turbulence based on ideas of dynamical geometry and topological conservation laws is developed. In terms of the quantum field theory, this is another example of the duality between a fluctuating vector field and fluctuating geometry. Some new exact solutions of Navier-Stokes and Euler equations are found. The loop equation suggested in the early 90-ties is investigated in detail. The loop equation plays the same role in our theory as the Boltzmann kinetic equation in statistical physics. It has the form of the Schrödinger equation with a complex Hamiltonian in loop space. The viscosity plays the role of Planck's constant. Strong turbulence corresponds to the WKB limit of the loop equation. In this limit, we find a fixed point of the loop equation we call Kelvinon. Kelvinon has a conserved circulation for a fixed loop in space, generalizing Kelvin's theorem. Clebsch field of this solution has nontrivial topology with two winding numbers. These topological conservation laws allow us to compute the PDF of circulation in a WKB limit (large circulation in the viscosity units). This PDF perfectly matches the results of numerical simulations of the conventional forced Navier-Stokes equation.

A new approach to strong turbulence based on ideas of dynamical geometry and topological conservation laws is developed. In terms of the quantum field theory, this is another example of the duality between a fluctuating vector field and fluctuating geometry. Some new exact solutions of Navier-Stokes and Euler equations are found. The loop equation suggested in the early 90-ties is investigated in detail. The loop equation plays the same role in our theory as the Boltzmann kinetic equation in statistical physics. It has the form of the Schrödinger equation with a complex Hamiltonian in loop space. The viscosity plays the role of Planck's constant. Strong turbulence corresponds to the WKB limit of the loop equation. In this limit, we find a fixed point of the loop equation we call Kelvinon. Kelvinon has a conserved circulation for a fixed loop in space, generalizing Kelvin's theorem. Clebsch field of this solution has nontrivial topology with two winding numbers. These topological conservation laws allow us to compute the PDF of circulation in a WKB limit (large circulation in the viscosity units). This PDF perfectly matches the results of numerical simulations of the conventional forced Navier-Stokes equation.