New equation for lagrangian incompressible fluid flows applied to turbulence

Theoretical developments in the field of Lagrangian turbulence are made difficult by the fact that equations governing the evolution of lagrangian flows are implicit contrary to eulerian flows. In this article, an {\it exact} explicit equation for incompressible lagrangian fluid flows at high-Reynolds number is constructed. The method to arrive at the equation of motion and the proof that it describes the motion of an incompressible fluid are provided. A truncated version of this new equation is used to show how the lagrangian turbulent spectrum ($E_\mathrm{lag}(\omega)$) could be inferred. This exercise showed a complex interrelation between the stirring force field and the flow itself in the lagrangian turbulence framework whereas the stirring is not affected by the flow in the eulerian point of view. The result is that $E_\mathrm{lag}(\omega)\propto \varepsilon\,\omega^{-2}$ seems independent upon the way the fluid is stirred in the inertial range for a given dissipated power $\varepsilon$. It also showed that $E_\mathrm{lag}(\omega)\sim \omega^{-s}$ with $0\leq s\leq 1/2$ (depending on the stirring) at low-$\omega$ and $\sim\omega^{-4}$ in the viscous range at high-$\omega$.