Abstract
We develop the Stochastic Closure Theory for Lagrangian Turbulence. This theory has already been
developed for homogeneous and boundary turbulence, by the authors, here we extend it to
Lagrangian Turbulence. We compute the structure functions of turbulence and their scaling, in a
lag variable, and compare the results with simulations of the structure function at high Reynolds
numbers. Interestingly the results validate the heuristic arguments, at small values of the lag, the
scaling is Lagrangian and follows from the Lagrangian Stochastic Navier-Stokes Equation. There is
no intermittency because vortices have not had time to form. This region is then followed by a
pass-over region to the same scaling as in homogeneous turbulence, the Kolmogorov-Obukhov
scaling with intermittency, as in the Eulerian Stochastic Closure Theory. Now the vortices have
formed, this causes intermittency, and the turbulence is homogeneous although usually not
isotropic. The pass-over does not have a dip when the log-log derivatives are plotted, as previously
thought. This turns out to be an artifact of using the log derivative of the second structure
functions, in fact when higher order structure functions are used one gets a bump not a dip. At the
end, for large values of the lag, the analogy with boundary turbulence contributes a new scaling,
1/k in Fourier space. This scaling is caused by the so-called detached eddies that shrink and spin up
in (angular velocity) as a skater on ice. These vortices were first noticed for boundary turbulence,
thus both homogeneous and boundary turbulence contribute to the resolution of Lagrangian
turbulence. This is a joint work with and Luiza Angheluta.