Temporal multiscaling in hydrodynamic turbulence

On the basis of the Navier-Stokes equations, we develop the high Reynolds number statistical theory of different-time, many-point spatial correlation functions of velocity differences. We find that their time dependence is not scale invariant: n-order correlation functions exhibit infinitely many distinct decorrelation times that are characterized by anomalous dynamical scaling exponents. We derive exact scaling relations that bridge all these dynamical exponents to the static anomalous exponents ${\mathrm{\zeta }}_{\mathrm{q}}$ of the standard structure functions. We propose a representation of the time dependence using the Legendre-transform formalism of multifractals that automatically reproduces all the newly found bridge relationships.