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We study a turbulence closure model in which the fractional Laplacian$(-\Delta)^\alpha$ of the velocity field represents the turbulence diffusivity.We investigate the energy spectrum of the model by applying Pao's energytransfer theory. For the case $\alpha=1/3$, the corresponding power law of theenergy spectrum in the inertial range has a correction exponent on the regularKolmogorov -5/3 scaling exponent. For this case, this model representsRichardson's particle pair-distance superdiffusion of a fully developedhomogeneous turbulent flow as well as L\'evy jumps that lead to thesuperdiffusion. For other values of $\alpha$, the power law of the energyspectrum is consistent with the regular Kolmogorov -5/3 scaling exponent. Wealso propose and study a modular time-stepping algorithm in semi-discretizedform. The algorithm is minimally intrusive to a given legacy code for solvingNavier-Stokes equations by decoupling the local part and nonlocal part of theequations for the unknowns. We prove the algorithm is unconditionally stableand unconditionally, first-order convergent. We also derive error estimates forfull discretizations of the model which, in addition to the time steppingalgorithm, involves a finite element spatial discretization and a domaintruncation approximation to the range of the fractional Laplacian.

Analysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson pair dispersion