Influence of the dispersive and dissipative scales α and β on the energy spectrum of the Navier-Stokes αβ equations

Lundgren's vortex model for the intermittent fine structure of high-Reynolds-number turbulence is applied to the Navier-Stokes alpha beta equations and specialized to the Navier-Stokes alpha equations. The Navier-Stokes alpha beta equations involve dispersive and dissipative length scales alpha and beta, respectively. Setting beta equal to alpha reduces the Navier-Stokes alpha beta equations to the Navier-Stokes alpha equations. For the Navier-Stokes alpha equations, the energy spectrum is found to obey Kolmogorov's -5/3 law in a range of wave numbers identical to that determined by Lundgren for the Navier-Stokes equations. For the Navier-Stokes alpha beta equations, Kolmogorov's -5/3 law is also recovered. However, granted that beta <alpha, the range of wave numbers for which this law holds is extended by a factor of alpha/beta. This suggests that simulations based on the Navier-Stokes alpha beta equations may have the potential to resolve features smaller than those obtainable using the Navier-Stokes alpha equations.