A generic model for transport in turbulent shear flows

Turbulence regulation by large-scale shear flows is crucial for a predictive modeling of transport in plasma. In this paper the suppression of turbulent transport by large-scale flows is studied numerically by measuring the turbulent diffusion D t and scalar amplitude 〈 n ′ 2 〉 of decaying passive scalar fields n ′ advected by various turbulent flows. Both uniform flows and shear flows are shown to suppress turbulence causing the quenching in transport and turbulence amplitude. The uniform flows U 0 = Λ y with the advection rate Λ in the case of a finite correlated forcing with τ F = 1 gives rise to the advection/sweeping effect which suppresses D t , 〈 u ′ 2 〉 and 〈 n ′ 2 〉 as ∝ Λ - 2 for Λ >> τ F - 1 . In contrast, no influence of the uniform flow is found in the case of a short correlated forcing τ F → 0 due to Galilean invariance. For the shear flow U 0 = Ω sin x y ∧ ( Ω = constant shearing rate) with the appropriate choice of the forcing ( τ F → 0 ) the nature of transport suppression is shown to crucially depend on the properties of the turbulence. Specifically, for prescribed turbulence with a short correlation time τ c = τ F ≪ Ω - 1 , the turbulence statistics scale as D t ∝ Ω - 0 . 02 , 〈 n ′ 2 〉 ∝ Ω - 0 . 62 and cross-phase cos θ ∝ Ω 0 . 29 . For consistently evolved turbulence with a finite correlation time τ c ≥ Ω - 1 , turbulence statistics are suppressed more strongly as D t ∝ Ω - 1 . 75 , 〈 n ′ 2 〉 ∝ Ω - 2 . 41 , 〈 u ' x 2 〉 ∝ Ω - 0 . 65 and 〈 ω ′ 2 〉 ∝ Ω - 0 . 50 . A novel renormalization scheme is then introduced to rescale our results into the regime within which the kinetic energy and enstrophy are unchanged by shear flow. This allows our numerical results to closely match previous analytical predictions [E. Kim, Mod. Phys. Lett. B 18, 1 (2004)] and to understand different experimental scalings observed in laboratory plasmas. Furthermore, D t is found to be related to 〈 n ′ 2 〉 by 〈 n ′ 2 〉 ∝ D t / D Ω , where D Ω ∝ Ω 2 3 is the shear accelerated diffusion of n ′ with an interesting scaling cos θ ∝ D t D Ω .