Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number
Résumé
We present direct numerical simulations (DNS) of unforced stratified turbulence with the objective of testing the strongly stratified turbulence theory. According to this theory the characteristic vertical scale of the turbulence is given by ℓv∼uh/N , where uh is the horizontal velocity scale and N the Brunt–Väisälä frequency. Combined with the hypothesis of the energy dissipation rate scaling as ϵ∼u3h/ℓh , this theory predicts inertial range scalings for the horizontal spectrum of horizontal kinetic energy and of potential energy, according to E(kh)∝k−5/3h . We begin by presenting a scaling analysis of the horizontal vorticity equation from which we recover the result regarding the vertical scale, ℓv∼uh/N , highlighting in the process the important dynamical role of large-scale vertical shear of horizontal velocity. We then present the results from decaying DNS, which show a good agreement with aspects of the theory. In particular, the vertical Froude number is found to reach a constant plateau in time, of the form Frv=uh/(Nℓv)=C with C=O(1) in all the runs. The derivation of the dissipation scaling ϵ∼u3h/ℓh at low Reynolds number in the context of decaying stratified turbulence highlights that the same scaling holds at high R=ReFr2h≫1 as well as at low R≪1 , which is known (see Brethouwer et al., J. Fluid Mech., vol. 585, 2007, pp. 343–368) but not sufficiently emphasized in recent literature. We find evidence in our DNS of the dissipation scaling holding at R=O(1) , which we interpret as being in the viscous regime. We also find ϵk∼u3h/ℓh and ϵp∼u3h/ℓh (with ϵ=ϵk+ϵp ), in our high-resolution run at earlier times corresponding to R=O(10) , which is in the transition between the strongly stratified and the viscous regimes. The horizontal spectrum of horizontal kinetic energy collapses in time using the scaling Eh(kh)=C1ϵ2/3kk−5/3h and the horizontal potential energy spectrum is well described by Ep(kh)=C2ϵpϵ−1/3kk−5/3h . The presence of an inertial range in the horizontal direction is confirmed by the constancy of the energy flux spectrum over narrow ranges of kh . However, the vertical energy spectrum is found to differ significantly from the expected Eh(kv)∼N2k−3v scaling, showing that Frv is not of order unity on a scale-by-scale basis, thus providing motivation for further investigation of the vertical structure of stratified turbulence.
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