Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number

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 l(v) similar to u(h)/N, where u(h) is the horizontal velocity scale and N the Brunt-Vaisala frequency. Combined with the hypothesis of the energy dissipation rate scaling as epsilon similar to u(h)(3)/l(h), this theory predicts inertial range scalings for the horizontal spectrum of horizontal kinetic energy and of potential energy, according to E(k(h)) proportional to k(h)(-5/3). We begin by presenting a scaling analysis of the horizontal vorticity equation from which we recover the result regarding the vertical scale, l(v) similar to u(h)/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 Fr-v = u(h)/(Nl(v))= C with C = O(1) in all the runs. The derivation of the dissipation scaling epsilon similar to u(h)(3)/l(h) at low Reynolds number in the context of decaying stratified turbulence highlights that the same scaling holds at high R = ReFrh2 >> 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 epsilon(k) similar to u(h)(3)/l(h) and epsilon(p) similar to u(h)(3)/l(h) (with epsilon = epsilon(k) + epsilon(p)), 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 E-h(k(h)) = C-1 epsilon(2/3)/k(h)(-5/3) and the horizontal potential energy spectrum is well described by E-p(k(h)) = C-2 epsilon(p)epsilon(-1/3)(k)k(h)(-5/3). The presence of an inertial range in the horizontal direction is confirmed by the constancy of the energy flux spectrum over narrow ranges of k(h). However, the vertical energy spectrum is found to differ significantly from the expected E-h(k(v)) similar to N(2)k(v)(-3) scaling, showing that Fr-v is not of order unity on a scale-by-scale basis, thus providing motivation for further investigation of the vertical structure of stratified turbulence.