Vertical spectra of stratified turbulence at large horizontal scales

Stably stratified turbulence is investigated with the aim of increasing our limited understanding of the vertical structure of this type of turbulent flow. For strongly stratified turbulence there is a theoretical prediction that the energy spectra in the vertical direction of gravity are very steep, possessing the well-known form E-h(k(v)) alpha N(2)kv(-3) , where N is the Brunt-Vaisala frequency and kv is the vertical wave number, but supporting evidence from experiments and numerical simulations is lacking. We conduct direct numerical simulation (DNS) with uniform background stratification and forcing at large scales. In order to consider the large anisotropic scales only, the vertical energy spectra are decomposed into large-scale vertical spectra E-large(k(v)) and small-scale vertical spectra E-small(k(v)) using a horizontal demarcation scale. We find that this approach gives results that are in close agreement with E-large(k(v)) alpha N(2)k(v)(-3) for the DNS runs performed. This result holds approximately over the wave-number range k(b) <= k(v) <= k(oz), where kb is the buoyancy wave number and koz is the Ozmidov wave number, in agreement with theory. Similarly, large-scale vertical spectra of potential energy are found to be E-p,E-large(k(v)) alpha N(2)k(v)(-3) , over a narrower range of wave numbers. The evidence supports the existence of a scale-by-scale balance between inertia and buoyancy occurring in strongly stratified turbulence at large horizontal scales. Finally, the current results are put in the context of ocean turbulence by making a comparison with measurements of vertical shear spectra made in the ocean interior.