Spectral analysis of the transition to turbulence from a dipole in stratified fluid

We investigate the spectral properties of the turbulence generated during the nonlinear evolution of a Lamb–Chaplygin dipole in a stratified fluid for a high Reynolds number Re= 280.167em 000 and a wide range of horizontal Froude number Fₕ ∈ [0. 0225~0. 135] and buoyancy Reynolds number R= ReFₕ ² ∈ [14~510] . The numerical simulations use a weak hyperviscosity and are therefore almost direct numerical simulations (DNS). After the nonlinear development of the zigzag instability, both shear and gravitational instabilities develop and lead to a transition to small scales. A spectral analysis shows that this transition is dominated by two kinds of transfer: first, the shear instability induces a direct non-local transfer toward horizontal wavelengths of the order of the buoyancy scale Lb = U/ N , where U is the characteristic horizontal velocity of the dipole and N the Brunt–Väisälä frequency; second, the destabilization of the Kelvin–Helmholtz billows and the gravitational instability lead to small-scale weakly stratified turbulence. The horizontal spectrum of kinetic energy exhibits a ε K 2/ 3 kₕ^- 5/ 3 power law (where kₕ is the horizontal wavenumber and ε K is the dissipation rate of kinetic energy) from kb = 2π / Lb to the dissipative scales, with an energy deficit between the integral scale and kb and an excess around kb . The vertical spectrum of kinetic energy can be expressed as E(kz )= CN N² kz^- 3 + Cε K 2/ 3 kz^- 5/ 3 where CN and C are two constants of order unity and kz is the vertical wavenumber. It is therefore very steep near the buoyancy scale with an N² kz^- 3 shape and approaches the ε K 2/ 3 kz^- 5/ 3 spectrum for kz > kₒ , kₒ being the Ozmidov wavenumber, which is the cross-over between the two scaling laws. A decomposition of the vertical spectra depending on the horizontal wavenumber value shows that the N² kz^- 3 spectrum is associated with large horizontal scales kₕ < kb and the ε K 2/ 3 kz^- 5/ 3 spectrum with the scales kₕ > kb .