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= 28 000 and a wide range of horizontal Froude number F_hin and buoyancy Reynolds number R= ReF_h². 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 L_b = 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 K^2/3 k_h5/3 power law (where k_h is the horizontal wavenumber and K is the dissipation rate of kinetic energy) from k_b = 2πL_b to the dissipative scales, with an energy deficit between the integral scale and k_b and an excess around k_b. The vertical spectrum of kinetic energy can be expressed as E(k_z)= C N N²k_z-3+ CK ^2/3k_z-^5/3} where CN and C are two constants of order unity and k_z is the vertical wavenumber. It is therefore very steep near the buoyancy scale with an N²k_z-3 shape and approaches the K^2/3k_z-^5/3 spectrum for k _z> k_o, k_o 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 N²k_z-3 spectrum is associated with large horizontal scales k_h <k_band the K^2/3k_z- ^5/3 spectrum with the scales k_hgt;k_b.