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

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* Corresponding author 1 LadHyX - Laboratoire d-hydrodynamique

Abstract : 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-h epsilon 0.0225 0.135 and buoyancy Reynolds number R = ReFh2 epsilon 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 L-b = U-N, where U is the characteristic horizontal velocity of the dipole and N the Brunt-Vaisala 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 epsilon2-3Kkh-5-3 power law where kh is the horizontal wavenumber and epsilonK is the dissipation rate of kinetic energy from kb = 2 pi-L-b 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 Ekz = CNN2kz-3 + C epsilon2-3Kkz-5-3 where C-N 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 N2kz-3 shape and approaches the epsilon2-3Kkz-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 N2kz-3 spectrum is associated with large horizontal scales vertical bar khvertical bar < kb and the epsilon2-3Kkz-5-3 spectrum with the scales vertical bar khvertical bar > kb.

Author: ** Pierre Augier - Jean-Marc Chomaz - Paul Billant - **

Source: https://hal.archives-ouvertes.fr/