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

Abstract

AbstractWe 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 $Re= 28\hspace{0.167em} 000$ and a wide range of horizontal Froude number ${F}_{h} \in [0. 0225~0. 135] $ and buoyancy Reynolds number $\mathscr{R}= Re{{F}_{h} }^{2} \in [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–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 ${{\varepsilon }_{K} }^{2/ 3} { k}_{h}^{\ensuremath{-} 5/ 3} $ power law (where ${k}_{h} $ is the horizontal wavenumber and ${\varepsilon }_{K} $ is the dissipation rate of kinetic energy) from ${k}_{b} = 2\lrm{\pi} / {L}_{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} {N}^{2} { k}_{z}^{\ensuremath{-} 3} + C{{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ where ${C}_{N} $ 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 ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ shape and approaches the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum for ${k}_{z} \gt {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 ${N}^{2} { k}_{z}^{\ensuremath{-} 3} $ spectrum is associated with large horizontal scales $\vert {\mathbi{k}}_{h} \vert \lt {k}_{b} $ and the ${{\varepsilon }_{K} }^{2/ 3} { k}_{z}^{\ensuremath{-} 5/ 3} $ spectrum with the scales $\vert {\mathbi{k}}_{h} \vert \gt {k}_{b} $.