Non-linear evolution of the horizontal shear instability in stratified rotating fluids under the complete Coriolis acceleration

TL;DR

This work investigates the nonlinear evolution of horizontal shear instability in a stratified, rotating fluid under the full Coriolis acceleration using direct numerical simulations with two-dimensional horizontal perturbations but three velocity components. By varying the non-dimensional Brunt–Väisälä frequency $N$ and the non-traditional Coriolis parameter $ ilde{f}$ at fixed $Re=2000$ and $Sc=1$, it identifies four nonlinear regimes (traditional, traditional-like, mixed, non-traditional) and shows that strong stratification recovers the traditional dynamics via a transformation. The study analyzes energy and enstrophy budgets to explain how vertical velocity and buoyancy perturbations drive secondary instabilities, leading to small-scale turbulence in certain regimes. In the strongly stratified limit, a scaling framework is derived showing that the nonlinear dynamics can be mapped to the traditional case, enabling predictive relations for horizontal/vertical energies and buoyancy, with implications for momentum and heat transport in stellar and planetary contexts.

Abstract

This paper investigates the non-linear dynamics of horizontal shear instability in an incompressible, stratified and rotating fluid in the non-traditional $f$-plane, i.e. with the full Coriolis acceleration, using direct numerical simulations. The study is restricted to two-dimensional horizontal perturbations. It is therefore independent of the vertical (traditional) Coriolis parameter. However, the flow has three velocity components due to the horizontal (non-traditional) Coriolis parameter. Three different scenarios of non-linear evolution of the shear instability are identified, depending on the non-dimensional Brunt-Väisälä frequency $N$ and the non-dimensional non-traditional Coriolis parameter $\tilde{f}$ (non-dimensionalized by the maximum shear), in the range $\tilde{f}<N$ for fixed Reynolds and Schmidt numbers $Re=2000$, $Sc=1$. When the stratification is strong $N\gg 1$, the shear instability generates stable Kelvin-Helmholtz billows like in the traditional limit $\tilde{f}=0$. Furthermore, when $N\gg1$, the governing equations for any $\tilde{f}$ can be transformed into those for $\tilde{f}=0$. This enables us to directly predict the characteristics of the flow depending on $\tilde{f}$ and $N$. When $N$ is around unity and $\tilde{f}$ is above a threshold, the primary Kelvin-Helmholtz vortex is destabilised by secondary instabilities but it remains coherent. For weaker stratification, $N\leqslant0.5$ and $\tilde{f}$ large enough, secondary instabilities develop vigorously and destroy the primary vortex into small-scales turbulence. Concomitantly, the enstrophy rises to high values by stretching/tilting as in fully three-dimensional flows. A local analysis of the flow prior to the onset of secondary instabilities reveals that the Fjortoft necessary condition for instability is satisfied, suggesting that they correspond to shear instabilities.

Figures (22)

• Figure 1: (a) Sketch of the local Cartesian frame at the colatitude $\theta$ in a sphere rotating at rate $\Omega_{0}$. (b) Sketch of the horizontal shear flow in a local Cartesian frame. The colour gradient from yellow to orange illustrates the increasing density as $z$ decreases.
• Figure 2: (a) Growth rate $\sigma$ as a function of the wavenumber $k_{x}$ for different parameters $(N, \tilde{f})$, coming from Parketal2021 and from the cases listed in table \ref{['tab_ref_simu']}. The lines represent the results of Parketal2021 in the inviscid limit, whereas the symbols show the present results for $Re=2000$ and $Sc=1$. (b) Rescaled growth rate $\sigma\sqrt{\alpha}$ as a function of the rescaled wavenumber $k_{x}\sqrt{\alpha}$, where $\alpha=N^{2}/(\tilde{f}^{2}+N^{2})$.
• Figure 3: Map of the simulations in the parameter space $(N,\tilde{f})$ for $Re=2000$ and $Sc=1$. Grey, green, blue and red symbols represent the simulations where "Traditional", "Traditional-like", "Mixed" and "Non-traditional" behaviours have been observed, respectively. The black dashed line corresponds to the line $N=\tilde{f}$ delimiting the explored region in the present paper. The blue and red dotted lines indicate approximately the limits of the "mixed" domain.
• Figure 4: Total vertical vorticity ($\xi=\partial_x v-\partial_y u$) at, (a), $t=60$, (b), $t=67$, (c), $t=78$, (d), $t=200$ for $\tilde{f}=0$, $Re=2000$ and $Sc=1$ (Traditional evolution).
• Figure 5: Same as figure \ref{['Quatre_instants_Cas_trad']} but for $(N, \tilde{f})=(2, 1.5)$ and (a), $t=51$, (b), $t=58$, (c), $t=70$, (d), $t=200$ (traditional-like evolution).The $y$-axis has been cropped compared to the original computational domain.