Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and vertical stratified temperature
Abstract
The most important features that distinguish fluid flow in the atmosphere and ocean are the effects of rotation and stratification, which are described by the Boussinesq equations with rotation. This paper examines the stability threshold at high Reynolds number Re for the 3D Boussinesq equations with rotation near Couette flow and a vertical stratified temperature. For the zero modes, the combined effects of rotation and stratification sustain velocity-temperature oscillations with frequencies bounded away from zero. In the stable rotational regime, rotation suppresses the linear growth associated with the lift-up effect, while stratification sustains oscillations in the double zero modes. Although the zero modes dynamics are not fully dispersive, we exploit the structure of the oscillatory phases and cancellations in the equations to obtain a modified energy estimate for the entire zero modes system. Combining this estimate with enhanced dissipation and inviscid damping of the nonzero modes, we establish a bound for the stability threshold, with order Re−2/3(1+logRe)−1−a0 for any fixed a0∈(0,1/2).
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- Stability threshold of the Couette flow in 2D Navier-Stokes Boussinesq equations on 𝕋 × [−1, 1] and ℝ × [−1, 1]
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BibTeX
@misc{huang2026stability,
title = {{Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and vertical stratified temperature}},
author = {Wenting Huang and Zekai Luo and Ying Sun and Jiahong Wu and Xiaojing Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00964},
eprint = {2611.00964},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00964},
}
RIS
TY - UNPB
TI - Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and vertical stratified temperature
AU - Wenting Huang
AU - Zekai Luo
AU - Ying Sun
AU - Jiahong Wu
AU - Xiaojing Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00964
UR - https://arxiv.org/abs/2611.00964
AB - The most important features that distinguish fluid flow in the atmosphere and ocean are the effects of rotation and stratification, which are described by the Boussinesq equations with rotation. This paper examines the stability threshold at high Reynolds number Re for the 3D Boussinesq equations with rotation near Couette flow and a vertical stratified temperature. For the zero modes, the combined effects of rotation and stratification sustain velocity-temperature oscillations with frequencies bounded away from zero. In the stable rotational regime, rotation suppresses the linear growth associated with the lift-up effect, while stratification sustains oscillations in the double zero modes. Although the zero modes dynamics are not fully dispersive, we exploit the structure of the oscillatory phases and cancellations in the equations to obtain a modified energy estimate for the entire zero modes system. Combining this estimate with enhanced dissipation and inviscid damping of the nonzero modes, we establish a bound for the stability threshold, with order Re^−2/3(1+logRe)^−1−a_0 for any fixed a_0∈(0,1/2).
ER -