Stability of 3D Stratified Plane Couette Flow: A New Lift-up Mechanism and Potential Vorticity
Abstract
Since the pioneering work of Taylor and Goldstein in 1931 laid the foundation for the study of parallel stratified shear flows, non-parallel configurations have received comparatively little attention. We study the stability of the 3D Boussinesq system near the plane Couette flow V*=(y,0,0) under the vertically stratified background η*=αz. The shear and stratification directions are perpendicular. This geometry produces two structural effects that are absent in the parallel configuration: an additional nonlocal coupling in the non-zero-mode system, which obstructs a directly useful symmetrization, and a wave-shear coupled lift-up mechanism for the zero modes. The Lp transient growth due to the lift-up effect can be suppressed either by sufficiently strong stratification (for p>2) or under suitable assumptions on the initial data (for p≥ 2). To study the nonzero-mode system, we introduce a reformulation based on the linearized potential vorticity (PV), a fundamental quantity in geophysical fluid dynamics, to recover a better energy structure. We establish the inviscid damping of u1,≠ and u2,≠. We further apply the linearized PV formulation to study the linear stability of the tilted Couette flow. Finally, we use this formulation to prove the nonlinear stability for perturbations on 𝕋×ℝ×𝕋.
Related
- Vanishing transport noise enhances stability of stratified Couette flow in the 3D Boussinesq system
- The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on 𝕋×ℝ2
- Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and vertical stratified temperature
- Nonlinear Inviscid Damping of Couette Flow with Vorticity Perturbations Supported up to the Boundary
- PV-Wave Decomposition and Combined Wave Effects for Stratified Couette Flows
BibTeX
@misc{yao2026stability,
title = {{Stability of 3D Stratified Plane Couette Flow: A New Lift-up Mechanism and Potential Vorticity}},
author = {Yiting Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33960},
eprint = {2609.33960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33960},
}
RIS
TY - UNPB
TI - Stability of 3D Stratified Plane Couette Flow: A New Lift-up Mechanism and Potential Vorticity
AU - Yiting Yao
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.33960
UR - https://arxiv.org/abs/2609.33960
AB - Since the pioneering work of Taylor and Goldstein in 1931 laid the foundation for the study of parallel stratified shear flows, non-parallel configurations have received comparatively little attention. We study the stability of the 3D Boussinesq system near the plane Couette flow V^*=(y,0,0) under the vertically stratified background η^*=αz. The shear and stratification directions are perpendicular. This geometry produces two structural effects that are absent in the parallel configuration: an additional nonlocal coupling in the non-zero-mode system, which obstructs a directly useful symmetrization, and a wave-shear coupled lift-up mechanism for the zero modes. The L^p transient growth due to the lift-up effect can be suppressed either by sufficiently strong stratification (for p>2) or under suitable assumptions on the initial data (for p≥ 2). To study the nonzero-mode system, we introduce a reformulation based on the linearized potential vorticity (PV), a fundamental quantity in geophysical fluid dynamics, to recover a better energy structure. We establish the inviscid damping of u_1,≠ and u_2,≠. We further apply the linearized PV formulation to study the linear stability of the tilted Couette flow. Finally, we use this formulation to prove the nonlinear stability for perturbations on 𝕋×ℝ×𝕋.
ER -