Stationary Structures of the Boussinesq equations
Abstract
We investigate the steady configurations supported by the inviscid two-dimensional Boussinesq equations in the periodic channel. We show steady laminar velocity fields (closed non-contractible fluid trajectories) coupled to non-decreasing density stratifications are necessarily stratified shear flows. The direction of density monotonicity along level sets is in fact fundamental: in striking contrast with the Euler equations, we exhibit laminar non-stagnant non-shear stationary solutions in the periodic channel. We also provide distinct flexibility and rigidity results (laminar persistence or appearance of islands for nearby non-shear solutions) close to stratified shear flows according to their stagnation points. As a by-product of our analysis, we establish a uniform Crandall-Rabinowitz bifurcation theorem that might be of independent interest.
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BibTeX
@misc{nualart2026stationary,
title = {{Stationary Structures of the Boussinesq equations}},
author = {Marc Nualart and Jaemin Park and Javier Peñafiel-Tomás},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40334},
eprint = {2610.40334},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40334},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stationary Structures of the Boussinesq equations
AU - Marc Nualart
AU - Jaemin Park
AU - Javier Peñafiel-Tomás
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40334
UR - https://arxiv.org/abs/2610.40334
AB - We investigate the steady configurations supported by the inviscid two-dimensional Boussinesq equations in the periodic channel. We show steady laminar velocity fields (closed non-contractible fluid trajectories) coupled to non-decreasing density stratifications are necessarily stratified shear flows. The direction of density monotonicity along level sets is in fact fundamental: in striking contrast with the Euler equations, we exhibit laminar non-stagnant non-shear stationary solutions in the periodic channel. We also provide distinct flexibility and rigidity results (laminar persistence or appearance of islands for nearby non-shear solutions) close to stratified shear flows according to their stagnation points. As a by-product of our analysis, we establish a uniform Crandall-Rabinowitz bifurcation theorem that might be of independent interest.
ER -