Relaxing the sharp density stratification and columnar motion assumptions in layered shallow water systems

Adim, Mahieddine; Bianchini, Roberta; Duchêne, Vincent

Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1597-1626

We rigorously justify the bilayer shallow-water system as an approximation to the hydrostatic Euler equations in situations where the flow is density-stratified with close-to-piecewise constant density profiles, and close-to-columnar velocity profiles. Our theory accommodates with continuous stratification, so that admissible deviations from bilayer profiles are not pointwise small. This leads us to define refined approximate solutions that are able to describe at first order the flow in the pycnocline. Because the hydrostatic Euler equations are not known to enjoy suitable stability estimates, we rely on thickness-diffusivity contributions proposed by Gent and McWilliams. Our strategy also applies to one-layer and multilayer frameworks.

Nous justifions rigoureusement les équations de Saint-Venant bicouche en tant que modèle pour les équations d’Euler hydrostatiques dans les situations où l’écoulement est stratifié avec des profils de densité et de vitesse approximativement constants par morceaux. Notre théorie accepte des profils continus, de sorte que les écarts admissibles par rapport aux profils parfaitement bicouches ne sont pas petits pour la distance uniforme. Cela nous amène à définir des solutions approchantes raffinées capables de décrire au premier ordre l’écoulement dans la pycnocline. En l’absence d’estimations de stabilité appropriées sur les équations d’Euler hydrostatiques, nous nous appuyons sur des contributions de diffusivité sur les variables d’épaisseur proposées par Gent et McWilliams. Notre stratégie s’applique également aux équations de Saint-Venant monocouche et multicouche.

Classification : 35L03, 35Q35, 76B55, 76E17
Keywords: Internal waves, Hydrostatic models, Sharp stratification limit, Columnar motion
Mots-clés : Ondes internes, Modèles hydrostatiques, limite d’interface, écoulements en colonne

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