Abstract
We consider a steady, geophysical 2D fluid in a domain, and focus on its western boundary layer, which is formally governed by a variant of the Prandtl equation. By using the von Mises change of variables, we show that this equation is well-posed under the assumptions that the trace of the interior stream function is large, and variations in the coastline profile are moderate.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Note that here y is the tangential variable and \(\xi \) is the rescaled normal variable, in contrast with the usual convention in the study of the Prandtl equation. We have made this choice to stick to the geophysical setting, in which u is the East-West component of the velocity and v its North-South component.
References
Barcilon, V., Constantin, P., Titi, E.S.: Existence of solutions to the Stommel–Charney model of the Gulf Stream. SIAM J. Math. Anal. 19(6), 1355–1364 (1988)
Bresch, D., Colin, T.: Some remarks on the derivation of the Sverdrup relation. J. Math. Fluid Mech. 4(2), 95–108 (2002)
Bresch, D., Gérard-Varet, D.: Roughness-induced effects on the quasi-geostrophic model. Commun. Math. Phys. 253(1), 81–119 (2005)
Charney, J.G.: The Gulf Stream as an inertial boundary layer. Proc. Natl. Acad. Sci. U.S.A. 41(10), 731 (1955)
Dalibard, A.-L., Saint-Raymond, L.: Mathematical Study of Degenerate Boundary Layers: A Large Scale Ocean Circulation Problem, vol. 253. American Mathematical Society, Providence (2018)
Desjardins, B., Grenier, E.: On the homogeneous model of wind-driven ocean circulation. SIAM J. Appl. Math. 60(1), 43–60 (2000)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Inc., Englewood Cliffs (1964)
Ierley, G.R., Ruehr, O.G.: Analytic and numerical solutions of a nonlinear boundary-layer problem. Stud. Appl. Math. 75(1), 1–36 (1986)
Lagrée, P.-Y.: Interactive Boundary Layer (IBL). Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances. Volume 523 of CISM Courses and Lectures, pp. 247–286. Springer, New York (2010)
Oleĭnik, O.A.: On the system of Prandtl equations in boundary-layer theory. Dokl. Akad. Nauk SSSR 150, 28–31 (1963)
Oleinik, O.A., Samokhin, V.N.: Mathematical Models in Boundary Layer Theory, Volume 15 of Applied Mathematics and Mathematical Computation. Chapman & Hall/CRC, Boca Raton (1999)
Pedlosky, J.: Ocean Circulation Theory. Springer, Berlin (1996)
Smith, C.J., Mallier, R.: Removing the separation singularity in a barotropic ocean with bottom friction. In: Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods (BAIL 2002), Vol. 166. pp. 281–290 (2004)
Wang, X., Wang, Y.-G.: Well-posedness and blowup of the geophysical boundary layer problem. arXiv preprint arXiv:1903.07016 (2019)
Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program Grant Agreement No. 637653, project BLOC “Mathematical Study of Boundary Layers in Oceanic Motion”. The authors have also been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interests.
Additional information
Communicated by D. Gerard-Varet.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dalibard, AL., Paddick, M. An Existence Result for the Steady Rotating Prandtl Equation. J. Math. Fluid Mech. 23, 13 (2021). https://doi.org/10.1007/s00021-020-00524-4
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-020-00524-4