A note on the lower boundary condition for Ekman flows

Abstract

We show that the assumption of a stress-free boundary condition at a finite intermediate depth, namely, at the bottom of the Ekman layer, in the analysis of wind-driven ocean flows necessarily leads to an unphysical current profile. Indeed, if the z-derivative of the fluid velocity vanishes at a given depth, then this depth necessarily corresponds to a minimum of the velocity profile, with the velocity increasing beneath it. Using a WKB ansatz based on the small variations of the ocean's water density at great depths, we also argue that a no-slip condition at the bottom of the ocean, if sufficiently deep, still effectively implies (up to a very small error) the orthogonality of the Ekman transport and the wind-stress.

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BibTeX RIS

BibTeX

@misc{puntini2026note,
  title = {{A note on the lower boundary condition for Ekman flows}},
  author = {Christian Puntini and Luigi Roberti},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.29261},
  eprint = {2607.29261},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.29261},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A note on the lower boundary condition for Ekman flows
AU  - Christian Puntini
AU  - Luigi Roberti
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2607.29261
UR  - https://arxiv.org/abs/2607.29261
AB  - We show that the assumption of a stress-free boundary condition at a finite intermediate depth, namely, at the bottom of the Ekman layer, in the analysis of wind-driven ocean flows necessarily leads to an unphysical current profile. Indeed, if the z-derivative of the fluid velocity vanishes at a given depth, then this depth necessarily corresponds to a minimum of the velocity profile, with the velocity increasing beneath it. Using a WKB ansatz based on the small variations of the ocean's water density at great depths, we also argue that a no-slip condition at the bottom of the ocean, if sufficiently deep, still effectively implies (up to a very small error) the orthogonality of the Ekman transport and the wind-stress.
ER  -

Abstract

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