Wave-Driven Surface Ocean Currents

Definitions and mathematical statement

The supplied problem concerns wind-generated ocean waves and surface ocean currents in the setting of atmosphere--ocean interaction. Here wave-field evolution refers descriptively to the evolution of the wind-wave field, and surface ocean currents to the associated ocean circulation near the surface. The record does not give mathematical definitions of either quantity or specify an averaging convention.

Research problem. Explain mathematically how the evolution of the wind-wave field drives surface ocean currents, accounting for the interaction between waves and currents. The supplied background describes waves and currents as linked through several mechanisms. Its dated literature assessment identifies a general theory of two-way interaction as the outstanding objective; the explicit source question asks specifically about the influence of evolving waves on currents.

Unresolved formulation. No governing equations, spatial domain, initial or boundary conditions, forcing assumptions, physical parameter regime, or criterion for a satisfactory explanation are supplied. In particular, the record does not distinguish Eulerian currents, Lagrangian transport, and averaged velocity fields. Consequently, it does not determine a precise quantified conjecture or a specific mathematical problem to prove. Choosing such a formulation would require additional source evidence.

The supplied assessment mentions radiation stress, Stokes drift, Langmuir circulation, and phase-averaged models as addressing important mechanisms. These are contextual qualifications, not prescribed components of a specified model or independently verified claims of completeness.

Short English statement

Explain how evolving wind waves drive surface ocean currents within atmosphere--ocean interaction. The supplied curation also calls for a general theory of two-way wave--current interaction. The record leaves the model, current definition, assumptions, and mathematical success criteria unspecified, so this remains a research question requiring further formulation.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30005236$, title Wave-Driven Surface Ocean Currents, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2022.
• Supplied report citation: Multiscale Wave-Turbulence Dynamics in the Atmosphere and Ocean (2023), Oberwolfach Reports, pp. 2467--2510, DOI 10.4171/owr/2022/42. https://doi.org/10.4171/owr/2022/42. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature reference: A consistent phase-averaged model. https://arxiv.org/abs/2602.21976.
• Supplied literature reference: Mellor, wave-current interaction. https://doi.org/10.1002/2016JC011768.
• Supplied literature reference: Oberwolfach Report 42/2022. https://ems.press/content/serial-article-files/46979.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that a general two-way theory remains open despite mechanisms addressed by radiation stress, Stokes drift, Langmuir circulation, and phase-averaged models. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.