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We propose to model physical effects at the sharp density interface between atmosphere and ocean with the help of diffuse interface approaches for multiphase flows with variable densities. We use the thermodynamical consistent variable density model proposed in Abels et al. (Mathematical Models and Methods in Applied Sciences 22:1150013, 2012). This results in a Cahn–Hilliard-/Navier–Stokes-type system which we complement with tangential Dirichlet boundary conditions to incorporate the effect of wind in the atmosphere. Wind is responsible for waves at the surface of the ocean, whose dynamics have an important impact on the \(\textit{CO}_{2}\)—exchange between ocean and atmosphere. We tackle this mathematical model numerically with fully adaptive and integrated numerical schemes tailored to the simulation of variable density multiphase flows governed by diffuse interface models. Here, fully adaptive, integrated, efficient, and reliable means that the mesh resolution is chosen by the numerical algorithm according to a prescribed error tolerance in the a posteriori error control on the basis of residual-based error indicators, which allow to estimate the true error from below (efficient) and from above (reliable). Our approach is based on the work of Hintermuller et al. (Journal of Computational Physics 235:810–827, 2013), Garcke et al. (Applied Numerical Mathematics 99:151–171, 2016), where a fully adaptive efficient and reliable numerical method for the simulation of two-dimensional multiphase flows with variable densities is developed. In a first step, we incorporate the stimulation of surface waves via appropriate volume forcing.

Diffuse Interface Approaches in Atmosphere and Ocean—Modeling and Numerical Implementation