  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 variable-density model
proposed in \cite{m6:AbelsGarckeGruen_CHNSmodell}. 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 $CO_2-$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, {\it 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 {\it 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
\cite{m6:HintermuellerHinzeKahle_adaptiveCHNS,m6:GarckeHinzeKahle_CHNS_AGG_linearStableTimeDisc},
where a fully adaptive efficient and reliable numerical method for the
simulation of two-dimensional multiphase flows with variable densities is
developed. We incorporate the stimulation of surface waves via appropriate
boundary conditions.
