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In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn– Hilliard–Navier–Stokes equations in the free flow region and Cahn–Hilliard–Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions. 

Abstract Mechanisms behind the phenomenon of Arctic amplification are widely discussed. To contribute to this debate, the (AC) 3 project was established in 2016 ( www.ac3-tr.de/ ). It comprises modeling and data analysis efforts as well as observational elements. The project has assembled a wealth of ground-based, airborne, shipborne, and satellite data of physical, chemical, and meteorological properties of the Arctic atmosphere, cryosphere, and upper ocean that are available for the Arctic climate research community. Short-term changes and indications of long-term trends in Arctic climate parameters have been detected using existing and new data. For example, a distinct atmospheric moistening, an increase of regional storm activities, an amplified winter warming in the Svalbard and North Pole regions, and a decrease of sea ice thickness in the Fram Strait and of snow depth on sea ice have been identified. A positive trend of tropospheric bromine monoxide (BrO) column densities during polar spring was verified. Local marine/biogenic sources for cloud condensation nuclei and ice nucleating particles were found. Atmospheric–ocean and radiative transfer models were advanced by applying new parameterizations of surface albedo, cloud droplet activation, convective plumes and related processes over leads, and turbulent transfer coefficients for stable surface layers. Four modes of the surface radiative energy budget were explored and reproduced by simulations. To advance the future synthesis of the results, cross-cutting activities are being developed aiming to answer key questions in four focus areas: lapse rate feedback, surface processes, Arctic mixed-phase clouds, and airmass transport and transformation.