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This research is funded by the Marine, Coastal and Polar Systems (PACES II) of the Hermann von Helmholtz-Gemeinschaft Deutscher Forschungszentren e.V.. K.S. is funded by the European Commission Erasmus Mundus Masters Course in Environmental Sciences, Policy and Management (MESPOM). C.L., O.K. and K.K. received support from the "Modular System for Shelves and Coasts" (MOSSCO) grant provided by the Bundesministerium für Bildung und Forschung under agreements 03F0667A and 03F0667B; O.K. and K.W. are also supported by the DFG priority programme 1704 "Flexibility matters: Interplay between trait diversity and ecological dynamics using aquatic communities as model system" (DynaTrait) under grant agreement KE 1970/1-1. K.K. is furthermore supported by the DFG Collaborative Research Center "Energy Transfers in Atmosphere and Ocean" TRR181. We thank all co-developers of the model coupling framework MOSSCO, foremost M. Hassan Nasermoaddeli and Richard Hofmeister. The authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at Forschungszentrum Jülich. We are grateful to the open source community that provided many of the tools used in this study, including but not limited to the communities developing ESMF, FABM, GETM and GOTM.

The state of the art of the numerics of hydrostatic structured-grid coastal ocean models is reviewed here. First, some fundamental differences in the hydrodynamics of the coastal ocean, such as the large surface elevation variation compared to the mean water depth, are contrasted against large scale ocean dynamics. Then the hydrodynamic equations as they are used in coastal ocean models as well as in large scale ocean models are presented, including parameterisations for turbulent transports. As steps towards discretisation, coordinate transformations and spatial discretisations based on a finite-volume approach are discussed with focus on the specific requirements for coastal ocean models. As in large scale ocean models, splitting of internal and external modes is essential also for coastal ocean models, but specific care is needed when drying & flooding of intertidal flats is included. As one obvious characteristic of coastal ocean models, open boundaries occur and need to be treated in a way that correct model forcing from outside is transmitted to the model domain without reflecting waves from the inside. Here, also new developments in two-way nesting are presented. Single processes such as internal inertia-gravity waves, advection and turbulence closure models are discussed with focus on the coastal scales. Some overview on existing hydrostatic structured-grid coastal ocean models is given, including their extensions towards non-hydrostatic models. Finally, an outlook on future perspectives is made.