Knut Klingbeil

The Greenland Ice Sheet is a major contributor to global sea level rise. While surface melting is driven by the atmosphere, oceanic processes melt the floating glacier tongues in northern Greenland from below. Because direct observations beneath these tongues are limited, numerical models are crucial for a detailed understanding of ice–ocean interactions. To study the oceanic melting of Greenland's largest floating ice tongue and the circulation induced by meltwater, we developed a high-resolution three-dimensional model of the 79° North Glacier fjord (500 m horizontal resolution, 100 adaptive vertical layers). Our simulation reveals that melting at the ice–ocean interface is driven by three distinct subglacial plumes with different signatures in temperature–salinity space. The paths of these buoyant gravity currents are set by the ice topography, particularly basal channels in the ice shelf, and the Coriolis effect. One plume flows around cone-like features in the ice base with dimensions comparable to the Rossby radius, suggesting that these cones may have formed through plume-induced melting. At about 100 to 200 m depth, the plumes detach from the ice and export meltwater out of the fjord toward the open ocean. Heat for melting is supplied by a dense bottom plume flowing into the glacier cavity across the sill at the fjord entrance. Downstream of the sill, hydraulic control leads to enhanced mixing between plume and ambient water, cooling the inflow and reducing the amount of heat that reaches the glacier base. Our model resolves these details of the plumes in the ice cavity, improving the understanding of ocean-driven melt below glacier tongues.

Quantifying interactions between distinct water bodies is critical yet challenging for understanding ocean and estuarine dynamics. We introduce the coincidence index (CI), defined as the product of two arbitrary tracers (, ), to diagnose where two water masses coexist and mix. By deriving a transport equation for the CI, we identify a single active production term, , that pinpoints physical mixing locations, while advection and diffusion only redistribute CI. We apply this method in two numerical models: Oceananigans for a Large‐Eddy simulation of river confluence, and General Estuarine Transport Model for the analysis of the interaction of water masses from the Elbe and Weser river plumes in the German Bight. In both, the CI accurately highlights mixing interfaces; for example, the CI pattern from the river confluence simulation matches observations from the Lancang River. In the German Bight, the CI reveals a narrow, persistent interface between river plumes, with vertical diffusion dominating production. Artificial numerical production of CI, arising from the discretization of advection, is also diagnosed but found to be less significant. The CI method thus enables precise, quantitative diagnostics of water mass mixing from laboratory to realistic estuarine systems, advancing our understanding of transport and mixing processes central to aquatic environments.

This review paper presents, explains and discusses major aspects of estuarine mixing which is defined as the destruction of salinity variance. Due to the large amounts of brackish water in estuaries produced by mixing of fresh river discharge and salty ocean water, mixing is one major characteristic of what is an estuary. In this review, mixing is quantified locally as well as on estuary-wide scales. Diagnostics of integrated mixing are given for estuarine volumes bounded by transects as well as isohalines (surfaces of constant salinity) moving with the flow. It is shown how entrainment across a moving isohaline surface depends on gradients of turbulent salt flux and mixing per salinity class. Various relations are derived that link estuarine salt mixing to other estuarine properties such as the freshwater discharge and the bulk estuarine circulation. For estuaries bounded towards the ocean by a fixed transect, the Knudsen mixing law is explained, where estuarine mixing is the product of the Knudsen salinities of inflowing and outflowing water masses and the river discharge. When the estuarine volume is bounded by a moving isohaline surface of salinity S, mixing inside the estuary is simply the product of S² and the river discharge. Major processes that drive estuarine mixing are presented on various time scales (tidal, fortnightly, weather and discharge time scales) and spatial scales (channel-shoal interaction, mixing fronts). As underlying methods for the quantification of mixing, observational concepts, as well as numerical modelling methods such as consistent turbulence closure modelling and numerical mixing analyses are presented. As an outlook, some future perspectives are sketched. Many of the concepts presented in this review are illustrated using simulation results from a numerical model setup of the Elbe River estuary.

River plumes play an essential role in the transport of terrestrially derived materials (like nutrients, sediments, pollutants, etc.) into the coastal ocean. Quantifying the cross‐shore transport in river plumes can help to better understand the contribution of river‐borne substances to marine biogeochemical cycles and to parameterize these processes in global ocean models which are usually too coarse to resolve individual rivers. It is known that besides external factors (like runoff, latitude, wind, and tides) also internal estuarine processes like salt mixing affect the exchange flow between an estuary and the coastal ocean. A theoretical framework to separate the plume and the estuary mixing in isohaline coordinates is presented. An idealized coastal ocean model setup resolving the whole plume‐estuary continuum is used to validate the theoretical relation and to study the link between the estuarine pre‐conditioning and the cross‐shore export of river water under different forcing scenarios. It is found that the most effective cross‐shore transport of river water happens under moderately upwelling favorable wind conditions and weak tidal forcing. This scenario is characterized by relatively weak estuarine mixing, strong stratification, and little interaction between the surface and bottom boundary layers such that a thin layer of buoyant river water can extend far into the ocean. We conclude that reduced estuarine mixing is indicative of an enhanced accumulation of fresh water near the shore, but is not directly related to the cross‐shore transport in river plumes.

This review paper presents, explains and discusses major aspects of estuarine mixing which is defined as the destruction of salinity variance. Due to the large amounts of brackish water in estuaries produced by mixing of fresh river discharge and salty ocean water, mixing is one major characteristic of what is an estuary. In this review, mixing is quantified locally as well as on estuary-wide scales. Diagnostics of integrated mixing are given for estuarine volumes bounded by transects as well as isohalines (surfaces of constant salinity) moving with the flow. It is shown how entrainment across a moving isohaline surface depends on gradients of turbulent salt flux and mixing per salinity class. Various relations are derived that link estuarine salt mixing to other estuarine properties such as the freshwater discharge and the bulk estuarine circulation. For estuaries bounded towards the ocean by a fixed transect, the Knudsen mixing law is explained, where estuarine mixing is the product of the Knudsen salinities of inflowing and outflowing water masses and the river discharge. When the estuarine volume is bounded by a moving isohaline surface of salinity S, mixing inside the estuary is simply the product of S² and the river discharge. Major processes that drive estuarine mixing are presented on various time scales (tidal, fortnightly, weather and discharge time scales) and spatial scales (channel-shoal interaction, mixing fronts). As underlying methods for the quantification of mixing, observational concepts, as well as numerical modelling methods such as consistent turbulence closure modelling and numerical mixing analyses are presented. As an outlook, some future perspectives are sketched. Many of the concepts presented in this review are illustrated using simulation results from a numerical model setup of the Elbe River estuary.

This review paper presents, explains and discusses major aspects of estuarine mixing which is defined as the destruction of salinity variance. Due to the large amounts of brackish water in estuaries produced by mixing of fresh river discharge and salty ocean water, mixing is one major characteristic of what is an estuary. In this review, mixing is quantified locally as well as on estuary-wide scales. Diagnostics of integrated mixing are given for estuarine volumes bounded by transects as well as isohalines (surfaces of constant salinity) moving with the flow. It is shown how entrainment across a moving isohaline surface depends on gradients of turbulent salt flux and mixing per salinity class. Various relations are derived that link estuarine salt mixing to other estuarine properties such as the freshwater discharge and the bulk estuarine circulation. For estuaries bounded towards the ocean by a fixed transect, the Knudsen mixing law is explained, where estuarine mixing is the product of the Knudsen salinities of inflowing and outflowing water masses and the river discharge. When the estuarine volume is bounded by a moving isohaline surface of salinity S, mixing inside the estuary is simply the product of S² and the river discharge. Major processes that drive estuarine mixing are presented on various time scales (tidal, fortnightly, weather and discharge time scales) and spatial scales (channel-shoal interaction, mixing fronts). As underlying methods for the quantification of mixing, observational concepts, as well as numerical modelling methods such as consistent turbulence closure modelling and numerical mixing analyses are presented. As an outlook, some future perspectives are sketched. Many of the concepts presented in this review are illustrated using simulation results from a numerical model setup of the Elbe River estuary.

This review paper presents, explains and discusses major aspects of estuarine mixing which is defined as the destruction of salinity variance. Due to the large amounts of brackish water in estuaries produced by mixing of fresh river discharge and salty ocean water, mixing is one major characteristic of what is an estuary. In this review, mixing is quantified locally as well as on estuary-wide scales. Diagnostics of integrated mixing are given for estuarine volumes bounded by transects as well as isohalines (surfaces of constant salinity) moving with the flow. It is shown how entrainment across a moving isohaline surface depends on gradients of turbulent salt flux and mixing per salinity class. Various relations are derived that link estuarine salt mixing to other estuarine properties such as the freshwater discharge and the bulk estuarine circulation. For estuaries bounded towards the ocean by a fixed transect, the Knudsen mixing law is explained, where estuarine mixing is the product of the Knudsen salinities of inflowing and outflowing water masses and the river discharge. When the estuarine volume is bounded by a moving isohaline surface of salinity S, mixing inside the estuary is simply the product of S² and the river discharge. Major processes that drive estuarine mixing are presented on various time scales (tidal, fortnightly, weather and discharge time scales) and spatial scales (channel-shoal interaction, mixing fronts). As underlying methods for the quantification of mixing, observational concepts, as well as numerical modelling methods such as consistent turbulence closure modelling and numerical mixing analyses are presented. As an outlook, some future perspectives are sketched. Many of the concepts presented in this review are illustrated using simulation results from a numerical model setup of the Elbe River estuary.

Mixing and Water Mass Transformations (WMT) in the ocean are key to the dynamics, ranging from small-scale dissipation to large-scale overturning circulations. Therefore, the quantification of mixing and WMT in ocean models is of fundamental importance and enables a more detailed analysis of oceanic processes. In this presentation different diagnostic approaches developed during the last years are reviewed and compared. These diagnostic methods also offer to investigate the contributions originating from discretization errors in the numerical transport schemes, causing spurious numerical mixing and spurious overturning circulations. Subtleties of the methods and recent refinements will be presented to provide an accurate analysis framework that is consistent with analytical theories and the discrete model equations.

This review paper presents, explains and discusses major aspects of estuarine mixing which is defined as the destruction of salinity variance. Due to the large amounts of brackish water in estuaries produced by mixing of fresh river discharge and salty ocean water, mixing is one major characteristic of what is an estuary. In this review, mixing is quantified locally as well as on estuary-wide scales. Diagnostics of integrated mixing are given for estuarine volumes bounded by transects as well as isohalines (surfaces of constant salinity) moving with the flow. It is shown how entrainment across a moving isohaline surface depends on gradients of turbulent salt flux and mixing per salinity class. Various relations are derived that link estuarine salt mixing to other estuarine properties such as the freshwater discharge and the bulk estuarine circulation. For estuaries bounded towards the ocean by a fixed transect, the Knudsen mixing law is explained, where estuarine mixing is the product of the Knudsen salinities of inflowing and outflowing water masses and the river discharge. When the estuarine volume is bounded by a moving isohaline surface of salinity S, mixing inside the estuary is simply the product of S² and the river discharge. Major processes that drive estuarine mixing are presented on various time scales (tidal, fortnightly, weather and discharge time scales) and spatial scales (channel-shoal interaction, mixing fronts). As underlying methods for the quantification of mixing, observational concepts, as well as numerical modelling methods such as consistent turbulence closure modelling and numerical mixing analyses are presented. As an outlook, some future perspectives are sketched. Many of the concepts presented in this review are illustrated using simulation results from a numerical model setup of the Elbe River estuary.

The Greenland Ice Sheet is a major contributor to global sea level rise. While surface melting is driven by the atmosphere, oceanic processes melt the floating glacier tongues in northern Greenland from below. Because direct observations beneath these tongues are limited, numerical models are crucial for a detailed understanding of ice–ocean interactions. To study the oceanic melting of Greenland's largest floating ice tongue and the circulation induced by meltwater, we developed a high-resolution three-dimensional model of the 79° North Glacier fjord. Our simulation reveals that basal melting is driven by three distinct subglacial plumes with different signatures in temperature–salinity space. The paths of these buoyant gravity currents are set by the ice topography, particularly subglacial channels, and the Coriolis effect. One plume flows around cone-like features in the ice base with dimensions comparable to the Rossby radius, suggesting that the ice cones might be formed by the plume through subglacial melting. At about 100 m to 200 m depth, the plumes detach from the ice and export meltwater out of the fjord toward the open ocean. Heat for melting is supplied by a dense bottom plume flowing into the glacier cavity across the sill at the fjord entrance. Downstream of the sill, hydraulic control leads to enhanced mixing between plume and ambient water, cooling the inflow and reducing the amount of heat that reaches the glacier base. Our model resolves these details of the plumes in the ice cavity, improving the understanding of ocean-driven melt below glacier tongues.