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For the primitive equations of large-scale atmosphere and ocean dynamics, we study the problem of determining by means of a variational data assimilation algorithm initial conditions that generate strong solutions which minimize the distance to a given set of time-distributed observations. We suggest a modification of the adjoint algorithm whose novel elements is to use norms in the variational cost functional that reflects the H 1 -regularity of strong solutions of the primitive equations. For such a cost functional, we prove the existence of minima and a first-order adjoint condition for strong solutions that provides the basis for computing these minima. We prove the local convergence of a gradient-based descent algorithm to optimal initial conditions using the second-order adjoint primitive equations. The algorithmic modifications due to the H 1 -norms are straightforwardly to implement into a variational algorithm that employs the standard L 2 -metrics.

The non‐local model of mixing based on internal wave breaking, IDEMIX, is implemented as an enhancement of a turbulent kinetic energy closure model in three non‐eddy resolving general circulation ocean models that differ in the discretization and choice of computational grids. In IDEMIX internal wave energy is generated by an energy flux resulting from near‐inertial waves induced by wind forcing at the surface, and at the bottom, by an energy flux that parameterizes the transfer of energy between baroclinic and barotropic tides. In all model simulations with IDEMIX, the mixing work is increased compared to the reference solutions without IDEMIX, reaching values in better agreement with finestructure observations. Furthermore, the horizontal structure of the mixing work is more realistic as a consequence of the heterogeneous forcing functions. All models with IDEMIX simulate deeper thermocline depths related to stronger shallow overturning cells in the Indo‐Pacific. In the North Atlantic, deeper mixed layers in simulations with IDEMIX are associated with an increased Atlantic overturning circulation and an increase of northward heat transports toward more realistic values. The response of the deep Indo‐Pacific overturning circulation and the weak bottom cell of the Atlantic to the inclusion of IDEMIX is incoherent between the models, suggesting that additional unidentified processes and numerical mixing may confound the analysis. Applying different tidal forcing functions leads to simulation differences that are small compared to differences between the different models or between simulations with IDEMIX and without IDEMIX. Plain Language Summary Waves in the ocean interior play a fundamental role for ocean dynamics since they can carry energy over long distances and, once they break, lead to turbulent mixing. This turbulent mixing can cause dense water masses to rise from the deep ocean with a direct impact on large‐scale currents. The wave dynamics occur on spatial scales that cannot be resolved in global ocean or climate models. To account for these processes, we apply the new parameterization IDEMIX that describes internal wave generation, propagation, and mixing. Using three different ocean models with and without IDEMIX ensures that we can identify model‐specific effects of the parameterization and discriminate them from those independent of the model. We find that the simulated mixing patterns agree better with observations once IDEMIX is applied. Large‐scale currents and the vertical temperature distribution are substantially affected by the internal wave parameterization. Whether this leads to an improved agreement with observed currents and water mass properties depends on the specific model and on numerical effects. In most cases, simulations with IDEMIX are not very sensitive to details of how the internal wave model is driven by tidal energy input. Key Points the IDEMIX closure for a consistent representation of internal wave‐induced mixing is evaluated in three state‐of‐the‐art ocean models only in simulations with IDEMIX can the models reproduce the magnitude and spatial variability of the observed mixing work most changes with IDEMIX can be attributed to stronger mixing, but some effects are confounded by other processes and numerical mixing

Correctly capturing the transition to turbulence in a barotropic instability requires fine spatial resolution. To reduce computational cost, we propose a dynamic super-resolution approach where a transient simulation on a coarse mesh is frequently corrected using a U-net-type neural network. For the nonlinear shallow water equations, we demonstrate that a simulation with the Icosahedral Nonhydrostatic ocean model with a 20 km resolution plus dynamic super-resolution trained on a 2.5km resolution achieves discretization errors comparable to a simulation with 10 km resolution. The neural network, originally developed for image-based super-resolution in post-processing, is trained to compute the difference between solutions on both meshes and is used to correct the coarse mesh solution every 12 h. We show that the ML-corrected coarse solution correctly maintains a balanced flow and captures the transition to turbulence in line with the higher resolution simulation. After an 8 d simulation, the L 2 -error of the corrected run is similar to a simulation run on a finer mesh. While mass is conserved in the corrected runs, we observe some spurious generation of kinetic energy.

We study the problem of determining through a variational data assimilation approach the initial condition for a coupled set of nonlinear partial differential equations from which a model trajectory emerges in agreement with a given set of time-distributed observations. The partial differential equations describe an idealized coupled atmospheric–ocean model on a rotating torus. The model consist of the viscous shallow-water equations in geophysical scaling that represents the large-scale atmospheric dynamics coupled via a simplified but physically plausible coupling to a model that represents the large-scale ocean dynamics and consists of the incompressible two-dimensional Navier–Stokes equations and an advection–diffusion equation. We propose a variational algorithm (4D-Var) of the coupled data assimilation problem that is solvable and computable. This algorithm relies on the use of a variational cost functional that is tailored to the regularity of the coupled model as well as to the regularity of the observations by means of derivative-based norms. We support this proposal by developing regularity results for an idealized coupled atmospheric–ocean model using the concept of classical solutions. Based on these results we formulate a suitable cost functional. For this cost functional we prove the existence of optimal initial conditions in the sense of minimizers of the cost functional and characterize the minimizers by a first-order necessary condition involving the coupled adjoint equations. We prove local convergence of a gradient-based descent algorithm to optimal initial conditions using second-order adjoint information. Instrumental for our results is the use of suitable Sobolev norms instead of the standard Lebesgue norms in the cost functional. The index of the actual Sobolev space provides an additional scale selective mechanisms in the variational algorithm.

We consider the hydrostatic Boussinesq equations of global ocean dynamics, also known as the “primitive equations”, coupled to advection–diffusion equations for temperature and salt. The system of equations is closed by an equation of state that expresses density as a function of temperature, salinity and pressure. The equation of state TEOS-10, the official description of seawater and ice properties in marine science of the Intergovernmental Oceanographic Commission, is the most accurate equations of state with respect to ocean observation and rests on the firm theoretical foundation of the Gibbs formalism of thermodynamics. We study several specifications of the TEOS-10 equation of state that comply with the assumption underlying the primitive equations. These equations of state take the form of high-order polynomials or rational functions of temperature, salinity and pressure. The ocean primitive equations with a nonlinear equation of state describe richer dynamical phenomena than the system with a linear equation of state. We prove well-posedness for the ocean primitive equations with nonlinear thermodynamics in the Sobolev space H 1 . The proof rests upon the fundamental work of Cao and Titi (Ann. Math. 166:245–267, 2007) and also on the results of Kukavica and Ziane (Nonlinearity 20:2739–2753, 2007). Alternative and older nonlinear equations of state are also considered. Our results narrow the gap between the mathematical analysis of the ocean primitive equations and the equations underlying numerical ocean models used in ocean and climate science.

Grids of variable resolution are of great interest in atmosphere and ocean modeling as they offer a route to higher local resolution and improved solutions. On the other hand there are changes in grid resolution considered to be problematic because of the errors they create between coarse and fine parts of a grid due to reflection and scattering of waves. On complex multidimensional domains these errors resist theoretical investigation and demand numerical experiments. With a low-order hybrid continuous/discontinuous finite-element model of the inviscid and viscous shallow-water equations a numerical study is carried out that investigates the influence of grid refinement on critical features such as wave propagation, turbulent cascades, and the representation of geostrophic balance. The refinement technique the authors use is static h refinement, where additional grid cells are inserted in regions of interest known a priori. The numerical tests include planar and spherical geometry as well as flows with boundaries and are chosen to address the impact of abrupt changes in resolution or the influence of the shape of the transition zone. For the specific finite-element model under investigation, the simulations suggest that grid refinement does not deteriorate geostrophic balance and turbulent cascades and the shape of mesh transition zones appears to be less important than expected. However, the results show that the static local refinement is able to reduce the local error, but not necessarily the global error and convergence properties with resolution are changed. The relatively simple tests already illustrate that grid refinement has to go along with a simultaneous change of the parameterization schemes.

In the era of exascale computing, machines with unprecedented computing power are available. Making efficient use of these massively parallel machines, with millions of cores, presents a new challenge. Multi-level and multi-dimensional parallelism will be needed to meet this challenge.Coarse-grained component concurrency provides an additional parallelism dimension that complements typically used parallelization methods such as domain decomposition and loop-level shared-memory approaches. While these parallelization methods are data-parallel techniques, and they decompose the data space, component concurrency is a function-parallel technique, and it decomposes the algorithmic space. This additional dimension of parallelism allows us to extend scalability beyond the limits set by established parallelization techniques. It also offers a way to maintain performance (by using more compute power) when the model complexity is increased by adding components, such as biogeochemistry or ice sheet models. Furthermore, concurrency allows each component to run on different hardware, thus leveraging the usage of heterogeneous hardware configurations.In this work we study the characteristics of component concurrency and analyse its behaviour in a general context. The analysis shows that component concurrency increases the “parallel workload”, improving the scalability under certain conditions. These generic considerations are complemented by an analysis of a specific case, namely the coarse-grained concurrency in the multi-level parallelism context of two components of the ICON modelling system: the ICON ocean model ICON-O and the marine biogeochemistry model HAMOCC. The additional computational cost incurred by the biogeochemistry module is about 3 times that of the ICON-O ocean stand alone model, and data parallelization techniques (domain decomposition and loop-level shared-memory parallelization) present a scaling limit that impedes the computational performance of the combined ICON-O–HAMOCC model. Scaling experiments, with and without concurrency, show that component concurrency extends the scaling, in cases doubling the parallel efficiency. The experiments' scaling results are in agreement with the theoretical analysis.

We develop a new Earth System model configuration framed into the ICON architecture, which provides the baseline for the next generation of climate predictions and projections (hereafter ICON XPP – where XPP stands for eXtended Predictions and Projections). ICON XPP comprises the atmospheric component of the numerical weather prediction (ICON NWP), the ICON ocean and land surface components, and an ensemble-variational data assimilation system, all adjusted to an Earth System model for pursuing climate research and operational climate forecasting. Two baseline configurations are presented: (1) a 160 km atmosphere and a 40 km ocean resolution, and (2) 80 km atmosphere and 20 km ocean resolution. A CMIP DECK (Diagnostic, Evaluation and Characterization of Klima) experimentation framework is used for a first evaluation. ICON XPP depicts the basic properties of the coupled climate. The pre-industrial climate shows a top-of-atmosphere balanced radiation budget and a mean global near-surface temperature of 13.8–14.0 °C. The ocean shows circulation strengths in the range of the observed values, such as the AMOC at 16–18 Sv and the flows through the common passages. The current climate is characterized by a trend in the global mean temperature of ∼ 1.2 °C since the 1850s, similar to reference datasets. Regionally, the hydroclimate differs greatly from observed conditions. For example, the inter-tropical convergence zone (ITCZ) has a double peak and a wet southern subtropical branch across the oceans. Further, the Southern Ocean sea surface temperature has a strong positive mean bias with temperatures up to 5 °C higher than observations. Dynamical processes, such as El Niño/Southern Oscillation (ENSO) performs similarly to CMIP6-like coupled models. Tropical waves and the Madden-Julian Oscillation are well captured, and the 40 km atmospheric configuration has a spontaneous weak quasi-biennial oscillation. The atmospheric dynamics in the northern extra-tropics of both configurations represent well the position of the jet stream as well as the influences of the transient momentum transports and their feedbacks on the jet stream. Overall, ICON XPP performs similarly to climate models performed in CMIP6 making it a good basis for climate forecasts and projections, and climate research.

This paper evaluates barotropic tides simulated by a newly developed multi-layer ocean general circulation, ICON-O, and assesses processes and model configurations that can impact the quality of the simulated tides. Such an investigation is crucial for applications addressing internal tides that are much more difficult to evaluate than the barotropic tides. Although not specially tuned for tides and not constrained by any observations, ICON-O is capable of producing the main features of the open-ocean barotropic tides as described by the geographical distributions of amplitude, phase, and amphidromic points. An error analysis shows, however, that the open-ocean tides simulated by ICON-O are less accurate than those simulated by two other ocean general circulation models (OGCMs), especially when not properly adjusting the time step and the parameters used in the time-stepping scheme. Based on a suite of tidal experiments, we show that an increase in horizontal resolution only improves tides in shallow waters. Relevant for using ICON-O with its telescoping grid capacity, we show that spatial inhomogeneity does not deteriorate the quality of the simulated tides. We further show that implementing a parameterization of topographic wave drag improves the quality of the simulated tides in deep ocean independent of the model configuration used, whereas the implementation of a self-attraction and loading (SAL) parameterization in a low-resolution (40 km) version of ICON-O degrades the quality of tides in shallow ocean. Finally, we show that the quality of tides simulated by ICON-O with low resolution (40 km) can be significantly improved by adjusting the time step or the parameters in the time-stepping scheme used for obtaining the model solution.

Four-dimensional variational data assimilation (4D-Var) produces unavoidable inaccuracies in the models initial state vector. In this paper the authors investigate a novel variational error estimation method to calculate these inaccuracies. The impacts of model, background, and observational errors on the state estimate produced by 4D-Var are analyzed by applying the variational error estimation method. The structure of the method is similar to the conventional 4D-Var, with the differences in that (i) instead of observations it assimilates observational errors, and (ii) the original model equations (used in 4D-Var as constraints) are first linearized with respect to a small perturbation in the initial state vector and then used as the constraints. The authors then carry out a proof-of-concept study and validate the reliability of this method through multiple twin experiments on the basis of a 2D shallow-water model. All required differentiated models were generated by means of algorithmic differentiation directly from the nonlinear model source code. The experiments reveal that the suggested method works well in a wide range of assimilation windows and types of observational and model errors and can be recommended for error estimation and prediction in data assimilation.