This paper provides a comprehensive derivation of the total energy equations for the atmospheric components of Earth System Models (ESMs). The assumptions and approximations made in this derivation are motivated and discussed. In particular, it is emphasized that closing the energy budget is conceptually challenging and hard to achieve in practice without resorting to ad hoc fixers. As a concrete example, the energy budget terms are diagnosed in a realistic climate simulation using a global atmosphere model. The largest total energy errors in this example are spurious dynamical core energy dissipation, thermodynamic inconsistencies (e.g., coupling parameterizations with the host model) and missing processes/terms associated with falling precipitation and evaporation (e.g., enthalpy flux between components). The latter two errors are not, in general, reduced by increasing horizontal resolution. They are due to incomplete thermodynamic and dynamic formulations. Future research directions are proposed to reconcile and improve thermodynamics formulations and conservation principles.

The 3rd-order upstream advection scheme for scalars on the Voronoi C-grid, once introduced by Skamarock and Gassmann (2011), is applied to horizontal momentum advection.A prerequisite is that the 2nd-order momentum advection is available in advection form for a trivariate coordinate system, so that the higher order terms can be formulated as an add-on.Three key ingredients for a successful application are (i) the determination of the advecting velocity, (ii) the determination of directional Laplacians of wind components and (iii) the determination of the upstream direction.The scheme is tested in two settings, a shallow water framework on the regular hexagonal mesh and the baroclinic wave test on the sphere, where the mesh is slightly deformed.In both cases, the trailing ripples and waves known to represent dispersion errors are impressively reduced.If they are not removed, they can lead to spurious excitation of gravity waves or wavy vorticity patterns.After upscale error growth, they can no longer be identified as a result of numerical errors.The effects of the 3rd-order upstream add-on and a Smagorinsky diffusion are compared.The Smagorinsky model reduces the amplitude of the mentioned waves, but does not erase them.With regard to the dissipation properties, the Smagorinsky diffusion is in accordance with the 2nd law of thermodynamics and dissipation is locally only positive.In contrast, dissipation can be locally negative in runs with the 3rd-order upstream add-on.Therefore, physical and numerical requirements cannot be fulfilled simultaneously.

Higher order upwind biased advection schemes are often used for potential temperature advection in dynamical cores of atmospheric models. The inherent diffusive and anti-diffusive fluxes are interpreted here as the effect of irreversible sub-gridscale dynamics. For those, total energy conservation and positive internal entropy production must be guaranteed. As a consequence of energy conservation, the pressure gradient term should be formulated in Exner pressure form. The presence of local antidiffusive fluxes in potential temperature advection schemes foils the validity of the second law of thermodynamics. Due to this failure, a spurious wind acceleration into the wrong direction is locally induced via the pressure gradient term. When correcting the advection scheme to be more entropically consistent, the spurious acceleration is avoided, but two side effects come to the fore: (i) the overall accuracy of the advection scheme decreases and (ii) the now purely diffusive fluxes become more discontinuous compared to the original ones, which leads to more sudden body forces in the momentum equation. Therefore the amplitudes of excited gravity waves from jets and fronts increase compared to the original formulation with inherent local antidiffusive fluxes.The means used for supporting the argumentation line are theoretical arguments concerning total energy conservation and internal entropy production, pure advection tests, one-dimensional advection-dynamics interaction tests and evaluation of runs with a global atmospheric dry dynamical core.

For the protection of people and society against harm and health threats -- especially for the COVID-19 pandemic -- a variety of different disciplines needs to be involved. The data collection of very basic and health-related data of individuals in today's highly mobile society does help to plan, protect, and identify next steps health authorities and governments can, shall, or need to plan for or even implement. Thus, every individual, every human, and every inhabitant of the world is the key player -- very different to many past crises'. And since the individual is involved -- all individuals -- his/her (a) health and (b) privacy shall be considered in a very carefully crafted balance, not overruling one aspect with another one or even prioritizing certain aspects. Privacy remains the key. Thus, the solution of the current pandemic's data collection can be based on a fully privacy-preserving application, which can be used by individuals on their mobile devices, such as smartphones, while maintaining at the same time their privacy. Additionally, respective data collected in such a fully distributed setting does help to confine the pandemic and can be achieved in a democratic and very open, but still and especially privacy-protecting world. Therefore, the WeTrace approach and application as described in this paper utilizes the Bluetooth Low Energy (BTE) communication channel, many modern mobile devices offer, where asymmetric cryptography is being applied to allows for the decyphering of a message for that destination it had been intended for. Since literally every other potential participant only listens to random data, even a brute force attack will not succeed. WeTrace and its Open Source implementation is the only known approach so far, which ensures that any receiver of a message knows that this is for him/her, but does not know who the original sender was.

The Ertel's potential vorticity (EPV) budget equation does not see the contribution of an inactive EPV flux component backward difference theta x backward difference B because it drops out when taking the divergence. A part of the actual EPV flux can always be interpreted as such an inactive component and is thus likewise shed from the EPV budget equation. The deviation from this inactive EPV flux is called the active EPV flux and the associated wind is called the active wind. The horizontal active wind is comparable to the ageostrophic wind. The vertical active wind component is similar to the isentropic displacement vertical wind. In contrast to the actual wind, the vertical active wind does not vanish at the surface, because the inactive wind blows along isentropes, which may intersect the ground. Transformed governing equations are derived as functions of the active wind components. The terms on the right of the transformed equations can be scrutinized with respect to their effects on the evolution of the atmospheric state. An idealized baroclinic wave in a dry atmosphere is discussed with focus on the fronts and the generation or depletion of kinetic energy. Since the vertical inactive wind does not necessarily vanish at the surface, the arising vertical active wind is responsible for the cooling (raising of isentropes) and the warming (sinking of isentropes) in the different regions of a cyclone. The new method allows for a unique separation of gravity waves and vortical modes. This facilitates the analysis of gravity wave generation and propagation from jets and fronts.

The atmosphere is a forced-dissipative system. It has to export more entropy than it imports in a steady state. Therefore, the entropy inflow and outflow have to be distinguished from the internal entropy production, which has to be positive in the mean on long timescales. This principle does not only hold for the whole atmosphere, but also for subsystems like individual grid boxes in a numerical model. However, the constraint of positive internal entropy production was not taken into account when developing contemporary subgrid-scale parameterization schemes for atmospheric models. Some of these schemes suit automatically into this framework; some do not. This article discusses the current understanding and scientific discussion of this topic and illustrates possible future development paths.

The ICON-IAP model (Icosahedral non-hydrostatic model at Insitute of Atmospheric Physics) is a non-hydrostatic compressible atmospheric general circulation model. Its design is published in several papers. Gassmann (2011) discusses why a hexagonal C-grid is better suited than a triangular C-grid. The main point is that the overspecified horizontal velocity components must remain linear dependent for all forcing terms in the horizontal velocity equations. This perspective is refined and rounded up in Gassmann (2018), where all linearization variants of the vorticity flux term and the momentum diffusion operator are discussed in depth. Scientific questions remain here the correct treatment of deformed meshes in comparison to regular meshes, where all triangles and hexagons are equal in shape. The advection of scalar variables on the hexagonal C-grid on the sphere is described in Skamarock and Gassmann (2011). Finally, the full model description including three-dimensionality with its terrain-following coordinates and time stepping is documented in Gassmann (2013). The model equations reside of the moist formulation outlined in Gassmann and Herzog (2008).

This article discusses the generalized Coriolis and friction terms on the hexagonal C‐grid from two perspectives: (a) within the linearized discretized momentum equations on an equilateral grid, and (b) as nonlinear terms on a distorted mesh.The discrete linearized momentum equations are formulated using a trivariate coordinate system. The tendencies of the different forcing terms for each wind component must be linearly dependent. This constraint determines unique discretizations for each term. The linearized vorticity flux term around a zonal mean current requires only the four rhombus potential vorticities (PVs) next to an edge. The vector Laplacian must be formulated with the vorticity on vertices defined as the average of three rhombus vorticities.A modified generalized Coriolis term is defined on the deformed mesh. The baroclinic wave test on the sphere does not reveal any sign of nonlinear Hollingsworth instability, even though it is demonstrated that the vector‐invariant form and the advective form of momentum advection are not equivalent.Physical constraints determine the shape of the stress tensor. These are invariance to the addition of solid‐body rotation and a resulting positive‐definite dissipation rate. An appropriate stress‐tensor formulation does not deliver Laplacian momentum diffusion in the linear case. On the deformed mesh, parts of this stress tensor are obtained by a least‐squares reconstruction of wind gradients. This approach avoids spurious deformations diagnosed for constant flow in the vicinity of pentagon cells.

Numerical formulations of turbulent heat fluxes must lead to positive energy dissipation of energy contained in the resolved scales and positive internal entropy production. Current parametrization approaches deliver positive dissipation rates only for free convection, not for forced convection. This contribution explains how positive dissipation rates are achieved by a new formulation of subgrid‐scale terms in the case of stable stratification. This is of importance for the numerical realization of the breakdown of gravity waves. A turbulent atmosphere tends to an isentropic stratification, because in addition to turbulent heat diffusion, pressure work leads to expansion when air is rising and contraction when air is sinking. This pressure work remains completely subgrid‐scale when the atmosphere is unstably stratified. When the atmosphere is stably stratified, this pressure work has to be done by the outer environment. Therefore, a turbulent pressure gradient term is introduced in the vertical momentum equation and the equations account for the irreversible energy conversion from resolved kinetic energy into the model's internal energy. Numerical experiments for breaking gravity waves in the mesosphere highlight the different behaviour of new and conventional approaches. The conventional approach leads to a deepening of the wave amplitudes, whereas the new approach supports wave overturning. The observed foliate structure of very sharp inversion layers is indeed simulated with the new approach for stable stratification. In contrast to the traditional setting, the new scheme does not evolve into persistent non‐physical wave structures on long time‐scales.

Numerical weather, climate, or Earth system models involve the coupling of components. At a broad level, these componentscanbeclassifiedastheresolvedfluiddynamics,unresolvedfluiddynamicalaspects(i.e.,thoserepresentedby physicalparameterizationssuchassubgrid-scalemixing),andnonfluiddynamicalaspectssuchasradiationandmicro-physicalprocesses.Typically,eachcomponentisdeveloped,atleastinitially,independently.Oncedevelopmentismature, the components are coupled to deliver a model of the required complexity. The implementation of the coupling can haveasignificantimpactonthemodel.Astheerrorassociatedwitheachcomponentdecreases,theerrorsintroducedby the coupling will eventually dominate. Hence, any improvement in one of the components is unlikely to improve the performanceoftheoverallsystem.Thechallengesassociatedwithcombiningthecomponentstocreateacoherentmodel are here termedphysics–dynamics coupling. The issue goesbeyondthe coupling betweenthe parameterizations andthe resolved fluid dynamics. This paper highlights recent progress and some of the current challenges. It focuses on three objectives:toillustratethephenomenologyofthecouplingproblemwithreferencestoexamplesintheliterature,toshow howtheproblemcanbeanalyzed,andtocreateawarenessoftheissueacrossthedisciplinesandspecializations.Thetopics addressed are different ways of advancing full models in time, approaches to understanding the role of the coupling and evaluation of approaches, coupling ocean and atmosphere models, thermodynamic compatibility between model components,andemergingissuessuchasthosethatariseasmodelresolutionsincreaseand/ormodelsusevariableresolutions.

Geophysical models of the atmosphere and ocean invariably involve parameterizations. These represent two distinct areas: Subgrid processes that the model cannot resolve, and diabatic sources in the equations, due to radiation for example. Hence, coupling between these physics parameterizations and the resolved fluid dynamics and also between the dynamics of the air and water, is necessary. In this paper weather and climate models are used to illustrate the problems. Nevertheless the same applies to other geophysical models. This coupling is an important aspect of geophysical models. However, often model development is strictly segregated into either physics or dynamics. As a consequence, this area has many unanswered questions. Recent developments in the design of dynamical cores, extended process physics and predicted future changes of the computational infrastructure are increasing complexity. This paper reviews the state-of-the-art of the physics-dynamics coupling in geophysical models, surveys the analysis techniques, and illustrates open questions in this field. This paper focuses on two objectives: To illustrate the phenomenology of the coupling problem with references to examples in the literature and to show how the problem can be analysed. Proposals are made on how to advance the understanding and upcoming challenges with emerging modeling strategies. This paper is of interest to model developers who aim to improve the models and have to make choices on and test new implementations, to users who have to understand choices presented to them and finally users of outputs, who have to distinguish physical features from numerical problems in the model data.

Geophysical models of the atmosphere and ocean invariably involve parameterizations. These represent two distinct areas: Subgrid processes that the model cannot resolve, and diabatic sources in the equations, due to radiation for example. Hence, coupling between these physics parameterizations and the resolved fluid dynamics and also between the dynamics of the air and water, is necessary. In this paper weather and climate models are used to illustrate the problems. Nevertheless the same applies to other geophysical models. This coupling is an important aspect of geophysical models. However, often model development is strictly segregated into either physics or dynamics. As a consequence, this area has many unanswered questions. Recent developments in the design of dynamical cores, extended process physics and predicted future changes of the computational infrastructure are increasing complexity. This paper reviews the state-of-the-art of the physics-dynamics coupling in geophysical models, surveys the analysis techniques, and illustrates open questions in this field. This paper focuses on two objectives: To illustrate the phenomenology of the coupling problem with references to examples in the literature and to show how the problem can be analysed. Proposals are made on how to advance the understanding and upcoming challenges with emerging modeling strategies. This paper is of interest to model developers who aim to improve the models and have to make choices on and test new implementations, to users who have to understand choices presented to them and finally users of outputs, who have to distinguish physical features from numerical problems in the model data.

Numerical models of the atmosphere should fulfil fundamental physical laws. The second law of thermodynamics is associated with positive local entropy production and dissipation of available energy. In order to guarantee this positivity in numerical simulations, subgrid‐scale turbulent fluxes of heat, water vapour and momentum are required to depend on numerically resolved gradients in a unique way. The task of parametrization remains to deliver phenomenological coefficients. Inspecting commonly used parametrizations for subgrid fluxes, we find that some of them obey the second law of thermodynamics and some do not. The conforming approaches are Smagorinsky momentum diffusion, phase changes and sedimentation fluxes for hydrometeors. Conventional turbulent heat‐flux parametrizations do not conform with the second law. A new water‐vapour flux formulation is derived from the requirement of locally positive entropy production. The conventional and new water‐vapour fluxes are compared using high‐resolution radiosonde data. Conventional water‐vapour fluxes are wrong by up to 10% and exhibit a negative bias. Two numerical tests (the Boulder windstorm test case and a convective boundary‐layer experiment) are performed with the Icosahedral Nonhydrostatic model at the Institute for Atmospheric Physics (ICON–IAP). The experiments compare conventional and entropy‐consistent heat‐flux parametrizations. Both test cases indicate that negative thermal dissipation can occur for the conventional heat flux. Obviously, the additional energy made available by this negative dissipation to the resolved turbulence is later on dissipated by friction, so that the total dissipation is again comparable, at least for the boundary‐layer experiment.

We demonstrate how efficient r-adapted grids for the prediction of tropical cyclone (TC) tracks can be constructed with the help of goal-oriented error estimates. The binary interaction of TCs in a barotropic model is used as a test case. We perform a linear sensitivity analysis for this problem to evaluate the contribution of each grid cell to an error measure correlated with the cyclone positions. This information allows us to estimate the local grid resolution required to minimize the TC position error. An algorithm involving the solution of a Poisson problem is employed to compute how grid points should be moved such that the desired local resolution is achieved. A hexagonal shallow-water version of the next-generation numerical weather prediction and climate model ICON is used to perform model runs on these adapted grids. The results show that for adequately chosen grid adaptation parameters, the accuracy of the track prediction can be maintained even when a coarser grid is used in regions for which the estimated error contribution is low. Accurate track predictions are obtained only when a grid with high resolution consisting of cells with nearly constant size and regular shape covers the part of the domain where the estimated error contribution is large. The number of grid points required to achieve a certain accuracy in the track prediction can be decreased substantially with our approach.

This study describes a new global non-hydrostatic dynamical core (ICON-IAP: Icosahedral Nonhydrostatic model at the Institute for Atmospheric Physics) on a hexagonal C-grid which is designed to conserve mass and energy. Energy conservation is achieved by discretizing the antisymmetric Poisson bracket which mimics correct energy conversions between the different kinds of energy (kinetic, potential, internal). Because of the bracket structure this is even possible in a complicated numerical environment with (i) the occurrence of terrain-following coordinates with all the metric terms in it, (ii) the horizontal C-grid staggering on the Voronoi mesh and the complications induced by the need for an acceptable stationary geostrophic mode, and (iii) the necessity for avoiding Hollingsworth instability. The model is equipped with a Smagorinsky-type nonlinear horizontal diffusion. The associated dissipative heating is accounted for by the application of the discrete product rule for derivatives. The time integration scheme is explicit in the horizontal and implicit in the vertical. In order to ensure energy conservation, the Exner pressure has to be off-centred in the vertical velocity equation and extrapolated in the horizontal velocity equation. Test simulations are performed for small-scale and global-scale flows. A test simulation of linear non-hydrostatic flow over a rough mountain range shows the theoretically expected gravity wave propagation. The baroclinic wave test is extended to 40 days in order to check the Lorenz energy cycle. The model exhibits excellent energy conservation properties even in this strongly nonlinear and dissipative case. The HeldSuarez test confirms the reliability of the model over even longer time-scales. Copyright (c) 2012 Royal Meteorological Society