Explore the vast range of titles available.

We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014, https://doi.org/10.1175/MWR-D-14-00069.1) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) vertical differencing, which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term‐by‐term balance in the kinetic, internal, and potential energy budgets, ensuring an energy‐consistent discretization up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit‐explicit time stepping algorithm. Plain Language Summary Energy consistent discretizations have proven useful in guiding the development of numerical methods for simulating fluid dynamics. They ensure that the discrete method does not have any spurious sources of energy, which can lead to unstable and unrealistic simulations. Here we provide an energy consistent discretization of the equations used by global models of the Earth's atmosphere. The discretization is written in terms of standard variables in spherical coordinates and supports a wide variety of terrain following vertical coordinates. It can be used with any horizontal discretization that has a discrete version of the integration‐by‐parts identity. Key Points We give a discrete Hamiltonian formulation of the nonhydrostatic equations in primitive variables The formulation supports mass or height terrain following coordinates The Lorenz staggered vertical discretization obeys a derivative product rule

This study investigates inherent numerical dissipation due to upwind fluxes and reconstruction strategies for collocated Finite‐Volume integration of the Euler equations. Idealized supercell simulations are used without any explicit dissipation. Flux terms are split into: mass flux, pressure, and advected quantities. They are computed with the following upwind strategies: central, advectively upwind, and acoustically upwind. This is performed for third and ninth‐order‐accurate reconstructions with and without Weighted Essentially Non‐Oscillatory limiting. Acoustic‐only upwinding for pressure and mass flux terms and advective‐only upwinding for advected quantities is the most flexible simplification found. It reduces data movement and computations. Assuming a constant speed of sound in acoustic upwinding gives similar results to using the true speed of sound. Dissipation from upwind adapts automatically to grid spacing, time step, reconstruction accuracy, and flow smoothness. While stability is maintained even at 21st‐order spatial accuracy, there is a limit to the spatial order of accuracy for which upwinding alone can create a realizable solution in the conditions of this study. Convex combinations of upwind and central solutions for flux terms also reduced dissipation, but as the central proportion grows, solutions become physically unrealizable. The range of length scales of the kinetic energy spectra can be extended along k−5/3 to smaller spatial scales by reducing dissipation either with higher‐order reconstructions or using convex combinations of upwind and central fluxes. However, not all extensions of the length scale range along k−5/3 exhibit physically realizable solutions, even though the spectra appear to be physical. Plain Language Summary Integrating the Euler equations that govern certain regimes of fluid motion forward in time must be performed carefully. Subtle choices in the numerical algorithm can lead to instability or solutions that are unphysical. “Upwinding” is an algorithmic choice where the developer prefers information that comes from the upwind direction, and it is known to lead to smoother and more stable solutions in general. This study explores the effects on stability and physical realism of using different upwind preferences for different variables in the fluxes of fluid properties between adjacent cells in a discrete 3‐D grid. Sound waves are a part of the equations of atmospheric motion. They propagate through time on the edge of an expanding sphere, which makes determining a single “upwind” direction more complex. One can transform the equations into an equivalent form called “characteristic” form, though, that simplifies this definition. That transformation couples the variables together tightly and requires extra computations and data movement across a large parallel machine. This paper shows that there are simplifications to this fully upwind characteristic approach that are just as physically realistic and stable but reduce the number of computations and volume of data transfers significantly. Key Points There are simplifications to a fully upwind Riemann solver that require fewer computations and less data movement Upwinding according to advective versus acoustic dynamics makes a large difference in stability One can use convex combinations of central and upwind fluxes to obtain physically realizable solutions with less dissipation

The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar characteristics: conservation laws, inertia-gravity and Rossby waves, and a (quasi-) balanced state. In order to obtain realistic simulation results, it is desirable that numerical models have discrete analogues of these properties. Two prototypical examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07) Z-grid total energy and potential enstrophy conserving scheme. Unfortunately, the AL81 scheme is restricted to logically square, orthogonal grids, and the S07 scheme is restricted to uniform square grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to arbitrary orthogonal spherical polygonal grids in a manner that allows for both total energy and potential enstrophy conservation, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos, and others) and discrete exterior calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp, and others). Detailed results of the schemes applied to standard test cases are deferred to part 2 of this series of papers.

We present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize analytically the emergence of gaps in the dispersion relation at specific wavenumbers, when they exist, and compute their specific locations. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. We also show that the Fourier analysis generates mathematical artifacts, and we explain how to remove them through a branch selection procedure conducted by analysis of eigenvectors and associated reconstructed solutions. The higher frequency eigenmodes, named erratic in this study, are also investigated analytically and numerically.

We investigate mechanisms governing moist energy exchanges at the atmosphere-ocean interface in global Earth system models. The goal of this work is to overcome deficiencies like energy fixers and unphysical thermodynamic formulations and designs that are commonly used in modern models. For example, while the ocean surface evaporation is one of the most significant climatological drivers, its representation in numerical models may not be physically accurate. In particular, existing schemes give an incorrect atmospheric air temperature tendency during evaporation events. To remedy this, starting from first principles, we develop a new mechanism for the ocean-atmosphere moist energy transfers. It utilizes consistent thermodynamics of water species, distributes latent heat of evaporation in a physically plausible way, and avoids reliance on artificial energy fixers. The temperature and water mass tendencies are used to formulate a set of ordinary differential equations (ODEs) representing a simple box model of ocean-air exchange. We investigate the properties of the ODEs representing the proposed mechanism and compare them against those derived from the current designs of the Energy Exascale Earth System Model (E3SM). The proposed simplified box model highlights the advantages of our approach in capturing physically appropriate atmospheric temperature changes during evaporation while conserving energy.

Motivated by reducing errors in the energy budget related to enthalpy fluxes within the Energy Exascale Earth System Model (E3SM), we study several physics–dynamics coupling approaches. Using idealized physics, a moist rising bubble test case, and the E3SM's nonhydrostatic dynamical core, we consider unapproximated and approximated thermodynamics applied at constant pressure or constant volume. With the standard dynamics and physics time-split implementation, we describe how the constant-pressure and constant-volume approaches use different mechanisms to transform physics tendencies into dynamical motion and show that only the constant-volume approach is consistent with the underlying equations. Using time step convergence studies, we show that the two approaches both converge but to slightly different solutions. We reproduce the large inconsistencies between the energy flux internal to the model and the energy flux of precipitation when using approximate thermodynamics, which can only be removed by considering variable latent heats, both when computing the latent heating from phase change and when applying this heating to update the temperature. Finally, we show that in the nonhydrostatic case, for physics applied at constant pressure, the general relation that enthalpy is locally conserved no longer holds. In this case, the conserved quantity is enthalpy plus an additional term proportional to the difference between hydrostatic pressure and full pressure.

This paper presents (Lagrangian) variational formulations for single and multicomponent semi-compressible fluids with both reversible (entropy-conserving) and irreversible (entropy-generating) processes. Semi-compressible fluids are useful in describing low-Mach dynamics, since they are soundproof. These models find wide use in many areas of fluid dynamics, including both geophysical and astrophysical fluid dynamics. Specifically, the Boussinesq, anelastic and pseudoincompressible equations are developed through a unified treatment valid for arbitrary Riemannian manifolds, thermodynamic potentials and geopotentials. By design, these formulations obey the 1st and 2nd laws of thermodynamics, ensuring their thermodynamic consistency. This general approach extends and unifies existing work, and helps clarify the thermodynamics of semi-compressible fluids. To further this goal, evolution equations are presented for a wide range of thermodynamic variables: entropy density \\(s\\), specific entropy \\(\\), buoyancy \\(b\\), temperature \\(T\\), potential temperature \\(\\) and a generic entropic variable \\(\\); along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation is developed for all three equation sets in the case of reversible dynamics, and for the Boussinesq/anelastic equations in the case of irreversible dynamics; and some discussion is given of the difficulty in formulating the elliptic equation for the pseudoincompressible equations with irreversible dynamics.

Some existing approaches to modeling the thermodynamics of moist air make approximations that break \\(thermodynamic consistency\\), such that the resulting thermodynamics do not obey the 1st and 2nd laws or have other inconsistencies. Recently, an approach to avoid such inconsistency has been suggested: the use of \\(thermodynamic potentials\\) in terms of their \\(natural variables\\), from which all thermodynamic quantities and relationships are derived. In this paper, we develop this approach for \\(unapproximated\\) moist air thermodynamics and two widely used approximations: the constant \\(\\) approximation and the dry heat capacities approximation. The consistent constant \\(\\) approximation is particularly attractive because it leads to, when using virtual potential temperature \\(_v\\) as the thermodynamic variable, adiabatic dynamics that depend only on total mass, independent of the breakdown between water forms. Additionally, a wide variety of material from different sources in the literature on thermodynamics in atmospheric modelling is brought together. It is hoped that this paper provides a comprehensive reference for the use of thermodynamic potentials in atmospheric modelling, especially for the three systems considered here.

Much attention has recently been devoted to data-based computing of evolution of physical systems. In such approaches, information about data points from past trajectories in phase space is used to reconstruct the equations of motion and to predict future solutions that have not been observed before. However, in many cases, the available data does not correspond to the variables that define the system's phase space. We focus our attention on the important example of dissipative dynamical systems. In that case, the phase space consists of coordinates, momenta and entropies; however, the momenta and entropies cannot, in general, be observed directly. To address this difficulty, we develop an efficient data-based computing framework based exclusively on observable variables, by constructing a novel approach based on the thermodynamic Lagrangian, and constructing neural networks that respect the thermodynamics and guarantees the non-decreasing entropy evolution. We show that our network can provide an efficient description of phase space evolution based on a limited number of data points and a relatively small number of parameters in the system.

TRiSK-type numerical schemes are widely used in both atmospheric and oceanic dynamical cores, due to their discrete analogues of important properties such as energy conservation and steady geostrophic modes. In this work, we show that these numerical methods are best understood as a discrete exterior calculus (DEC) scheme applied to a Hamiltonian formulation of the rotating shallow water equations based on split exterior calculus. This comprehensive description of the differential geometric underpinnings of TRiSK-type schemes completes the one started in Thuburn2012,Eldred2017, and provides a new understanding of certain operators in TRiSK-type schemes as discrete wedge products and topological pairings from split exterior calculus. All known TRiSK-type schemes in the literature are shown to fit inside this general framework, by identifying the (implicit) choices made for various DEC operators by the different schemes. In doing so, unexplored choices and combinations are identified that might offer the possibility of fixing known issues with TRiSK-type schemes such as operator accuracy and Hollingsworth instability.