The time integration scheme is probably one of the most fundamental choices in the development of an ocean model. In this paper, we investigate several time integration schemes when applied to the shallow water equations. This set of equations is accurate enough for the modeling of a shallow ocean and is also relevant to study as it is the one solved for the barotropic (i.e. vertically averaged) component of a three dimensional ocean model. We analyze different time stepping algorithms for the linearized shallow water equations. High order explicit schemes are accurate but the time step is constrained by the Courant-Friedrichs-Lewy stability condition. Implicit schemes can be unconditionally stable but, in practice lack accuracy when used with large time steps. In this paper we propose a detailed comparison of such classical schemes with exponential integrators. The accuracy and the computational costs are analyzed in different configurations.
Some existing approaches to modelling the thermodynamics of moist air make approximations that break thermodynamic consistency, such that the resulting thermodynamics does not obey the first and second laws or has 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 (equations of state) are derived. In this article, we develop this approach for unapproximated moist-air thermodynamics and two widely used approximations: the constant-κ approximation and the dry heat capacities approximation. The (consistent) constant-κ approximation is particularly attractive because it leads to, with the appropriate choice of thermodynamic variable, adiabatic dynamics that depend only on total mass and are 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 article provides a comprehensive reference for the use of thermodynamic potentials in atmospheric modelling, especially for the three systems considered here.
This SAND report documents CIS Late Start LDRD Project 22-0311, "Differential geometric approaches to momentum-based formulations for fluids". The project primarily developed geometric mechanics formulations for momentum-based descriptions of nonrelativistic fluids, utilizing a differential geometry/exterior calculus treatment of momentum and a space+time splitting. Specifically, the full suite of geometric mechanics formulations (variational/Lagrangian, Lie-Poisson Hamiltonian and Curl-Form Hamiltonian) were developed in terms of exterior calculus using vector-bundle valued differential forms. This was done for a fairly general version of semi-direct product theory sufficient to cover a wide range of both neutral and charged fluid models, including compressible Euler, magnetohydrodynamics and Euler-Maxwell. As a secondary goal, this project also explored the connection between geometric mechanics formulations and the more traditional Godunov form (a hyperbolic system of conservation laws). Unfortunately, this stage did not produce anything particularly interesting, due to unforeseen technical difficulties. There are two publications related to this work currently in preparation, and this work will be presented at SIAM CSE 23, at which the PI is organizing a mini-symposium on geometric mechanics formulations and structure-preserving discretizations for fluids. The logical next step is to utilize the exterior calculus based understanding of momentum coupled with geometric mechanics formulations to develop (novel) structure-preserving discretizations of momentum. This is the main subject of a successful FY23 CIS LDRD "Structure-preserving discretizations for momentum-based formulations of fluids".
This SAND report documents Exploratory Express LDRD Project 223790, "Structure-preserving numerical discretizations for domains with boundaries", which developed a method to incorporate consistent treatment of domain boundaries and arbitrary boundary conditions in discrete exterior calculus (DEC) for arbitrary polygonal (2D) and tensor-product structure prism (3D) grids. The new DEC required the development of novel discrete exterior derivatives, boundary operators, wedge products and Hodge stars. This was accomplished through the use of boundary extension and the blending of known 2D operators on the interior with 1D operators on the boundary. The Hodge star was based on the Voronoi Hodge star, and retained the limitation of a triangular circumcentric primal or dual grid along with low-order accuracy. In addition to the new DEC, two related software packages were written: one for the study of DEC operators on arbitrary polygonal and polyhedral grids using both symbolic and numerical approaches and one for a (thermal) shallow water testbed using TRiSK-type numerics. Immediately relevant (already funded, through CANGA) followup work is the development of a high-order, geometrically flexible Hodge star and structure-preserving, high-order, oscillation-limiting transport operators (using WENO) for n-forms on arbitrary 2D and 3D grids. This will provide all of the machinery required for a high-order version of TRiSK with boundaries on arbitrary 2D and tensor-product 3D grids, which is applicable to both the atmospheric (CRM in E3SM-MMF) and oceanic (MPAS-O) components of E3SM.
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 thermodynamicvariables: entropy density s, specific entropy η, buoyancy b, temperature T, potential temperature O and a generic entropic variable Χ; along with a general definition of buoyancy valid for all three semicompressible models and arbitrary geopotentials. Finally, the elliptic equation for the pressure perturbation (the Lagrange multiplier that enforces semicompressibility) 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 it for the pseudoincompressible equations with irreversible dynamics.
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 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.