To define a conserved energy for an atmosphere with phase changes of water (such as vapor and liquid), motivation in the past has come from generalizations of dry energies—in particular, from gravitational potential energy ρgz. Here a new definition of moist energy is introduced, and it generalizes another form of dry potential energy, proportional to θ2, which is valuable since it is manifestly quadratic and positive definite. The moist potential energy here is piecewise quadratic and can be decomposed into three parts, proportional to bu2Hu, bs2Hs, and M2Hu, which represent, respectively, buoyant energies and a moist latent energy that is released upon a change of phase. The Heaviside functions Hu and Hs indicate the unsaturated and saturated phases, respectively. The M2 energy is also associated with an additional eigenmode that arises for a moist atmosphere but not a dry atmosphere. Both the Boussinesq and anelastic equations are examined, and similar energy decompositions are shown in both cases, although the anelastic energy is not quadratic. Extensions that include cloud microphysics are also discussed, such as the Kessler warm-rain scheme. As an application, empirical orthogonal function (EOF) analysis is considered, using a piecewise quadratic moist energy as a weighted energy in contrast to the standard L2 energy. By incorporating information about phase changes into the energy, the leading EOF modes become fundamentally different and capture the variability of the cloud layer rather than the dry subcloud layer.

With large eddy simulations (LES) and/or cloud-resolving models (CRMs), it is now possible to simultaneously simulate shallow and deep convection. However, using traditional methods, the computational expense is typically very large, due to the small grid spacings needed to resolve shallow clouds. Here, the main purpose is to present a method that is computationally less expensive by a factor of roughly 10 to 50. Unlike traditional grid stretching of only the vertical z grid spacing, the present method involves expansion of the grid spacing in all coordinate directions (x,y,z) and time t. A ˝ne grid spacing of O(10)-O(100) m can be used near the surface to resolve boundary layer turbulence, and the grid spacing expands to be O(1000) m at higher altitudes, which reduces computational cost while still resolving deep convection. Example simulations are conducted with a simpli˝ed LES/CRM in 2D to verify the theoretical cost savings.

Abstract. Strongly coupled nonlinear phenomena such as those described by Earth system models (ESMs) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESMs is to remap or interpolate component solution fields defined on their computational mesh to another mesh with a different combinatorial structure and decomposition, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESMs. However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations is conducted based on a progression of scalar fields from idealized and smooth to more general climate data with strong discontinuities and strict bounds. We examine four remapping algorithms with distinct design approaches, namely ESMF Regrid (Hill et al., 2004), TempestRemap (Ullrich and Taylor, 2015), generalized moving least squares (GMLS) (Trask and Kuberry, 2020) with post-processing filters, and WLS-ENOR (Li et al., 2020). By repeated iterative application of the high-order remapping methods to the test fields, we verify the accuracy of each scheme in terms of their observed convergence order for smooth data and determine the bounded error propagation using challenging, realistic field data on both uniform and regionally refined mesh cases. In addition to retaining high-order accuracy under idealized conditions, the methods also demonstrate robust remapping performance when dealing with non-smooth data. There is a failure to maintain monotonicity in the traditional L2-minimization approaches used in ESMF and TempestRemap, in contrast to stable recovery through nonlinear filters used in both meshless GMLS and hybrid mesh-based WLS-ENOR schemes. Local feature preservation analysis indicates that high-order methods perform better than low-order dissipative schemes for all test cases. The behavior of these remappers remains consistent when applied on regionally refined meshes, indicating mesh-invariant implementations. The MIRA intercomparison protocol proposed in this paper and the detailed comparison of the four algorithms demonstrate that the new schemes, namely GMLS and WLS-ENOR, are competitive compared to standard conservative minimization methods requiring computation of mesh intersections. The work presented in this paper provides a foundation that can be extended to include complex field definitions, realistic mesh topologies, and spectral element discretizations, thereby allowing for a more complete analysis of production-ready remapping packages.

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