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For over 60 years, an approximation involving spherical geopotentials has underlain the representation of gravity in global numerical models of Earth's atmosphere and oceans. This article explores how departures from sphericity can be allowed by assuming spheroidal geopotentials instead. A route to more accurate model formulation is indicated, and the theoretical basis of the classical spherical‐geopotential approximation is illuminated too. An overview—from the time of Newton to the present—is given of the development of zeroth‐ and first‐order approximations to the geopotential field in terms of two small non‐dimensional parameters. The importance of using geopotential coordinate systems in atmospheric and oceanic models is emphasized. Early suggestions for such systems using non‐spherical coordinates involved qualitatively inappropriate choices of ellipsoids derived from families of confocal ellipses. Specific examples of appropriate ellipsoid choices are considered before presentation of the recently developed Geophysically Realistic, Ellipsoidal, Analytically Tractable (GREAT) system. This is based on a suitably constructed geopotential field approximation, first order accurate, from which—without further approximation—may be analytically derived equations for geopotential surfaces and surfaces orthogonal to them. GREAT coordinates satisfy stated desiderata for geopotential coordinate systems and are applicable both above and below Earth's geoid (assumed to coincide with the WGS 84 [2004, https://gis‐lab.info/docs/nima‐tr8350.2‐wgs84fin.pdf] reference ellipsoid to an excellent approximation). GREAT‐coordinate analysis provides justification for the classical spherical‐geopotential approximation: It is revealed as a mathematical limit, but not a physically realizable one. Attention is drawn to a certain partially spherical limit that is realizable physically. Plain Language Summary Gravity is by far the dominant external force in the equations of motion for Earth's atmosphere and oceans. It is crucially important that it be adequately represented in global atmospheric and oceanic models for climate and weather prediction. In principle, one simply applies Newton's inverse square law of gravitational attraction to represent gravity in the equations of motion. In practice, this is far too complicated to do exactly—due principally to the rotating Earth being closer to spheroidal than spherical in shape, with an inhomogeneous mass distribution—thereby necessitating approximation. The dominant nature of gravity means that forecast accuracy is enhanced if one constructs an orthogonal coordinate system to integrate the equations of motion, whereby gravity only acts in the vertical and not in the horizontal. This is termed a geopotential coordinate system and, to further complicate matters, its construction is intrinsically coupled to a suitable approximation of gravity. We first review the basic concepts to represent gravity in global atmospheric and oceanic models. Next, we discuss the importance and principles of geopotential coordinates for modeling purposes. Various geopotential coordinate systems of varying accuracy are then compared. Finally, we outline some possible developments for representing gravity in future models. Key Points We review the derivation from first principles of geopotential approximations to represent gravity in global atmospheric and oceanic models Various geopotential coordinate systems of varying accuracy are compared, leading to one that satisfies all desiderata for such a system The low‐order, ubiquitous, classical, spherical‐geopotential approximation is shown to be the asymptotic limit of a more accurate one

The rotation of Earth breaks time-reversal and reflection symmetries in an opposite sense north and south of the equator, leading to a topological origin for certain atmospheric and oceanic equatorial waves. Away from the equator, the rotating shallow-water and stably stratified primitive equations exhibit Poincaré inertia–gravity waves that have nontrivial topology as evidenced by their strict superinertial time scale and a phase singularity in frequency–wavevector space. This nontrivial topology then predicts, via the principle of bulk-interface correspondence, the existence of two equatorial waves along the equatorial interface, the Kelvin and Yanai waves. To directly test the nontrivial topology of Poincaré-gravity waves in observations, we examine ERA5 data and study cross correlations between the wind velocity and geopotential height of the midlatitude stratosphere at the 50 hPa height. We find the predicted vortex and antivortex in the relative phase of the geopotential height and velocity at the high frequencies of the waves. By contrast, lower-frequency planetary waves are found to have trivial topology also as expected from theory. These results demonstrate a new way to understand stratospheric waves and provide a new qualitative tool to investigate waves in other components of the climate system.

This article introduces an ensemble clustering tool developed at the Weather Prediction Center (WPC) to assist forecasters in the preparation of medium-range (3–7 day) forecasts. Effectively incorporating ensemble data into an operational forecasting process, like that used at WPC, can be challenging given time constraints and data infrastructure limitations. Often forecasters do not have time to view the large number of constituent members of an ensemble forecast, so they settle for viewing the ensemble’s mean and spread. This ignores the useful information about forecast uncertainty and the range of possible forecast outcomes that an ensemble forecast can provide. Ensemble clustering could be a solution to this problem as it can reduce a large ensemble forecast down to the most prevalent forecast scenarios. Forecasters can then quickly view these ensemble clusters to better understand and communicate forecast uncertainty and the range of possible forecast outcomes. The ensemble clustering tool developed at WPC is a variation of fuzzy clustering where operationally available ensemble members with similar 500-hPa geopotential height forecasts are grouped into four clusters. A representative case from 15 February 2021 is presented to demonstrate the clustering methodology and the overall utility of this new ensemble clustering tool. Cumulative verification statistics show that one of the four forecast scenarios identified by this ensemble clustering tool routinely outperforms all the available ensemble mean and deterministic forecasts.

In this paper we present the analytical derivation of a local water mass transformation (WMT) framework for an individual water column. We exactly formulate the mapping of the governing equations from geopotential coordinates to an arbitrary tracer space. Unique definitions for the local effective vertical diasurface fluxes are given. In tracer space we derive new relations between the local diatracer fluxes and the mixing per tracer class. The key relation between the effective vertical diatracer velocity and the mixing per tracer class directly formulates how the overturning circulation is linked to local tracer variance dissipation. Horizontal integration of the governing equations in tracer space and the relations between the diatracer quantities finally recovers the well-known integral WMT formulations.

Using a 3-km regional ensemble prediction system (EPS), this study tested a three-dimensional (3D) rescaling mask for initial condition (IC) perturbation. Whether the 3D mask-based EPS improves ensemble forecasts over current two-dimensional (2D) mask-based EPS has been evaluated in three aspects: ensemble mean, spread, and probability. The forecasts of wind, temperature, geopotential height, sea level pressure, and precipitation were examined for a summer month (1–28 July 2018) and a winter month (1–27 February 2019) over a region in North China. The EPS was run twice per day (initiated at 0000 and 1200 UTC) to 36 h in forecast length, providing 56 warm-season forecast cases and 54 cold-season cases for verification. The warm and cold seasons are verified separately for comparison. The study found the following: 1) The vertical profile of IC perturbation becomes closer to that of analysis uncertainty with the 3D rescaling mask. 2) Ensemble performance is significantly improved in all three aspects. The biggest improvement is in the ensemble spread, followed by the probabilistic forecast, and the least improvement is in the ensemble mean forecast. Larger improvements are seen in the warm season than in the cold season. 3) More improvement is in the shorter time range (<24 h) than in the longer range. 4) Surface and lower-level variables are improved more than upper-level ones. 5) The underlying mechanism for the improvement has been investigated. Convective instability is found to be responsible for the spread increment and, thus, overall ensemble forecast improvement. Therefore, using a 3D rescaling mask is recommended for an EPS to increase its utility especially for shorter time range and surface weather elements.

Here, a nonhydrostatic alternative scheme (NAS) is proposed for the grey zone where the nonhydrostatic impact on the atmosphere is evident but not large enough to justify the necessity to include an implicit nonhydrostatic solver in an atmospheric dynamical core. The NAS is designed to replace this solver, which can be incorporated into any hydrostatic models so that existing well-developed hydrostatic models can effectively serve for a longer time. Recent advances in machine learning (ML) provide a potential tool for capturing the main complicated nonlinear-nonhydrostatic relationship. In this study, an ML approach called a neural network (NN) was adopted to select leading input features and develop the NAS. The NNs were trained and evaluated with 12-day simulation results of dry baroclinic-wave tests by the Weather Research and Forecasting (WRF) model. The forward time difference of the nonhydrostatic tendency was used as the target variable, and the five selected features were the nonhydrostatic tendency at the last time step, and four hydrostatic variables at the current step including geopotential height, pressure in two different forms, and potential temperature, respectively. Finally, a practical NAS was developed with these features and trained layer by layer at a 20-km horizontal resolution, which can accurately reproduce the temporal variation and vertical distribution of the nonhydrostatic tendency. Corrected by the NN-based NAS, the improved hydrostatic solver at different horizontal resolutions can run stably for at least one month and effectively reduce most of the nonhydrostatic errors in terms of system bias, anomaly root-mean-square error, and the error of the wave spatial pattern, which proves the feasibility and superiority of this scheme.

This article studies the growth of the prediction error over lead time in a schematic model of atmospheric transport. Inspired by the Lorenz (2005) system, we mimic an atmospheric variable in one dimension, which can be decomposed into three spatiotemporal scales. We identify parameter values that provide spatiotemporal scaling and chaotic behavior. Instead of exponential growth of the forecast error over time, we observe a more complex behavior. We test a power law and the quadratic hypothesis for the scale-dependent error growth. The power law is valid for the first days of the growth, and with an included saturation effect, we extend its validity to the entire period of growth. The theory explaining the parameters of the power law is confirmed. Although the quadratic hypothesis cannot be completely rejected and could serve as a first guess, the hypothesis's parameters are not theoretically justifiable in the model. In addition, we study the initial error growth for the ECMWF forecast system (500 hPa geopotential height) over the 1986 to 2011 period. For these data, it is impossible to assess which of the error growth descriptions is more appropriate, but the extended power law, which is theoretically substantiated and valid for the Lorenz system, provides an excellent fit to the average initial error growth of the ECMWF forecast system. Fitting the parameters, we conclude that there is an intrinsic limit of predictability after 22 d.

This study focuses on analysing the impact of deterministic modifications of the Stokes kernel and terrain correction methods for precise geoid determination using the Stokes-Helmert method over a sophisticated topography. Three deterministic modification methods of Stokes’s kernel (Wong-Gore, Vaníček-Kleusberg, and Featherstone-Evans-Olliver) are tried to minimize the truncation error emanating from the non-availability of gravity data all over the Earth by utilizing two independent satellite only global geopotential models. In parallel to the modified Stokes kernel functions, two terrain correction techniques, i.e., spatial-spectral combined method with mass-prisms and spatial method with mass-cylinders, have also been examined to assess their combined effects on geoid heights over the Konya Closed Basin in Türkiye. The developed geoid models are validated with GNSS-levelling data and inter-compared pixel-wise. The numerical results show that although the overall statistical values depict consistent precision for various combinations of TCs, Stokes kernel modifiers, and GGMs, a holistic validation-comparison analysis reveals significant variations in view of the cm-precise geoid.

The moist static energy (MSE) budget is widely used to understand moist atmospheric thermodynamics. However, the budget is not exact, and the accuracy of the approximations that yield it has not been examined rigorously in the context of large-scale tropical motions (horizontal scales ≥ 1000 km). A scale analysis shows that these approximations are most accurate in systems whose latent energy anomalies are considerably larger than the geopotential and kinetic energy anomalies. This condition is satisfied in systems that exhibit phase speeds and horizontal winds on the order of 10 m s −1 or less. Results from a power spectral analysis of data from the DYNAMO field campaign and ERA5 qualitatively agree with the scaling, although they indicate that the neglected terms are smaller than what the scaling suggests. A linear regression analysis of the MJO events that occurred during DYNAMO yields results that support these findings. It is suggested that the MSE budget is accurate in the tropics because motions within these latitudes are constrained to exhibit small fluctuations in geopotential and kinetic energy as a result of weak temperature gradient (WTG) balance.

The Kolmogorov–Smirnov goodness-of-fit test is used in many applications for testing normality in climate research. This note shows that the test usually leads to systematic and drastic errors. When the mean and the standard deviation are estimated, it is much too conservative in the sense that its p values are strongly biased upward. One may think that this is a small sample problem, but it is not. There is a correction of the Kolmogorov–Smirnov test by Lilliefors, which is in fact sometimes confused with the original Kolmogorov–Smirnov test. Both the Jarque–Bera and the Shapiro–Wilk tests for normality are good alternatives to the Kolmogorov–Smirnov test. A power comparison of eight different tests has been undertaken, favoring the Jarque–Bera and the Shapiro–Wilk tests. The Jarque–Bera and the Kolmogorov–Smirnov tests are also applied to a monthly mean dataset of geopotential height at 500 hPa. The two tests give very different results and illustrate the danger of using the Kolmogorov–Smirnov test.