Semi-Implicit Formulations of the Navier–Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling

This paper reviews nonhydrostatic atmospheric models for research and NWP. Classification of nonhydrostatic atmospheric models and numerical methods to treat sound waves are described with their relative advantages. The current operational nonhydrostatic NWP models at various forecast centers and community nonhydrostatic models for research are reviewed. Brief history and development of the JMA nonhydrostatic model, a community mesoscale model for research and NWP in Japan, is introduced. Current status and near future plans of the operational nonhydrostatic mesoscale model at JMA are presented.

Numerical integration of the compressible nonhydrostatic equations using semi-implicit techniques is complicated by the need to solve a Helmholtz equation at each time step. The authors present an accurate and efficient technique for solving the Helmholtz equation using a conjugate-residual (CR) algorithm that is accelerated by ADI preconditioners. These preconditioned CR solvers possess four distinct advantages over most other solvers that have been used with the Helmholtz equations that arise in compressible nonhydrostatic semi-implicit atmospheric models: the preconditioned CR methods 1) can solve Helmholtz equations containing variable coefficients, alleviating the need to prescribe a reference state in order to simplify the elliptic problem; 2) transparently include the cross-derivative terms arising from terrain transformations; 3) are efficient and accurate for nonhydrostatic models used across a broad range of scales, from cloud scales to synoptic-global scales; and 4) are easy to formulate and program. These features of the CR solver allow semi-implicit formulations that are unconstrained by the form of the Helmholtz equations, and the authors propose a formulation that is more consistent than those most often used in that it includes implicit treatment of all terms associated with the pressure gradients and divergence. This formulation is stable for nonhydrostatic-scale simulations involving steep terrain, whereas the more common semi-implicit formulation is not. The ADI preconditioners are presented for use in simulations of both hydrostatic and nonhydrostatic scale flows. These simulations demonstrate the efficiency and accuracy of the preconditioned CR method and the overall stability of the model formulation. The simulations also suggest a general convergence criteria for the iterative algorithm in terms of the solution divergence.

Global nonhydrostatic atmospheric models are becoming increasingly important for studying the climates of planets and exoplanets. However, such models suffer from computational difficulties due to the large aspect ratio between the horizontal and vertical directions. To overcome this problem, we developed a global model using a vertically implicit correction (VIC) scheme in which the integration time step is no longer limited by the vertical propagation of acoustic waves. We proved that our model, based on the Athena++ framework and its extension for planetary atmospheres—SNAP (Simulating Nonhydrostatic Atmospheres on Planets), rigorously conserves mass and energy in finite-volume simulations. We found that traditional numerical stabilizers such as hyperviscosity and divergence damping are not needed when using the VIC scheme, which greatly simplifies the numerical implementation and improves stability. We present simulation results ranging from 1D linear waves to 3D global circulations with and without the VIC scheme. These tests demonstrate that our formulation correctly tracks local turbulent motions, produces Kelvin–Helmholtz instability, and generates a super-rotating jet on hot Jupiters. Employing this VIC scheme improves the computational efficiency of global simulations by more than two orders of magnitude compared to an explicit model and facilitates the capability of simulating a wide range of planetary atmospheres both regionally and globally.

Accurate prediction of intensity and track of typhoon is crucial to mitigate typhoon disasters. To improve typhoon forecast, the "Global 7-km mesh nonhydrostatic model inter-comparison project for improving typhoon forecast (TYMIP-G7)" has been conducted under the JAMSTEC Earth Simulator Strategic Project with Special Support. In the project, ensemble of the 5-day typhoon forecast experiments in September and October 2013 were conducted by using three nonhydrostatic global atmospheric models. Preliminary results indicate that the nonhydrostatic models significantly improve the forecast skill in terms of typhoon intensity. However, the atmospheric models used in TYMIP-G7 are not fully coupled with ocean. This study aims to investigate the impact of ocean coupling on typhoon simulation by using a nonhydrostatic global ocean-atmosphere coupled general circulation model. It was found that the forecast error of typhoon intensity ranged from - 10 hPa to 10 hPa at the forecast time of 60 hours in the coupled model. We also found that there is no marked difference in the modeled typhoon intensity at the forecast lead time of 36 hours between the coupled model and standalone atmospheric model; however, the typhoon intensity was weakened after the forecast lead time of 36 hours in the coupled model compared to the standalone model. On the other hand, the forecast error of typhoon track was not changed by ocean coupling, indicating that large scale atmospheric circulations were not changed by ocean coupling in the 5-day simulation.

The elliptic problems in semi-implicit nonhydrostatic atmospheric models are difficult. Typically, they are poorly conditioned, nonseparable, contain cross-derivative terms, and are often nonsymmetric. Here, the resulting linear system is solved using a preconditioned Krylov subspace method—the generalized conjugate residual (GCR) algorithm. A horizontal spectral preconditioner is developed as an alternative to a more standard and much simpler line Jacobi relaxation scheme. However, the efficacy of the spectral preconditioner requires neglecting the cross-derivative terms and homogenization (e.g., averaging) metric coefficients over the computational domain. Because such a compromise causes a substantial departure of the preconditioner from the original elliptic operator, it is not obvious a priori whether it leads to a competitive solver. The robustness of the proposed approach over a broad range of representative meteorological applications is evaluated, in the context of a three-time-level sem...

<p>We will present our atmospheric physics contribution to the New Refined Observations of Climate Change from Spaceborne Gravity Missions (NEROGRAV) research group, which is funded by the German Research Foundation (DFG). The goal of NEROGRAV is to develop new analysis methods and modeling approaches to improve the current data analysis of the GRACE and GRACE-FO missions. Our contribution to this research unit is to is to (re)analyze data fields on horizontal grid sizes below 10 km from non-hydrostatic regional atmospheric models: the 1995-2019 COSMO-REA6 regional reanalysis and the July 2021 ICON-EU/D2 analyses. From these models, the mass densities of dry air and all water phases (gaseous, liquid clouds and rain, icy snow, sleet, and hail) are available in each of the 40-50 vertical layers, so that the total local and column-by-column atmospheric mass density can be calculated without using the hydrostatic assumption.</p><p>Current non-hydrostatic, high-resolution atmospheric models run under some idealizations (e.g., assumption of a perfect sphere, spatially constant gravitational acceleration, approximation of a flat atmosphere) that are inconsistent with the real measurements of the GRACE/GRACE-FO missions. Furthermore, high-resolution atmospheric model data are currently only available within the CORDEX-EU domain, which covers the North Atlantic-European region, or an even smaller domain. The results to be presented will answer the following questions: (1) Do the model idealizations cause discrepancies with respect to the GRACE/GRACE-FO measurements that are larger than the sum of the actual measurement uncertainties and the uncertainties resulting from the remaining dealiasing models? (2) Can the idealizations be corrected to achieve better agreement with the GRACE/GRACE-FO data without sacrificing the underlying model dynamics, e.g. by introducing additional gradients in the models? (3) Do extreme weather events, such as heavy rainfall observed in the Ruhr/Ahr/Erft/Maas basin in July 2021, cause mass variations that exceeds typical GRACE/GRACE-FO uncertainties? And finally, (4) does high-resolution but regional atmospheric mass variability leave a statistically significant fingerprint in the global loading coefficients?</p>

The Marine and Atmospheric Research System (MARS) for simulation of the Caspian Sea meteorological characteristics is presented. It includes computation of the atmospheric forcing with the regional non-hydrostatic atmosphere model Weather Research and Forecasting model (WRF), as well as computation of currents, sea level, temperature, salinity and sea ice with the model of marine circulation INMOM (Institute of Numerical Mathematics Ocean Model), and the computation of wind wave parameters using the Russian wind-wave model (RWWM). The results on verification of the hydrometeorological characteristics simulated with MARS are presented. Also, the retrospective simulation of the thermohydrodynamic characteristics for the Caspian Sea was performed with MARS for the ice-free period 2003–2013. The important features of the Caspian Sea circulation are shown. The wind currents contribute the most significantly to the Caspian Sea circulation. The upwelling effect in the eastern part of Middle Caspian is caused by regional northerly winds. It should be noted that where the outputs of different models for North and Middle Caspian are in good agreement, there is significant difference in the estimates of the currents in South Caspian. The wind fields computed in MARS using the non-hydrostatic model WRF and those obtained by the formulas of geostrophic wind inferred from the pressure field from the same model are in good agreement in Middle and North Caspian. However, they differ significantly for South Caspian. This is especially noticeable at the eastern and southern coasts, where the wind computed in WRF is directed transversely to the isobars. It is shown that the wind field and the current field are interconnected. For the Caspian Sea, one should use non-hydrostatic atmospheric models such as WRF or COSMO for wind computation.

A study of spectral element and discontinuous Galerkin methods for the Navier–Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases

Camera visualization of cloud fields simulated by non-hydrostatic atmospheric models

A new method for representing topography on a Cartesian grid is applied to a two-dimensional nonhydrostatic atmospheric model to achieve highly precise simulations over steep terrain. The shaved cell method based on finite-volume discretization is used along with a cell-combining approach in which small cut cells are combined with neighboring cells either vertically or horizontally. A unique staggered arrangement of variables enables quite simple computations of momentum equations by avoiding the evaluation of surface pressure and reducing the computational cost of combining cells for the velocity variables. The method successfully reproduces flows over a wide range of slopes, including steep slopes where significant errors are observed in a model using conventional terrain-following coordinates. The advantage of horizontal cell combination on extremely steep slopes is also demonstrated.

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally explicit – vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods.

This study aims to enhance our understanding of the bora-driven dense water dynamics in the Adriatic Sea using different state-of-the-art modelling approaches during the 2014–15 period. Practically, we analyse and compare the results of four different simulations: the latest reanalysis product for the Mediterranean Sea, the recently evaluated fine-resolution atmosphere-ocean Adriatic Sea climate model and the long-time running Adriatic Sea atmosphere-ocean forecast model used in both hindcast and data assimilation (with 4-day cycles) modes. As a first step, we evaluate the resolved physics in each simulation by focusing on the performance of the models. Then, we derive the general conditions in the ocean and the atmosphere during the investigated period. Finally, we analyse in detail the numerical reproduction of the dense water dynamics as seen by the four simulations. This study confirms that kilometre-scale atmosphere-ocean approach, non-hydrostatic atmospheric models, fine vertical resolutions in both atmosphere and ocean and proper location and forcing of the open boundary conditions are prerequisites for appropriate modelling of the ocean circulation in the Adriatic basin, which then may be improved by a data assimilation method. As proof, the 31-year long evaluation run of the Adriatic Sea climate model which meets these requirements is found to be able to outperform most aspects of the reanalysis product, the short-term hindcast and the data assimilated simulation, in reproducing the dense water dynamics in the Adriatic Sea.

Two nonhydrostatic effects that have been neglected traditionally in the hydrostatic primitiveequation models are studied in this article. One such effect is due to the vertical acceleration in the vertical equation of motion. The other is due to a Coriolis term involving 2Ω cos (latitude), where Ω is the rate of earth’s rotation, and is referred to here as a cos (latitude) Coriolis term. Cos (latitude) Coriolis terms appear in the vertical and zonal equations of motion. The questions to be investigated are: (1) what are the dynamical consequences of these two nonhydrostatic effects, (2) how the roles of cos (latitude) Coriolis terms can be compared with sin (latitude) Coriolis terms, (3) which nonhydrostatic effect is likely more important, and (4) should these effects be included in atmospheric modeling for describing what kind of motions? These questions are studied quantitatively through a normal mode analysis of compressible, and stratified atmosphere with rotation on a tangent plane in a three-dimensional space that is open horizontally, but bounded by two rigid horizontal planes in the vertical. Numerical results are presented for an isothermal model. Considering the current trend of numerical modeling in permitting to use finer resolutions and to extend the top of model atmosphere higher, it is prudent to include both nonhydrostatic effects in the dynamical core of next generation atmospheric models for all scales of motions.

Improvements on nonhydrostatic atmospheric models such as MM5 in the last few decades have enhanced our understanding of the precipitation mechanism affected by topography and nonlinear dynamics of the atmosphere. This study addresses the use of such a regional atmospheric model to estimate physical maximum precipitation rates for the next generation of flood management strategies under evolving climate conditions. First, 48 significant historical storm events were selected based on the continuous reconstructed precipitation conditions on the Feather, Yuba, and American River watersheds in California. Then, the boundary conditions of the numerical atmospheric model were modified with the fully saturated atmospheric layers (100% relative humidity) to generate the atmospheric conditions that maximize the precipitation over the three watersheds. Surprisingly, maximizing the atmospheric moisture supply at the model boundary does not always increase the precipitation in the watersheds of interest. A rain-shadow effect of the topography seemed to be intensified by the abundance of the atmospheric moisture in some cases. Consequently, although the linkage between the precipitable water in the atmosphere and the ground precipitation is generally proportional, the alignment of the topography and the wind field can modulate their relationship. Finally, a methodology to maximize the steady-state precipitation rate was discussed to characterize the conceptual continuous heavy storm event in the Feather, Yuba, and American River basins.

<strong class="journal-contentHeaderColor">Abstract.</strong> This study aims to enhance our understanding of the bora-driven dense water dynamics in the Adriatic Sea using different state-of-the-art modelling approaches during the 2014&ndash;15 period. Practically, we analyse and compare the results of four different simulations: the latest reanalysis product for the Mediterranean Sea, the recently evaluated fine-resolution atmosphere-ocean Adriatic Sea climate model and the long-time running Adriatic Sea atmosphere-ocean forecast model used in both hindcast and data assimilation (with 4-day cycles) modes. As a first step, we evaluate the resolved physics in each simulation by focusing on the performance of the models. Then, we derive the general conditions in the ocean and the atmosphere during the investigated period. Finally, we analyse in detail the numerical reproduction of the dense water dynamics as seen by the four simulations. This study confirms that kilometre-scale atmosphere-ocean approach, non-hydrostatic atmospheric models, fine vertical resolutions in both atmosphere and ocean and proper location and forcing of the open boundary conditions are prerequisites for appropriate modelling of the ocean circulation in the Adriatic basin, which then may be improved by a data assimilation method. As proof, the 31-year long evaluation run of the Adriatic Sea climate model which meets these requirements is found to be able to outperform most aspects of the reanalysis product, the short-term hindcast and the data assimilated simulation, in reproducing the dense water dynamics in the Adriatic Sea.