Baroclinic Wave Test Research Articles

A mountain-induced moist baroclinic wave test case for the dynamical cores of atmospheric general circulation models

Abstract. Idealized test cases for the dynamical cores of atmospheric general circulation models are informative tools to assess the accuracy of the numerical designs and investigate the general characteristics of atmospheric motions. A new test case is introduced that is built upon a baroclinically unstable base state with an added orographic barrier. The topography is analytically prescribed and acts as a trigger of both baroclinic Rossby waves and inertia–gravity waves on a rotating, regular-sized planet. Both dry and idealized moist configurations are suggested. The latter utilizes the Kessler warm-rain precipitation scheme. The test case enhances the complexity of the existing test suite hierarchy and focuses on the impacts of two midlatitudinal mountain ridges on the circulation. Selected simulation examples from four dynamical cores are shown. These are the Spectral Element and Finite Volume dynamical cores, which are part of the National Center for Atmospheric Research (NCAR) Community Earth System Model (CESM), versions 2.1.3 and 2.2, and the Cubed-Sphere Finite Volume dynamical cores, which is new to CESM version 2.2. In addition, the Model for Prediction Across Scales (MPAS) is tested. The overall flow patterns agree well in the four dynamical cores, but the details can vary greatly. The examples highlight the broad palette of use cases for the test case and reveal physics–dynamics coupling issues.

The Tangent-Linear and Adjoint Models of the NEPTUNE Dynamical Core

An approximate tangent-linear model and its adjoint have been developed for the dynamical core of the NEPTUNE model (Navy Environmental Prediction sysTem Utilizing a Nonhydrostatic Engine). NEPTUNE is a spectral element solver for the compressible, nonhydrostatic, deep atmosphere equations of motion on a cubed sphere grid. The tangent-linear model (TLM) and adjoint model (ADJ) were developed to enable 4D-Var-based forecast initialization systems for NEPTUNE by interfacing with driver software, such as JEDI (Joint Effort for Data assimilation Integration), but they are also useful tools for studies of linear instability and forecast sensitivity. Jablonowski and Williamson’s baroclinic wave test is used to illustrate the accuracy and characteristics of the TLM and ADJ for sensitivity studies and variational assimilation.

A Fully Compressible Nonhydrostatic Deep-Atmosphere Equations Solver for MPAS

AbstractA solver for the nonhydrostatic deep-atmosphere equations of motion is described that extends the capabilities of the Model for Prediction Across Scales-Atmosphere (MPAS-A) beyond the existing shallow-atmosphere equations solver. The discretization and additional terms within this extension maintain the C-grid staggering, hybrid height vertical coordinate, and spherical centroidal Voronoi mesh used by MPAS, and also preserve the solver’s conservation properties. Idealized baroclinic wave test results, using Earth-radius and reduced-radius sphere configurations, verify the correctness of the solver and compare well with published results from other models. For these test cases, the time evolution of the maximum horizontal wind speed, and the total energy and its components, are presented as additional solution metrics that may allow for further discrimination in model comparisons. The test case solutions are found to be sensitive to the configuration of dissipation mechanisms in MPAS-A, and many of the differences among models in previously published test case solutions appear to arise because of their differing dissipation configurations. For the deep-atmosphere reduced-radius sphere test case, small-scale noise in the numerical solution was found to arise from the analytic initialization that contains unstable lapse rates in the tropical lower troposphere. By adjusting a parameter in this initialization, the instability is removed and the unphysical large-scale overturning no longer occurs. Inclusion of the deep-atmosphere capability in the MPAS-A solver increases the dry dynamics cost by less than 5% on CPU-based architectures, and configuration of either the shallow- or deep-atmosphere equations is controlled by a simple switch.

Configuration and evaluation of a global unstructured mesh atmospheric model (GRIST-A20.9) based on the variable-resolution approach

Abstract. Targeting a long-term effort towards a variable-resolution (VR) global weather and climate model, this study systematically configures and evaluates an unstructured mesh atmospheric model based on the multiresolution approach. The model performance is examined from dry dynamics to simple physics and full physics scenarios. In the dry baroclinic wave test, the VR model reproduces comparable fine-scale structures in the refined regions as a fine-resolution quasi-uniform (QU) mesh model. The mesh transition zone does not adversely affect the wave pattern. Regional kinetic energy spectra show that the fine-scale resolving ability improves as the fine resolution increases. Compared to a QU counterpart that has equivalent degrees of freedom, the VR model tends to increase the global errors, but the errors can be reduced when the resolution of the coarse region is increased. The performance over the coarse region is generally close to that of a low-resolution QU counterpart. Two multi-region refinement approaches, the hierarchical and polycentric refinement modes, further validate the model performance under the multiresolution refinement. Activating hyperdiffusion for horizontal velocity is helpful with respect to VR modeling. An idealized tropical cyclone test is further used to examine its ability to resolve fine-scale structures. In the simple physics environment, the VR model can have the tropical cyclone stably pass the transition zone in various configurations. A series of sensitivity tests examines the model performance in a hierarchical refinement mode. The simulations exhibit consistency even when the VR mesh is slightly perturbed by one of the three parameters that control the density function. The tropical cyclone, starting from the second refinement region and passing through the inner transition zone, gets intensified and covers a smaller area in the refined regions. Such variations are consistent with the behavior that one may observe when uniformly refining the QU mesh. In the full physics environment with a highly variable mesh that reaches sub-10 km resolution, the VR model also produces a reasonable evolution for the tropical cyclone. The explicit diffusion shows its usefulness in terms of suppressing some unrealistic isolated-scale structures that are far away from the initial vortex and does not adversely affect the physically important object. The fine-scale structure is determined mainly by the fine-resolution area, although the systems may have larger differences before they move into the fine-resolution area. Altogether, this work demonstrates that the multiresolution configuration is a reliable and economic alternative to high-resolution global modeling. The adverse impact due to mesh transition and the coarse region can be controlled well.

Discretization of generalized Coriolis and friction terms on the deformed hexagonal C‐grid

This article discusses the generalized Coriolis and friction terms on the hexagonal C‐grid from two perspectives: (a) within the linearized discretized momentum equations on an equilateral grid, and (b) as nonlinear terms on a distorted mesh.The discrete linearized momentum equations are formulated using a trivariate coordinate system. The tendencies of the different forcing terms for each wind component must be linearly dependent. This constraint determines unique discretizations for each term. The linearized vorticity flux term around a zonal mean current requires only the four rhombus potential vorticities (PVs) next to an edge. The vector Laplacian must be formulated with the vorticity on vertices defined as the average of three rhombus vorticities.A modified generalized Coriolis term is defined on the deformed mesh. The baroclinic wave test on the sphere does not reveal any sign of nonlinear Hollingsworth instability, even though it is demonstrated that the vector‐invariant form and the advective form of momentum advection are not equivalent.Physical constraints determine the shape of the stress tensor. These are invariance to the addition of solid‐body rotation and a resulting positive‐definite dissipation rate. An appropriate stress‐tensor formulation does not deliver Laplacian momentum diffusion in the linear case. On the deformed mesh, parts of this stress tensor are obtained by a least‐squares reconstruction of wind gradients. This approach avoids spurious deformations diagnosed for constant flow in the vicinity of pentagon cells.

Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models

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.

Strong scaling for numerical weather prediction at petascale with the atmospheric model NUMA

Numerical weather prediction (NWP) has proven to be computationally challenging due to its inherent multiscale nature. Currently, the highest resolution global NWP models use a horizontal resolution of 9 km. At this resolution, many important processes in the atmosphere are not resolved. Needless to say, this introduces errors. In order to increase the resolution of NWP models, highly scalable atmospheric models are needed. The non-hydrostatic unified model of the atmosphere (NUMA), developed by the authors at the Naval Postgraduate School, was designed to achieve this purpose. NUMA is used by the Naval Research Laboratory, Monterey as the engine inside its next generation weather prediction system NEPTUNE. NUMA solves the fully compressible Navier–Stokes equations by means of high-order Galerkin methods (both spectral element as well as discontinuous Galerkin methods can be used). NUMA is capable of running middle and upper atmosphere simulations since it does not make use of the shallow-atmosphere approximation. This article presents the performance analysis and optimization of the spectral element version of NUMA. The performance at different optimization stages is analyzed using a theoretical performance model as well as measurements via hardware counters. Machine-independent optimization is compared to machine-specific optimization using Blue Gene (BG)/Q vector intrinsics. The best portable version of the main computations was found to be about two times slower than the best non-portable version. By using vector intrinsics, the main computations reach 1.2 PFlops on the entire IBM Blue Gene supercomputer Mira (12% of the theoretical peak performance). The article also presents scalability studies for two idealized test cases that are relevant for NWP applications. The atmospheric model NUMA delivers an excellent strong scaling efficiency of 99% on the entire supercomputer Mira using a mesh with 1.8 billion grid points. This allows running a global forecast of a baroclinic wave test case at a 3-km uniform horizontal resolution and double precision within the time frame required for operational weather prediction.

A Semi-Implicit Modification to the Lorenz N-Cycle Scheme and Its Application for Integration of Meteorological Equations

Abstract The Lorenz N-cycle is an economical time integration scheme that requires only one function evaluation per time step and a minimal memory footprint, but yet possesses a high order of accuracy. Despite these advantages, it has remained less commonly used in meteorological applications, partly because of its lack of semi-implicit formulation. In this paper, a novel semi-implicit modification to the Lorenz N-cycle is proposed. The advantage of the proposed new scheme is that it preserves the economical memory use of the original explicit scheme. Unlike the traditional Robert–Asselin (RA) filtered semi-implicit leapfrog scheme whose formal accuracy is only of first order, the new scheme has second-order accuracy if it adopts the Crank–Nicolson scheme for the implicit part. A linear stability analysis based on a univariate split-frequency oscillation equation suggests that the 4-cycle is more stable than other choices of N. Numerical experiments performed using the dynamical core of the Simplified Parameterizations Primitive Equation Dynamics (SPEEDY) atmospheric general circulation model under the framework of the Jablonowski–Williamson baroclinic wave test case confirms that the new scheme in fact has second-order accuracy and is more accurate than the traditional RA-filtered leapfrog scheme. The experiments also give evidence for Lorenz’s claim that the explicit 4-cycle scheme can be improved by running its two “isomeric” versions in alternating sequences. Unlike the explicit scheme, however, the proposed semi-implicit scheme is not improved by alternation of the two versions.

A Semi-Implicit Version of the MPAS-Atmosphere Dynamical Core

Abstract An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge–Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank–Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%–20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

A proposed baroclinic wave test case for deep‐ and shallow‐atmosphere dynamical cores

AbstractIdealised studies of key dynamical features of the atmosphere provide insight into the behaviour of atmospheric models. A very important, well understood, aspect of midlatitude dynamics is baroclinic instability. This can be idealised by perturbing a vertically sheared basic state in geostrophic and hydrostatic balance. An unstable wave mode then results with exponential growth (due to linear dynamics) in time until, eventually, nonlinear effects dominate and the wave breaks.A new, unified, idealised baroclinic instability test case is proposed. This improves on previous ones in three ways. First, it is suitable for both deep‐ and shallow‐atmosphere models. Second, the constant surface pressure and zero surface geopotential of the basic state makes it particularly well‐suited for models employing a pressure‐ or height‐based vertical coordinate. Third, the wave triggering mechanism selectively perturbs the rotational component of the flow; this, together with a vertical tapering, significantly improves dynamic balance.

Evaluation of Global Atmospheric Solvers Using Extensions of the Jablonowski and Williamson Baroclinic Wave Test Case

Abstract The hydrostatic and nonhydrostatic atmospheric solvers within the Model for Prediction Across Scales (MPAS) are tested using an extension of Jablonowski and Williamson baroclinic wave test case that includes moisture. This study uses the dry test case to verify the correctness of the solver formulation and coding by comparing results from the two different MPAS solvers and from the global version of the Advanced Research Weather Research and Forecasting Model (ARW-WRF). A normal mode initialization is used in the Jablonowski and Williamson test, and the most unstable mode is found to be wavenumber 9. The three solvers produce very similar normal mode structures and nonlinear baroclinic wave evolutions. Solutions produced using MPAS variable-resolution meshes are quite similar to the results from the quasi-uniform mesh with equivalent resolution. Importantly, the small-scale flow features are better resolved in the fine-resolution region and there is no apparent wave distortion in the fine-to-coarse mesh transition region, thus demonstrating the potential value of MPAS for multiscale flow simulation.

The ICON-1.2 hydrostatic atmospheric dynamical core on triangular grids – Part 1: Formulation and performance of the baseline version

Abstract. As part of a broader effort to develop next-generation models for numerical weather prediction and climate applications, a hydrostatic atmospheric dynamical core is developed as an intermediate step to evaluate a finite-difference discretization of the primitive equations on spherical icosahedral grids. Based on the need for mass-conserving discretizations for multi-resolution modelling as well as scalability and efficiency on massively parallel computing architectures, the dynamical core is built on triangular C-grids using relatively small discretization stencils. This paper presents the formulation and performance of the baseline version of the new dynamical core, focusing on properties of the numerical solutions in the setting of globally uniform resolution. Theoretical analysis reveals that the discrete divergence operator defined on a single triangular cell using the Gauss theorem is only first-order accurate, and introduces grid-scale noise to the discrete model. The noise can be suppressed by fourth-order hyper-diffusion of the horizontal wind field using a time-step and grid-size-dependent diffusion coefficient, at the expense of stronger damping than in the reference spectral model. A series of idealized tests of different complexity are performed. In the deterministic baroclinic wave test, solutions from the new dynamical core show the expected sensitivity to horizontal resolution, and converge to the reference solution at R2B6 (35 km grid spacing). In a dry climate test, the dynamical core correctly reproduces key features of the meridional heat and momentum transport by baroclinic eddies. In the aqua-planet simulations at 140 km resolution, the new model is able to reproduce the same equatorial wave propagation characteristics as in the reference spectral model, including the sensitivity of such characteristics to the meridional sea surface temperature profile. These results suggest that the triangular-C discretization provides a reasonable basis for further development. The main issues that need to be addressed are the grid-scale noise from the divergence operator which requires strong damping, and a phase error of the baroclinic wave at medium and low resolutions.

A global hexagonal C‐grid non‐hydrostatic dynamical core (ICON‐IAP) designed for energetic consistency

AbstractThis study describes a new global non‐hydrostatic dynamical core (ICON‐IAP: Icosahedral Nonhydrostatic model at the Institute for Atmospheric Physics) on a hexagonal C‐grid which is designed to conserve mass and energy. Energy conservation is achieved by discretizing the antisymmetric Poisson bracket which mimics correct energy conversions between the different kinds of energy (kinetic, potential, internal). Because of the bracket structure this is even possible in a complicated numerical environment with (i) the occurrence of terrain‐following coordinates with all the metric terms in it, (ii) the horizontal C‐grid staggering on the Voronoi mesh and the complications induced by the need for an acceptable stationary geostrophic mode, and (iii) the necessity for avoiding Hollingsworth instability. The model is equipped with a Smagorinsky‐type nonlinear horizontal diffusion. The associated dissipative heating is accounted for by the application of the discrete product rule for derivatives. The time integration scheme is explicit in the horizontal and implicit in the vertical. In order to ensure energy conservation, the Exner pressure has to be off‐centred in the vertical velocity equation and extrapolated in the horizontal velocity equation.Test simulations are performed for small‐scale and global‐scale flows. A test simulation of linear non‐hydrostatic flow over a rough mountain range shows the theoretically expected gravity wave propagation. The baroclinic wave test is extended to 40 days in order to check the Lorenz energy cycle. The model exhibits excellent energy conservation properties even in this strongly nonlinear and dissipative case. The Held–Suarez test confirms the reliability of the model over even longer time‐scales. Copyright © 2012 Royal Meteorological Society

Comparison of different order Adams-Bashforth methods in an atmospheric general circulation model

The Asselin-Robert time filter used in the leapfrog scheme does degrade the accuracy of calculations. As an attractive alternative to leapfrog time differencing, the second-order Adams-Bashforth method is not subject to time splitting instability and keeps excellent calculation accuracy. A second-order Adams-Bashforth model has been developed, which represents better stability, excellent convergence and improved simulation of prognostic variables. Based on these results, the higher-order Adams-Bashforth methods are developed on the basis of NCAR (National Center for Atmospheric Research) CAM 3.1 (Community Atmosphere Model 3.1) and the characteristics of dynamical cores are analyzed in this paper. By using Lorenz nonlinear convective equations, the filtered leapfrog scheme shows an excellent pattern for eliminating 2Δt wave solutions after 20 steps but represents less computational solution accuracy. The fourth-order Adams-Bashforth method is closely converged to the exact solution and provides a reference against which other methods may be compared. Thus, the Adams-Bashforth methods produce more accurate and convergent solution with differencing order increasing. The Held-Suarez idealized test is carried out to demonstrate that all methods have similar climate states to the results of many other global models for long-term integration. Besides, higher-order methods perform better in mass conservation and exhibit improvement in simulating tropospheric westerly jets, which is likely equivalent to the advantages of increasing horizontal resolutions. Based on the idealized baroclinic wave test, a better capability of the higher-order method in maintaining simulation stability is convinced. Furthermore, after the baroclinic wave is triggered through overlaying the steady-state initial conditions with the zonal perturbation, the higher-order method has a better ability in the simulation of baroclinic wave perturbation.

A Stability Analysis of Divergence Damping on a Latitude–Longitude Grid

The dynamical core of an atmospheric general circulation model is engineered to satisfy a delicate balance between numerical stability, computational cost, and an accurate representation of the equations of motion. It generally contains either explicitly added or inherent numerical diffusion mechanisms to control the buildup of energy or enstrophy at the smallest scales. The diffusion fosters computational stability and is sometimes also viewed as a substitute for unresolved subgrid-scale processes. A particular form of explicitly added diffusion is horizontal divergence damping. In this paper a von Neumann stability analysis of horizontal divergence damping on a latitude–longitude grid is performed. Stability restrictions are derived for the damping coefficients of both second- and fourth-order divergence damping. The accuracy of the theoretical analysis is verified through the use of idealized dynamical core test cases that include the simulation of gravity waves and a baroclinic wave. The tests are applied to the finite-volume dynamical core of NCAR’s Community Atmosphere Model version 5 (CAM5). Investigation of the amplification factor for the divergence damping mechanisms explains how small-scale meridional waves found in a baroclinic wave test case are not eliminated by the damping.

The development of a nonhydrostatic global spectral model

With the development of numerical weather prediction technology, the traditional global hydrostatic models used in many countries of the world for operational weather forecasting and numerical simulations of general circulation have become more and more unfit for high-impact weather prediction. To address this, it is important to invest in the development of global nonhydrostatic models. Few existing nonhydrostatic global models use consistently the grid finite difference scheme for the primitive equations of dynamical cores, which can subsequently degrade the accuracy of the calculations. A new nonhydrostatic global spectral model, which utilizes the Eulerian spectral method, is developed here from NCAR Community Atmosphere Model 3.0 (CAM3.0). Using Janjic’s hydrostatic/nonhydrostatic method, a global nonhydrostatic spectral method for the primitive equations has been formulated and developed. In order to retain the integrity of the nonhydrostatic equations, the atmospheric curvature correction and eccentricity correction are considered. In this paper, the Held-Suarez idealized test and an idealized baroclinic wave test are first carried out, which shows that the nonhydrostatic global spectral model has similar climate states to the results of many other global models for long-term idealized integration, as well as better simulation ability for short-term idealized integration. Then, a real case experiment is conducted using the new dynamical core with the full physical parameterizations of subgrid-scale physical processes. The 10-day numerical integration indicates a decrease in systematic error and a better simulation of zonal wind, temperature, and 500-hPa height.

Rotated Versions of the Jablonowski Steady‐State and Baroclinic Wave Test Cases: A Dynamical Core Intercomparison

The Jablonowski test case is widely used for debugging and evaluating the numerical characteristics of global dynamical cores that describe the fluid dynamics component of Atmospheric General Circulation Models. The test is defined in terms of a steady‐state solution to the equations of motion and an overlaid perturbation that triggers a baroclinically unstable wave. The steady‐state initial conditions are zonally symmetric. Therefore, the test case design has the potential to favor models that are built upon regular latitude‐longitude or Gaussian grids. Here we suggest rotating the computational grid so that the balanced flow is no longer aligned with the computational grid latitudes. Ideally the simulations should be invariant under rotation of the computational grid. Note that the test case only requires an adjustment of the Coriolis parameter in the model code. The rotated test case has been exercised by six dynamical cores. In addition, two of the models have been tested with different vertical coordinates resulting in a total of eight model variants. The models are built with different computational grids (regular latitude‐longitude, cubed‐sphere, icosahedral hexagonal/triangular) and use very different numerical schemes. The test‐case is a useful tool for debugging, assessing the degree of anisotropy in the numerical methods and grids, and evaluating the numerical treatment of the pole points since the rotated test case directs the flow directly over the geographical poles. Special treatments such as polar filters are therefore more exposed in this rotated test case.

A mass‐conservative version of the semi‐implicit semi‐Lagrangian HIRLAM

AbstractA mass‐conservative version of the semi‐implicit semi‐Lagrangian High‐Resolution Limited‐Area Model (HIRLAM) is presented. The explicit continuity equation is solved with the so‐called cell‐integrated semi‐Lagrangian (CISL) method. To allow for long time steps, the CISL scheme is coupled with a recently developed semi‐implicit time‐stepping scheme that involves the same non‐complicated elliptic equation as in HIRLAM. Contrarily to the traditional semi‐Lagrangian method, the trajectories are backward in the horizontal and forward in the vertical, i.e. cells moving with the flow depart from model layers and arrive in a regular column, and their vertical displacements are computed from continuity of mass and hydrostatic balance in the arrival column. This involves just two‐dimensional upstream integrals and allows for a Lagrangian discretization of the energy conversion term in the thermodynamic equation.Preliminary validation of the new model version is performed using an idealized baroclinic wave test case. The accuracy of the new formulation of HIRLAM is comparable to the reference version though it is slightly more diffusive. A main finding is that the new discretization of the energy conversion term leads to more accurate simulations compared to the traditional ‘Eulerian’ treatment. Copyright © 2008 Royal Meteorological Society