Shallow-atmosphere Approximation Research Articles

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.

Benchmark experiments with global climate models applicable to extrasolar gas giant planets in the shallow atmosphere approximation

The growing field of exoplanetary atmospheric modelling has seen little work on standardised benchmark tests for its models, limiting understanding of the dependence of results on specific models and conditions. With spatially resolved observations as yet difficult to obtain, such a test is invaluable. Although an intercomparison test for models of tidally locked gas giant planets has previously been suggested and carried out, the data provided were limited in terms of comparability. Here, the shallow PUMA model is subjected to such a test, and detailed statistics produced to facilitate comparison, with both time means and the associated standard deviations displayed, removing the time dependence and providing a measure of the variability. Model runs have been analysed to determine the variability between resolutions, and the effect of resolution on the energy spectra studied. Superrotation is a robust and reproducible feature at all resolutions.

The Importance of the Nontraditional Coriolis Terms in Large-Scale Motions in the Tropics Forced by Prescribed Cumulus Heating

AbstractIn meteorological dynamics, the shallow-atmosphere approximation is generally used in the momentum equation, together with the “traditional approximation.” In the traditional approximation, two cosine Coriolis terms [hereafter called nontraditional Coriolis terms (NCTs)] and some metric terms are omitted. However, some studies have suggested that the omission of the NCTs may not be appropriate for conditions near the equator. Therefore, this paper investigates the effect of the NCTs on large-scale motions forced by local positive-only heating mimicking cumulus convection.The authors use the linearized quasihydrostatic equation system on an equatorial β plane. Prescribed heating is assumed to move eastward with the slow phase speed of an intraseasonal period. Results of scale analysis and numerical calculations show that differences with and without the NCTs (hereafter, contributions) exhibit the following four features. (i) Whereas contributions to horizontal divergence and vertical velocity are small, the NCTs have large effects on horizontal velocity (vertical vorticity) and perturbations of pressure, potential temperature, and density. (ii) Contributions to horizontal velocity and pressure perturbation have equivalent barotropic structure. (iii) Contributions show east–west asymmetric patterns, with large contributions appearing at the western side of the forcing. (iv) Contributions to vertical vorticity and pressure perturbation are extremely large when the meridional gradient of heating is large. These features can be comprehensively explained by the tilting of the meridional component of the planetary vorticity, which is caused by the meridional gradient of heating.

MCore: A non-hydrostatic atmospheric dynamical core utilizing high-order finite-volume methods

This paper presents a new atmospheric dynamical core which uses a high-order upwind finite-volume scheme of Godunov type for discretizing the non-hydrostatic equations of motion on the sphere under the shallow-atmosphere approximation. The model is formulated on the cubed-sphere in order to avoid polar singularities. An operator-split Runge–Kutta–Rosenbrock scheme is used to couple the horizontally explicit and vertically implicit discretizations so as to maintain accuracy in time and space and enforce a global CFL condition which is only restricted by the horizontal grid spacing and wave speed. The Rosenbrock approach is linearly implicit and so requires only one matrix solve per column per time step. Using a modified version of the low-speed AUSM+-up Riemann solver allows us to construct the vertical Jacobian matrix analytically, and so significantly improve the model efficiency. This model is tested against a series of typical atmospheric flow problems to verify accuracy and consistency. The test results reveal that this approach is stable, accurate and effective at maintaining sharp gradients in the flow.

Treatment of vector equations in deep‐atmosphere, semi‐Lagrangian models. II: Kinematic equation

AbstractThe rotation matrix formalism described in Part I of this study is here applied to the deep‐atmosphere form of the vector kinematic equation and the determination of semi‐Lagrangian ‘departure points’ (locations at the current time of fluid particles that arrive at model gridpoints at the forecast time). Following a brief review of published methods for determining departure points in models that incorporate the restrictive shallow‐atmosphere approximation, a certain discrete form of the kinematic equation is used to illustrate application of the rotation matrix technique in spherical coordinates in the deep‐atmosphere case. This discrete form–referred to as ‘doubly implicit’ –involves velocities at arrival gridpoints as well as at departure points but has several advantages over alternatives that involve velocities at a point or points along the intervening trajectory. A local Cartesian transformation method for finding departure points emerges from the rotation matrix approach applied in spherical polar coordinates. In spheroidal coordinates, a local transformation method also emerges, but the relevant Cartesian origins lie along the axis of symmetry of the spheroids rather than at their common centre. Modifications for the spherical polar case under the shallow‐atmosphere approximation are obtained by a constraint method; apart from order‐unity multiplicative factors, the resulting departure‐point equations are of the form expected given the shallow‐atmosphere momentum component equations found in Part I. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.

Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi‐hydrostatic and non‐hydrostatic

AbstractWe study global atmosphere models that are at least as accurate as the hydrostatic primitive equations (HPEs), reviewing known results and reporting some new ones. The HPEs make spherical geopotential and shallow atmosphere approximations in addition to the hydrostatic approximation. As is well known, a consistent application of the shallow atmosphere approximation requires omission of those Coriolis terms that vary as the cosine of latitude and of certain other terms in the components of the momentum equation. An approximate model is here regarded as consistent if it formally preserves conservation principles for axial angular momentum, energy and potential vorticity, and (following R. Müller) if its momentum component equations have Lagrange's form. Within these criteria, four consistent approximate global models, including the HPEs themselves, are identified in a height‐coordinate framework. The four models, each of which includes the spherical geopotential approximation, correspond to whether the shallow atmosphere and hydrostatic (or quasi‐hydrostatic) approximations are individually made or not made. Restrictions on representing the spatial variation of apparent gravity occur. Solution methods and the situation in a pressure‐coordinate framework are discussed. © Crown copyright 2005.

Normal modes of deep atmospheres. II: f–F‐plane geometry

AbstractNumerical results have shown that, for a rotating spherical atmosphere with terrestrial parameters, making the shallow‐atmosphere approximation has only a very small impact on the frequency of linear, unforced normal modes, compared with those obtained from the full, unapproximated equations. In nearly all cases the normal‐mode frequencies are smaller in magnitude when the shallow‐atmosphere approximation is relaxed. Relaxing the approximation was also found to have only a small impact on the structures of normal modes, with the exception of long‐zonal‐wavelength internal acoustic modes. These results are particularly surprising in the tropics where the inclusion of the F = 2Ω cosϕ Coriolis terms (which are dropped in the shallow‐atmosphere approximation) might be expected to dominate the usual f=2Ωsinϕ Coriolis terms. The complexity of the full equations, however, prevents analysis of why this insensitivity to the extra terms arises.In this paper normal modes are examined under the f–F‐plane approximation and compared with those on the more usual f‐plane. The resulting equations are more amenable to analysis than the full equation set, and analytic expressions for the dispersion relation and for the normal‐mode structures are obtained for the particular case of an isothermal reference profile. This simplified geometry allows the effects of the F Coriolis terms to be examined while eliminating the geometrical effects of relaxing the shallow‐atmosphere approximation, giving some insight into the relative importance of the two types of effect as well as the physical mechanisms at work. The F Coriolis terms are found to be responsible for the structural changes to long‐zonal‐wavelength internal acoustic modes, and can also affect extremely shallow and extremely deep gravity modes. However, these terms are found to have only a small effect on normal‐mode frequencies, and geometrical effects, rather than these Coriolis terms, are responsible for the systematic reduction in the magnitude of normal‐mode frequencies in a deep spherical atmosphere.In planar geometry the inclusion of the F terms gives rise to a new kind of normal mode in addition to the usual Rossby, gravity, and acoustic modes. The new modes are inertial in character, have frequency very close to f, and have extremely strong vertical tilt. © Royal Meteorological Society, 2002. N. Wood's and A. Staniforth's contributions are Crown copyright.

Normal modes of deep atmospheres. I: Spherical geometry

AbstractNumerical weather‐ and climate‐prediction models have traditionally applied the hydrostatic approximation and also, in particular, the shallow‐atmosphere approximation. In addition, and probably as a result, studies of the normal modes of the atmosphere too have made the shallow‐atmosphere approximation. The approximation appears to be based on simple scaling arguments. Here, the forms of the unforced, linear normal modes for the deep atmosphere on a sphere are considered and compared with those of the shallow atmosphere. Also, the impact of ignoring the vertical variation of gravity is investigated. For terrestrial parameters, it is found that relaxing either or both of these approximations has very little impact on the spatial form of the energetically significant components of most normal modes. The frequencies too are only slightly changed. However, relaxing the shallow‐atmosphere approximation does lead to significant changes in the tropical structure of internal acoustic modes. Relaxing the shallow‐atmosphere approximation also leads to non‐zero vertical‐velocity and potential‐temperature fields for external acoustic and Rossby modes; these fields are identically zero when the shallow‐atmosphere approximation is made.For a finite‐difference numerical model to be able to represent well the behaviour of the free atmosphere it must be able to capture accurately the structures of the normal modes. Therefore, the structures of normal modes can have implications for the choice of prognostic variables and grid staggering. In particular, the vertical structure of normal modes suggests that density and temperature should be analytically eliminated in favour of pressure and potential temperature as the prognostic thermodynamic variables, and that potential temperature and vertical velocity should be staggered in the vertical with respect to the other dynamic prognostic variables, the so‐called Charney– Phillips grid. © Royal Meteorological Society, 2002. N. Wood's and A. Staniforth's contributions are Crown copyright.

Effects of eddy viscosity and thermal conduction and Coriolis force in the dynamics of gravity wave driven fluctuations in the OH nightglow

Recently, Walterscheid et al. (1987) have described a dynamical‐chemical model of wave‐driven fluctuations in the OH nightglow which incorporated a five‐reaction photochemical scheme and the dynamics of linearized acoustic‐gravity waves in an isothermal, motionless atmosphere. The intensity oscillation (δI) about the time‐averaged intensity (I0) and the temperature oscillation (δT) about the time‐averaged temperature (T0) were related by means of the complex ratio η ≡ (δI/I0)/(δT/T0). One of the main conclusions of their work was that the inclusion of dynamical effects is absolutely essential for a valid assessment of η at any wave period. In this paper the model of Walterscheid et al. (1987) is modified to include in the gravity wave dynamics the effects of eddy viscosity, eddy thermal conduction, and Coriolis force (with the shallow atmosphere approximation), and calculations are performed for the same nominal case as used by these previous authors (i.e., λx = 100 km and atmospheric conditions pertinent to 83 km altitude), but only gravity wave periods are considered. It is found that for wave periods greater than some 2 or 3 hours the value of η is greatly modified by the inclusion of eddy thermal conduction. Although when acting alone the eddy viscosity is relatively unimportant, it significantly modifies the results when acting in conjunction with the eddy thermal conduction. The inclusion of the Coriolis force is found to be insignificant at any wave period. Although it is for the longest‐period waves that the values of η are most modified by the inclusion of dissipation, this dissipation may be severe enough to place an observational constraint on such waves. Results of Walterscheid et al. (1987) suggest that η is virtually independent of horizontal wavelength (λx), but it is indicated here that the inclusion of dissipation is likely to make η highly dependent on λx and to complicate comparisons which have been made between observation and theory. The effects of varying the Prandtl number are also discussed.