Weather prediction models or generally speaking atmospheric general circulation models are the discrete, numerical representatives of the underlying governing physical laws. The following two web sites provide first insight into the concepts of a weather prediction model and give some hints concerning its complex structures.
Numerical weather prediction models at the German Weather Services (DWD), Offenbach, Germany Forecasting by computer at the European Centre for Medium-Range Weather Forecasts (ECMWF), Reading, England
Atmospheric models consist of several components which describe the state of the atmosphere. Important model components are the dynamics package, the so-called dynamical core, and the physics package which strongly interact in a non-linear fashion. The dynamical core contains the large-scale adiabatic part of a model (the discretized governing equations) and is explicitly resolved on the underlying grid, whereas the physics is characterized by diabatic, subgrid-scale processes. These physical processes such as radiation, clouds, friction and boundary layer interactions play an important role in the general circulation. However, their characterisic spatial scales are too small to be resoved on a typical GCM grid with grid spacings of order 50 km or wider. Therefore, the overall effects of the small-scale processes are estimated via so-called physical parameterizations. These are often derived empirically.
The interaction of the model components in a full GCM makes it difficult or even impossible to decide which phenomena are caused by which model component. Each attempt to gather information on a specific model component - so for example information on the dynamical core of a model - is influenced by the impact of the physical parameterizations. This difficulty is alleviated when testing the dynamical core in isolation. This can ve viewed as a 'unit test before coupling the dynamics to the physics parametrization suite. Dry dynamical core tests give valuable information about the characteristics of the numerical discretizations, such as the diffusive behavior, and are especially useful as a begugging tool during the model development phases. In addition, dynamical cores can be intercompared which provides information about the uncertainties in the numerical solutions.
Two groups of dynamical core tests need to be distinguished. First, dynamical cores can be tested with short deterministic test cases which typically cover a simulation period of about 10-30 days. Second, dynamical cores can be tested in a climate mode that assesses the statistics of the model simulations. These runs span a multi-year time period and are typically run with simple prescribed forcings (Rayleigh friction and Newtonian temperature relaxation) that replace the complex physics suite. Two such 'climate' forcings have been suggested in the literature. They are known as the Held-Suarez test and the Boer-Denis test which are briefly described below.
Deterministic Dynamical Core TestsShort deterministic test cases for dynamical cores have become prominient over the last few years. The key to their widespread use is that they need to be 'easy-to-implement'. Short dynamical core test cases start from prescribed initial conditions that are ideally provided in analytic form. The simulations are run in a 'forecast mode' for 10-30 day time periods and are either compared to analytic solutions (if available) or high-resolution reference solutions. Examples of dry dynamical core test cases are the Jablonowski-Williamson steady-state and baroclinic wave test (Jablonowski and Williamson, QJ 2006; Williamson et al., MWR 2009; Lauritzen et al., JAMES 2010), 3D advection, Rossby-Haurwitz waves, mountain-generated Rossby waves or gravity waves (Jablonowski et al. 2008, Ullrich and Jablonowski, JCP 2012). We also suggest a moist tropical cyclone test case in aqua-planet mode that can either be run in a full-physics or simple-physics setup (Reed and Jablonowski, JAMES 2012). These simulations are initialized with a weak axisymmetric vortex that is embedded into tropical environmental conditions (Reed and Jablonowski, MWR 2011). The vortex then evolves into a tropical cyclone over 10 simulation days. This collection and examples of the model simulations are provided in:
- Jablonowski, C. and D. L. Williamson (2006): A baroclinic instability test case for atmospheric model dynamical cores , Quart. J. Roy. Meteorol. Soc., Vol. 132, 2943-2975
- Jablonowski, C. and D. L. Williamson (2006b): A Baroclinic Wave Test Case for Dynamical Cores of General Circulation Models: Model Intercomparisons , NCAR Technical Note NCAR/TN-469+STR, Boulder, CO, 89 pp.
- Jablonowski, C., P. H. Lauritzen, R. Nair and M. Taylor (2008): Idealized test cases for the dynamical cores of Atmospheric General Circulation Models: A proposal for the NCAR ASP 2008 summer colloquiumi, Manuscript May/29/2008, to be submitted as an NCAR Technical Report and journal paper
- Jablonowski, C., P. H. Lauritzen, M. A. Taylor and R. Nair (2008): An intercomparison of 10 atmospheric model dynamical cores, Presentation at AGU Fall Meeting, San Francisco, CA, USA, December 15-19, 2008
- Jablonowski, C., P. H. Lauritzen, R. Nair and M. Taylor (2009): A test suite for GCMs: An intercomparison of 10 atmospheric dynamical cores, Presentation at the Workshop on Solutions to Partial Differential Equations on the Sphere, Santa Fe, USA, April 27-30, 2009
- Williamson, D. L., J. Olson and C. Jablonowski (2009): Two dynamical core formulation flaws exposed by a baroclinic instability test case, Mon. Wea. Rev., Vol. 137, 790-796
- Lauritzen, P. H., C. Jablonowski, M. A. Taylor and R. D. Nair (2010): Rotated versions of the Jablonowski steady-state and baroclinic wave test cases: A dynamical core intercomparison, J. Adv. Model. Earth Syst., Vol. 2, Art. #15, 34 pp, Article Spotlight in Dec. 2009
- Reed, K. A. and C. Jablonowski (2011a): An Analytic Vortex Initialization Technique for Idealized Tropical Cyclone Studies in AGCMs, Mon. Wea. Rev., Vol. 139, 689-710
- Reed, K. A. and C. Jablonowski (2011b): Impact of physical parameterizations on idealized tropical cyclones in the Community Atmosphere Model, Geophys. Res. Lett., Vol. 38, L04805
- Reed, K. A. and C. Jablonowski (2011c): Assessing the Uncertainty of Tropical Cyclone Simulations in NCAR's Community Atmosphere Model, J. Adv. Model. Earth Syst., Vol. 3, Art. 2011MS000076, 16 pp.
- Reed, K. A. and C. Jablonowski (2012): Idealized tropical cyclone simulations of intermediate complexity: a test case for AGCMs, J. Adv. Model. Earth Syst., in press
- Ullrich, P. A. and C. Jablonowski (2012): MCore: A Non-hydrostatic Atmospheric Dynamical Core Utilizing High-Order Finite-Volume Methods, J. Comput. Phys., revised (Jan. 2012)
Additional information is available on the following web pages that provide
Held-Suarez TestThis test of the dynamical core has been designed by Isaac Held (GFDL, Princeton) and Max Suarez (NASA) who published the test method in 1994. The article is available online from AMS:
Held, I. M. and M. J. Suarez (1994): A proposal for the intercomparison of the dynamical cores of atmospheric general circulation models, Bull. Am. Meteorol. Soc. 73, 1825-1830 .
The Held-Suarez test evaluates the dynamical core in a climate mode. 1200-day integrations are required that allow the assessment of climate statistics like the zonal-mean time-mean general ciculation and the Eddy fluxes. The basic idea behind the test method is to replace the complex physics package with simplified physics. This simplified physics (or more precisely idealized forcing) consists of a temperature relaxation function and Rayleigh friction for the wind in lower levels. Using this forcing a dynamical core can be tested on its own or can be compared with other dynamical cores because the dynamically induced circulation is no longer influenced by interactions with the physical parameterizations.
A variation of the Held-Suarez test has been developed by D. L. Williamson, J. G. Olson and B.A. Boville, NCAR, Boulder, USA, in 1998 and is here referenced as the Held-Suarez-Williamson test. Williamson et al. (1998) modified the Held-Suarez temperature forcing function in the upper atmosphere (above 100hPa) to test the model behavior in the stratosphere and mesosphere. This change becomes important when using vertical high resolution models since the Held-Suarez forcing provides an isothermal, stable temperature profile in the upper atmosphere which keeps the stratosphere and mesosphere passive.
The Held-Suarez-Williamson test method has been published in:
Williamson, D. L., J. G. Olson and B.A. Boville (1998), "A Comparison of semi-Lagrangian and Eulerian tropical climate simulations", Monthly Weather Review 126:1001-1012 .
Boer-Denis TestThe second method to test the dynamical core in a 'climate mode' was introduced at the first 'Workshop on Dynamical Cores' in Breckenridge, Colorado, USA, in 1996. One year later this test method was published. The article in available from Springer:
Boer, G. J. and B. Denis, (1997), "Numerical convergence of the dynamics of a GCM", Clim. Dyn. 13:359-374 .
The Boer-Denis test is similar to the Held-Suarez test. The physics package is replaced with an idealized forcing mechanism. These forcing functions are based on two prescribed temperature and heating profiles as well as a friction term that slows down the wind at lower levels.
Ideas for new dynamical core test cases and the results of model intercomparisons are discussed at the 'Partial Differential Equations on the Sphere (PDEs on the Sphere)' workshops that take place every 18-24 months. The list of the PDEs on the Sphere Workshops since 1998 can be found below. In addition, I am one of the initiators of the 'Dynamical Core Model Intercomparison Project (DCMIP)' which is based on our experience with two summer schools. In 2008, we conducted a hands-on model intercomparison workshop and summer school at NCAR that focused on dry dynamical core tests (Jablonowski et al. 2008). In the summer of 2012, we will conduct a DCMIP summer school with a focus on non-hydrostatic dynamical cores and their interactions with simplified moisture processes.
Dynamical Core Model Intercomparison Project (DCMIP) and Summer Schools:
PDEs on the Sphere Workshops
- 2008 Colloquium on 'Numerical Techniques for Global Atmospheric Models', hosted by NCAR's Advanced Study Program (ASP), Boulder, CO, June 2008
- 2012 DCMIP Summer School on Future-Generation Non-Hydrostatic Weather and Climate Models, hosted by NCAR's Computational & Information Systems Laboratory (CISL), Boulder, CO, 7/30-8/10/2012
- Breckenridge, Colorado, USA, June 1996
- Gatlinburg, Tennessee, USA, April 1998
- San Francisco, California, USA, November 1999
- Montreal, Quebec, Canada, May 2001
- Toronto, Ontario, Canada, August 2002
- Yokohama, Japan, July 2004
- Monterey, California, USA, June 2006
- Exeter, United Kingdom, September 2007
- Santa Fe, New Mexico, April 2009
- Potsdam, Germany, August 2010
- Cambridge, U.K., Isaac Newton Institute for Mathematical Sciences, September 2012
3D advection with prescribed winds:
Figure: Selected results of the dynamical core intercomparison project conducted during the 2008 NCAR ASP summer school. The names of the dynamical cores are specified in the headers. The figure shows a cross section of an advected slotted ellipse after 12 days that was transported up and down and around the sphere via an analytically prescribed 3D wind field. The initial condition and reference solution can be used to assess the characteristics of the advection schemes in the participating models denoted by the acronyms. The horizontal grid spacings are approximately 110 km with 60 levels in the vertical direction (vertical grid spacing is 250 m).
Figure: Surface pressure field at day 9 of the baroclinic instability test case simulated with 9 different dynamical cores. The tests starts with balanced initial conditions that are overlaid by a Gaussian hill perturbation. The grid imprint of the cubed-sphere and icosahedral grids can be seen in the Southern Hemisphere (GEOS-FVCUBE, GME, ICON, OLAM). Spectral ringing appears in CAM-EUL and HOMME. The baroclinic wave test is documented in Jablonowski and Williamson (QJ, 2006) and in the Jablonowski and Williamson NCAR Technical Report TN-469+STR (2006).
Overview of the modelsThe dynamical cores of three different general circulation models have been tested using the proposal of Held-Suarez. The models involved in this investigation are global weather prediction models that are or have been used operationally at the German Weather Center (DWD, Offenbach, Germany) and the European Centre for Medium-Range Weather Forecasts (ECMWF, Reading, England). The table below provides an overview of these GCMs and their numerical designs.
GME (DWD) GM (DWD) IFS (ECMWF) Model type Grid point model Spectral model Spectral model Grid Spherical icosahedral grid Gaussian grid Reduced Gaussian grid Horizontal discretization finite differences, 2nd order spectral
Horizontal resolution ni=64 (approx. 110 km) T106 (approx. 125 km) T106 (approx. 125 km) Vertical resolution hybrid
Model top 10 hPa 10 hPa 10 hPa Prognostic variables
zonal wind u
meridional wind v
surface pressure ps
natural logarithm (ps)
natural logarithm (ps)
Advection scheme Eulerian Eulerian Semi-Lagrange Time stepping scheme semi-implicit
Time step 400 s 900 s 2700 s Diffusion linear, 4th order linear, 4th order linear, 4th order Numerical properties of the global weather prediction models GME Version 1.7 (DWD model), GM Version 1.15 (DWD model) and IFS cycle 18 (ECMWF model).
Comparison of the gridsAn important - and most obvious - difference between the three models is the different underlying grid structure. In contrast to the two spectral models GM (DWD) and IFS (ECMWF) that use a quasi-regular Gaussian or reduced Gaussian grid, the DWD model GME is based on an irregular, spherical icosahedral grid. This grid structure has been chosen in order to avoid the so-called 'pole problem' (convergence of the meridians near the poles) that is present in regular latitude-longitude grids.
The design of the spherical icosahedral grid is demonstrated below. An icosahedron is a geometric construct that consists of 20 identical triangles which touch the surrounding sphere at 12 points. This grid represents an icosahedral grid at the resolution ni=1 and can now be continuously refined. Each refinement step divides each side of the icosahedral triangles into two, so that the number of refinements 'ni' can be used to indicate the grid resolution. The following figures illustrate the structure of the icosahedral grid at the resolution ni=1, ni=2 and ni=4 (from left to right).
In contrast, a Gaussian grid represents a quasi regular latitude-longitude grid and its principle grid structure is shown in the figure below.
Convergence analysis: The DWD model GME
University of Michigan, Ann Arbor, MI, USA
Collaborators: Paul A. Ullrich, James Kent
National Center for Atmospheric Research (NCAR), Boulder, CO, USA
Collaborators: David L. Williamson, Peter H. Lauritzen, Ram D. Nair
Sandia National Laboratories (SNL), Albuquerque, NM, USA
Collaborator: Mark A. Taylor
German Weather Service DWD, Offenbach, Germany (in German , in English )
Collaborator: Dipl.-Met. Detlev Majewski
European Centre for Medium-Range Weather Forecasts ECMWF, Reading, England
Collaborators: Mariano Hortal (now at the Spanish Weather Service) & Clive Temperton (retired)