Short deterministic and long climate simulations

Christiane Jablonowski (cjablono@umich.edu)

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-calleddynamical core, and thephysicspackage 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 thephysicsis 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 Tests

Short 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 theJablonowski-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 amoist tropical cyclone test case in aqua-planet modethat 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., Vol. 4, M04001, doi:10.1029/2011MS000099

- Ullrich, P. A. and C. Jablonowski (2012): MCore: A Non-hydrostatic Atmospheric Dynamical Core Utilizing High-Order Finite-Volume Methods, J. Comput. Phys., Vol. 231, 5078-5108

- Kent, J., P. A. Ullrich and C. Jablonowski (2014): Dynamical Core Model Intercomparison Project: Tracer Transport Test Cases, Quart. J. Roy. Meteorol. Soc., Vol. 140, 1279-1293

- Ullrich, P. A., T. Melvin, C. Jablonowski and A. Staniforth (2014): A proposed baroclinic wave test case for deep- and shallow-atmosphere dynamical cores, Quart. J. Roy. Meteorol. Soc., in press

Additional information is available on the following web pages that provide

- high-resolution reference solutions and the uncertainty estimates for the baroclinic wave test case (Jablonowski and Williamson, QJ 2006).
- an overview of our dynamical core research at the University of Michigan.
.## Held-Suarez Test

This 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. Thissimplified physics(or more preciselyidealized 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 Test

The 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.

Workshops and the Dynamical Core Model Intercomparison Project (DCMIP)

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:

- 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
PDEs on the Sphere Workshops

- 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
- Boulder, Colorado, NCAR, April 2014

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).

Baroclinic wave:

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 models

The 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 typeGrid point model Spectral model Spectral model GridSpherical icosahedral grid Gaussian grid Reduced Gaussian grid Horizontal discretizationfinite differences, 2nd order spectral

triangular truncationspectral

triangular truncationHorizontal resolutionni=64 (approx. 110 km) T106 (approx. 125 km) T106 (approx. 125 km) Vertical resolutionhybrid

19 levelshybrid

19 levelshybrid

31 levelsModel top10 hPa 10 hPa 10 hPa Prognostic variables(dry model)zonal wind u

meridional wind v

temperature T

surface pressure p_{s}relative vorticity

horizontal divergence

temperature T

natural logarithm (p_{s})relative vorticity

horizontal divergence

temperature T

natural logarithm (p_{s})Advection schemeEulerian Eulerian Semi-Lagrange Time stepping schemesemi-implicit

3-time-levelsemi-implicit

3-time-levelsemi-implicit

2-time-levelTime step400 s 900 s 2700 s Diffusionlinear, 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 grids

An 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 theicosahedral gridat the resolutionni=1, ni=2 and ni=4(from left to right).In contrast, a

Gaussian gridrepresents a quasi regular latitude-longitude grid and its principle grid structure is shown in the figure below.

Jablonowski, C. (1998): Test der Dynamik zweier globaler Wettervorhersagemodelle des Deutschen Wetterdienstes: Der Held-Suarez Test, Diploma Thesis, Metorological Institute of the University of Bonn, Germany, September 1998, 151 pp.

The document is written in German and the translated title reads 'Test of the Dynamics of two global Weather Prediction Models of the German Weather Service: The Held-Suarez Test'. While the text might be hard to read for a non-German speaker, the figures are self-explanatory and can easily be compared to other sources in the literature.

## Convergence analysis: The DWD model GME

- In January 1998 the dynamical core of the ECMWF model IFS at T63 resolution
has been tested with an increased vertical resolution. Instead of the operational
31 vertical levels 50 model levels have been used to improve the vertical
resolution especially in the upper atmosphere (stratosphere and lower mesosphere
which lie between 11 km - approximately 50 km). The model top has been
set to a pressure level of 0.1 hPa. This approach has led to interesting
and new model results that are still under investigation.

The vertical high resolution dynamical core test produced a long-term oscillation in the tropical upper atmosphere that resembles the well known meteorological phenomenon of a QBO (quasi-biennial oscillation). The QBO phenomenon is characterized by a change of the major zonal wind regime in the equatorial region of the stratosphere. In nature, it is observed that easterly and westerly wind components change in a specific way approximately every 22-30 months. These wave structures have been well-met by the dynamical core test, but the model simulations are dominated by a longer time period.

This QBO-like oscillation of the IFS dynamical core has been an impressive and unexpected result. The underlying figure shows a cross section of the mean zonal wind along the equator and demonstrates the change in the zonal wind characteristics over a time period of 4800 days (approx. 13 years). Units are m/s.

- The dynamical core tests at DWD have been initialized with artificial
initial conditions and generate during the first 200-250 days of the long-term
simulations a fully developed global circulation. The following snapshots
provide insight into the beginning circulation on day 70. The underlying
figures show the GME and GM
surface pressure fields and the GME
and GM temperature fields near the
surface. In addition, the GME vertical
velocity field demonstrates the different updraft and downdraft regions
at 500hPa.

As far as the model GME is concerned it becomes obvious that the whole circulation is dominated by a wave number 5 structure which is connected to the icosahedral grid. This wave number 5 signal vanishes after model day 130 when baroclinic instability processes break the symmetry of the data.

Compare these images to those produced by I. Held, one of the authors of the dynamical core test method. He presents a snapshot of the temperature field (lower panel) and vertical motion (upper panel) in the lower troposphere (from the idealized atmospheric model (at 'T106' resolution) described in "A Proposal for the Intercomparison of the Dynamical Cores of Atmospheric General Circulation Models").

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)