Adaptive Mesh Refinement (AMR) techniques provide an attractive framework for atmospheric flows since they allow an improved resolution in limited regions without requiring a fine grid resolution throughout the entire model domain. The model regions at high resolution are kept at a minimum and can be individually tailored towards the research problem associated withatmospheric model simulations.
A solution-adaptive grid is a virtual necessity for resolving a problem with different length scales. In order to avoid under-resolving high-gradient regions in the problem, or conversely, over-resolving low-gradient regions at the expense of more critical regions, solution adaptation is a powerful tool saving several orders of magnitude in computing resources for many problems. Climate and weather models, or generally speaking computational fluid dynamics (CFD) codes, are among the many applications that are characterized by  multiscale phenomena and their resulting interactions.
For instance, large-scale weather systems such as midlatitude cyclones drive small-scale frontal zones, thunderstorms or rain events. These small-scale features may then influence the larger scale if, as an example, evaporation processes and turbulence at the surface trigger sensible and latent heat fluxes. But although today's atmospheric general circulation models (GCMs), and in particular weather prediction codes, are already capable of uniformly resolving horizontal scales of order $20 km$ (e.g. the model IFS of the European Centre for Medium-Range Weather Forecasts), the atmospheric motions of interest span many more scales than those captured in a fixed resolution model run. The widely varying spatial and temporal scales, in addition to the nonlinearity of the dynamical system, raise an interesting and challenging modeling problem. Solving such a problem more efficiently and accurately requires variable resolution.

The development of the adaptive dynamical core for weather and climate research is based on the so-called Lin-Rood finite-volume dynamical core that has been designed at the NASA Goddard Space Flight Center (NASA/GSFC) in the late 1990's. This hydrostatic global dynamics package in flux form is built upon the Lin and Rood 1996 advection algorithm, which utilizes advanced oscillation-free numerical approaches to solving the transport equation. In particular,  Lin and Rood 1996 extended a Godunov-type methodology to multiple dimensions and made use of 2nd order van Leer-type (van Leer 1974, van Leer 1979) and 3rd order piecewise parabolic (PPM) methods (Colella and Woodward 1984, Carpenter et al. 1990). In 1997, the advection scheme became the fundamental building block of a shallow water code (Lin and Rood 1997, Lin 1997) which then led to the development of the current 3D, primitive-equation (PE) based, finite-volume dynamics package (Lin 2004).

Today, the 3D dynamical core is used operationally for data assimilation applications at NASA/GSFC and, in 2001, it was included in NCAR's climate-prediction system CCSM, the Community Climate System Model. This climate prediction system contains the Community Atmosphere Model, CAM, with physics and dynamics components. In particular, the finite-volume dynamical core is now one of the three standard dynamics modules in CAM.

Additional details of the FV dynamics package can be found in the tutorial 'The Lin-Rood Finite Volume (FV) Dynamical Core'.

The adaptive dynamical core has been run in two configurations: the full 3D hydrostatic dynamical core on the sphere and the corresponding 2D shallow water model that has been extracted out of the 3D version (Jablonowski 2004,Jablonowski et al. 2006). In general, the shallow water system can be considered a 1-level version of the 3D dynamical core. This shallow water setup serves as an ideal testbed for the horizontal discretization and the 2D adaptive-mesh strategy. It further allows the efficient and quick testing of interpolation routines at fine-coarse grid interfaces.

Both static and dynamic adaptation strategies have been tested. Static adaptations can be used to vary the resolution in pre-defined regions of interest. This includes static refinements near mountain ranges or static coarsenings in the longitudinal direction for the implementation of a so-called reduced grid in polar regions. Dynamic adaptations are based on flow characteristics and guided by refinement criteria that detect user-defined features of interest during a simulation. In particular, flow-based refinement criteria, such as vorticity or gradient indicators, have been tested. Refinements and coarsenings then occur according to pre-defined threshold values.

An example of an adaptive passive advection test on the sphere is shown below. The figure shows the initial conditions for the shallow water standard test case 1 with a 90^o rotation angle (see Williamson et a. 1992 for the test specifications). Here the adapted blocks track a cosine bell as it is transported once around the sphere. Note that each self-similar blocks contains 9x6 grid points in lon x lat direction so that the finest grid resolution in this example corresponds to a 0.625^o x 0.625^o grid. The maximum number of refinement levels is set to 3.



The adaptation criterion is based on a simple threshold assessment. A block is refined as soon as the height of the cosine bell exceeds a user-determined threshold value at (at least) one grid point within the block. On the other hand, a block gets coarsened if the height of the cosine bell in the block no longer meets the criterion. The corresponding mpeg movie (4.1 MB) shows that the cosine bell is successfully captured as indicated by the overlaid block distribution. There are no visible distortions of the height field as the cosine bell approaches, passes over and leaves the poles. The increased resolution clearly helps preserve the shape and peak amplitude.

A second example of a dynamically adapted flow field is presented in the next figure. It shows an idealized flow over a single mountain at model day 10 (test case 5, Williamson et a. 1992). Here the adaptations are guided by the absolute value of the geopotential height gradient. The refined regions pick out the strong gradient regimes that are associated with the evolving wave train behind the mountain.

The 3D adaptive finite-volume dynamical core is assessed with a statically adaptive setup that utilizes prior knowledge of the developing flow field. The model is evaluated with the Jablonowski-Williamson baroclinic wave test case for dynamical cores of GCMs (Jablonowski and Williamson 2006). The flow field is characterized by a baroclinic wave train in the Northern Hemisphere that is triggered by a small perturbation superimposed upon the smooth initial zonal wind field.

The adapted blocks are used to refine the pre-determined storm track in the midlatitudes (Northern Hemisphere) with increasing horizontal resolutions. The refinements are confined to the horizontal directions so that the whole vertical column is refined in the event of refinement requests. Refinements are selected at the beginning of the forecast, which is followed by an analytical reinitialization step.

Here the analysis of the baroclinic wave train is focused on the surface pressure field. The two figures below show the surface pressure patterns for an adaptive model setup with (a) 1 and (b) 3 refinement levels at model day 8. The adapted blocks are overlaid that illustrate the chosen band structure of the refinement approach. It can clearly be seen that the representation of the baroclinic wave improves significantly with an increasing number of refinement levels. The baroclinic wave intensifies as expected in the fine grid adaptive run and is rather damped on the coarser grid.

This research project was part of my Ph.D. thesis. For further information, please download the thesis below and feel free to contact me. The references to the corresponding papers will follow shortly.

Carpenter, R. L., K. K. Droegemeier, P. R. Woodward and C. E> Hane, Application of the Piecewise Parabolic Method to Meteorological Modeling, Mon. Wea. Rev., 118, 586-612, 1990.

Colella, P. and P. R. Woodward, The Piecewise Parabolic Method (PPM) for Gas-Dynamical Simulations, J. Comput. Phys., 54, 174-201, 1984.

Jablonowski, C., Adaptive Grids in Weather and Climate Modeling, Ph.D. dissertation, University of Michigan, Ann Arbor, MI, 2004 (download the pdf version, 8MB),

Jablonowski, C., M. Herzog, J. E. Penner, R. C. Oehmke, Q. F. Stout, B. van Leer and K. G. Powell, Block-Structured Adaptive Grids on the Sphere: Advection Experiments, Mon. Wea. Rev. in press (Dec. 2006) (download the pdf version, 1MB),

Jablonowski, C. and D. L. Williamson, A Baroclinic Instability Test Case for Atmospheric Model Dynamical Cores, Quarterly J. Roy. Met. Soc., accepted, 2006

Lin, S.-J., A Finite-Volume Integration Method for Computing the Pressure Forces in General Vertical Coordinates, Quart. J. Roy. Meteor. Soc., 123, 1749-1762, 1997.

Lin, S.-J., A "Vertically Lagrangian" Finite-Volume Dynamical Core for Global Models, Mon. Wea. Rev., 132, 2293-2307, 2004.

Lin, S.-J. and R. B. Rood, Multidimensional Flux-Form Semi-Lagrangian Scheme, Mon. Wea. Rev., 124, 2046-2070, 1996.

Lin, S.-J. and R. B. Rood, An Explicit Flux-Form Semi-Lagrangian Shallow Water Model on the Sphere, Quart. J. Roy. Meteor. Soc., 123, 2477-22498, 1997.

Oehmke, R. C. and Q. F. Stout, Parallel Adaptive Blocks on a Sphere, in Proc. 11th SIAM Conference on Parallel Processing for Scientific Computing, 2001, CD-ROM. Also available online at http://www.eecs.umich.edu/~qstout/pap/SIAMPP01.ps

Oehmke, R. C., High Performance Dynamic Array Structures, Ph.D. Dissertation, University of Michigan, Ann Arbor, 2004, Department of Electrical Engineering and Computer Science, 93 pp.

van Leer, B., Towards the Ultimate Conservative Difference Scheme. II. Monotonicity and Conservation Combined in a Second-Order Scheme, J. Comput. Phys., 14, 361-370, 1974.

van Leer, B., Towards the Ultimate Conservative Difference Scheme. IV. A New Approach to Numerical Convection, J. Comput. Phys., 23, 276-299, 1977.

Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob and P. N. Swarztrauber, A Standard Test Set for Numerical Approximations to the Shallow Water Equations in Spherical Geometry, J. Comput. Phys., 102, 211-224, 1992

The Lin-Rood Finite Volume (FV) Dynamical Core: Tutorial

Dr. Michael Herzog, Geophysical Fluid Dynamics Laboratory (GFDL), Princeton

Dr. Robert C. Oehmke, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, (since 9/2006 NCAR, Boulder, CO)

Prof. Joyce E. Penner, Department of Atmospheric, Oceanic & Space Sciences, University of Michigan, Ann Arbor

Prof. Kenneth Powell, Department of Aerospace Engineering, University of Michigan, Ann Arbor

Prof. Quentin F. Stout, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor

Prof. Bram van Leer, Department of Aerospace Engineering, University of Michigan, Ann Arbor