Development of an atmospheric climate model with self-adapting grid and physics

Transcription

1 Intitute of Phyic Publihing Journal of Phyic: Conference Serie 16 (2005) doi: / /16/1/049 SciDAC 2005 Development of an atmopheric climate model with elf-adapting grid and phyic Joyce E. Penner 1, Michael Herzog 2, Chritiane Jablonowki 3, Bram van Leer 1, Robert C. Oehmke 1, Quentin F. Stout 1, and Kenneth G. Powell 1 1 Univerity of Michigan 2 Geophyical Fluid Dynamic Laboratory 3 National Center for Atmopheric Reearch penner@umich.edu Abtract. An adaptive grid dynamical core for a global atmopheric climate model ha been developed. Adaptation allow a mooth tranition from hydrotatic to non-hydrotatic phyic at mall reolution. The adaptation ue a parallel program library for block-wie adaptive grid on the phere. Thi library alo upport the ue of a reduced grid with coarer reolution in the longitudinal direction a the pole are approached. Thi permit the ue of a longer time tep ince the CFL number retriction (CFL < 1) in a regular longitude-latitude grid i mot evere in the zonal direction at high latitude. Several tet how that our modelling procedure are table and accurate. 1. Introduction The goal of thi reearch project i to develop adaptive grid technique for future climate model and weather prediction. Thi approach will lead to new inight into mall-cale and large-cale flow interaction that are unreolved by current uniform-grid imulation. Adaptive meh refinement (AMR) technique provide an attractive framework for atmopheric motion ince they allow improved horizontal reolution in a limited region without requiring a fine grid reolution throughout the entire model domain. Therefore, the model domain to be reolved with higher reolution i kept at a minimum, greatly reducing computer memory and peed requirement. Adaptive grid technique have been developed for a parallel verion of the NASA / NCAR Finite- Volume Community Climate Model. Thi global hydrotatic model i baed on NCAR phyic and the o-called Lin-Rood finite-volume dynamical core 1 that provide highly efficient algorithm for high performance computing. Thi reearch project i characterized by an interdiciplinary approach involving atmopheric cience, computer cience and mathematical/numerical apect. The work i done in cloe collaboration between the Atmopheric Science, Computer Science and Aeropace Engineering Department at the Univerity of Michigan, NCAR, and NASA. 2. The Adaptive Grid Library The newly developed verion of the NASA finite-volume dynamical core with elf-adaptive grid ue a general purpoe library of module to implement the block-wie adaptive grid on the phere. A 2005 IOP Publihing Ltd 353

2 354 indicated in Figure 1, the grid i ubdivided horizontally into elf-imilar block that contain an identical number of grid point per block. In the event of a refinement a block i plit into four new block, thereby doubling the patial reolution; coarening revere thi proce. Here the patial reolution of adjacent block i only allowed to differ by a factor of two. On the phere, a regular longitude-latitude grid ha been adopted. In addition, an initial reduced grid etup can be elected (Figure 2). In cae the reduced grid i elected, the longitudinal reolution in polar region i coarened which alleviate the convergence of the meridian at the pole. Refine Coaren Figure 1: Schematic view of the refinement and coarening principle with 2 refinement level and 3 3 grid cell per block. (a) (b) (c) Figure 2: Ditribution of grid point and block over the phere in an orthographic projection centered at (45 N, 0 ). The reolution i (a) non-adapted cae, (b) reduced grid cae, (c) reduced grid howing adaptation. The parallel adaptive grid library provide the functionality neceary to implement the grid data tructure by providing the following ervice: 1) the creation of the initial pherical grid including etting up neighbor connection, and ditributing block acro proceor, 2) the update of ghot cell tranparent to proceor location, 3) the adaptation of block including providing communication for uer-data update, maintaining neighbor connection and reditributing block. The library wa deigned for flexibility and efficiency, allowing the uer to arrange their data however they wih. Uer-defined ubroutine need to be provided that pecify the algorithm for plit and join, and ghot

3 355 cell operation. Thee routine include the interpolation and averaging procedure for the initialization of new block and the data exchange algorithm for neighboring block at both identical and varying reolution. 3. The Hydrotatic Dynamical Core Statically and dynamically adaptive grid have been uccefully implemented and teted in 2D hallow water imulation and 3D hydrotatic dynamical core run on the phere. Figure 3 how an example of a 2D hallow water imulation at model day 10. The depicted geopotential height field i characterized by a lee-ide wave that i induced by an idealized mountain. Here a combination of tatically and dynamically refined block i preented. The dynamic adaptation track the evolution of the wave by a gradient-baed adaptation criterion. Other adaptation criteria that are, for example, baed on vorticity have alo been uccefully applied. Tet with the adaptive 3D hydrotatic dynamical core ugget that adaptation are a viable option for future modeling tudie. 3D idealized experiment with locally refined reolution along torm track have been performed. The reult how that the developing torm ytem are predicted accurately without the need for a fine reolution in the entire model domain Latitude Longitude Geopotential height [m] Figure 3: Geopotential height field at day The Non-Hydrotatic Dynamical Core One of the mot important advance needed in global climate model i the development of model that can reliably treat convection. At the preent time, convection i a ub-grid proce that mut be parameterized. The explicit treatment of convection require patial reolution at which the hydrotatic aumption i no longer valid. Therefore we have developed a non-hydrotatic code which moothly replace the hydrotatic treatment when required. In the non-hydrotatic formulation, a ma-baed Lagrangian vertical coordinate replace the preure-baed Lagrangian vertical coordinate of the hydrotatic code. In thi ytem, the ma continuity equation i: π ( π ) + ( vπ ) + = 0 t (1)

4 356 d where = i the vertical velocity and π = ρg z i the vertically integrated ma per unit area. dt Our Lagrangian vertical coordinate i defined by = 0, o that the lat term in (1) i zero. Prognotic equation for the denity anomaly and the vertical velocity anomaly are olved. Thu, the anomaly (from the hydrotatic equilibrium olution) of the denity, ρ, and the anomaly of the vertical velocity, w, are predicted from: ρ + ( ρ v )= 2 ρ ρ t π t ( π w ) g Φ v + ρ 2 π g w ( Φ ) v (2) + ( π w v ) g P = 0 (3) where ρ, w, and Φ are the hydrotatic olution for the denity, vertical velocity and geopotential, and v i the horizontal wind vector. The full denity ρ = ρ + ρ i ued to predict the full geopotential, Φ, from which the non-hydrotatic geopotential anomaly i derived, and the hydrotatic vertical velocity w i derived from the temporal evolution of Φ. P i calculated from the hydrotatic and non-hydrotatic denitie, ρ and ρ Upper boundary condition Along the characteritic d( P ρc w) equal zero. Here, P i the non-hydrotatic preure anomaly, ρ the denity, c the peed of ound and w the vertical velocity. Auming equilibrium and contant denity, the above characteritic reduce in one dimenion to a condition for the vertical gradient: z = 1 P ρc z (4) The ue of Equ. (4) for the upper boundary condition account for the coupling between the anomalie for preure and vertical velocity at the model top. However, the atmophere i not in equilibrium and the vertical denity gradient i far from being zero. Starting from an initial nonhydrotatic denity (or preure) perturbation, we imulated the hydrotatic adjutment proce applying equation (4) at the model top. The model failed to reach hydrotatic equilibrium, and trong reflection occurred at the model top. The model wa numerically untable. To take into account the tranitional character of the atmophere, we formulated an upper boundary condition baed on time derivative: t = 1 P ρc t (5) With thi condition the model i much better behaved. Since equation (5) aume contant denity and doen't do a full backward integration along the characteritic, mall reflection of ound wave till occur at the model top (ee Figure 4a). To enure that hydrotatic equilibrium i alway reached, we added a damping term for P in equation (5): t = 1 P P ρc t τ (5a)

5 357 where τ i a time contant on the order of everal minute. Thi procedure ha greatly reduced reflection at the model top, but ha not entirely eliminated them (Figure 4b). Defining the vertical velocity at the cell interface in the vertical direction, rather than at the cell center, facilitate the implementation of a emi-implicit time-tepping algorithm, which remain table at longer time tep and damp the fat-traveling ound wave (Figure 4c). (a) (b) (c) Figure 4: Propagation of a denity anomaly in the non-hydrotatic code. The anomaly i initiated near 5 km and propagate vertically where it i reflected at the upper boundary. (a) how an explicit time tep procedure with no damping. (b) how the explicit model with damping added. (c) how an implicit verion with no damping. 5. Concluion Thi project i aimed at developing a climate model that elf-adjut the grid reolution and the complexity of the phyic model to the actual atmopheric flow condition. To accomplih thi, we have implemented a fully 3-dimenional non-hydrotatic model within a hydrotatic code while uing a block-tructured grid that allow for the implementation of maller grid reolution within both the hydrotatic and non-hydrotatic portion of the grid. Several tet of the model how that it i table and can moothly tranition from hydrotatic to non-hydrotatic flow. One of the major building block of thi project i a parallel adaptive grid library which i currently under development at the Univerity of Michigan under the direction of co-i Profeor Stout. Thi MPI-baed communication library manage the block-tructured data layout, handle the ghot cell update among neighboring block and plit a block a refinement occur. The current functionality provide routine for a reduced grid deign. In a reduced grid, the reolution i coarened in the longitudinal direction a the pole i approached. Thi often allow the ue of a longer time tep ince the CFL number retriction (CFL < 1) in a regular longitude-latitude grid i mot evere in the zonal direction at high latitude. We have recently ubmitted a paper decribing the ue of thi library in the hydrotatic portion of our code for advection. 2 Reference [1] Lin, S.-J. and R. B. Rood, 1996: Multidimenional flux-form emi-lagrangian tranport cheme, Mon. Wea. Rev., 124, [2] Jablonowki, C., M. Herzog, J.E. Penner, R.C. Oehmke, Q.F. Stout, B. van Leer, and K.G. Powell, 2005: Block-Structured Adaptive Grid on the Sphere: Advection Experiment, Mon. Wea. Rev., ubmitted.