| The trends are witnessed to use the nonhydrostatic and fully elastic dynamicalcore for simulation and numerical weather forecast. The high resolution simulation ofatmospheric flows over complex orography is the main difficult numerical problem.Through the error analysis for pressure gradient force (PGF) and the development fornumerical model dynamical framework, the main conclusions are summarized asfollows:(1) The errors in several kinds of PGF computing methods have been diagnosedunder very steep slope with high-resolution used in meso-scale model by ideal fieldtests. Tests show that PGF errors can reach 10-2 m/s2 for classical method, 10-3 m/s2for average classical method and the highest precision 10-4m/s2 for Corby's schemeunder ideal meso-scale parameters. It is also indicated that computing results of PGFin theσterrain-following vertical coordinates will be deteriorated slightly and theerrors will not be converged under the conditions of steep slopes if the modelresolutions increase in vertical direction.(2) A revised scheme has been put forward in the conditions of high-resolutionmeso-scale model. It is based on the techniques of interpolating to isobaric level tocalculate PGF after finding precise geopotentials through integrating hydrostaticequation calculations with high vertical resolution. The ideal field tests show that thePGF errors decreased greatly under the steep topography slopes in the meso-scalemodel and the highest precision can reach 10-6 m/s2 under the typical meso-scaleparameters. The calculation errors come mainly from integrals of truncation errors ofhydrostatic equation for geopotentials in vertical direction when number of verticallayers is less than 100. When the number of vertical layers is more than 100,truncation errors of differencing in horizontal discretizations will also play animportant role. The most significant feature is that the errors of PGF will reducedramatically with vertical resolution increase and the revised scheme on PGF will beconverged with the increasing vertical resolutions under the conditions of the idealatmospheric field.(3) A dynamical core of numerical model of nonstagger (A-grid) and no referenceprofile is introduced with the nonhydrostatic and fully elastic atmosphere equations. Asemi lagrangian time discretization is used for time integration of the advection termswhile the horizontal explicit and vertical implicit are adopted for time integration ofthe pressure gradient force. The advantages of the scheme are: 1) the reference profileimpact on dynamical core is taken away otherwise than most of the nonhydrostaticdynamical cores using different reference profile. 2) the horizontal explicit andvertical implicit time stepping makes the scheme more simple than 3D implicit timediscretization because there are no need to solve 3D Helmholtz equations. 3)nonstagger A-grid mesh is much more simple than stagger mesh (such as C-grid)when the equations are carded out for spatial discretization. At the same time, thecalculation is also reduced for semi lagrangian interpolations under A-grid mesh. Theprimary tests with idealized field show that the dynamical core has the ability to simulate nonlinear flow motions, such as density flow and rising thermal, as well aspropagation of gravity waves in horizontal and topographic mountain waves.(4) GRAPES (Global/Regional Assimilation and PrEdiction System) modelis a new generation numerical weather model with non-hydrostatic multi-scalecharacterization. The GRAPES dynamical core is fully compressible withnon-hydrostatic/hydrostatic switch and large time step by semi-implicit andsemi-lagrangian technique. This paper reviews briefly the basic prognostic equations,the scheme for time step integration and differencing discretizations used byGRAPES. It also describes the key technique methods in GRAPES framework andproblems during the model development, including GCR solvers for pressure equation,lagrangian interpolation and vector discretizations, as well as designing philosophy ofthe dynamical core. In addition, the important issues during developing dynamicalcore, such as physical model used in framework, vertical coordinate, implicit orexplicit time stepping, are discussed. The three idealized tests, which are calledbalanced flow, density flow and the bell-pattern mountain separately, have beencarded out in order to understand and verify the GRAPES model's ability to simulateand forecast weather systems, epically for middle and small systems. The balancedflow test shows that the GRAPES dynamical core preserve pattern and stable duringtime integration under the condition of geostrophic balance. The density currentsuggests that the model framework has the capability to simulate nonlinear stream andits continual evolution in case of fine scale. And the last case, bell-pattern mountaintrial, verifies that the model has the ability to simulate propogation in horizontal andvertical direction when the flow runs across an isolate mountain. The idealized testsabove imply primarily that the GRAPES multi-scale dynamical core has the potentialto simulate synoptic-scale and meso-scale systems. To verify whether the GRAPESdynamical core can take as a framework of AGCM, the long time integration has beencarried out with GRAPES dynamical core following benchmark test similar to thetechnique introduced by Held and Suarez. And the results indicate that it is feasibleusing the GRAPES frame as a dynamical core for AGCM and climate investigation.(5) The revised PGF scheme has been applied to the nonhydrostatic and fullyelastic dynamical core to test the impact under the steep slope. The results show thenew PGF scheme is better than before with 3% errors of the old scheme and theperformances are stable. The idea applying the revised PGF scheme to GRAPESmodel has been brought forward and a balanced flow test has been carded out withreasonable results.
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