In this study, kinetic energy budget equations of rotational and divergent flow in pressure coordinates are derived on terrain-following coordinates. The new formulation explicitly shows the terrain effects and can be...In this study, kinetic energy budget equations of rotational and divergent flow in pressure coordinates are derived on terrain-following coordinates. The new formulation explicitly shows the terrain effects and can be applied directly to model-simulated dynamic and thermodynamic fields on the model's original vertical grid. Such application eliminates interpolation error and avoids errors in virtual weather systems in mountainous areas. These advantages and their significance are demonstrated by a numerical study in terrain-following coordinates of a developing vortex after it moves over the Tibetan Plateau in China.展开更多
With increasing resolution in numerical weather prediction （NWP） models, the model topography can be described with finer resolution and includes steeper slopes. Consequently, negative effects of the traditional ter...With increasing resolution in numerical weather prediction （NWP） models, the model topography can be described with finer resolution and includes steeper slopes. Consequently, negative effects of the traditional terrain-following vertical coordinate on high-resolution numerical simulations become more distinct due to larger errors in the pressure gradient force （PGF） calculation and associated distortions of the gravity wave along the coordinate surface. A series of numerical experiments have been conducted in this study, including idealized test cases of gravity wave simulation over a complex mountain, error analysis of the PGF estimation over a real topography, and a suite of real-data test cases. The GRAPES-Meso model is utilized with four different coordinates, i.e., the traditional terrain-following vertical coordinate proposed by Gal-Chen and Somerville （hereinafter referred to as the Gal.C.S coordinate）, the one-scale smoothed level （SLEVE1）, the two-scale smoothed level （SLEVE2）, and the COSINE （COS） coordinates. The results of the gravity wave simulation indicate that the GRAPES-Meso model generally can reproduce the mountain-induced gravity waves, which are consistent with the analytic solution. However, the shapes, vertical structures, and intensities Of the waves are better simulated with the SLEVE2 coordinate than with the other three coordinates. The model with the COS coordinate also performs well, except at lower levels where it is not as effective as the SLEVE2 coordinate in suppressing the PGF errors. In contrast, the gravity waves simulated in both the Gal.C.S and SLEVE1 coordinates are relatively distorted. The estimated PGF errors in a rest atmosphere over the real complex topography are much smaller （even disappear at the middle and upper levels） in the GRAPES-Meso model using the SLEVE2 and COS coordinates than those using the Gal.C.S and SLEVE1 coordinates. The results of the real-data test cases conducted over a one-month period suggest that the three modified vertical coordinates （SLEVE1, SLEVE2, and COS coordinates） give better results than the traditional Gal.C.S coordinate in terms of forecasting bias and root mean square error, and forecasting anomaly correlation coefficients. In conclusion, the SLEVE2 coordinate is proved to be the best option for the GRAPES-Meso model.展开更多
The basic terrain-following(BTF)coordinate simplifies the lower boundary conditions of a numerical model but leads to numerical error and instability on steep terrain.Hybrid terrain-following(HTF)coordinates with smoo...The basic terrain-following(BTF)coordinate simplifies the lower boundary conditions of a numerical model but leads to numerical error and instability on steep terrain.Hybrid terrain-following(HTF)coordinates with smooth slopes of vertical layers(slopeVL)generally overcome this difficulty.Therefore,the HTF coordinate becomes very desirable for atmospheric and oceanic numerical models.However,improper vertical layering in HTF coordinates may also increase the incidence of error.Except for the slopeVL of an HTF coordinate,this study further optimizes the HTF coordinate focusing on the thickness of vertical layers(thickVL).Four HTF coordinates(HTF1–HTF4)with similar slopeVL but different vertical transition methods of thickVL are designed,and the relationship between thickVL and numerical errors in each coordinate is compared in the classic idealized thermal convection[two-dimensional(2D)rising bubble]experiment over steep terrain.The errors of potential temperatureθand vertical velocity w are reduced most,by approximately 70%and 40%,respectively,in the HTF1 coordinate,with a monotonic increase in thickVL according to the increasing height;however,the errors ofθincreased in all the other HTF coordinates,with nonmonotonic thickVLs.Furthermore,analyses of the errors of vertical pressure gradient force(VPGF)show that due to the interpolation errors of thickVL,the inflection points in the vertical transition of thickVL induce the initial VPGF errors;therefore,the HTF1 coordinate with a monotonic increase in thickVL has the smallest errors among all the coordinates.More importantly,the temporal evolution of VPGF errors manifests top-type VPGF errors that propagate upward gradually during the time integration.Only the HTF1 and HTF4 coordinates with a monotonic increase in thickVL near the top of the terrain can suppress this propagation.This optimized HTF coordinate(i.e.,HTF1)can be a reference for designing a vertical thickVL in a numerical model.展开更多
A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrain...A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrainfollowing grid. The MCV algorithm defines two types of moments: the point-wise value (PV) and the volume-integrated average (VIA). The unknowns (PV values) are defined at the solution points within each cell and are updated through the time evolution formulations derived from the governing equations. Rigorous numerical conservation is ensured by a constraint on the VIA moment through the flux form formulation. The 3D atmospheric dynamic core reported in this paper is based on a three-point MCV method and has some advantages in comparison with other existing methods, such as uniform third-order accuracy, a compact stencil, and algorithmic simplicity. To check the performance of the 3D nonhydrostatic dynamic core, various benchmark test cases are performed. All the numerical results show that the present dynamic core is very competitive when compared to other existing advanced models, and thus lays the foundation for further developing global atmospheric models in the near future.展开更多
This study shows a new way to implement terrain-following s-coordinate in a numerical model,which does not lead to the well-known"pressure gradient force(PGF)"problem.First,the causes of the PGF problemare a...This study shows a new way to implement terrain-following s-coordinate in a numerical model,which does not lead to the well-known"pressure gradient force(PGF)"problem.First,the causes of the PGF problemare analyzedwith existing methods that are categorized into two different types based on the causes.Then,the new method that bypasses the PGF problem all together is proposed.By comparing these threemethods and analyzing the expression of the scalar gradient in a curvilinear coordinate system,this study finds out that only when using the covariant scalar equations of s-coordinate will the PGF computational form have one term in each momentum component equation,thereby avoiding the PGF problem completely.A convenient way of implementing the covariant scalar equations of s-coordinate in a numerical atmospheric model is illustrated,which is to set corresponding parameters in the scalar equations of the Cartesian coordinate.Finally,two idealized experimentsmanifest that the PGF calculated with the new method is more accurate than using the classic one.This method can be used for oceanic models as well,and needs to be tested in both the atmospheric and oceanic models.展开更多
A three-dimensional hydrodynamic model is presented which combines a terrain-following vertical coordinate with a horizontally orthogonal curvilinear coordinate system to fit the complex bottom topography and coastlin...A three-dimensional hydrodynamic model is presented which combines a terrain-following vertical coordinate with a horizontally orthogonal curvilinear coordinate system to fit the complex bottom topography and coastlines near estuaries, continental shelves, and harbors. To solve the governing equations more efficiently, we improve the alternating direction implicit method, which is extensively used in the numerical modeling of horizontal two-dimensional shallow water equations, and extend it to a three-dimensional tidal model with relatively little computational effort. Through several test cases and realistic applications, as presented in the paper, it can be demonstrated that the model is capable of simulating the periodic to-and-fro currents, wind-driven flow, Ekman spirals, and tidal currents in the near-shore region.展开更多
基金supported by the Key Program of the Chinese Academy of Sciences(No.KZZD-EW-05-01)the Supporting Program for Science and Technological Research of China(No.2008BAC37B01)+1 种基金the National Basic Research Program of China(Nos.2012CB417201 and 2009CB421505)the National Natural Sciences Foundation of China(Nos.41205033 and 41175056)
文摘In this study, kinetic energy budget equations of rotational and divergent flow in pressure coordinates are derived on terrain-following coordinates. The new formulation explicitly shows the terrain effects and can be applied directly to model-simulated dynamic and thermodynamic fields on the model's original vertical grid. Such application eliminates interpolation error and avoids errors in virtual weather systems in mountainous areas. These advantages and their significance are demonstrated by a numerical study in terrain-following coordinates of a developing vortex after it moves over the Tibetan Plateau in China.
基金Supported by the National(Key)Basic Research and Development(973)Program of China(2013CB430106)National Natural Science Foundation of China(41375108)National Science and Technology Support Program of China(2012BAC22B01)
文摘With increasing resolution in numerical weather prediction （NWP） models, the model topography can be described with finer resolution and includes steeper slopes. Consequently, negative effects of the traditional terrain-following vertical coordinate on high-resolution numerical simulations become more distinct due to larger errors in the pressure gradient force （PGF） calculation and associated distortions of the gravity wave along the coordinate surface. A series of numerical experiments have been conducted in this study, including idealized test cases of gravity wave simulation over a complex mountain, error analysis of the PGF estimation over a real topography, and a suite of real-data test cases. The GRAPES-Meso model is utilized with four different coordinates, i.e., the traditional terrain-following vertical coordinate proposed by Gal-Chen and Somerville （hereinafter referred to as the Gal.C.S coordinate）, the one-scale smoothed level （SLEVE1）, the two-scale smoothed level （SLEVE2）, and the COSINE （COS） coordinates. The results of the gravity wave simulation indicate that the GRAPES-Meso model generally can reproduce the mountain-induced gravity waves, which are consistent with the analytic solution. However, the shapes, vertical structures, and intensities Of the waves are better simulated with the SLEVE2 coordinate than with the other three coordinates. The model with the COS coordinate also performs well, except at lower levels where it is not as effective as the SLEVE2 coordinate in suppressing the PGF errors. In contrast, the gravity waves simulated in both the Gal.C.S and SLEVE1 coordinates are relatively distorted. The estimated PGF errors in a rest atmosphere over the real complex topography are much smaller （even disappear at the middle and upper levels） in the GRAPES-Meso model using the SLEVE2 and COS coordinates than those using the Gal.C.S and SLEVE1 coordinates. The results of the real-data test cases conducted over a one-month period suggest that the three modified vertical coordinates （SLEVE1, SLEVE2, and COS coordinates） give better results than the traditional Gal.C.S coordinate in terms of forecasting bias and root mean square error, and forecasting anomaly correlation coefficients. In conclusion, the SLEVE2 coordinate is proved to be the best option for the GRAPES-Meso model.
基金Supported by the National Natural Science Foundation of China(42230606)14th Five-Year Plan Basic Research Program of Institute of Atmospheric Physics,Chinese Academy of Sciences(E268081801)National Key Research and Development Program of China(2017YFA0603901)。
文摘The basic terrain-following(BTF)coordinate simplifies the lower boundary conditions of a numerical model but leads to numerical error and instability on steep terrain.Hybrid terrain-following(HTF)coordinates with smooth slopes of vertical layers(slopeVL)generally overcome this difficulty.Therefore,the HTF coordinate becomes very desirable for atmospheric and oceanic numerical models.However,improper vertical layering in HTF coordinates may also increase the incidence of error.Except for the slopeVL of an HTF coordinate,this study further optimizes the HTF coordinate focusing on the thickness of vertical layers(thickVL).Four HTF coordinates(HTF1–HTF4)with similar slopeVL but different vertical transition methods of thickVL are designed,and the relationship between thickVL and numerical errors in each coordinate is compared in the classic idealized thermal convection[two-dimensional(2D)rising bubble]experiment over steep terrain.The errors of potential temperatureθand vertical velocity w are reduced most,by approximately 70%and 40%,respectively,in the HTF1 coordinate,with a monotonic increase in thickVL according to the increasing height;however,the errors ofθincreased in all the other HTF coordinates,with nonmonotonic thickVLs.Furthermore,analyses of the errors of vertical pressure gradient force(VPGF)show that due to the interpolation errors of thickVL,the inflection points in the vertical transition of thickVL induce the initial VPGF errors;therefore,the HTF1 coordinate with a monotonic increase in thickVL has the smallest errors among all the coordinates.More importantly,the temporal evolution of VPGF errors manifests top-type VPGF errors that propagate upward gradually during the time integration.Only the HTF1 and HTF4 coordinates with a monotonic increase in thickVL near the top of the terrain can suppress this propagation.This optimized HTF coordinate(i.e.,HTF1)can be a reference for designing a vertical thickVL in a numerical model.
基金supported by the National Key Research and Development Program of China (Grant Nos. 2017YFC1501901 and 2017YFA0603901)the Beijing Natural Science Foundation (Grant No. JQ18001)
文摘A 3D compressible nonhydrostatic dynamic core based on a three-point multi-moment constrained finite-volume (MCV) method is developed by extending the previous 2D nonhydrostatic atmospheric dynamics to 3D on a terrainfollowing grid. The MCV algorithm defines two types of moments: the point-wise value (PV) and the volume-integrated average (VIA). The unknowns (PV values) are defined at the solution points within each cell and are updated through the time evolution formulations derived from the governing equations. Rigorous numerical conservation is ensured by a constraint on the VIA moment through the flux form formulation. The 3D atmospheric dynamic core reported in this paper is based on a three-point MCV method and has some advantages in comparison with other existing methods, such as uniform third-order accuracy, a compact stencil, and algorithmic simplicity. To check the performance of the 3D nonhydrostatic dynamic core, various benchmark test cases are performed. All the numerical results show that the present dynamic core is very competitive when compared to other existing advanced models, and thus lays the foundation for further developing global atmospheric models in the near future.
基金supported by the Knowledge Innovation Program of the Chinese Academy of Sciences(KZCX2-YW-Q11-04)the National Basic Research Program of China(973 Program,Grant No.2011CB309704)The second author was supported by the National Natural Science Foundation of China(NSFC)under Grant No.40875022,41175064 and 40633016.
文摘This study shows a new way to implement terrain-following s-coordinate in a numerical model,which does not lead to the well-known"pressure gradient force(PGF)"problem.First,the causes of the PGF problemare analyzedwith existing methods that are categorized into two different types based on the causes.Then,the new method that bypasses the PGF problem all together is proposed.By comparing these threemethods and analyzing the expression of the scalar gradient in a curvilinear coordinate system,this study finds out that only when using the covariant scalar equations of s-coordinate will the PGF computational form have one term in each momentum component equation,thereby avoiding the PGF problem completely.A convenient way of implementing the covariant scalar equations of s-coordinate in a numerical atmospheric model is illustrated,which is to set corresponding parameters in the scalar equations of the Cartesian coordinate.Finally,two idealized experimentsmanifest that the PGF calculated with the new method is more accurate than using the classic one.This method can be used for oceanic models as well,and needs to be tested in both the atmospheric and oceanic models.
基金We appreciate the detailed suggestions and comments provided by the editor and the anonymous reviewers. Several research programs supported the work presented in this article： the National Basic Research Program of China （No. 2015CB954100）, the National Natural Science Foundation of China （Grant No. 41306078）, and the Priority Academic Program Development of Jiangsu Higher Education Institutions （Grant No. 1411109012）.
文摘A three-dimensional hydrodynamic model is presented which combines a terrain-following vertical coordinate with a horizontally orthogonal curvilinear coordinate system to fit the complex bottom topography and coastlines near estuaries, continental shelves, and harbors. To solve the governing equations more efficiently, we improve the alternating direction implicit method, which is extensively used in the numerical modeling of horizontal two-dimensional shallow water equations, and extend it to a three-dimensional tidal model with relatively little computational effort. Through several test cases and realistic applications, as presented in the paper, it can be demonstrated that the model is capable of simulating the periodic to-and-fro currents, wind-driven flow, Ekman spirals, and tidal currents in the near-shore region.
