Kinetic Energy Budget Equations of Rotational and Divergent Flow in Terrain-following Coordinates

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.

Characteristics of pressure gradient force errors in a terrain-following coordinate

A terrain-following coordinate （a-coordinate） in which the computational form of pressure gradient force （PGF） is two-term （the so-called classic method） has significant PGF errors near steep terrain. Using the covariant equations of the a-coordinate to create a one-term PGF （the covariant method） can reduce the PGF errors. This study investigates the factors inducing the PGF errors of these two methods, through geometric analysis and idealized experiments. The geometric analysis first demonstrates that the terrain slope and the vertical pressure gradient can induce the PGF errors of the classic method, and then generalize the effect of the terrain slope to the effect of the slope of each vertical layer （φ）. More importantly, a new factor, the direction of PGF （a）, is proposed by the geometric analysis, and the effects of φ and a are quantified by tan φ.tan a. When tan φ.tan a is greater than 1/9 or smaller than -10/9, the two terms of PGF of the classic method are of the same order but opposite in sign, and then the PGF errors of the classic method are large. Finally, the effects of three factors on inducing the PGF errors of the classic method are validated by a series of idealized experiments using various terrain types and pressure fields. The experimental results also demonstrate that the PGF errors of the covariant method are affected little by the three factors.

Optimized Vertical Layers for the Hybrid Terrain-Following Coordinate Minimizing Numerical Errors in a 2D Rising Bubble Experiment near Steep Terrain

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.

Effects of Terrain-Following Vertical Coordinates on High-Resolution NWP Simulations

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.

A new three-dimensional terrain-following tidal model of free-surface flows

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.

A 3D Nonhydrostatic Compressible Atmospheric Dynamic Core by Multi-moment Constrained Finite Volume Method

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.

Primary safe criterion of earth-brushing flight for flying vehicle over digital surface model

In modern terrain-following guidance it is an important index for flight vehicle to cruise about safely and normally. On the basis of a constructing method of digital surface model (DSM), the definition, classification and scale analysis of an isolated obstacle threatening flight safety of terrain-following guidance are made. When the interval of vertical-and cross-sections on DSM is 12. 5 m, the proportion of isolated obstacles to the data amount of DSM model to be loaded is optimal. The main factors influencing the lowest flying height in terrain-following guidance are analyzed, and a primary safe criterion of the lowest flying height over DSM model is proposed. According to their test errors, the lowest flying height over 1:10 000 DSM model can reach 40. 5 m^45. 0 m in terrain-following guidance. It is shown from the simulation results of a typical urban district that the proposed models and methods are reasonable and feasible.

A New Approach to Implement Sigma Coordinate in a Numerical Model

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.

A Method to Identify Flight Obstacles on Digital Surface Model

In modern low-altitude terrain-following guidance, a constructing method of the digital surface model (DSM) is presented in the paper to reduce the threat to flying vehicles of tall surface features for safe flight. The relationship between an isolated obstacle size and the intervals of vertical- and cross-section in the DSM model is established. The definition and classification of isolated obstacles are proposed, and a method for determining such isolated obstacles in the DSM model is given. The simulation of a typical urban district shows that when the vertical- and cross-section DSM intervals are between 3 m and 25 m, the threat to terrain-following flight at low-altitude is reduced greatly, and the amount of data required by the DSM model for monitoring in real time a flying vehicle is also smaller. Experiments show that the optimal results are for an interval of 12.5 m in the vertical- and cross-sections in the DSM model, with a 1:10 000 DSM scale grade.