Improvement of the Semi-Lagrangian Advection Scheme in the GRAPES Model:Theoretical Analysis and Idealized Tests

ABSTRACT The Global/Regional Assimilation and PrEdiction System （GRAPES） is the newgeneration numerical weather predic- tion （NWP） system developed by the China Meteorological Administration. It is a fully compressible non-hydrostatical global/regional unified model that uses a traditional semi-Lagrangian advection scheme with cubic Lagrangian interpola tion （referred to as the SL_CL scheme）. The SL_CL scheme has been used in many operational NWP models, but there are still some deficiencies, such as the damping effects due to the interpolation and the relatively low accuracy. Based on Reich＇s semi-Lagrangian advection scheme （referred to as the R2007 scheme）, the Re_R2007 scheme that uses the low- and high-order B-spline function for interpolation at the departure point, is developed in this paper. One- and two-dimensional idealized tests in the rectangular coordinate system with uniform grid cells were conducted to compare the Re..R2007 scheme and the SL_CL scheme. The numerical results showed that： （1） the damping effects were remarkably reduced with the Re_R2007 scheme; and （2） the normalized errors of the Re_R2007 scheme were about 7.5 and 3 times smaller than those of the SL_CL scheme in one- and two-dimensional tests, respectively, indicating the higher accuracy of the Re..R2007 scheme. Furthermore, two solid-body rotation tests were conducted in the latitude-longitude spherical coordinate system with non uniform grid cells, which also verified the Re_R2007 scheme＇s advantages. Finally, in comparison with other global advection schemes, the Re_R2007 scheme was competitive in terms of accuracy and flow independence. An encouraging possibility for the application of the Re_R2007 scheme to the GRAPES model is provided.

Implementation of the Semi-Lagrangian Advection Scheme on a Quasi-Uniform Overset Grid on a Sphere

The semi-Lagrangian advection scheme is implemented on a new quasi-uniform overset （Yin-Yang） grid on the sphere. The Yin-Yang grid is a newly developed grid system in spherical geometry with two perpendicularly-oriented latitude-longitude grid components （called Yin and Yang respectively） that overlapp each other, and this effectively avoids the coordinate singularity and the grid convergence near the poles. In this overset grid, the way of transferring data between the Yin and Yang components is the key to maintaining the accuracy and robustness in numerical solutions. A numerical interpolation for boundary data exchange, which maintains the accuracy of the original advection scheme and is computationally efficient, is given in this paper. A standard test of the solid-body advection proposed by Williamson is carried out on the Yin-Yang grid. Numerical results show that the quasi-uniform Yin-Yang grid can get around the problems near the poles, and the numerical accuracy in the original semi-Lagrangian scheme is effectively maintained in the Yin-Yang grid.

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

Study of Wildebeest Foraging Processes Using Advection Diffusion Equation: Case of the Serengeti Ecosystem in Tanzania

We studied the foraging processes of wildebeest using an advection-diffusion equation. We equipped the model with data collected between 1999 and 2007 from the Serengeti ecosystem from 18 GPS-collared wildebeest. Results analysis show that wildebeest foraging behavior can be explained by advective and diffusive parameters in a heterogeneous habitat like the Serengeti ecosystem.

Application of a Shape-Preserving Advection Scheme to the Moisture Equation in an E-grid Regional Forecast Model

This paper presents a methodology which is very useful to design shape-preserving advection finite difference scheme on general E-grid horizontal arrangement of variables through introducing a two-step shape-preserving positive definite advection scheme in the moisture equation of the LASG-REM (LASG regional E-grid eta-coordinate forecast model). By trial-forecasting six local heavy raincases, the efficiency of the shape-preserving advection scheme in practical application has been examined. The LASG-REM with the shape-preserving advection scheme has a good forecasting ability for local precipitation.

Diagnosis of a Moist Thermodynamic Advection Parameter in Heavy-Rainfall Events

A moist thermodynamic advection parameter, defined as an absolute value of the dot product of hori- zontal gradients of three-dimensional potential temperature advection and general potential temperature, is introduced to diagnose frontal heavy rainfall events in the north of China. It is shown that the parameter is closely related to observed 6-h accumulative surface rainfall and simulated cloud hydrometeors. Since the parameter is capable of describing the typical vertical structural characteristics of dynamic, thermodynamic and water vapor fields above a strong precipitation region near the front surface, it may serve as a physical tracker to detect precipitable weather systems near to a front. A tendency equation of the parameter was derived in Cartesian coordinates and calculated with the simulation output data of a heavy rainfall event. Results revealed that the advection of the parameter by the three-dimensional velocity vector, the covariance of potential temperature advection by local change of the velocity vector and general potential temperature, and the interaction between potential temperature advection and the source or sink of general potential temperature, accounted for local change in the parameter. This indicated that the parameter was determined by a combination of dynamic processes and cloud microphysical processes.

A Two-Step Shape-Preserving Advection Scheme

This paper proposes a new two-step non-oscillatory shape-preserving positive definite finite difference advection transport scheme, which merges the advantages of small dispersion error in the simple first-order upstream scheme and small dissipation error in the simple second-order Lax-Wendroff scheme and is completely different from most of present positive definite advection schemes which are based on revising the upstream scheme results. The proposed scheme is much less time consuming than present shape-preserving or non-oscillatory advection transport schemes and produces results which are comparable to the results obtained from the present more complicated schemes. Elementary tests are also presented to examine the behavior of the scheme.

Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients

In a one-dimensional advection-diffusion equation with temporally dependent coefficients three cases may arise: solute dispersion parameter is time dependent while the flow domain transporting the solutes is uniform, the former is uniform and the latter is time dependent and lastly the both parameters are time dependent. In the present work analytical solutions are obtained for the last case, studying the dispersion of continuous input point sources of uniform and increasing nature in an initially solute free semi-infinite domain. The solutions for the first two cases and for uniform dispersion along uniform flow are derived as particular cases. The dispersion parameter is not proportional to the velocity of the flow. The Laplace transformation technique is used. New space and time variables are introduced to get the solutions. The solutions in all possible combinations of increasing/decreasing temporal dependence are compared with each other with the help of graphs. It has been observed that the concentration attenuation with position and time is the fastest in case of decreasing dispersion in accelerating flow field.

Role of Horizontal Density Advection in Seasonal Deepening of the Mixed Layer in the Subtropical Southeast Pacific

The mechanisms behind the seasonal deepening of the mixed layer （ML） in the subtropical Southeast Pacific were investigated using the monthly Argo data from 2004 to 2012. The region with a deep ML （more than 175 m） was found in the region of （22°-30°S, 105°-90°W）, reaching its maximum depth （-200 m） near （27°-28°S, 100°W） in September. The relative importance of horizontal density advection in determining the maximum ML location is discussed qualitatively. Downward Ekman pumping is key to determining the eastern boundary of the deep ML region. In addition, zonal density advection by the subtropical countercurrent （STCC） in the subtropical Southwest Pacific determines its western boundary, by carrying lighter water to strengthen the stratification and form a ＂shallow tongue＂ of ML depth to block the westward extension of the deep ML in the STCC region. The temperature advection by the STCC is the main source for large heat loss from the subtropical Southwest Pacific. Finally, the combined effect of net surface heat flux and meridional density advection by the subtropical gyre determines the northern and southern boundaries of the deep ML region： the ocean heat loss at the surface gradually increases from 22~S to 35~S, while the meridional density advection by the subtropical gyre strengthens the strat- ification south of the maximum ML depth and weakens the stratification to the north. The freshwater flux contribution to deepening the ML during austral winter is limited. The results are useful for understanding the role of ocean dynamics in the ML formation in the subtropical Southeast Pacific.

Energetics of lateral eddy diffusion/advection: Part III. Energetics of horizontal and isopycnal diffusion/ advection

Gravitational Potential Energy （GPE） change due to horizontal/isopycnal eddy diffusion and advection is examined. Horizontal/isopycnal eddy diffusion is conceptually separated into two steps： stirring and sub scale diffusion. GPE changes associated with these two steps are analyzed. In addition, GPE changes due to stirring and subscale diffusion associated with horizontal/isopycnal advection in the Eulerian coordinates are analyzed. These formulae are applied to the SODA data for the world oceans. Our analysis indicates that horizontal/isopycnal advection in Eulerian coordinates can introduce large artificial diffusion in the model. It is shown that GPE source/sink in isopycnal coordinates is closely linked to physical property distribution, such as temperature, salinity and velocity. In comparison with z-coordinates, GPE source/sink due to stir ring/cabbeling associated with isopycnal diffusion/advection is much smaller. Although isopycnal coordi nates may be a better choice in terms of handling lateral diffusion, advection terms in the traditional Eule rian coordinates can produce artificial source of GPE due to cabbeling associated with advection. Reducing such numerical errors remains a grand challenge.

RETROSPECTIVE TIME INTEGRAL SCHEME AND ITS APPLICATIONS TO THE ADVECTION EQUATION

To put more information into a difference scheme of a differential equation for making an accurate prediction, a new kind of time integration scheme, known as the retrospective (RT) scheme, is proposed on the basis of the memorial dynamics. Stability criteria of the scheme for an advection equation in certain conditions are derived mathematically. The computations for the advection equation have been conducted with its RT scheme. It is shown that the accuracy of the scheme is much higher than that of the leapfrog (LF) difference scheme.

Study of Effects of Beta Term and Nonlinear Advection on the Structure of Tropical Cyclones

The effects of the Beta term on the typhoon structure are examined within the linear framework in terms of an analytical method of 2-D Fourier representation and numerical experiments by a Beta-plane quasi-geostrophic barotropic model. Results show that the joint effects of the difference of Rossby phase velocities and the dispersion of typhoon energy keep the maximum wind velocity reasonably evolving rather than irrestrictively increasing. On the one hand, the nonlinear advection accelerates typhoon vortex damping, and on the other, the high pressure system formed downstream due to energy dispersion makes it easy to maintain.

Revisiting mesoscale eddy genesis mechanism of nonlinear advection in a marginal ice zone

A three-dimensional（3-D） ocean model is coupled with a two-dimensional（2-D） sea ice model, to revisit a nonlinear advection mechanism, one of the most important mesoscale eddy genesis mechanisms in the marginal ice zone. Two-dimensional ocean model simulations suggest nonlinear advection mechanism is more important when the water gets shallower. Instead of considering the ocean as barotropic fluid in the 2-D ocean model, the 3-D ocean model allows the sea ice to affect the current directly in the surface layer via ocean-ice interaction. It is found that both mesoscale eddy and sea surface elevation are sensitive to changes in a water depth in the 3-D simulations. The vertical profile of a current velocity in 3-D experiments suggests that when the water depth gets shallower, the current move faster in each layer, which makes the sea surface elevation be nearly inverse proportional to the water depth with the same wind forcing during the same time. It is also found that because of the vertical motion, the magnitude of variations in the sea surface elevation in the 3-D simulations is very small,being only 1% of the change in the 2-D simulations. And it seems the vertical motion to be the essential reason for the differences between the 3-D and 2-D experiments.

Energetics of lateral eddy diffusion/advection: Part IV. Energetics of diffusion/advection in sigma coordinates and other coordinates

Gravitational potential energy （GPE） source and sink due to stirring and cabbeling associated with sigma dif fusion/ advection is analyzed. It is shown that GPE source and sink is too big, and they are not closely linked to physical property distribution, such as temperature, salinity and velocity. Although the most frequently quoted advantage of sigma coordinate models are their capability of dealing with topography; the exces sive amount of GPE source and sink due to stirring and cabbeling associated with sigma diffusion/advec tion diagnosed from our analysis raises a very serious question whether the way lateral diffusion/advection simulated in the sigma coordinates model is physically acceptable. GPE source and sink in three coordinates is dramatically different in their magnitude and patterns. Overall, in terms of simulating lateral eddy diffu sion and advection isopycnal coordinates is the best choice and sigma coordinates is the worst. The physical reason of the excessive GPE source and sink in sigma coordinates is further explored in details. However, even in the isopycnal coordinates, simulation based on the Eulerian coordinates can be contaminated by the numerical errors associated with the advection terms.

Consistency Problem with Tracer Advection in the Atmospheric Model GAMIL

The radon transport test, which is a widely used test case for atmospheric transport models, is carried out to evaluate the tracer advection schemes in the Grid-Point Atmospheric Model of IAP-LASG （GAMIL）. Two of the three available schemes in the model are found to be associated with significant biases in the polar regions and in the upper part of the atmosphere, which implies potentially large errors in the simulation of ozone-like tracers. Theoretical analyses show that inconsistency exists between the advection schemes and the discrete continuity equation in the dynamical core of GAMIL and consequently leads to spurious sources and sinks in the tracer transport equation. The impact of this type of inconsistency is demonstrated by idealized tests and identified as the cause of the aforementioned biases. Other potential effects of this inconsistency are also discussed. Results of this study provide some hints for choosing suitable advection schemes in the GAMIL model. At least for the polax-region-concentrated atmospheric components and the closely correlated chemical species, the Flux-Form Semi-Lagrangian advection scheme produces more reasonable simulations of the large-scale transport processes without significantly increasing the computational expense.

Implementation of a Conservative Two-step Shape-Preserving Advection Scheme on a Spherical Icosahedral Hexagonal Geodesic Grid

An Eulerian flux-form advection scheme, called the Two-step Shape-Preserving Advection Scheme （TSPAS）, was generalized and implemented on a spherical icosahedral hexagonal grid （also referred to as a geodesic grid） to solve the transport equation. The C grid discretization was used for the spatial discretization. To implement TSPAS on an unstructured grid, the original finite-difference scheme was further generalized. The two-step integration utilizes a combination of two separate schemes （a low-order monotone scheme and a high-order scheme that typically cannot ensure monotonicity） to calculate the fluxes at the cell walls （one scheme corresponds to one cell wall）. The choice between these two schemes for each edge depends on a pre-updated scalar value using slightly increased fluxes. After the determination of an appropriate scheme, the final integration at a target cell is achieved by summing the fluxes that are computed by the different schemes. The conservative and shape-preserving properties of the generalized scheme are demonstrated. Numerical experiments are conducted at several horizontal resolutions. TSPAS is compared with the Flux Corrected Transport （FCT） approach to demonstrate the differences between the two methods, and several transport tests are performed to examine the accuracy, efficiency and robustness of the two schemes.

Seasonal variability of zonal heat advection in the mixed layer of the tropical Pacific

Zonal heat advection （ZHA） plays an important role in the variability of the thermal structure in the tropical Pacific Ocean, especially in the western Pacific warm pool （WPWP）. Using the Simple Ocean Data Assimilation （SODA） Version 2.02/4 for the period 1958-2007, this paper presents a detailed analysis of the climatological and seasonal ZHA in the tropical Pacific Ocean. Climatologically, ZHA shows a zonal- band spatial pattern associated with equatorial currents and contributes to forming the irregular eastern boundary of the WPWP （EBWP）. Seasonal variation of ZHA with a positive peak from February to July is most prominent in the Nifio3.4 region, where the EBWP is located. The physical mechanism of the seasonal cycle in this region is examined. The mean advection of anomalous temperature, anomalous advection of mean temperature and eddy advection account for 31%, 51%, and 18% of the total seasonal variations, respectively. This suggests that seasonal changes of the South Equatorial Current induced by variability of the trade winds are the dominant contributor to the anomalous advection of mean temperature and hence, the seasonality of ZHA. Heat budget analysis shows that ZHA and surface heat flux make comparable contributions to the seasonal heat variation in the Nifio3.4 region, and that ZHA cools the upper ocean throughout the calendar year except in late boreal spring. The connection between ZHA and EBWP is further explored and a statistical relationship between EBWP, ZHA and surface heat flux is established based on least squares fitting.

含平流(advection)的热单温吸积流的径向结构

本文在小粘滞参数（α＝０．１）、低吸积率的条件下，借助含平流的热吸积流的自相似解，考虑非热正负电子对产生，采用分两个区域分别求解然后对接的方法，研究了含平流的热单温吸积流的径向结构．证实了热吸积流所具有的一些显著特征和基本性质，同时得到一些新的结果：含平流的热吸积流存在一个临界半径ｒｃｒ；明确指出含平流的热吸积流的辐射冷却率与中心天体的质量成平方反比关系；提出正负电子对过程对含平流的热吸积流的辐射过程有显著影响．

Numerical Analysis of Novel Coronavirus (2019-nCov) Pandemic Model with Advection

Recently,the world is facing the terror of the novel corona-virus,termed as COVID-19.Various health institutes and researchers are continuously striving to control this pandemic.In this article,the SEIAR(susceptible,exposed,infected,symptomatically infected,asymptomatically infected and recovered)infection model of COVID-19 with a constant rate of advection is studied for the disease propagation.A simple model of the disease is extended to an advection model by accommodating the advection process and some appropriate parameters in the system.The continuous model is transposed into a discrete numerical model by discretizing the domains,finitely.To analyze the disease dynamics,a structure preserving non-standard finite difference scheme is designed.Two steady states of the continuous system are described i.e.,virus free steady state and virus existing steady state.Graphical results show that both the steady states of the numerical design coincide with the fixed points of the continuous SEIAR model.Positivity of the state variables is ensured by applying the M-matrix theory.A result for the positivity property is established.For the proposed numerical design,two different types of the stability are investigated.Nonlinear stability and linear stability for the projected scheme is examined by applying some standard results.Von Neuman stability test is applied to ensure linear stability.The reproductive number is described and its pivotal role in stability analysis is also discussed.Consistency and convergence of the numerical model is also studied.Numerical graphs are presented via computer simulations to prove the worth and efficiency of the quarantine factor is explored graphically,which is helpful in controlling the disease dynamics.In the end,the conclusion of the study is also rendered.

A Survey of the Implementation of Numerical Schemes for Linear Advection Equation

The interpolation method in a semi-Lagrangian scheme is decisive to its performance. Given the number of grid points one is considering to use for the interpolation, it does not necessarily follow that maximum formal accuracy should give the best results. For the advection equation, the driving force of this method is the method of the characteristics, which accounts for the flow of information in the model equation. This leads naturally to an interpolation problem since the foot point is not in general located on a grid point. We use another interpolation scheme that will allow achieving the high order for the box initial condition.