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.

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.

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.

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.

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.

RETROSPECTIVE TIME INTEGRAL SCHEME AND ITS APPLICATIONS TO THE ADVECTION EQUATION

To put more information into a difference scheme of a differentialequation 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 memorialdynamics. Stability criteria of the scheme for an advection equation in certain condi-tions are derived mathematically. The computations for the advection equation havebeen conducted with its RT scheme. It is shown that the accuracy of the scheme ismuch higher than that of the leapfrog (LF) difference scheme.

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.

A high-order splitting scheme for the advection diffusion equation

A high order splitting scheme for the advection diffusion equation of pollutants is proposed in this paper. The multidimensional advection diffusion equation is splitted into several one dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth order spatial accuracy. Several typically pure advection and advection diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can be efficiently solved with little programming effort.

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.

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 subscale 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 stirring/cabbeling associated with isopycnal diffusion/advection is much smaller. Although isopycnal coordinates may be a better choice in terms of handling lateral diffusion, advection terms in the traditional Eulerian coordinates can produce artificial source of GPE due to cabbeling associated with advection. Reducing such numerical errors remains a grand challenge.

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

The Global/Regional Assimilation and PrEdiction System(GRAPES) is the new-generation numerical weather prediction(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 interpolation(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 nonuniform 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.

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 stratification 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.

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．

Backward Calculation Based on the Advection and Diffusion of Oil Spills on the Sea Surface

In the light of the problem of oil pollution brought about by ships, in this paper we present the concept of backward tracing oil spills. In the course of backward calculation of the two-dimensional convection & diffusion equation, on the one hand, the advection term itself has the strong unilateral property, which means information in the upper reaches is transmitted downstream via the advection term; on the other hand, because of the opposite direction of calculation, it is essential for information to be conveyed upstream by means of the advection term. In addition, unlike that in the forward calculation, the diffusion term in the backward calculation is prone to accumulate errors, and thus renders the whole scheme unstable. Therefore, we adopt the central difference to deal with both the convectional term and the diffusion term. By examining two practical examples (1) under the unlimited boundary condition, and (2) under the limited boundary condition, it is proven that this method could achieve fundamentally satisfactory results not only in the open ocean but also in the closed or semi-closed bay.

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 diffusion/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 excessive amount of GPE source and sink due to stirring and cabbeling associated with sigma diffusion/advection 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 diffusion 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

氡运输测试，是为大气的运输模型的一个广泛地使用的测试用例，被执行在 IAP-LASG (GAMIL ) 的格子点大气的模型评估 tracer 移流计划。在模型的三个可得到的计划中的二个被发现在极的区域并且在空气的上面的部分与重要偏爱被联系，它在像臭氧的 tracers 的模拟暗示潜在地大的错误。矛盾在 GAMIL 并且因而的动态核心在移流计划和分离连续性方程之间存在的理论分析表演导致假来源并且在 tracer 运输方程下沉。这类矛盾的影响被理想化的测试表明并且作为上述的偏爱的原因识别了。这个矛盾的另外的潜在的效果也被讨论。这研究的结果为在 GAMIL 模型选择合适的移流计划提供一些提示。至少为 polar-region-concentrated 大气的部件和密切相关的化学种类，没有显著地增加计算开销，流动形式 Semi-Lagrangian 移流计划生产大规模运输过程的更合理的模拟。

The Influence of Weakly-Nonlinear Vertical Advection on the Wind Field of PBL with Large-Scale Orography

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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 Ni o3.4 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 Ni o3.4 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.

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.