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.

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.

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 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.

The Influence of Advection Schemes and Turbulence Closure Models on Drag Coefficient Calculation Around a Circular Cylinder at High Reynolds Number

Different advection schemes and two-equation turbulence closure models based on eddy viscosity concept are used to compute the drag coefficient around a circular cylinder at high Reynolds number (106).The numerical results from these simulations are compared with each other and with experimental data in order to evaluate the performance of different combinations of advection scheme and two-equation turbulence model.The separate contributions from form drag and friction drag are also ana-lyzed.The computational results show that the widely used standard k-ε turbulence closure is not suitable for such kind of study,while the other two-equation turbulence closure models produce acceptable results.The influence of the different advection schemes on the final results are small compared to that produced by the choice of turbulence closure method. The present study serves as a reference for the choice of advection schemes and turbulence closure models for more complex numerical simulation of the flow around a circular cylinder at high Reynolds number.

Evaluation of a Lagrangian advection scheme for cloud droplet diffusion growth with a maritime shallow cumulus cloud case

A Lagrangian advection scheme(LAS)for solving cloud drop diffusion growth was previously proposed(in 2020)and validated with simulations of cloud droplet spectra with a one-and-a-half dimensional(1.5D)cloud bin model for a deep convection case.The simulation results were improved with the new scheme over the original Eulerian scheme.In the present study,the authors simulated rain embryo formation with the LAS for a maritime shallow cumulus cloud case from the RICO(Rain in Cumulus over the Ocean)campaign.The model used to simulate the case was the same 1.5D cloud bin model coupled with the LAS.Comparing the model simulation results with aircraft observation data,the authors conclude that both the general microphysical properties and the detailed cloud droplet spectra are well captured.The LAS is robust and reliable for the simulation of rain embryo formation.

AN UNSPLIT LAGRANGIAN ADVECTION SCHEME FOR VOLUME OF FLUID METHOD

An Unsplit Lagrangian Advection （ULA） scheme for Volume Of Fluid （VOF） method is presented in this article. The ULA scheme is developed based on an algorithm of Piecewise Linear Interface Construction （PLIC）. The volume fluxes between cells are calculated through solving the new equation of the linear interfaces in cells in the ULA scheme. The fluxes flowing out from one cell is the inflow fluxes for another cell. In this way the whole fluid volume is conserved strictly without using any redistribution algorithms. The ULA scheme is based on two-dimensional structured rectangular mesh and may be extended to three-dimensional structured mesh with more geometrical efforts. The results from three widely-used benchmark tests show that the ULA scheme can achieve the accuracy higher than Split Lagaragian Advection （SLA） scheme and the Flux-Corrected Transport （FCT） algorithm.

NUMERICAL SIMULATIONS OF SEA ICE WITH DIFFERENT ADVECTION SCHEMES

Numerical simulations are carried out for sea ice with four different advection schemes to study their effects on the simulation results. The sea ice model employed here is the Sea Ice Simulator （SIS） of the Geophysical Fluid Dynamics Laboratory （GFDL） Modular Ocean Model version 4b （MOM4b） and the four advection schemes are, the upwind scheme originally used in the SIS, the Multi-Dimensional Positive Advection （MDPA） scheme, the Incremental Remapping Scheme （IRS） and the Two Step Shape Preserving （TSSP） scheme. The latter three schemes are newly introduced. To consider the interactions between sea ice and ocean, a mixed layer ocean model is introduced and coupled to the SIS. The coupled model uses a tri-polar coordinate with 120x65 grids, covering the whole earth globe, in the horizontal plane. Simulation results in the northern high latitudes are analyzed. In all simulations, the model reproduces the seasonal variation of sea ice in the northern high latitudes well. Compared with the results from the observation, the sea ice model produces some extra sea ice coverage in the Greenland Sea and Barents Sea in winter due to the exclusion of ocean current effects and the smaller simulated sea ice thickness in the Arctic basin. There are similar features among the results obtained with the introduced three advection schemes. The simulated sea ice thickness with the three newly introduced schemes are all smaller than that of the upwind scheme and the simulated sea ice velocities of movement are all smaller than that of the upwind scheme. There are more similarities shared in the results obtained with the MPDA and TSSP schemes.

An MPI+OpenACC-Based PRM Scalar Advection Scheme in the GRAPES Model over a Cluster with Multiple CPUs and GPUs

A moisture advection scheme is an essential module of a numerical weather/climate model representing the horizontal transport of water vapor.The Piecewise Rational Method(PRM) scalar advection scheme in the Global/Regional Assimilation and Prediction System(GRAPES) solves the moisture flux advection equation based on PRM.Computation of the scalar advection involves boundary exchange,and computation of higher bandwidth requirements is complicated and time-consuming in GRAPES.Recently,Graphics Processing Units(GPUs) have been widely used to solve scientific and engineering computing problems owing to advancements in GPU hardware and related programming models such as CUDA/OpenCL and Open Accelerator(OpenACC).Herein,we present an accelerated PRM scalar advection scheme with Message Passing Interface(MPI) and OpenACC to fully exploit GPUs’ power over a cluster with multiple Central Processing Units(CPUs) and GPUs,together with optimization of various parameters such as minimizing data transfer,memory coalescing,exposing more parallelism,and overlapping computation with data transfers.Results show that about 3.5 times speedup is obtained for the entire model running at medium resolution with double precision when comparing the scheme’s elapsed time on a node with two GPUs(NVIDIA P100) and two 16-core CPUs(Intel Gold 6142).Further,results obtained from experiments of a higher resolution model with multiple GPUs show excellent scalability.

UPWIND SCHEME FOR IDEAL 2-D MHD FLOWS BASED ON UNSTRUCTURED MESH

An upwind scheme based on the unstructured mesh is developed to solve ideal 2-D magnetohydrodynamics （MHD） equations. The inviscid fluxes are approximated by using the modified advection upstream splitting method （AUSM） scheme, and a 5-stage explicit Runge-Kutta scheme is adopted in the time integration. To avoid the influence of the magnetic field divergence created during the simulation, the hyperbolic divergence cleaning method is introduced. The shock-capturing properties of the method are verified by solving the MHD shock-tube problem. Then the 2-D nozzle flow with the magnetic field is numerically simulated on the unstructured mesh. Computational results demonstrate the effects of the magnetic field and agree well with those from references.

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.

Viscous liquid sloshing damping in cylindrical container using a volume of fluid method

Liquid sloshing is a kind of very complicated free surface flow and exists widely in many fields.In order to calculate liquid sloshing damping precisely a volume of fluid method based on finite volume scheme is used to simulate free surface flows in partly filled cylindrical containers.A numerical method is pre-sented to simulate the movement of the free surface flow,in which a piecewise linear interface con-struction scheme and an unsplit Lagrangian advection scheme instead of Eulerian advection scheme are used.The damping performance of liquid sloshing in cylindrical containers under fundamental sloshing mode is investigated.There are four factors determining the surface-wave damping:free surface,boundary-layer,interior fluid and contact line.In order to study different contributions from these four factors to whole damping,several examples are simulated.No-slip and slip wall boundary conditions on both side wall and bottom wall of the cylindrical containers are studied to compare with the published results obtained by solving Stokes equations.In the present method the first three main factors can be considered.The simulation results show that the boundary-layer damping contribution increases while the interior fluid damping contribution decreases with increase of Reynolds number.