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Feng Xiao

Abstract

A class of semi-Lagrangian schemes has been derived for solving the advection equation. Compared with other semi-Lagrangian-type schemes, the presented schemes require less computational stencils for interpolation construction. Besides the dependent variable itself, its spatial derivatives are also evaluated based on a Lagrangian invariant solution. This makes estimating the first-order derivatives from the values of the dependent variable at neighboring grid points unnecessary. The resulting numerical formula appears spatially compact and only one mesh cell is needed for constructing the interpolation profile.

The 2D basic formulation is based on a rational interpolation function. It shows an ability to prevent numerical oscillation. Some variants of the scheme can be easily obtained by minor modifications. It is easy to get the desired numerical properties such as diffusion reduction, oscillation suppression, or (more strongly) monotonicity with the presented schemes.

Grid refinement analysis shows that all the schemes presented in this paper have convergence factors larger than 2 based on an l 2 norm.

The presented schemes need some extra memory space to store the derivatives of the interpolation function, but do appear competitive with other conventional semi-Lagrangian methods based on Hermite interpolants, in terms of arithmetic operation counts.

Parallel implementation shows that the presented schemes are easily portable to a parallel environment with distributed memory architecture and data communications take place only on the cells on the boundaries of the parallel partition.

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Chungang Chen, Juzhong Bin, and Feng Xiao

Abstract

A third-order numerical model is developed for global advection transport computation. The multimoment constrained finite-volume scheme has been implemented to the hexagonal geodesic grid for spherical geometry. Two kinds of moments (i.e., point value and volume-integrated average) are used as the constraint conditions to derive the time evolution equations to update the computational variables, which are the values defined at the specified points over each mesh element in the present model. The numerical model has rigorous numerical conservation and third-order accuracy. One of the major merits of the present method is that it does not explicitly involve numerical quadrature, which leads to great convenience in accurately computing curved geometry and source terms. The present paper provides an accurate and practical formulation for advection calculation in the hexagonal-type geodesic grid.

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Xiao-Feng Li, Jingjing Yu, and Yun Li

Abstract

Rainfall over northern Australia (NA) in austral summer is the largest water source of Australia. Previous studies have suggested a strong zonal-dipole trend pattern in austral summer rainfall since 1950, with rainfall increasing in northwest Australia (NWA) but decreasing in northeast Australia (NEA). The dynamics of rainfall increase in NWA was linked to sea surface temperature (SST) in the south Indian Ocean and the rainfall decrease in NEA was associated with SST in the northeast Indian Ocean.

This study reports that, in contrast to a zonal-dipole trend pattern, a dominant wetting pattern over NA has recently been observed in the post-1979 satellite era. The recent NA rainfall increase also manifests as the first leading mode of summer rainfall variability over the Australian continent. Further investigation reveals that SST in the tropical western Pacific (TWP) has replaced the SST in the south and northeast Indian Ocean as the controlling factor responsible for the recent NA rainfall increase. Direct thermal forcing by increasing TWP SST gives rise to an anomalous Gill-type cyclone centered around NA, leading to anomalously high rainfall. As such, the increasing SST in the TWP induces over 50% of the observed rainfall wetting trend over NA. The increased rainfall in turn induces land surface cooling in NA. This mechanism can be confirmed with results obtained from sensitivity experiments of a numerical spectral atmospheric general circulation model. Thus, increasing SST in the TWP has contributed much of the recent summer rainfall increase and consequently the surface cooling over NA.

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Chungang Chen, Feng Xiao, and Xingliang Li

Abstract

An adaptive global shallow-water model is proposed on cubed-sphere grid using the multimoment finite volume scheme and the Berger–Oliger adaptive mesh refinement (AMR) algorithm that was originally designed for a Cartesian grid. On each patch of the cubed-sphere grid, the curvilinear coordinates are constructed in a way that the Berger–Oliger algorithm can be applied directly. Moreover, an algorithm to transfer data across neighboring patches is proposed to establish a practical integrated framework for global AMR computation on the cubed-sphere grid.

The multimoment finite volume scheme is adopted as the fluid solver and is essentially beneficial to the implementation of AMR on the cubed-sphere grid. The multimoment interpolation based on both volume-integrated average (VIA) and point value (PV) provides the compact reconstruction that makes the present scheme very attractive not only in dealing with the artificial boundaries between different patches but also in the coarse–fine interpolations required in the AMR computations. The single-cell-based reconstruction avoids involving more than two nesting levels during interpolations. The reconstruction profile of constrained interpolation profile–conservative semi-Lagrangian scheme with third-order polynomial function (CIP-CSL3) is adopted where the slope parameter provides a flexible and convenient switching to get the desired numerical properties, such as high-order (fourth order) accuracy, monotonicity, and positive definiteness.

Numerical experiments with typical benchmark tests for both advection equation and shallow-water equations are carried out to evaluate the proposed model. The numerical errors and the corresponding CPU times of numerical experiments on uniform and adaptive meshes verify the performance of the proposed model. Compared to the uniformly refined grid, the AMR technique is able to achieve the similar numerical accuracy with less computational cost.

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Xiao-Feng Li, Jianping Li, and Yun Li

Abstract

The middle–lower valley of the Yangtze River (MLY), located in the middle of eastern China, has been one of the largest economic centers of China since ancient times. Winter precipitation variability over the MLY is important for China because of its significant influence on the local economy. However, few studies have focused on the long-term variability of winter precipitation over the MLY. This study reports a significant wetting trend over the MLY in winter during the three decades since the late 1970s, forming a “mid-east-China winter wetting” pattern, which has become an important feature of precipitation change under the weakening East Asian winter monsoon. This wetting trend in the MLY also implies the poleward extension of the precipitation belts of southern China.

Further investigation reveals that the increasing sea surface temperature (SST) in the tropical Indian Ocean (TIO) is the dominant factor responsible for recent increases in precipitation over the MLY. The thermal forcing driven by warming of the TIO SST gives rise to an anomalous cyclonic circulation along the coast of eastern China. This transports more water vapor onto the Chinese mainland, shifts and causes anomalous convergence over the MLY, and generates the increase in precipitation there. As such, the increasing SST in the TIO induces over 80% of the observed wetting trend over the MLY. This mechanism was verified by results obtained from two sets of sensitivity experiments using a numerical spectral atmospheric general circulation model. Thus, increasing SST in the TIO has made a dominant contribution to the recent winter precipitation increase over the MLY.

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Xingliang Li, Chungang Chen, Xueshun Shen, and Feng Xiao

Abstract

The two-dimensional nonhydrostatic compressible dynamical core for the atmosphere has been developed by using a new nodal-type high-order conservative method, the so-called multimoment constrained finite-volume (MCV) method. Different from the conventional finite-volume method, the predicted variables (unknowns) in an MCV scheme are the values at the solution points distributed within each mesh cell. The time evolution equations to update the unknown point values are derived from a set of constraint conditions based on the multimoment concept, where the constraint on the volume-integrated average (VIA) for each mesh cell is cast into a flux form and thus guarantees rigorously the numerical conservation. Two important features make the MCV method particularly attractive as an accurate and practical numerical framework for atmospheric and oceanic modeling. 1) The predicted variables are the nodal values at the solution points that can be flexibly located within a mesh cell (equidistant solution points are used in the present model). It is computationally efficient and provides great convenience in dealing with complex geometry and source terms. 2) High-order and physically consistent formulations can be built by choosing proper constraints in view of not only numerical accuracy and efficiency but also underlying physics. In this paper the authors present a dynamical core that uses the third- and the fourth-order MCV schemes. They have verified the numerical outputs of both schemes by widely used standard benchmark tests and obtained competitive results. The present numerical core provides a promising and practical framework for further development of nonhydrostatic compressible atmospheric models.

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Yuzhang Che, Chungang Chen, Feng Xiao, Xingliang Li, and Xueshun Shen

Abstract

A new multimoment global shallow-water model on the cubed sphere is proposed by adopting a two-stage fourth-order Runge–Kutta time integration. Through calculating the values of predicted variables at half time step t = t n + (1/2)Δt by a second-order formulation, a fourth-order scheme can be derived using only two stages within one time step. This time integration method is implemented in our multimoment global shallow-water model to build and validate a new and more efficient numerical integration framework for dynamical cores. As the key task, the numerical formulation for evaluating the derivatives in time has been developed through the Cauchy–Kowalewski procedure and the spatial discretization of the multimoment finite-volume method, which ensures fourth-order accuracy in both time and space. Several major benchmark tests are used to verify the proposed numerical framework in comparison with the existing four-stage fourth-order Runge–Kutta method, which is based on the method of lines framework. The two-stage fourth-order scheme saves about 30% of the computational cost in comparison with the four-stage Runge–Kutta scheme for global advection and shallow-water models. The proposed two-stage fourth-order framework offers a new option to develop high-performance time marching strategy of practical significance in dynamical cores for atmospheric and oceanic models.

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Xindong Peng, Feng Xiao, Takashi Yabe, and Keiji Tani

Abstract

A semi-Lagrangian-type advection scheme, cubic-interpolated pseudoparticle (CIP) method is implemented to the fifth-generation Pennsylvania State University–NCAR Mesoscale Model (MM5, version 3.4). A dimensional splitting CIP algorithm, with spatial third-order and temporal second-order accuracy, is derived to compute the advection in the MM5. The modified model is evaluated with ideal tests and real case studies in comparing with the leapfrog scheme, which was originally employed in the MM5. The CIP method appears remarkably superior to the leapfrog scheme in respect to both dissipative and dispersive errors, especially when discontinuities or large gradients exist in the advected quantity. Two real cases of severe mesoscale phenomena were simulated by using both the CIP scheme and the leapfrog scheme. In the advection dominant regions, the CIP shows remarkable advantages in capturing the detail structures of the predicted field. As computations with high resolution become more and more popular in experimental and/or operational modeling, implementing more accurate advection in numerical models, as is done in the present study, will be increasingly demanded.

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Xindong Peng, Feng Xiao, Wataru Ohfuchi, and Hiromitsu Fuchigami

Abstract

A conservative semi-Lagrangian scheme with rational function for interpolation is implemented in spherical geometry and tested in an atmospheric general circulation model (AGCM). The new scheme, different from the conventional semi-Lagrangian method, is conservative and oscillation free. By introducing polar mixing and a time split computation of divergence, the scheme can compute advection transport correctly over the polar regions. Idealized advection tests with various velocity fields were carried out to demonstrate numerical accuracy and conservation in comparison with the spectral schemes. The impact of the advection computation on water vapor circulation in an AGCM is also investigated with numerical simulations on the Earth Simulator. Both pure advection tests and general circulation experiments show that the presented scheme is effective in improving the tracer transport property and the precipitation field in comparison with the leapfrog-spectral method.

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Xingliang Li, Dehui Chen, Xindong Peng, Keiko Takahashi, and Feng Xiao

Abstract

A numerical model for shallow-water equations has been built and tested on the Yin–Yang overset spherical grid. A high-order multimoment finite-volume method is used for the spatial discretization in which two kinds of so-called moments of the physical field [i.e., the volume integrated average (VIA) and the point value (PV)] are treated as the model variables and updated separately in time. In the present model, the PV is computed by the semi-implicit semi-Lagrangian formulation, whereas the VIA is predicted in time via a flux-based finite-volume method and is numerically conserved on each component grid. The concept of including an extra moment (i.e., the volume-integrated value) to enforce the numerical conservativeness provides a general methodology and applies to the existing semi-implicit semi-Lagrangian formulations. Based on both VIA and PV, the high-order interpolation reconstruction can only be done over a single grid cell, which then minimizes the overlapping zone between the Yin and Yang components and effectively reduces the numerical errors introduced in the interpolation required to communicate the data between the two components. The present model completely gets around the singularity and grid convergence in the polar regions of the conventional longitude–latitude grid. Being an issue demanding further investigation, the high-order interpolation across the overlapping region of the Yin–Yang grid in the current model does not rigorously guarantee the numerical conservativeness. Nevertheless, these numerical tests show that the global conservation error in the present model is negligibly small. The model has competitive accuracy and efficiency.

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