An invariant theory of the linearized shallow water equations with rotation and its application to a sphere and a plane

The system of linearized shallow water equations is formulated in this paper on any rotating and smooth surface M in terms of differential geometry. The system decouples into two separate equations: a scalar one for the height deviation and a vector one for the velocity field. For low and high frequencies these equations yield asymptotic equations whose solutions are the generalizations of the Poincare and Rossby waves to smooth surface. The application of these equations to the β-plane yields both new and previously known equations for the height deviation and for the velocity components. The application of the equations to the rotating spherical Earth shows that the meridional amplitudes of Poincare and Rossby waves are both described by the prolate angular spheroidal wave functions. The asymptotic and the power series expansions of the eigenvalues of these functions yield new approximations for the dispersion relations of these waves on a sphere. The new dispersion relations are very accurate in the physically relevant range of the single nondimensional model parameter – the square of the nondimensional gravity waves’ phase speed. The invariant formulation can also be applied to other surfaces that are of geophysical interest such as an oblate ellipsoid of revolution.     

浅水波方程在全球上的变时间步长积分

本文讨论了浅水波方程在全球上的变时间步长(以下简称变步长)积分。并讨论了两步Lax-Wendroff的变时间步长(以下简称TLW变步长)积分,发现积分结果对平滑系数κ很敏感,必须选择合适。井将常数时间步长(以下简称定步长)的积分结果和两种变步长的积分结果进行比较。发现用实际资料作预报,变步长法的预报均方根误差比定步长法的小。还发现变步长的24小时预报对近极地系统移动偏慢的问题比定步长的有较大的改善。变步长积分的计算稳定性也优于定步长。     

The shallow water equations on the sphere and their Lagrange-Galerkin-solution

Summary ?The shallow water equations are formulated on the sphere in a three-dimensional coordinate system with the aid of tangential velocity components and differential operators. We introduce a modified semi-Lagrangian scheme for the discretization in time. The discretization in space is solved by linear finite elements. The grids we use are regular refinements of a macro triangulation which itself is derived from a highly symmetric polyeder also known as a bucky or soccer ball. The good numerical results show that this combination is a promising approach. The numerical algorithm is stable and its strength is the conservation of mass and energy. Received April 13, 2001; Revised December 18, 2001     

预估-校正的半隐式半拉格朗日时间积分方案及其在CMA-GFS模式中的应用

CMA-GFS采用的是传统的二时间层半隐式半拉格朗日时间积分方案(SISL).拉格朗日平流速度和非线性项需要采用时间外插进行计算,在急流轴附近等梯度大值区会造成计算不稳定,甚至积分中断现象.文中通过构造预估-校正半隐式半拉格朗日时间积分方案(SISL/P-C),以减少时间外插的影响;半隐式系数由原来的0.72减小到0....     

Revisiting well-posed boundary conditions for the shallow water equations

We derive a general form of well-posed open boundary conditions for the two-dimensional shallow water equations by using the energy method. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. The boundary conditions are also discussed for a rectangular domain. We compare the results with a number of often used open boundary conditions and show that they are a subset of the derived general form.     

The pseudospectral approximation applied to the shallow water equations on a sphere

Abstract

The Radar data Analysis, Processing and Interactive Display (RAPID) system, developed by McGill University researchers, synthesizes spherical coordinate radar data onto Cartesian maps displaying Constant Altitude Plan Position Indicator (CAPPI) reflectivity, Vertically Integrated Liquid water content (VIL), and other radar‐based parameters. In this study, Carvel radar (53.34°N, 114.09°W) data from July 2000 were processed using McGill's RAPID software. Specifically, we compared observations of severe convection, as identified by selected radar‐based reflectivity parameters, with surface severe weather reports and atmospheric sounding data. July 2000 was characterized by frequent severe thunderstorm activity over central Alberta; there were seven days with golfball‐sized hail, and two days with confirmed tornadoes. The VIL, upper level VIL (UVIL), and the maximum reflectivity at 7 km (Z7) were employed to quantify the strength and frequency of storms within a 120‐km radial distance of Carvel. For each day, the intensity of the convection was quantified by counting the total number of 1‐km² pixels in the study area that exceeded the severe thresholds for VIL, UVIL and Z7. The severe thunderstorm algorithms were found to be very effective at correctly identifying the observed severe thunderstorm events. All three radar parameters indicated a diurnal cycle, with severe convection starting after noon and peaking between 16:00 and 18:00 Local Daylight Time. A positive correlation was evident between the observed storm severity and the daily UVIL and Z7 pixel counts. The daily UVIL and Z7 pixel counts were also positively correlated with the Convective Available Potential Energy (CAPE) calculated from proximity soundings.     

A note on the performance of the multiply-upstream semi-Lagrangian advection schemes for one-dimensional nonlinear momentum conservation equation

Summary Two-time-level multiply-upstream semi-Lagrangian schemes were examined in the case of the self-advecting, one-dimensional nonlinear momentum conservation equation. The shock formation process was analyzed. It is pointed out that the shocks cannot be created in the truncated systems satisfying the Pudykiewicz, Benoit and Staniforth criterion.The numerical integrations were restricted to 12 h. It was shown that, at least in the sub-CFL range, increased complexity of the scheme can compensate reduced horizontal resolution. A considerable sensitivity of the schemes with respect to the time step was detected. In the super-CFL mode, several windows on various time scales were found within which the Pudykiewicz, Benoit and Staniforth criterion was satisfied. The time step of 1.44 times the maximum time step allowed by the CFL criterion was used in the semi-Lagrangian runs.The super-CFL, semi-Lagrangian solutions were diverging progressively from the sub-CFL ones as the forecasts advanced. This was also reflected in the energy spectra.Unacceptably large energy losses were encountered in the super-CFL, semi-Lagrangian runs. Most of these losses could be explained by the reduced mean wind speed, i.e., the amplitude of the zero wavenumber wave. At the same time, the energy content in the shorter waves increased. In a more complex model, such a situation would resemble a loss of zonal, and an increase of transient eddy kinetic energy.A trajectory error measure was defined as the maximum absolute value of the distance between the actual arriving point of the particle originating at the estimated departure point, and the grid point assumed to be the arrival point in the semi-Lagrangian procedure. In contrast to the sub-CFL regime, this measure could reach a considerable fraction of the grid distance in the computations with the super-CFL time steps.In the physical system considered, the trajectories are determined only by the velocities at the departure points. With the semi-Lagrangian schemes the distances traveled by the particles are estimated on the basis of the velocities at the points downstream with respect to the departure points. Thus, unless the solution is smooth (in space and time) on the scales of the extrapolation distances/times, the upstream extrapolation does not promise the convergence of the solution.With 16 Figures     

长江荆江分蓄洪区历史演变、前景和风险管理

用部分溃坝的波流与堰流相交法模拟淦河流域溃口点的流量。以溃口点的流量、DEM为基础数据, 利用GIS的暴雨洪涝淹没模型, 对2010年7月14日淦河流域由强降水引发的溃口式洪水淹没过程进行模拟, 并利用实际灾情对模拟结果进行检验分析。模拟结果表明:随着洪水演进, 淦河流域的淹没面积不断地增大, 其中0.5~1 m水深段的淹没面积增长最快; 14日18时, 洪水到达任窝村, 20时淹没至马桥镇, 22时淹没至严洲村。灾情调查检验结果显示, 对于洪水到达时间和地点, 淹没模型模拟值与实况值较为吻合, 表明该模型在溃口式洪水淹没过程方面具有较好的模拟效果。

     

The effect of ekman suction on a flow over a shallow topography in the beta plane

The effect of bottom Ekman layer suction on a homogeneous, constant depth, eastwards, low Rossby number flow over a shallow bottom topography in the beta plane is studied. The governing vorticity equation is obtained by expanding the velocities in the continuity and momentum equations in powers of the Rossby number, ?, and matching the vertical velocity with the vertical velocity at the outer edge of the bottom Ekman layer obtained from the Ekman layer solution. The suction effect is then linearized using an Oseen approxiamation and the resulting linear model is solved using Fourier transforms with the requirement that the solution behave like a vortex near the origin which is equivalent to the effect of an isolated bump, i.e., a Green's function solution is obtained. An analytical solution is thus, obtained in integral form and then numerically integrated. The effect of Ekman suction is found to be a damping of the downstream Rossby waves in a distance of order $\text{2√2U/f}{}_0\text{E}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, an increased upstream influence, and a counterclockwise rotation of the closed streamline region about the origin. It is pointed out that the vortex solutions can be superimposed in order to obtain the solution for flow over topographies of finite horizontal text. This technique was used to compute the flow over a right circular cylinder. The results agree favorably with the experimental results of McCartney (1975).     

Vortex erosion in a shallow water model of the polar vortex

The erosion of a model stratospheric polar vortex in response to bottom boundary forcing is investigated numerically. Stripping of filaments of air from the polar vortex has been implicated in the occurrence of stratospheric sudden warmings (SSWs) but it is not understood in detail what factors determine the rate and amount of stripping. Here a shallow water vortex forced by topography is used to investigate the factors initiating stripping and whether this leads the vortex to undergo an SSW. It is found that the amplitude of topographic forcing must exceed some threshold (of order 200–450 m) in order for significant stripping to occur. For larger forcing amplitudes significant stripping occurs, but not as an instantaneous response to the forcing; rather, the forcing appears to initiate a process that ultimately results in stripping several tens of days later. There appears to be no simple quantitative relationship between the amount of mass stripped and the topography amplitude. However, at least over the early stages of the experiments, there is a good correlation between the amount of mass stripped and the global integral of wave activity, which may be interpreted as a measure of the accumulated topographic forcing. Finally there does not appear to be a simple correspondence between amount of mass stripped and the occurrence of an SSW.     

发展方程差分格式的构造和应用

本文把许多拟线性方程归结为算子形式的“发展方程”:F/t+AF=0.证明了带有非负算子A的“强隐式(即1/2≤θ≤1)格式”是绝对隐定的;而带有反对称算子A的“弱隐式和显式(即0≤θ≤1/2)格式”是绝对不稳定的。文中又以一维非线性平流方程为例,具体地构造了具有非负算子A的三种差分格式和相应的带人工耗散项的稳定格式,给出一个计算实例,说明后一类格式对消除虚假的寄生波也是很有效的,适于计算有间断的问题。     

Effects of optimized root water uptake parameterization schemes on water and heat flux simulation in a maize agroecosystem

As root water uptake (RWU) is an important link in the water and heat exchange between plants and ambient air, improving its parameterization is key to enhancing the performance of land surface model simulations. Although different types of RWU functions have been adopted in land surface models, there is no evidence as to which scheme most applicable to maize farmland ecosystems. Based on the 2007–09 data collected at the farmland ecosystem field station in Jinzhou, the RWU function in the Common Land Model (CoLM) was optimized with scheme options in light of factors determining whether roots absorb water from a certain soil layer (W_x) and whether the baseline cumulative root efficiency required for maximum plant transpiration (W_c) is reached. The sensibility of the parameters of the optimization scheme was investigated, and then the effects of the optimized RWU function on water and heat flux simulation were evaluated. The results indicate that the model simulation was not sensitive to W_x but was significantly impacted by W_c. With the original model, soil humidity was somewhat underestimated for precipitation-free days; soil temperature was simulated with obvious interannual and seasonal differences and remarkable underestimations for the maize late-growth stage; and sensible and latent heat fluxes were overestimated and underestimated, respectively, for years with relatively less precipitation, and both were simulated with high accuracy for years with relatively more precipitation. The optimized RWU process resulted in a significant improvement of CoLM’s performance in simulating soil humidity, temperature, sensible heat, and latent heat, for dry years. In conclusion, the optimized RWU scheme available for the CoLM model is applicable to the simulation of water and heat flux for maize farmland ecosystems in arid areas.     

Drag and bulk transfer coefficients associated with a shallow water surface

During September, 1971, measurements of Reynolds stress and sensible heat flux were made over a shallow coral reef off the coast of Papua. The water varied (with the tide) from 0 to 2.5 m in depth. Two independent measurements of the momentum flux confirmed that the surface deviated only slightly from aerodynamically smooth. The independent determinations of the sensible heat flux are consistent with this result, indicating a much lower Stanton number than would be expected over deeper water (viz., 1.1 × 10^–3 rather than about 1.5 × 10^–3).The near smoothness of the surface appears to be associated with either the attenuation of small surface waves resulting from the presence of a surface film or their inability to grow in the absence of longer wave components. Although it is not clear which of these effects was predominant, the data show that great care should be taken in extrapolating shallow water results to open ocean conditions, particularly when it is conceded that surface films may exist in either situation.Now with Argonne National Laboratory, Argonne, Illinois, U.S.A.Now with CSIRO Division of Mineral Physics, Sydney, N.S.W., Australia.     

“准拉格朗日法”和“欧拉法”的数学一致性与三次插值函数算法时间积分方案

从预报方程组通式和欧拉算符出发,用泰勒级数展开,推导出时空间微商余项为二阶、四阶完全预报方程组。可以证明,同阶时空间微商余项的准拉格朗日法和向前差分欧拉法时间积分方案,具有相同的物理意义和数学一致性,相比之下,传统中央差欧拉法时间积分方案是"简单格式"与"增大不可预测计算误差"并存。进而讨论用"三次插值函数"实现二阶时空间微商余项准拉格朗日法、或同阶向前差分欧拉法(可"二选一"),它们应该分别替代"双线性插值"准拉格朗日法和传统时空间中央差欧拉法,因前二者时空间微商余项及计算精度高于后二者。所以,三次插值函数算法可将准拉格朗日法和欧拉法时间积分方案、以及CFL判据统一起来。由于三次插值函数具有对原函数"变量场"的数学定律"收敛性"和二阶可导"最优性",且一次"三次插值函数"运算,即具对网格变量场二阶可导拟合"等价性":不仅拟合变量场斜率、还拟合其曲率和挠率。并因周期"三次插值函数",可作全球变量场"三次插值函数"二阶可导拟合,实现全球"三次"数值模式,并且可按变量场曲率判断,作变量场局域或单点平滑,保持"三次"模式时间积分的稳定性。     

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.     

The role of shallow convection in the water and energy cycles of the atmosphere

The Canadian Centre for Climate Modelling and Analysis atmospheric general circulation model (AGCM4) is used to study the role of shallow convection in the hydrologic and energy cycles of the atmosphere. Sensitivity tests with AGCM4 show a marked effect of the parameterization of shallow convection in the model. In particular, including the parameterization of shallow convection produces considerably enhanced vertical mixing and decreased stratiform cloud amounts in the lower subtropical atmosphere over the oceans. The differences in simulated stratiform cloud amounts are associated with a change in the globally averaged outgoing shortwave radiative flux at the top of the atmosphere of about 11 W m⁻². Additionally, precipitation rates are considerably reduced for stratiform clouds and enhanced for convective clouds in the subtropics, if the parameterization of shallow convection is included in the model. Additional tests show that the simulated responses in cloud amounts and precipitation to the treatment of shallow convection are robust. Additional simulations with modified closures for deep convection and other changes to the treatment of convection in the model still lead to similar responses of the model results.     

The relationship between nonconservative schemes and initial values of nonlinear evolution equations

For the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensionalshallow water wave equation as an example, the necessary conditions of computational stability are given.Based on numerical tests, the relationship between the nonlinear computational stability and the construc-tion of difference schemes, as well as the form of initial values, is further discussed. It is proved throughboth theoretical analysis and numerical tests that if the construction of difference schemes is definite, thecomputational stability of nonconservative schemes is decided by the form of initial values.