A global spectral element model for poisson equations and advective flow over a sphere

A global spherical Fourier–Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre–Gauss–Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity,with only one element. Then, the partial differential equations are solved on the longitude–latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. Highprecision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.     

On expansion of analytical functions in ultraspherical polynomials on the sphere

An algorithm is proposed for computing two-dimensional Fourier series in a spherical system of coordinates over a set of orthogonal ultraspherical polynomials whose special cases are the Legendre polynomials and Chebyshev polynomials of the first and second kind. The series uniformly converge at all points of the sphere including the poles. Unlike traditional spectral expansions on the sphere, they explicitly contain additional terms that characterize an odd component of a desired analytical function relative to the poles. It is shown that the expansion in the small vicinity of the poles (polar caps) is simplified because of the closeness to zero of the Fourier series terms responsible for the approximation of the components of the function that are odd relative to the poles. As the equatorial zone is approached, the value of the components of the desired function unsymmetrical relative to the poles increases and becomes comparable to the contribution of the symmetrical components. The new method is used for a special case of the spectral approximation of a continuous scalar analytical function with spherical harmonics used as the orthogonal basis. It is shown that double Fourier series in this case give an extension of the traditional spectral method. An alternative is to construct double Fourier series in associated Chebyshev polynomials of the first and second kind. An example of the spectral approximation of an analytical function on the sphere is presented.     

移极旋转坐标系中的大气动力学方程

本文仅研究无粘、无地形和绝热运动假定下的大气动力学问题，用分析力学的方法求得了固定在地球上的移极旋转坐标系， 如直角坐标系， 球坐标系和柱坐标系中的第二类 Lagrange方程。从而求得普遍的大气动力学方程。所谓的视示力 —— Coriolis力和离心力 —— 与其它各种力统一处理。当地球自转轴不为坐标系的z轴时，不存在大气运动的对称和反对称性。特别值得提出的是以往很多学者利用柱坐标系来数值模拟台风(飓风)的轴对称和非轴对称性时所使用的基本方程是含糊的， 本文给出了准确的基本方程。     

A new large-eddy simulation model for simulating air flow and warm clouds above highly complex terrain. Part I: The dry model

This paper presents the dry version of a new large-eddy simulation (LES) model, which is designed to simulate air flow and clouds above highly complex terrain. The model is three-dimensional and nonhydrostatic, and the governing equations are sound filtered by use of the anelastic approximation. A fractional step method is applied to solve the equations on a staggered Cartesian grid. Arbitrarily steep and complex orography can be accounted for through the method of viscous topography. The dynamical model core is validated by comparing the results for a spreading density current against a benchmark solution. The model accuracy is further assessed through the simulation of turbulent flow across a quasi two-dimensional ridge. The results are compared with wind-tunnel data. The method of viscous topography is not restricted to moderately sloped terrain. Compared to models using curvilinear grids, it allows this model to be applied to a much wider range of flows. This is illustrated through the simulation of an atmospheric boundary-layer flow over a surface mounted cube. The results show that the dry model version is able to accurately represent the complex flow in the vicinity of three-dimensional obstacles. It is concluded that the method of viscous topography was successfully implemented into a micrometeorological LES model. As will be shown in Part II, this allows the detailed study of clouds in highly complex terrain.     

守恒型有理函数插值半拉格朗日平流方案及性能分析

通过多种理想试验对正定、保形守恒型有理函数插值半拉格朗日平流方案分别在平面直角坐标以及阴阳网格球面坐标中进行了计算性能分析,并采用多种误差模对守恒型有理函数插值半拉格朗日平流方案的网格收敛性进行评估。结果表明,采用分段有理函数插值的守恒型半拉格朗日平流方案可以有效消除不连续分布处的数值振荡、保证正定性,物理场平滑分布时维持1-2阶收敛速度;而在不连续点或大梯度区域以及应用分维技术的多维算法都会通过有理函数的降阶特性,影响平流计算的收敛阶数,并且,在球面坐标中受球面曲率的影响,守恒型有理函数插值半拉格朗日平流算法的网格收敛速度有所降低。     

DESIGN OF A THREE-DIMENSIONAL GLOBAL ADIABATIC SPECTRAL PRIMITIVE EQUATION MODEL*

A three-dimensional(3D) global adiabatic spectral primitive equation model has been designed.The main features are as follows.(1) Adoption of spherical harmonics and Tschebyscheff polynomials as the basis functions in the horizontal and vertical respectively,but the unknowns in the spectral equations are two-dimensional;(2) Inclusion of the tropopause,which may vary with time and space;(3) Suggestion of a spectral method for representing the vertical structure of the atmosphere applicable to the unsmoothed profile case;(4) In consideration of nonlinear vertical aliasing a technique is proposed to avoid it and nonlinear computational instability.Based on real data forecasts up to 48 hours have been performed.The results show that the statistical verifications with the model are superior on the average to those with the T42L9 used operationally before 1995 at NMC of China at the same mean resolution.     

Evaluation of horizontal gradients in sigma‐coordinate shallow water models

Abstract

Evaluating horizontal gradients in three‐dimensional shallow water models that use bottom‐following sigma coordinates can lead to large errors near steep bathymetry. A technique that has been proposed to minimize this problem involves computing horizontal gradients in cartesian coordinates, while treating all other terms in a sigma coordinate framework. We study this technique through both truncation error analysis and numerical experimentation, and compare it to the direct application of sigma coordinates. While the Cartesian coordinate method has better convergence properties and generally smaller truncation errors when horizontal gradients are zero, the sigma coordinate method can be more accurate in other physically relevant situations. Also, the Cartesian coordinate method introduces significant numerical diffusion of variable sign near the bottom (where physical diffusion is particularly small), thus potentially leading to instabilities. Overall, we consider the sigma coordinates to be the best approach.     

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.     

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.     

非静力平衡条件下的垂直坐标变换及湿空气动力学方程组

为了解决目前湿空气动力学中一些悬而未决的问题， 拓宽物理量垂直坐标系的应用范围，本文讨论了非静力平衡条件下的垂直坐标转换，给出了不同坐标系的转换公式和动力学方程组，最后运用它们建立了湿空气动力学方程组。     

Errors in estimating spherical harmonic coefficients from partially sampled GCM output

 This study describes a method of calculating the mean squared error (MSE) incurred when estimating the spherical harmonic coefficients of a climatological field that is sampled at a small network of points. The method can also be applied to the coefficients of any other set of orthonormal basis functions that are defined on the sphere. It, therefore, provides a formalism that can be applied in a variety of contexts, such as in climate change detection, where inferences are attempted about fingerprint coefficients that are imperfectly estimated from observational data. By incorporating the fingerprint as part of a set of basis functions, the methodology can be used to estimate the sampling error in the fingerprint coefficient. The MSE is expressed in terms of the spherical harmonics (or other orthonormal expansion) of the empirical orthogonal functions (EOFs), the locations of the points in the network and a set of weights that are applied at these points. The weights are optimised by minimising the expected MSE. The method is applied to a number of network configurations using monthly-mean screen temperature and 500 mb height simulated by the Canadian Climate Centre 2nd generation general circulation model in an ensemble of six 10-year simulations. In comparison with uniform weighting, optimal weighting can reduce the MSE by an order of magnitude or more for some spherical harmonic coefficients and some network configurations. Also, the MSEs vary seasonally for each network. In particular, the relative MSE of low order spherical harmonic coefficients is found to be larger in DJF than in JJA. We demonstrate how MSEs improve with increasing network density and identify graphically, the coefficients that can be estimated reliably with each network configuration. Received: 19 April 1996 / Accepted: 10 April 1997     

The atmospheric boundary layer over a wavy surface

The wavy area of north-west Bohemia (Czechoslovakia) is simulated by a cylindrical surface model. A curvilinear orthogonal system of coordinates along the model surface is introduced. The hydrodynamical equations of motion are transformed into this system of coordinates. By applying boundary-layer assumptions, the equations of motion for the atmospheric boundary layer (ABL) above the model are derived. The equations and boundary conditions show an equivalence of the ABL above the model with that above a flat surface with external pressure gradient.     

Rossby waves in a rotating fluid of spheroidal configuration

The ray method is used to study quasi-geostrophic waves in a thin layer of incompressible, inviscid fluid of constant density both in a rotating spheroidal shell and on a rotating spheroid bounded above by a free surface. Asymptotic approximations to solutions of both the time-dependent and time-reduced problems are found; the dispersion relation obtained has the form of the Rossby-Haurwitz formula when the shell is spherical, and is asymptotically equivalent to that found by Longuet-Higgins (1965) for the free surface problem on a sphere. The approximation is first applied to free oscillations in a rotating spherical shell and Longuet-Higgins' (1964) results are rederived. Spheroidal shells in which the shell thickness depends only upon the latitude are studied next and a necessary and sufficient condition for oscillations to occur is obtained.     

DESIGN OF A THREE-DIMENSIONAL GLOBAL ADIABATIC SPECTRAL PRIMITIVE EQUATION MODEL

A three-dimensional(3D)global adiabatic spectral primitive equation model has beendesigned.The main features are as follows.(1)Adoption of spherical harmonics and Tschebyscheff polynomials as the basis functions inthe horizontal and vertical respectively,but the unknowns in the spectral equations are two-dimensional;(2)Inclusion of the tropopause,which may vary with time and space;(3)Suggestion of a spectral method for representing the vertical structure of the atmosphereapplicable to the unsmoothed profile case;(4)In consideration of nonlinear vertical aliasing a technique is proposed to avoid it andnonlinear computational instability.Based on real data forecasts up to 48 hours have been performed.The results show that thestatistical verifications with the model are superior on the average to those with the T42L9 usedoperationally before 1995 at NMC of China at the same mean resolution.     

在大气运动中考虑重力加速度空间变化的问题

给出了在旋转坐标系中考虑重力加速度g的空间变化的大气控制方程组,并证明了大气总质量和总能量的守恒性,以及和取常值g时得到的结果相一致的动、位能和动、内能之间的转换关系.还讨论了在球坐标系中应用方程组时可能出现的困难,给出了在高度近似下在该系中考虑g的空间变化的方案,它可以用来建立完全弹性的非静力模式.     

A Mixed Spectral-Integration Model for Neutral Mean Wind Flow Over Hills

A linear model for neutral surface-layer flow over orography is presented. The Reynolds-Averaged Navier-Stokes and E– turbulence closure equations are expressed in a terrain-following coordinate system created from a simple analytical expression in the Fourier domain. The perturbation equations are solved spectrally horizontally and by numerical integration vertically. Non-dimensional solutions are stored in look-up tables for quick re-use. Model results are compared to measurements, as well as other authors’ flow models in three test cases. The model is implemented and tested in two-dimensional space; the equations for a full three-dimensional version are presented.     

球坐标中原始运动方程的自由振动

利用球坐标中的原始运动方程组,讨论了大气的自由振动问题。把东、西风平均纬向风速廓线,以及与之相适应的位势高度廓线作为大气的基本状态,对线性化方程组,采用数值方法求出了特征波动解。比较了四阶差分和二阶差分的特征解,肯定了前者有较好的结果。     

The solitary wave of barotropic atmosphere on a sphere

In this paper, we have investigated large scale disturbance in barotropic atmosphere on a sphere. It demonstrates that: considering nonlinear effects of interaction between waves in barotropic vorticity equation, the wave packet of the disturbance is governed by the famous equation-nonlinear Schrodinger equation. For the solitary wave, two fac-tors are very important: one is spherical effect of the disturbance and the other is meridional shear of blocking high. In comparison with the results of local Cartesian coordinates, the former factor is the individuality of spherical soliton.     

ALow-order Model of Two-dimensional Fluid Dynamics on the Surface of a Sphere

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Application of the Energy-Casimir Method to the Study of Mesoscale Disturbance Stability

Taking into account the effect of moisture, we derive a three-dimensional pseudoenergy wave-activity relation for moist atmosphere from the primitive zonal momentum and total energy equations in Cartesian coordinates by using the energy-Casimir method. In the derivation, a Casimir function is introduced, which is a single-wlue function of virtual potential temperature. Since the pseudoenergy wave-activity relation is constructed in the ageostrophic and nonhydrostatic dynamical framework, it may be applicable to diagnosing the stability of mesoscale disturbance systems in a steady-stratified atmosphere. The theoretical analysis shows that the wave-activity relation takes a nonconservative form in which the pseudoenergy wave-activity density is composed of perturbation kinetic energy, available potential energy, and buoyant energy. The local change of pseudoenergy wave-activity density depends on the combined effects of zonal basic flow shear, Coriolis force work and wave-activity source or sink as well as wave-activity flux divergence. The diagnosis shows that horizontal distribution and temporal trend of pseudoenergy wave-activity density are similar to those of the observed 6-h accumulated surface rainfall. This suggests that the pseudoenergy wave-activity density is capable of representing the dynamical and thermodynamic features of mesoscale precipitable systems in the mid-lower troposphere, so it is closely related to the observed surface rainfall. The calculation of the terms in the wave-activity relation reveals that the wave-activity flux divergence shares a similar temporal trend with the local change of pseudoenergy wave-activity density and the observed surface rainfall. Although the terms of zonal basic flow shear and Coriolis force contribute to the local change of pseudoenergy wave-activity density, the contribution from the wave-activity flux divergence is much more significant.