Computing Streamfunction and Velocity Potential in a Limited Domain of Arbitrary Shape. Part II: Numerical Methods and Test Experiments

Built on the integral formulas in Part I,numerical methods are developed for computing velocity potential and streamfunction in a limited domain.When there is no inner boundary(around a data hole) inside the domain,the total solution is the sum of the internally and externally induced parts.For the internally induced part,three numerical schemes(grid-staggering,local-nesting and piecewise continuous integration) are designed to deal with the singularity of the Green’s function encountered in numerical calculations.For the externally induced part,by setting the velocity potential(or streamfunction) component to zero,the other component of the solution can be computed in two ways:(1) Solve for the density function from its boundary integral equation and then construct the solution from the boundary integral of the density function.(2) Use the Cauchy integral to construct the solution directly.The boundary integral can be discretized on a uniform grid along the boundary.By using local-nesting(or piecewise continuous integration),the scheme is refined to enhance the discretization accuracy of the boundary integral around each corner point(or along the entire boundary).When the domain is not free of data holes,the total solution contains a data-hole-induced part,and the Cauchy integral method is extended to construct the externally induced solution with irregular external and internal boundaries.An automated algorithm is designed to facilitate the integrations along the irregular external and internal boundaries.Numerical experiments are performed to evaluate the accuracy and efficiency of each scheme relative to others.     

有限区域风速场求解流函数和速度势场的有效方案

流函数和速度势是表示风场的一种变量, 在数值天气预报模式和分析、同化方案中经常使用, 通常可以用风速分量场求解Poisson方程得到。对于有限区域系统, 往往采用差分方法, 但由于存在边界问题, 用计算所得到的流函数和速度势场重建风速场, 在边界附近经常出现明显的偏差。基于差分方法、利用有限区域风速场求解流函数和速度势场的基本方法和特点的分析, 在Arakawa A网格分布的有限区域, 设计了一种用差分方法求解流函数和速度势场的有效方案。在该有效方案中, 通过将有限区域向外扩展二圈, 风速场线性外推, 改进计算边界风速值和边界定解条件的效果; 尽可能使用协调、一致的差分格式, 提高求解精度; 最后利用一种增量订正迭代方法, 迭代2~3次就可以获得令人满意的结果。实例试验的对比、检验显示, 用该方案计算求得的流函数和速度势场重建风速场, 具有非常高的精度。     

Hybrid Methods for Computing the Streamfunction and Velocity Potential for Complex Flow Fields over Mesoscale Domains

Three types of previously used numerical methods are revisited for computing the streamfunction ψ and velocity potential χ from the horizontal velocity v in limited domains. The first type, called the SOR-based method, uses a classical successive over-relaxation (SOR) scheme to compute ψ (or χ) first with an arbitrary boundary condition (BC) and then χ (or ψ) with the BC derived from v . The second type, called the spectral method, uses spectral formulations to construct the inner part of (ψ, χ)—the inversion of (vorticity, divergence) with a homogeneous BC, and then the remaining harmonic part of (ψ, χ) with BCs from v . The third type, called the integral method, uses integral formulas to compute the internally induced (ψ, χ)—the inversion of domain-internal (vorticity, divergence) using the free-space Green?s function without BCs and then the remaining harmonic ψ (or χ) with BCs from v minus the internally-induced part. Although these methods have previously been successfully applied to flows in large-scale and synoptic-scale domains, their accuracy is compromised when applied to complex flows over mesoscale domains, as shown in this paper. To resolve this problem, two hybrid approaches, the integral-SOR method and the integral-spectral method, are developed by combining the first step of the integral method with the second step adopted from the SOR-based and spectral methods, respectively. Upon testing these methods on real-case complex flows, the integral-SOR method is significantly more accurate than the integral-spectral method, noting that the latter is still generally more accurate than the three previously-used methods. The integral-SOR method is recommended for future applications and diagnostic studies of complex flows.     

Subsidence fronts

Approximate analytical solution of the coupled momentum and heat transfer equations are obtained, using an integral method, for natural convection boundary layers which are produced by a step change of surface temperature. The accuracy of the laminar solution is shown to be good, especially at high Prandtl Number (Pr). The corresponding solutions for the turbulent fluid yield a model of a boundary layer in which the horizontal flow strengthens downstream and is fed by downward motion from above. We call this system, which could possibly simulate the initiation of a sea breeze circulation, a subsidence front. Interesting properties of the solution include the control of the boundary-layer development by the product (PrS), whereS is the slope of the boundary layer, and also the generation of log-linear velocity and temperature profiles by heat flux divergence.     

基于Endlich迭代调整的全球流函数和速度势计算及风场分解与重建

本文将有限区域求解流函数和速度势的Endlich迭代调整方法(E-T方法)推广至全球区域,并与常用Guass-Seidel迭代求解Poisson方程方法(G-P方法)进行了比较。结果表明:E-T方法适应于全球流函数和速度势的计算,其由内向外调整,不需严格考虑边界条件,可消除边界对计算结果的影响;E-T方法能准确分解和重建原始风场,而由于极地边界和差分格式影响,G-P方法求得的旋转风和辐散风之和不能准确重建原始风场,尤其是经向为固定边界时,两极地附近重建风场误差非常明显;E-T方法与G-P方法另一重要差异是前者利用流函数和速度势与风场偏微分关系直接进行迭代求解,不需计算涡度和散度,不但保证了重建风场的准确性,还防止了涡度和散度计算误差带来的二次污染。     

Recent Developments in Computing Streamfunction and Velocity Potential in a Limited Domain

After first reviewing historical and current difficulties in solving streamfunction and velocity potential in a limited domain,and describing recent developments in obtaining accurate solutions in a limited domain with arbitrary shape,a newly proposed approach is introduced and its application to a torrential rain event is reported.The results show that the newly developed method has advantages in capturing mesoscale information,compared with horizontal winds.     

斜压性对地转动量Ekman流的影响——理论分析

本文研究了斜压效应对地转动量Ekman流的影响。利用两变量奇异摄动方法求得了边界层中风场及顶部垂直速度的前二级一致有效渐近解析解,解中明显地反映了斜压情形地转风随高度变化(即热成风)的影响,尤其是其中一级近似解完全由热成风影响所致。在边界层顶垂直速度的解中导出了三种由斜压效应引起的Ekmon抽吸新物理因子,即热成风形变、热成风涡度及热成风涡度交叉项等抽吸因子。分析表明,这些因子只在具有水平温度梯度不均匀的系统(譬如锋区)中方能出现。文中还对锋区内上述三种抽吸因子的动力特征作了具体的分析,指出在锋区这样的强斜压系统中,此三种抽吸因子的贡献是显著的。下一文中,我们将利用本文所得理论解对斜压效应进行具体的定量计算。      

Uniqueness of solutions and conservation laws for the quasigeostrophic model

In this study we investigate which conditions are needed to assure uniqueness of solutions of the barotropic QG model, and, how these conditions are related to the conservation laws of mass, vorticity and energy. Uniqueness and conservation laws are analyzed for a simply connected domain (a closed basin) and for a doubly connected domain (a periodic or re-entrant channel).For the multiply connected domain we find, besides the model proposed by McWilliams in 1977, another consistent model whose solutions satisfy the conservation laws. The additional conditions in our model are: (a) the sum of the circulations around all closed solid walls is time-independent; (b) the value of the QG streamfunction (Φ) is the same at all closed walls. McWilliams' model and ours are not equivalent.As a simple application we study the free Rossby normal modes in a channel. For a non-zonal channel on a β-plane there are solutions (modes) that are independent of the coordinate along the channel. These are used to compare the modes and frequencies obtained from three different, but well posed, models. Solutions that are independent of the along-channel coordinate do not exist for the planetary geostrophic and for the shallow water equations on a β-plane.     

山地边界层中场变量的动力学研究

采用与实测较接近的二次函数来表达Ｅｋｍａｎ层中的湍流粘性系数Ｋ，在圆形气压场条件下，求得了山地上空边界层中的风速，进而求得散度、涡度和垂直速度等场变量随高度的分布。并作图分析了这些场变量的一些动力学特征。改进了以往在求解析解时，略去运动方程中湍流粘性力项中的关于高度的一阶导数项，以及取山坡面上风速为零作下边界条件等欠合理欠精确的做法。所求得的风速、散度、涡度和垂直速度均用简单的初等函数表示出来，有助于边界层参数化和深化对边界层动力学的认识。     

The interaction of ellipsoidal vortices with background shear flows in a stratified fluid

We show that an arbitrarily oriented ellipsoid of uniform potential vorticity, embedded in a background flow described by a quadratic streamfunction, is an exact solution of the quasigeostrophic equations governing motion in a uniformly stratified, unbounded fluid. This type of flow includes plane horizontal shear and strain as well as uniform vertical shear of a unidirectional horizontal flow. We derive ordinary differential equations describing the motion of such a vortex and discuss some aspects of their solutions. We note the existence of steady states (solutions in which the vortex is in equilibrium with the background flow), of periodic solutions near these steady states, of non-periodic trajectories which nervertheless remain in the vicinity of the steady states, and of solutions which represent the shearing out of the vortex by the background flow. We try to use this information to propose partial answers to the question of when a given horizontal or vertical shear flow is likely to destroy a vortex and when a vortex might survive external shear and strain.     

The Ekman boundary layer over orography: An analysis of vertical motion

A model of the planetary boundary layer is used to determine the field of vertical motion over large-scale orography. This model represents Ekman boundary-layer dynamics modified by the inclusion of accelerations of the geostrophic wind under the geostrophic momentum approximation. The orography is represented by a circular mountain. The inviscid solution is provided by the sum of a constant translation and a steady, uniform potential vorticity, anticyclonic vortex. The boundary-layer solution vanishes on the mountain, but is matched to the inviscid solution as the top of the boundary layer is approached. The vertical velocity field at the top of the boundary layer is determined by integration of the continuity equation. The field of motion is largely determined by descent from above into the anticyclonic circulation, as in the classical Ekman model. Contributions that arise from the inclusion of accelerations are associated with boundary-layer advection and ageostrophic divergence that produce vorticity tendencies. Finally, the boundary-layer vertical motion is shown to be comparable in magnitude to the vertical motion forced by inviscid flow over the orography, although the distributions of each are significantly different. Effects of mountain asymmetry and a changing pressure field, that can be treated more fully by numerical model simulations, are not considered in the present study.On leave at the University of Colorado, 1990.     

INERTIAL THEORY FOR OCEANIC CROSS-EQUATORIAL JET

In this paper, a theory for inertial boundary layer is developed to study the northward forcedand the southward induced cross-equatorial jets by easterlies in the Southern Hemisphere in westand east boundaries respectively. The solutions of the nonlinear potential vorticity equation and theBernoulli equation show that subtle changes of the physical parameters can produce different equi-librium states and the velocity of these jets all can reach 10°m/s which are comparable to observa-tions. Simulations show that the induced current in the east boundary is very important to the for-mation of the local eastward equatorial current in the east boundary. This can be used to study thecharacteristics of the local surface eastward equatorial current during E1 Nino periods,     

Simple applications of McWilliams' “A note on a consistent quasigeostrophic model in a multiply connected domain”

We have solved several linear quasigeostrophic problems in singly and multiply connected domains to illustrate the use and effects of correct boundary conditions which balance circulation changes with forcing along each boundary. The results with these boundary conditions differ markedly from those obtained by setting the streamfunction to zero on the walls either when the fluid is forced by a time-dependent wind stress curl or when the fluid oscillates in a free basin mode in a domain which contains interior islands.     

Midlatitude unstable air-sea interaction with atmospheric transient eddy dynamical forcing in an analytical coupled model

While recent observational studies have shown the critical role of atmospheric transient eddy (TE) activities in midlatitude unstable air-sea interaction, there is still a lack of a theoretical framework characterizing such an interaction. In this study, an analytical coupled air-sea model with inclusion of the TE dynamical forcing is developed to investigate the role of such a forcing in midlatitude unstable air-sea interaction. In this model, the atmosphere is governed by a barotropic quasi-geostrophic potential vorticity equation forced by surface diabatic heating and TE vorticity forcing. The ocean is governed by a baroclinic Rossby wave equation driven by wind stress. Sea surface temperature (SST) is determined by mixing layer physics. Based on detailed observational analyses, a parameterized linear relationship between TE vorticity forcing and meridional second-order derivative of SST is proposed to close the equations. Analytical solutions of the coupled model show that the midlatitude air-sea interaction with atmospheric TE dynamical forcing can destabilize the oceanic Rossby wave within a wide range of wavelengths. For the most unstable growing mode, characteristic atmospheric streamfunction anomalies are nearly in phase with their oceanic counterparts and both have a northeastward phase shift relative to SST anomalies, as the observed. Although both surface diabatic heating and TE vorticity forcing can lead to unstable air-sea interaction, the latter has a dominant contribution to the unstable growth. Sensitivity analyses further show that the growth rate of the unstable coupled mode is also influenced by the background zonal wind and the air–sea coupling strength. Such an unstable air-sea interaction provides a key positive feedback mechanism for midlatitude coupled climate variabilities.

     

边界层过程对暴雨影响的敏感性试验

用MM5v3模式对江淮暴雨进行了计入与不计入边界层和计入与不计入地面通量的敏感性试验,结果指出二者的影响是一致的,地面通量是边界层影响的主要部分。一般说,不计边界层影响,则湿位涡、水汽通量散度、涡度、低空急流等分布发生变化,其中心位置改变,强度也有改变(大部分减弱),造成总的降水减弱和降水位置的改变。边界层正是通过湿位涡、水汽通量散度、涡度、低空急流等因素的综合影响来影响暴雨,并影响暴雨的落区和强度。     

用自由模对1998年夏季阻塞形势的诊断分析

签于大气自由模和阻塞高压的重要性 ,首先简要介绍阻塞高压与中国夏季降水的关系及 1998年夏季 5 0 0hPa异常环流场 ;然后利用 2 1波菱形截断的谱方法求出无粘性、无强迫正压涡度方程的自由解 ,即和实际大气流场十分接近的大气自由模 ,并讨论其特征 ;最后利用大气自由模的特征诊断分析了实际大气的环流特征 ,并重点用其分析了 1998年夏季的大气环流异常形势。结果表明大气自由模能很好地反映大气环流大尺度特征 ,而略去了中、小尺度涡旋 ;大气自由模的流场和绝对涡度场之间的散点图反映了 (q ,Ψ )的函数关系是分段线性或非线性的 ,尽管实际流场与绝对涡度场之间的点阵图反映两者之间成不同斜率的线性函数关系 ,但点阵比较发散 ,不能突出某些重要系统 ;1998年夏季实际大气自由模态突出了大气环流的重要特征 ,例如 ,越赤道气流、西风急流、副热带高压、阻塞高压及两极的边界效应等 ;通过自由模态的特征可以识别出纬向流及阻塞高压的持续异常 ,利用自由模的稳定性判断阻塞高压的稳定性 ,不稳定模态可以维持一段较长的时间 ,且由于这些不稳定的模存在 ,才使 1998年 6月中旬到下旬的中高纬阻塞高压稳定维持。     

BAROCLINITY AND NONLINEAR EKMAN PUMPING DYNAMICS

With the Ekman momentum approximation,the influence of atmospheric baroclinity on the dynamics of boundarylayer is studied.Some new results are obtained.These results show that the atmospheric baroclinity plays an importantrole in altering the horizontal velocity of Ekman boundary layer and its angle with the horizontal wind velocity compo-nent near the surface.There are three different physical factors affecting the nonlinear Ekman suction,the vertical mo-tion at the top of boundary layer:first,barotropic geostrophic relative vorticity at the ground;second,the thermal windvorticity induced by the baroclinity;and third,the nonlinear interaction between the barotropic geostrophic relativevorticity and the baroclinic thermal wind vorticity.These results may provide a better physical basis for theparameterization of boundary layer and the interpretation of the numerical modeling results.     

中尺度涡旋的雷达探测和分析

应用单部多普勒天气雷达观测资料对出现在京津冀地区的中尺度涡旋进行了分析，分析结果表明:发展较强的中小尺度天气系统往往与风场的旋转或辐合辐散有关，气旋性涡度场对积云对流活动有明显的组织和增强作用，是降雨增强的一种判据。     

Regional Climate Effects of Internally and Externally Mixed Aerosols over China

A regional climate model is employed to simulate the aerosols(dust,sulfate,black carbon,and organic carbon) and their direct effect on the climate over China.The emphasis is on the direct radiative forcing due to the change in mixing state of aerosols.The results show that direct radiative forcing is significantly different between externally and internally mixed aerosols.At the top of the atmosphere(TOA),the radiative forcing of externally mixed aerosols is larger than that of internally mixed ones,especially in the Tarim desert region where the difference is about 0.7 W m 2.At the surface,however,the situation becomes opposite,especially in the Sichuan basin where the difference is about-1.4 W m 2.Nonetheless,either externally or internally mixed aerosols in China can result in a significant cooling effect,except for the warming in South China in winter and the slight warming in North China in February.The cooling effect induced by externally mixed aerosols is weaker than that induced by internally mixed aerosols,and this is more obvious in spring and winter than in summer and autumn.In spring and summer,the inhibiting effect of externally mixed aerosols on precipitation is less than that of internally mixed aerosols,whereas in autumn and winter the difference is not obvious.     

Haurwitz solutions of the nonlinear shallow-water equations for small froude number

Summary In this note we find special solutions of the nonlinear shallow-water equations. From the first integrals of the potential vorticity and energy equations for steady flow we derive a single equation in the streamfunction. In the limiting case of very small Froude number, that equation has no solutions corresponding to gravity waves. Under a nonlinear transformation of dependent variable, it becomes a linear equation whose solutions are related to Haurwitz's solutions for nondivergent barotropic flow in spherical surfaces. The distinguishing feature of these solutions is that the streamlines coincide with contours of the free surface: thus, although the height of the free surface varies, the motions of the fluid are horizontal, and the flow is nondivergent.The solutions are easily modified to correspond to Rossby waves propagating eastward or westward without change of shape.The National Center for Atmospheric Research is sponsored by the National Science Foundation.This paper is dedicated to the memory of my scientific mentor and old friend, Bernhard Haurwitz.