大气数值预报模式中高度地形追随坐标的设计研究：理论分析与理想试验

采用一种统一的地形追随坐标的形式,对Gal-Chen & Somerville(简称Gal.C.S坐标)、平缓坐标(smoothed level vertical coordinate,简称SLEVE坐标)等几种典型的高度地形追随坐标进行了气压梯度力计算误差影响和二维质量平流试验的理论分析,并与一种新提出的高度地形追随坐标——三角函数平缓坐标(简称COS坐标)进行比较.气压梯度力计算误差分析结果显示,与Gal.C.S坐标相比,单尺度平缓坐标(简称SLEVE1坐标)、双尺度平缓坐标(简称SLEVE2坐标)和COS坐标在减小气压梯度力计算误差上有不同程度的改进,SLEVE2坐标和COS坐标比其他两种坐标更具优势,衰减系数b和坐标转换的雅可比项对减小误差起决定性作用.二维质量平流试验也有类似的结果,与无地形的参考试验结果相比,COS坐标的质量输送计算误差最小,且经优化的COS坐标的质量输送计算误差几乎和参考计算误差完全重合,在4种坐标中最优.     

逐层平滑地形的平缓地形追随坐标在高分辨率GRAPES模式中的应用研究

高分辨率GRAPES模式如3 km模式对地形的识别程度更高,模式中各高度坐标面可识别的地形坡度也更大,地形作用带来的气压梯度力计算误差和平流输送误差更突出。平缓地形追随坐标可以通过多种方式衰减坐标面上的地形影响进而减小这些计算误差。选择一种逐层平滑地形的平缓地形追随坐标,基于GRAPES-3km模式进行理想试验和批量模拟试验。试验结果显示:逐层平滑地形的平缓地形追随坐标相对其他平缓地形追随坐标对地形重力波模拟更接近解析值;24 h滚动预报月连续模拟试验中逐层平滑地形的平缓地形追随坐标一定程度上能降低高层月平均的温度场、风场的模拟误差,月平均的降水评分也有所提高。逐层平滑地形的平缓地形追随坐标应用于GRAPES-3km模式有较好的模拟效果。      

一种改进的平缓-混合地形追随坐标在GRAPES中尺度模式中的应用研究

平缓-混合地形追随坐标（T-F坐标）可以减小坐标面上的地形影响带来的各种计算误差。以余弦三角函数为基函数的平缓-混合坐标（COS坐标）高层坐标面水平,计算误差较小,但是低层坐标面之间的厚度较薄,增大了计算误差,给模式稳定性及模拟效果带来较大的影响。设计一种改进的COS坐标,使低层坐标面垂直分布更加均匀,应用于GRAPES-Meso模式进行理想试验和实际模拟试验。结果表明,改进的COS坐标相对COS坐标,中高层计算误差相当,低层地形作用衰减的垂直变化更加均匀,减小了计算误差,提高了计算稳定性;地形重力波试验结果显示,改进的COS坐标重力波破碎相对COS坐标有一定缓解,更接近解析值;批量模拟试验结果显示,改进的COS坐标各个层次上的月平均模拟偏差比单尺度双曲函数平缓-混合坐标（简称SLEVE1坐标）更小,均方根误差减小,距平相关系数增大。改进的COS坐标有效地解决了COS坐标的计算问题,提高了模式预报效果。      

Pressure gradient errors in a covariant method of implementing the σ-coordinate: idealized experiments and geometric analysis

本文针对经典σ坐标的气压梯度误差(PGF误差),采用多种地形展开理想试验,对比经典σ坐标的经典方案和协变方案的PGF误差。结果表明:计算空间中,协变方案始终能减小经典方案的误差,地形越陡,效果越明显。然而,几何分析和理想试验均表明:协变方案仅能减小计算空间的误差,不能减小物理空间的误差;相比经典方案,正交地形追随坐标能同时减小计算空间和物理空间的误差。     

地形跟随坐标下的中尺度模式气压梯度力计算误差分析及其改进方案

地形跟随坐标系中水平气压梯度力的算法一直是困扰数值模式发展的关键问题之一。目前数值模式中使用的方法只能在天气尺度的模式中部分缓解气压梯度力的计算误差问题。在高分辨率中尺度模式中，随着地形坡度的进一步加大，气压梯度力的计算误差问题更加突出。作者通过理想场的计算分析了几种主要气压梯度力算法的误差，结果显示在中尺度模式分辨率下，计算的水平气压梯度力不但不收敛于真值，而且随着地形坡度的加大或模式分辨率的提高，计算误差逐渐增大。作者提出了基于静力方程订正的回插等压面改进方案，理想场的计算结果表明该方案的计算误差可显著减小，在典型中尺度模式参数的设置下计算精度可达10-6m/s2。其最大特点是随着模式分辨率的提高，该方案的计算误差将逐步收敛到零。     

地形追随垂直运动方程在南疆极端暴雨中的诊断分析

针对2021年6月15～17日发生在昆仑山脉北坡的南疆极端暴雨过程,本文综合考虑地形对暴雨发生、发展的作用后,利用地形追随坐标控制方程并采用Boussinesq近似推导建立了地形追随坐标的非静力平衡广义垂直运动方程。诊断结果表明,经向气压梯度力耦合经向散度项（项一）、垂直气压梯度力耦合纬向散度项（项二）和非绝热加热经向梯度项（项三）是激发暴雨垂直运动发展演变的三个主要强迫项。项一体现了偏北风逐渐增强,在昆仑山脉的阻挡下导致经向辐合增强,触发了垂直上升运动。经向气流辐合始终是对流活动最主要的强迫过程,其次为纬向气流辐合。在地形追随坐标形式下,经向和垂直气压梯度能够增强项一和项二。对流发展阶段,水汽辐合与非绝热加热过程增强了非绝热加热经向梯度,促进了垂直上升运动发展。在地形的影响下,对流层中高层西风过山气流波动特征明显。重力波活动导致的高层辐散进一步促进了山脉迎风坡对流活动。经向和纬向气流辐合、非绝热加热过程以及重力波活动等多个因素共同造成了此次南疆极端暴雨。     

数值预报模式中双三次曲面地形与水平气压梯度力计算

利用C3连续双三次曲面拟合了全球数值模式地形曲面;讨论构建了有复杂地形数值模式引入地形追随高度坐标((z)坐标)后,同时引入包含定常斜率、曲率和挠率的双三次曲面地形,又进一步讨论了双三次曲面地形模式大气的水平气压梯度力计算问题.结果表明,对(z)坐标模式大气的压、温、湿场,通过做经、纬向三次样条拟合,求得地形斜率“静力平衡”气压差,从而插值(反演)任一水平面(海平面)上的气压场,同时可以求得时变的参考大气,则计算水平气压梯度(力)的精度,完全依赖于插值(反演)对应的水平面(海平面)气压场的计算精度.并指出,理论上可按三次样条的曲率判断,做变量场(地形)的局域或单点平滑.     

Characteristics of pressure gradient force errors in a terrain-following coordinate

"地形追随坐标系中气压梯度力误差的特征分析"一文通过几何分析和理想实验,对比了地形追随坐标系两种方案(经典方案和协变方案)中气压梯度力(PGF)误差的特征。结果表明:(1)经典方案的PGF误差受"垂直气压梯度","气压梯度的方向(α)","垂直层的坡度(φ)"三者影响,垂直气压梯度越大,气压梯度与水平方向的夹角越大,垂直层坡度越大,误差越大;(2)协变方案的PGF误差不受上述三因子影响。此外,通过定义参数TT(TT=tanφ·tanα)能定量分析经典方案的PGF误差。     

有地形模式中气压梯度力误差扣除法

：在现有有地形的数值模式中，陡峭地形区气压梯度力的计算存在一个普遍问题，即计算精度较高的格式比较繁琐而费时，简单省时的格式又不精确和不稳定。为解决这个问题，作者等在最近提出了一种方法，称为气压梯度力的误差扣除法。该方法假定：气压梯度力的计算误差主要来自地形而与气压形势关系较小。用理想场对该方法进行检验后表明，这一方法是成功的。本文则用气候模式对作者提出的有地形数值模式中气压梯度力的误差扣除法进行了模拟检验。所用模式是作者等使用多年的Ｐ－混合坐标系５层模式，选用了四种气压梯度力的计算格式，即ＤＤＤ格式、Ｃｏｒｂｙ格式、平均温度格式和经典中央差格式。比较了这四种格式在有无误差扣除时的模拟结果，发现：对于计算精度较高的格式，如ＤＤＤ格式、Ｃｏｒｂｙ格式及平均温度格式，有无误差扣除的结果相差不大，但误差扣除法仍可在一定程度上改善模拟效果。对于计算精度差的格式，如经典中央差格式，在无误差扣除时计算不稳定，得不到模拟结果，进行误差扣除后，从根本上提高了其计算精度，因而也提高了计算的稳定性，达到了较满意的模拟效果。而且与其它格式的模拟结果相当接近。本文提出的误差扣除法可同时用于格点模式和谱模式。     

非静力中尺度高分辨率模式模拟中的垂直坐标影响研究

近几年来,随着高性能计算机技术的高速发展,甚高分辨率的中尺度数值预报模式业务应用已成为可能,与之相关的一系列模式技术的新问题也随之提了出来,垂直坐标系的影响就是其中之一。文中借助于美国新一代数值预报模式WRF(WeatherReseachandForecast),比较了非静力中尺度模式高分辨率模拟应用的垂直坐标影响问题。研究表明,当选用几何高度(z)和气压(p)来构造地形追随坐标时,低层两坐标引起的误差基本一样,中高层高度地形追随坐标引起的误差小于气压地形追随坐标;而且分辨率越高差异愈大。高分辨率模拟结果也表明,这种差异趋势是存在的;此外,本文对天气过程的预报要素场进行了相关的分析。     

Effects of terrain-following vertical coordinates on high-resolution NWP simulations

With increasing resolution in numerical weather prediction(NWP)models,the model topography can be described with finer resolution and includes steeper slopes.Consequently,negative effects of the traditional terrain-following vertical coordinate on high-resolution numerical simulations become more distinct due to larger errors in the pressure gradient force(PGF)calculation and associated distortions of the gravity wave along the coordinate surface.A series of numerical experiments have been conducted in this study,including idealized test cases of gravity wave simulation over a complex mountain,error analysis of the PGP estimation over a real topography,and a suite of real-data test cases.The GRAPES-Meso model is utilized with four different coordinates,i.e.,the traditional terrain-following vertical coordinate proposed by Gal-Chen and Somerville(hereinafter referred to as the Gal.C.S coordinate),the one-scale smoothed level(SLEVE1),the two-scale smoothed level(SLEVE2),and the COSINE(COS)coordinates.The results of the gravity wave simulation indicate that the GRAPES-Meso model generally can reproduce the mountain-induced gravity waves,which are consistent with the analytic solution.However,the shapes,vertical structures,and intensities of the waves are better simulated with the SLEVE2 coordinate than with the other three coordinates.The model with the COS coordinate also performs well,except at lower levels where it is not as effective as the SLEVE2 coordinate in suppressing the PGF errors.In contrast,the gravity waves simulated in both the Gal.C.S and SLEVE1 coordinates are relatively distorted.The estimated PGF errors in a rest atmosphere over the real complex topography are much smaller(even disappear at the middle and upper levels)in the GRAPES-Meso model using the SLEVE2 and COS coordinates than those using the Gal.C.S and SLEVE1 coordinates.The results of the real-data test cases conducted over a one-month period suggest that the three modified vertical coordinates(SLEVE1,SLEVE2,and COS coordinates)give better results than the traditional Gal.C.S coordinate in terms of forecasting bias and root mean square error,and forecasting anomaly correlation coefficients.In conclusion,the SLEVE2 coordinate is proved to be the best option for the GRAPES-Meso model.     

Testing a radiative upper boundar condition in a nonlinear model with hybrid vertical coordinate

Summary The application of a radiative upper boundary condition (RUBC) in a mesoscale numerical weather prediction (NWP) model with hybrid vertical coordinate is presented. Results of two- and three-dimensional numerical simulations are discussed. Starting from earlier work by Klemp and Durran (1983) and Bougeault (1983) the radiative upper boundary condition is formally derived for a hybrid vertical coordinate. The basic assumptions include hydrostaticity, linearity, neglect of Coriolis effects and restriction to internal gravity waves. The resulting RUBC is global in space and local in time. In a second step. the RUBC is tested in a twodimensional vertical-plane version of the NWP model, in which essential properties of the full three-dimensional model have been preserved. Gravity wave experiments clearly show the superiority of the RUBC over the commonly used lid-type upper boundary condition. For the setting with an isolated bell-shaped mountain with resolution-independent steepness, the RUBC tends to work more effectively with increasing horizontal resolution. At the same time, the application of a radiative instead of a lid-type, and thus reflecting, upper boundary condition appears to become more important with decreasing mesh width. Finally, the RUBC is introduced into the full three-dimensional NWP model. This requires further approximations. In particular for a limited-area model, the geopotential field at the uppermost model level needs to be bi-periodic. Here, a linear detrending technique is applied. First results for a weather situation with strong northwesterly flow towards the Alps show that application of the RUBC drastically reduces the development of unrealistic standing, hydrostatic mountain waves, which become apprent as distinct mesoscale ridge-trough structures in the simulation with the lid-type upper boundary. Implications of the RUBC on the time-stepping procedure of the NWP model are also discussed. In the experiments whown, the additional RUBC-terms are treated explicitly.     

水平气压梯度力差分格式误差的定量分析

通过分析发现地形坐标系中气压梯度力差分格式的计算误差应分为二类，其中，第二类误差在水平坐标面倾斜时出现。这种误差可理解为将水平坐标面上的气压(或位势高度)插回到等高面(或等压面)上的插值误差。 构造气压梯度力差分格式的目的正是为了减小这种第二类误差，而不是其他误差。 误差分析表明静力扣除法和Corby 格式的第二类误差都很小，比一般方法的误差小约一个量级。分析还表明第二类误差产生的主要原因是由于气压随高度非线性变化。依据这一分析思路，本文改进了#As(z)#a坐标系中的静力扣除法。 改进后的方法比改进前的误差小约一个量级。     

Blocking in periodic valleys

Results are presented from a study of blocked flow (practically stagnant or recirculating light winds) in periodic valleys in thermally stably stratified ambient conditions. Inviscid and turbulent diffusion cases were modelled numerically to clarify the effects of turbulence on the blocking. The reflection of gravity waves from the top boundary of the hydrostatic model atmosphere was avoided by employing the radiation condition given by Klemp and Durran (1983). The dissipative numerical results are compared with new laboratory experiments which utilized the technique of Baines and Hoinka (1985) to simulate a semi-infinitely deep region.A criterion for the occurrence of blocked flow cannot be defined for the inviscid case except when the Froude number, Fr, based on the peak-to-trough ridge amplitude is less than about 0.4: then blocking is clearly identifiable before wave-breaking occurs. Breaking of waves is evident for Fr as large as 0.75, in agreement with analytical results given by Lilly and Klemp (1979).At small Froude number (Fr 0.5) in the dissipative flow simulations, blocked flow (stagnation) is present in the valleys, but a lee rotor (complete stagnation) is not evident. For order unity Froude numbers, blocking is a wave phenomenon, resulting from wave steepening and overturning or turbulent mixing. A finite thickness is brought to rest or participates in a recirculating flow when it first appears. A strong upward flow appears ahead of the rotor in the valleys, and the downslope wind over the windward side of the valleys is strengthened. Thus the present study shows that conditions for the onset of a rotor, and of stagnant flow, in periodic valleys are different.When blocked flow exists, the amplitudes of gravity waves in the upper layer are only 15% (Fr = 0.3) to 80% (Fr = 1.5) of those given by linear theory; this is supported by observations.     

地形区两种典型气压梯度力计算方法的比较

试验比较了两种较为典型的地形区气压梯度力计算方法:一是先在准水平的等压面上计算，再垂直插值到地形坐标面上的方法，二是差-微-差方法。初始化分析和预报结果检验都表明，在减轻地形影响上，前者的优势在大气上层，后者则在中下层。     

Diagnosis of the forcing of inertial-gravity waves in a severe convection system

The non-hydrostatic wave equation set in Cartesian coordinates is rearranged to gain insight into wave generation in a mesoscale severe convection system. The wave equation is characterized by a wave operator on the lhs, and forcing involving three terms—linear and nonlinear terms, and diabatic heating—on the rhs. The equation was applied to a case of severe convection that occurred in East China. The calculation with simulation data showed that the diabatic forcing and linear and nonlinear forcing presented large magnitude at different altitudes in the severe convection region. Further analysis revealed the diabatic forcing due to condensational latent heating had an important influence on the generation of gravity waves in the middle and lower levels. The linear forcing resulting from the Laplacian of potential-temperature linear forcing was dominant in the middle and upper levels. The nonlinear forcing was determined by the Laplacian of potential-temperature nonlinear forcing. Therefore, the forcing of gravity waves was closely associated with the thermodynamic processes in the severe convection case. The reason may be that, besides the vertical component of pressure gradient force, the vertical oscillation of atmospheric particles was dominated by the buoyancy for inertial gravity waves. The latent heating and potential-temperature linear and nonlinear forcing played an important role in the buoyancy tendency. Consequently, these thermodynamic elements influenced the evolution of inertial-gravity waves.     

数值模式中气压梯度力的算法试验

本文在差微差一致性二阶精度的坐标变换公式的基础上,推导出了气压梯度力计算的新公式。同时,根据静力扣除法原理,提出了初值静力扣除法,用以计算有地形数值模式中的气压梯度力。将这两种新的方法与二阶计算精度的回插法和局地等温法(Corby格式)相比后发现:新方法(文中称为经典修正法)与回插法精度相近,但可节省计算时间,比局地等温法精度高。静力扣除法则效果较差,但其原因可能在于地转风垂直插值方法有误差。本文的试验模式是上海台风研究所的业务数值预报模式,文中对模式及若干技术处理也作了简要介绍。     

A note on hydraulic theory of internal bores

Mass and momentum conservation across an internal bore, together with an assumption that energy is dissipated in both fluid layers, yields a range of possible bore speeds. The upper speed limit is that given by Wood and Simpson [Wood, I.R., Simpson, J.E., 1984. Jumps in layered miscible fluids. J. Fluid Mech., 140: 215–231.] who assume no energy dissipation in the contracting layer, while the lower limit is that of Klemp et al. [Klemp, J.B., Rotunno, R., Skamarock, W.C., 1997. On the propagation of internal bores. J. Fluid Mech., 331: 81–106] who assume no energy dissipation in the expanding layer. The two bore speeds agree to within a few percent, except when the expanding layer is shallow upstream and the internal bore propagates as a gravity current.