﻿ 非静力AREM模式设计及其数值模拟Ⅰ:非静力框架设计 非静力AREM模式设计及其数值模拟Ⅰ:非静力框架设计
 大气科学  2018, Vol. 42 Issue (6): 1286-1296 PDF

1 中国科学院大气物理研究所大气科学和地球流体力学数值模拟国家重点实验室(LASG), 北京 100029
2 地理信息工程国家重点实验室, 西安 710054

Design of Non-hydrostatic AREM Model and Its Numerical Simulation Part Ⅰ: Design of Non-hydrostatic Dynamic Core
CHENG Rui1,2, YU Rucong1, XU Youping1, WANG Bin1
1 State Key Laboratory of Numerial Modeling for Atmospheric Sciences and Geophysical Fluid Dynemics(LASG) Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029
2 State Key Laboratory of Geo-Information Engineering, Xi'an 710054
Abstract: The Advanced Regional Eta-coordinate Model (AREM) is featured as a useful tool to simulate and forecast meso-scale systems such as torrential rainfall and typhoons in China. However, the hydrostatic approximation has curbed its further development, which is more and more noticeable with the increasingly high model resolution. In this paper, the authors present a non-hydrostatic extension to AREM through the consideration of higher-order correctness due to the vertical acceleration. The AREM non-hydrostatic dynamics employs a primitive Euler equation system of motion and effectively uses the deduction of standard stratification and IAP (Institute of Atmospheric Physics) transformation of the current hydrostatic model. Also, the non-hydrostatic version of AREM is formulated in the spherical colatitude-longitude mesh and discretized in the Arakawa-E grid and a vertical η coordinate system. The prognostic equations are split into two parts, that is, the quasi-hydrostatic system and non-hydrostatic system, which facilitates efficient integration of the dynamic core. The sound wave associated with the non-hydrostatic system is calculated through the Thomas algorithm and iteration method. This approach to non-hydrostatic modeling is favorable for the preservation of advantages of hydrostatic AREM. The non-hydrostatic and hydrostatic frames agree well with each other in terms of governing equations and modeling results when the non-hydrostatic core is degraded to the hydrostatic one. In part Ⅱ of this paper, the non-hydrostatic AREM will be verified through idealized and real-data numerical experiments.
Keywords: AREM model    Non-hydrostatic dynamic core    E-grid    ETA coordinate    IAP transformation
1 引言

MM5（Pennsylvania State University-NCAR Mesoscale Model）的非静力版本（Dudhia, 1993）构建出发点即要能与静力版本兼容和一致。一些计算模块如水汽处理、平流、扩散、辐射、边界层、陆面、对流参数化等经过很少改动即可在非静力模式中得到使用。这种作法的优势在于能够尽可能地保留静力平衡模式的优势，特别是在较大尺度非静力模拟之时。Dudhia（1993）对静力框架的修正主要包括：其一、增加半隐式声波处理，替换气压—动量计算关系；其二、增加垂直运动和气压扰动预报方程。

2 非静力方程组 2.1 基本动力方程组

2.1.1 $\eta$坐标定义

 $\eta =\sigma \cdot {{\eta }_\text{s}}=\frac{\pi -{{\pi }_\text{t}}}{{{\pi }_\text{s}}-{{\pi }_\text{t}}}{{\eta }_\text{s}},$ (1)

 ${{\eta }_\text{s}}=\frac{{{\pi }_\text{rf}}({{z}_\text{s}})-{{\pi }_\text{t}}}{{{\pi }_\text{rf}}({{z}_\text{b}})-{{\pi }_\text{t}}},$ (2)

 ${{P}^{2}}=\frac{{{\pi }_\text{s}}-{{\pi }_\text{t}}}{{{\eta }_\text{s}}},$ (3)

 $\eta =\frac{\pi -{{\pi }_\text{t}}}{{{P}^{2}}}.$ (4)

2.1.2 静力平衡方程推导

 $\frac{\partial \mathit{\Phi } }{\partial \pi }=-\alpha,$ (5)

 $\frac{\partial \mathit{\Phi } }{\partial \eta }={{P}^{2}}\frac{\partial \mathit{\Phi } }{\partial \pi }=-{{P}^{2}}\frac{RT}{p},$ (6)

2.1.3 垂直动量方程推导

 $\frac{dw}{dt}=-\alpha \frac{\partial p}{\partial z}-g=-\alpha g\frac{\partial p}{\partial \mathit{\Phi } }-g=-g+g\frac{\partial p}{\partial \pi },$ (7)

 $\varepsilon =\frac{\text{1}}{g}\frac{\text{d}w}{\text{d}t},$ (8)

 $\frac{\partial p}{\partial \pi }=\text{1}+\varepsilon .$ (9)
2.1.4 非静力连续方程
 $w=\frac{\text{d}z}{\text{d}t}=\frac{\text{1}}{g}\frac{\text{d}\mathit{\Phi } }{\text{d}t}=\frac{\text{1}}{g}\left(\frac{\partial \mathit{\Phi } }{\partial t}+V\cdot {{\nabla }_{\eta }}\mathit{\Phi } + \dot{\eta }\frac{\partial \mathit{\Phi } }{\partial \eta } \right),$ (10)

2.1.5 非静力框架下静力气压表达的连续方程推导

 ${{\left[ \frac{\partial }{\partial t}\left( \rho \frac{\partial z}{\partial s} \right) \right]}_{s}}+{{\nabla }_{s}}\cdot \left( \rho \boldsymbol {V}\frac{\partial z}{\partial s} \right)+\frac{\partial \left( \rho \dot{s}\frac{\partial z}{\partial s} \right)}{\partial s}=\text{0},$ (11)

 $\frac{\partial }{\partial t}\left(\rho \frac{\partial \mathit{\Phi } }{\partial \eta } \right)+{{\nabla }_{\eta }}\cdot \left(\rho V\frac{\partial \mathit{\Phi } }{\partial \eta } \right)+\frac{\partial \left(\rho \dot{\eta }\frac{\partial \mathit{\Phi } }{\partial \eta } \right)}{\partial \eta }=\text{0}.$ (12)

$\partial \mathit{\Phi } /\partial \eta =-{{P}^{\text{2}}}\alpha$代入上式，可得

 $\frac{\partial {{P}^{\text{2}}}}{\partial t}+{{\nabla }_{\eta }}\cdot ({{P}^{\text{2}}}\boldsymbol {V})+\frac{\partial ({{P}^{\text{2}}}\dot{\eta })}{\partial \eta }=\text{0}.$ (13)

 $\frac{\partial {{P}^{\text{2}}}}{\partial t}=-\frac{\text{1}}{{{\eta }_{s}}}\int{{{\nabla }_{\eta }}\cdot ({{P}^{\text{2}}}\boldsymbol {V})\text{d}\eta }.$ (14)

 ${{P}^{\text{2}}}\dot{\eta }=-\eta \frac{\partial {{P}^{\text{2}}}}{\partial t}-\int{{{\nabla }_{\eta }}\cdot ({{P}^{\text{2}}}\boldsymbol {V})\text{d}\eta }.$ (15)
2.1.6 气压梯度力推导
 \begin{align} &{{\nabla }_{z}}p={{\nabla }_{\eta }}p-\frac{\partial p}{\partial \pi }{{P}^{\text{2}}}\frac{\partial \eta }{\partial z}{{\nabla }_{\eta }}z={{\nabla }_{\eta }}p-\left(\text{1}+\varepsilon \right){{P}^{\text{2}}}\frac{\partial \eta }{\partial \mathit{\Phi } }{{\nabla }_{\eta }}\mathit{\Phi } \\ &\;\;\;\;\;\;\;\;={{\nabla }_{\eta }}p+\left(\text{1}+\varepsilon \right)\frac{\text{1}}{\alpha }{{\nabla }_{\eta }}\mathit{\Phi }, \\ \end{align} (16)

 $\frac{\text{d}\mathit{\boldsymbol{V}}}{\text{d}t}=-\left(\text{1}+\varepsilon \right){{\nabla }_{\eta }}\mathit{\Phi }-\alpha {{\nabla }_{\eta }}p-f\mathit{\boldsymbol{k}}\times \mathit{\boldsymbol{V}},$ (17)

2.1.7 热力学方程推导

 ${{c}_{p}}\frac{\text{d}T}{\text{d}t}=\alpha \frac{\text{d}p}{\text{d}t},$ (18)

 \begin{align} &\frac{\partial T}{\partial t}=-\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}T-\dot{\eta }\frac{\partial T}{\partial \eta }+\frac{\alpha }{{{c}_{p}}}\frac{\text{d}p}{\text{d}t}=-\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}T-\dot{\eta }\frac{\partial T}{\partial \eta }+ \\ &\;\;\;\;\;\frac{\alpha }{{{c}_{p}}}\omega =-\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}T-\dot{\eta }\frac{\partial T}{\partial \eta }+\frac{\alpha }{{{c}_{p}}}({{\omega }_{\text{1}}}+{{\omega }_{\text{2}}}), \\ \end{align} (19)

 $p=p\left[ x, y, \pi \left(x, y, z, t \right), t \right],$ (20)

 $\frac{\partial p}{\partial t}=\frac{\partial p}{\partial {{t}_{\pi }}}+{{\frac{\partial p}{\partial \pi }}_{t}}\cdot \frac{\partial \pi }{\partial t}=\frac{\partial p}{\partial {{t}_{\pi }}}+(\text{1}+\varepsilon)\frac{\partial \pi }{\partial t}.$ (21)

 $\dot{\eta }\frac{\partial p}{\partial \eta }=\dot{\eta }\frac{\partial p}{\partial \pi }\frac{\partial \pi }{\partial \eta }=\dot{\eta }(\text{1}+\varepsilon)\frac{\partial \pi }{\partial \eta },$ (22)

 \begin{align} &\frac{\text{d}p}{\text{d}t}=\frac{\partial p}{\partial t}+\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}p+\dot{\eta }\frac{\partial p}{\partial \eta }=\frac{\partial p}{\partial {{t}_{\pi }}}+(1+\varepsilon)\frac{\partial \pi }{\partial t}+ \\ &\;\;\;\;\;\;\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}p+\dot{\eta }(1+\varepsilon)\frac{\partial \pi }{\partial \eta }={{\omega }_{1}}+{{\omega }_{2}}, \\ \end{align} (23)

 ${{\omega }_{\text{1}}}=(\text{1}+\varepsilon)\frac{\partial \pi }{\partial t}+\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}p+\dot{\eta }(\text{1}+\varepsilon)\frac{\partial \pi }{\partial \eta },$ (24)
 ${{\omega }_{\text{2}}}=\frac{\partial p}{\partial {{t}_{\pi }}}.$ (25)

${{\omega }_{\text{1}}}$还可以进一步变形为

 \begin{align} &{{\omega }_{\text{1}}}=(\text{1}+\varepsilon)\left(\frac{\partial \pi }{\partial t}+\dot{\eta }\frac{\partial \pi }{\partial \eta } \right)+\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}p \\ &\text{ }=(\text{1}+\varepsilon)\left[ \frac{\partial \pi }{\partial t}-\eta \frac{\partial {{P}^{\text{2}}}}{\partial t}-\int{{{\nabla }_{\eta }}\cdot ({{P}^{\text{2}}}\mathit{\boldsymbol{V}})d\eta } \right]+\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}p \\ &\text{ }=\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}p-\left(\text{1}+\varepsilon \right)\int{{{\nabla }_{\eta }}\cdot ({{P}^{\text{2}}}\mathit{\boldsymbol{V}})d\eta .} \\ \end{align} (26)

 \begin{align} & {{\left(\frac{\partial T}{\partial t} \right)}_{\text{1}}}=-\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}T-\dot{\eta }\frac{\partial T}{\partial \eta }+\frac{\alpha {{\omega }_{1}}}{{{c}_{p}}}=-\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}T- \\ & \dot{\eta }\frac{\partial T}{\partial \eta }+\frac{\alpha }{{{c}_{p}}}\left[ \mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}p-(\text{1}+\varepsilon)\int{{{\nabla }_{\eta }}\cdot \left({{P}^{\text{2}}}\mathit{\boldsymbol{V}} \right)\text{d}\eta } \right], \\ \end{align} (27)
 ${{\left(\frac{\partial T}{\partial t} \right)}_{\text{2}}}=\frac{\alpha {{\omega }_{2}}}{{{c}_{p}}}=\frac{\alpha }{{{c}_{p}}}{{\left(\frac{\partial p}{\partial t} \right)}_{\pi }}.$ (28)
2.1.8 高阶订正参数定义
 $\varepsilon =\frac{\text{d}w}{g\text{d}t}=\frac{\text{1}}{g}\left(\frac{\partial w}{\partial t}+\mathit{\boldsymbol{V}}\cdot {{\nabla }_{\eta }}w+ \dot{\eta }\frac{\partial w}{\partial \eta } \right).$ (29)
2.1.9 气体状态方程
 $\alpha =\frac{RT}{p}.$ (30)
2.2 标准层结扣除及扰动方程组 2.2.1 扣除假设

 \left\{ \begin{align} &\tilde{T}=\tilde{T}(p), \\ &\tilde{\mathit{\Phi } }=\tilde{\mathit{\Phi } }(p), \\ \end{align} \right. (31)

 $P\eta \left(U\frac{\partial \text{ln}{{P}^{\text{2}}}}{a\text{sin}\theta \partial \lambda }+V\frac{\partial \text{ln}{{P}^{\text{2}}}}{a\partial \theta } \right),$ (79)

 \begin{align} &\left(\frac{P\tilde{c}}{\pi }+\frac{R{\Pi }'}{{{c}_{p}}\pi } \right)\left({{\omega }_{1}}+{{\omega }_{2}} \right)=S\left(\tilde{c}+\frac{R{\Pi }'}{{{c}_{p}}P} \right)\frac{{{\omega }_{1}}+{{\omega }_{2}}}{P\eta }= \\ &\text{ }S\left(\tilde{c}+\frac{R{\Pi }'}{{{c}_{p}}P} \right)\left(-\frac{1}{P\eta }\int_{0}^{\eta }{\frac{1}{a\sin \theta }\left(\frac{\partial PU}{\partial \lambda }+\frac{\partial PV\sin \theta }{\partial \theta } \right)\cdot } \right. \\ &\text{ }\left. d\eta +U\frac{\partial \ln {{P}^{2}}}{a\sin \theta \partial \lambda }+V\frac{\partial \ln {{P}^{2}}}{a\partial \theta } \right). \\ \end{align} (80)

 ${{D}_{xy}}=\frac{\text{1}}{a\text{sin}\theta }\left(\frac{\partial PU}{\partial \lambda }+\frac{\partial PV\text{sin}\theta }{\partial \theta } \right),$ (81)

 \begin{align} & \left(\frac{P\tilde{c}}{\pi }+\frac{R\mathit{{\Pi }'}}{{{c}_{p}}\pi } \right)\left({{\omega }_{\text{1}}}+{{\omega }_{2}} \right)=S\left(\tilde{c}+\frac{R\mathit{{\Pi }'}}{{{c}_{p}}P} \right)\cdot \\ & \left(-\frac{\text{1}}{P\eta }\int{{{D}_{xy}}\text{d}\eta +U\frac{\partial \text{ln}{{P}^{\text{2}}}}{a\text{sin}\theta \partial \lambda }}+V\frac{\partial \text{ln}{{P}^{\text{2}}}}{a\partial \theta } \right). \\ \end{align} (82)

5.4 静力平衡方程

 $\frac{\partial \mathit{{\Phi }'}}{\partial \eta }=-{{P}^{\text{2}}}\frac{R{T}'}{\pi }=-\frac{S\tilde{c}\mathit{{\Pi }'}}{P\eta }.$ (83)

 图 2 模式积分24 h的500 hPa风场矢量对比：（a）非静力退化为静力平衡框架（黑色箭头）、原静力平衡框架（彩色箭头）；（b）非静力框架（彩色箭头） Figure 2 Comparison of 24-hour simulations of 500-hPa wind vectors: (a) The hydrostatic core degraded from the non-hydrostatic one (black vectors), the original hydrostatic core (colored vectors), (b) the non-hydrostatic core (colored vectors)
6 总结与结论

AREM非静力框架的构造和程序实现充分考虑了原静力平衡框架的设计优势，也结合了国际上主流非静力模式设计的成功经验。主要有以下特点：

（1）该非静力框架是通过对静力平衡近似进行高阶松弛而建立的。与静力平衡框架相比，新发展的非静力框架主要区别在于完善了第三运动方程，定义了高阶订正参数和非静力连续方程。同时，水平动量方程中气压梯度力发生变化，热力学能量方程中非平流项发生变化。

（2）设计AREM非静力框架时仍然较好地使用了标准层结扣除和IAP变换，从而使静力平衡框架和非静力框架之间的转换与比较更加便利，并减小了由于两框架差别项过多而引起的计算困难。通过理论推导和模拟比对，可以发现：在静力近似下，AREM非静力框架退化为与原静力平衡框架一致的形式。

（3）AREM非静力框架依然基于$\eta$坐标和E网格进行空间离散，变量水平分布类似于静力平衡模式，但重新设计了变量的垂直交错分布形式。仍然采用具有二阶精度的中央差分形式进行空间离散，其中水平平流采用半格距差分形式。对于适应快过程采用Mesinger（1977）提出的经济格式，对于平流和扩散过程采用前差格式进行积分计算。

（4）基于对热力学方程非平流项的分析，采用时间两部离散技术进行积分计算从而可以单独求解声波，提高了模式运行效率。本文通过“追赶法”及迭代法对声波进行解算。

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