﻿ 一次梅雨锋暴雨过程中多尺度能量相互作用的研究Ⅰ.理论分析 一次梅雨锋暴雨过程中多尺度能量相互作用的研究Ⅰ.理论分析
 大气科学  2018, Vol. 42 Issue (5): 1109-1118 PDF

1 南京信息工程大学气象灾害教育部重点实验室/气候与环境变化国际合作联合实验室/气象灾害预报预警与评估协同创新中心, 南京 210044
2 中国科学院大气物理研究所云降水物理与强风暴重点实验室, 北京 100029
3 上海海洋气象台, 上海 201306
4 浙江大学地球科学学院, 杭州 310027

The Study of Multi-scale Energy Interactions during a Meiyu Front Rainstorm. Part Ⅰ: Theoretical Analysis
SHEN Xinyong1,2, SHA Sha1,3, LI Xiaofan4
1 Key Laboratory of Meteorological Disaster, Ministry of Education/Joint International Research Laboratory of Climate and Environment Change/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science and Technology, Nanjing 210044
2 Key Laboratory of Cloud-Precipitation Physics and Severe Storms, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029
3 Shanghai Marine Meteorological Center, Shanghai 201306
4 School of Earth Sciences, Zhejiang University, Hangzhou, Zhejiang 310027
Abstract: This paper is the beginning part of the discussion on the multi-scale energy interactions in the Meiyu front rainstorm. The kinetic energy equations and potential energy equations of large scale background field, meso-α scale and meso-micro-β scale are derived from motion equations and thermodynamic equations in the z coordinate system using the anelastic approximation by dividing the basic physical quantity into three scales for the purpose to analyze the multi-scale energy interactions. Various energy conversions are presented in the energy equations. The kinetic energy equations mainly include kinetic energy transportation or conversion, the work of horizontal pressure gradient force, vertical perturbation pressure gradient force, buoyancy, Coriolis force and friction. The potential energy equations mainly include potential energy transportation or conversion, and the effects of buoyancy and diabetic heating. Among them, the effect of buoyancy is substantially the energy conversion term between the potential energy and kinetic energy, and thus is the most critical energy conversion term in the process of rainstorm development. In the future study, the energy equations will be applied to the Meiyu front rainstorm, and the physical mechanism for the diagnosed energy interaction, which affects the development and disappearance of rainstorm, will be given.
Keywords: Large scale background field      Meso-α scale      Meso-micro-β scale      Kinetic energy and potential energy equation      Interaction
1 引言

1955年，Lorenz（1955）把总势能与最小总势能的差值定义为位能，随密度或者温度水平梯度的存在而存在。在绝热过程中，位能和动能的总和为常数。洛伦兹还将大气物理量分解成平均场和扰动场，当低纬的暖空气和高纬的冷空气相遇，平均场会将部分能量转换给扰动场，具体为平均位能将部分能量转换给扰动位能，扰动位能再将部分能量转换给扰动动能，最后扰动动能将能量转换给平均动能，这就是著名的洛伦兹能量循环。1957年，Saltsman（1957）通过对行星尺度的运动采用傅里叶级数分析来定义运动的尺度，再进行各个尺度能量方程的推导，推导的能量方程中包含特定波数域中涡旋气流和平均气流的位能的产生、释放和向动能的能量转换等。1983年，Plumb（1983）在转换欧拉平均（Andrews and McIntyre, 1976）的基础上从原始方程出发对能量方程的推导做出了一些改进，推导出的能量方程中不仅包含平均动能向涡动动能转换之类的项，还包含了能量通量这样的输送项。1984年，Kanzawa（1984）也在转换欧拉平均的基础上从准地转方程出发作出进一步的研究，发现平均位能和扰动位能之间不存在能量转换。2000年以前，不同的学者在不同的条件下从欧拉观点出发推导了各种能量方程。这之后，Iwasaki（2001）从拉格朗日经圈环流的角度出发在等熵坐标系中推导位能和动能变率方程来解释波流相互作用，研究表明在经度—高度剖面上没有平均动能和扰动动能的能量转换；进一步研究发现平均气流波数增加，平均动能向扰动位能转换能量，平均气流波数减小，扰动位能向扰动动能转换能量。2010年，Murakami（2010）提出了一个新的局地能量诊断方案，与之前不同的是，能量方程不仅包含平均能量方程和扰动能量方程，还包含了一个新的能量方程，即相互作用能量方程。当采取适当平均之后，相互作用能量可能会消失，但相互作用能量的方程不会消失，并且将相互作用能量通量和两类能量转换项有机地建立关系，丰富和完善了洛伦兹经典的能量循环图。这几十年以来，几乎所有学者都是以平均场和扰动场为研究对象去探讨能量之间的相互作用或能量循环。很少有学者探讨三个尺度及其以上之间的能量学特征，2011年，Hsu et al.（2011）p坐标系中推导了三个尺度下的动能方程，用于研究夏季季节内震荡和天气尺度扰动之间的相互作用，这是能量学的一大突破。

2 能量方程的推导 2.1 三个原始的运动方程和热力学方程

z坐标系中的原始方程（出发方程）：

 $\frac{{{\rm{d}}u}}{{{\rm{d}}t}} = - \frac{1}{\rho }\frac{{\partial p}}{{\partial x}} + fv + \upsilon (\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} + \frac{{{\partial ^2}u}}{{\partial {z^2}}}),$ (1)
 $\frac{{{\rm{d}}v}}{{{\rm{d}}t}} = - \frac{1}{\rho }\frac{{\partial p}}{{\partial y}} - fu + \upsilon (\frac{{{\partial ^2}v}}{{\partial {x^2}}} + \frac{{{\partial ^2}v}}{{\partial {y^2}}} + \frac{{{\partial ^2}v}}{{\partial {z^2}}}),$ (2)
 $\frac{{{\rm{d}}w}}{{{\rm{d}}t}} = - \frac{1}{\rho }\frac{{\partial p}}{{\partial z}} - g + \upsilon (\frac{{{\partial ^2}w}}{{\partial {x^2}}} + \frac{{{\partial ^2}w}}{{\partial {y^2}}} + \frac{{{\partial ^2}w}}{{\partial {z^2}}}),$ (3)
 $\frac{{{\rm{d}}\theta }}{{{\rm{d}}t}} = \frac{\theta }{{{c_{\rm{p}}}T}}\dot Q,$ (4)

 $p = \rho RT,$ (5)
 $\theta =T{{\left( \frac{1000}{p} \right)}^{{R}/{{{c}_{p}}}\;}},$ (6)

 $\frac{{\tilde \rho + \rho '}}{{\overline \rho }} = \frac{{{c_v}}}{{{c_p}}}\frac{{\tilde p + p'}}{{\overline p }} - \frac{{\tilde \theta + \theta '}}{{\overline \theta }},$ (7)

 $\begin{array}{l} \frac{{{\rm{d}}w}}{{{\rm{d}}t}} = - \frac{1}{{\overline \rho }}\frac{{\partial (\tilde p + p')}}{{\partial z}} - \frac{{\tilde \rho + \rho '}}{{\overline \rho }}g + \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}}\frac{{\partial (\tilde p + p')}}{{\partial z}} = \\ - \frac{1}{{\overline \rho }}\frac{{\partial (\tilde p + p')}}{{\partial z}} + \frac{{\tilde \theta + \theta '}}{{\overline \theta }}g - \frac{{{c_v}}}{{{c_p}}}\frac{{\tilde p + p'}}{{\overline p }}g + \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}}\frac{{\partial (\tilde p + p')}}{{\partial z}}. \end{array}$ (8)
2.2 动能方程的推导 2.2.1 计算u′×公式(1)+v′×公式(2)+w′×公式(8)，求${\partial {K}'}/{\partial \ t}\;$
 $\begin{array}{*{35}{l}} \partial {K}'/\partial t={{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}+{{a}_{5}}+{{a}_{6}}+{{a}_{7}}+{{a}_{8}}+{{a}_{9}}+ \\ {{a}_{10}}+{{a}_{11}}+{{a}_{12}}+{{a}_{13}}+{{a}_{14}}+{{a}_{15}}+{{a}_{16}}+{{a}_{17}}+{{a}_{18}}+ \\ {{a}_{19}}+{{a}_{20}}+{{a}_{21}}, \\ \end{array}$ (9)
 ${a_1} = - \left[ {u'(\frac{{\partial \overline u }}{{\partial t}} + \frac{{\partial \tilde u}}{{\partial t}}) + v'(\frac{{\partial \overline v }}{{\partial t}} + \frac{{\partial \tilde v}}{{\partial t}}) + w'(\frac{{\partial \overline w }}{{\partial t}} + \frac{{\partial \tilde w}}{{\partial t}})} \right],$ (10)
 ${a_2} = - \left(\begin{array}{l} u'\overline u \frac{{\partial \overline u }}{{\partial x}} + v'\overline u \frac{{\partial \overline v }}{{\partial x}} + w'\overline u \frac{{\partial \overline w }}{{\partial x}} + u'\overline v \frac{{\partial \overline u }}{{\partial y}} + v'\overline v \frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}w'\overline v \frac{{\partial \overline w }}{{\partial y}} + u'\overline w \frac{{\partial \overline u }}{{\partial z}} + v'\overline w \frac{{\partial \overline v }}{{\partial z}} + w'\overline w \frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (11)
 ${a_3} = - \left(\begin{array}{l} u'\tilde u\frac{{\partial \overline u }}{{\partial x}} + v'\tilde u\frac{{\partial \overline v }}{{\partial x}} + w'\tilde u\frac{{\partial \overline w }}{{\partial x}} + u'\tilde v\frac{{\partial \overline u }}{{\partial y}} + v'\tilde v\frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}w'\tilde v\frac{{\partial \overline w }}{{\partial y}} + u'\tilde w\frac{{\partial \overline u }}{{\partial z}} + v'\tilde w\frac{{\partial \overline v }}{{\partial z}} + w'\tilde w\frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (12)
 ${a_4} = - \left(\begin{array}{l} u'u'\frac{{\partial \overline u }}{{\partial x}} + v'u'\frac{{\partial \overline v }}{{\partial x}} + w'u'\frac{{\partial \overline w }}{{\partial x}} + u'v'\frac{{\partial \overline u }}{{\partial y}} + v'v'\frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}w'v'\frac{{\partial \overline w }}{{\partial y}} + u'w'\frac{{\partial \overline u }}{{\partial z}} + v'w'\frac{{\partial \overline v }}{{\partial z}} + w'w'\frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (13)
 ${a_5} = - \left(\begin{array}{l} u'\overline u \frac{{\partial \tilde u}}{{\partial x}} + v'\overline u \frac{{\partial \tilde v}}{{\partial x}} + w'\overline u \frac{{\partial \tilde w}}{{\partial x}} + u'\overline v \frac{{\partial \tilde u}}{{\partial y}} + v'\overline v \frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}w'\overline v \frac{{\partial \tilde w}}{{\partial y}} + u'\overline w \frac{{\partial \tilde u}}{{\partial z}} + v'\overline w \frac{{\partial \tilde v}}{{\partial z}} + w'\overline w \frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (14)
 ${a_6} = - \left(\begin{array}{l} \pi u'\tilde u\frac{{\partial \tilde u}}{{\partial x}} + v'\tilde u\frac{{\partial \tilde v}}{{\partial x}} + w'\tilde u\frac{{\partial \tilde w}}{{\partial x}} + u'\tilde v\frac{{\partial \tilde u}}{{\partial y}} + v'\tilde v\frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}w'\tilde v\frac{{\partial \tilde w}}{{\partial y}} + u'\tilde w\frac{{\partial \tilde u}}{{\partial z}} + v'\tilde w\frac{{\partial \tilde v}}{{\partial z}} + w'\tilde w\frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (15)
 ${a_7} = - \left(\begin{array}{l} u'u'\frac{{\partial \tilde u}}{{\partial x}} + v'u'\frac{{\partial \tilde v}}{{\partial x}} + w'u'\frac{{\partial \tilde w}}{{\partial x}} + u'v'\frac{{\partial \tilde u}}{{\partial y}} + v'v'\frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}w'v'\frac{{\partial \tilde w}}{{\partial y}} + u'w'\frac{{\partial \tilde u}}{{\partial z}} + v'w'\frac{{\partial \tilde v}}{{\partial z}} + w'w'\frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (16)
 ${a_8} = - \left(\begin{array}{l} u'\overline u \frac{{\partial u'}}{{\partial x}} + v'\overline u \frac{{\partial v'}}{{\partial x}} + w'\overline u \frac{{\partial w'}}{{\partial x}} + u'\overline v \frac{{\partial u'}}{{\partial y}} + v'\overline v \frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}w'\overline v \frac{{\partial w'}}{{\partial y}} + u'\overline w \frac{{\partial u'}}{{\partial z}} + v'\overline w \frac{{\partial v'}}{{\partial z}} + w'\overline w \frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (17)
 ${a_9} = - \left(\begin{array}{l} u'\tilde u\frac{{\partial u'}}{{\partial x}} + v'\tilde u\frac{{\partial v'}}{{\partial x}} + w'\tilde u\frac{{\partial w'}}{{\partial x}} + u'\tilde v\frac{{\partial u'}}{{\partial y}} + v'\tilde v\frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}w'\tilde v\frac{{\partial w'}}{{\partial y}} + u'\tilde w\frac{{\partial u'}}{{\partial z}} + v'\tilde w\frac{{\partial v'}}{{\partial z}} + w'\tilde w\frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (18)
 ${a_{10}} = - \left(\begin{array}{l} u'u'\frac{{\partial u'}}{{\partial x}} + v'u'\frac{{\partial v'}}{{\partial x}} + w'u'\frac{{\partial w'}}{{\partial x}} + u'v'\frac{{\partial u'}}{{\partial y}} + v'v'\frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}w'v'\frac{{\partial w'}}{{\partial y}} + u'w'\frac{{\partial u'}}{{\partial z}} + v'w'\frac{{\partial v'}}{{\partial z}} + w'w'\frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (19)
 ${a_{11}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({u'\frac{{\partial \overline p }}{{\partial x}} + v'\frac{{\partial \overline p }}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({u'\frac{{\partial \overline p }}{{\partial x}} + v'\frac{{\partial \overline p }}{{\partial y}}} \right)} \right],$ (20)
 ${a_{12}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({u'\frac{{\partial \tilde p}}{{\partial x}} + v'\frac{{\partial \tilde p}}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({u'\frac{{\partial \tilde p}}{{\partial x}} + v'\frac{{\partial \tilde p}}{{\partial y}}} \right)} \right],$ (21)
 $\begin{array}{l} {a_{13}} = \\ - \left[ {\frac{1}{{\overline \rho }} \cdot \left({u'\frac{{\partial p'}}{{\partial x}} + v'\frac{{\partial p'}}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({u'\frac{{\partial p'}}{{\partial x}} + v'\frac{{\partial p'}}{{\partial y}}} \right)} \right] \end{array},$ (22)
 ${a_{14}} = - \left[ {w'\left({\frac{1}{{\overline \rho }} - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}}} \right)\frac{{\partial (\tilde p + p')}}{{\partial z}}} \right],$ (23)
 ${a_{15}} = \frac{g}{{\overline \theta }}w'\tilde \theta ,$ (24)
 ${a_{16}} = \frac{g}{{\overline \theta }}w'\theta ',$ (25)
 ${a_{17}} = - \left({w'\frac{{{c_v}}}{{{c_p}}}\frac{{\tilde p + p'}}{{\overline p }}g} \right),$ (26)
 ${a_{18}} = f(u'\overline v + u'\tilde v - \overline u v' - \tilde uv'),$ (27)
 ${a_{19}} = \upsilon \left[ \begin{array}{l} u'\left({\frac{{{\partial ^2}\overline u }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline u }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline u }}{{\partial {z^2}}}} \right) + v'\left({\frac{{{\partial ^2}\overline v }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline v }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline v }}{{\partial {z^2}}}} \right) + \\ {\rm{ }}w'\left({\frac{{{\partial ^2}\overline w }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline w }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline w }}{{\partial {z^2}}}} \right) \end{array} \right],$ (28)
 ${a_{20}} = \upsilon \left[ \begin{array}{l} u'\left({\frac{{{\partial ^2}\tilde u}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde u}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde u}}{{\partial {z^2}}}} \right) + v'\left({\frac{{{\partial ^2}\tilde v}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde v}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde v}}{{\partial {z^2}}}} \right) + \\ {\rm{ }}w'\left({\frac{{{\partial ^2}\tilde w}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde w}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde w}}{{\partial {z^2}}}} \right) \end{array} \right],$ (29)
 ${a_{21}} = \upsilon \left[ \begin{array}{l} u'\left({\frac{{{\partial ^2}u'}}{{\partial {x^2}}} + \frac{{{\partial ^2}u'}}{{\partial {y^2}}} + \frac{{{\partial ^2}u'}}{{\partial {z^2}}}} \right) + v'\left({\frac{{{\partial ^2}v'}}{{\partial {x^2}}} + \frac{{{\partial ^2}v'}}{{\partial {y^2}}} + \frac{{{\partial ^2}v'}}{{\partial {z^2}}}} \right) + \\ {\rm{ }}w'\left({\frac{{{\partial ^2}w'}}{{\partial {x^2}}} + \frac{{{\partial ^2}w'}}{{\partial {y^2}}} + \frac{{{\partial ^2}w'}}{{\partial {z^2}}}} \right) \end{array} \right],$ (30)

 $\begin{array}{l} \partial {K}'/\partial t= {I_\beta } + {T_K}_{\;(\beta, L)} + {T_K}_{\;(\beta, \alpha)} + {T_K}_{\;(\beta, \beta)} + {H_\beta } + {V_\beta } + \\ {B_{(\alpha, \beta)}} + {B_{(\beta, \beta)}} + {S_\beta } + {C_\beta } + {F_\beta }. \end{array}$ (31)

2.2.2 计算$\tilde{u}\times 公式(1)+\tilde{v}\times 公式(2)+\tilde{w}\times 公式(8)$，求${\partial \tilde{K}}/{\partial \ t}\;$
 $\begin{array}{l} \partial \tilde{K}\text{/}\partial \ t = {b_1} + {b_2} + {b_3} + {b_4} + {b_5} + {b_6} + {b_7} + {b_8} + {b_9} + {b_{10}} + {b_{11}} + \\ {b_{12}} + {b_{13}} + {b_{14}} + {b_{15}} + {b_{16}} + {b_{17}} + {b_{18}} + {b_{19}} + {b_{20}} + {b_{21}}, \end{array}$ (32)
 ${b_1} = - \left[ {\tilde u(\frac{{\partial \overline u }}{{\partial t}} + \frac{{\partial u'}}{{\partial t}}) + \tilde v(\frac{{\partial \overline v }}{{\partial t}} + \frac{{\partial v'}}{{\partial t}}) + \tilde w(\frac{{\partial \overline w }}{{\partial t}} + \frac{{\partial w'}}{{\partial t}})} \right],$ (33)
 ${b_2} = - \left(\begin{array}{l} \tilde u\overline u \frac{{\partial \overline u }}{{\partial x}} + \tilde v\overline u \frac{{\partial \overline v }}{{\partial x}} + \tilde w\overline u \frac{{\partial \overline w }}{{\partial x}} + \tilde u\overline v \frac{{\partial \overline u }}{{\partial y}} + \tilde v\overline v \frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}\tilde w\overline v \frac{{\partial \overline w }}{{\partial y}} + \tilde u\overline w \frac{{\partial \overline u }}{{\partial z}} + \tilde v\overline w \frac{{\partial \overline v }}{{\partial z}} + \tilde w\overline w \frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (34)
 ${b_3} = - \left(\begin{array}{l} \tilde u\tilde u\frac{{\partial \overline u }}{{\partial x}} + \tilde v\tilde u\frac{{\partial \overline v }}{{\partial x}} + \tilde w\tilde u\frac{{\partial \overline w }}{{\partial x}} + \tilde u\tilde v\frac{{\partial \overline u }}{{\partial y}} + \tilde v\tilde v\frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}\tilde w\tilde v\frac{{\partial \overline w }}{{\partial y}} + \tilde u\tilde w\frac{{\partial \overline u }}{{\partial z}} + \tilde v\tilde w\frac{{\partial \overline v }}{{\partial z}} + \tilde w\tilde w\frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (35)
 ${b_4} = - \left(\begin{array}{l} \tilde uu'\frac{{\partial \overline u }}{{\partial x}} + \tilde vu'\frac{{\partial \overline v }}{{\partial x}} + \tilde wu'\frac{{\partial \overline w }}{{\partial x}} + \tilde uv'\frac{{\partial \overline u }}{{\partial y}} + \tilde vv'\frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}\tilde wv'\frac{{\partial \overline w }}{{\partial y}} + \tilde uw'\frac{{\partial \overline u }}{{\partial z}} + \tilde vw'\frac{{\partial \overline v }}{{\partial z}} + \tilde ww'\frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (36)
 ${b_5} = - \left(\begin{array}{l} \tilde u\overline u \frac{{\partial \tilde u}}{{\partial x}} + \tilde v\overline u \frac{{\partial \tilde v}}{{\partial x}} + \tilde w\overline u \frac{{\partial \tilde w}}{{\partial x}} + \tilde u\overline v \frac{{\partial \tilde u}}{{\partial y}} + \tilde v\overline v \frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}\tilde w\overline v \frac{{\partial \tilde w}}{{\partial y}} + \tilde u\overline w \frac{{\partial \tilde u}}{{\partial z}} + \tilde v\overline w \frac{{\partial \tilde v}}{{\partial z}} + \tilde w\overline w \frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (37)
 ${b_6} = - \left(\begin{array}{l} \tilde u\tilde u\frac{{\partial \tilde u}}{{\partial x}} + \tilde v\tilde u\frac{{\partial \tilde v}}{{\partial x}} + \tilde w\tilde u\frac{{\partial \tilde w}}{{\partial x}} + \tilde u\tilde v\frac{{\partial \tilde u}}{{\partial y}} + \tilde v\tilde v\frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}\tilde w\tilde v\frac{{\partial \tilde w}}{{\partial y}} + \tilde u\tilde w\frac{{\partial \tilde u}}{{\partial z}} + \tilde v\tilde w\frac{{\partial \tilde v}}{{\partial z}} + \tilde w\tilde w\frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (38)
 ${b_7} = - \left(\begin{array}{l} \tilde uu'\frac{{\partial \tilde u}}{{\partial x}} + \tilde vu'\frac{{\partial \tilde v}}{{\partial x}} + \tilde wu'\frac{{\partial \tilde w}}{{\partial x}} + \tilde uv'\frac{{\partial \tilde u}}{{\partial y}} + \tilde vv'\frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}\tilde wv'\frac{{\partial \tilde w}}{{\partial y}} + \tilde uw'\frac{{\partial \tilde u}}{{\partial z}} + \tilde vw'\frac{{\partial \tilde v}}{{\partial z}} + \tilde ww'\frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (39)
 ${b_8} = - \left(\begin{array}{l} \tilde u\overline u \frac{{\partial u'}}{{\partial x}} + \tilde v\overline u \frac{{\partial v'}}{{\partial x}} + \tilde w\overline u \frac{{\partial w'}}{{\partial x}} + \tilde u\overline v \frac{{\partial u'}}{{\partial y}} + \tilde v\overline v \frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}\tilde w\overline v \frac{{\partial w'}}{{\partial y}} + \tilde u\overline w \frac{{\partial u'}}{{\partial z}} + \tilde v\overline w \frac{{\partial v'}}{{\partial z}} + \tilde w\overline w \frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (40)
 ${b_9} = - \left(\begin{array}{l} \tilde u\tilde u\frac{{\partial u'}}{{\partial x}} + \tilde v\tilde u\frac{{\partial v'}}{{\partial x}} + \tilde w\tilde u\frac{{\partial w'}}{{\partial x}} + \tilde u\tilde v\frac{{\partial u'}}{{\partial y}} + \tilde v\tilde v\frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}\tilde w\tilde v\frac{{\partial w'}}{{\partial y}} + \tilde u\tilde w\frac{{\partial u'}}{{\partial z}} + \tilde v\tilde w\frac{{\partial v'}}{{\partial z}} + \tilde w\tilde w\frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (41)
 ${b_{10}} = - \left(\begin{array}{l} \tilde uu'\frac{{\partial u'}}{{\partial x}} + \tilde vu'\frac{{\partial v'}}{{\partial x}} + \tilde wu'\frac{{\partial w'}}{{\partial x}} + \tilde uv'\frac{{\partial u'}}{{\partial y}} + \tilde vv'\frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}\tilde wv'\frac{{\partial w'}}{{\partial y}} + \tilde uw'\frac{{\partial u'}}{{\partial z}} + \tilde vw'\frac{{\partial v'}}{{\partial z}} + \tilde ww'\frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (42)
 ${b_{11}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({\tilde u\frac{{\partial \overline p }}{{\partial x}} + \tilde v\frac{{\partial \overline p }}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({\tilde u\frac{{\partial \overline p }}{{\partial x}} + \tilde v\frac{{\partial \overline p }}{{\partial y}}} \right)} \right],$ (43)
 ${b_{12}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({\tilde u\frac{{\partial \tilde p}}{{\partial x}} + \tilde v\frac{{\partial \tilde p}}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({\tilde u\frac{{\partial \tilde p}}{{\partial x}} + \tilde v\frac{{\partial \tilde p}}{{\partial y}}} \right)} \right],$ (44)
 ${b_{13}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({\tilde u\frac{{\partial p'}}{{\partial x}} + \tilde v\frac{{\partial p'}}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({\tilde u\frac{{\partial p'}}{{\partial x}} + \tilde v\frac{{\partial p'}}{{\partial y}}} \right)} \right],$ (45)
 ${b_{14}} = - \left[ {\tilde w\left({\frac{1}{{\overline \rho }} - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}}} \right)\frac{{\partial (\tilde p + p')}}{{\partial z}}} \right],$ (46)
 ${b_{15}} = \frac{g}{{\overline \theta }}\tilde w\tilde \theta ,$ (47)
 ${b_{16}} = \frac{g}{{\overline \theta }}\tilde w\theta ',$ (48)
 ${b_{17}} = - \left({\tilde w\frac{{{c_v}}}{{{c_p}}}\frac{{\tilde p + p'}}{{\overline p }}g} \right),$ (49)
 ${b_{18}} = f(\tilde u\overline v + \tilde u\tilde v - \overline u \tilde v - \tilde u\tilde v),$ (50)
 ${b_{19}} = \upsilon \left[ \begin{array}{l} \tilde u\left({\frac{{{\partial ^2}\overline u }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline u }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline u }}{{\partial {z^2}}}} \right) + \tilde v\left({\frac{{{\partial ^2}\overline v }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline v }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline v }}{{\partial {z^2}}}} \right) + \\ {\rm{ }}\tilde w\left({\frac{{{\partial ^2}\overline w }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline w }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline w }}{{\partial {z^2}}}} \right) \end{array} \right],$ (51)
 ${b_{20}} = \upsilon \left[ \begin{array}{l} \tilde u\left({\frac{{{\partial ^2}\tilde u}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde u}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde u}}{{\partial {z^2}}}} \right) + \tilde v\left({\frac{{{\partial ^2}\tilde v}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde v}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde v}}{{\partial {z^2}}}} \right) + \\ {\rm{ }}\tilde w\left({\frac{{{\partial ^2}\tilde w}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde w}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde w}}{{\partial {z^2}}}} \right) \end{array} \right],$ (52)
 ${b_{21}} = \upsilon \left[ \begin{array}{l} \tilde u\left({\frac{{{\partial ^2}u'}}{{\partial {x^2}}} + \frac{{{\partial ^2}u'}}{{\partial {y^2}}} + \frac{{{\partial ^2}u'}}{{\partial {z^2}}}} \right) + \tilde v\left({\frac{{{\partial ^2}v'}}{{\partial {x^2}}} + \frac{{{\partial ^2}v'}}{{\partial {y^2}}} + \frac{{{\partial ^2}v'}}{{\partial {z^2}}}} \right) + \\ \tilde w\left({\frac{{{\partial ^2}w'}}{{\partial {x^2}}} + \frac{{{\partial ^2}w'}}{{\partial {y^2}}} + \frac{{{\partial ^2}w'}}{{\partial {z^2}}}} \right) \end{array} \right].$ (53)

 $\begin{array}{l} \partial \tilde{K}\text{/}\partial \ t = {I_\alpha } + {T_K}_{\;(\alpha, L)} + {T_K}_{\;(\alpha, \alpha)} + {T_K}_{\;(\alpha, \beta)} + {H_\alpha } + \\ {V_\alpha } + {B_{(\alpha, \alpha)}} + {B_{(\beta, \alpha)}} + {S_\alpha } + {C_\alpha } + {F_\alpha }. \end{array}$ (54)

2.2.3 计算$\overline{u}\times 公式(1)+\overline{v}\times 公式(2)+\overline{w}\times 公式(8)$，求${\partial \overline{K}}/{\partial \ t}\;$
 $\begin{array}{l} \partial \tilde{K}\text{/}\partial \ t = {c_1} + {c_2} + {c_3} + {c_4} + {c_5} + {c_6} + {c_7} + {c_8} + {c_9} + {c_{10}} + \\ {c_{11}} + {c_{12}} + {c_{13}} + {c_{14}} + {c_{15}} + {c_{16}} + {c_{17}} + {c_{18}} + {c_{19}} + {c_{20}} + {c_{21}}, \end{array}$ (55)
 ${c_1} = - \left[ {\overline u (\frac{{\partial \tilde u}}{{\partial t}} + \frac{{\partial u'}}{{\partial t}}) + \overline v (\frac{{\partial \tilde v}}{{\partial t}} + \frac{{\partial v'}}{{\partial t}}) + \overline w (\frac{{\partial \tilde w}}{{\partial t}} + \frac{{\partial w'}}{{\partial t}})} \right],$ (56)
 ${c_2} = - \left(\begin{array}{l} \overline u \overline u \frac{{\partial \overline u }}{{\partial x}} + \overline v \overline u \frac{{\partial \overline v }}{{\partial x}} + \overline w \overline u \frac{{\partial \overline w }}{{\partial x}} + \overline u \overline v \frac{{\partial \overline u }}{{\partial y}} + \overline v \overline v \frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}\overline w \overline v \frac{{\partial \overline w }}{{\partial y}} + \overline u \overline w \frac{{\partial \overline u }}{{\partial z}} + \overline v \overline w \frac{{\partial \overline v }}{{\partial z}} + \overline w \overline w \frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (57)
 ${c_3} = - \left(\begin{array}{l} \overline u \tilde u\frac{{\partial \overline u }}{{\partial x}} + \overline v \tilde u\frac{{\partial \overline v }}{{\partial x}} + \overline w \tilde u\frac{{\partial \overline w }}{{\partial x}} + \overline u \tilde v\frac{{\partial \overline u }}{{\partial y}} + \overline v \tilde v\frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}\overline w \tilde v\frac{{\partial \overline w }}{{\partial y}} + \overline u \tilde w\frac{{\partial \overline u }}{{\partial z}} + \overline v \tilde w\frac{{\partial \overline v }}{{\partial z}} + \overline w \tilde w\frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (58)
 ${c_4} = - \left(\begin{array}{l} \overline u u'\frac{{\partial \overline u }}{{\partial x}} + \overline v u'\frac{{\partial \overline v }}{{\partial x}} + \overline w u'\frac{{\partial \overline w }}{{\partial x}} + \overline u v'\frac{{\partial \overline u }}{{\partial y}} + \overline v v'\frac{{\partial \overline v }}{{\partial y}} + \\ {\rm{ }}\overline w v'\frac{{\partial \overline w }}{{\partial y}} + \overline u w'\frac{{\partial \overline u }}{{\partial z}} + \overline v w'\frac{{\partial \overline v }}{{\partial z}} + \overline w w'\frac{{\partial \overline w }}{{\partial z}} \end{array} \right),$ (59)
 ${c_5} = - \left(\begin{array}{l} \overline u \overline u \frac{{\partial \tilde u}}{{\partial x}} + \overline v \overline u \frac{{\partial \tilde v}}{{\partial x}} + \overline w \overline u \frac{{\partial \tilde w}}{{\partial x}} + \overline u \overline v \frac{{\partial \tilde u}}{{\partial y}} + \overline v \overline v \frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}\overline w \overline v \frac{{\partial \tilde w}}{{\partial y}} + \overline u \overline w \frac{{\partial \tilde u}}{{\partial z}} + \overline v \overline w \frac{{\partial \tilde v}}{{\partial z}} + \overline w \overline w \frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (60)
 ${c_6} = - \left(\begin{array}{l} \overline u \tilde u\frac{{\partial \tilde u}}{{\partial x}} + \overline v \tilde u\frac{{\partial \tilde v}}{{\partial x}} + \overline w \tilde u\frac{{\partial \tilde w}}{{\partial x}} + \overline u \tilde v\frac{{\partial \tilde u}}{{\partial y}} + \overline v \tilde v\frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}\overline w \tilde v\frac{{\partial \tilde w}}{{\partial y}} + \overline u \tilde w\frac{{\partial \tilde u}}{{\partial z}} + \overline v \tilde w\frac{{\partial \tilde v}}{{\partial z}} + \overline w \tilde w\frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (61)
 ${c_7} = - \left(\begin{array}{l} \overline u u'\frac{{\partial \tilde u}}{{\partial x}} + \overline v u'\frac{{\partial \tilde v}}{{\partial x}} + \overline w u'\frac{{\partial \tilde w}}{{\partial x}} + \overline u v'\frac{{\partial \tilde u}}{{\partial y}} + \overline v v'\frac{{\partial \tilde v}}{{\partial y}} + \\ {\rm{ }}\overline w v'\frac{{\partial \tilde w}}{{\partial y}} + \overline u w'\frac{{\partial \tilde u}}{{\partial z}} + \overline v w'\frac{{\partial \tilde v}}{{\partial z}} + \overline w w'\frac{{\partial \tilde w}}{{\partial z}} \end{array} \right),$ (62)
 ${c_8} = - \left(\begin{array}{l} \overline u \overline u \frac{{\partial u'}}{{\partial x}} + \overline v \overline u \frac{{\partial v'}}{{\partial x}} + \overline w \overline u \frac{{\partial w'}}{{\partial x}} + \overline u \overline v \frac{{\partial u'}}{{\partial y}} + \overline v \overline v \frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}\overline w \overline v \frac{{\partial w'}}{{\partial y}} + \overline u \overline w \frac{{\partial u'}}{{\partial z}} + \overline v \overline w \frac{{\partial v'}}{{\partial z}} + \overline w \overline w \frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (63)
 ${c_9} = - \left(\begin{array}{l} \overline u \tilde u\frac{{\partial u'}}{{\partial x}} + \overline v \tilde u\frac{{\partial v'}}{{\partial x}} + \overline w \tilde u\frac{{\partial w'}}{{\partial x}} + \overline u \tilde v\frac{{\partial u'}}{{\partial y}} + \overline v \tilde v\frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}\overline w \tilde v\frac{{\partial w'}}{{\partial y}} + \overline u \tilde w\frac{{\partial u'}}{{\partial z}} + \overline v \tilde w\frac{{\partial v'}}{{\partial z}} + \overline w \tilde w\frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (64)
 ${c_{10}} = - \left(\begin{array}{l} \overline u u'\frac{{\partial u'}}{{\partial x}} + \overline v u'\frac{{\partial v'}}{{\partial x}} + \overline w u'\frac{{\partial w'}}{{\partial x}} + \overline u v'\frac{{\partial u'}}{{\partial y}} + \overline v v'\frac{{\partial v'}}{{\partial y}} + \\ {\rm{ }}\overline w v'\frac{{\partial w'}}{{\partial y}} + \overline u w'\frac{{\partial u'}}{{\partial z}} + \overline v w'\frac{{\partial v'}}{{\partial z}} + \overline w w'\frac{{\partial w'}}{{\partial z}} \end{array} \right),$ (65)
 ${c_{11}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({\overline u \frac{{\partial \overline p }}{{\partial x}} + \overline v \frac{{\partial \overline p }}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({\overline u \frac{{\partial \overline p }}{{\partial x}} + \overline v \frac{{\partial \overline p }}{{\partial y}}} \right)} \right],$ (66)
 ${c_{12}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({\overline u \frac{{\partial \tilde p}}{{\partial x}} + \overline v \frac{{\partial \tilde p}}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({\overline u \frac{{\partial \tilde p}}{{\partial x}} + \overline v \frac{{\partial \tilde p}}{{\partial y}}} \right)} \right],$ (67)
 ${c_{13}} = - \left[ {\frac{1}{{\overline \rho }} \cdot \left({\overline u \frac{{\partial p'}}{{\partial x}} + \overline v \frac{{\partial p'}}{{\partial y}}} \right) - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}} \cdot \left({\overline u \frac{{\partial p'}}{{\partial x}} + \overline v \frac{{\partial p'}}{{\partial y}}} \right)} \right],$ (68)
 ${c_{14}} = - \left[ {\overline w \left({\frac{1}{{\overline \rho }} - \frac{{\tilde \rho + \rho '}}{{{{\overline \rho }^2}}}} \right)\frac{{\partial (\tilde p + p')}}{{\partial z}}} \right],$ (69)
 ${c_{15}} = \frac{g}{{\overline \theta }}\overline w \tilde \theta ,$ (70)
 ${c_{16}} = \frac{g}{{\overline \theta }}\overline w \theta ',$ (71)
 ${c_{17}} = - \left({\overline w \frac{{{c_v}}}{{{c_p}}}\frac{{\tilde p + p'}}{{\overline p }}g} \right),$ (72)
 ${c_{18}} = f(\overline u \overline v + \overline u \tilde v - \overline u \overline v - \tilde u\overline v),$ (73)
 ${c_{19}} = \upsilon \left[ \begin{array}{l} \overline u \left({\frac{{{\partial ^2}\overline u }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline u }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline u }}{{\partial {z^2}}}} \right) + \overline v \left({\frac{{{\partial ^2}\overline v }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline v }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline v }}{{\partial {z^2}}}} \right) + \\ {\rm{ }}\overline w \left({\frac{{{\partial ^2}\overline w }}{{\partial {x^2}}} + \frac{{{\partial ^2}\overline w }}{{\partial {y^2}}} + \frac{{{\partial ^2}\overline w }}{{\partial {z^2}}}} \right) \end{array} \right],$ (74)
 ${c_{20}} = \upsilon \left[ \begin{array}{l} \overline u \left({\frac{{{\partial ^2}\tilde u}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde u}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde u}}{{\partial {z^2}}}} \right) + \overline v \left({\frac{{{\partial ^2}\tilde v}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde v}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde v}}{{\partial {z^2}}}} \right) + \\ {\rm{ }}\overline w \left({\frac{{{\partial ^2}\tilde w}}{{\partial {x^2}}} + \frac{{{\partial ^2}\tilde w}}{{\partial {y^2}}} + \frac{{{\partial ^2}\tilde w}}{{\partial {z^2}}}} \right) \end{array} \right],$ (75)
 ${c_{21}} = \upsilon \left[ \begin{array}{l} \overline u \left({\frac{{{\partial ^2}u'}}{{\partial {x^2}}} + \frac{{{\partial ^2}u'}}{{\partial {y^2}}} + \frac{{{\partial ^2}u'}}{{\partial {z^2}}}} \right) + \overline v \left({\frac{{{\partial ^2}v'}}{{\partial {x^2}}} + \frac{{{\partial ^2}v'}}{{\partial {y^2}}} + \frac{{{\partial ^2}v'}}{{\partial {z^2}}}} \right) + \\ {\rm{ }}\overline w \left({\frac{{{\partial ^2}w'}}{{\partial {x^2}}} + \frac{{{\partial ^2}w'}}{{\partial {y^2}}} + \frac{{{\partial ^2}w'}}{{\partial {z^2}}}} \right) \end{array} \right].$ (76)

 $\begin{array}{l} \partial \bar{K}/\partial t= {I_L} + {T_K}_{\;(L, L)} + {T_K}_{\;(L, \alpha)} + {T_K}_{\;(L, \beta)} + {H_L} + \\ {V_L} + {B_{(\alpha, L)}} + {B_{(\beta, L)}} + {S_L} + {C_L} + {F_L}. \end{array}$ (77)

2.3 位能方程的推导 2.3.1 计算${\theta }'\times 公式(4)$，求${\partial {A}'}/{\partial \ t}\;$
 $\begin{array}{l} \partial {A}'/\partial \ t= \frac{\partial }{{\partial \;t}}\left({\frac{\gamma }{2}{{\theta '}^2}} \right) = {d_1} + {d_2} + {d_3} + {d_4} + \\ {d_5} + {d_6} + {d_7} + {d_8}, \end{array}$ (78)
 ${d_1} = - \left[ {\gamma \;\theta '\frac{{\partial \left({\overline \theta + \tilde \theta } \right)}}{{\partial t}}} \right],$ (79)
 ${d_2} = - \gamma \left(\begin{array}{l} \theta '\;\overline u \frac{{\partial \overline \theta }}{{\partial x}}\; + \theta '\;\overline v \frac{{\partial \overline \theta }}{{\partial y}} + \theta '\;\tilde u\frac{{\partial \overline \theta }}{{\partial x}} + \\ {\rm{ }}\theta '\;\tilde v\frac{{\partial \overline \theta }}{{\partial y}} + \theta '\;u'\frac{{\partial \overline \theta }}{{\partial x}} + \theta '\;v'\frac{{\partial \overline \theta }}{{\partial y}} \end{array} \right),$ (80)
 ${d_3} = - \gamma \left(\begin{array}{l} \theta '\;\overline u \frac{{\partial \tilde \theta }}{{\partial x}}\; + \theta '\;\overline v \frac{{\partial \tilde \theta }}{{\partial y}} + \theta '\;\tilde u\frac{{\partial \tilde \theta }}{{\partial x}} + \\ {\rm{ }}\theta '\;\tilde v\frac{{\partial \tilde \theta }}{{\partial y}} + \theta '\;u'\frac{{\partial \tilde \theta }}{{\partial x}} + \theta '\;v'\frac{{\partial \tilde \theta }}{{\partial y}} \end{array} \right),$ (81)
 ${d_4} = - \gamma \left(\begin{array}{l} \theta '\;\overline u \frac{{\partial \theta '}}{{\partial x}}\; + \theta '\;\overline v \frac{{\partial \theta '}}{{\partial y}} + \theta '\;\tilde u\frac{{\partial \theta '}}{{\partial x}} + \\ {\rm{ }}\theta '\;\tilde v\frac{{\partial \theta '}}{{\partial y}} + \theta '\;u'\frac{{\partial \theta '}}{{\partial x}} + \theta '\;v'\frac{{\partial \theta '}}{{\partial y}} \end{array} \right),$ (82)
 ${d_5} = - \left({\frac{{\theta '}}{{\overline \theta }}\overline w g} \right),$ (83)
 ${d_6} = - \left({\frac{{\theta '}}{{\overline \theta }}\tilde wg} \right),$ (84)
 ${d_7} = - \left({\frac{{\theta '}}{{\overline \theta }}w'g} \right),$ (85)
 ${d_8} = \gamma \frac{{\theta '}}{{{c_p}}}{\left({\frac{p}{{{p_0}}}} \right)^{R/{{c}_{p}}}}\left({\overline Q + \tilde Q + Q'} \right),$ (86)

 $\begin{array}{l} \partial {A}'/\partial \ t= \frac{\partial }{{\partial \;t}}\left({\frac{\gamma }{2}{{\theta '}^2}} \right) = {P_{\;\beta }} + {T_P}_{\;(\beta, L)} + {T_P}_{\;(\beta, \alpha)} + \\ {T_P}_{\;(\beta, \beta)} - {B_{(\beta, L)}} - {B_{(\beta, \alpha)}} - {B_{(\beta, \beta)}} + {D_\beta }. \end{array}$ (87)

2.3.2 计算$\tilde{\theta }\times 公式(4)$，求${\partial \tilde{A}}/{\partial \ t}\;$
 $\partial \tilde{A}/\partial \ t= \frac{\partial }{{\partial \;t}}\left({\frac{\gamma }{2}{{\tilde \theta }^2}} \right) = {e_1} + {e_2} + {e_3} + {e_4} + {e_5} + {e_6} + {e_7} + {e_8},$ (88)
 ${e_1} = - \left[ {\gamma \;\tilde \theta \frac{{\partial \left({\overline \theta + \theta '} \right)}}{{\partial t}}} \right],$ (89)
 ${e_2} = - \gamma \left(\begin{array}{l} \tilde \theta \;\overline u \frac{{\partial \overline \theta }}{{\partial x}}\; + \tilde \theta \;\overline v \frac{{\partial \overline \theta }}{{\partial y}} + \tilde \theta \;\tilde u\frac{{\partial \overline \theta }}{{\partial x}} + \tilde \theta \;\tilde v\frac{{\partial \overline \theta }}{{\partial y}} + \\ {\rm{ }}\tilde \theta \;u'\frac{{\partial \overline \theta }}{{\partial x}} + \tilde \theta \;v'\frac{{\partial \overline \theta }}{{\partial y}} \end{array} \right),$ (90)
 ${e_3} = - \gamma \left(\begin{array}{l} \tilde \theta \;\overline u \frac{{\partial \tilde \theta }}{{\partial x}}\; + \tilde \theta \;\overline v \frac{{\partial \tilde \theta }}{{\partial y}} + \tilde \theta \;\tilde u\frac{{\partial \tilde \theta }}{{\partial x}} + \tilde \theta \;\tilde v\frac{{\partial \tilde \theta }}{{\partial y}} + \\ {\rm{ }}\tilde \theta \;u'\frac{{\partial \tilde \theta }}{{\partial x}} + \tilde \theta \;v'\frac{{\partial \tilde \theta }}{{\partial y}} \end{array} \right),$ (91)
 ${e_4} = - \gamma \left(\begin{array}{l} \tilde \theta \;\overline u \frac{{\partial \theta '}}{{\partial x}}\; + \tilde \theta \;\overline v \frac{{\partial \theta '}}{{\partial y}} + \tilde \theta \;\tilde u\frac{{\partial \theta '}}{{\partial x}} + \tilde \theta \;\tilde v\frac{{\partial \theta '}}{{\partial y}} + \\ \;\;\;\tilde \theta \;u'\frac{{\partial \theta '}}{{\partial x}} + \tilde \theta \;v'\frac{{\partial \theta '}}{{\partial y}} \end{array} \right),$ (92)
 ${e_5} = - \left({\frac{{\tilde \theta }}{{\overline \theta }}\overline w g} \right),$ (93)
 ${e_6} = - \left({\frac{{\tilde \theta }}{{\overline \theta }}\tilde wg} \right),$ (94)
 ${e_7} = - \left({\frac{{\tilde \theta }}{{\overline \theta }}w'g} \right),$ (95)
 ${e_8} = \gamma \frac{{\tilde \theta }}{{{c_p}}}{\left({\frac{p}{{{p_0}}}} \right)^{R/{{c}_{p}}}}\left({\overline Q + \tilde Q + Q'} \right),$ (96)

 $\begin{array}{l} \partial \tilde{A}/\partial \ t= \frac{\partial }{{\partial \;t}}\left({\frac{\gamma }{2}{{\tilde \theta }^2}} \right) = {P_{\;\alpha }} + {T_P}_{\;\;(\alpha, L)} + {T_P}_{\;\;(\alpha, \alpha)} + {T_P}_{\;\;(\alpha, \beta)} - \\ {B_{\;(\alpha, L)}} - {B_{\;(\alpha, \alpha)}} - {B_{\;(\alpha, \beta)}} + {D_{\;\alpha }}. \end{array}$ (97)

2.3.3 计算$\tilde{\theta }\times (4)$，求${\partial \overline{A}}/{\partial \ t}\;$
 $\begin{array}{l} \partial \bar{A}/\partial \ t = \frac{\partial }{{\partial \;t}}\left({\frac{\gamma }{2}{{\overline \theta }^2}} \right) = {f_1} + {f_2} + {f_3} + {f_4} + {f_5} + \\ {f_6} + {f_7} + {f_8}, \end{array}$ (98)
 ${f_1} = - \left[ {\gamma \;\overline \theta \frac{{\partial \left({\tilde \theta + \theta '} \right)}}{{\partial t}}} \right],$ (99)
 ${f_2} = - \gamma \left[ \begin{array}{l} \overline \theta \;\overline u \frac{{\partial \overline \theta }}{{\partial x}}\; + \overline \theta \;\overline v \frac{{\partial \overline \theta }}{{\partial y}} + \overline \theta \;\tilde u\frac{{\partial \overline \theta }}{{\partial x}} + \overline \theta \;\tilde v\frac{{\partial \overline \theta }}{{\partial y}} + \\ \;\;\;\overline \theta \;u'\frac{{\partial \overline \theta }}{{\partial x}} + \overline \theta \;v'\frac{{\partial \overline \theta }}{{\partial y}} \end{array} \right],$ (100)
 ${f_3} = - \gamma \left[ \begin{array}{l} \overline \theta \;\overline u \frac{{\partial \tilde \theta }}{{\partial x}}\; + \overline \theta \;\overline v \frac{{\partial \tilde \theta }}{{\partial y}} + \overline \theta \;\tilde u\frac{{\partial \tilde \theta }}{{\partial x}} + \overline \theta \;\tilde v\frac{{\partial \tilde \theta }}{{\partial y}} + \\ \;\;\;\;\overline \theta \;u'\frac{{\partial \tilde \theta }}{{\partial x}} + \overline \theta \;v'\frac{{\partial \tilde \theta }}{{\partial y}} \end{array} \right],$ (101)
 ${f_4} = - \gamma \left[ \begin{array}{l} \overline \theta \;\overline u \frac{{\partial \theta '}}{{\partial x}}\; + \overline \theta \;\overline v \frac{{\partial \theta '}}{{\partial y}} + \overline \theta \;\tilde u\frac{{\partial \theta '}}{{\partial x}} + \overline \theta \;\tilde v\frac{{\partial \theta '}}{{\partial y}} + \\ \;\;\;\;\overline \theta \;u'\frac{{\partial \theta '}}{{\partial x}} + \overline \theta \;v'\frac{{\partial \theta '}}{{\partial y}} \end{array} \right],$ (102)
 ${f_5} = - \left({\frac{{\overline \theta }}{{\overline \theta }}\overline w g} \right),$ (103)
 ${f_6} = - \left({\frac{{\overline \theta }}{{\overline \theta }}\tilde wg} \right),$ (104)
 ${f_7} = - \left({\frac{{\overline \theta }}{{\overline \theta }}w'g} \right),$ (105)
 ${f_8} = \gamma \frac{{\overline \theta }}{{{c_p}}}{\left({\frac{p}{{{p_0}}}} \right)^{R/{{c}_{p}}}}\left({\overline Q + \tilde Q + Q'} \right),$ (106)

 $\begin{array}{l} \partial \bar{A}/\partial \ t = \frac{\partial }{{\partial \;t}}\left({\frac{\gamma }{2}{{\overline \theta }^2}} \right) = {P_{\;L}} + {T_P}_{\;\;(L, L)} + {T_P}_{\;\;(L, \alpha)} + {T_P}_{\;\;(L, \beta)} - \\ {B_{\;\;(L, L)}} - {B_{\;\;(L, \alpha)}} - {B_{\;\;(L, \beta)}} + {D_{\;L}}. \end{array}$ (107)

3 结论与讨论

 图 1 三种尺度动能和位能之间转换示意图 Figure 1 Energy conversions between kinetic energy and potential energy between three scales

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