# 動力氣象學/大氣運動基本方程組

## 運動方程

• ${\displaystyle {\frac {d_{a}{\vec {V}}_{a}}{dt}}=\sum _{i}{\vec {F}}_{i}}$

• ${\displaystyle {\frac {d_{a}{\vec {V}}_{a}}{dt}}={\frac {d{\vec {V}}}{dt}}+{\frac {d_{e}{\vec {V}}}{dt}}={\frac {d{\vec {V}}}{dt}}-\Omega ^{2}{\vec {R}}+2{\vec {\Omega }}\times {\vec {V}}=\sum _{i}{\vec {F}}_{i}}$

• ${\displaystyle {\frac {d_{e}{\vec {V}}}{dt}}=-\Omega ^{2}{\vec {R}}+2\Omega \land {\vec {V}}=\sum _{i}{\vec {F}}_{i}}$

• ${\displaystyle {\frac {d{\vec {V}}}{dt}}=\sum _{i}{\vec {F}}_{i}+\Omega ^{2}{\vec {R}}-2\Omega \land {\vec {V}}}$

## 連續方程

### 拉格朗日觀點

• ${\displaystyle {\frac {d\rho }{dt}}+\rho \nabla \cdot {\vec {V}}=0}$

• ${\displaystyle {\frac {d\rho }{dt}}}$：氣團密度隨體變化率
• ${\displaystyle \nabla \cdot {\vec {V}}={\frac {1}{\delta \tau }}{\frac {d\delta \tau }{d{\dot {\tau }}}}}$：氣團體積的相對變化率

### 歐拉觀點

• ${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {V}})=0}$

• ${\displaystyle {\frac {\partial \rho }{\partial t}}}$：固定空間密度的局地變化率——單位時間單位空間體積（固定）內的質量變化
• ${\displaystyle \nabla \cdot (\rho {\vec {V}})}$：單位時間單位空間體積內流體質量的流入流出量

## 熱流量方程

• ${\displaystyle C_{V}{\frac {dT}{dt}}+P{\frac {d\alpha }{dt}}={\dot {Q}}}$

• ${\displaystyle p\alpha =RT\rightarrow {\frac {d}{dt}}(P\alpha )={\frac {d}{dt}}(RT)}$

${\displaystyle P{\frac {d\alpha }{dt}}+\alpha {\frac {dP}{dt}}=R{\frac {dT}{dt}}}$

${\displaystyle (C_{V}+AR){\frac {dT}{dt}}-\alpha {\frac {dP}{dt}}={\dot {Q}}\Rightarrow C_{p}{\frac {dT}{dt}}-\alpha \omega ={\dot {Q}}}$

${\displaystyle {\begin{cases}{\frac {d{\vec {V}}}{dt}}={\vec {g}}-{\frac {1}{\rho }}\nabla p-2{\vec {\Omega }}\land {\vec {V}}+{\vec {F_{\gamma }}}\\{\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {V}})=0\\p=\rho RT\\C_{p}{\frac {dT}{dt}}-\alpha \omega ={\dot {Q}}\end{cases}}}$