気象学 : 大気の移動方程式 : 移動方程式

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移動方程式の行程

作成: 2022-09-20
更新: 2022-10-08

$P_x( 0 ) = P0_x \\ P_y( 0 ) = P0_y \\ P_z( 0 ) = P0_z \\ \ \\ v_x( 0 ) = v0_x \\ v_y( 0 ) = v0_y \\ v_z( 0 ) = v0_z \\$

$t \rightarrow t + \Delta t$

$P_x( t + \Delta t ) = P_x( t ) - v_x( t )\ cos\bigl( sin^{-1}\bigl( \frac{P_z( t )}{R}\ \bigr) \bigr)\ \Delta t \\ P_y( t + \Delta t ) = P_y( t ) + v_y( t )\ cos\bigl( sin^{-1}\bigl( \frac{P_y( t )}{R}\ \bigr) \bigr)\ \Delta t \\ P_z( t + \Delta t ) = P_z( t ) + v_z( t )\ cos\bigl( sin^{-1}\bigl( \frac{P_z( t )}{R}\ \bigr) \bigr)\ \Delta t \\ \ \\ v_x( t + \Delta t ) = v_x( t ) + a_x(t)\ \Delta t \\ v_y( t + \Delta t ) = v_y( t ) + a_y(t)\ \Delta t \\ v_z( t + \Delta t ) = v_z( t ) + a_z(t)\ \Delta t \\$

$P(t),\ {\bf{v}}(t)$

$P( t + \Delta t )$

${\bf{v}}( t + \Delta t )$

$( P(t), {\bf{v}}(t) )$

$P( t + \Delta t ),\ {\bf{v}}( t + \Delta t )$

$P( ( t + \Delta t ) + \Delta t' )$

${\bf{v}}( ( t + \Delta t ) + \Delta t' )$

$( P( t + \Delta t), {\bf{v}}( t + \Delta t) )$

$[ t,\ t + \Delta t ]$

$P(t),\ {\bf{v}}(t)$ の固定座標から開始
$( P(t), {\bf{v}}(t) )$ -回転角 $\alpha(t)$ を求める
$P(t),\ {\bf{v}}(t)$ の固定座標を $( P(t), {\bf{v}}(t) )$ -座標に変換
$P(t),\ {\bf{v}}(t)$ に対する加速度 ${\bf{a}}(t)$ を $( P(t), {\bf{v}}(t) )$ -座標で計算
$P(t + \Delta t)$ を $( P(t), {\bf{v}}(t) )$ -座標で計算
${\bf{v}}(t + \Delta t)$ を $( P(t), {\bf{v}}(t) )$ -座標で計算
$P(t + \Delta t),\ {\bf{v}}(t + \Delta t)$ の $( P(t), {\bf{v}}(t) )$ -座標を固定座標に変換

1. $P(t),\ {\bf{v}}(t)$ の固定座標から開始

$P(t) = (P_\hat{x}(t),\ P_\hat{y}(t),\ P_\hat{z}(t) ) \\ {\bf{v}}(t) = ( v_\hat{x}(t),\ v_\hat{y}(t),\ v_\hat{z}(t) ) \\$

2. $( P(t), {\bf{v}}(t) )$ -回転角 $\alpha(t)$ を求める

$P_{\hat{z}} = 0 \Longrightarrow \ \alpha = sin^{-1} \bigl( \frac{ P_{\hat{y}} }{ R } \bigr) \\ \ \\ P_{\hat{z}} > 0 \Longrightarrow \ \alpha = sin^{-1} \bigl( \frac{ A }{ \sqrt{ A^2 + B^2 } } \bigr) \\ P_{\hat{z}} < 0 \Longrightarrow \ \alpha = - sin^{-1} \bigl( \frac{ A }{ \sqrt{ A^2 + B^2 } } \bigr)$

$\quad A = P_{\hat{x}}^2\ v_{\hat{y}} - P_{\hat{x}}\ P_{\hat{y}}\ v_{\hat{x}} - R^2\ v_{\hat{y}} \\ \quad B = P_{\hat{x}}\ P_{\hat{y}}\ v_{\hat{y}} - P_{\hat{y}}^2\ v_{\hat{x}} + R^2\ v_{\hat{x}} \\$

3. $P(t),\ {\bf{v}}(t)$ の固定座標を $( P(t), {\bf{v}}(t) )$ -座標に変換

$P_x(t) = P_\hat{x}(t)\ cos( \alpha(t) ) + P_\hat{y}(t)\ sin( \alpha(t) ) \\ P_y(t) = - P_\hat{x}(t)\ sin( \alpha(t) ) + P_\hat{y}(t)\ cos( \alpha(t) ) \\ P_z(t) = P_\hat{z}(t) \\ \ \\ v_x(t) = v_\hat{x}(t)\ cos( \alpha(t) ) + v_\hat{y}(t)\ sin( \alpha(t) ) \\ v_y(t) = - v_\hat{x}(t)\ sin( \alpha(t) ) + v_\hat{y}(t)\ cos( \alpha(t) ) \\ v_z(t) = v_\hat{z}(t) \\$

4. $P(t),\ {\bf{v}}(t)$ に対する加速度 ${\bf{a}}(t)$ を $( P(t), {\bf{v}}(t) )$ -座標で計算

${\bf{a}}(t) = ( a_x(t),\ a_y(t),\ a_z(t) ) \\ v(t) = | {\bf{v}}(t) |$

$P_x = R,\ P_y = P_z = 0,\ v_x = v_z = 0$

$S$

$a_x(t) = - \frac{v(t)^2}{R} + R \Omega^2 \\ a_y(t) = 0 \\ a_z(t) = 0 \\$

$P_y = 0,\ v_y = 0$

$S$

$a_x(t) = - \frac{v(t)^2 }{R^2}\ P_x(t) + \Omega^2 \ P_x(t) \\ a_y(t) = 0 \\ a_z(t) = - \frac{v(t)^2}{R^2}\ P_z(t) \\$

$P_x = R,\ P_y = P_z = 0$

$P$

$a_x(t) = - \frac{v(t)^2}{R}\ + R \Omega^2 \\ a_y(t) = 0 \\ a_z(t) = 0 \\$

$a_x(t) = - \frac{v(t)^2}{R^2}\ P_x(t) + \Omega^2\ P_x(t) \\ a_y(t) = - \frac{v(t)^2}{R^2}\ P_y(t) + \Omega^2 \ P_y(t) \\ a_z(t) = - \frac{v(t)^2}{R^2}\ P_z(t) \\$

5. $P(t + \Delta t)$ を $( P(t), {\bf{v}}(t) )$ -座標で計算

$P(t)_x = R,\ P(t)_y = P(t)_z = 0,\ v(t)_x = v(t)_z = 0$

$S$

$P(t + \Delta t)_x = R\ cos\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) \\ P(t + \Delta t)_y = R\ sin\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) \\ P(t + \Delta t)_z = 0$

$P(t)_y = 0,\ v(t)_y = 0$

$S$

$P(t + \Delta t)_x = P(t)_x\ cos\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) - P(t)_z\ sin\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) \\ P(t + \Delta t)_y = 0 \\ P(t + \Delta t)_z = P(t)_z\ cos\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) + P(t)_x\ sin\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr)$

$P(t)_x = R,\ P(t)_y = P(t)_z = 0$

$P$

$P(t + \Delta t)_y = R\ \frac{ v(t)_y }{ v }\ \bigl( sin( \frac{ v(t)\ \Delta t }{ R } ) \bigr)\ \\ P(t + \Delta t)_z = R\ \frac{ v(t)_z }{ v }\ \bigl( sin( \frac{ v(t)\ \Delta t }{ R } ) \bigr)\ \\$

$P(t + \Delta t)_x = P(t)_x\ cos\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) - \sqrt{ R^2 - P(t)_x^2 }\ \ \ \ sin\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) \\ P(t + \Delta t)_y = P(t)_y\ cos\bigl( \frac{ v(t)\ \Delta t }{ R } ) + \frac{ P(t)_x\ P(t)_y }{ \sqrt{ R^2 - P(t)_x^2 } }\ \ \ \ sin\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) \\ P(t + \Delta t)_z = P(t)_z\ cos\bigl( \frac{ v(t)\ \Delta t }{ R } ) + \frac{ P(t)_z\ P(t)_x }{ \sqrt{ R^2 - P(t)_x^2 } }\ \ \ \ sin\bigl( \frac{ v(t)\ \Delta t }{ R } \bigr) \\$

6. ${\bf{v}}(t + \Delta t)$ を $( P(t), {\bf{v}}(t) )$ -座標で計算

$v'_x = v(t)_x + a(t)_x\ \Delta t \\ v'_y = v(t)_y + a(t)_y\ \Delta t \\ v'_z = v(t)_z + a(t)_z\ \Delta t \\$

$v' = \sqrt{ {v'_x}^2 + {v'_y}^2 + {v'_z}^2 } \\ \theta = sin^{-1}( \frac{ P(t + \Delta t)_x\ v'_x + P(t + \Delta t)_y\ v'_y + P(t + \Delta t)_z\ v'_z }{ R\ v' }\ \ ) \\$

$v(t + \Delta t)_x = v'_x\ cos( \theta ) \\ v(t + \Delta t)_y = v'_y\ cos( \theta ) \\ v(t + \Delta t)_z = v'_z\ cos( \theta ) \\$

7. $P(t + \Delta t),\ {\bf{v}}(t + \Delta t)$ の $( P(t), {\bf{v}}(t) )$ -座標を固定座標に変換

$P(t + \Delta t)_\hat{x} = P(t + \Delta t)_x\ cos( \alpha ) - P(t + \Delta t)_y\ sin( \alpha ) \\ P(t + \Delta t)_\hat{y} = P(t + \Delta t)_x\ sin( \alpha ) + P(t + \Delta t)_y\ cos( \alpha ) \\ P(t + \Delta t)_\hat{z} = P(t + \Delta t)_z \\ \ \\ v(t + \Delta t)_\hat{x} = v(t + \Delta t)_x\ cos( \alpha ) - v(t + \Delta t)_y\ sin( \alpha ) \\ v(t + \Delta t)_\hat{y} = v(t + \Delta t)_x\ sin( \alpha ) + v(t + \Delta t)_y\ cos( \alpha ) \\ v(t + \Delta t)_\hat{z} = v(t + \Delta t)_z \\$