< ol > <李> < p >压力坐标通常使用< span class = " math-container " > \ω< / span >美元的垂直速度分量,而不是< span class = " math-container " > w < / span >美元如笛卡尔坐标。李李< / p > < / > < > < p >获得压力坐标系中的速度组件:< / p > < /李> < / ol > < p >使用物质导数可以与垂直速度分量和我将在下面描述怎么做。< / p > < p >开头两个坐标系统类< span = " math-container " >美元(x, y, z, t) $ < / span >和<跨类= " math-container " >美元(x, y, p、t) $ < / span >与垂直坐标<跨类= " math-container " > z z =美元美元(t, x, y, p) < / span >和<跨类= " math-container " > $ p = p (t, x, y, z) < / span >美元。一些变量的导数< span class = " math-container " > < / span >美元美元对其他变量类< span = " math-container " > < / span >加元美元(可以<跨类= " math-container " > $ x $ < / span >, < span class = " math-container " > $ y $ < / span >或<跨类= " math-container " > t < / span >)美元在一个系统中其他相关如下:< / p > < p > <跨类= " math-container " > \开始{方程}\离开(\压裂{\部分}{部分c \} \右)_z = \离开(\压裂{\部分}{部分c \} \右)_p + \压裂{\部分}{\部分p} \离开(\压裂{\部分p}{部分c \} \右)_z。结束\{方程}< / span >的垂直坐标认为< span class = " math-container " > \开始{方程}\压裂{\部分}{\部分p} = \压裂{\部分}{部分z \} \压裂{\部分z}{\部分p}, \{方程}< / span >结束,因此,我们发现<跨类= " math-container " > \开始{方程}\离开(\压裂{\部分}{\部分c} \右)_z = \离开(\压裂{\部分}{\部分c} \右)_p + \压裂{\部分}{部分z \} \压裂{\部分z}{\部分p} \离开(\压裂{\部分p}{\部分c} \右)_z。结束\{方程}< / span > < / p > < p >这个(2 d)梯度写道:< / p > < p > <跨类= " math-container " > \开始{方程}\ nabla_z = \ nabla_p + \压裂{\部分z}{\部分p} \压裂{\部分}{部分z \} \ nabla_z p。\{方程}< / span >结束时间导数是微不足道的,因为它遵循从上面的一般形式。Now we can write the material derivative in pressure coordinates as follows: \begin{equation} \begin{split} \left( \frac{\text{D}}{\text{D} t} \right)_p &= \left( \frac{\partial}{\partial t} \right)_p+ \vec{u} \cdot \nabla_p + \omega \frac{\partial}{\partial p}\\ &= \left( \frac{\partial }{\partial t} \right)_z - \frac{\partial z}{\partial p}\left( \frac{\partial p}{\partial t} \right)_z \frac{\partial }{\partial z} + \vec{u} \cdot \left[ \nabla_z - \frac{\partial z}{\partial p} \nabla_z p \frac{\partial }{\partial z} \right] + \omega \frac{\partial z}{\partial p}\frac{\partial}{\partial z} \\ &= \left( \frac{\partial }{\partial t} \right)_z + \vec{u} \cdot \nabla_z + \left[\omega - \left(\frac{\partial p}{\partial t}\right)_z - \vec{u} \cdot \nabla_z p \right] \frac{\partial z}{\partial p} \frac{\partial}{\partial z} . \end{split} \end{equation} Hence, \begin{equation} w = \left[\omega - \left(\frac{\partial p}{\partial t}\right)_z - \vec{u} \cdot \nabla_z p \right] \frac{\partial z}{\partial p} \end{equation} and \begin{equation} \omega =\left( \frac{\partial p }{\partial t}\right)_z + \vec{u} \cdot \nabla_z p + w \frac{\partial p}{\partial z}. \end{equation} What typically follows is the use of the hydrostatic approximation in the last term. If you do a (large) scale analysis you will find that $-w\rho g = w \partial p/\partial z$ is the dominating term and thus, you can approximate $\omega = -w\rho g$. However, based on your sketch I think it would not add to an understanding to just say $\omega = 0$, since $w = 0$. Why it makes sense to use pressure coordinates I tried to answer here, although I want to add that in a theoretical context it can be much more useful to apply e.g. sigma coordinates (terrain following coordinates) to simplify the treatment of boundary conditions.