一般 \(\sigma\) 坐标转换方程为

\[\sigma = \frac{z-\eta}{D} = \frac{z-\eta}{H+\eta}
\]

转换后水深 z 范围由原始的 \([-H, \eta]\) 变为 \(\sigma \in [-1,0]\)，将这组坐标转换定义为 X = G(x)，即

\[G(x,y,z) = (x^*, y^*, \sigma^*)
\]

\(\sigma\) 坐标转换后碰到最主要问题就是，在原始 z 坐标方程中各个微分项如何变化？

以水平压力梯度计算为例，设函数 \(F(x,y,z)\) 为 z 坐标内压力场，F*为对应 \(\sigma\) 坐标内函数形式，即

\[\begin{array}{c}
\left[x,y,z \right] = G(x^*, y^*, \sigma^*) \cr
F(x,y,z) = F^*(x^*,y^*,\sigma^*)
\end{array}\]

F 在z坐标内水平梯度为 \(\nabla F = (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y})\)，我们来看其中 x 分量的计算

\[\frac{\partial F}{\partial x} = lim \frac{F(x_0+\Delta x, y_0, z_0) - F(x_0, y_0, z_0)}{\Delta x}
\]

为了在 \(\sigma\) 坐标内表示上述极限形式方程，就应该寻找函数 \(F(x_0+\Delta x, y_0, z_0)\) 与 \(F(x_0, y_0, z_0)\) 在 \(\sigma\) 坐标内表示形式。

若

\[\begin{array}{c}
\left[x,y,z \right] = G(x^*, y^*, \sigma^*) \cr
F(x,y,z) = F^*(x^*,y^*,\sigma^*)
\end{array}\]

那么当 z 坐标内此点 x 坐标有增量 \(\Delta x^*\) 时，对应压力场的值

\[\begin{array}{c}
\left[x+\Delta x,y,z \right] = G(x^*+\Delta x^*, y^*, \sigma^*+\Delta \sigma^*) \cr
F(x+\Delta x,y,z) = F^*(x^*+\Delta x^*, y^*, \sigma^*+\Delta \sigma^*)
\end{array}\]

为什么 x 增加增量后，对应 \(\sigma\) 坐标内垂向坐标 \(\sigma^*\) 也会产生增量？这个增量大小 \(\Delta \sigma\) 是多少？

首先我们看 \(\sigma\) 坐标转换方程

\[\sigma^*(x,y) = \frac{z-\eta}{D}=\frac{z-\eta(x,y,t)}{H(x,y)+\eta(x,y,t)}
\]

由于 \(\eta, H\) 都是 x 和 y 的函数，因此对应的 \(\sigma\) 也是与 x 和 y 相关的。所以若 \(x_0\) 增加增量 \(\Delta x\) 后，对应 \(\sigma^*\) 也会产生增量

\[\Delta \sigma^* = \frac{\partial \sigma^*}{\partial x} \Delta x
\]

其中 \(\frac{\partial \sigma^*}{\partial x}\) 可以根据 \(\sigma^*\) 表达式计算

\[\frac{\partial \sigma^*}{\partial x} = \frac{-\frac{\partial \eta}{
\partial x}D - (z-\eta)\frac{\partial D}{
\partial x} }{D^2} = -\left( \frac{1}{D}\frac{\partial \eta}{
\partial x} + \frac{\sigma}{D}\frac{\partial D}{
\partial x} \right)\]

同时利用

\[\begin{array}{l}
F^*(x^*+\Delta x^*, y^*, \sigma^*+\Delta \sigma^*) \cr
=F^*(x^*+\Delta x^*, y^*, \sigma^*+\Delta \sigma^*) -F^*(x^*+\Delta x^*, y^*, \sigma^*) + F^*(x^*+\Delta x^*, y^*, \sigma^*) \cr
= \frac{\partial F^*}{\partial \sigma^*}\Delta \sigma^* + F^*(x^*+\Delta x, y^*, \sigma^*) \cr
\end{array}\]

\[\begin{array}{l}
\frac{\partial F}{\partial x} = \frac{\frac{\partial F^*}{\partial \sigma^*}\Delta \sigma^* + F^*(x^*+\Delta x^*, y^*, \sigma^*) - F^*(x^*, y^*, \sigma^*) }{\Delta x} \cr
= \frac{\partial F^*}{\partial \sigma^*}\frac{\partial \sigma^*}{\partial x} + \frac{\partial F^*}{\partial x} \cr
= \frac{\partial F^*}{\partial \sigma^*}\frac{\partial \sigma^*}{\partial x} + \frac{\partial F^*}{\partial x^*}\frac{\partial x^*}{\partial x}
\end{array}\]

故 \(\nabla F\) 在 \(\sigma\) 坐标内表示形式为

\[\begin{array}{l}
(\frac{\partial F^*}{\partial \sigma^*}\frac{\partial \sigma^*}{\partial x} + \frac{\partial F^*}{\partial x}, \frac{\partial F^*}{\partial \sigma^*}\frac{\partial \sigma^*}{\partial y} + \frac{\partial F^*}{\partial y})\end{array}\]

将上面的 \(\frac{\partial \sigma^*}{\partial x}\) 代入便可得到完整表达形式。与笛卡尔坐标系差别很大的根本原因就是 \(\sigma\) 也是 x 和 y 的函数。