In studies of rotating fluid flows around a relativistic star, the vertical height, or (half)-thickness, of the flow is usually obtained from the vertical component of the Euler equation.
The Euler equation: $$u^\mu\nabla_\mu u^\nu + \frac{1}{P+\rho}(g^{\mu\nu}+u^\mu u^\nu)\frac{\partial P}{\partial x^\mu}=0$$ Here, $\nabla_\mu$ is the covariant derivative and $\nu$ is the free index.
Usually, the vertical Euler equation is written in cylindrical coordinates. In this case, $\nu=z$ is chosen and we obtain a $\frac{\partial P}{\partial z}$ term. Then, in case of a thin flow the following approximation is used: $$\frac{\partial P}{\partial z}\simeq\frac{P}{H}.$$
Question: If I use spherical coordinates instead of cylindrical coordinates, then I need to choose $\nu=\theta$ and this would give me a $\frac{\partial P}{\partial \theta}$ term. How can I choose an approximation in this case similar to the case with cylindrical coordinates? Certainly, I cannot write $\frac{\partial P}{\partial \theta}\simeq\frac{P}{\theta}$, and I need somehow to express this in terms of the height $H$ of the flow.