The Euler equations for steady fluid flow $\mathbf{u}$ in a rapidly rotating system can be written

$$\rho \mathbf{f} \times \mathbf{u}=-\nabla p+\rho \mathbf{g},$$

where $\rho$ is the density of the fluid, $p$ is its pressure, $\mathbf{g}$ is the acceleration due to gravity and $\mathbf{f}=(0,0, f)$ is the constant Coriolis parameter in a Cartesian frame of reference $(x, y, z)$, with $z$ pointing vertically upwards.

Fluid occupies a layer of slowly-varying height $h(x, y)$. Given that the pressure $p=p_{0}$ is constant at $z=h$ and that the flow is approximately horizontal with components $\mathbf{u}=(u, v, 0)$, show that the contours of $h$ are streamlines of the horizontal flow. What is the leading-order horizontal volume flux of fluid between two locations at which $h=h_{0}$ and $h=h_{0}+\Delta h$, where $\Delta h \ll h_{0}$ ?

Identify the dimensions of all the quantities involved in your expression for the volume flux and show that your expression is dimensionally consistent.