I'd like to find the pressure difference between two points $X$ and $Y$ on the Earth's surface given that wind blows steadily with speed $U$ from (say) West to East. Here's a picture: ($z_i$ is the height above sea-level in metres)
I have seen
- The Bernoulli equation for potential flow:
$\rho \frac{ \partial \phi}{\partial t} + \frac{1}{2} \rho \vec{u}^2 + p + \rho g z = f(t)$
and
- the Euler equation for a rotating fluid:
$\frac{D\vec{u}}{Dt} + 2 \rho \vec{\Omega} \times \vec{u} = -\nabla p + \rho \vec{g}$
Attempt 1:
If I neglect the rotation of the Earth and use Bernoulli, then I get
$p_X + \rho g z_1 = p_Y + \rho g z_2$,
since
- if the flow is steady the potential function $\phi$ is independent of $t$, so $\partial \phi/\partial t=0$
- the $\frac{1}{2} \rho \vec{u}^2$ terms cancel.
This gives a pressure difference of $\rho g(z_1-z_2)$ (where $\rho$ is the air density).
This solution doesn't seem right as I haven't used the extra information about the wind speed.
Attempt 2:
If we further assume that the air is incompressible ($\nabla U = 0$) then $Du/Dt = 0$.
If I can write $\vec{\Omega} \times \vec{u}$ in terms of known quantities, then I can compute $\nabla p$ from the Euler equation and I'm essentially done.
However, I'm not sure how to handle the $\vec{\Omega} \times \vec{u}$ term.
Any ideas? I think I'm overcomplicating this.
