Pressure difference between two points on the Earth's surface

Question

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.

Answer 1

Attempt 1 is right. If you want to see a contribution from the Coriolis term, you should make the wind blow into or out of the page rather than along the direction from X to Y.