Why does relative vorticity in spherical coordinates have an extra term as compared to the usual cartesian derivation?

Question

With $u, v$ denoting the components of wind velocity along zonal (x) and meridional (y) direction, atmospheric (relative) vorticity is usually defined by

$$\zeta_z = \frac{\partial v}{\partial x} -\frac{\partial u}{\partial y} $$

This is the z-Component of the 3D-Vorticity, defined by

$$\vec \zeta = \vec \nabla \times \vec v$$

However, atmospheric flow is not on a flat earth but on a sphere. E.g. a westerly wind follows the curved earth's surface and is better described in spherical coordinates. There, the vertical component of the 3D-vorticity is obtained by carrying out the curl operation in spherical coordinates (see here). This would give

$$\zeta_z = \frac{1}{r \sin \theta } \left[ \frac{\partial }{\partial \theta} (u \sin \theta ) + \frac{\partial v}{\partial \varphi}\right]$$

$u, v$ are the local horizontal velocities along the unity vectors $\vec e_\varphi$ (zonal) and $-\vec e_\theta$ (meridonal). Note that here latitude $\theta $ is measured from pole and not, as usual, from equator.

By expanding this expression, we get

$$\zeta_z = \frac{1}{r \sin \theta } \left[ \sin \theta \frac{\partial u }{\partial \theta} + u \frac{\partial \sin \theta }{\partial \theta} + \frac{\partial v}{\partial \varphi}\right]$$

On the local tangential plane we have $dx = r \sin \theta d \varphi $ and $dy = -r d \theta $

so the derivatves with respect to $\varphi$ and $\theta$ can be replaced by derivates with respect to x and y. This gives finally:

$$\zeta_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} + \frac{u}{r} \cdot \frac{\cos \theta}{\sin\theta} $$

which is clearly not in accordance with the first equation: There is an extra term proportional to u. Why do we have this discrepancy and why is this not mentioned in standard textbooks?

Answer 1

The best way to see this extra term is to consider a bug moving along a latitude circle at constant speed $|u|$ in a constant direction arounf the poles. If the bug is moving along a geodesic then there is no vorticity, but the latitude circle is not a geodesic except at the Equator ($\cos\theta=0$). So movement along any other latitude circle will entail some vorticity, even with the vector components remaining constant.

The vorticity term with no component gradients has two additional features. First, we see that it is an odd function of displacement from the Equator; the Northern and Southern Hemispheres give opposite signs. This corresponds to the latitude circle deviating in opposite ways from the geodesic in the two hemispheres. If the "bug" in our example were to be taken as following the rotation of Earth itself, it would appear to be turning clockwise if it were seen from the South Pole (so it woukd be in the Southern Hemisphere) but counterclockwise if seen from the North Pole.

Second, the term goes to infinity as we approach either pole ($\sin\theta\to0$). The closer our bug gets to a pole, tge more angular velocity it would need to attain a given nonzero target $u$ value, and the vorticity in the absence of gradients follows suit. Natural motions near the pole a spherical body (including Earth) either have decreased speed in the latitude direction (balancing out the increased $\cot\theta$ factor) or quickly move away from the pole (the vorticity is then balanced out by nonzero derivative terms).