= Solution
In the horizontal rotating-fluid <equation of motion>, scale advective acceleration as $U^2/L$ and <Coriolis acceleration> as $|f|U$, where $f=2\Omega\sin\phi$. Their ratio is the <Rossby number>,
$$
\boxed{\mathrm{Ro}=\frac{U}{|f|L}.}
$$
A small <Rossby number> means that rotation strongly constrains the flow; with slow evolution, weak friction and a suitable pressure field it permits <geostrophic balance>. Local time dependence introduces the separate condition $1/(|f|t_{\rm evol})\ll1$. A synoptic midlatitude cyclone with $U\sim10\,\mathrm{m\,s^{-1}}$, $L\sim1000\,\mathrm{km}$ and $|f|\sim10^{-4}\,\mathrm{s^{-1}}$ has $\mathrm{Ro}\sim0.1$. A tornado with $U\sim50\,\mathrm{m\,s^{-1}}$ and $L\sim1\,\mathrm{km}$ has $\mathrm{Ro}\sim500$; its rotational motion does not mean that the Earth's rotation dominates its local dynamics.
The vertical <relative vorticity> is $\zeta=\partial_xv-\partial_yu\sim U/L$, while planetary <vorticity> is $f$. Thus $|\zeta/f|$ is of the order of the <Rossby number>, although it is not an identity for arbitrary velocity fields: an irrotational strain can have nonzero $U/L$ but zero $\zeta$. Absolute <vorticity> is $f+\zeta$.
On large, slowly evolving midlatitude scales, the horizontal <pressure gradient force> is mainly balanced by the <Coriolis force>, which is perpendicular to velocity. Consequently the wind is nearly parallel to <atmospheric isobars>, rather than accelerating directly into a low or out of a high. Curvature adds a <gradient wind> correction, and surface friction permits the cross-isobar component considered in part (d).
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