= Solution
A horizontal <pressure gradient force> accelerates air from higher toward lower <pressure>. In a rotating frame the <Coriolis force> deflects the flow. Away from strong friction, at large spatial scales and small <Rossby number>, a slowly changing flow can reach <geostrophic balance>:
$$
f\widehat{\mathbf z}\times\mathbf u_g=-\frac1\rho\nabla_h p,\qquad \boxed{\mathbf u_g=\frac1{\rho f}\widehat{\mathbf z}\times\nabla_h p},
$$
where $f=2\Omega\sin\varphi$ is the <Coriolis parameter>. The <geostrophic wind> runs along <atmospheric isobars>, with speed $|\nabla_h p|/(\rho|f|)$. Closely spaced <atmospheric isobars> therefore indicate stronger <geostrophic wind> when <density> and latitude are comparable. In the northern hemisphere lower <pressure> lies to the left of this wind, producing counterclockwise circulation around a low; the sense reverses in the southern hemisphere.
A curved flow also has <centripetal acceleration>. For cyclonic circular flow in the northern hemisphere, the <gradient wind> relation is
$$
\frac{v^2}{r}+fv=\frac1\rho\frac{dp}{dr}.
$$
Near the ground, friction reduces wind speed and weakens its Coriolis contribution, so air crosses <atmospheric isobars> toward lower <pressure>. The simple geostrophic formula fails near the equator, where $f$ becomes small, and where rapid changes or tight curvature make neglected acceleration important. Vertically, large-scale motion is often approximated by <hydrostatic balance>, $dp/dz=-\rho g$.
<Bernoulli's principle> can relate speed changes to <pressure> changes along an appropriate inviscid streamline, but its simple dynamic-pressure relation is not a replacement for the rotating force balance governing a broad atmospheric pressure map.
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