Why the wind blows along the isobars, not across them. Air starts to fall down the pressure gradient from high to low — but the Coriolis veer turns it until the pressure push and the Coriolis force exactly cancel. What is left runs parallel to the isobars. Rendered, not quoted.
SOURCE Buys Ballot, C.H.D., “Note sur le rapport de l’intensité et de la direction du vent…”, Comptes rendus de l’Académie des sciences, 9 Nov 1857, pp.765–768 — no stable open link (Ferrel 1856 has theoretical priority); cite author/title/year. law overview.
Large-scale, slow air feels two horizontal forces. The pressure-gradient force
pushes from high toward low with magnitude (1/ρ)·dP/dn. The Coriolis
force deflects the moving air to the right (northern hemisphere), magnitude f·v,
where f = 2Ωsinφ.
At balance the two are equal and opposite: f·v = (1/ρ)·dP/dn.
Solving gives the geostrophic wind v₀ = (1/ρf)·dP/dn. Because Coriolis
acts sideways to the motion, the balanced flow is perpendicular to the pressure gradient
— it runs along the isobars.
The turning that makes this balance possible is the Coriolis effect. Geostrophic balance
is f v = (1/ρ) dP/dn, so the flow parallels the isobars with low pressure on the left
in the north — the-coriolis-effect supplying the veer that balances the pressure push, and
Buys Ballot’s law naming the geometry.
neighbour → the-coriolis-effect · this → the-geostrophic-balance
Re-runs the balance checks live and confirms the geometry. If Red’s tamper (6) makes the wind cross the isobars, this flips red at once.
Air density ρ = 1.2 kg/m³, Coriolis parameter f = 1×10⁻⁴ s⁻¹
(mid-latitude), and a pressure gradient dP/dn = 1×10⁻³ Pa/m
(≈ 1 hPa per 100 km) pointing from low toward high.
Geostrophic wind speed and direction, proven perpendicular to the gradient with low on the left:
“Wind is just air falling from high to low pressure — straight down the gradient.”
That is the initial impulse, not the steady flow. On a rotating planet the deflection accumulates until the wind runs 90° away from that push. Ignore Coriolis and every weather map is wrong: you would predict wind crossing the isobars instead of circling the lows and highs.
Faster spin (larger f) means a stronger wind.
→ Opposite: v₀ ∝ 1/f. Larger f means a weaker geostrophic wind
for the same gradient; near the equator f→0 and the balance breaks down entirely. AMBER
Geostrophic balance holds everywhere, always.
→ Only for large-scale, slow flow (small Rossby number). Tight, fast systems — tornadoes, tropical cyclone cores — add centrifugal terms and deviate. AMBER
Force the wind to blow down the pressure gradient — across the isobars, as if Coriolis did not exist. Buys Ballot’s law and the along-isobar flow both break; the Witness (7) catches it.