The Weather Equation - Numberphile

TL;DR

The quasi-geostrophic omega equation diagnoses atmospheric vertical velocity from vorticity and temperature advection, letting forecasters identify where low and high pressure systems develop.

Key Points

• 1.The QG omega equation is diagnostic, not predictive. It reveals current vertical velocity from vorticity and thermal advection but says nothing about future atmospheric states.
• 2.Omega (ω) is vertical velocity in pressure coordinates, measured in Pascals per second. It relates to geometric vertical velocity W via the hydrostatic equation: ω ≈ −ρgW, scaled by air density.
• 3.Measuring vertical wind directly is nearly impossible. Errors from sparse horizontal wind measurements are the same magnitude as the vertical motion itself, making equations the only reliable diagnostic tool.
• 4.Geostrophic balance is the equilibrium between pressure gradient force and Coriolis force. It explains why winds blow parallel to isobars around low pressure systems rather than straight into them.
• 5.The Rossby number (Ro = V/fL) quantifies how geostrophic a flow is. Synoptic-scale weather systems yield Ro ≈ 0.1 (near-geostrophic), while a tropical cyclone eye yields Ro ≈ 1,000 (cyclostrophic, not geostrophic).
• 6.Planetary vorticity f (Coriolis parameter) at Exeter (50°N) ≈ 1.1 × 10⁻⁴ per second. Calculated as f = 2Ω sin(latitude), where Ω = 7.292 × 10⁻⁵ rad/s.
• 7.Pure geostrophic wind has zero divergence, making vertical velocity incalculable from it alone. This is why the 'quasi' prefix exists — the full wind must be retained in select terms to preserve vertical motion information.
• 8.The QG system enabled the first workable numerical weather forecast models in the early 1950s. While the broader equation set is now obsolete, the omega equation remains actively used by operational forecasters for conceptual diagnosis.
• 9.Forecasters use the equation qualitatively, not numerically. Positive vorticity advection or warm temperature advection both indicate ascent (negative ω), signaling developing low pressure, storms, or named weather systems.