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Wind and Air: Geostrophic Balance and Limits

Derive a pressure-driven wind with consistent signs and a scale analysis.

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5 questions0m 0s

Prerequisites

Rotating reference frames, vector momentum equations, partial derivatives, and nondimensional scaling.

Learn the skill

Use x east, y north, constant density rho, and Northern Hemisphere f > 0. In steady frictionless horizontal geostrophic balance, -f v = -(1/rho) dp/dx and f u = -(1/rho) dp/dy. Thus u = -(dp/dy)/(rho f), v = (dp/dx)/(rho f). The approximation requires suitable large scales and small Rossby number Ro = U/(f L).

Worked example

For rho = 1 kg/m^3, f = 0.0001 s^-1, dp/dx = 0.002 Pa/m and dp/dy = 0, v = 20 m/s north and u = 0. The westward pressure force balances eastward Coriolis acceleration. This is not a boundary-layer or equatorial forecast.

For straight isobars in the Northern Hemisphere, a northward geostrophic wind balances a westward pressure-gradient acceleration against an eastward Coriolis acceleration. This excludes friction and uses nonzero positive f.
For straight isobars in the Northern Hemisphere, a northward geostrophic wind balances a westward pressure-gradient acceleration against an eastward Coriolis acceleration. This excludes friction and uses nonzero positive f.
Question 1 What is v for the numerical example?
Question 2 If dp/dx = 0 and dp/dy = 0.001 Pa/m with the same rho and f, what is u?
Question 3 For U = 10 m/s, f = 0.0001 s^-1 and L = 1000000 m, Ro is what?
Question 4 Why does this formula become inappropriate near the equator?
Question 5 Which setting needs important corrections to this ideal balance?

Further inquiry

Derive both horizontal wind components from the rotating-frame momentum equations. Nondimensionalize the advective term to recover Rossby number, then discuss friction, curved flow and the f approaching zero limit. Explain why a small Rossby number supports an approximation rather than proving every neglected term vanishes.

Review criteria

  • Keep axes, f sign, pressure-gradient signs and units explicit.
  • Compare acceleration terms with a dimensionless scale argument.
  • Identify where steady frictionless geostrophic balance is insufficient.