Atmospheric Tower Profile

What is modelled

Wind slows near the ground because of terrain roughness and increases with height; temperature falls with height at a chosen lapse rate (it can also rise in a night-time inversion); pressure follows from the weight of the air above. The tower is treated as a slender cylinder of constant illustrative diameter, so wind pressure times diameter gives a load per metre, and integrating it over height gives the force and the overturning moment at the base.

Equations

• Power-law wind profile u(z) = u10 (z / 10 m)α with the Counihan exponent α = 0.24 + 0.096 log10 z0 + 0.016 (log10 z0)².
• Temperature T(z) = Ts - Γ z, surface temperature with a daily cycle peaking at 15:00.
• Hydrostatic pressure p = p0 (1 - Γ z / Ts)g / (R Γ), density ρ = p / (R T).
• Dynamic pressure q = ½ ρ u², load per metre q Cd D, base moment M = ∫ q Cd D z dz.
• Vortex shedding f = St u / D with St = 0.2.

Checks run in the page

• Hydrostatic integration against the closed-form pressure at the crown.
• The standard atmosphere against tabulated pressure and temperature at 1000 m.
• Power law against the logarithmic law between 10 and 100 m.
• Moment from slicing the tower against the closed form.

Honest simplifications

The power law is an engineering fit that is best in the lowest few hundred metres; above that, real winds depend on stability and the geostrophic wind. The gust factor is a simple turbulence-intensity rule, the tower is a uniform cylinder with an invented diameter, and there is no dynamic response, no shielding and no stability correction. The 3D lab's own sway rule is shown only for comparison in scene units.

• Choose Calm night inversion: temperature rises with height and the wind changes very little with height.
• Choose City terrain and compare the crown wind with the Default preset: the exponent grows with roughness.
• Choose Storm over open sea and watch the base moment grow with the square of the wind.