Wind Shear⚠ unverified

Understanding the Parameters

• Reference height $h_0 = 10\ \text{m}$ — not arbitrary: 10 m is the WMO standard anemometer height, so virtually every published wind climatology, airport METAR, and wind-resource map is quoted there. The default exists so that a station reading drops straight in.

• The exponent $\alpha$ — the only interesting parameter. It encodes surface roughness: rougher ground drags more momentum out of the low-level wind, making the profile steeper. Representative values:

Terrain	$\alpha$
Smooth sea, ice, open water	0.10
Open flat country, short grass (default)	0.14
Crops, scattered hedges	0.20
Wooded country, suburbs	0.25
City centre, tall buildings	0.33+

$\alpha$ also rises at night under stable stratification (a nocturnal inversion decouples the surface from the air above, so wind aloft accelerates while the surface goes calm) and falls in unstable daytime convection, which mixes momentum down. A single fixed $\alpha$ therefore hides real diurnal variation — sometimes a factor of two.

• Why $\alpha$ is small — and why that matters — a small exponent means a strongly concave profile: the wind gains fast near the ground, then flattens. Between 10 m and 100 m (a tenfold climb) the wind rises only by $10^{0.14} = 1.38$, i.e. 38 %. That modest-sounding figure is decisive in wind energy: power scales as $V^3$, so a 38 % speed gain is a $1.38^3 = 2.6\times$ power gain. This single fact is why wind turbines are mounted on tall towers, and why tower height is often a bigger design lever than rotor diameter.

• Altitude $h$ — mind the domain — the formula is meant for the surface layer, roughly the first 100–200 m. It has two failure modes at the extremes. As $h \to 0$ it gives $V \to 0$, which sounds right but is wrong in detail — the true profile approaches zero logarithmically, not as a power. And it grows without bound as $h$ increases, whereas real wind speed tops out at the gradient height (300–600 m) where surface friction stops mattering and the flow reaches the geostrophic balance of pressure gradient against Coriolis. Extrapolating this equation to cruise altitude is meaningless. The default $h = 1.0\ \text{m}$ is a placeholder that returns $V_0 \times 10^{-0.14} = 0.72\,V_0$ — below the reference, as expected.

• What this is not — in aviation, "wind shear" is a hazard: microbursts, gust fronts, and low-level jets, where wind changes sharply over seconds and metres and can strip an aircraft of airspeed on approach. That is a transient, three-dimensional phenomenon. This calculator gives only the steady mean profile, and will tell you nothing about a downdraft.

Derivation (Approaching a Proof)

There is no first-principles derivation. The power law is a curve fit — a convenient function that happens to match observed profiles over a limited range. Being clear about that is more useful than dressing it up. But its legitimacy comes from its close agreement with a profile that is derivable, so the honest account is to show the real one and then the fit.

The physically-grounded result: the log law. In the surface layer, momentum is transported downward by turbulent eddies. The controlling quantity is the friction velocity $u_\ast$, defined from the surface shear stress $\tau_0$ by $u_\ast \equiv \sqrt{\tau_0/\rho}$. Prandtl's mixing-length argument observes that the only length scale available to an eddy near a wall is its distance from that wall, so the eddy size $\ell = \kappa z$ grows linearly with height ($\kappa \approx 0.41$, the von Kármán constant). Dimensional analysis then forces the velocity gradient — built from the only velocity scale $u_\ast$ and the only length scale $\kappa z$ — to be

$$\frac{dV}{dz} = \frac{u_\ast}{\kappa z}.$$

Integrating from the roughness length $z_0$ (the height at which the extrapolated wind vanishes) gives the logarithmic wind profile:

$$V(z) = \frac{u_\ast}{\kappa}\ln\!\left(\frac{z}{z_0}\right).$$

This is derived — from turbulence physics and dimensional reasoning — and it is what meteorology actually uses for neutral conditions.

So why use a power law at all? Because it is far more convenient: it needs no $u_\ast$ (which is not measured at airports) and it scales directly from one anemometer reading. And over the range of interest the two agree well. Sketch the equivalence: take the log law at two heights and ask what $\alpha$ makes the power law match. Setting

$$\frac{V(h)}{V(h_0)} = \frac{\ln(h/z_0)}{\ln(h_0/z_0)} \stackrel{!}{=} \left(\frac{h}{h_0}\right)^{\alpha},$$

and taking logarithms,

$$\alpha = \frac{\ln\!\left[\ln(h/z_0)\big/\ln(h_0/z_0)\right]}{\ln(h/h_0)}.$$

The right-hand side is not constant — it drifts with $h$ — which is exactly why the power law is an approximation rather than an identity. But over a decade or so of height it varies slowly enough to treat as fixed, and it reproduces the roughness dependence: a common engineering shorthand is $\alpha \approx 1/\ln(h_r/z_0)$ for a reference height $h_r$. With open-country $z_0 \approx 0.03\ \text{m}$ and $h_r = 10\ \text{m}$: $\alpha \approx 1/\ln(333) \approx 0.17$ — the right order, close to the classic $1/7 = 0.143$.

Where $1/7$ comes from. The value is inherited from pipe flow, not meteorology. Ludwig Prandtl and Theodore von Kármán found that turbulent velocity profiles in smooth pipes fit $u/u_{max} = (y/R)^{1/7}$ over a wide Reynolds range. The atmospheric boundary layer, being another turbulent shear flow over a wall, was found to fit the same exponent tolerably well over open terrain — a borrowed empirical coincidence, not a derivation. That $1/7$ has survived a century says more about its convenience than its rigour.

Dimensional check. $h/h_0 = \text{m}/\text{m}$ is dimensionless, so raising it to $\alpha$ leaves it dimensionless (as it must — a dimensional quantity raised to a fractional power would be meaningless), and $V$ inherits the m/s of $V_0$ ✓. Note this check also requires $\alpha$ to be dimensionless, as labelled.

History and Development

• Hellmann (1910s). The German meteorologist Gustav Hellmann published systematic measurements of how wind speed varies with height near the ground, and the power-law exponent still carries his name in the European literature.

• Borrowed from pipes (1920s–30s). Prandtl's mixing-length theory (1925) and the Prandtl–von Kármán work on turbulent pipe flow produced both the log law and the $1/7$ power fit. The log law is the rigorous result; the power law was the practical one, and the atmospheric community adopted the pipe-flow exponent wholesale.

• Monin–Obukhov similarity (1954). Andrei Monin and Alexander Obukhov extended the log law to non-neutral conditions, adding stability corrections through the Obukhov length. This is the modern rigorous framework and it explains why $\alpha$ varies with time of day — something the bare power law can only absorb into a fudged exponent.

• Wind energy revives it (1970s– ). The power law's second life came with wind-turbine siting after the 1970s oil shocks. Resource assessment needs to extrapolate a 10 m station record to an 80–120 m hub, over decades of data, at thousands of candidate sites — and the power law's one-parameter simplicity beat the log law's need for $u_\ast$ and $z_0$. IEC 61400-1, the wind-turbine design standard, specifies $\alpha = 0.2$ for the normal wind profile model used in load calculations. Every wind farm built since is, in part, a bet on this equation.

• The aviation hazard is a different story. The Eastern Air Lines Flight 66 (JFK, 1975) and Delta Air Lines Flight 191 (DFW, 1985) microburst accidents drove Ted Fujita's identification of the downburst and led to the FAA's terminal Doppler weather radar and airborne predictive windshear systems. That programme addressed the transient shear this steady profile deliberately ignores — a useful reminder that the same two words name two very different problems.

Related Concepts: ISA Density, ISA Temperature, Dynamic Pressure, Reynolds Number Atm, Drag Force, Lift Force

Notes: Empirical correlation, not a derived law — no ISA standing; the physically-derived neutral profile is the log law $V = (u_\ast/\kappa)\ln(z/z_0)$, which this approximates over ~a decade of height. $\alpha \approx 0.14$ ($1/7$) is borrowed from turbulent pipe flow (Prandtl/von Kármán), not derived for the atmosphere; it encodes terrain roughness (0.10 sea → 0.33+ city) and also rises under stable nocturnal stratification. Valid in the surface layer only (~first 100–200 m); real wind caps at the gradient height (300–600 m), so extrapolation to flight altitude is meaningless. $h_0 = 10\ \text{m}$ is the WMO standard anemometer height. Power $\propto V^3$, so a 38 % speed gain from 10→100 m is a 2.6× power gain — the reason turbine towers are tall (IEC 61400-1 uses $\alpha = 0.2$). Not the aviation microburst hazard, which is a transient this steady profile does not model.