Near-surface atmospheric dynamics and turbulent transport
2026-02-27
On a cold, clear morning in the winter of 2008, photographer Christian Steiness flew over the Horns Rev offshore wind farm in the North Sea and captured one of the most reproduced images in wind energy history: 80 turbines each trailing a distinct plume of condensation, the entire farm a visible map of its own turbulent wake. The photograph made tangible something engineers already knew but the public rarely sees — that a wind turbine doesn’t just extract energy from the air, it transforms the air, leaving kilometres of disturbed flow that rob downwind turbines of their resource.
The atmospheric boundary layer is the lowest one to two kilometres of the troposphere, the region where wind interacts directly with the Earth’s surface. It is not calm. Every tree, building, and hillside injects turbulence into the flow. The boundary layer is also where all our wind turbines stand — hub heights now routinely reach 100–150 m — and where the fuel for wind power is found or squandered.
The physics that governs wind in this layer is more nuanced than the smooth logarithmic profiles often drawn in textbooks. Stability matters enormously: on a sunny afternoon, solar heating destabilises the lower atmosphere and turbulence mixes momentum aggressively downward, boosting hub-height winds. On a clear, calm night, radiative cooling stabilises the surface, turbulence suppresses, and a nocturnal jet can form above the rotor disk, decoupled from the measured surface wind. Monin-Obukhov similarity theory provides the framework for predicting all of this from a surface anemometer and temperature sensors. This model builds that framework up from first principles — friction velocity, roughness length, stability corrections — and shows why a wind developer who ignores boundary layer physics leaves significant energy yield on the table.
How much wind power is available at 100 m hub height given surface observations?
Atmospheric Boundary Layer (ABL):
Lowest 1-2 km of atmosphere directly influenced by surface.
Characteristics: - Wind shear (speed increases with height) - Turbulence (eddies mix momentum, heat, moisture) - Diurnal cycle (day/night differences)
Surface layer:
Lowest 10% of ABL (~50-100 m).
Constant flux layer: Momentum, heat fluxes approximately constant.
Applications: - Wind energy (turbine siting) - Air quality (pollution dispersion) - Aviation (turbulence, windshear) - Agriculture (evapotranspiration) - Building design (wind loads)
Neutral conditions (no buoyancy):
u(z) = \frac{u_*}{\kappa} \ln\left(\frac{z}{z_0}\right)
Where: - u(z) = wind speed at height z (m/s) - u_* = friction velocity (m/s) - \kappa = von Kármán constant (0.4) - z_0 = roughness length (m)
Friction velocity:
u_* = \sqrt{\frac{\tau_0}{\rho}}
Where: - \tau_0 = surface shear stress (Pa) - \rho = air density (kg/m³)
Roughness length:
Depends on surface: - Water (smooth): z_0 = 0.0001 m - Grass: z_0 = 0.01-0.05 m - Crops: z_0 = 0.1-0.2 m - Forest: $z_0 = 0.5-2 m - Urban: $z_0 = 1-3 m
Empirical approximation:
\frac{u(z)}{u_{ref}} = \left(\frac{z}{z_{ref}}\right)^\alpha
Where: - u_{ref} = wind at reference height z_{ref} - \alpha = power law exponent (~0.14-0.43)
Typical: \alpha = 1/7 (seventh-root law)
Example: u(10m) = 5 m/s, find u(100m)
u(100) = 5 \times \left(\frac{100}{10}\right)^{1/7} = 5 \times 10^{0.143} = 5 \times 1.39 = 6.95 \text{ m/s}
40% increase from 10 to 100 m!
Buoyancy modifies profile:
Unstable (daytime, surface warmer): - Enhanced turbulence - More mixing - Winds increase less rapidly with height
Stable (nighttime, surface cooler): - Suppressed turbulence - Less mixing
- Strong wind shear (jet formation possible)
Monin-Obukhov length:
L = -\frac{u_*^3 T}{\kappa g \overline{w'\theta'}}
Where: - T = temperature (K) - g = 9.81 m/s² - \overline{w'\theta'} = kinematic heat flux (K·m/s)
L > 0: Stable
L < 0: Unstable
|L| \to \infty: Neutral
From reference height to hub height:
Given: u(z_1), want u(z_2)
Log profile:
\frac{u(z_2)}{u(z_1)} = \frac{\ln(z_2/z_0)}{\ln(z_1/z_0)}
Need z_0!
Estimate from measurements at two heights:
z_0 = z_1 \exp\left(-\frac{\kappa u_1}{u_*}\right)
Or assume typical z_0 for terrain.
Power law (simpler):
u(z_2) = u(z_1) \left(\frac{z_2}{z_1}\right)^\alpha
Available power in wind:
P = \frac{1}{2} \rho A u^3
Where: - P = power (W) - A = rotor swept area (m²) - u = wind speed (m/s)
Cube law: Power ∝ u^3
Example: 5 m/s → 7 m/s
\frac{P_2}{P_1} = \left(\frac{7}{5}\right)^3 = 1.4^3 = 2.74
2.7× more power from 40% wind increase!
Wind power density:
\frac{P}{A} = \frac{1}{2} \rho u^3 (W/m²)
For \rho = 1.2 kg/m³, u = 7 m/s:
\frac{P}{A} = 0.6 \times 343 = 206 \text{ W/m}^2
Turbine efficiency: Betz limit = 59% maximum
Actual: 35-45% typical
Per unit mass:
TKE = \frac{1}{2}(\overline{u'^2} + \overline{v'^2} + \overline{w'^2})
Where primes denote turbulent fluctuations.
Typical surface layer:
TKE \approx 2.5 u_*^2 (neutral)
TKE controls: - Dispersion rates - Turbulence intensity (important for turbines) - Aviation hazards
Turbulence intensity:
TI = \frac{\sigma_u}{\bar{u}}
Where \sigma_u = \sqrt{\overline{u'^2}}
Low turbulence (<10%): Smooth flow, less turbine stress
High turbulence (>20%): Gusty, higher fatigue loads
Problem: Extrapolate wind for turbine siting.
Measurements: - u(10m) = 6 m/s (anemometer) - Surface: Agricultural (crops) - Conditions: Neutral (midday, moderate wind)
Turbine specifications: - Hub height: 80 m - Rotor diameter: 90 m - Cut-in speed: 3 m/s - Rated speed: 12 m/s
Calculate hub height wind and power density.
Step 1: Estimate roughness length
Crops: z_0 \approx 0.15 m (typical)
Step 2: Friction velocity
From log profile:
u(10) = \frac{u_*}{0.4} \ln\left(\frac{10}{0.15}\right)
$6 = \frac{u_*}{0.4} \ln(66.7) = \frac{u_*}{0.4} \times 4.20$
$6 = 10.5 u_*$
u_* = 0.571 \text{ m/s}
Step 3: Hub height wind (log profile)
u(80) = \frac{0.571}{0.4} \ln\left(\frac{80}{0.15}\right)
= 1.43 \times \ln(533) = 1.43 \times 6.28 = 8.98 \text{ m/s}
Hub wind: ~9 m/s
Step 4: Verification with power law
u(80) = 6 \times \left(\frac{80}{10}\right)^{1/7} = 6 \times 8^{0.143} = 6 \times 1.35 = 8.1 \text{ m/s}
Similar result (power law slightly underestimates)
Step 5: Wind power density
\frac{P}{A} = 0.6 \times 9^3 = 0.6 \times 729 = 437 \text{ W/m}^2
Step 6: Turbine power
Rotor area:
A = \pi r^2 = \pi \times 45^2 = 6362 \text{ m}^2
Available power:
P_{avail} = 437 \times 6362 = 2.78 \text{ MW}
Actual power (40% efficiency):
P_{actual} = 0.40 \times 2.78 = 1.11 \text{ MW}
Step 7: Assessment
9 m/s < 12 m/s (rated) → Below rated power
9 m/s > 3 m/s (cut-in) → Turbine operational
Good site (average 9 m/s excellent for wind energy)
Below is an interactive boundary layer wind profile simulator.
<label>
10m wind speed (m/s):
<input type="range" id="wind-10m" min="2" max="15" step="0.5" value="6">
<span id="wind-val">6</span>
</label>
<label>
Surface roughness:
<select id="roughness">
<option value="0.0001">Water (smooth)</option>
<option value="0.03">Grass/farmland</option>
<option value="0.15" selected>Crops</option>
<option value="1.0">Forest</option>
<option value="2.0">Urban</option>
</select>
</label>
<label>
Stability:
<select id="stability">
<option value="unstable">Unstable (day)</option>
<option value="neutral" selected>Neutral</option>
<option value="stable">Stable (night)</option>
</select>
</label>
<div class="wind-info">
<p><strong>80m hub wind:</strong> <span id="hub-wind">--</span> m/s</p>
<p><strong>Friction velocity:</strong> <span id="friction-vel">--</span> m/s</p>
<p><strong>Power density (80m):</strong> <span id="power-density">--</span> W/m²</p>
<p><strong>Wind class:</strong> <span id="wind-class">--</span></p>
</div><canvas id="wind-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas>Observations: - Logarithmic increase of wind with height - Rougher surfaces show steeper wind shear - Unstable conditions show more gradual profiles - Stable conditions show stronger wind shear - Hub height winds 30-50% higher than 10m - Power density increases dramatically with height (cube law)
Key insights: - Wind power highly sensitive to hub height - Surface roughness critically affects available wind - Atmospheric stability modifies wind profiles - 80m hub height standard balances power and cost
Site assessment:
Economic viability:
Class 4+ (>6.5 m/s @ 80m) generally viable
Class 6-7: Excellent returns
Capacity factor:
Fraction of time producing rated power.
Typical: 25-45%
Pollution transport:
Unstable: Rapid vertical mixing, dilutes pollutants quickly
Stable: Limited mixing, pollutants concentrate near surface
Inversion: Extreme stability, traps pollutants (smog events)
Gaussian plume model:
Uses u_*, L to determine dispersion parameters.
Critical for: - Stack design - Emergency response - Air quality forecasts
Low-level windshear:
Wind change <1000 ft altitude.
Microburst: Extreme case (downdraft)
Boundary layer effects:
Strong shear during stable conditions (night, early morning)
Affects: - Takeoff/landing performance - Turbulence encounters
Hills/mountains violate flat terrain assumption.
Speedup over ridges: 20-50% local acceleration
Valley channeling: Flows decouple from geostrophic wind
Solution: CFD models (WRF, WINDNINJA) or mesoscale models
Nocturnal acceleration: Winds peak 100-500m at night
Profile inverted: Maximum wind aloft, not at surface
Example - Great Plains:
Jets 20-30 m/s common (500 mb)
Implications: - Enhanced turbine production overnight - But also increased turbulence - Fatigue loads
Transition zones: Abrupt roughness changes
Internal boundary layer: New ABL grows downwind
Typically 10-20× roughness transition distance
Forest edge z_0 = 1 m → fetch = 10-20 m for adjustment
Wind turbines in cold climates:
Icing on blades reduces efficiency, adds weight
Detection: Power curve degradation
Mitigation: Heated blades (expensive), shutdown protocols
LES resolves large turbulent eddies directly.
Grid: 1-10 m resolution
Subgrid: Parameterized (small eddies)
Applications: - Wind farm optimization (wake effects) - Urban meteorology (heat islands) - Complex terrain wind assessment
Computational cost: High (supercomputers)
Example - Wind farm wake:
Downstream turbines: 10-40% power reduction from upstream wakes
LES optimizes spacing: 5-7× rotor diameter typical
Instantaneous value:
u = \bar{u} + u'
Where: - \bar{u} = time-averaged (mean) - u' = fluctuation (turbulent)
Properties:
\overline{u'} = 0 (by definition)
\overline{u'^2} > 0 (variance)
Momentum flux from turbulence:
\tau = -\rho \overline{u'w'}
Where w' = vertical velocity fluctuation.
Surface layer: \tau \approx \rho u_*^2
Controls: Wind profile shape