modelling Level 4

Boundary Layer Turbulence and Wind Profiles

How does wind speed change with height and how strong is turbulent mixing? The atmospheric boundary layer exhibits logarithmic wind profiles and turbulent transport driven by surface friction and buoyancy. This model derives surface layer equations, implements Monin-Obukhov similarity theory, demonstrates turbulent kinetic energy calculations, and applications to wind power and dispersion.

Prerequisites: logarithmic wind profile, friction velocity, monin obukhov, turbulent kinetic energy

27 Feb 2026 Updated 12 Mar 2026 11 min read

1. The Question

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)

2. The Conceptual Model

Logarithmic Wind Profile

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

Power Law (Simplified)

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!

Stability Effects

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

3. Building the Mathematical Model

Wind Extrapolation

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):

\[u(z_2) = u(z_1) \left(\frac{z_2}{z_1}\right)^\alpha\]

Wind Power Density

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

Turbulent Kinetic Energy (TKE)

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

4. Worked Example by Hand

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.

Solution

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)

5. Computational Implementation

Below is an interactive boundary layer wind profile simulator.

10m wind speed (m/s): 6 Surface roughness: Stability:

80m hub wind: -- m/s

Friction velocity: -- m/s

Power density (80m): -- W/m²

Wind class: --

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

6. Interpretation

Wind Energy Siting

Site assessment:

Measure: 10m anemometer, 1 year minimum
Extrapolate: To hub height (log profile or power law)
Calculate: Average wind, power density
Classify: Wind resource class

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%

Atmospheric Dispersion

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

Aviation

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

7. What Could Go Wrong?

Complex Terrain

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

Low-Level Jets

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

Forest Edge Effects

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

Icing

Wind turbines in cold climates:

Icing on blades reduces efficiency, adds weight

Detection: Power curve degradation

Mitigation: Heated blades (expensive), shutdown protocols

8. Extension: Large Eddy Simulation

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

9. Math Refresher: Reynolds Decomposition

Turbulent Flow

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)

Reynolds Stress

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

Summary

Logarithmic wind profile describes speed increase with height in neutral boundary layer
Friction velocity u_* characterizes surface stress and turbulent momentum flux
Roughness length z_0 varies from 0.0001m (water) to 2m (urban/forest)
Power law provides simplified extrapolation with exponent typically 1/7
Wind power density proportional to velocity cubed making hub height critical
Monin-Obukhov theory incorporates stability effects modifying wind profiles
Atmospheric stability controls turbulent mixing and dispersion rates
Wind resource Class 4+ (>6.5 m/s @ 80m) required for economic viability
Applications span wind energy, pollution dispersion, aviation safety
Complex terrain and low-level jets violate flat-surface assumptions requiring advanced modelling