Wind Profiles and Atmospheric Turbulence
modelling Level 3

Wind Profiles and Atmospheric Turbulence

Wind speed increases with height above the ground. Near the surface, turbulent eddies mix momentum downward, creating a logarithmic wind profile. Surface roughness controls the shape—forests slow wind more than grassland. This model derives the log law and introduces roughness length and friction velocity.

Prerequisites: logarithmic profile, boundary layer, shear stress, turbulent transport

Updated 11 min read

1. The Question

Why does wind speed increase with height, and why does the rate of increase depend on the surface?

Measure wind speed at multiple heights:

  • At ground level: Nearly calm (friction with surface)
  • At 2 m: Light breeze
  • At 10 m: Moderate wind
  • At 100 m: Strong wind

The shape of this wind profile depends on the surface:

  • Ocean: Smooth → wind increases rapidly with height
  • Grassland: Moderate roughness → intermediate profile
  • Forest: Rough → wind increases slowly with height

The mathematical question: What is the equation for wind speed as a function of height, and how does surface roughness enter the model?

2. The Conceptual Model

The Atmospheric Boundary Layer

The atmospheric boundary layer (ABL) is the lowest ~1 km of atmosphere where surface friction matters.

Three sublayers:

  1. Surface layer (0–50 m):
    • Wind profile dominated by surface friction
    • Logarithmic profile
    • Our focus
  2. Ekman layer (50 m–1 km):
    • Coriolis force matters
    • Wind turns with height
    • Beyond our scope
  3. Free atmosphere (> 1 km):
    • Friction negligible
    • Geostrophic balance

Turbulent Momentum Transfer

Wind creates turbulent eddies that mix momentum downward.

Shear stress (momentum flux):

\[\tau = -\rho \overline{u'w'}\]

Where:

  • $\tau$ = shear stress (N/m² or Pa)
  • $\rho$ = air density (kg/m³)
  • $\overline{u’w’}$ = covariance of horizontal and vertical wind fluctuations
  • Overbar = time average, prime = fluctuation from mean

Near the surface, shear stress is approximately constant with height (constant stress layer).

Friction velocity ($u_*$):

\[u_* = \sqrt{\frac{\tau}{\rho}}\]

This is a velocity scale (m/s) characterizing turbulence intensity, not the actual wind speed.

The Logarithmic Law

In the surface layer, wind speed increases logarithmically with height:

\[u(z) = \frac{u_*}{k} \ln\left(\frac{z}{z_0}\right)\]

Where:

  • $u(z)$ = wind speed at height $z$ (m/s)
  • $u_*$ = friction velocity (m/s)
  • $k = 0.41$ = von Kármán constant (dimensionless)
  • $z_0$ = roughness length (m)

Roughness length ($z_0$) is where the logarithmic profile extrapolates to zero wind speed.

3. Building the Mathematical Model

Derivation from Mixing Length Theory

Prandtl’s mixing length hypothesis:

Turbulent eddies have a characteristic size $\ell$ (mixing length). They transport momentum proportional to:

\[\tau \propto \rho \ell^2 \left(\frac{du}{dz}\right)^2\]

Near the surface, mixing length is proportional to height:

\[\ell = kz\]

Where $k$ is the von Kármán constant.

Combine:

\[\tau = \rho (kz)^2 \left(\frac{du}{dz}\right)^2\]

In the constant stress layer, $\tau$ is constant. Define $u_*^2 = \tau/\rho$:

\[u_*^2 = (kz)^2 \left(\frac{du}{dz}\right)^2\] \[\frac{du}{dz} = \frac{u_*}{kz}\]

Integrate:

\[u(z) = \frac{u_*}{k} \ln(z) + C\]

Boundary condition: At $z = z_0$, $u = 0$ (roughness length is where wind extrapolates to zero).

$$0 = \frac{u_}{k} \ln(z_0) + C \implies C = -\frac{u_}{k}\ln(z_0)$$

Final form:

\[u(z) = \frac{u_*}{k} \left[\ln(z) - \ln(z_0)\right] = \frac{u_*}{k} \ln\left(\frac{z}{z_0}\right)\]

This is the logarithmic wind profile or log law.

Roughness Length

Physical meaning: Height where the logarithmic profile extrapolates to zero wind.

Typical values:

Surface Type $z_0$ (m)
Open ocean, ice 0.0001–0.001
Snow surface 0.001–0.005
Bare soil, sand 0.001–0.01
Short grass 0.01–0.05
Crops 0.05–0.15
Shrubland 0.1–0.3
Deciduous forest 0.5–2.0
Conifer forest 1.0–3.0
Urban areas 0.5–2.0

Rule of thumb: $z_0 \approx 0.1 h$, where $h$ is vegetation height.

Calculating Friction Velocity

If wind speed is known at two heights, solve for $u_*$:

\[u_2 - u_1 = \frac{u_*}{k} \ln\left(\frac{z_2}{z_1}\right)\] \[u_* = \frac{k(u_2 - u_1)}{\ln(z_2/z_1)}\]

Example: $u(2\text{ m}) = 3$ m/s, $u(10\text{ m}) = 5$ m/s:

\[u_* = \frac{0.41 \times (5 - 3)}{\ln(10/2)} = \frac{0.82}{1.61} = 0.51 \text{ m/s}\]

Zero-Plane Displacement

For tall, dense vegetation (forests, crops), wind appears to originate from a height $d$ above the ground (zero-plane displacement).

Modified log law:

\[u(z) = \frac{u_*}{k} \ln\left(\frac{z - d}{z_0}\right)\]

Typical values:

  • $d \approx 0.7h$ (where $h$ is canopy height)
  • $z_0 \approx 0.1h$

Example: 20 m forest:

  • $d = 14$ m
  • $z_0 = 2$ m

4. Worked Example by Hand

Problem: A grassland has roughness length $z_0 = 0.03$ m. Wind speed at 10 m height is 8 m/s.

(a) Calculate friction velocity $u_*$.
(b) Predict wind speed at 2 m height.
(c) At what height is wind speed 12 m/s?

Solution

(a) Friction velocity

From log law:

\[u(10) = \frac{u_*}{k} \ln\left(\frac{10}{z_0}\right)\]

$$8 = \frac{u_*}{0.41} \ln\left(\frac{10}{0.03}\right)$$

$$8 = \frac{u_*}{0.41} \ln(333.3)$$

$$8 = \frac{u_*}{0.41} \times 5.81$$

\[u_* = \frac{8 \times 0.41}{5.81} = 0.565 \text{ m/s}\]

(b) Wind speed at 2 m

\[u(2) = \frac{0.565}{0.41} \ln\left(\frac{2}{0.03}\right)\] \[= 1.378 \times \ln(66.7)\] \[= 1.378 \times 4.20 = 5.79 \text{ m/s}\]

(c) Height for 12 m/s

$$12 = \frac{0.565}{0.41} \ln\left(\frac{z}{0.03}\right)$$

$$12 = 1.378 \ln\left(\frac{z}{0.03}\right)$$

\[\ln\left(\frac{z}{0.03}\right) = \frac{12}{1.378} = 8.71\] \[\frac{z}{0.03} = e^{8.71} = 6063\] \[z = 0.03 \times 6063 = 182 \text{ m}\]

Wind speed reaches 12 m/s at 182 m height.

5. Computational Implementation

Below is an interactive wind profile simulator.

Friction velocity (u*): m/s

Wind at 2 m: m/s

Wind at 50 m: m/s

10m/2m ratio:

Try this:

  • Switch to ocean: Very smooth → wind increases rapidly with height
  • Switch to forest: Very rough → wind increases slowly, high z₀
  • Toggle log scale: On log-height axis, profile becomes a straight line (proof it’s logarithmic)
  • Increase reference wind: Friction velocity increases proportionally
  • Notice: Rougher surfaces have lower wind speeds at all heights for same reference wind

Key insight: The logarithmic profile is universal — only $u_*$ and $z_0$ change between surfaces.

6. Interpretation

Why Logarithmic?

The log profile emerges from:

  1. Turbulent mixing dominates (not molecular viscosity)
  2. Constant stress with height near surface
  3. Mixing length proportional to height ($\ell = kz$)

These three assumptions lead uniquely to the logarithmic form.

Roughness and Drag

Surface drag is parameterized by $z_0$:

  • Large $z_0$ → strong drag → more turbulence → slower wind near surface
  • Small $z_0$ → weak drag → less turbulence → faster wind near surface

Momentum flux to surface:

\[\tau = \rho u_*^2\]

Higher $u_*$ (stronger wind or rougher surface) → more momentum transferred to ground.

Wind Energy

Wind power increases with cube of wind speed:

\[P = \frac{1}{2}\rho A u^3\]

Why turbines are tall: At 100 m vs. 10 m:

For grassland ($z_0 = 0.03$ m), if $u(10) = 8$ m/s:

\[u(100) = \frac{u_*}{k}\ln(100/0.03) = \frac{0.565}{0.41}\times 8.11 = 11.2 \text{ m/s}\] \[\frac{P(100)}{P(10)} = \left(\frac{11.2}{8}\right)^3 = 2.2\]

2.2× more power at 100 m!

Urban Heat Island

Cities have large $z_0$ (buildings) → more turbulent mixing → enhanced heat/moisture transport from surface to atmosphere → affects local climate.

7. What Could Go Wrong?

Assuming Neutral Stability

Our log law assumes neutral stability (no buoyancy effects).

Unstable (convective, daytime):

  • Warm surface → rising air
  • Enhanced mixing
  • Wind profile steeper than log law

Stable (nighttime, inversion):

  • Cool surface → suppressed turbulence
  • Reduced mixing
  • Wind profile less steep

Monin-Obukhov similarity theory corrects for stability.

Applying Below Canopy

The log law applies above vegetation, not within it.

Inside a canopy:

  • Wind decreases exponentially with depth
  • Different profile shape
  • Multiple length scales

Ignoring Atmospheric Pressure Gradients

The log law assumes horizontally uniform flow.

Real atmosphere has:

  • Pressure gradients → acceleration
  • Terrain variation → flow separation, channeling
  • Thermal effects → local circulations

Constant Roughness

Real surfaces have heterogeneous roughness:

  • Patchy vegetation
  • Urban/rural transitions
  • Fetch effects (distance upwind to equilibrium)

Internal boundary layers form at roughness transitions.

8. Extension: Heat and Moisture Profiles

The same turbulent eddies that transport momentum also transport heat and moisture.

Temperature profile (analog to wind):

\[T(z) - T_s = \frac{H}{\rho c_p u_* k} \ln\left(\frac{z}{z_h}\right)\]

Where:

  • $H$ = sensible heat flux (from Model 18)
  • $z_h$ = roughness length for heat (typically $z_h < z_0$)

Moisture profile:

\[q(z) - q_s = \frac{LE}{L_v \rho u_* k} \ln\left(\frac{z}{z_q}\right)\]

Where:

  • $q$ = specific humidity
  • $z_q$ = roughness length for moisture

These profiles enable bulk transfer formulas used in climate models to calculate surface fluxes from atmospheric conditions.

9. Math Refresher: The Natural Logarithm

Definition

The natural logarithm is the inverse of the exponential function:

\[y = e^x \iff x = \ln(y)\]

Where $e = 2.71828…$ (Euler’s number).

Properties

\[\ln(ab) = \ln(a) + \ln(b)\] \[\ln(a/b) = \ln(a) - \ln(b)\] \[\ln(a^b) = b\ln(a)\] \[\ln(1) = 0, \quad \ln(e) = 1\]

Why It Appears in Wind Profiles

The differential equation:

\[\frac{du}{dz} = \frac{u_*}{kz}\]

Has solution:

\[u = \frac{u_*}{k}\ln(z) + C\]

Separating variables:

\[du = \frac{u_*}{k}\frac{dz}{z}\]

Integrate:

\[\int du = \int \frac{u_*}{k}\frac{dz}{z}\] \[u = \frac{u_*}{k}\ln(z) + C\]

The $1/z$ term on the right forces a logarithm on the left.

Summary

  • Logarithmic wind profile: $u(z) = \frac{u_*}{k}\ln(z/z_0)$
  • Friction velocity ($u_*$): Characterizes turbulence intensity
  • von Kármán constant: $k = 0.41$ (universal)
  • Roughness length ($z_0$): Where profile extrapolates to zero wind
  • Ocean: $z_0 \sim 0.0002$ m; Grassland: $z_0 \sim 0.03$ m; Forest: $z_0 \sim 1.5$ m
  • Rougher surfaces → slower wind at all heights
  • Log law emerges from turbulent mixing with mixing length $\ell = kz$
  • Wind increases as $\ln(z)$ — very rapid increase near surface, slower aloft
  • Same principles apply to heat and moisture transport

References