The logarithmic wind profile

The logarithmic wind profile, often referred to as the log law, is a theoretical model describing the vertical distribution of mean horizontal wind speed within the atmospheric surface layer. This layer is typically the lowest 10–100 metres of the atmosphere, where turbulent fluxes of momentum, heat, and moisture are approximately constant with height.

The log law arises from the concept of turbulent shear stress and the assumption of a constant momentum flux in the surface layer. Air moving over a surface experiences friction, creating a shear stress that opposes the flow. This stress is transferred downwards through turbulence. In a neutrally stratified atmosphere (where temperature does not significantly change with height, preventing buoyancy-driven turbulence), the velocity profile adjusts to maintain this constant momentum flux.

Mathematically, the relationship is expressed as:

U(z) = (u_* / κ) * ln((z - d) / z₀)

Where:

• U(z) is the mean wind speed at height z.
• u_* is the friction velocity, representing the turbulent shear stress.
• κ (kappa) is the von Kármán constant, an empirical constant.
• z is the height above the ground.
• d is the displacement height, accounting for the effective ground level over tall obstacles.
• z₀ is the roughness length, characterising the aerodynamic roughness of the surface.

This equation indicates that wind speed increases logarithmically with height above the effective surface, which is a slower rate of increase than a linear relationship.

The friction velocity (u_*) is a measure of the turbulent shear stress at the surface. It is not a true velocity in the sense of a mean wind speed, but rather a scaling velocity that characterises the intensity of turbulence and the rate of momentum transfer from the atmosphere to the surface. A higher friction velocity indicates greater surface drag and more intense turbulence.

The von Kármán constant (κ) is an empirical constant used in the log law. It represents the ratio of the mixing length to the height above the surface in the turbulent surface layer. Commonly cited values for κ range from 0.40 to 0.41. For most meteorological and engineering applications, a value of κ = 0.4 is widely adopted.

These two parameters, u_* and κ, are crucial for defining the shape of the logarithmic wind profile. While κ is considered universal, u_* is highly dependent on the surface roughness and the mean wind speed. It is often determined indirectly from measurements of the wind profile itself or from direct measurements of turbulent fluxes using eddy covariance techniques.

Understanding u_* allows for the calculation of the wind profile from a single reference measurement and a known roughness length, provided the atmospheric conditions are neutrally stable. The Wind Agent's Shear Glass (chart id: glass) implicitly accounts for these principles by presenting height-matched wind speeds from a numerical model that resolves the boundary layer physics.

A critical condition for the direct application of the simple logarithmic wind profile is neutral atmospheric stability. Neutral stability occurs when the vertical temperature gradient in the atmosphere is adiabatic, meaning there is no net buoyancy force to either enhance or suppress vertical mixing. In such conditions, turbulence is primarily mechanically generated by wind shear, and the momentum flux is approximately constant with height.

In reality, the atmosphere is rarely perfectly neutral. During daytime with solar heating, the surface warms, creating an unstable atmosphere where warm air rises. This convective turbulence enhances vertical mixing, making the wind profile flatter (less shear) than predicted by the log law. Conversely, during clear nights, the surface cools, creating a stable atmosphere where a temperature inversion can form. This suppresses vertical mixing, leading to stronger wind shear and often a low-level jet, where wind speeds peak at a few tens of metres above the ground before decreasing or levelling off higher up.

When stability deviates from neutral, modifications to the log law are necessary, incorporating stability functions. These functions adjust the profile to account for the additional (or suppressed) turbulence due to buoyancy. The Wind Agent's Shear Glass (chart id: glass) and shear heatmap (chart id: shear_heatmap) show the model's prediction of wind at various heights, implicitly incorporating stability effects that a simple log law cannot capture.

Let's calculate the wind speed at 50 metres, given a known speed at 10 metres, assuming neutral stability and an open grassland surface. We will use the log law:

U(z) = (u_* / κ) * ln((z - d) / z₀)

First, we need to determine the roughness length (z₀) and assume a displacement height (d).

Assumptions:

• Surface: Open grassland.
• Roughness length (z₀): Commonly cited as 0.03 metres.
• Displacement height (d): 0 metres (for short vegetation).
• Von Kármán constant (κ): 0.4.
• Reference height (z₁): 10 metres.
• Reference wind speed (U(z₁)): 8 m/s.
• Target height (z₂): 50 metres.

Step 1: Calculate friction velocity (u_*) from the 10 m reference. U(z₁) = (u_* / κ) * ln(z₁ / z₀) 8 m/s = (u_* / 0.4) * ln(10 m / 0.03 m) 8 = (u_* / 0.4) * ln(333.33) 8 = (u_* / 0.4) * 5.81 u_* = (8 * 0.4) / 5.81 u_* ≈ 3.2 / 5.81 ≈ 0.551 m/s

Step 2: Calculate wind speed at 50 m (U(z₂)). U(z₂) = (u_* / κ) * ln(z₂ / z₀) U(50 m) = (0.551 / 0.4) * ln(50 m / 0.03 m) U(50 m) = 1.3775 * ln(1666.67) U(50 m) = 1.3775 * 7.42 U(50 m) ≈ 10.22 m/s

So, if the wind speed at 10 m is 8 m/s over open grassland, the speed at 50 m would be approximately 10.22 m/s under neutral conditions. This demonstrates the increase in wind speed with height due to reduced surface friction.

For surfaces with significant obstacles, such as forests, urban areas, or dense crops, the effective level where wind speed becomes zero is not the ground surface itself. Instead, the wind profile effectively starts above the average height of these obstacles. This elevated effective surface is accounted for by the displacement height (d) in the log law.

The displacement height represents the average height at which the momentum is absorbed by the obstacles. For example, over a forest, the wind speed within the canopy is significantly reduced, and the logarithmic profile typically applies above the canopy top. The displacement height is commonly estimated as a fraction of the average obstacle height, often cited as approximately 0.7 * h (where h is the average height of the obstacles).

Examples of typical displacement heights:

• Short grass: d ≈ 0 m
• Tall crops (e.g., maize): d ≈ 0.5 - 1.5 m
• Forests: d ≈ 0.7 * canopy height (e.g., for a 20 m forest, d ≈ 14 m)
• Urban areas: d ≈ 0.6 - 0.8 * average building height

Incorporating d shifts the entire wind profile upwards. Without it, the log law would underestimate wind speeds at higher levels above dense canopies or buildings, as it would incorrectly assume the friction layer extends all the way to the ground. The Wind Agent's models inherently account for complex terrain and urban effects through their detailed land surface schemes, which influence the modelled wind profiles.

While the logarithmic wind profile is a powerful tool, it has limitations and breaks down under certain conditions:

1. Non-neutral stability: As discussed, the log law is strictly valid only under neutral atmospheric conditions. In unstable (convective) or stable (inversion) conditions, buoyancy forces significantly alter turbulence and momentum transfer, requiring more complex stability-corrected profile functions.
2. Above the surface layer: The assumption of constant momentum flux with height, fundamental to the log law, holds primarily within the atmospheric surface layer (typically the lowest 10–100 m). Above this, in the outer part of the boundary layer, the Coriolis force and pressure gradient become more influential, and the log law is no longer applicable.
3. Complex terrain: The log law assumes a horizontally homogeneous and flat surface. Over hills, valleys, cliffs, or highly heterogeneous surfaces (e.g., a patchwork of forest and open land), the flow becomes complex, with separation, reattachment, and wake effects that the simple log law cannot describe. Numerical models are essential for such scenarios.
4. Very low wind speeds: At very low wind speeds, mechanical turbulence diminishes, and the flow can become laminar or intermittently turbulent, making the assumptions of fully developed turbulence, on which the log law relies, invalid.

For these reasons, while the log law provides a foundational understanding, operational forecasts from The Wind Agent use sophisticated numerical models that integrate boundary layer schemes capable of handling varying stability, terrain, and surface types to provide a more accurate representation of the wind profile.

Both the logarithmic wind profile (log law) and the power law are commonly used to describe how wind speed changes with height, but they originate from different theoretical bases and have distinct applications.

Logarithmic Law:

• Theoretical Basis: Derived from turbulent boundary layer theory, assuming constant momentum flux in the surface layer under neutral stability.
• Parameters: Requires roughness length (z₀), displacement height (d), friction velocity (u_*), and the von Kármán constant (κ).
• Accuracy: Generally considered more physically robust for the atmospheric surface layer under neutral conditions, especially close to the surface.
• Complexity: More complex to apply as it requires u_* and z₀, which can be difficult to determine.

Power Law:

• Theoretical Basis: An empirical relationship, often derived as a simplification of the log law or from curve fitting to observed data.
• Parameters: Requires a reference wind speed at a reference height and a single shear exponent (α).
• Accuracy: Simpler to use and often provides a reasonable fit over a broader range of heights (within the entire boundary layer) and conditions, but less physically grounded.
• Complexity: Simpler to apply, as it only needs the shear exponent, which can be estimated or chosen from typical values.

In practice, the power law is often preferred for its simplicity in engineering applications, particularly when only an approximate profile is needed or when detailed surface characteristics are unknown. The log law is typically favoured in micrometeorological research and for more precise applications within the surface layer. The Wind Agent uses height-matched model data (chart id: glass) that implicitly follows the complex physics, avoiding the need for users to choose between these simplified laws.