The power law wind profile: a simple model for wind shear

The wind speed generally increases with height above the ground due to the effect of surface friction. This phenomenon is known as wind shear. The power law provides a straightforward empirical relationship to model this change:

U(z₂) = U(z₁) * (z₂ / z₁)^α

Where:

• U(z₂) is the wind speed at height z₂.
• U(z₁) is the wind speed at a reference height z₁.
• α (alpha) is the shear exponent, a dimensionless value that characterises how quickly wind speed increases with height.

This formula is particularly useful for estimating wind speeds at heights relevant to wind turbines, cranes, or other tall structures, where direct measurements may not be available. It simplifies the complex physics of the atmospheric boundary layer into a single, adjustable parameter, making it practical for many engineering and operational applications. However, it is an approximation and does not account for all meteorological complexities, such as changes in atmospheric stability or specific terrain features beyond a general roughness classification. The power law is an empirical model. Its accuracy depends heavily on the appropriate selection of the shear exponent (α) for the specific conditions and terrain. It is not a fundamental physical law.

The shear exponent α is not a constant; it varies significantly with the characteristics of the underlying terrain and the atmospheric conditions. Over very smooth surfaces, like open water, the friction is minimal, leading to a smaller α value and less pronounced shear. Conversely, over rough terrain with many obstacles (e.g., forests, urban areas), the friction is much greater, resulting in a larger α and stronger shear.

Commonly cited typical values for α include:

These values are general guidelines. The user's own site-specific studies or detailed meteorological analyses should always govern the selection of α for critical operations. For instance, a wind farm might conduct extensive mast measurements to derive a highly specific α for its location. When using the power law, consider the roughness of the terrain upwind of your location. A change in wind direction can mean a change in the effective terrain roughness and thus the appropriate α.

Atmospheric stability significantly influences the shear exponent α. Stability refers to the atmosphere's tendency to resist or enhance vertical motion, which in turn affects turbulence and momentum transfer.

• Unstable (convective) conditions: During sunny days, particularly over land, the ground heats up and transfers heat to the air, causing parcels of air to rise. This leads to strong vertical mixing and turbulence, which tends to distribute momentum more evenly through the lower atmosphere. Consequently, the wind shear is reduced, and α values are generally lower (e.g., 0.05 – 0.10).
• Neutral conditions: These occur when there is little or no net heat exchange between the surface and the air, often during overcast and conditions with significant air movement. Vertical mixing is primarily mechanically driven by friction. This is when the typical α values for various terrain types (as listed previously) are most applicable.
• Stable (nocturnal) conditions: On clear nights, especially with light air movement, the ground cools rapidly, leading to a temperature inversion where cooler, denser air lies beneath warmer air. This suppresses vertical mixing, trapping momentum near the surface. Shear can become very strong, leading to higher α values (e.g., 0.30 – 0.50 or even higher). This can result in a significant increase in wind speed just tens of metres above the surface, a phenomenon known as a low-level jet.

The Wind Agent's Shear Glass (chart: glass) implicitly accounts for stability by showing model-derived wind speeds at multiple heights (10/80/120/180 m), rather than relying on a single, fixed α. This chart shows how wind shear (the change in wind speed with height) can vary significantly across hours, often driven by changes in atmospheric stability. The Wind Agent uses multi-level model outputs (10, 80, 120, 180 m) rather than a single power law exponent. This provides a more dynamic and accurate representation of shear, especially during stable or unstable periods.

If wind speed measurements are available at two different heights, z₁ and z₂, the shear exponent α can be calculated directly. This is particularly useful for site-specific analysis, where a general α value might not accurately represent local conditions.

The formula to derive α is a rearrangement of the power law equation:

α = ln(U₂/U₁) / ln(z₂/z₁)

Where:

• U₁ is the wind speed measured at height z₁.
• U₂ is the wind speed measured at height z₂.
• ln denotes the natural logarithm.

Worked Example: Assume a meteorological mast measures a wind speed of 6 m/s at 10 metres (z₁) and 8.5 m/s at 50 metres (z₂).

1. Calculate the ratio of speeds: U₂/U₁ = 8.5 / 6 = 1.4167
2. Calculate the ratio of heights: z₂/z₁ = 50 / 10 = 5
3. Apply the formula for α: α = ln(1.4167) / ln(5) α = 0.3483 / 1.6094 α ≈ 0.216

This calculated α of approximately 0.216 suggests conditions typical of farmland with hedges or a suburban environment, which aligns with the commonly cited ranges. This method allows for a more precise, site-specific α to be determined, improving the accuracy of subsequent height extrapolations.

The shear exponent α is typically much lower over open sea compared to land. This difference is primarily due to the significantly lower surface roughness of water.

• Low Surface Roughness: The surface of the open ocean is generally much smoother than land surfaces. While waves do create some roughness, their effect on the atmospheric boundary layer is less pronounced than the obstacles found on land (e.g., trees, buildings, hills). This reduced friction means less drag on the air flowing over the surface.
• Reduced Mechanical Turbulence: With less surface friction, there is less mechanical turbulence generated. Turbulence acts to mix momentum vertically, effectively reducing the shear. Over the sea, this mixing is less intense, allowing the wind speed to increase more gradually with height.
• Thermal Stability: The ocean's temperature changes more slowly than land, leading to more frequent neutral or slightly stable atmospheric conditions, especially in temperate zones. This tends to suppress strong convective mixing that can occur over land, further contributing to lower α values.

For example, α values over open sea are commonly cited in the range of 0.08 to 0.12, whereas over rough land, they can easily exceed 0.25. This has significant implications for offshore wind energy, where turbines can capture stronger, more consistent winds at lower hub heights compared to onshore locations, for a given reference speed at 10 metres.

The power law is a fundamental tool in wind energy for estimating wind speeds at the hub height of wind turbines. Turbines are designed to operate most efficiently at specific wind speeds, and their hub heights can range from 80 metres to over 180 metres. Direct measurements at these heights are often not available during initial site assessment.

Using the power law, a short-term measurement campaign at a lower height (e.g., 10 m or 30 m) can be extrapolated to the proposed hub height. For instance, if a site assessment measures an average wind speed of 7 m/s at 30 m, and the proposed turbine has a hub height of 120 m, with a typical α of 0.14 for open grassland:

U(120m) = 7 m/s * (120 / 30)^0.14 U(120m) = 7 m/s * (4)^0.14 U(120m) = 7 m/s * 1.22 U(120m) ≈ 8.54 m/s

This estimated hub-height speed is crucial for calculating the potential energy yield of a turbine and for making economic decisions about a wind farm project. However, it is essential to recognise that this is an estimate. The Wind Agent's Shear Glass (chart: glass) provides modelled wind speeds at standard turbine heights (80, 120, 180 m) directly from the forecast model, offering a more robust estimate than a single power law calculation, as it accounts for varying α due to stability and other factors.

While the power law is a useful tool, its accuracy diminishes significantly when extrapolating wind speeds over large height differences. This is due to several factors:

1. Non-linear Profile: The actual wind profile in the atmospheric boundary layer is often not perfectly described by a simple power law, especially at very low or very high altitudes within the layer. The α value itself can change with height.
2. Boundary Layer Depth: The power law assumes that the wind profile continues to increase with height throughout the boundary layer. However, above a certain height (the top of the boundary layer, typically 500-1500 m), the wind speed approaches the geostrophic wind and the shear becomes negligible or even reverses.
3. Changes in α: As discussed, α varies with atmospheric stability and terrain roughness. A single α value derived from a lower height might not be representative of the shear conditions much higher up, where different atmospheric processes might dominate.

For example, extrapolating a 10 m measurement to 200 m using a single α can introduce considerable error, particularly during stable nocturnal conditions where a low-level jet might exist, causing the actual speed at 200 m to be much higher than predicted by a constant α. For critical applications, such as large crane operations or wind turbine commissioning, relying solely on a power law extrapolation over large height differences without verification from multi-level measurements or advanced numerical weather prediction models is not recommended. The Wind Agent's Shear Glass provides modelled data up to 180 m, reducing the need for extensive extrapolation.