Wind Speed Estimation and the Monin-Obukhov Similarity Theory
Accurately predicting wind speed at various heights is critical for fields ranging from meteorology to structural engineering. Because wind interacts with the Earth's surface, its velocity changes as you move upward from the ground. To quantify this change, scientists use specific mathematical formulations that account for surface friction, obstacles, and atmospheric stability.
The Fundamental Wind Speed Equation
The mean wind speed (uz) at a specific height (z) above the ground is estimated using a formula that integrates several environmental variables. The general equation is expressed as:
uz = (u* / κ) [ ln((z - d) / z0) + ψ(z, z0, L) ]
In this formulation, u* represents the friction velocity (m/s), and κ is the Von Kármán constant, which is approximately 0.41. The term ψ is a stability term based on the Obukhov length (L) from the Monin-Obukhov similarity theory—a framework used to describe the vertical profile of wind in the surface layer of the atmosphere.
Under conditions of neutral stability, where the ratio z/L equals zero, the stability term (ψ) is removed. This simplifies the equation to:
uz = (u* / κ) [ ln((z - d) / z0) ]
[ไม่มีภาพประกอบ]Understanding Surface Variables
Zero-Plane Displacement (d)
Zero-plane displacement is the height above the ground where the mean wind speed effectively becomes zero due to the presence of obstacles like buildings or trees. This value is typically estimated as 2/3 to 3/4 of the average height of the obstacles. For instance, if a forest canopy has an average height of 30 meters, the zero-plane displacement (d) would be estimated at 20 meters.
Roughness Length (z0)
Roughness length is a corrective measure used to account for how the texture of the terrain affects wind flow. Because this value is subjective and varies by source, it is usually provided in tabular formats based on the terrain description.
| Terrain Description | Roughness Length (z0) Range (meters) |
|---|---|
| Very Flat (Snow, Desert) | 0.001 – 0.005 |
| Open Terrain (Grassland) | 0.01 – 0.05 |
| Cropland | 0.1 – 0.25 |
| Brush/Forest | 0.5 – 1.0 |
| Suburban | 0.1 – 0.5 |
| Dense Urban | 1.0 – 5.0 |
Calculating Wind Speed Between Two Heights
In many practical applications, it is necessary to estimate the wind speed at a target height (z2) based on a known wind speed at a different height (z1). By rearranging the primary formula, the following relationship is used:
u(z2) = u(z1) * [ ln((z2 - d) / z0) / ln((z1 - d) / z0) ]
This allows engineers and researchers to extrapolate wind data to the specific height of a structure or sensor without needing to calculate friction velocity directly.
Key Facts
- The Von Kármán constant (κ) is approximately 0.41.
- Zero-plane displacement is usually 66% to 75% of the average obstacle height.
- Roughness length varies significantly by terrain, from 0.001m in deserts to 5m in dense urban areas.
- Neutral stability occurs when z/L = 0, simplifying the wind speed calculation.
- The Monin-Obukhov similarity theory provides the basis for the stability term (ψ) in wind profiles.
Frequently Asked Questions
What is the purpose of the roughness length (z0)?
Roughness length is used as a corrective measure to account for the friction and turbulence created by the surface of the terrain, which slows down wind flow near the ground.
How is zero-plane displacement calculated for a forest?
It is approximated as 2/3 to 3/4 of the average canopy height. For a 30m canopy, the displacement is roughly 20m.
When does the stability term (ψ) drop out of the equation?
The stability term is omitted under neutral stability conditions, specifically when the ratio of height to Obukhov length (z/L) is zero.
How can I find the wind speed at a new height if I already have a measurement from another height?
You can use the rearranged ratio formula that divides the natural log of the target height's adjusted displacement by the natural log of the known height's adjusted displacement, then multiply by the known wind speed.
What is the difference between suburban and dense urban roughness lengths?
Suburban terrain typically has a roughness length between 0.1 and 0.5 meters, whereas dense urban areas have a much higher range of 1 to 5 meters due to larger and more frequent obstacles.