Higher-order mean velocity profile in the convective atmospheric boundary layer

This paper derives a higher-order mean velocity profile for the convective atmospheric boundary layer using matched asymptotic expansions and field data from the M$^2$HATS campaign, demonstrating excellent agreement with measurements and validating the convective logarithmic friction law to at least second order while accounting for deviations from standard similarity theories.

The Gist

Imagine the atmosphere near the ground as a giant, invisible river of air flowing over the Earth's surface. This "river" is called the Atmospheric Boundary Layer (ABL). Just like a river flowing over rocks, the air near the ground gets turbulent, swirling and churning.

Scientists have long tried to predict exactly how fast the wind blows at different heights. They have a famous, simple rule for this called the "Log Law" (like a logarithmic curve). Think of this Log Law as a basic map: "If you go up a little bit, the wind gets a little faster."

However, this basic map isn't perfect. It's like a GPS that works great in a straight city street but gets confused when you hit a steep hill or a bumpy dirt road. In the real world, the air is heated by the sun (creating "convection," like boiling water), and the ground isn't perfectly smooth. These factors make the wind profile deviate from the simple Log Law.

This paper by Tong and his team is like upgrading that GPS to a high-definition, 3D navigation system.

The Problem: The "One-Size-Fits-All" Map Failed

For decades, scientists used the Log Law and a related theory (Monin-Obukhov Similarity) to predict wind speeds. These theories work well in "ideal" conditions. But in the real world, especially on hot, sunny days when the air is rising vigorously (the "Convective" boundary layer), the wind doesn't follow the simple rules.

The old maps couldn't explain why the wind was behaving strangely or how to fix the prediction. They were missing the "higher-order" details—the subtle corrections needed for real-world bumps and hills.

The Solution: A "Russian Doll" Approach

The authors used a mathematical technique called Matched Asymptotic Expansions. To understand this, imagine the atmosphere isn't just one big layer, but three nested layers, like Russian nesting dolls:

1. The Outer Layer (The Big Picture): High up, near the top of the boundary layer. Here, the wind is influenced by the overall size of the layer and the heat rising from the ground.
2. The Middle Layer (The Transition): Closer to the ground, where the "Log Law" usually applies, but things start getting messy.
3. The Inner Layer (The Roughness): Right at the surface, where trees, buildings, and grass create friction.

The authors realized that to get a perfect prediction, you can't just look at one layer. You have to mathematically "stitch" these three layers together. They treated the atmosphere like a puzzle where the pieces overlap. By matching the edges of these layers, they found the "glue" that connects them.

The "Secret Ingredients" (Small Parameters)

In their math, they found three "small parameters" (tiny numbers that act like dials) that control how the wind behaves:

• How hot is the ground? (Represented by $L$, the Obukhov length).
• How tall is the boundary layer? (Represented by $z_i$).
• How rough is the ground? (Represented by $h_0$, like the height of grass or buildings).

The old theories ignored these dials or assumed they were zero. The new theory says: "No, these dials matter!" They derived new equations that show exactly how turning these dials changes the wind speed.

The Field Test: The M2HATS Campaign

You can't just do math on a whiteboard; you need to check it against reality. The team went to Tonopah, Nevada, for a massive field experiment called M2HATS.

• They set up towers with wind sensors (like giant weather vanes) at different heights.
• They used a Doppler Lidar (a laser radar) to "see" the wind speed all the way up to 3 kilometers high.
• They collected data over many days, capturing different weather conditions.

The Results: A Perfect Fit

When they plugged their new, complex "high-definition" equations into the real-world data, it was a match made in heaven.

• The Old Way: The simple Log Law had errors, especially when the wind was very turbulent.
• The New Way: Their higher-order equations predicted the wind speed with excellent accuracy.

They even managed to calculate a fundamental constant of nature called the von Kármán constant (which relates wind speed to height) more accurately than before. They found that previous measurements were slightly off because they were trying to fit a simple curve to a complex reality. By accounting for the "bumps" (the higher-order terms), they found the true value.

Why Does This Matter?

Think of this new profile as a better blueprint for the invisible river of air. This is crucial for:

• Wind Energy: Placing wind turbines at the exact height where they catch the most wind without breaking.
• Weather Forecasting: Predicting storms and pollution dispersion more accurately.
• Airplane Safety: Understanding how wind shear affects takeoff and landing.

The Takeaway

In simple terms, this paper says: "The old rule for wind speed was a good sketch, but it was missing the details. We used advanced math to stitch together the different layers of the atmosphere, tested it with lasers and towers in the desert, and now we have a much more accurate, detailed map of how the wind really blows."

It's the difference between a rough sketch of a coastline and a satellite image that shows every cove and inlet.

Technical Summary

1. Problem Statement

The mean velocity profile is a fundamental characteristic of the Atmospheric Boundary Layer (ABL) with critical applications in friction drag estimation, pollutant transport, weather prediction, and wind energy. While the Monin-Obukhov Similarity Theory (MOST) successfully predicts the leading-order behavior (the log law in the surface layer and local-free-convection scaling in the mixed layer), it fails to account for:

• Deviations from the leading-order scaling when the height $z$ is comparable to the Obukhov length $L$ (i.e., $z \sim -L$).
• The dependence of the mean profile on the ratio of the inversion height to the Obukhov length ($z_i/L$) and the roughness height ($h_0/L$).
• Systematic departures from the MOST profile observed in field measurements that cannot be explained by measurement uncertainty alone.

Previous attempts to address these deviations relied on empirical dimensional analysis, which could not determine the functional form or the relative importance of these additional parameters.

2. Methodology

The authors employ the method of matched asymptotic expansions based on the governing equations of fluid dynamics (Reynolds-stress, potential-temperature flux, and variance budget equations) rather than purely empirical fitting.

• Three-Layer Structure: Building on Tong & Ding (2020), the Convective Boundary Layer (CBL) is modeled as having three distinct scaling layers:
  1. Outer Layer (Mixed Layer): Scaled by the inversion height $z_i$ and mixed-layer velocity $w_*$.
  2. Inner-Outer Layer (Surface Layer/MOST layer): Scaled by the Obukhov length $L$ and friction velocity $u_*$.
  3. Inner-Inner Layer (Roughness Layer): Scaled by the roughness height $h_0$ and $u_*$.
• Perturbation Analysis: The governing equations are non-dimensionalized for each layer. The analysis identifies small perturbation parameters that drive higher-order effects:
  • $(-z_i/L)^{-4/3}$ and $(-z_i/L)^{-2/3}$: Representing the effects of mean shear production of velocity variance and unsteadiness.
  • $-h_0/L$: Representing the effects of buoyancy production of vertical velocity variance.
• Asymptotic Matching:
  • Outer $\leftrightarrow$ Inner-Outer: Matching these layers yields higher-order corrections to the local-free-convection scaling.
  • Inner-Outer $\leftrightarrow$ Inner-Inner: Matching these layers yields higher-order corrections to the logarithmic law.
• Data-Driven Coefficient Determination: The theoretical expansions contain universal, non-dimensional coefficients. These were quantified using high-quality field data from the M2HATS (Multipoint Monin-Obukhov Similarity Horizontal Array Turbulence Study) campaign in Tonopah, Nevada (2023).
  • Technique: An iterative multi-step regression procedure was used, combining linear regression with ridge regression (to stabilize coefficients against multicollinearity) and bootstrap resampling (to quantify statistical uncertainty).

3. Key Contributions

1. Derivation of Higher-Order Profiles: The paper derives explicit analytical expressions for the mean velocity profile in the CBL that include second-order terms. These terms account for deviations from the log law and local-free-convection scaling due to finite $z_i/L$ and $h_0/L$ values.
2. Validation of the Convective Friction Law: The study confirms that the convective logarithmic friction law derived by Tong & Ding (2020), $U_m/u_* + C = \frac{1}{\kappa} \ln(-L/h_0)$, is valid to at least the second order. Unlike channel or pipe flows where the friction law is only a leading-order approximation, the CBL friction law appears robust across higher orders.
3. Re-evaluation of the von Kármán Constant ($\kappa$): By fitting the full higher-order expansion rather than just the leading-order log law, the authors isolate the true logarithmic term. They derive a value of $\kappa \approx 0.344$, which is lower than many recent field measurements (often $\sim 0.4$) but consistent with Businger et al. (1971). The paper argues that previous overestimations of $\kappa$ resulted from fitting the log law to data that included negative second-order corrections.
4. Quantification of Universal Coefficients: The study provides specific numerical values for the expansion coefficients (e.g., $A = -4.37$, $E = -1.58$, $D = 0.57$, $G = -0.23$ for the local-free-convection layer; $\kappa = 0.344$, $C' = -4.841$ for the log layer).

4. Key Results

• Local-Free-Convection Layer: The velocity defect profile is expressed as:
  $$\frac{U - U_m}{u_*} = A(-z/L)^{-1/3} + E(-z/L)^{-5/3} + G(-z/L)^{-3} + D \epsilon'_3 (-z/L)^{1/3} + \dots$$
  Comparisons with M2HATS data show excellent agreement, with the higher-order terms effectively capturing the scatter and deviations from the pure $(-z/L)^{-1/3}$ scaling.
• Log-Law Layer: The profile is expressed as:
  $$\frac{U}{u_*} = \frac{1}{\kappa} \ln\left(\frac{z}{h_0}\right) + C + C'\left(-\frac{z}{L}\right) + C'\alpha\left(-\frac{z}{L}\right)^2$$
  The inclusion of the linear and quadratic terms in $(-z/L)$ allows the model to fit data up to $z \sim -L$, a region where the pure log law fails.
• Friction Law Validation: The predicted friction law aligns perfectly with the measured $U_m/u_*$ values across a wide range of stability conditions, confirming that no additional parameters beyond $L/h_0$ are required for the second-order description.

5. Significance

• Theoretical Advancement: This work moves beyond empirical curve-fitting by deriving the functional form of velocity profiles directly from the governing equations, identifying the physical origins of deviations (shear production, unsteadiness, buoyancy).
• Improved Accuracy: The higher-order asymptotic profiles offer significantly improved accuracy over empirical profiles, particularly in the transition regions where $z \sim -L$ or $z \sim h_0$.
• Practical Application: The refined coefficients and the validated friction law provide more reliable inputs for numerical weather prediction models, wind energy site assessments, and atmospheric dispersion modeling.
• Resolution of Discrepancies: The study offers a plausible explanation for the long-standing discrepancy in the reported values of the von Kármán constant, suggesting that previous high values were artifacts of neglecting higher-order terms in the fitting process.