[Paper Review] Neutral stability height correction for ocean winds

Ross N. Hoffman|arXiv (Cornell University)|Jul 7, 2011

Ocean Waves and Remote Sensing2 references3 citations

TL;DR

This paper proposes a simple, accurate analytical approximation for estimating ocean surface roughness length ($z_0$) under neutral stability conditions, enabling precise neutral stability height correction for ocean wind observations. The method uses a linear fit to the relationship between $y = \log(z/z_0)$ and $\gamma = (ak^2V^2)/(gz)$, yielding a closed-form expression for $z_0$ that is accurate for 1–30 m/s winds and serves as an excellent initial guess for Newton iteration, reducing height correction errors to a few percent.

ABSTRACT

Adjusting ocean wind observations to a standard height, usually 10 m, requires the use of a boundary layer model, and knowledge of the thermodynamical variables. Height adjustment is complicated by the fact that a necessary parameter, the roughness height, cannot be given in a closed form solution. If only the wind and reporting height are known, the best that can be done is to assume neutral stability. The determination of roughness height is analyzed and a simple approximation (used by Atlas et al. 2011) is derived in detail. This approximation is accurate for winds in the range of 1 - 30 m/s for neutral stratification and would be an excellent initial estimate for a Newton iteration to determine the roughness height precisely, whether or not neutral stability is assumed.

Motivation & Objective

• To develop a reliable, closed-form approximation for ocean surface roughness length ($z_0$) when only wind speed and measurement height are known.
• To improve the accuracy of neutral stability height correction for oceanic wind observations by replacing constant $z_0$ assumptions with wind-speed-dependent estimates.
• To provide an initial estimate for $z_0$ that enables rapid convergence in Newton-Raphson iteration for precise $z_0$ determination, even under non-neutral stability.
• To reduce systematic errors in wind height correction—previously up to 5–15%—by accounting for $z_0$ variation with wind speed.

Proposed method

• Derives the relationship $z_0 = \frac{a}{g} C_{dn} |V|^2$ using the Charnock formula and neutral drag coefficient $C_{dn} = \left[ \frac{k}{\log(z/z_0)} \right]^2$, linking $z_0$ to wind speed $|V|$ and height $z$.
• Introduces the auxiliary variable $\gamma = \frac{ak^2}{gz} |V|^2$, which links $z_0$ to measurable quantities via $\gamma = y^2 e^{-y}$, where $y = \log(z/z_0)$.
• Employs a linear fit $y = c_0 + c_1 \log(\gamma)$ with $c_0 = 3.7$, $c_1 = -1.165$ for $\gamma$ values corresponding to $|V| \leq 40$ m/s, derived from numerical data over $j = 6$ to $30$ in log-space $z_0$ values.
• Constructs the final approximation $\hat{z}_0 = z \exp[-(c_0 + c_1 \log(\gamma))]$, providing a direct, non-iterative estimate of $z_0$ from $|V|$ and $z$.
• Validates the method by comparing $\hat{z}_0$ to exact values, showing small errors in $z_0$ but negligible impact on wind speed ratios (e.g., $|V_{10}/V_{19.5}|$ differs by only 0.001 in the table).

Experimental results

Research questions

• RQ1 Can a simple, closed-form approximation for ocean surface roughness length ($z_0$) be derived that depends on wind speed and height under neutral stability?
• RQ2 How accurate is this approximation for typical oceanic wind speeds (1–30 m/s) when compared to exact iterative solutions?
• RQ3 Can this approximation serve as an effective initial guess for Newton-Raphson iteration to solve for $z_0$ precisely, even when stability is not neutral?
• RQ4 What is the impact of using a wind-speed-dependent $z_0$ on the accuracy of 10 m neutral wind speed correction compared to assuming a constant $z_0$?

Key findings

• The proposed approximation $\hat{z}_0 = z \exp[-(3.7 - 1.165 \log(\gamma))]$, with $\gamma = \frac{ak^2}{gz} |V|^2$, provides a highly accurate estimate of $z_0$ for neutral stability and wind speeds between 1 and 30 m/s.
• The method reduces height correction errors from up to 15% (when using constant $z_0$) to just a few percent by accounting for $z_0$ variation with wind speed.
• The approximation yields wind speed ratios (e.g., $|V_{10}/V_{19.5}|$) that differ by less than 0.001 from exact values, confirming its practical accuracy despite small $z_0$ errors.
• The approximation is robust enough to serve as an excellent initial estimate for Newton-Raphson iteration, which converges to machine precision in 2–4 steps.
• The derived coefficients ($c_0 = 3.7$, $c_1 = -1.165$) are independent of $z$, $a$, or $g$, making the method universally applicable for any height when $|V|$ is known.
• The method is particularly valuable in operational settings where thermodynamic data for stability are unavailable, as it enables accurate neutral stability correction with minimal assumptions.