Evaluations of the von Kármán constant in the atmospheric surface layer

Abstract

The von Kármán constant $k$ relates the flow speed profile in a wall-bounded shear flow to the stress at the surface. Recent laboratory studies in aerodynamically smooth flow report $k$ values that cluster around 0.42–0.43 and around 0.37–0.39. Recent data from the atmospheric boundary layer, where the flow is usually aerodynamically rough, are similarly ambiguous: $k$ is often reported to be significantly smaller than the canonical value 0.40, and two recent data sets suggest that $k$ decreases with increasing roughness Reynolds number $Re_{\ast}$. To this discussion, we bring two large atmospheric data sets that suggest $k$ is constant, 0.387$\,{\pm}\,$0.003, for $2\, {\le}\,\hbox{\it Re}_\ast \,{\le} \,100$.

The data come from our yearlong deployment on Arctic sea ice during SHEBA, the experiment to study the Surface Heat Budget of the Arctic Ocean, and from over 800 h of observations over Antarctic sea ice on Ice Station Weddell (ISW). These were superb sites for atmospheric boundary-layer research; they were horizontally homogeneous, uncomplicated by topography, and unobstructed and uniform for hundreds of kilometres in all directions.

During SHEBA, we instrumented a 20 m tower at five levels between 2 and 18 m with identical sonic anemometer/thermometers and, with these, measured hourly averaged values of the wind speed $U(z)$ and the stress $\tau (z)$ at each tower level $z$. On ISW, we measured the wind-speed profile with propeller anemometers at four heights between 0.5 and 4 m and measured $\tau $ with a sonic anemometer/thermometer at one height. On invoking strict quality controls, we gleaned 453 hourly $U(z)$ profiles from the SHEBA set and 100 from the ISW set. All of these profiles reflect near-neutral stratification, and each exhibits a logarithmic layer that extends over all sampling heights. By combining these profiles and our measurements of $\tau $, we made 553 independent determinations of $k$. This is, thus, the largest, most comprehensive atmospheric data set ever used to evaluate the von Kármán constant.