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Evaluations of the von Kármán constant in the atmospheric surface layer

  • EDGAR L ANDREAS (a1), KERRY J. CLAFFEY (a1), RACHEL E. JORDAN (a1), CHRISTOPHER W. FAIRALL (a2), PETER S. GUEST (a3), P. OLA G. PERSSON (a2) (a4) and ANDREY A. GRACHEV (a2) (a4)...
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.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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