Friction law and turbulent properties in a laboratory Ekman boundary layer

We use spin-up/spin-down laboratory experiments to study the neutrally stratified Ekman boundary layer. The experiments are performed in the 13 m diameter, 1 m deep Coriolis rotating tank of the LEGI in Grenoble, France. A global flow rotation is produced by an initial change in the tank rotation speed. It then slowly decays under the effect of Ekman friction, evolving from the turbulent state to the laminar state. It is checked that the Ekman layer itself remains in a quasi-steady state during this decay. The velocity is measured by Particle Imaging Velocimetry (PIV) at two scales: the global rotation in a horizontal plane, and the vertical profile inside the boundary layer, where the three velocity components are obtained by stereoscopic PIV. The friction law is obtained by relating the decay rate of the bulk velocity to the velocity itself. This method is justified by the fact that this bulk velocity is independent of height beyond the top of the boundary layer (a few cm), as expected from the Taylor-Proudman theorem for rotating fluids. The local measurements inside the boundary layer provide profiles of the mean velocity and Reynolds stress components, in particular the cross-isobar angle between the interior and near surface velocities. In the laminar regime, good agreement is obtained with the classical Ekman's theory, which validates the method. In the turbulent regime, the results are found consistent with the classical Atmospheric Boundary Layer (ABL) model based on the von Karman logarithmic layer. Our experiments therefore indicate that this theory, in principle valid for very large Reynolds numbers, is already relevant close to the transitional regimes. A fit of the empirical coefficients A and B appearing in this theory yields A = 3.3 and B = 3.0. Extrapolating the results to the atmospheric case gives a friction velocity u_* about 12% higher than the traditional fit for the ABL. We may safely deduce that for the oceanic bottom boundary layer, corresponding to lower Reynolds numbers than the atmosphere, our result provides a correct estimate within 10%. The previous laboratory results of Caldwell et al. [“A laboratory study of the turbulent Ekman layer,” Geophys. Fluid Dyn. 3, 125–160 (1972) https://doi.org/10.1080/03091927208236078] provided frictions velocities about 20% higher than in our experiments, and slightly higher cross-isobar angles. We attribute this difference to the higher vortical Rossby number Ro_t in those experiments, and maybe also to roughness effects. We take into account the effect of this vortical Rossby number within the framework of the Ekman layer (Ro_t → 0) by replacing the tank rotation rate by the fluid rotation rate.