On the scaling of stress-driven entrainment experiments

The entrainment experiments of Kato & Phillips (1969) and Kantha, Phillips & Azad (1977) (hereafter KP and KPA) are analysed to demonstrate a more general and effective scaling of the entrainment observations. The preferred scaling is \[ V^{-1} dh/dt = E(R_v), \] where h is the mixed-layer depth, V is the mean velocity of the mixed layer, R v = B / V 2 and B is the total mixed-layer buoyancy. This scaling effectively collapses entrainment data taken at various h/L , where L is the tank width, and in cases in which the interior is density stratified (KP) or homogeneous (KPA). The entrainment law E ( R v ) is computed from the KP and KPA observations using the conservation equations for mean momentum and buoyancy. A side-wall drag term is included in the momentum conservation equation. In the range 0·5 R v E ≃ 5 × 10 −4 R −4 v . This is very similar to the entrainment law followed by a surface half-jet (Ellison & Turner 1959) and by the wind-driven ocean surface mixed layer (Price, Mooers & Van Leer 1978). The analysis shows that, when forcing is steady, R v is quasi-steady and, provided that side-wall drag is not large, R v ≃ 0·6 over a wide range of R T = B/U 2 * , where U * is the friction velocity of the imposed stress. In the absence of side-wall drag (vanishing h/L ) the conservation of momentum then leads to U −1 * dh / dt = n (0·6) ½ R −½ T , where n = ½ or 1 if the interior is linearly stratified or homogeneous. The KP, KPA data show this dependence throughout the range 17 R T