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Abstract

This is our response to a comment by Walter Eifler on our paper `A simple model for
the short-time evolution of near-surface current and temperature profiles' (arXiv:physics/0503186,
accepted for publication in Deep-Sea Research II). Although Eifler raises genuine
issues regarding our model's validity and applicability, we are nevertheless of the
opinion that it is of value for the short-term evolution of the upper-ocean profiles
of current and temperature. The fact that the effective eddy viscosity tends to infinity
for infinite time under a steady wind stress may not be surprising. It can be interpreted
as a vertical shift of the eddy viscosity profile and an increase in the size of the
dominant turbulent eddies under the assumed conditions of small stratification and
infinite water depth.

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 < 1·0, which includes nearly all of the KP, KPA data, 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 < 160 where the effect of side-wall drag is negligible or can be removed by a linear extrapolation. This result, together with the form and magnitude of the observed side-wall effect, suggests that mean momentum conservation is a key constraint upon the entrainment rate in the KP, KPA experiments.