A Numerical Investigation of Mixed-Layer Dynamics

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Abstract
The structure of the stratified turbulent upper mixed layer of the ocean has been numerically investigated by using the turbulence closure model of Gibson and Launder, under the action of an impulsive wind stress τ0 and zero surface heat flux. The values of buoyancy and Coriolis frequencies assumed are N = 0.94 × 10−2 s−1 and f = 10−4 s−1, respectively. The solutions indicate that the turbulent diffusion terms, small in general, transfer kinetic energy downward, although its effect on the deepening is negligible. Let ti be the time in inertial periods and u* be the friction velocity. Then for 0.05 < ti < 0.3, the rate of increase of potential energy in the water column varies as ∂(PE)/∂t ∝ t½, rising to a maximum of ∼1.1u*3 and implying a mixed layer depth h∝t½ as in the Pollard-Rhines-Thompson (PRT) model. For 1 < ti < 6, ∂(PE)/∂t decreases only slightly from a quasi-steady value of ∂(PE)/∂t ≈ 0.25u*3, implying a deepening rate slightly smaller than the Kraus-Turner h ∝ t⅓. The reason for this difference in behavior for the two time ranges is the separation of the flow into a depth-independent inertial oscillation and a quasi-steady shearing flow that carries almost all the turbulent stresses in the water column. The mechanism for deepening is always the lifting of heavier mass by the locally generated turbulence at the base of the mixed layer. For very large times (ti > 12), ∂(PE)/∂t drops sharply, and no deepening was detected with a vertical resolution of 1 m. The assumption necessary to derive the PRT energy equation, namely, that the depth-integrated dissipation nearly balances τ0·(U0 − Û),where U0 is the surface velocity and Û the depth-averaged velocity, is approximately valid. For ti < 0.25, the PRT bulk Richardson number criterion is equivalent to a local critical gradient Richardson number criterion, and is due to the self-similarity of the solutions and the consequent thickening of the “interface.” The self-similarity breaks down for larger times, either because of the Coriolis forces becoming more important or because of the appearance of a sharp interface due to a nonlinear mechanism, whichever is earlier. An imposition of a kinetic energy input at the sea surface, so as to simulate the wind-wave flux, has certain desirable features.

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