## Abstract

The dependence of mixing efficiency on time-varying forcing is studied by direct numerical simulation (DNS) of Kelvin–Helmholtz (KH) instability. Time-dependent forcing fields are designed to reproduce a wavelike oscillation by solving the equations of motion in a tilted coordinate frame and allowing the tilt angle to vary in time. Mixing efficiency Γ* _{c}* is defined as the ratio of potential energy gain to dissipation, both averaged over one forcing cycle and first examined via parameters characterizing waves: the minimum Richardson number Ri

_{min}and the normalized frequency of the forcing

*ω*/

*N*. The effect of Reynolds number Re

_{0}and the initial random disturbance amplitude

*b*are also examined. In the experiments presented, Γ

*varies between 0.21 and 0.36 and is controlled by the timing of two events: the emergence of KH billows and the arrival of the deceleration of the mean shear by the wavelike forcing. Here, Γ*

_{c}*is higher than a canonical value of 0.2 when the deceleration phase of the forcing suppresses the less efficient turbulence after breakdown of KH billows. However, when Ri*

_{c}_{min}and

*ω*/

*N*are small, KH billows start to develop before Ri

_{min}is achieved. Therefore, the forcing accelerates the mean shear and thereby sustains turbulence after the breakdown of KH billows. The canonical value is then reproduced in the DNS. Although larger values of Re

_{0}and

*b*intensify the development of KH billows and modify Γ

*, this effect is less significant when forcing fields act to sustain turbulence. The time-averaged Thorpe scale and Ozmidov scale are also used to see how mixing is modified by forcing fields and compared with past microstructure measurements. It is found that DNS also corresponds to past observations if the forcing accelerates the mean shear to sustain turbulence.*

_{c}*Corresponding author address:* Ryuichiro Inoue, Ocean Physics Department, Applied Physics Laboratory, University of Washington, 1013 NE 40th St., Seattle, WA 98105-6698. Email: rinoue@apl.washington.edu