  While various parameterizations of vertical turbulent fluxes at different
levels of complexity have been proposed, each has its own limitations. For
example, simple first-order closure schemes such as the K-Profile
Parameterization (KPP) lack energetic constraints; two-equation models like
$k-\epsilon$ directly solve an equation for the turbulent kinetic energy but do
not account for non-local fluxes, and high-order closures that include the
non-local transport terms are computationally expensive. To address these, here
we extend the Assumed-Distribution Higher-Order Closure (ADC) framework
originally proposed for the atmospheric boundary layer and apply it to the
ocean surface boundary layer (OSBL). By assuming a probability distribution
function relationship between the vertical velocity and tracers, all
second-order and higher-order moments are exactly constructed and turbulence
closure is achieved in the ADC scheme. In addition, the ADC parameterization
scheme has full energetic constraints. We have tested the ADC scheme against a
combination of large eddy simulation (LES), KPP, and $k-\epsilon$ for surface
buoyancy-driven convective mixing and found that the ADC scheme is robust with
different vertical resolutions and compares well to the LES results.
