A Second-Moment Closure Model of Langmuir Turbulence

The Reynolds stress equation is modified to include the Craik-Leibovich vortex force, arising from the interaction of the phase-averaged surface wave Stokes drift u(S) with upper-ocean turbulence. An algebraic second-moment closure of the Reynolds stress equation yields an algebraic Reynolds stress model (ARSM) that requires a component of the vertical momentum flux to be directed down the gradient partial derivative(z)u(S) of the Stokes drift, in addition to the conventional component down the gradient partial derivative(z)(u) over bar of the ensemble-averaged Eulerian velocity. For vertical w' and horizontal u' component fluctuations, the momentum flux must be closed using the form (u'w') over bar=-K-M partial derivative(z)(u) over bar -K-M(S)partial derivative(z)u(S), where the coefficient K-M(S) is generally distinct from the eddy viscosity K-M or eddy diffusivity K-H. Rational expressions for the stability functions S-M = K-M/(ql), S-M(S) = K-M(S)/(ql), and S-H = K-H/(ql) are derived for use in second-moment closure models where the turbulent velocity q and length l scales are dynamically modeled by prognostic equations for q(2) and q(2)l. The resulting second-moment closure (SMC) includes the significant effects of the vortex force in the stability functions, in addition to source terms contributing to the q(2) and q(2)l equations. Additional changes are made to the way in which l is limited by proximity to boundaries or by stratification. The new SMC model is tuned to, and compared with, a suite of steady-state large-eddy simulation (LES) solutions representing a wide range of oceanic wind and wave forcing conditions. Comparisons with LES show the modified SMC captures important processes of Langmuir turbulence, but not without notable defects that may limit model generality.