Resonant interaction of large-scale Langmuir circulation and thermoclinic internal waves

Frequently, the density of the upper ocean becomes very nearly uniform with depth as a consequence of the mixing accomplished by Langmuir cells or other vortical motions. The sharp density gradient—termed the thermocline—which develops at the base of this ‘mixed layer’ arrests the deepening of the cells into the abyssal fluid. Using a two-layer model to capture the essential features of these typical upper ocean conditions, we perform a theoretical investigation of the slow temporal dynamics of large-scale Langmuir cells when displacements of the thermocline are permitted. All rotational motion is assumed to be confined to the upper layer by the discrete stratification; the laminar, abyssal fluid is modeled as a semi-infinite, inviscid layer. We focus on the nonlinear interaction of stationary, near-critical Langmuir cells residing in the upper layer with internal waves propagating in opposite directions along the interface. We further restrict attention to ‘thermoclinic waves’ having a wavelength twice that of the Langmuir circulation field and traveling perpendicular to the axes of the Langmuir vortices, for such waves can resonantly interact with the Langmuir cells. Exploiting a small wavenumber limit, we derive a set of ordinary differential equations governing the evolution of the amplitudes of the modes comprising the resonant triad. In particular, we obtain analytical expressions for the coefficients in the resonant interaction equations. The coefficients are significant over a restricted, but physically relevant parameter regime, indicating that the resonant interaction may be a powerful mechanism for the transfer of energy between Langmuir cells and thermoclinic waves of the appropriate wavelength and orientation. Analytical and numerical integrations of the resonant interaction equations reveal interesting phenomenology including internal wave reflection from Langmuir cells, transient instability of Langmuir cells to internal wave disturbances (with concomitant velocity reversals in the cellular motion), and internal wave ‘bursts’ excited by Langmuir cells intensifying above a compliant thermocline.