Collaborative Research: Nonlinear Interactions between Surface and Internal Gravity Waves in the Ocean

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The action of the wind on the ocean surface transfers momentum, serves as a source of kinetic energy for both turbulence and waves and is a controlling factor of the structure of the ocean surface boundary layer. When wind blows over the ocean, the ocean responds in two significant ways. The first is that viscous shear stresses are communicated below the sea surface via transfer of turbulent momentum. The second is that the sea surface acquires an undulatory character as the pressure differences across the leading and trailing faces of surface waves transfer momentum via form drag. While the undulatory character of the ocean surface can be painfully obvious to the ocean scientist onboard a ship, effects such as mixing associated with surface gravity wave breaking, instabilities of the surface wave shear, and coupling of the surface wave?s Stokes drift with wind driven shear to form Langmuir circulations are ignored in standard ocean surface boundary layer models. Missing as well from these models is a representation of relatively high frequency internal waves that could be prone to breaking immediately below the mixed layer. This project seeks to broaden understanding of the connectivity between ocean surface boundary conditions and upper ocean mixing. The highly nonlinear multi-layer model and the numerical and analytical tools to be developed in this project could be useful for a wide range of physical problems in the upper ocean including shear instabilities and coherent features such as Langmuir vortices. It is also important to stress that small changes in the interior mixing coefficients in the tropical oceans can have an immense feedback on Sea Surface Temperature and, therefore, other physical quantities, including convection and precipitation, of climatological significance. This project will provide training in first principles understanding of nonlinear waves and their interactions to a graduate student and a post-doc in applied mathematics. Research training through their active participation in this cross-disciplinary collaboration will provide them a unique opportunity to broaden their research experience in physical oceanography and improve their understanding of the interplay between the two disciplines. Ocean observations and basic physical considerations point towards a paradigm of greatly enhanced transfers at high wind speeds, trapping in the upper ocean at buoyancy frequency turning points that allows a nonlinear equilibration process and interaction with lower frequency shear that promotes enhanced internal wave dissipation. This paradigm demands consideration of something more sophisticated than a resonant analysis. The objective of this project is to understand the role of surface gravity waves resulting in the nonlinear excitation of internal gravity waves and assessing the internal waves propensity for mixing the upper ocean. Three possible parameter regimes are proposed. At low wind speeds, transfers tend to be from the background Internal Gravity Wave (IGW) field to the Surface Gravity Wave (SGW) field. At high wind speeds, current theoretical predictions of SGW-IGW transfer rates are proportional to wind speed, i.e. a very sensitive dependence upon wind speed. Transfers reverse sign and energy is transferred from the SGW field to the IGW field. The change in sign denoting the transition from low-wind to high-wind conditions coincides with gale force wind conditions. Extrapolating such dependencies to gale force, let alone hurricane force, invalidates the validity of the nonlinear theory. A likely third parameter regime coincides with the breakdown of this theory. The proposed research consists of three coordinated efforts. The first is an observational study with the objectives of documenting the vertical structure of upper ocean turbulent dissipation relative to standard mixed-layer schemes and estimates of SGW-IGW transfers rates, and documenting the relationship of high frequency internal wave variability to wind and wave conditions. The second of the three efforts is to develop a highly nonlinear model for a multi-layer system, focusing on the three-layer (well-mixed upper, relatively thin transitional, and deep lower layers) case, without any limitations on wavelength scales, and to perform a numerical study, to investigate both resonant and non-resonant SGW-IGW interactions at finite amplitude. Questions of the onset of internal wave breaking and transition layer mixing will be addressed. The third effort is to construct a self-consistent finite amplitude analytic description of nonlinear SGW-IGW interactions using the proposed layered formulation. The proposed third approach is, by taking advantage of the canonical Hamiltonian structure of the model, to investigate the equilibration of the IGW field with SGW variability and how this equilibration changes at finite amplitude. Then, the numerical and analytic studies will be cross-validated and compared with the ocean observations.