OCE-0648284 A widely used vertical configuration for describing processes confined to the uppermost part of the ocean, say above the thermocline, is the reduced-gravity configuration. The PI contributed to demonstrate that the presence of the freely moving interface with the quiescent, infinitely deep fluid changes the classical horizontally-rigid-boundary baroclinic instability paradigm in many respects. In particular, the incorporation of free-boundary effects provided theoretical support for the claimed central role played by baroclinic instability waves in providing controls for mixed layer re-stratification and deep-ocean convection. A fundamental aspect of Hamiltonian instability theory is that it provides means for setting a priori bounds on the amplitude of those waves. Largely unexplored remains the study of their accuracy and utility in constructing eddy closures. Furthermore, it is unknown the extent to which closure-model predictions agree with the a priori instability saturation bounds, which make no assumption on the nature of the initial perturbation. Objectives. The proposed research effort will center on making assessments of the accuracy of conservation-law-based rigorous upper bounds on instability saturation, particularly in a thin layer confined near the ocean surface. The study will explore the use of a priori information on instability saturation in the construction of a transient-eddy parametrization. Intellectual Merit. The proposed study will be carried out using Hamiltonian ocean models, which, possessing Lyapunov stable equilibria, allow the derivation of a priori bounds on the growth of perturbations on unstable states. These are bounds on the ergodicity of the systems, which can be quantified in terms of the metric defined by the wave-enstrophy variance. Structure-preserving low-order approximations of the parent infinite-dimensional Hamiltonian set of equations will be derived to perform weakly (and not so weakly) nonlinear analysis, which will give a deeper insight into basic physics of finite-amplitude instability. "Structurepreserving approximation" is understood to be one that preserves the explicit symmetries of the parent infinite-dimensional system, i.e. those leading to the conservation laws of energy and momentum via Noether's theorem, and as best as possible the implicit symmetries, which relate to the existence of Casimir invariants. "Low-order approximation" is understood to be a finite-dimensional model that includes those (Fourier) components that allow the description of some aspects of nonlinear instability. The rigorous bounds will be used to quantify how well these aspects are represented. In addition to extending the validity of the low-dimensional model analysis, direct nonlinear numerical simulations will be performed to make an assessment of the accuracy of these saturation bounds as predictors of eddy amplitudes. These simulations will be carried out using recently proposed geometric integration methods. The robustness of the bounds beyond conservative theory will be investigated by performing forced dissipative simulations. The study will also seek to use the bounds to develop an eddy closure; it is expected that the weakly nonlinear analysis will provide guidance in performing this task. Broader Impacts. This project will increase our knowledge of physical processes that occur in the uppermost layer of the ocean, where most of the biomass concentrates and through which atmosphere ocean coupling takes place. The proposed research thus has implications for biology and climate, and consequently is beneficial for society. The proposed investigation effort will contribute to the career development of the PI who is employed as a 100% soft-money scientist. The PI is a female Hispanic, which will broaden the participation of underrepresented groups in ocean sciences. The proposed work offers interesting possibilities to further the education of graduate students. Funds are budgeted to incorporate one PhD student in the project.
