A Kinematic–Dynamic 3D Model for Density-Driven Ocean Flows: Construction, Global Well-Posedness, and Dynamics

Differential buoyancy sources at an ocean surface may induce a density-driven flow that joins faster flow components to create a multi-scale, 3D flow. Potential temperature and salinity are active tracers that determine the ocean’s potential density: their distribution strongly affects the density-driven component, while the overall flow affects their distribution. We present a robust framework that allows one to study the effects of a general prescribed 3D flow on a density-driven velocity component through temperature and salinity transport, by constructing a modular 3D model of intermediate complexity. The model contains an incompressible velocity that couples two advection–diffusion equations for the two tracers. Instead of solving the Navier–Stokes equations for the velocity, we consider a prescribed flow composed of several spatially predetermined modes. One of these modes models the density-driven flow: its spatial form describes a density-driven flow structure and its strength is determined dynamically by averaged density differences. The other modes are completely predetermined, consisting of any incompressible, possibly unsteady, 3D flow, e.g., as determined by kinematic models, observations, or simulations. The result is a hybrid kinematic–dynamic model, formulated as a nonlinear, weakly coupled system of two non-local PDEs. We prove its well-posedness in the sense of Hadamard and obtain a priori rigorous bounds regarding analytical solutions. When the relevant Rayleigh number is small enough, we show, both rigorously and numerically, that for all initial conditions, the corresponding solutions converge to a unique steady state. Motivated by the Atlantic Meridional Overturning Circulation, the model’s relevance to oceanic systems is demonstrated by tuning the parameters to mimic the North Atlantic ocean. We show that in one limit the model may recover a simplified oceanic box model, including a bi-stable regime, and in another limit a kinematic model of oceanic chaotic advection, suggesting it can be utilized to study spatially dependent feedback processes in the ocean.