  Differential buoyancy surface sources in the ocean 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 3D
flow on a density-driven velocity component, by constructing a modular
observation-based 3D model of intermediate complexity. The model contains an
incompressible velocity that couples two advection-diffusion equations, for
temperature and salinity. Instead of solving the Navier-Stokes equations for
the velocity, we consider a flow composed of several temporally separated,
spatially predetermined modes. One of these modes models the density-driven
flow: its spatial form describes the density-driven flow structure and its
strength is determined dynamically by average 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 model is a non-linear, weakly coupled system of two non-local
PDEs. We prove its well-posedness in the sense of Hadamard, and obtain rigorous
bounds regarding analytical solutions. The model's relevance to oceanic systems
is demonstrated by tuning the model to mimic the North Atlantic ocean's
dynamics. In one limit the model recovers a simplified oceanic box model 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.

Geophysics;Analysis of PDEs;Fluid Dynamics