D8 Numerical Modeling and Assimilation

Topic

Numerical ocean models integrate the primitive equations of the ocean—derived from the Navier-Stokes equations, alongside thermodynamics and the seawater equation of state—across three-dimensional grids covering the entire ocean; using these, they simultaneously simulate general circulation, thermohaline circulation, tides, and oceanic mesoscale features. Today, the highest-resolution models reach a grid spacing of one-twelfth of a degree (approximately eight kilometers)—fine enough to explicitly resolve the mesoscale rather than having to approximate it. Several such numerical ocean models exist, developed at various centers in Europe and the United States; each is designed for a specific domain—such as the global ocean, a particular basin, or a coastal zone—meaning that no single numerical ocean model is the best for every application.

Alongside these circulation models, wave models address a different physical problem: they integrate the wave action equation on two-dimensional grids, simulating the generation, propagation, and eventual dissipation of the full wave spectrum that constitutes the sea state at any given moment. Consequently, these wave models complement ocean circulation models rather than replacing them.

Ocean data assimilation combines the observations described in previous documents—autonomous floats, satellite measurements, and hydrographic cruises—with these numerical ocean models to derive the best possible estimate of the ocean's actual state. The four-dimensional variational method minimizes a cost function that weights the discrepancy between the model and the observations over a specific time window, seeking the initial conditions that best explain those observations; The ensemble Kalman filter, in contrast, evolves an entire set of distinct realizations of the same model in parallel and updates that entire set whenever new observations arrive. In both cases, the final result is the analysis—the best estimate of the ocean state at that moment and the starting point for any subsequent prediction; it is precisely this combination of observations and model that makes the analysis consistently more accurate than either component alone.

Ocean reanalyses apply this same data assimilation process but look backward in time over periods spanning several decades. They retrospectively combine all available historical observations with the numerical model, thereby reconstructing a coherent, continuous history of the ocean state—even for eras with far fewer observations than today. Consequently, for those earlier decades, ocean reanalyses serve as the only truly homogeneous source of ocean data available.

Ocean digital twins bring this same concept into the present: they are virtual representations of the ocean, updated in real-time or near real-time, that combine high-resolution numerical ocean models with data assimilation and machine learning systems to generate operational forecasts. Within these digital twins, machine learning is applied to emulate climate models that are otherwise too computationally expensive to run directly, to detect ocean eddies, to classify water masses, and to improve parameterizations of small-scale processes that the model grid itself cannot resolve; thus, these digital twins combine almost all the previously described techniques into a single system.

The expected behavior of this entire chain—from the numerical ocean model to the digital twin that updates it in real-time—is for the model solution to converge toward the observations whenever the data assimilation process functions correctly. This convergence is measured by comparing two errors: the background error—the difference between the model and observations prior to assimilation—and the analysis error—that same difference after assimilation. It is precisely the fact that the second error is much smaller than the first that confirms that ocean data assimilation genuinely improved the state of the ocean as described by the model.

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