Inside an ocean model: grids, physics, forcing, and boundaries

In the video, Fernand Assene opened up an ocean model to show its building blocks. This reading gathers them in one place, so you can see how a model is assembled and why each choice matters. The recurring theme: every part of a model is a choice, and each choice trades off realism, the processes you want to capture, and computational cost.

The grid: turning a continuous ocean into points

The real ocean is continuous, but a computer can only solve equations at a finite number of locations. The grid (or mesh) is the set of points where the model computes its variables – and the central challenge is representing a curved, complex coastline on a discrete mesh.

There are two broad families:

• Structured grids have a regular, organised pattern. The simplest is a regular latitude-longitude grid – rectangular cells, straightforward mathematics – but its cells shrink badly toward the poles, which is inefficient for global models. A curvilinear grid deforms this regular pattern so the grid lines follow the natural geometry of the domain, such as a coastline or a continental shelf, while keeping the mathematical structure intact.
• Unstructured grids abandon a regular pattern entirely, filling the domain with triangles of any size. Resolution can then vary continuously – kilometres offshore, down to metres inside a harbour or tidal channel – all in one simulation. The trade-off is that unstructured models are harder to run efficiently on a computer.

Concentrating resolution where it matters

Because high resolution everywhere is expensive, models use strategies to put detail only where it’s needed:

• Progressive stretching (“Swiss cross”) smoothly shrinks a structured grid’s cells toward a region of interest and grows them again away from it
• Nesting embeds a finer “child” grid inside a coarser “parent” grid, so a regional model inherits its edges from a larger-scale one. Information can flow one way (parent to child) or both ways
• Unstructured refinement lets triangle size vary freely across the mesh – the most flexible option, and the most demanding

The vertical dimension

A model must also divide the ocean vertically, and this choice strongly affects how well it represents mixing and flow near the seabed. Three approaches:

• Z-levels are flat layers at fixed depths – simple and efficient, and the most widely used, but they struggle to represent steep bottom topography cleanly
• Terrain-following (sigma) coordinates follow the shape of the seafloor, squeezed together in shallow water and stretched in deep water. Excellent for coastal work, though they can introduce some artificial mixing
• Isopycnic coordinates follow surfaces of constant density – ideal for tracking how temperature, salinity, and chemicals travel through the ocean, since in reality they move mostly along density surfaces

Many modern operational models use hybrid vertical coordinates, combining the strengths of two or more of these in different parts of the water column.

Two steps: the physics, then the numerics

Every ocean model is built in two distinct steps:

• Step 1 is physical – writing down the governing equations (the primitive equations for momentum, heat, salt, and mass) that describe what we want to simulate
• Step 2 is numerical – replacing those continuous equations with discrete values on the grid, and advancing them forward in time

That second step comes with a fundamental constraint. To stay stable, the model’s time step must be short enough that information never jumps across more than one grid cell at a time. This is the CFL condition, and its consequence shapes everything: finer grids require shorter time steps, which drives up the computational cost dramatically. It’s the deep reason resolution is never free.

Forcing: what drives the model

A model doesn’t move on its own – it is driven by forcing, the external inputs applied at its boundaries, usually changing over time:

• At the surface: wind stress (which drives currents), heat fluxes (solar heating and exchange with the air), and freshwater from evaporation and rain
• At the lateral edges: river runoff, large-scale temperature, salinity and currents, and tides
• At the seafloor: friction, and in some regions geothermal inputs

These come from observations, atmospheric reanalyses, or weather forecasts, and are applied at every time step. Their quality is a primary control on accuracy: even a perfect model produces poor results if forced with inaccurate winds or heat fluxes.

Boundary conditions: closing the domain

Finally, a model needs boundary conditions – the values prescribed at the edges of its domain – to have a single, well-defined solution. There are three kinds:

• Initial conditions define the full 3D ocean state (temperature, salinity, currents, sea level) at time zero. Starting from a good estimate – often a reanalysis such as GLORYS12 or a climatology – matters a lot: a poor start leaves the model needing a long “spin-up” to settle into a realistic state
• Surface and bottom conditions let atmospheric forcing enter at the top and friction act at the seabed
• Lateral boundaries come in two types: closed boundaries (coastlines, where water cannot flow through land) and open boundaries (where the domain is artificially cut off, needing special treatments so waves and currents can leave and enter without spurious reflections)

For the curious: a few deeper choices

The lecture also touched on some more technical decisions, included here for those who want them – they are not essential to the rest of the week:

• Where variables sit in a cell (Arakawa grids). Storing every variable at the same point turns out to behave poorly for rotating flows, so models stagger them. The Arakawa C-grid – velocities on the cell faces, scalars at the centre – is the modern standard because it mimics how rotating ocean flows naturally behave.
• How space derivatives are computed. Three method families exist: finite difference (simple, efficient, suited to structured grids), finite volume (naturally conserving mass and tracers, works on any grid), and finite element (the most flexible, and the heaviest to compute).

Key takeaways

• A model turns the continuous ocean into a finite grid – structured or unstructured – with strategies like nesting to concentrate resolution where it’s needed
• The vertical can be divided by fixed depth (z-level), by seabed shape (sigma), or by density (isopycnic), often combined as hybrid coordinates
• Building a model is two steps – the physics (equations), then the numerics (solving them on a grid); the CFL condition ties finer grids to shorter time steps and higher cost
• Forcing drives the model from its boundaries, and its quality directly limits accuracy
• Boundary conditions – initial, surface/bottom, and lateral (closed and open) – close the domain so the model has a single, well-defined solution