Parameterisation

The grid stops at a certain size. Everything smaller than the grid box is represented by a formula, and those formulas — not the equations of motion — are where models diverge most from one another.

What the grid cannot hold

A numerical weather model divides the atmosphere into boxes. At the coarser end, a global model might run at roughly ten kilometres horizontal resolution; at the fine end, a regional model pressed to its operational limit might manage a kilometre or less. Either way, the grid has an edge. Anything that happens on a smaller scale — a convective cell boiling up on a July afternoon, the turbulent transfer of heat from a sun-warmed field, the microphysics of ice crystal formation inside a cloud — simply does not fit in a single grid box, let alone a fraction of one. It cannot be computed directly from the resolved equations of motion. It has to be represented: described by a formula that relates the averaged, box-scale quantities the model does know to the effects the sub-grid process would have produced if the model could see it. That representation is parameterisation, and it is where the art enters an enterprise that otherwise looks entirely like physics and arithmetic.

The term carries no mystery once you see what it is doing. A parameterisation scheme takes a physical process — convection, say — and asks: given what I know about the temperature, humidity, and wind in this grid box, what flux of heat and moisture would convection realistically produce here? The answer is written as a formula with tunable constants — parameters — that have been set by comparing the scheme's output against real observations or against high-resolution simulations that explicitly resolve the process in question. Adjust a parameter and the scheme's behaviour shifts. Several of those parameters carry genuine uncertainty; different modelling centres make different choices, and those choices propagate forward into the forecast.

Where the schemes live and what they do

The largest family of parameterisation schemes concerns clouds and convection, because clouds are among the most consequential things the atmosphere does and among the hardest to put on a grid. Deep convection — the vigorous overturning that drives thunderstorms — occurs on scales of one to ten kilometres, well below the grid spacing of any global model and below the grid spacing of many regional ones. A convection scheme must decide, from box-scale signals alone, whether deep convection is likely to trigger, how much heating and drying it will cause in the column, and what fraction of the moisture will fall as rain at the surface. The scheme devised by Arakawa and Schubert in the early 1970s and its many descendants approach this by imagining a statistical ensemble of plumes rising within the column; different centres use different closure assumptions about what sets the strength of those plumes, and those assumptions produce measurably different precipitation patterns even when the large-scale dynamics are nearly identical.

Boundary-layer turbulence is a second major domain. The lowest kilometre of the atmosphere is continuously churned by friction with the surface and by heating from below. That turbulence mixes heat, moisture, and momentum in ways that control surface temperatures, humidity, and the triggering of convection — yet a single large eddy might be tens to hundreds of metres across, invisible to the model. Parameterisation schemes for the boundary layer usually work by computing turbulent fluxes from local gradients, with corrections for stability: a strongly stable night-time boundary layer mixes very little; an unstable daytime one mixes vigorously. Getting the stability treatment right affects overnight minimum temperatures, fog formation, and the timing of the afternoon convective maximum.

Radiation is handled by schemes that calculate how solar and infrared energy moves through a column of atmosphere loaded with water vapour, cloud water, cloud ice, ozone, and aerosols. The spectrum should properly be integrated over millions of wavelength intervals; a model does it over a few dozen, carefully chosen to be representative. Aerosol loading matters here: how much dust is in the column, how much sea salt, how absorbing the particles are — all of this enters through parameters that are uncertain and geographically variable.

Land-surface schemes form a fourth category. Soil type, vegetation cover, soil moisture, snow depth, and albedo all control how the surface partitions incoming energy between warming the ground, evaporating water, and heating the air. A grid box covers many square kilometres of heterogeneous terrain; the scheme must collapse that variety into a single answer. The treatment of soil moisture in particular has a known influence on summer precipitation forecasts in continental interiors, where the boundary between wet and dry soil can shift the location of convective initiation by hundreds of kilometres.

The tuning problem

Because parameterisation schemes contain adjustable constants, models can be and are tuned: parameters are varied until the model's climatological behaviour matches observations as closely as possible. This is not dishonest; it is a recognition that fundamental uncertainty exists and that observational constraints are a legitimate way to narrow it. But tuning introduces a subtle difficulty. A scheme tuned against global averages may perform less well in specific regimes — over the tropical oceans, say, or in polar winter. Compensating errors can hide inside a well-tuned model: two schemes each slightly wrong in ways that cancel. Change one scheme and the cancellation breaks down, which is why switching from one convection scheme to another in an otherwise identical model can degrade performance in a region that had nothing to do with convection.

The ECMWF ↗, whose global model has long set a benchmark for medium-range forecasting, publishes detailed documentation of its parameterisation choices alongside verification statistics — a practice that makes differences between centres legible and enables the community to track where schemes succeed and where they do not. The World Meteorological Organization ↗ coordinates comparisons between operational centres, and the discipline of model verification exists partly to separate genuine improvements in dynamics from regressions hidden under better tuning.

The direction of travel

Increased computing power is steadily pushing model grids finer. At one kilometre, deep convection can begin to be explicitly resolved rather than parameterised, and several national centres now run kilometre-scale forecasts operationally over limited areas. But the gain at one scale reveals a new edge: at one kilometre, microphysics — the collision, coalescence, and freezing of individual droplets and crystals — is still sub-grid and still parameterised. The boundary between what is resolved and what is represented moves without disappearing. Parameterisation is not a temporary scaffolding to be removed when computers grow fast enough; it is a permanent feature of representing a continuous, fractal atmosphere on a finite grid. The schemes will keep getting better because the observations that constrain them keep getting richer, and because the physical understanding written into them keeps deepening. But the gap between the grid and reality will always need something to fill it.