Geophysical Fluid Dynamics

Vilhelm Bjerknes stated the programme in 1904. The atmosphere obeys known equations; measure its present state and integrate. He also knew what stood in the way: enough observations, and an impossible quantity of arithmetic. Lewis Fry Richardson attempted the arithmetic by hand during the First World War, for a single six-hour forecast, and got a pressure change of 145 millibars where the real change was almost nothing — an account of which belongs to numerical methods for partial differential equations.

Jule Charney found the reason in 1948, and it was not arithmetic error. The full equations support sound waves and other fast oscillations that carry almost no energy but dominate the rate of change at any instant. Richardson's initial pressures and winds, taken from independent measurements, were not in geostrophic balance with each other, and the imbalance rang the atmosphere like a bell. Charney derived a filtered system — the quasi-geostrophic equations — that keeps the slow weather-bearing motions and removes the fast ones. Those were the equations integrated on the ENIAC in 1950, for the first successful numerical forecast.

Then the programme met a limit of a different kind. Edward Lorenz, studying a drastically simplified convection model, found that trajectories starting from almost identical states diverge exponentially — described under chaos theory. For the atmosphere the doubling time of an error is a day or two, so an initial uncertainty of a per cent becomes total within a fortnight regardless of model quality. This is why forecasts are now issued as ensembles: fifty runs from slightly different initial states, reported as probabilities. The practical skill horizon has moved from about three days in 1980 to nearly ten today, and it cannot be pushed past roughly two weeks.

Three calculations, each a line or two, fix the characteristic scales that a forecast model has to resolve.

The geostrophic wind. The Coriolis parameter is $f = 2\Omega\sin\varphi$. With the Earth's rotation rate $\Omega = 7.292\times10^{-5}\ \mathrm{s^{-1}}$, at latitude 45°:

$$f = 2(7.292\times10^{-5})(0.707) = 1.03\times10^{-4}\ \mathrm{s^{-1}}.$$

Geostrophic balance sets $f V = (1/\rho)\,\partial p/\partial n$. A typical weather map has isobars 1 hPa apart every 100 km, that is $10^{-3}$ Pa/m, and air density is about 1.2 kg/m³:

$$V = \frac{1}{\rho f}\frac{\partial p}{\partial n} = \frac{10^{-3}}{(1.2)(1.03\times10^{-4})} = 8.1\ \mathrm{m/s}.$$

About 8 m/s, or 16 knots — which is what such a map's isobar spacing means to a forecaster, derived from nothing but the rotation rate of the planet.

The Rossby number. Whether rotation matters at all is the ratio of inertial to Coriolis terms, $\mathrm{Ro} = U/fL$. For a weather system, $U = 10$ m/s and $L = 1000$ km:

$$\mathrm{Ro} = \frac{10}{(10^{-4})(10^{6})} = 0.1.$$

Rotation dominates. Now the bath. Water draining from a tub has $U \approx 0.3$ m/s over $L \approx 0.3$ m:

$$\mathrm{Ro} = \frac{0.3}{(10^{-4})(0.3)} = 10^{4}.$$

The Coriolis force is ten thousand times too weak to matter, and the direction a bath drains is set by its shape and by how the water was disturbed. The folk claim is wrong by four orders of magnitude, and the Rossby number says exactly how wrong.

The deformation radius. The scale at which rotation and stratification balance is $L_R = NH/f$, where $N$ is the buoyancy frequency — the rate at which a displaced parcel oscillates — and $H$ the depth of the fluid. For the atmosphere, $N \approx 10^{-2}\ \mathrm{s^{-1}}$ and $H \approx 10$ km:

$$L_R = \frac{(10^{-2})(10^{4})}{10^{-4}} = 10^{6}\ \mathrm{m} = 1000\ \mathrm{km}.$$

That is the size of a weather system, and it is not a coincidence: the instability that creates mid-latitude cyclones grows fastest at this scale, so the atmosphere makes storms a thousand kilometres across because of its depth, its stratification and the rotation rate of the Earth. For the ocean, $N$ is larger but $H$ is smaller and the result is around 50 km, which is why ocean eddies are twenty times smaller than atmospheric ones — and why resolving them in a global model is twenty times harder.

The distinction between weather and climate is the distinction between a trajectory and a statistical attractor, and it is why a chaotic system can be unpredictable next month and predictable next century. What cannot be computed is the part of the system below the grid scale. A global climate model has cells tens of kilometres across; a cumulus cloud is a kilometre, and the droplets that determine whether it reflects or traps radiation are microns. Those processes are represented by parameterisations fitted to present-day observations, and the leading uncertainty in how much the planet warms — the open problem above — is whether those fits hold in a climate that has changed.

Henry Stommel's 1961 paper is the other thing this field contributed to that argument, and it required only two boxes and two equations. A circulation driven by both temperature and salinity carries the salinity contrast that drives it, which makes the system self-reinforcing and permits two stable states under identical forcing. A gradual change in forcing can therefore produce an abrupt switch that does not reverse when the forcing does. Whether the Atlantic overturning circulation is near such a threshold is among the most consequential open questions in the earth sciences, and the mathematical structure behind it is the same bistability that dynamical systems studies in the abstract.