Chapter 2 · A short history of numerical weather prediction

Part I — What predictability means Knob: spurious wind, extrapolation interval, mean depth live notebook

Overview

In 1922 Lewis Fry Richardson published a six-hour weather forecast computed by hand from the governing equations. His answer was a surface-pressure change of 145 hPa [citation needed]. The observed change was essentially nothing, and nothing comparable has ever been observed anywhere on Earth.

The equations were right, and so, as far as anyone has been able to check, was the arithmetic. So what went wrong? The usual one-line answer — his initial conditions were unbalanced — is true and not quite the point. This chapter builds the smallest system that contains Richardson’s problem and measures what happened, and the answer is sharper than the slogan: the tendency he computed was real, and his error was extrapolating an oscillation.

The model

One-dimensional rotating shallow water on a periodic domain:

\partial_t u + u\,\partial_x u - f v = -g\,\partial_x h, \qquad \partial_t v + u\,\partial_x v + f u = 0, \qquad \partial_t h + \partial_x (h u) = 0 .

This is the least a model can contain and still have Richardson’s difficulty: a slow balanced mode that carries the weather, and a fast inertia-gravity mode that carries none of it. Derivatives are spectral, so the linear dispersion relation is exact to round-off rather than accurate to the order of a stencil — the measured frequencies sit on $\omega^2 = f^2 + gHk^2$ to 0.05 %.

Two features of that relation drive everything. There is a floor — no inertia-gravity wave is slower than $f$ , about 17 hours at mid-latitudes — and $\omega$ grows without bound with $k$ , so refining the grid makes the system stiffer. With $H = 8000$ m, the external mode Richardson’s equations carried, $c = \sqrt{gH} = 280$ m/s and the Rossby radius is 2800 km: comparable to a synoptic weather system, so the balanced and unbalanced parts of an analysis are not separated by scale and cannot be told apart by eye.

In this system the balanced state is exactly steady. With $u \equiv 0$ and $fv = g\,\partial_x h$ every tendency vanishes identically — $|\partial_t u|$ is $4\times10^{-19}$ , which is round-off, and a 30 m/s jet sits there with a six-hour drift of exactly zero. That is one dimension being kind: in two dimensions, with $f$ varying and advection able to feed back, no exact balanced manifold exists, which is why nonlinear normal-mode initialisation and digital filtering had to be invented.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

The tendency was real; extrapolating it was the error

Break the balance the way an observing system breaks it — with a spurious divergent wind. That is the right way to break it, because $\partial_t h = -\partial_x(hu)$ is a flux divergence, so the pressure tendency responds only to the divergent flow.

One metre per second of spurious divergence forces a pressure tendency of 9.18 hPa per hour, exactly linear in the wind (normalised tendencies 0.5000, 1.0000, 2.0000, 5.0000 at 0.5, 1, 2 and 5 m/s). So Richardson’s 145 hPa needs a spurious divergent wind of about 2.6 m/s, if the tendency is multiplied by six hours as he did. That is a small error in a 1910 wind analysis, and inside the uncertainty of a modern analysis in a data-sparse region. The surprising thing is not that his answer was large but how little it took.

And the multiplication is the mistake. At 1 m/s the tendency extrapolated over six hours gives 55.1 hPa; the model’s actual six-hour change is 1.28 hPa, a factor of 43 smaller. The tendency oscillates, at the inertia-gravity period of 2.45 hours, so

\Delta p(T) = \frac{A}{\omega}\sin \omega T

is bounded by $A/\omega$ for ever while the extrapolation $AT$ grows without limit. Measured: the largest excursion anywhere in 24 hours is 3.684 hPa against a predicted bound of 3.587 hPa — a ratio of 1.027. The extrapolation error is of order $\omega T$ , about a factor of ten once the forecast is long compared with a gravity-wave period.

That is the useful form of the lesson, because it says what to do: filter in time, or start from a state with no fast mode.

A small pressure tendency is not a balanced state

There is a trap here that survives into modern practice. Since the height equation is a flux divergence, it sees only the divergent wind. Break the balance in the rotational wind instead — scale $v$ down, leaving $u = 0$ — and the initial pressure tendency is not small. It is exactly zero. The forecast is ruined anyway: removing the rotational wind entirely gives a six-hour pressure change of 29.3 hPa, and removing a tenth of it gives 2.86 hPa, with the diagnostic reading zero throughout.

So “the pressure tendency is small” and “the state is balanced” are different statements, and the first does not imply the second. Pressure-tendency and divergence-based balance checks are still used and are blind to exactly this error. What catches it is checking the wind against the mass field — which is what geostrophic balance is, and what a variational analysis enforces through $\mathbf{B}$ (chapter 18).

It also explains why Richardson’s failure was legible. He broke the divergent wind, so his diagnostic screamed. An analysis that breaks the rotational wind produces a forecast that is quietly wrong, which is worse.

The timestep, and why filtering was worth a computer

The fast mode is a problem a second time, for reasons of arithmetic rather than physics. An explicit scheme is stable only while information moves less than about a grid cell per step, and the relevant speed is the fastest signal in the system — the external gravity wave at 280 m/s, not the wind at 30.

Measured, the threshold is $C\,\Delta x/c$ with $C$ between 0.843 and 0.911 across a factor of four in $c$ — an 8 % spread, no fitted constant beyond the scheme’s own Courant number. And the wind barely enters: quadrupling the jet moves the threshold a few per cent, while quadrupling $c$ quarters it. The timestep is set by the wave, not by the weather — and the wave carries no forecast information at all.

Hence filtering: equations from which the fast mode has been removed analytically. The barotropic vorticity equation $\partial_t \zeta + J(\psi,\zeta) = 0$ has no gravity waves by construction, because it has no divergence and no free surface. It is what Charney, Fjørtoft and von Neumann integrated on ENIAC in 1950 [citation needed], and over 400 steps it conserves energy and enstrophy to better than $10^{-6}$ %. At the external mode with a 30 m/s jet, a six-hour forecast costs 170 steps unfiltered against 16 filtered — a factor of 10.5, which on a machine doing a few hundred operations a second was the difference between a forecast and none.

Filtering then dominated for about a decade and lost. It had to: the barotropic vorticity equation cannot represent fronts or convection or anything else living in the divergent flow. What replaced it was the primitive equations again — Richardson’s equations — with the imbalance handled at the analysis step instead of by amputating the physics. That is the line from this chapter to chapters 18–20: the modern answer to Richardson is not a better equation set but a better initial condition.

Exercises

The extrapolation error is $\omega T$ . Read the ratio off the sliders at 1 hour and at 24 hours. Where would the two curves cross, and what would have to be true of the atmosphere for a six-hour extrapolation to be safe?
Reduce the depth to 500 m, roughly an internal rather than external mode. What happens to the Rossby radius, and would Richardson’s arithmetic have worked on a model carrying only that mode?
The rotational imbalance is invisible to the pressure tendency. Propose a diagnostic that would catch it using only fields Richardson had, and say whether he could have computed it by hand.
The timestep goes as $1/c$ with $c = \sqrt{gH}$ . If you halve $\Delta x$ , what happens to the total arithmetic for a fixed forecast length — and is that better or worse than the factor filtering buys?