The real butterfly effect and maggoty apples

Lorenz’s 1963 paper cannot address that question—and hence the notion of the butterfly effect as Lorenz intended it to mean in his 1972 talk—because the 1963 model equations do not describe how fluid flows at different spatial scales interact. In fact, in his 1972 talk, Lorenz was informally discussing results from a highly technical paper he had published in 1969 in the Swedish journal Tellus. The abstract of the paper, titled “The predictability of a flow which possesses many scales of motion,” begins as follows:

The last clause of the sentence is worth reading a couple of times, because it is so surprising. Lorenz is describing chaotic unpredictability in the extreme. That type of unpredictability is much greater than that in his 1963 model of chaos. In the early model, you can predict as far ahead as you like by making the initial error sufficiently small. From a mathematical standpoint, Lorenz’s 1963 model has the property that the evolved state depends continuously on the initial state. As the initial state tends to the true state, so, too, does the forecast state.

On the basis of the Navier–Stokes partial differential equations, Lorenz’s 1969 paper describes systems that do not plausibly have that continuity property. Indeed, the limit of vanishing initial error, which I’ll discuss in more detail below, is what’s known as a singular limit.

To better appreciate what Lorenz proposed in his 1969 paper, suppose that we can observe the initial state of the atmosphere perfectly, with no errors or gaps. That does not mean that we can forecast perfectly, because to make a forecast of the weather, you must assimilate observations into a computational weather model, thus creating a set of initial conditions for the model.

The weather model approximates the Navier–Stokes and other relevant atmospheric equations using a finite, 3D array of so-called gridboxes. Collectively, the gridboxes cover the whole atmosphere and oceans. (Some models use finite sets of orthogonal functions, such as spherical harmonics, but that doesn’t change the argument.) Inside a gridbox, the weather model erroneously assumes that the atmosphere is completely homogeneous. The horizontal size of each gridbox in the very best global weather-forecast models is currently around 10 km.

Next, let’s suppose that we can make accurate weather forecasts of low-pressure systems on average up to seven days ahead with our weather model. In the idealized case of perfect observations, the source of error that limits the forecast’s accuracy lies in the gridbox-homogeneity assumption. Hence, it is reasonable to ask (our employers) for a bigger computer that would allow the weather equations to be integrated with a gridbox half the size. The incorrect homogeneity assumption would then be restricted to scales smaller than before by a factor of two.

Would that factor of two double the range of forecast accuracy from 7 days to 14 days? In his 1969 paper, Lorenz argues that it does not. The errors associated with small scales that were unresolved in the old model but are subsequently resolved in the new one would grow faster than errors in the smallest scales resolved in the old model. For example, if the error-doubling time of the newly resolved scales was half the error-doubling time of the previously resolved scales—meaning that the errors grow twice as fast—the predictability time with the new weather model will only increase by a factor of (1 + ½), which is significantly less than a factor of two.

Indeed, if later still we could afford a computer that would allow a further halving of the size of the gridboxes, the predictability time would only be increased from (1 + ½) × 7 days to (1 + ½ + ¼) × 7 days. If you carried on like that—halving the gridbox an infinite number of times—the predictability time would not be infinite. Rather, it would be (1 + ½ + ¼ + ⅛ + 1/16 …) × 7, or 14 days. With infinitesimally small gridboxes, forecasters would have increased the predictability time of the original model by only a factor of two. (The existence of that finite limit is consistent with the Kolmogorov energy spectrum for 3D fluid turbulence.)

But that sounds contradictory. After an infinite number of gridbox halvings, the (now infinitely powerful) computer represents the Navier–Stokes equations precisely. And because those equations are completely deterministic, we should be able to forecast infinitely far ahead.

To understand what accounts for the short forecast range, imagine having a bucket of apples that contain maggots. If you bite into an apple and discover half a maggot, then you have eaten half a maggot—an unpleasant experience. However, if you bite into an apple and discover a quarter of a maggot, then that’s even worse because you have eaten three-quarters of a maggot. More generally, if you bite into an apple and discover 1/n of a maggot, you have eaten 1 − 1/n of a maggot.

The larger the value of n, the greater the fraction of the maggot you have eaten, and the more unpleasant the experience. You might therefore imagine that the limit n = ∞ of a sequence of such apple bitings describes the most unpleasant experience. But it doesn’t. If you bite into an apple and discover no maggot, you may not have eaten a maggot at all! (A tiny maggot fraction is qualitatively different from no maggot.)

That example, first described by theoretical physicist Michael Berry, is known as a singular limit (see his Reference Frame, Physics Today, May 2002, page 10). Such limits abound in physics. For example, blackbody radiators never experience a UV catastrophe—the prediction that the intensity of their emitted radiation goes to infinity as wavelength decreases—provided that Planck’s constant h remains nonzero (no matter how small it is). Set h precisely to zero, however, and the classical Rayleigh–Jeans spectrum diverges.

In another example, as long as a fluid’s viscosity remains nonzero, it is able to generate aerodynamic lift across an airfoil, no matter how small the viscosity may be. If viscosity is set to zero, however, the boundary condition across the airfoil qualitatively changes. The lifting force of a 3D body in incompressible, inviscid, irrotational flow is zero, a phenomenon known as d’Alembert’s paradox.

There is also a singular limit at the heart of what I call the real butterfly effect.⁵ No matter how small the initial uncertainty, the butterfly effect limits predictability to a finite time horizon. Only when the initial uncertainty is identically zero can you potentially predict arbitrarily far ahead with the Navier–Stokes equations. That’s an unrealistic limit, of course. Is the singular predictability limit a rigorous mathematical property of the Navier–Stokes equations? No one knows. The problem of whether solutions depend continuously on initial conditions is related to the unsolved Clay Mathematics Institute Millennium Prize Problem concerning the existence of smooth, unique solutions to the Navier–Stokes equations.

That is not to say that Lorenz’s more famous 1963 model of chaos has nothing useful to say about the predictability of weather. I have used the model on many occasions to demonstrate that the predictability of a nonlinear system is not a fixed quantity. It varies from one initial condition to another, as shown in figure 2. Hence, although the average predictability of day-to-day weather may be around two weeks, it can sometimes be longer and sometimes shorter than that. Meteorologists can estimate such flow-dependent predictability by running ensembles of forecasts—typically 50 are run from almost but not quite identical initial conditions. When the atmosphere is in a predictable state, the ensemble forecast spread will be relatively small. When the atmosphere is in an unpredictable state, the spread will be relatively large.

Ensemble prediction has transformed weather forecasting over recent years. For example, it determines the probability of precipitation on your weather app. More importantly, it is changing the way in which humanitarian and disaster relief agencies respond to extreme weather events. In the past, the unreliability of deterministic predictions meant that they would typically wait for an extreme event to occur before sending in medicine, food, water, and emergency shelter to stricken regions. Now, on the basis of a cost-benefit analysis, those agencies predetermine a threshold probability for extreme weather. And if the ensemble-based forecast probabilities exceed the threshold, the agencies take what’s known as “anticipatory action,” sending in emergency supplies ahead of the weather event.

The real butterfly effect implies that although the governing partial differential equations are deterministic, any computational representation of the equations will be indeterministic. That’s not, however, the way weather and climate models have traditionally been formulated. The processes in such models that cannot be resolved explicitly—cloud formation, the flow of air over small mountains, and ocean mixing, for example—have been represented by deterministic parameterization formulas that mimic molecular viscosity and diffusion.

The real butterfly effect, however, implies that no consistent way to represent those subgrid processes by deterministic formulas exists. One way to alleviate the problem is to make the parameterization formulas in weather and climate models explicitly stochastic.^6,7 The first stochastic-parameterization scheme was introduced into a weather forecast model in 1999. And today, most weather models incorporate some form of stochastic parameterization.

Even so, many climate models—even those contributing to assessment reports from the Intergovernmental Panel on Climate Change—are still formulated with deterministic closure schemes. Such models are inconsistent with the Navier–Stokes equations’ scaling symmetries, which contributes to their (sometimes substantial) long-term systematic errors.⁸ Stochasticity can have unexpected effects in nonlinear models.⁹ Figure 3, for example, shows that adding noise to the Lorenz 1963 equations helps to stabilize the Lorenz-attractor regimes. The stabilizing effect is quite counterintuitive until you realize that the model makes transitions from one regime to the other in small regions of state space. Those transitions can be disrupted (and thus the regimes stabilized) by small amounts of noise.

Artificial intelligence (AI) is now being used to make weather forecasts with levels of skill comparable to more traditional physics-based models. For both training and forecasting, those AI-based models still use sets of gridded, global atmospheric states, in which atmospheric observations have been assimilated into a global physics-based model. Can such AI forecast systems simulate the real butterfly effect?

To answer that question, Tobias Selz and George Craig (both at the German Aerospace Center in Oberpfaffenhofen) compared the growth of estimates of forecast uncertainty using AI and physics-based models last year.¹⁰ The estimate of the initial uncertainty was obtained by taking the difference between two randomly chosen members of an ensemble of data assimilations, which are used in ensemble weather forecasting. The members of the ensemble differ only in the precise values of the observations being assimilated into the model—the variations in those precise values being consistent with observational error.

By construction, the initial error for a weather forecast is spread across a range of scales—from weather systems with a horizontal wavelength of thousands of kilometers down to the model’s grid scale of 10 kilometers or so. The theory of data assimilation predicts that if the spacing between atmospheric observations is typically a few tens of kilometers, then observations do well at determining the large-scale initial weather patterns, with little error. On kilometer scales, however, the errors will become almost as large as it is possible for them to be. Small-scale errors in the initial conditions are thus almost saturated, while large-scale errors have plenty of opportunity to grow. Accordingly, errors grow almost immediately at the large scale but not at all at the small scale.

To study the real butterfly effect, Selz and Craig divided the initial-error field by a factor of 1000. Then, the small-scale errors were far from saturated. Because they grow so much faster than the large-scale errors, the errors should be dominated by the small scales. That is precisely what is seen when a physics-based model is used. And Selz and Craig used both a low-resolution and a high-resolution physics-based model to demonstrate it. Figure 4 shows the divergence of pairs of forecasts with small initial differences.

The high-resolution model did a much better job at simulating the rapid growth of the small-scale errors, but the low-resolution model was not completely hopeless; the growth was simply less dramatic. By contrast, the AI system completely failed to predict the growth of small-scale errors. That’s perhaps not surprising. In the real world, as I mentioned, the small-scale errors are already saturated at the initial time. The AI system never learns about the real butterfly effect from its training data. The results demonstrate that you must be cautious when applying AI to the weather-forecast problem; it does not contain the physics of the real butterfly effect.

As I discuss in my book The Primacy of Doubt,¹¹ studying the predictability of weather and climate reveals some deep and important properties of nonlinear systems. They are relevant to many problems in applied and fundamental science—in various fields, including social science and the foundations of quantum physics. In short, taking a rigorous approach to the science of uncertainty can help us improve our ability to both predict and understand our very chaotic world.

Tim Palmer is a Royal Society Research Professor in Climate Physics at the University of Oxford in the UK.