Last time, we saw that some mathematical systems are so sensitive to initial conditions that even very small uncertainties in their initial state can snowball, causing even very similar states to evolve very differently. The equations describing fluid turbulence are examples of such a system; Lorenz’s discovery of extreme sensitivity to initial conditions ended hopes for long term weather forecasting. Because the state of the weather can only be known so well, the small errors and uncertainties will quickly build up, rendering weather simulations useless for looking more than a few days ahead of time.
But Lorenz’s discovery doesn’t have much impact on climate modelling, contrary to the claims of some climate skuptix. Climate is not weather, and modelling is not forecasting.
Weather refers to the state of the atmosphere at a particular time and place: What temperature is it? Is it raining? How hard is the wind blowing, and in which direction? Climate, on the other hand, is defined in terms of the statistical behavior of these quantities:
“Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. [...] Climate change refers to a change in the state of the climate that can be identified (e.g., by using statistical tests) by changes in the mean and/or the variability of its properties, and that persists for an extended period, typically decades or longer. ” IPCC
Many climate skuptik talking points derive from confusing these two quantities, in much the same way that a gambler might win a few hands of poker and decide that they are on a roll.
Although it is generally not possible to predict a specific future state of the weather (there is no telling what temperature it will be in Oregon on December 21 2012), it is still possible to make statistical claims about the climate (it is very likely that Oregon’s December 2012 temperatures will be colder than its July 2012 temperatures). It is very likely that the reverse will be true in New Zealand. It is safe to conclude that precipitation will be more frequent in the Amazon than in the Sahara, even if you can’t tell exactly when and where that rain will fall.
In fact, Lorenz’s groundbreaking paper, ‘Deterministic Nonperiodic Flow’, would seem to endorse this sort of statistical approach to understanding fluid dynamics:
“Because instantaneous turbulent flow patterns are so irregular, attention is often confined to the statistics of turbulence, which, in contrast to the details of turbulence, often behave in a regular well-organized manner.” (Lorenz 1963)
Let’s take a closer look.
Fig. 1. Three solutions of the Lorenz equations, starting at virtually identical points. Although the solutions are similar at first, they rapidly decouple around T=12.
The Lorenz equations consist of three variables describing turbulent fluid flow (X,Y, and Z), and three controlling parameters (r, b, and s). The equations are differential equations, meaning that a variable is described in terms of how it changes over time- saying ‘Johnny is driving west at 60 miles per hour’ is a simple differential equation. In order to solve a DiffEq, you need an initial condition – “Johnny started in Chicago” is an initial condition; without knowing that, you can’t say where she will be after driving for three hours. Continue reading