San José State University

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Thayer Watkins
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The Nature of Climate
and Its Predictability

Some define the climate of a region in terms of its average weather over a number of years; i.e., the mean values of temperature, cloudiness, rainfall and so forth. This notion of climate is not adequate because the variability of weather is another essential aspect of the climate of a region. For example, Texas is noted for the variability of its weather.

The definition above could be extended to include not only mean values but also measures of variation such as ranges, extreme values and variances. However, what is being touched upon is that it is the probability distributions that define the climate of a region. These probability distributions of course depend upon the season and could be subject to other cycles and trends. They could also be subject to change due to exogenous circumstances such as volcanic eruptions.

The interesting question is whether these probability distributions of weather are completely general or whether they necessarily belong to a family of probability distributions and hence are describable by a limited set of parameters. Many but not all variables in nature have a so-called normal or bell-shaped distribution. Such distributions are completely specified by two parameters, the mean and variance. There is a central limit theorem which says that variables which are the sum of a large number of independent random variables will tend to have a normal distribution.

As an example of this matter consider the distribution of the changes in average global temperatures.

This is a reasonable approximation of a normal distribution. For more on the question of whether the distribution of changes in global average temperature is normal see Normality

Although the central limit theorem is quite general there are limits to its applicability. The independent random variables referred to in the previous sentence have to have finite variance.

There is a generalized central limit theorem which would suggest that the weather characteristics being the result of a large number of independent random variables would have a Levy-stable distribution. The normal distribution is one of those Levy-stable distributions. For an application of Levy-stable distributions to rainfall data see Rainfall in San Jose.

The probability distribution of weather changes cyclically with the season. This would not be construed as a change in climate. There are other cyclic changes in the probability distri) phenominon. butions of the weather that are part of climate and not a change in climate, Besides the seasonal cycle the other most obvious cycle is the diurnal (time of day) cycle. Another one is the ENSO (El Niño Southern Oscillation). More controversial is the Pacific Multidcadal Oscillation. It is an approximately sixty-year cycle that leads to a thirty-year upswing in average global temperature and a thirty-year downswing. The ignoring of this natural cycle has led to a great deal of confusion concerning climate change.

The Pacific Multidcadal Oscillation is evident in the average global temperature statistics from the early 19th century.

The average global temperature declined from about 1940 to 1975. From 1975 it started increasing. Global warming alarmists wanted to construe all of the increase in average global temperature as being due to the increase in CO2 in the atmosphere. The problem was that about the year 2000 the average global stopped increasing and remainded steady for more than a decade, despite the fact that the CO2 content of the atmoshere continued to increase. Some alarmists looked for ways to revise the procedure for estimate the average temperature of the Earth's surface. With 70 percent of the Earh's surface covered with oceans the global average is sensitive to the limited temperature readings of ships, there is a lot of room for revisions that would show an increase in global temperature since 2000. But the readings of atmosheric temperature from satellites do not show such an increase.

Ironically if the global warming alarmists accept the Pacific Multidecadal Oscillation then they would have an explanation for the absence of global temperature increases since 2000 and furthermore a stronger justification for the case they are trying to make. That is that the effect of increased CO2 has increased in recent years and if it had not the average global temperature would have started declining. Thus by being dishonest about the Pacific Multidecadal Oscillation ended up weakening their case.

The Climate Data Center for the National Oceanic and Atmospheric Administration in Asheville, North Carolina has the best store of climate data. In the 1990's this organization, under the direction of Thomas R. Karl, began considering questions concerning possible shifts in the distributions of weather variables in connection with the debate on global warming. The results published suggested an increase in extreme weather events. It would be desirable if this organization separated itself from the polemics concerning global warming and climate change and focused on fundamental matters such as the nature of the distributions and how to rigorously test for changes.

Chaos Theory

In the early 1960's the meteorologist and mathematician Edward Lorenz was investigating nonlinear mathematical systems as analogues of weather forecasting models. By fortuitous chance he discovered the mathematical phenomenon of chaos.

A chaotic system is one that exhibits an extreme sensitivity to initial conditions. Even though the value for one time is deterministically given by the value at a previous time, say

xt+1=f(xt),

the relationship can become so complex after a number of iterations that the later values as t→∞ appear to be random.

For example, consider the case of the logistics curve where f(x)=αx(1-x). For some values of α, xt will go to an equilibrium point and remain there forever.

Stochastic versus Deterministic Equilibrium

For systems characterized by a deterministic transformation of the type

xt+1=f(xt),

one thing of interest is equilibria; i.e., values of x such that x=f(x). There are some remarkable mathematical theorems, called fixed point theorems, having to do with the existence of such solutions. Repeating cycles can be included in such analysis by looking fixed points of iterations of the transformation. For example, two period cycles are fixed points of f(f(x)). (Equilibria are just special cases of a two period cycles.)

One can also look at how the distribution for xt get transformed into the distribution for xt+1. It is this perspective that is relevant in the matter of climate. A stable climate would be one that the distributions of weather variables at one point in time leads to the same distributions at later points in time. What is need is a fixed point theorem for distributions. Such a fixed point would constitute a stochastic equilibrium.

Analysis

One approach to climate theory would be to look for probability density functions such the distribution of the output of a transformation is the same as the distribution of the inputs.

For illustration consider first the simple case when the transformation y=f(x) is monotonic. The monotonicity of the transformation guarantees that there exists an inverse function x=f-1(y). If p(x) is the probability density function of x then the probability density function of y, q(y), is given by

q(y)dy = p(f-1(y))dx
or, equivalently
q(f(x)|f'(x)| = p(x)

This means that q(y)=p(x)|f'(x)|, and thus q(y) is the same as f(x) if and only if f(x)=±1.

This applies only to monotonic transformations.

For the logistic function y=α[x−x²] and thus dy/dx=α[1−2x] and thus

q(αx(1-x)) = p(x)/|α(1-2x)|

If we are looking for the distribution of x such that y has the same distribution then p(x) must satisfy the functional equation

p(αx(1-x)) = p(x)/|α(1-2x)|
or, equivalently
p(αx(1-x)) = p(x)/(2|α(½-x)|

Functional equations are generally hard to solve.

Ergodicity

In statistical mechanics a system will, starting from a specific point, go through all of the possible configurations. Furthermore the average of characteristic of the system as it follows the time path from any initial starting point is the same and is equal to the average for probability distribution, usually called the ensemble, for the system.

This is what is assumed without any theoretical justification in climate model projections. That is to say, the projecters run their models and presume that some average of the projections will be the same as the average for the future distribution of the weather variables. This might be so, and then again it might not be so.

This question was investigated by Edward Lorenz before his death in 2008 at age 90. His work on chaos theory established why weather could not be accurately predicted beyond a week to ten days. Given this limit there is a serious question of whether future climate can be predicted. Lorenz' answer on the predictability of climate was a tentative affirmative. His paper is included in the published proceedings of a conference on the question of predictability, The Predictability of the Weather and Climate (2006) edited by Tim Palmer and Renate Hagedorn.


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