- What is Environmental Science?
- Why does mathematics work in the real world?
- Simple theories and complex models

When models took over from theories

What is Environmental Science?

Most of the natural sciences have something to say about the environment.
Since the 1950s however, in step with rising public concerns over environmental problems,
the term `Environmental Science' has come to describe a discipline
with a distinct character of its own.
Many environmental problems are now seen to be caused by
our having focused on just one of the web of processes that operate in the biosphere,
whilst ignoring the rest--often with disastrous consequences.
For example, the use of pesticides and herbicides led to the
*Green Revolution* of the 1950s and 1960s,
but their unforseen impacts on the food chain have caused massive damage.
So the first characteristic of Environmental Science is that it is `holistic';
bringing together traditional disciplines like
Biology, Meteorology, Hydrology, Chemistry, Physics and many more,
to understand the workings of the environment as a whole.

A direct effect of the public concern with the environment
is the increased demand for
*Environmental Impact Assessments* (*EIA*) for major and minor developments.
The Environmental Science that is called on to provide these *EIA*s has to be
immediate, predictive and prepared to cope with the untidiness and clutter of the real world.
The environmental researcher may have to provide a ``best bet'' answer
without the luxury of full understanding of the system that is under scrutiny.
The starkest example of this is the science of
Climate Change--the
*Greenhouse Effect*.
This not only encompasses almost all of the natural sciences,
in predicting Climate Change and the biosphere's response to it,
but also has had to guide international policy decisions at the same time
as climate scientists were striving to reduce the uncertainty of their predictions.

We can define environmental science then by its characteristic marks: it is interdisciplinary; it is prepared to deal with real rather than ideal situations; and it is predictive. What rôle does mathematics play in a science like this? Well, at a certain level it provides a natural tool kit, allowing us to organise observations with statistical methods and to manipulate the averages we obtain. But it is much more interesting to ask whether it also plays the deeper and rather mysterious rôle that it plays in the quantitative natural sciences.

Why does mathematics work in the real world?

The `` unreasonable effectiveness of mathematics in the natural sciences'' has long been a source of intense interest amongst scientists with a philosophical bent[ 4]. Why does mathematics seem to be the language in which nature is written? There are many examples of mathematicians developing their theories as exercises in formal logic; many years later these theories prove to be indispensable in formalising a new concept in physics. One example is the non-Euclidean geometries of Bolyai, Lobachevsky and Riemann. These were regarded as so pathologically contrary to common-sense, when they were discovered in the nineteenth century, that the great mathematician Gauss ^ concealed his own independent discovery of the field. Seventy years later Riemannian geometry was the language in which Einstein ^ wrote his general theory of relativity. Experiment has since confirmed, as his theory predicts--and non-Euclidean geometry allows--that space on the scale of galaxies is curved, not flat as our earthly experience suggests.

Great scientists from the time of Galileo ^, Huygens ^ and Newton ^ have gone much further than merely using mathematics as the language in which to couch their theories. The Nobel Prize winning physicist, Paul Dirac ^ wrote[ 1] in 1939 that:

``The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with remarkable success. This must be ascribed to some mathematical quality in Nature; [such] a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important rôle in Nature's scheme.''

In other words, if a natural phenomenon can be couched in mathematical terms, then mathematical manipulation will often yield results that also correspond to observable reality but may not have even been suspected up to that point.

Does this remarkable property of mathematics continue to hold in the untidy field that we have claimed for Environmental Science, and have we any right to expect that it should? Well, at a certain disciplinary level it certainly does. An example will show how this occurs.

Simple theories and complex models

Predicting the spread of smoke or gas from a chimney stack, or some other industrial source, is an element of many environmental impact assessments. The mathematics of the way the smoke-plume spreads were first worked out[ 2] by the Cambridge physicist G.I.Taylor in 1921.

Taylor's theory predicted three things:

- 1.
- close to the chimney the smoke-plume spreads quite rapidly, in proportion to the distance from the chimney;
- 2.
- further downstream it spreads more slowly, as the square root of distance in fact; and
- 3.
- the average concentration,
or
*probability*of encountering a puff of smoke as one moves across the plume, follows a Gaussian (bell-shaped) curve (see Figure 1).

This theory of diffusion has been confirmed by innumerable experiments.
With modifications from Taylor's idealizations,
to account for the variation of the real atmosphere,
it is widely used to predict air-pollution.
This is called the ``
Gaussian Plume model''.
An Australian version called
*AUSPLUME*
is the standard used by most states'
*EPAs*.
Over in the USA, versions are even written into law.

G.I.Taylor's theory illustrates very well three characteristics of predictive theories applied to the environment:

- First of all, it deals with statistical measures, not instantaneous values of wind speed or smoke concentration but the average levels of fluctuation. Now the use of statistical techniques tends to smooth the irregular real-world phenomena into something that can be more easily handled by other mathematics.
- Secondly, the theory predicts only the probability of the occurrence of smoke at a particular place. Predicting the probability of an event rather than its exact appearance is the nature of environmental science.
- Finally, the theory was derived by writing
*differential equations*. These describe relationships between the rates-of-change of particular properties, in this case of statistical measures and probabilities. Then one solves the equations to obtain predictions that can be tested against experimental observations.

Modelling the spread of a chimney plume is a well-defined problem that barely meets the specifications of environmental science as we have defined them. At the other end of the complexity spectrum lies the problem of predicting the response of climate to increases in the infra-red absorbing `greenhouse gases' in the atmosphere.

The methods used to make such predictions have many of the characteristics of G.I.Taylor's plume-spread theory. But some important new elements are added. It is still a question of solving differential equations. To the ones that describe motion of the atmosphere and oceans on a global scale, as well as their absorption and response to solar heating, are added equations for the interaction of biosphere and climate.

The equations of atmospheric and ocean movement have been known for many decades. However we can not solve them analytically, as G.I.Taylor did for the smoke-plume. Instead we use digital computers to approximate their solution.

Now here is another aspect to the problem. Even using the fastest computers available, to solve these equations we must divide the world into a set of `grid-boxes'. The predictions we get are averages of properties within these boxes.

In Figure 2 we can see the relatively coarse way
that Australia is represented in the ``state of the art''
*CSIRO* 9-level climate model.
Resorting to solving the equations digitally brings two new elements into play:

- techniques of numerical analysis are used to ensure that the solution is accurate, at least to the resolution of the grid-boxes; and
- any important process that occurs on a finer scale than the grid-boxes must be parameterized; that is, represented in terms of grid-box averages. There are many such important processes; for example cloud formation, convection and rainfall.

Deciding how to parameterize processes like these absorbs the energy of many of the world's best climate scientists. Australia currently makes a disproportionately strong contribution. Nevertheless, even for processes whose mechanisms are quite well understood, the parameterizations are often crude and their inclusion means that our equations have departed somewhat from the well-tried exact equations of atmosphere and water movement.

The situation becomes even murkier when we come to processes that are not well understood. For example, evaporation of water from vegetation and the assimilation of Carbon Dioxide are key processes, since water vapour and are the two principal Greenhouse gases. Obviously we would like to know how the earth's vegetation will react to changing climate and include it in our predictions as well. At present our knowledge is insufficient even to write the necessary differential equations. Instead we use crude correlations with a limited set of experiments on plants and ecosystems grown with artificially raised levels.

1/28/1997