A heuristic for setting effective standards to ensure global environmental sustainability

Abstract

Most environmental professionals and decision-makers, and certainly the public at large, hold the view that the integrity of earth’s natural environment will be conserved for posterity and sustainable development achieved if all the nations rigorously enforced their environmental and emission standards. It is argued in this paper that this view, sincerely held by many as an “axiomatic truth,” is mistaken and misplaced. This is because as a biogeochemical entity the Earth has limited self-regenerative capacity (SRC) to cope with anthropogenic pollution, and all kinds of environmental problems ensue when that limit is exceeded. Indeed, mounting environmental problems now occurring on all fronts amply testify to the fact that the limit has already been exceeded. They also provide necessary and sufficient proof that environmental and emission standards have been woefully inadequate for protecting earth’s natural environment and life-support systems. It is argued that true global environmental sustainability will be achieved, paving the way to true global sustainable development, if and only if global environmental and emission standards are set so that global anthropogenic pollution does not exceed the limit of earth’s natural SRC to cope with such pollution. These and related issues are discussed in this paper. A simple mathematical model using basic mathematics is also presented to explain how the phenomenon of “positive feedback” works in some of the environmental problems to exacerbate environmental degradation and progressively to erode nature’s SRC.

Appendix: a simplistic mathematical model

Currently there are many different climate models with varying degrees of mathematical complexity. As an example, the Atmosphere General Circulation Model (AGCM), which is sophisticated and used for long-term climate projection, is based on a three-dimensional representation of the atmosphere coupled to the land surface and the cryosphere. However, an understanding of the mathematical algorithm of such sophisticated models demands knowledge of the numerical processing of complex, sometimes very complex, coupled differential equations in space and time, and so the average reader or student lacking such knowledge regards them as “magic black boxes.” This is unsatisfactory, to put it mildly. The model presented below is generic, instructive, and simplistic; in simple terms it seeks to explain the functional relationship between SRC and pollution burden, as well as the mechanism of positive feedback that characterises many of the environmental problems and phenomena.

1.1 Environmental problems with positive feedback

The time-dependent functions of Fig. 1, P(t) and Q(t), refer to the origin of co-ordinate axes at O. Function P(t) represents the total amount of a given anthropogenic pollutant accumulated in the “system” (defined as an environmental aspect, compartment or sink) up to time t, while function Q(t) represents the upper limit of the system’s natural SRC at time t to cope with that pollutant. For simplicity, it is assumed that no noticeable damage to the system’s natural SRC (and therefore no environmental damage) occurs as long as P(t) ≤ Q(t); and that progressive degradation of the system’s natural SRC (and therefore environment damage) occurs when P(t) > P ₁. Clearly, P ₁ is the natural SRC of the system in its pristine state (or a past state taken as benchmark).

Fig. 1

Schematic of an environmental problem with positive feedback

Point A in Fig. 1 marks a transition because for P(t) ≤ P ₁ the system’s natural SRC to cope with the pollutant remains intact; while for P(t) > P ₁ the accumulated amount of the pollutant exceeds that which the system’s natural SRC can cope with.

Mathematically, environmental problems of this kind belong to a class of “positive feedback” problems characterised by three distinct elements. In the generic problem, to which Fig. 1 refers, these elements are: the “system” represented by the affected environmental compartment(s) or aspect(s); the “input” represented by the pollutant emitted to the system; and the “output” (outcome) represented by the degradation of the system’s natural SRC to cope with the pollutant(s) in question. Problems of global warming and ocean acidification are typical examples of environmental problems with positive feedback.

With reference to Fig. 1, the modus operandi of “positive feedback” may be explained as follows. Consider the situation at time T ₁ when the total quantity of the accumulated pollutant, which the system’s natural SRC cannot cope with, is represented by BD. This amount therefore resides in the system to further corrupt and degrade its natural SRC by the amount represented by DE at time T ₂. This spiral of the system’s natural SRC degradation in this way continues until time T ₃ when it is totally exhausted.

Positive feedback, unlike negative feedback which attempts to maintain homeostasis by seeking to establish a state of equilibrium, tends to increase output (degradation of the system’s natural SRC in this case). Problems of positive and negative feedback occur in Control Engineering (e.g. Bishop & Dorf, 2004), Elasto-Hydrodynamics (e.g. Nath, 1982), and in Electrical Engineering (e.g. Carlson & Gisser, 1990) among others.

1.2 A simple mathematical model

Clearly, the functional form of P(t) for a given pollutant would be determined by how it grows in the system over time. As an illustration, let us assume that P(t) grows exponentially with time according to the equation

$$ P(t) = P_0 \,{\text{e}}^{kt} $$

(1)

in which P ₀ denotes the value of P(t) at time t = 0, k the growth constant, and e the base of natural logarithm.

In order to simplify the mathematics, the co-ordinate axes are now translated (without rotation) to the new origin at A (where now t = 0) so that Eq. 1 transforms to

$$ P(t) = P_1 ({\text{e}}^{kt} - 1) $$

(2)

$$ P_1 = P_0{\text{e}}^{kT} $$

(3)

Then, the rate of degradation of Q(t) at time t ≥ 0 can be expressed as (see Fig. 1)

$$ - {\text{d}}Q/{\text{d}}t = \beta ({\text{BC}} + {\text{CD}}) = \beta [P(t) + Q(t)] $$

(4)

or,

$$ {\text{d}}Q/{\text{d}}t + \beta Q = - \beta P_1 ({\text{e}}^{kt} - 1) $$

(5)

in which β is an overall constant assumed to represent progressive degradation of the environmental processes, which together constitute the system’s natural SRC, caused by the anthropogenic pollutant in question.

Solution of Eq. 5 gives

$$ Q(t) = - P_1 \left( {\frac{{\beta \,{\text{e}}^{kt} }} {{\lambda}}-1} \right) + C\,{\text{e}}^{ - \beta t} $$

(6)

in which C is a constant and

$$ \lambda = (k + \beta ) $$

(7)

Noting that t = 0 and Q(t) = 0 at origin A, the explicit form of Eq. 6 is now obtained as

$$ Q(t) = P_1 [1 - \Phi (t)] $$

(8)

in which

$$ \Phi (t) = \frac{{(k\,{\text{e}}^{ - \beta t} + \beta \,{\text{e}}^{kt} )}} {\lambda } $$

(9)

The minimum value of Φ(t) is unity, and it occurs at t = 0 for all positive values of β and k (k cannot be negative because that would imply exponentially decreasing P(t); that is, anthropogenic pollution decreases exponentially with time, and so the problem does not exist. And a negative value of β would imply improbably that anthropogenic pollutants have beneficial impacts on the system). Also, demonstrably, Φ(t) is greater than unity for t > 0 and all positive values of β and k. Consequently, Q(t) is negative (with A as the origin of co-ordinates) and increases in magnitude over time as shown in Fig. 1.

Although for simplicity of analysis β is assumed to be a constant, in reality it is unlikely to be so for two reasons. First, in most cases degradation occurs imperceptibly over time, and sharply just before catastrophic and often irreversible system failure occurs. Such behaviour is characteristic of many environmental problems with positive feedback, as well as of other systems such as the “Euler Strut” in Mathematical Theory of Elasticity. Lateral deflections of the strut, caused by an increasing axial load applied along its central axis, remain imperceptibly small until they become theoretically infinite when the strut suddenly collapses as the applied load becomes equal to its first (fundamental) “critical load.” Acidification of the oceans provides a typical environmental example. Over time the impacts of ocean acidity have been growing imperceptibly and relentlessly and, unless remedial measures are taken to address them soon, all marine life will be at very serious risk (e.g. Turley, 2005). And, when catastrophic and/or irreversible impacts occur, the system “collapses” and little or nothing can be done to restore it to its earlier state. This means that in many of the environmental problems β should be expressed as a temporal function. And second, as different environmental systems under different ambient conditions sometimes respond differently to the same pollutant, β is likely to be a spatial function too. The difficulty of defining a mathematical function of both space and time to describe β reliably is thus obvious to see, especially in the global context.

Environmental properties such as waste absorption capacity and resource depletion are also aggregated in β. When these and other properties vary spatially, a discretised version of the model may be used in which each discretised cell is assigned a value of β (or value(s) of the corresponding parameter(s) in a more sophisticated model) representing the average aggregate value(s) of those properties for that cell. Indeed, this is a great advantage of domain discretisation used in the AGCM and other sophisticated models.

As the function e^-βt decays rapidly with time, an approximate form of Eq. 8 is obtained as

$$ Q(t) = P_1 \left( 1-{\frac{{{\text{e}}^{kt} }} {{1 + k/\beta }}} \right) $$

(10)

Results of sensitivity tests using Eq. 8 show, as does Eq. 10, that when the ratio k/β is small compared to unity, Q(t) becomes insensitive to β. As Q(t) = -P ₁ at t = T ₃, we can show from Eq. 10 that when k/β is small compared to unity,

$$ T_3 = \frac{{\ln \,2}} {k} $$

(11)

It is interesting to note that if k is defined as the decay constant of a radioactive material, Eq. 11 gives the “half-life” of the radio-nuclides of that material.

1.3 Application to global warming

Subject to availability of necessary data, namely the values of P ₁, β and k, the above mathematical model can be applied to glean useful information on how the SRC of a natural environmental system is being affected by an anthropogenic pollutant.

Consider the application of the model to the increasingly threatening problem of global warming, for which we will assume that the total quantity of CO₂ (the main greenhouse gas) present in the “system” (atmosphere) at a given time is directly proportional to its concentration at that time.

Judging by the mounting evidence of the diverse impacts of global warming, such as melting of polar ice (Rapley, 2005), sea level rise (ISSC, 2005; McCarthy, 2005a), rise in global mean temperature (Hare, 2005; ISSC, 2005), glacier retreat (Yamada et al., 1996), ocean acidification (Royal Society, 2005; Turley, 2005), and slowing-down of the Atlantic “conveyor” (and the global thermohaline conveyor) without which north-western Europe would be colder than it is, it is abundantly clear that earth’s SRC to cope with the relentlessly rising greenhouse gas emissions is being seriously degraded. And, therefore that environmental and emission standards to control greenhouse gases have been woefully inadequate.

It is of particular concern that in 2004 the concentration of the main greenhouse gas, CO₂, in the atmosphere was 379 ppmv, rising annually by >2 ppmv. At this rate the vital threshold of 400 ppmv will be reached by around 2015 (Hare, 2005; ISSC, 2005; Royal Society, 2005; McCarthy, 2005b). At present global air temperature anomaly is already 0.8°C above the pre-industrial level. It is expected to climb to 1°C in the next 25 years, and to over 3°C after 2070 (Hare, 2005; ISSC, 2005). Above the 2°C level, the risks of abrupt, accelerated or runaway climate change, exacerbated by positive feedback, also increase with potentially catastrophic consequences for all life on earth (ISSC, 2005; McCarthy, 2005b).

Records going back to 1750 show how CO₂ concentration has been increasing relentlessly over the last 254 years—from 278 ppmv in 1750 to 379 ppmv in 2004—caused mainly by the burning of fossil fuels for economic development. As the overall growth trend is exponential (Houghton, Jenkins, & Ephraums, 1990), especially since around 1860 when the Industrial Revolution began in earnest, k can be calculated from Eq. 1. With P(t) = 379 ppmv (2004 value), P ₀ = 278 ppmv (1750 value), and t = 254 years, this equation gives k = 0.00122 per year.

As a typical application of the model, we will assume that the greenhouse gas pollution status of earth’s atmosphere in 1750 represented its pristine state. That is, T = 1750 in Fig. 1, and P ₁ ( = AT) now denotes the SRC of earth’s atmosphere in 1750. Clearly, the atmosphere would be truly sustainable if and only if P(t) < P ₁ at all future times (P(t) now represents the total quantity of the main greenhouse gas, CO₂, in the atmosphere at time t). Unfortunately, we know little or nothing about precisely what the global value of P ₁ is. We note, however, that pure air contains only 0.03% of CO₂ by volume. This is what nature has provided. Arguably therefore, any excess CO₂ which the Carbon Cycle cannot cope with will stay in the atmosphere and act as greenhouse gas. Also, let β now denote the overall global constant pertaining to atmospheric pollution.

The curve of Fig. 2, prepared using Eq. 8, shows typically how the natural SRC of earth’s atmosphere has been degrading since 1750 (subject of course to the limitations of the simplistic mathematical model used). The model predicts that in around the year 2310 ( = T ₃) earth’s natural SRC to cope with greenhouse gas emissions will be totally exhausted triggering catastrophic events that may be likened to the visitation of the Four Horsemen of the Apocalypse prophesied in the Book of Revelation in the Holy Bible.

Fig. 2

A hypothetical solution of Eq. 8 with β = 0.10 per year and k = 0.00122 per year