An Introductory Global CO2 Model

(with Companion Media Pack)

Anthony J McHugh, Graham W Griffiths, William E Schiesser

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814663052
• Topic: Ciencias biológicas
• Subtopic: Ciencia medioambiental

The increasing concentration of atmospheric CO 2 is now a problem of global concern. Although the consequences of atmospheric CO 2 are still evolving, there is compelling evidence that the global environmental system is undergoing profound changes as seen in the recent spike of phenomena: extreme heat waves, droughts, wildfires, melting glaciers, and rising sea levels. These global problems directly resulting from elevated atmospheric CO 2, will last for the foreseeable future, and will ultimately affect everyone.

The CO 2 problem is generally not well understood quantitatively by a general audience; for example, in respect of the increasing rate of CO 2 emissions, and the movement of carbon to other parts of Earth's environmental system, particularly the oceans with accompanying acidification. This book therefore presents an introductory global CO 2 mathematical model that gives some key numbers — for example, atmospheric CO 2 concentration in ppm and ocean pH as a function of time for the calendar years 1850 (preindustrial) to 2100 (a modest projection into the future). The model is based on seven ordinary differential equations (ODEs), and is intended as an introduction to some basic concepts and a starting point for more detailed study.

Quantitative insights into the CO 2 problem are provided by the model and can be executed, with postulated changes to parameters, by a modest computer. As basic calculus is the only required mathematical background, this model is accessible to high school students as well as beginning college and university students. The programming of the model is in Matlab and R, two basic, widely used scientific programming systems that are generally accessible and usable worldwide. This book can therefore also be useful to readers interested in Matlab and/or R programming, or a translation of one to the change.

Contents:

• Model Structure
• Model Equations
• Model Parameters
• Computer Routines
• Main Programs
• Plotting Routines
• ODE Routines
• Emissions Rate Routines
• Base Case Model Output
• Ocean Chemistry
• An Interpretation of Dimensionless ODEs
• Calculation of pH
• Global CO 2 Concentration

Readership: Students (undergraduate and above) and members of the public interested in/concerned with long-term environmental change. Key Features:

• A basic model for the global CO 2 problem
• Routines for the model in source code that can be used in a quantitative study
• (1) and (2) provide insights into the global CO 2 problem that are otherwise not available from a static description based on the past and current literature

1 INTRODUCTION

The anticipated changes in the Earth’s climate that are now widely discussed are due in large part to the accumulation of so-called greenhouse gases (GHGs) in the Earth’s atmosphere. The increase in GHGs causes a reduction in the re-radiation of energy from the Sun back into outer space. Since less energy leaves the Earth’s atmosphere, heating of the atmosphere results in a temperature rise. This temperature rise, so-called global warming, is in turn a driving force for climate change.

Carbon dioxide (CO₂) is the major GHG, with increasing levels arising primarily from the burning of fossil fuels. Thus, changes in CO₂ level or concentration in the Earth’s atmosphere is of paramount importance in understanding anticipated warming and climate change. A second aspect of CO₂ accumulation in the atmosphere that is not as generally recognized and appreciated as temperature rise is the accumulation of carbon (from CO₂) in the oceans that leads to ocean acidifcation. CO₂ dissolves in ocean water and undergoes a series of chemical changes that ultimately leads to increased hydrogen ion concentration, denoted subsequently as [H⁺], and thus acidification. This increase in [H⁺] is manifested as a decrease in pH; note that [H⁺] and pH move in opposite directions due to the basic relation¹

For example, if [H⁺] = 10⁻⁸, pH = 8 while if [H⁺] = 10⁻⁷, pH = 7.

As a point of notation, square brackets placed around a chemical formula, e.g., [H⁺], denotes a molar concentration. The units of concentration are typically mols/liter, millimols/liter or micromols/liter (where liter is a liter of aqueous solution). Since for H₂O one liter weighs one kilogram (kg) (because the density of H₂O is 1 g/milliliter = 1 g/cc with 1 liter = 1000 milliliters), these concentrations are also in reciprocal kg. For the purpose of using Eq. (1.1) to calculate pH,[H⁺] is in g mol/liter = kg mol/m³.

Also, mols are taken specifically as g mols. One g mol of a chemical quantity has a weight equal to its atomic or molecular weight in g. For example, H₂O has a molecular weight of 2(1) + 16 = 18 g. Thus, one g mol of H₂O is 18 g.

The causes for changing environmental CO₂ levels are complex and not completely understood. But increasing atmospheric CO₂ is clearly established through measurements over more than 50 years [8]. Enough is known about CO₂ accumulation² to begin to formulate quantitative descriptions of the various physical and chemical processes that determine CO₂ levels with the goal of projecting³ how atmospheric CO₂ levels might be expected to increase in the future. To this end, we describe here an introductory global CO₂ model that elucidates at a basic level the mechanisms which determine CO₂ buildup in the Earth’s atmosphere.

The model is necessarily a simplification of the physical and chemical processes at work that determine CO₂ levels. However, the model provides insight into CO₂ dynamics (the variation of CO₂ levels over time); specifically, it can be used to study the effect of various phenomena and parameters that determine CO₂ levels, and how they change with time.

In particular, the effect of variations in the rate of anthropogenic emissions can be assessed. This is accomplished by the numerical integration of a system of ordinary differential equations (ODEs) starting with known conditions in the past (for example, at 1850, but this starting year can easily be changed). The forward integration of the ODE model equations through time can be to an arbitrary point in the future (for example, to 2100, but this final year can also easily be changed). The details of the ODE model, and some representative output from the model, are discussed subsequently.

A major advantage of a computer-based mathematical model is the execution of the associated computer code⁴ for the solution of the model equations to observe the effect of postulated conditions, e.g., the rate of anthropogenic CO₂ emissions. Thus, although only a limited set of model outputs is considered here, the code can easily be executed for other conditions to observe the effect of the model structure and parameters on projected CO₂ levels. Ideally, this process should elucidate the most relevant and sensitive conditions that determine future CO₂ levels, and thereby give an indication of plausible future CO₂ changes.

The model focuses only on CO₂. It does not have a climate component and it has only a basic global warming component consisting of multiplication of atmospheric CO₂ (in ppm) by a temperature sensitivity. While the current levels of CO₂ are relatively well known from measurements, the net effect of CO₂ on the Earth’s climate is not well understood quantitatively at this point in time. Thus we have limited the model to CO₂ dynamics as a main determinant of global warming and climate, but we do not attempt to explain the resulting dynamics of anticipated global warming and climate change.

The model does, however, have as an output ocean H⁺ and pH as a function of time. As with atmospheric CO₂, the long-term effects of ocean acidification are not completely understood at this point in time. However, two relatively well established effects can be observed:

•The ocean pH is decreasing and this can be measured.

•The effect of acidification on coral is being observed. We take CaCO₃ as the main component of coral, and the basic chemistry relating CaCO₃ and H⁺ is considered toward the end of this discussion.

Of course, other important effects of acidification could be considered, for example, on the ocean biosphere, but they are not included in the model. No doubt these effects will be important and will be elucidated with future research.

¹log₁₀ indicates a base-10 logarithm. To briefly review, a number (real, nonnegative) can be expressed as c = b^a where a is the logarithm (log) of c to the base b (e.g., 10; e = 2.718282, the base of the “natural” logarithm system). An important application of logs is to facilitate the multiplication or division of numbers. For example, two numbers c₁ = b^a₁,c₂ = b^a₂ can b...