"Large-Scale Disasters as Dynamical Systems," M.E. Online Exclusive, August 2008

August 2008

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Large-Scale Disasters as Dynamical Systems

Storm, famine, conflict, or flood—extreme events require analysis and planning that advanced engineering can provide.

by Mohamed
Gad-el-Hak

What do a hurricane, global warming, and a space shuttle have in common? The first is a natural disaster, the second is possibly an anthropogenic disaster, and the last is a man-made flying machine. All three, however, are complex dynamical systems that, in principle, at least can be modeled mechanistically. Their future state can be reasonably predicted, and responses can be planned for them. Treating all natural and man-made disasters as dynamical systems, often nonlinear and likely possessing numerous, if not infinite, degrees of freedom, is both a challenge and an opportunity.¹

And now, a bonus question: What do an emergency medic, a fireman, a policeman, a logistician, a tactician, a sociologist, a government official, an engineer, and a scientist have in common? They all have important contributions to make in protecting lives and property from natural and man-made disasters. The medic, fireman, policeman, logistician, and tactician are among the first to respond when disaster strikes. The sociologist studies the reaction to and consequences of a disaster. The government official takes the blame when things do not go well in the aftermath.

Though it has not been a traditional field of endeavor for scientists and engineers, they can apply considerable analytical and problem-solving skills to the prediction, prevention, control, and mitigation of catastrophic events.

The laws of nature govern the evolution of any disaster. In some cases, as for example in weather-related disasters, first-principle laws could be written in the form of field equations, but exact solutions of these often-nonlinear, partial differential equations are impossible to obtain, particularly for turbulent flows, and heuristic models together with intensive use of supercomputers are necessary to proceed. In other cases—for example, earthquakes—the precise laws are not even known and prediction becomes more or less a black art. Management of any type of disaster is more art than science. Nevertheless, much can be done by science and engineering to mitigate the resulting pain and suffering.

These and many other ideas are examined in the book Large-Scale Disasters: Prediction, Control, and Mitigation, published earlier this year by Cambridge University Press. The book includes a proposal to establish a universal, quantitative metric that puts all natural and man-made disasters on a common scale. It also introduces issues of prediction, control, and mitigation of catastrophes.

Large-scale disasters have always been a fact of life, and they were a fact before there was life. They have plagued us since Homo sapiens set foot on this third planet from the sun. Frequent disasters struck the Earth before that, of course, even at its formation around 4.5 billion years ago.

The geological Earth that we know today is believed to be the result of agglomeration of the so-called planetesimals and subsequent impacts between bodies of similar mass.² The planet was left molten after each giant impact, and its outer crust was formed upon radiative cooling to space. Those were the “good” disasters.

A series of coordinated suicide attacks upon the United States were carried out on Tuesday, September 11, 2001, in which 19 hijackers took control of four domestic commercial airliners. In the haunting image above, United Airlines Flight 175 goes through the southern tower of the World Trade Center. This scene was captured on live TV because filming crews were already photographing the northern tower attacked 18 minutes earlier.

On the bad side, there have been several mass extinctions throughout the Earth’s history. The dinosaurs, along with around 70 percent of all species existing at the time, became extinct because a large meteorite struck the Earth 65 million years ago, and the resulting airborne dust partly blocked the sun, thus making it impossible for cold-blooded animals to survive. But if we concern ourselves with our own warm-blooded species, then starting 200,000 years ago, ice ages, famines, infections, and attacks from rival groups and animals were constant reminders of human vulnerability.

On the average, there are about three large-scale disasters that strike the Earth every single day, but only very few of those natural or man-made calamities make it to the news. The fact that 100 persons die in a week somewhere of starvation, say, is not a typical news story. But 100 humans perishing in an airplane crash will make CNN all day.

Disaster has been with us from the beginning. Humans have survived it all because we were programmed to.

From an evolutionary point of view, disasters bring out the best in us, except when there is a profound sense of injustice. It almost has to be that way. Humans survived ice ages, famines, infections, etc., not because we were strong or fast, but because in the state of extreme calamity, we tend to be resourceful and cooperative. Commenting on a unique museum’s exhibition, Caroline Ash writes in Science (Vol. 317, p. 1869, 28 September 2007), “Although the perpetual threat of disaster makes us fear the unexpected, our imaginations also prepare us to manage disorder and suffering. Consequently, given some period of peace after a catastrophe, societies rapidly regroup and act not only to assist their own members but also to help others, often far distant and quite anonymous, to rebuild their infrastructure and their faiths. The objects that help humans to be resilient are the subject of the exhibition Scénario Catastrophe, …, currently at the Musée d’ethnographie de Genève.”

In the Eye of the Beholder

There is no easy answer to the question whether a particular disaster is large or small. The injury of one person may be perceived as catastrophic by that person, or by his or her loved ones. What we need to consider, however, are the adverse effects of an event on a community or an ecosystem. The scale of a disaster is determined by the number of people affected or the extent of the geographical area involved. Large-scale disaster taxes the resources of local communities and central governments. Under the weight of a large-scale disaster, a community diverges substantially from its normal social structure. Return to normalcy is typically a slow process that depends on the severity, but not the duration, of a calamity as well as on the resources and efficiency of the recovery process.

Disasters can naturally occur, but humans can also cause their share of devastation. Natural disasters include earthquakes, wildfires, pandemics, or volcanic eruptions. Human actions can lead to economic collapse, war, oil slicks, or terrorist acts. While technological advances exponentially increase human prosperity, they have also provided man with more destructive power. Tyranny is a form of disaster, too. Man-made disasters caused the death of at least 200 million people during the 20th century.³

In October 2005, a large and complex area of low pressure developed over the western Atlantic and eastern Caribbean with several centers of thunderstorm activity. This area of disturbed weather southwest of Jamaica slowly organized on 15 October into tropical depression number 24. It reached tropical storm strength on 17 October, and was named Wilma, the first storm ever to use a “W” name since alphabetical naming began in 1950, and tying the 1933 record for most storms in a season. Moving slowly over warm water with little wind shear, tropical storm Wilma strengthened steadily and became a hurricane on 18 October. Pictured above is Wilma’s projected path from 5:26 am EDT, Friday, 21 October 2005, to early morning Wednesday, 26 October 2005. The 5-day forecast was reasonably accurate.

There is also the possibility of human actions causing a natural disaster to become more damaging than it would otherwise have been. An example of such anthropogenic calamity is the intense coral-reef mining off the Sri Lankan coast, which removed a natural barrier that could mitigate the force of waves. As a result, the 2004 Pacific tsunami devastated Sri Lanka much more than it would have otherwise. Another example is the soil erosion caused by overgrazing, farming, and deforestation. In April 2006, wind from the Gobi Desert dumped 300,000 tons of sand and dust on Beijing, China. Such gigantic dust tempests—exasperated by soil erosion—blow around the globe and make people sick, kill coral reefs, and melt mountain snow packs continents away. Examples like that incited the 1995 Nobel laureate and Dutch chemist Paul J. Crutzen to call the present geological period anthropocene to characterize humanity’s adverse effects on global climate and ecology; see http://www.mpch-mainz.mpg.de/~air/anthropocene/.

For the disaster’s magnitude, how large is large? One of the chapters of Large-Scale Disasters proposes a universal metric by which all types of disasters are sized in terms of the number of people affected or the extent of the geographical area involved. This quantitative scale applies to both natural and man-made disasters. The scale is logarithmic. Thus, moving up the scale requires an order of magnitude increase in the severity of the disaster as it adversely affects people or an ecosystem. Under this system, a disaster may affect only a geographical area without any direct and immediate impact on humans. For example, a wildfire in an uninhabited forest may have long-term adverse effects on the local and global ecosystem, although no human is immediately killed, injured, or dislocated as a result of the event.

The primary advantage of having a universal, quantitative classification scheme is that it gives officials a measure to guide them so that proper response can be mobilized and adjusted as warranted. The metric suggested applies to all types of disasters. It puts them on a common scale, which is more informative than the variety of scales currently used for different disaster types: the Saffir–Simpson scale for hurricanes, the Fujita scale for tornadoes, the Richter scale for earthquakes, and the recently introduced Northeast Snowfall Impact Scale (notable, significant, major, crippling, extreme) for the winter storms that occasionally strike the Northeast of the United States. Of course, the individual scales also have their utility; for example, knowing the range of wind speeds in a hurricane, as provided by the Saffir–Simpson scale, is a crucial piece of information to complement the number of casualties the proposed scale supplies. In fact, a prediction of wind speed allows estimation of potential damage to people and property. The proposed metric also applies to disasters, such as terrorist acts or droughts, where no quantitative scales are currently available to measure severity.

The scope of a disaster is determined if at least one of two criteria is met, relating to either the number of people displaced or harmed, or the area affected by the event. The proposed classification system is diagrammed in Figure 1.

Figure 1. Classification of disaster’s severity.

Under this system, if 70 persons were injured as a result of a wildfire that covered 20 sq. km, this would be considered Scope III, large disaster (the larger of the two categories II and III). On the other hand, if 70 persons were killed as a result of a wildfire that covered 2 sq. km, this would be considered Scope II, medium disaster. An extreme example, at least in the sense of even attempting to classify it, is the case of a country slightly larger than 1 million square kilometers with almost 80 million citizens tormented by a virtual police state. This man-made cataclysm is readily stigmatized by the highest classification, Scope V, gargantuan disaster.

In formulating all scales, including the proposed one, a certain degree of arbitrariness is unavoidable. The range of 10 to 100 persons associated with a Scope II disaster, for example, could very well be 20 to 80, or some other range. What is important is the relative comparison among various disaster degrees; a Scope IV disaster causes an order of magnitude more damage than a Scope III disaster, and so on. One could arbitrarily continue beyond five categories, always increasing the influenced number of people and geographic area by an order of magnitude, but it seems that any calamity adversely affecting more than 10,000 persons or 1,000 km² is so catastrophic that a single Scope V is adequate to classify it as a gargantuan disaster.

In the case of certain disasters, the scope can be predicted in advance to a certain degree of accuracy; otherwise, the scope can be estimated shortly after the calamity strikes with frequent updates as warranted. The magnitude of the disaster should determine the size of the first-responder contingency to be deployed: which hospitals to mobilize and to what extent; whether the military forces should be involved; what resources, such as food, water, medicine, and shelter, should be stockpiled and delivered to the stricken area, and so on. Predicting the scope should facilitate the subsequent recovery.

Facets of Large-Scale Disasters

If a disaster cannot be stopped, then predicting its occurrence, location, and severity could help make the best of a bad situation. Accurate prediction guides preparation for a calamity, even to evacuating large segments of the population out of harm’s way.

For certain disaster types, evolution equations can be formulated. Predictions can then be made to different degrees of success using heuristic models, empirical observations, and giant computers. Technology can predict the path and intensity of a hurricane, for example, much better than it can forecast a tornado.

Once disaster strikes, mitigating its adverse effects becomes the primary concern. How does society save lives, take care of the survivors’ needs, and protect property from any further damage? Dislocated people need shelter, water, food, and medicine. Both the physical and mental health of the survivors as well as relatives of the deceased can be severely jeopardized. Looting, price gouging, and other law-breaking activities need to be contained or eliminated. Hospitals need to prioritize and perhaps ration treatments. Roads need to be operable and free of landslides, debris, and traffic jams for the unhindered flow of first responders and supplies to the stricken area, and evacuees and ambulances away from it. This is not always possible, especially if the disaster damages most roads, as happened after the 2005 Kashmir earthquake. Buildings, bridges, and roads need to be rebuilt or repaired, and power, potable water, and sewage need to be restored.

The schematic in Figure 2 depicts the different facets of large-scale disasters. The important thing is to judiciously employ the finite resources available to improve the science of disaster prediction, and to artfully manage the resulting mess once it has occurred to minimize loss of life and property.

Figure 2. Different facets of a large-scale disaster.

The Science of Disaster Prediction and Control

Science can help in predicting the course of certain types of disaster. When, where, and how intensely will a severe weather phenomenon strike? Are the weather conditions favorable to extinguishing a particular wildfire? What is the probability of a particular volcano erupting? How about an earthquake striking a population center? How much air and water pollution is going to be caused by the addition of a factory cluster to a community? How would a toxic chemical or biological substance disperse in the atmosphere or in a body of water? Below a certain concentration, certain dangerous substances are harmless, and a safe zone could be established based on the dispersion forecast.

The ability to answer these and similar questions varies widely. Once formed, the course and intensity of a hurricane, which typically lasts from inception to dissipation for a couple of weeks, can be predicted about a week in advance. The path of the much smaller and short-lived, albeit more deadly, tornado can be predicted only about 15 minutes in advance, although weather conditions favoring its formation can be predicted a few hours ahead.

Earthquake prediction is far from satisfactory, but is seriously attempted nevertheless. The accuracy of predicting volcanic eruptions is somewhere in between those of earthquakes and severe weather. Patanè et al.⁷ reported on the ability of scientists to “see” inside Italy’s Mount Etna and forecast its eruption using seismic tomography, a technique similar to that used in computed tomography scans in the medical field. The method yields time photographs of the three-dimensional movement of rocks to detect their internal changes. The success of the technique is in no small part due to unusual circumstances. As Europe’s biggest volcano, Mount Etna is equipped with a high-quality monitoring system and seismic network, tools not readily available at most other volcanoes.

The Kashmir Earthquake—aka the Northern Pakistan Earthquake or South Asia Earthquake—of 2005 was a major seismological disturbance that occurred at 08:50:38 Pakistan Standard Time on 8 October 2005, with the epicenter in the Pakistan-administered region of the disputed territory of Kashmir in South Asia. It registered 7.6 on the Richter scale, making it a major earthquake similar in intensity to the 1935 Quetta Earthquake, the 2001 Gujarat Earthquake, and the 1906 San Francisco Earthquake.

Science and technology can also help control the severity of a disaster, but here the achievements to date are much less spectacular than those in prediction. Cloud seeding to avert drought is still far from being practical. On the other hand, employing scientific principles to combat a wildfire is doable. So is the development of scientifically based strategies to reduce air and water pollution, to moderate urban sprawl, to evacuate a large city, and to minimize the probability of accident for vehicles. Structures are designed to stand an earthquake of a given magnitude, wind of a given speed, and so on. Dams are constructed to moderate the flood–drought cycles of rivers, and dikes are erected to protect lands below sea level from the vagaries of the weather.

Storm drains, fire hydrants, fire-retardant materials, sprinkler systems, pollution control, simple hygiene, strict building codes, traffic rules and regulations in air, land, and sea are among the measures that society takes to mitigate or eliminate the harm of natural and man-made disasters.

Of course, there are limits. Much better fire safety will be achieved if a fire station is built on every city block, and earthquakes will claim fewer casualties if every house is built to withstand the strongest possible tremor. But, clearly, the cost of such efforts is prohibitive.

In contrast to the natural kind, man-made disasters are generally somewhat easier to control, but more difficult to predict. The war on terrorism is a case in point. Who could predict the behavior of a crazed suicide bomber? A civilized society spends its valuable resources on intelligence gathering, internal security, border control, and screening to prevent (control) such devious behavior, whose dynamics (i.e., time evolution) obviously cannot be distilled into a differential equation to be solved.

However, even in certain disastrous situations that depend on human behavior, predictions can sometimes be made. Crowd dynamics is a prime example. The behavior of a crowd in an emergency can, to some degree, be modeled and anticipated so that adequate escape or evacuation routes can be properly designed.⁸ Helbing et al.⁹ write on simulation of panic situations and other crowd disasters modeled as nonlinear dynamical systems.

The tragedy of the numerous man-made disasters is that they are all preventable, at least in principle. We cannot prevent a hurricane—at least, not yet—but using less fossil fuel and seeking alternative energy sources could at least slow down global warming trends. Conflict resolution strategies can be employed between nations to avert wars. A bit more humanity, common sense, and moderation, as well as a bit less greed, meanness, and zealotry, and the world will be a better place for having fewer man-made disasters.

To predict the behavior of disasters that involve (fluid) transport phenomena, such as severe weather, fire, and release of toxic substances, the governing equations can be formulated subject to some assumptions—the fewer, the better. Modeling is usually in the form of nonlinear partial differential equations with appropriate numbers of initial and boundary conditions. But those field equations are typically impossible to solve analytically, particularly if the fluid flow is turbulent, which unfortunately is the norm for the high Reynolds number flows encountered in the atmosphere and oceans.

Furthermore, initial and boundary conditions are required for both analytical and numerical solutions. Computers are not fast enough, so numerical integration of the instantaneous equations (direct numerical simulations) for high Reynolds number natural flows is computationally prohibitively expensive, if not outright impossible, at least for the foreseeable future. Modeling then comes to the rescue, but at a price.

Large-eddy simulations, spectral methods, probability density function models, and the more classical Reynolds-stress models are examples of such closure schemes that are not as computationally intensive as direct numerical simulations, but are not as reliable either. This type of second-tier modeling is phenomenological in nature and does not stem from first principles. The more heuristic the modeling is, the less accurate the expected results.

Eight decades after the 1906 “Great Earthquake” of San Francisco, another big one struck the region. This became known as the Loma Prieta Earthquake. At 5:04 pm, on 17 October 1989, a magnitude 7.1 earthquake on the Richter scale severely shook the San Francisco and Monterey Bay regions. The epicenter was located near Loma Prieta peak in the Santa Cruz Mountains, approximately 14 km northeast of Santa Cruz and 96 km south-southeast of San Francisco. The tremor lasted for 15 seconds and occurred when the crustal rocks comprising the Pacific and North American Plates abruptly slipped as much as 2 m along their common boundary—the San Andreas Fault system . The rupture initiated at a depth of 18 km and extended 35 km along the fault, but it did not break the surface of the Earth.

Together with massive ground, sea, and sky data to provide at least in part the initial and boundary conditions, the models are entered into supercomputers that come out with a forecast, be it a prediction of a severe thunderstorm that is yet to form, the future path and strength of an existing hurricane, or the impending concentration of a toxic gas that was released in a far away location some time in the past.

The important issue is to precisely state the assumptions needed to write the evolution equations, which are basically statements of the conservation of mass, momentum, and energy, in a certain form. The resulting equations and their eventual analytical or numerical solutions are only valid under those assumptions. This seemingly straightforward fact is often overlooked and wrong answers readily result when the situation we are trying to model is different from that assumed. For a succinct derivation of the conservation laws for a continuum, non-relativistic fluid, see Reference 10.

The prediction of weather-related disasters has met spectacular successes within the last few decades. The painstaking advances made in fluid mechanics in general and turbulence research in particular, together with the exponential growth of computer memory and speed, no doubt contributed immeasurably to those successes. A generation ago, the next day’s weather was hard to predict. Today, the 10-day forecast is available 24/7 on weather.com for almost any city in the world.

Size and Other Complexities

Despite their already complicated nature, other effects could further entangle the fluid transport equations. For example, geophysical flows occur on such large scales as to invalidate the inertial frame assumption typically made in small-scale flows. The Earth’s rotation affects these flows, and such things as centrifugal and Coriolis forces enter into the equations rewritten in a non-inertial frame of reference fixed with the rotating Earth. Oceanic and atmospheric flows are more often than not turbulent flows, and they span an enormous range of scales, from a few millimeters to thousands of kilometers.^11,12

Density stratification is important for many atmospheric and oceanic phenomena. Buoyancy forces are produced by density variations in a gravitational field, and those forces drive significant convection in natural flows.¹³ In the ocean, those forces are further complicated by the competing influences of temperature and salt.¹¹ The competition affects the large-scale global ocean circulation and, in turn, climate variability. For weak density variations, the Bousinessq approximation permits the use of the coupled incompressible flow equations, but more complexities are introduced in situations with strong density stratification, such as when strong heating and cooling are present. Complex topography further complicates convective flows in the ocean and atmosphere.

The air-sea interface governs many of the important transport phenomena in the ocean and atmosphere and plays a crucial role in determining the climate. The location of that interface is itself not known a priori and thus is the source of further complexity in the problem. Chemical reactions are obviously important in fires, but are even present in some atmospheric transport processes. When liquid water or ice is present in the air, two-phase treatment of the equations of motion may need to be considered, again complicating even the relevant numerical solutions.

But even in those complex situations, simplifying assumptions can be rationally made to facilitate solving the problem. Any symmetry in the problem must be exploited. If the mean quantities are time-independent, that too can be exploited.

An extreme example of simplification that surprisingly yields reasonable results occurs in swirling giants, such as oceanic whirlpools, hurricanes, and spiral galaxies. These can simply be modeled as a rotating, axisymmetric viscous core and an external inviscid vortex joined by a Burger’s vortex. The viscous core leads to a circumferential velocity proportional to the radius, and the inviscid vortex leads to a velocity proportional to 1/r. This model leads to surprisingly good results in some narrow sense for those exceedingly complex flows.

A cyclone’s pressure is the best indicator of its intensity since it can be precisely measured, whereas winds have to be estimated. The simple model above yields the maximum wind speed from measurements of the center pressure, the ambient pressure, and the size of the eye of the storm. It is still important to note that it is the difference in the hurricane’s pressure and that of its environment that actually gives it its strength. This difference in pressure is known as the “pressure gradient” and it is this change in pressure over a distance that causes wind. The bigger the gradient is, the faster the winds generated will be. If two cyclones have the same minimum pressure but one is in an area of higher ambient pressure than the other, that one is in fact stronger. The cyclone must be more intense to get its pressure commensurately lower, and its larger pressure gradient would make its winds faster.

Being Prepared

The prediction, control, and mitigation of natural and man-made disasters is a vast field of research that no one article or even a whole book can cover in any meaningful details. In the present brief, we defined what constitutes a large-scale disaster and introduced a metric to evaluate its scope. Basically, any natural or man-made event that adversely affects many humans or an expanded ecosystem is a large-scale disaster. The number of people displaced, injured, or killed, and the size of the area adversely affected determine the disaster’s scope. Large-scale disasters tax the resources of local communities and central governments.

Science can help predict different type of disasters and reduce their resulting adverse effects. There is much in common when it comes to preparing for the occasional calamity of whatever type and managing the resulting mess. Open lines of communication among all concerned, efficient command structure, supplies, material and financial resources, and above all fairness of the system are all needed to overcome the adversity and return the community to normalcy. But we must be prepared for the rare but inevitable disaster.

The last annus horribilis in particular has shown the importance of being prepared for large-scale disasters, and how the world can get together to help alleviate the resulting pain and suffering. In its own small way, the book from which this article is excerpted will better prepare scientists, engineers, first responders, and above all politicians to deal with man-made and natural disasters.

References

1. Bunde, A., Kropp, J., and Schellnhuber, H.J. (2002) The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes, Springer, Berlin, Germany.

2. Huppert, H.E. (2000) “Geological Fluid Mechanics,” in Perspectives in Fluid Dynamics: A Collective Introduction to Current Research, eds. G. K. Batchelor, H. K. Moffatt, and M. G. Worster, pp. 447–506, Cambridge University Press, London, United Kingdom.

3. de Boer, J., and van Remmen, J. (2003) Order in Chaos: Modelling Medical Disaster Management Using Emergo Metrics, LiberChem Publication Solution, Culemborg, The Netherlands.

4. Fischer, H.W. (2003) “The Sociology of Disaster: Definition, Research Questions, Measurements in a Post-September 11, 2001 Environment,” 98th Annual Meeting of the American Sociological Association, Atlanta, Georgia, 16–19 August 2003.

5. Fischer, H.W., III (2003) “The Sociology of Disaster: Definition, Research Questions, and Measurements. Continuation of the Discussion in a Post-September 11 Envornment,” Int. J. Mass Emergencies & Disasters 21, pp. 91–107.

6. Posner, R.A. (2004) Catastrophe: Risk and Response, Oxford University Press, Oxford, United Kingdom.

7. Patanè, D., de Gori, P., Chiarabba, C., and Bonaccorso, A. (2006) “Magma Ascent and the Pressurization of Mount Etna’s Volcanic System,” Science 299, pp. 2061–2063.

8. Adamatzky, A. (2005) Dynamics of Crowd-Minds: Patterns of Irrationality in Emotions, Beliefs and Actions, World Scientific, London, United Kingdom.

9. Helbing, D., Farkas, I.J., and Vicsek, T. (2002) “Crowd Disasters and Simulation of Panic Situations,” in The Science of Disasters: Climate Disruptions, Heart Attacks, and Market Crashes, eds. A. Bunde, J. Kropp and H. J. Schellnhuber, pp. 331–350, Springer, Berlin, Germany.

10. Gad-el-Hak, M. (2000) Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University Press, London, United Kingdom.

11. Garrett, C. (2000) “The Dynamic Ocean,” in Perspectives in Fluid Dynamics: A Collective Introduction to Current Research, eds. G. K. Batchelor, H. K. Moffatt and M. G. Worster, pp. 507–556, Cambridge University Press, London, United Kingdom.

12. McIntyre, M.E. (2000) “On Global-Scale Atmospheric Circulations,” in Perspectives in Fluid Dynamics: A Collective Introduction to Current Research, eds. G. K. Batchelor, H. K. Moffatt and M. G. Worster, pp. 557–624, Cambridge University Press, London, United Kingdom.

13. Linden, P.F. (2000) “Convection in the Environment,” in Perspectives in Fluid Dynamics: A Collective Introduction to Current Research, eds. G. K. Batchelor, H. K. Moffatt and M. G. Worster, pp. 289–345, Cambridge University Press, London, United Kingdom.

Mohamed Gad-el-Hak, an ASME Fellow, is the Inez Caudill Eminent Professor of Biomedical Engineering and chair of mechanical engineering at Virginia Commonwealth University in Richmond.