Hurricane Sandy Meets MathematicsA sensitivity to mathematics and its powerful analytical tools helps give insight into Sandy's devastation and how to minimize the consequences of future powerful storms...
In late October 2012 a large storm, dubbed Hurricane Sandy, struck New Jersey and New York causing large areas of devastation in New Jersey, New York City, Long Island, and the counties north of New York City. What was remarkable about the storm was it resulted in a power black out for the southern part of Manhattan Island, an area that included Wall Street, even though there were some individual buildings that retained power. This area of NYC involves some of the most important businesses in America, including the New York Stock Exchange. One major hospital lost power, and neither of its two backup systems could be employed. Patients had to be evacuated. Several NYC tunnels that had never had water problems with prior storms were knocked out completely. Many of these difficulties occurred because of a surge of water in the NYC harbor area where the two rivers that bound Manhattan Island meet. This surge was unusually high due to a combination of the storm itself and the timing of high tides when the storm arrived. In some areas a second coastal storm dropped a heavy wet snow on trees already weakened by Hurricane Sandy, leading to additional power outages and tree damage.
(Courtesy NOAA - National Oceanic and Atmospheric Administration)
Like many hurricanes, Sandy affected not only the U.S. mainland but countries in the Caribbean. The track below shows the way hurricanes sometimes double back on their prior track (the track depends not only on the pattern of the Earth's rotation but more localized considerations such as land masses in its track and ocean temperatures). In Hurricane Sandy's case some human storm forecasters as well as computer models predicted that the storm would head out into the Atlantic instead of turning inland. One model commonly used in Europe predicted the course of Sandy very accurately though there is no guarantee it will do as well with the next hurricane.
(Courtesy of Wikipedia)
A sensitivity to mathematics and its powerful analytical tools helps give insight into Sandy's devastation and how to minimize the consequences of future powerful storms as well as ideas about how to optimize the speed of recovery after a large storm or other natural disaster.
What Sandy wrought
During and after Hurricane Sandy residents of the Metropolitan New York area (where I live) were flooded with images and information from many sources, including just taking a walk in one's neighborhood. Seeing large trees down, either blocking a street or severely damaging a home, was very common and gave one a sense of the tremendous power of the storm.
(NYC Parks Department: Crotona Park, Bronx, New York; Photo credit: Malcolm Pinckney)
There were also many images of whole neighborhoods without power (estimates for the NY area were that over 6 million people lost their power, many for a period of nearly two weeks), downed trees, houses swept off their foundations, and water in subway stations and basements of houses and buildings. The Hugh Carey Tunnel (also known as the Brooklyn Battery Tunnel- see below) was filled with water for the first time in its history. (Use of the tunnel was not returned to normal until over a month after the storm.) The storm caused many deaths and large amounts of human misery.
(Courtesy of Wikipedia, via the Metropolitan Transportation Authority of New York)
The severity of the storm's damage and the extended outage of power to a large part of Manhattan set off discussions about what seemed to some people to be an increase in the number of severe storms and their consequences for a world fast growing in population, especially in urban areas. Hurricane Sandy occurred at a time when many countries are having to come to grips with the consequences of the Earth's sky-rocketing population. This growing human presence has led to lengthy discussions about what effect mankind is having on the future of the planet and on mankind itself. Has the current population of the Earth affected the world's climate? If so, what does this mean for the future?
(Graph generated with Wolfram Alpha)
There are three local maxima and three local minima in the picture above as well as the global maximum on the right (a little to the right of x = 4), and a global minimum on the left (a little to the left of x = -6). One can easily design examples of polynomials (the example above was generated using a polynomial) which are defined on an interval which includes its end points. where the largest value on the interval occurs at a value which is also a relative maximum.
It is instructive to look at the data below - time series diagrams of the number of hurricanes, major hurricanes, and accumulated cyclone energy prepared by the National Oceanic and Atmospheric Administration (NOAA), of which the National Hurricane Center (NHC) is a part. The data below runs from 1851 to the present and represents only information about storms in the "Atlantic Basin." In addition to the diagrams the National Hurricane Center has tables with this and related data. As you look at the data, do you see what you think are short term trends, long term trends, cyclic behavior, etc.?
To construct such diagrams one has to give a definition of "hurricane" and stick to it. Are hurricanes the same or different from cyclones and typhoons? Here are the definitions (in red text) used by the National Hurricane Center, where kt refers to knots:
Mathematics brings to the study of any subject an interest in classification and making distinctions between similar but different things. While each hurricane is distinct from every other hurricane, hurricanes have many ways in which they are similar to each other. Exploiting insights into the differences and similarities of storms, while of interest for its own sake, is also of interest for policy decisions. These include questions about tying the pattern of recent storms to the need or wisdom for building new homes on the coast to standards which can resist higher levels of flooding than was necessary in the past, etc.
Hurricane Sandy has spurred the asking of new questions, some of them mathematical in nature. For example, one might try to construct a mathematical model for how small trees versus large trees behave in winds that come consistently from one direction or are coming from rapidly changing directions. What effects would pruning the crowns of large mature trees have on preventing them from uprooting during a hurricane? What happens when tree roots and/or leaves are water soaked? While some of these questions have received attention in the past, perhaps there is new motivation and/or data for helping get answers to these types of questions.
Planning for storms before they occur
A well-known adage is that an ounce of prevention is worth a pound of cure. This thought applies to preparation for major storms as it does more generally. If one knows with high probability there will be major flooding in a certain area, then area residents might be required to leave their homes. This is a lot better plan than trying to rescue them at the 11th hour with helicopters or rescue teams in boats, when the people are on their roof tops with their cars floating away. If long-range forecasts for severe storms such as Sandy are available, then people in the path of the storm can be given reliable advice about how to protect themselves. Such actions range from boarding up windows and securing objects which might move in severe winds, to having adequate flashlights and batteries for the possibility of the loss of power, to evacuation. For some sizable communities evacuation is not a matter to be treated lightly because area roads may become strained by an evacuation order as well as there being issues of economic losses that occur when such action is taken. Tradeoffs between potential loss of life and economic losses put tremendous pressure on officials making decisions of this kind. In Italy recently seismologists were imprisoned after having been convicted for their role in what information was given to the public concerning the possibility of an earthquake, which, when it occurred, resulted in loss of property and life. To avoid this kind of thing from happening in the future, what should seismologists and meteorologists do when advising government officials about impending earthquakes, volcanic eruptions, or severe storms? Perhaps in part because of issues of this kind, the NYC subway and commuter rail systems were shut down quite a bit prior to the arrival of Hurricane Sandy. Although this inconvenienced a lot of people and "stranded" some who were unable to use public transportation to get where they needed to, it prevented the possibility of danger and/or loss of life to people who might have been trapped in a train within a tunnel when the tunnel was flooded. Assessing tradeoffs in risk and costs requires the building of complex models to determine what policies are the wisest.
Trees and wind
For past snowstorms and hurricanes that hit Long Island, while there has often been lots of damage and many power outages have occurred, the number of separate locales that had to be checked before power could be restored was relatively small. What happened with Hurricane Sandy was that the number of downed trees that caused separate "traumas" to the electrical network was very high. In this environment one needs some priority plan adaptable enough to sequence how to get things restored to normal, especially when the number of workers to carry out the restoration may be limited initially but would grow as more crews from distant locales became available to assist with the restoration efforts.
Where is the damage?
One of the reasons that recovery can be complex after a storm is that the power company may not even know the location of the causes of outages. For example, LIPA has a system where people can report power outages but the system does not collect much useful information. The company may not have an easy way to tell if the outage is at a single home or a block of homes. If a single home reports a power outage this does not mean that the whole block does not have an outage because other people may not be home when the outage occurs to report that they, too, do not have power. At night, if I lose power I can go outside and often tell whether other houses on my block have power or not, but during the day this is not always easy to ascertain. I have urged my neighbors to always report outages and not assume some other person will have done so but I am reasonably sure that not all people do this partly because they are not sure of the procedure and partly because it is a "nuisance." I am not even convinced, however, that the power company can put to use this extra information that there is a cluster of outages, though in an "ideal world" they should be able to use this information. In particular, they should also be able to use past information about outages. For example, my side of the block, for reasons I don't truly understand, loses power much more commonly than people across the street or on the next block.
Mathematics to the rescue?
Mathematics may seem a small part of what is involved in dealing with severe weather-induced events such as floods and hurricanes. This does not mean it might not be a good idea for mathematicians to look at ways they can improve the models that are used to predict hurricane formation and tracking as well as ideas for recovery after such events. The notion of informing the public that insight from mathematics might help is a good one. Furthermore power companies might want to look into better operations research methods for their disaster recovery plans.
Queuing models might be of use to help minimize such secondary problems as the emergence of long lines at gasoline stations due to problems with allocating fuel to stations and power issues at stations. People lined up at gas stations even without knowledge if the stations had gas or if they expected an imminent delivery. There were two types of customers: those wanting small amounts of gas for their home electric generators and/or to put into their car which had run out of fuel and people who wanted to top off their tanks, etc. Many people who did not need gas were nervous and were getting it anyway, preventing people who had low tanks from getting gas when stations ran out. Police, at great cost, were assigned to prevent flaring tempers at gasoline stations when they might have been better used elsewhere.
When planning for how a cluster of homes in an area could have power restored, it might be possible to improve the time and cost efficiency of restoring power to these customers by using ideas related to the traveling salesman problem and vehicle routing. The idea is that to provide service at a collection of locales starting and ending at a depot, one can find a route which minimizes the time of visiting the sites. Problems for power are more complex because one will probably need to have estimates for the time to be spent at each site which may be hard to predict. It might also help to have this model be dynamic so if some new nearby site becomes eligible for power (perhaps because it has been certified by an electrician that damage to basement electrical equipment due to flooding has been corrected), then one can schedule this additional service stop when there will be a truck nearby instead of putting the customer at the bottom of a queue.
None other than Leonardo da Vinci studied the "geometry of trees." He noticed that if one cuts a tree horizontally one typically gets a circle for the truck and a collection of circles for the places where the tree has started to branch. Da Vinci claimed that the sum of the diameters of the branches was equal to the sum of the diameters of the trunck. While, not surprisingly, this model represents a vast simplification of the complexities one sees in tree growth, recently attempts have been made to study the accuracy of what da Vinci claimed, as well as to make his claims more precise. Interest in this matter has to do with being able to choose trees for planting that minimize tree loss in areas subject to hurricanes. Other parts of mathematics might be of use to understand what happens to the roots of trees when they are in very wet soil caused by the extensive rains often associated with hurricanes. Mathematics might also have value in designing pruning strategies for areas with lots of trees which are hit often by hurricanes. Much work has been done in trying to understand the mechanics of tree motion in strong and/or variable wind environments.
Applied mathematicians are very clever in using existing tools to study wide ranges of problems. They may also be inspired to do new mathematics based on situations which arise from storms or floods.
Learning from the past
Hurricanes are complex storms and to understand them from "first principles" requires basic research in meteorology with the mathematical support of such subjects as fluid dynamics and numerical methods. The mathematical equations which explain what goes on in the Earth's atmosphere and how storms are "born" of a combination of effects due to the Earth's rotation, the nature of the oceans and its currents, and motion of the air, strains both theory and computer power. However, there is also insight to be obtained from comparing what has happened with predictions. There is "data" which shows how some storms of the past have not behaved as predicted. This can lead to the changing of the model that made the prediction by using additional information obtained from experience, but data can also be used in conjunction with an existing model to get further insight.
Anthes, R. Tropical cyclones: their evolution, structure and effects. American Meteorological Society, 1982.
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