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Arthur offers an opportunistic spin on hurricane math

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Hurricane Arthur is now exiting the North Carolina Outer Banks, and racing toward the Canadian Maritime provinces, enroute to its demise during the next few days (Fig. 1). Such is the case with these tropical weather systems. Born of tropical heat and humidity, they transport those ingredients north to cooler and drier climes. It’s their job as part of the overall global weather and climate system that is designed to keep the Earth-ocean-atmosphere systems in closer “balance.”

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Without humans and our structures getting in the way, hurricanes, typhoons and tropical cyclones (all the same weather feature, just known by different names in different parts of the world), would carry out their assigned mission without a care. They’d move around sand, mix ocean waters, carve out new inlets and fill in older ones and otherwise help shape the coastlines (as they have done for eons).

In other places, they would bring much needed rainfall to places that would otherwise be far too dry to sustain much life. After all, not every place in low latitudes is blessed with a lush, tropical landscape.

Arthur, however, has a put an opportunistic spin on the mathematics associated with tropical storms and hurricanes. Consider the following suite of math linkages to hurricanes (not an all-inclusive listing):

Arthur reached category 2 hurricane status late on Jul. 3, 2014 (and had winds estimated at 100 miles per hour early on Jul. 4). See the table defining the classification of hurricane strength based on wind. Note that these are based on the strongest one-minute sustained winds (NOT WIND GUSTS). Wind gusts (shorter period peaks) are often 10 to 20 percent higher than associated sustained winds.

Early this Jul. 4, the center of Hurricane Arthur was located near latitude 36.0 degrees North Latitude and 75.3 degrees West Longitude or about 20 miles to the east of Kitty Hawk, NC. According to the 5:00 A.M. E.D.T. National Hurricane Center (NHC) advisory today (Jul. 4, 2014), Arthur was moving toward the northeast at 23 miles per hour.

Latitude and longitude involve a grid of locations of a spherical Earth (Fig. 2). Latitude lines encircle the globe and run east-west and represent angular positions from the center of the Earth. Longitude lines (known as meridians) emanate from each pole and arc around the globe, much like the separations that run from the top to the bottom of a pumpkin. These locations are what allows our GPS navigation systems and similar devices to function properly. Without such systems, it would be harder for navigating airplanes and ships on their journeys.

The grid system is often portrayed on a flat map, such as this hurricane tracking chart (Fig. 3), with lines in a rectangular grid. Although this representation is often easier to visualize, it leads to distortion of areas and land shapes at higher latitudes.

Although Arthur seems to be relatively good-sized on satellite imagery (Fig. 4), that is only the cloud canopy as seen from atop the storm by geostationary weather satellites. Inside, hurricane force winds are concentrated in a very small region within 35 miles from the eye (or storm center). Tropical storm force winds extend 125 miles from the storm’s center. Fig. 5 shows the inner details as seen from a coastal weather radar. The storm is actually quite a bit smaller than satellite imagery alone would have us believe.

Note that the pattern of high-level clouds on the satellite image (Fig. 4) resembles the s-shaped logo often used to represent a hurricane on weather maps. How cool is that?

NHC also reported that a weather station in Duck, NC, just outside the eye wall of the storm, clocked a sustained wind of 74 miles per hour, with a gust of 84 miles per hour, early this morning.

Arthur landfalled on the North Carolina coast last evening, at around the time of high tide. The daily tide cycle is driven by a complex set of Sun-Moon-Earth gravitational interactions. Mathematically, it is possible to determine, far in advance, when high tide should occur at different places and how high the tide would be (without the effects of storms and winds). Typically the Mean Higher High Water (MHHW) is used as the frame of reference. However, it seems that hurricanes tend to landfall closer to high tide times than to low tide times. Several college students are working with a private weather company to assess this hypothesis.

Atop of tidal considerations, wind pushes water onto coastal and nearby lands, often creating both general flooding and forceful storm surge (Fig. 6). Thanks to the power of moving water (64 times more dense than air), storm surge can carry both water and water-borne debris ashore. These, in turn, can destroy buildings, down trees and create even more water-borne destructive material. The impacts can take place as water is pushed ashore or as it eventually races back to the ocean.

The surge can come from the ocean side of a barrier island, as well as from the sound side depending upon the wind direction, speed and fetch (persistence of winds from a particular direction). In Arthur’s case, several parts of the North Carolina Outer Banks received a double whammy, with surge coming from both directions, thanks to rapidly switching winds.

Rainfall, often measured in five to ten inch amounts (or more) over relatively short time periods), can lead to inland and river flooding. When the rainfall reaches coastal areas, it can add to any coastal flooding that occurred or is still in progress. Such heavy rainfall in mountainous areas (e.g., Appalachians and Caribbean Sea islands) can allow water to collect in narrow canyons, leading to rapidly rising, and potentially deadly, flash flood waters.

Wind direction around hurricanes completes a full, 360-degree circular pattern. Wind direction, reported as the direction from which the wind comes, is often expressed in compass directions (e.g., northeast). It can also be described in 10-degree increments and can be represented as compass points. East is shown as 90 degrees, south 180 degrees, west 270 degrees and north as 360 degrees. Signs on airport runways (Fig. 7) use the same numbering system (except that they show direction toward which an airplane is moving), sans the trailing zero (hence, 23 means 230 degrees or toward the southwest).

NHC public advisories and discussions often use this compass description to further refine storm motion. For example, this morning, Arthur was moving toward the northeast or 40 degrees.

Hurricane advisories also include both English and metric units throughout because the advisories are used by many nations. Not every nation is linked, as the U.S. is, to the English system. This allows for comparisons and conversions of units to better understand each system.

NHC provides probabilities and cumulative probabilities for wind speeds for specific locations in the storm’s path (Fig. 8). The probabilities are always greatest for the lighter wind categories. The easiest way to use these in real-time is to understand that the higher the probability, the greater the risk that storm, gale and/or hurricane force sustained winds will occur.

For most hurricanes (Arthur is a good example of an exception), the storm track history resembles a sideways parabola. Rotate the path as shown on the map of 2004 storms (Fig. 9), such that shape now looks like a mountain profile, and one has the shape taken by any projectile sent flying (e.g., golf ball, basketball).

Finally, evacuation considerations involve math, as well. How many people are at risk (i.e., flood levels at or near residential location)? How long will it take to evacuate these people safely? Can it be done during daylight hours only (a safer process)?

In some places, given population growth and road configurations, evacuation borders on the impossible, given the usual time frames for issuing hurricane watches and warnings.

As Arthur heads for the end of his lifetime, please keep this suite of mathematical considerations handy for use with the next storms to follow. Hurricane forecasters in both governmental and private sector realms have assessed the potential for different category storms this hurricane season (June 1 to November 30 in the Atlantic Ocean basin).

Accuweather forecast 2014

Colorado State University forecast 2014

NOAA forecast 2014

Note that these numbers don’t indicate how strong storms may be (only general categories) and they don’t provide any information about where and/or when such storms may affect land areas.

So, what about the age-old question, “When will I ever use math again?” One doesn’t have to look far for at least one important example.

© 2014 H. Michael Mogil

H. Michael Mogil is a Certified Consulting Meteorologist and self-professed math maven. He and his wife operate a mathematically-focused tutoring center in Naples, FL.

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