SL Math Internal Assessment: Wind Chill on Christmas Day

Tomoff 1 Alex Tomoff Mr. Abraham SL Math 7 January 2014 SL Math Internal Assessment: Wind Chill on Christmas Day Rationale: Often times in the winter months, after watching the weather segment on the local news, people plan outdoor activities because the forecasted temperatures do not seem that cold. It turns out that these forecasts are not entirely true as a result of a meteorological phenomenon called wind chill that makes air temperatures feel colder, and not realizing this can lead to serious health, travel, and social issues. On the other hand, the media may sometimes focus too much on the wind chill index, which causes people to misunderstand the serious meteorological and sociological implications at stake. Being interested in studying atmospheric science in college and later finding a career as a meteorologist, I decided to research and analyze the mathematical properties of wind chill. To do so, I analyzed two wind chill models and then collected historical temperature and wind speed data from previous Christmas Days in Cleveland, Ohio from 1973 to 2012 (I chose this day as my sample because it is considered one of the most significant holidays in the year). I chose Cleveland as I have lived in this city for most of my life where wind chill is apparent nearly every day in the winter. Its geographic location near the Great Lakes is meteorologically interesting and definitely affects the nature of wind chill. Introduction: The Antarctic explorer, Paul Siple, first coined the term wind chill in his study in 1939 called the Adaptation of the Explorer to the Climate of Antarctica (Oliver 805). Both he and

Tomoff 2 Charles Passel went on to research the nature of wind chill from 1939 to 1941 during World War II to create their Wind Chill Index, which would help American generals devise military plans that accounted for colder climates. The model for their Wind Chill Index was based on their observations of a plastic canister and the rate at which the water inside froze while suspended in the wind. Essentially, wind causes heat to be removed more quickly from a system that has a higher temperature than the surrounding environment. Using the explorers new formula and table to calculate wind chill, the National Weather Service (N.W.S.) eventually created the wind chill equivalent temperature. It was defined as the temperature at which the wind chill equivalent temperature would be the same in the complete absence of wind (Bluestein and Osczevski 1454). Sipel and Passel s first model was used by the N.W.S. from 1973 until 2001, and it is defined mathematically by the equation T(wind chill)=.0817(3.71v.5 + 5.81 -.25V)(T- 91.4) + 91.4, where T is temperature ( F) and V is the wind speed (mph) ( N.W.S.). Currently, wind chill is considered to be the rate of heat loss on the human body resulting from the combined effect of low temperature and wind (N.W.S.). It is mathematically modeled by the equation T(wc) = 35.74 +.06215T 35.75(V.16 ) +.4275T(V.16 ) (N.W.S.). It was not until 2001 that the United States, Canada, and the United Kingdom implemented a universal wind chill based on the heat transfer of a person at head-height (five feet) walking straight into wind at 1.4 meters per second. This helped the model apply more directly to a person s experience of the wind, as the old model was based on an inanimate object s heat transfer and suspended above head-height at thirty-three feet where wind speeds were likely much different (Steadman 676). I first analyzed the differences between the old and new models. After doing this, I aggregated the data from three different sample times on Christmas Day from 1973 to 2012 using archived weather data from a website called weatherunderground.com. Then, I calculated

Tomoff 3 wind chill using both models. Lastly, I found the averages of each cardinal, ordinal, and false wind direction s speed and wind chill and considered the implications of each direction s vector components to determine what causes the most serious wind chills. (The cardinal directions are North (0 ), East (90 ), South (180 ), and West (270 ). The ordinal directions are Northeast (45 ), Southeast (135 ), Southwest (225 ), and Northwest (315 ). The false directions are Northnortheast (22.5 ), East-northeast (67.5 ), East-southeast (112.5 ), South-southeast (157.5 ), South-southwest (202.5 ), West-southwest (247.5 ), West-northwest (292.5 ), and Northnorthwest (337.5 ). There are a total of sixteen different directions).

Tomoff 4 Models: To better understand how the old and new models differ, I graphed them using a program called GeoGebra. First, I decided to keep temperature constant at 10 F for both models in order to have only one unknown variable (wind speed). This resulted in two exponential functions: Wind chill ( F) new Wind speed (mph) old Stationary point (55.7, -38.8) Fig. 1. Old and New Model Graphs with Temperature Constant *Let the old model be f(x) and the new model be g(x). *f(x) and g(x) are the wind chill values ( F), and the x values are the wind speeds (mph). *Substitute any T values for 10 F in both models. In the old model, f(x), as the wind speed increases, the wind chill decreases on the interval (0, 55.7). This is because f (x) < 0. The function increases on the interval (55.7, + ) because f (x) > 0. These intervals were found by determining the derivative of the function for the old model, f(x), and then letting f (x) = 0 to solve for the x value of the stationary point.

Tomoff 5 First, rewrite f(x) as f(x) =.0817(3.71V.5 + 5.81 -.25V)(10 F -91.4) + 91.4 =-6.65((3.71V.5 + 5.81 -.25V)) + 91.4 = -24.7V.5 +1.66V +52.8. Using the power rule, f (x) = -12.4V -.5 +1.66. Let f (x) = 0 = -12.4V -.5 +1.66. Solve for V: -1.66 = -12.4V -.5 V.5 = 7.46. V = 55.7 mph = x coordinate of stationary point. Additionally, the stationary point is a minimum according to the first derivative test. That is, f (x) = 0, f(x) decreases on (0, 55.7) because f (x) < 0 (for example, f (1) = -10.7, which is negative), and f(x) increases on (55.7, ) because f (x) > 0 (for instance, f (100) =.42, which is positive). Therefore, there is a minimum at x = 55.7. The y coordinate of the stationary point is f(55.7) = -38.8. In the new model, g(x), as the wind speed increases, the wind chill decreases on the interval (0, + ). This is because g (x) is always < 0 when V > 0. The function never increases and there is no stationary point. This was determined by finding the derivative of the function for the new model, g(x), and then letting g (x) = 0 to try and solve for the x coordinate of a potential stationary point. Rewrite g(x) as g(x) = 35.74 +.06215 (10 mph) 35.75V.16 +.4275(10)( V.16 ) = 36.4-35.75V.16 + 4.275V.16 Using the power rule, g (x) = -5.72V -.84 +.684V -.84 Let g (x) = 0 = -5.72V -.84 +.684V -.84 Factor and solve for V: 0 = (V -.84 )(-5.036)

Tomoff 6-5.036 0 and 1 0. Since there is no stationary point, as g (x) is never greater than or equal to zero, g(x) must always be decreasing. It should be noted that the wind chill should theoretically never increase as wind speed increases. Understanding this concept helped me figure out why the new model is more efficient than the old model in taking account of wind speed in determining wind chill. The function g(x) never increases. On the other hand, f(x) begins to increase at wind speeds greater than 55.7 mph, when the wind chill should technically still be decreasing. Therefore, this supports that the new model is more reliable and practical.

Tomoff 7 Next, I decided to keep the wind speed constant at 10 mph with temperature as the variable for both models. Intersection point at (8.55, -10.1) new old Fig. 2. Old and New Model Graphs with Wind Speed Constant *Let the old model be line a and the new model be line f. *The y values are the wind chill values ( F) and x values are the temperatures ( F). *Substitute the V values for 10 mph in both models and rewrite the equations in slope intercept form. a: y =.0817(3.71(10 mph.5 ) + 5.81 -.25 (10 mph))(x - 91.4) + 91.4 =.0817(15.0)(x - 91.4) + 91.4 = 1.23x - 20.6 f: y = 35.74 +.06215(x) - 35.75(10 mph.16 ) +.4275(x)(10.16 ) =.680x -15.9

Tomoff 8 For both models, as temperature increases, the wind chill also increases as expected. Interestingly, these two variables represent a linear relationship as opposed to the exponential one with wind chill and wind speed. The old model has a greater slope (1.23) than the new model (.680). It is difficult to say which model is more accurate at predicting wind chills in regards to varying temperatures. Nevertheless, finding where the lines intersect will tell at what point the old and new models differ when accounting for temperature for wind chill: Set the old and new models equations equal to each other: 1.23x - 20.6 =.680x -15.9 Isolate x:.550x = 4.7 Solve for x: x = 8.55 Substitute x in for either model to find the y coordinate: y = 1.23(8.55) 20.6 = -10.1 The lines intersect at the point (8.55, -10.1). When the temperature is less than 8.55 F, the old model suggests that the wind chill would be colder. When the temperature is greater than 8.55 F, the new model suggests that the wind chill would feel colder. Data Collection: First, I gathered all of the temperature readings from Cleveland on December 25 from 1973 until 2012. I decided to take three temperature readings from each day: one in the morning, evening, and at night. Specifically, I chose the three sample times 10:00 a.m., 4:00 p.m., and 10:00 p.m. I chose to use the Imperial System of measurements because my research is relative to the Cleveland area, which uses these unit measurements. Also, note that if there was a calm wind report (0 mph), then there was no wind direction and the wind chills remained the same temperature as the actual temperature.

Tomoff 9 Data Tables 10:00 a.m. 4:00 p.m. 10:00 p.m.

Tomoff 10 Figures 3, 4 and 5. Raw Temperature and Wind Speed Readings from 1973 to 2012 supplied by Weather Underground, LLC, for Cleveland Burke, OH. Weather History for Cleveland Burke, OH. Weatherunderground. n.p., n.d. Web. 5 January 2014. Calculations: I calculated the old and new wind chill values using their respective models. Old Model: T(wc) =.0817(3.71V.5 + 5.81 -.25V)(T-91.4) + 91.4, where V is the wind s velocity (mph) and T is the temperature ( F). For instance, at 10 a.m. in 1973, T(wc) =.0817(3.71(21.9 mph).5 + 5.81 -.25(21.9 mph))(42.1 F -91.4) + 91.4 = 20.1 F. New Model: T(wc) = 35.74 +.06215T 35.75(V.16 ) +.4275T(V.16 ), where V is the wind s velocity (mph) and T is the temperature ( F). For instance, at 10 a.m. in 1973, T(wc) = 35.74 +.06215(42.1 F 35.75(21.9 mph.16 ) +.4275(42.1 F)(21.9 mph.16 ) = 32.8 F. I calculated the averages for speed and wind chill (new model only) for all sixteen directions of the wind. Average velocity was found by using the formula: V average = n n i=1 Vxi n, where Vxi is the summation of all the velocities and n is the frequency of winds i=1 from a particular direction. For instance, for the wind directions that came from the direction of 17.3+8.1+11.5+ 9.2 + 25.3+ 6.9 + 5.8+16.1+ 6.9 0, V average = 9 = 11.9 mph. The average wind chill was calculated in the same manner.

Tomoff 11 Here is a table that displays these calculations. Fig. 6. Average Wind Speeds and Wind Chills for Each Wind Direction. Knowing these averages that better represent the raw data, I can now determine some trends in the magnitude of each of the sixteen wind directions and their relation to wind chill values. To do this, I used filled radar charts with Microsoft Excel. Fig. 7. Average Wind Chill (New Model) Radian Chart. Fig. 8. Average Wind Speed Radian Chart.

Tomoff 12 These above charts suggest that the coldest wind chills usually come from the West and that the greatest wind speeds also tend to come from the west (1 = North, 5 = East, 9 = South, and 13 = West). The rings in the graphs represent the average wind chill or speed values. Vector Analysis Fig. 9. Horizontal and Vertical Components of Wind. Fig. 10. Radian Chart of Wind s Horizontal and Vertical Components. To calculate the x and y components of each wind direction s average speed, I used triangular trigonometry. For instance, in the upper right quadrant of the radar chart at position 2: (22.5 ), sin(22.5 ) = opposite hypotenuse = Vx, so Vx = 2.64 mph. 6.9 Likewise, cos(22.5 ) = adjacent hypotenuse = Vy, so Vy = 6.37 mph. 6.9

Tomoff 13 I used another radar chart to show the x and y components of each wind direction s average velocity overlaid on top of each other. It appears that the greatest horizontal components and vertical components still come from the west, or the left side of the radar chart. The maximum x component was 15.4 mph from 292.5, or west-northwest. The maximum y component was 16.5 mph from 337.5, or north-northwest. This correlation is interesting as it poses the question as to why the strongest winds and lowest wind chills occur when the winds have a westerly component. Conclusion and Reflection: I encountered a number of limitations while doing research for this investigation. I could not find any sufficient temperature and wind speed-readings in Cleveland before 1973. It is possible that different types of winds and wind chill values occurred more or less frequently earlier in the century. Additionally, readings from 11 p.m. to 7 a.m. were not always available, so I could not use any nighttime data. Also, I was forced to pick one day as my sample space because the amount of data collection for the entire winter would be rather difficult, but I also wanted to keep some consistency in the data I collected. I had originally picked Thanksgiving as my sample day, as it is one of the busiest travel days in the year. However, I soon realized it was never on the same date each year, which could lead to too much variability in the temperature and wind speed-readings. It becomes quite apparent why a new wind chill model was devised for public use in 2001 after looking at the scientific and mathematic issues of the old model. The way the old model behaves does not fit what is scientifically expected. The new model, on the other hand,

Tomoff 14 accounts for biological and technical implications regarding heat transfer. Furthermore, it has more support from computer modeling and numerous clinical tests (N.W.S.). The findings of the investigation propose that the strongest wind chills are associated with westerly winds. Meteorologically speaking, this is likely because of the typical path that low-pressure systems follow around the Great Lakes region. They travel from west to east and have winds that travel counterclockwise around their centers. Therefore, as these low-pressure systems approach Cleveland, the winds tend to increase first from the south. Then, as the systems pass Cleveland, the winds switch to more westerly and northerly directions. This tends to bring cold air from Lake Erie and Canada into the region. With this typical weather pattern, wind chill is always a prominent social and health issue in this area. Fig. 11. National Weather Service Windchill Chart. Wind Chill. The National Weather Service. Dodge City: National Weather Service Forecast Office, August 2001. Web. 30 November 2013. This table supplied by the National Weather Service shows how dangerous wind chills can be by determining the time it takes for frostbite to occur under varying wind and temperature

Tomoff 15 situations. The most dangerous wind chills amplify the cooling effects of the air and lead to health conditions such as frostbite and hypothermia. According to the American Center for Disease Control, nearly half of the 700 deaths that occur due to hypothermia each year are a result of cold weather. In conclusion, wind chill is a serious social and meteorological issue that has extensive mathematical properties, and it is extremely important for people to recognize its potentially dangerous implications during their daily lives.

Tomoff 16 Works Cited Bluestein and Osczevski. The New Wind Chill Equivalent Temperature Chart. Bulletin of the American Meteorological Society. 86.10 (2005): 1453-1458. Print. Oliver, John. The Encyclopedia of World Climatology. Dordrecht: Springer, 2005. Print. Steadman, R.G. Indices of Windchill of Clothed Persons. The American Meteorological Society. 10.4 (1971): 674-683. Print. Weather History for Cleveland Burke, OH. Weatherunderground. n.p., n.d. Web. 5 January 2014. Wind Chill. The National Weather Service. Dodge City: National Weather Service Forecast Office, August 2001. Web. 30 November 2013.

Tomoff 17 Reflection -Change to Christmas Limitations -couldn t go past 1973, couldn t get times from 11 p.m to 7 a.m. Sources http://www.wunderground.com/history/airport/kbkl/1973/12/25/dailyhistory.html?req_city= NA&req_state=NA&req_statename=NA -database Wind Chill Formula: T(wc) = 35.74 + 0.6215T - 35.75(V 0.16 ) + 0.4275T(V 0.16 ) http://voices.washingtonpost.com/capitalweathergang/2010/02/wind_chill_its_history_effect.htm l http://www.crh.noaa.gov/ddc/?n=windchill -definition of wind chill http://www.windfinder.com/wind/windspeed.htm bearings for meteorology http://www.teachingboxes.org/avc/content/an_introduction_to_wind_chill.htm -wind chill stuff Old Model: T(wc) =.0817(3.71V**.5 + 5.81 -.25V)(T-91.4) + 91.4

Tomoff 18 New Model: T(wc) = 35.74 +.06215T 35.75(V**.16) +.4275T(V**.16) Presentation format models Old Model: T(wc) =.0817(3.71V.5 + 5.81 -.25V)(T-91.4) + 91.4 New Model T(wc) = 35.74 +.06215T 35.75(V.16 ) +.4275T(V.16 ) Cell references: Old T(wc)=((((3.71*SQRT(C2))+5.81)-0.25*(C2))*(B2-91.4))*0.0817+91.4 New:=35.74+(0.6215*(B2))-(35.75*(C2^0.16))+(0.4275*(B2)*(C2^0.16)) Recalc. 6.9