Using the Golden Ratio as a Model for Tornadogenesis. George McGivern Brad Walton Dr. Mikhail Shvartsman

Transcription

1 Using the Golden Ratio as a Model for Tornadogenesis George McGivern Brad Walton Dr. Mikhail Shvartsman

2 1. Introduction: Intro to tornadoes and tornado forecasting. Problem and Goals: The main problems and goals we have sought out to accomplish 3. Results and Methodology: The progress we have made so far and how we accomplished it 4. Future Directions: Our future plans for the project 5. Bibliography 1. Introduction A tornado is a violent, rotating column of air spawned by powerful thunderstorms known as supercells. Tornadoes can effectively be viewed and studied through an understanding of thermodynamics. Energy for tornadoes is provided by latent heat caused by phase transition of water-vapor, the atmosphere is composed of both dry air and water-vapor which are viewed with the ideal gas laws, and tornadic systems take advantage of varying atmospheric conditions to utilize convection and rotating wind forces to produce the final product. Meteorologists analyze many factors that they believe play a role in the formation of a tornado. With most of these factors, it is difficult to distinguish just how much needs to be in play for successful tornadogenesis. Additionally, while we have many methods of measuring atmospheric conditions, such as Doppler radar, weather balloons, and other meteorological instruments, it can be difficult to monitor all of these near the ground where tornadoes would

3 actually touch down. Finally, because tornadoes are rare events, in order to effectively forecast a tornado, meteorologists must be able to predict at an accuracy of at least 1 r, where r is the probability of a tornado occurring. If they are unable to predict at this accuracy, then the forecast will be less than 50%, and will be less effective than a forecast which says no every time. As an example, we can assume the probability of a tornado occurring in a specific county is 0.01( r ). In this case, assuming a tornado will not occur regardless of the circumstance will only be inaccurate when a tornado does occur, 1 r ; in our example it will still be 99% accurate. In these circumstances in order for a forecast to be even minimally accurate, they must be able to predict with an accuracy greater than This degree of accuracy is not realistically attainable for meteorologists. For this reason, instead of focusing on a yes or no when forecasting, they instead gauge the atmospheric conditions and attempt to assess the risk a system carries for tornadogenesis. In order to measure this risk, they create indices which gather various tornado-causing measures and combine them while scaling them to a flag value that can be interpreted to give a snapshot of developing tornadic conditions. Over the past half century, meteorologists have created many indices, and their ability to forecast tornadoes has improved because of it. Nevertheless, there is room for improvement, and as researchers strive to refine the tornado indices, we are faced with the problem of determining which indices are most effective.. Problem and Goals This summer, our main objective was to find a method by which we can determine the effectiveness of the tornado indices. To accomplish this goal, we set out to find ways we can analyze these indices and relate them with each other. Additionally, our past knowledge of nature and math prompted us to explore the elements of tornadoes that relate to fractal symmetry and the golden ratio. During an early discussion about meteorological data, we were introduced to an

4 equation made by H. Cai that was used to relate wind vorticity, the curl of wind velocity (denoted by w), to length scale in radar ( r), which describes the optimal scaling of radar data to represent the accurate amount of rotation in a cyclone. For large tornadoes and small tornadoes the equation was: w = 1 r 1.6 (large) 1 w = r (small) We initially noticed that the exponent 1.6 in the large tornado equation was very close to that of the golden ratio, approximately In order to prove if this connection was more than just a coincidence, we set out to construct an equation that could further scale existing indices and to analyze how well they satisfy the length scale equation. In order to better understand the indices, we also sought out to understand the individual components of each index, specifically the calculated and derived values convective available potential energy (CAPE) and storm relative helicity (SRH). 3.Results and Methodology From the beginning we were aware of the similarities between a tornado and the golden spiral, a visual model used to illustrate the ratios of fibonacci terms. The golden spiral is found naturally in many places, such as seashells, flowers, cells, and even the shape of some galaxies, but the similarity to the radar image of a hook echo was what caught our eye. This was the first thing that indicated that there may be a relationship between tornadogenesis and the golden ratio. Furthermore, when observing radar soundings of tornadic supercells, the golden spiral often centered on the actual location of the tornado when we tried to align it with the hook echo. These similarities prompted us to apply past knowledge of the golden ratio into an equation that could relate tornado indices to the vorticity-length scale equation. One important characteristic of the golden ratio is that its square is one more than itself (approx..618), and its inverse is one less than itself (approx ). In order to show this algebraically, we can imagine the equation: x = x + 1

5 We can solve this equation using the quadratic formula to give us a positive exact answer of Additionally, there are other numbers that share properties with the golden ratio, and by trying out different equations like x = x+ or x = x+3, we can find more numbers that look 8 13 similar (+ and 3+ ). We can go further by creating an equation as: x ) x n = ( n b and by taking the limit as x approaches infinity, we are left with: b+ 4+b x = When using this equation, note that a b value of 1 will result in the golden ratio. And a b value of 0 will yield 1. In order to fully utilize this equation, we must use it along with a tornado index. While there were many indices available, we decided to focus on the energy helicity index (EHI), due to its relative simplicity. CAP E SRH E HI = * 160,000 EHI multiplies CAPE and SRH and scales them to create an index. An EHI around 5 indicates conditions that can form large tornadoes, and an EHI around 1 indicates the minimum conditions needed to form a tornado. CAPE and SRH can also be considered indices, but they are calculated from measured values. CAPE measures the energy a parcel of air would contain while moving vertically through the atmosphere. To calculate CAPE over a layer of the atmosphere, we use the following formula: C AP E = h h 0 (T ap T e) T e g Where h 0 and h represent the boundaries of the atmospheric layer, T ap represents the temperature of an individual air parcel and T e represents the environmental temperature, and g represents the force of gravity.

6 SRH measures the potential for cyclonic rotation of a storm, and can be calculated by: S RH = h dv [v * k x ]dz dz h 0 Where h 0 and h represent the boundaries of the atmospheric layer, v represents a velocity vector, dv k represents a unit vector parallel to the z-axis, and represents the change in velocity as height changes with dz representing the change in height. One resource at our disposal that can help us collect relevant data for these two values are skew-t log-p charts. These charts show data for numerous measures of the atmosphere, including environmental temperature, wind velocity, wind direction, and more. We can access these skew-t charts through sounding archives, and we are able to find data from weather stations across the country within a 1-3 hour time frame. This data will be essential to use with calculating EHI values that we can run through our constructed equation. dz Now, going back to Cai s vorticity equation. In his research, he found that with big tornadoes, wind vorticity related to to length scale as: w = 1 r 1.6 And with smaller tornadoes, data indicated that an approximate relationship was: w = r 1 In order to connect this equation with our own constructed equation, we will assume that w = 1 r m This allows us to set the result of our equation equal to an m value that can be plugged into Cai s equation. In theory, if we can find a way to produce the golden ratio as an m value for big tornadoes, and an m value of 1 for smaller tornadoes with our constructed equation, we can see if Cai s equation remains accurate for all the m values in between. In order to get this result from our constructed equation, we can use a tornado index to determine the difference between large and small tornadoes. To achieve this, we can utilize a simple scaling function to take EHI values as an input.

7 b = ( h 1 )/4 By plugging EHI into the h value, we can manipulate the EHI value into a b value that satisfies the goal of our equation; if the EHI is 5 (big tornado), the equation will scale to a b value of 1, which will result in our constructed equation giving us the golden ration, and an EHI of 1 (small tornado) will give a b value of 0, and our constructed equation will give us a 1. We can simplify our constructed equation by plugging this function into our b value, giving us: (h 1)/4+ 4+((h 1)/4) m = Which can be further simplified to give us: m = h 1+ h h+65 8 With this final equation, we hope to use EHI values to test our interpretation of Cai s equation. 4.Future Direction In the future, we hope to use sounding data to establish a method that will allow us to see if our constructed equation can effectively be used to predict an optimal length-scale. We will also want to tweak the formula so that we can use it with other tornado indices, taking note of which indices work best. We hope that this analysis will prove to be an effective way to help distinguish advantages that certain indices hold over others. Additionally, we hope to make progress in learning various software programs that will allow us to simulate and visualize atmospheric conditions, which can further our understanding of tornado dynamics, as well as creating models for statistical analysis. 5.Bibliography H. Cai, 005: Comparison between tornadic and nontornadic mesocyclones using the vorticity (pseudovorticity) line technique. Doswell, C. A. III, and D. M. Schultz, 006: On the use of indices and parameters in forecasting severe storms. Electronic J. Severe Storms Meteor., 1, 1 14 A. Chorin and J. E. Marsden, 000: A Mathematical Introduction to Fluid Mechanics. 1,1-46

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