1. The mathematics of rain and flooding: Learning about volumes, ratios, and fractals D. Quesada School of Science, Technology and Engineering Management St. Thomas University, Miami Gardens, FL 33054

2. Composition of the Atmosphere

3. Formation of current Earth’s Atmosphere: Out-gassing As millions of years passed, the constant outpouring of gases from the hot interior – known as Out-gassing – provided a rich supply of water vapor, which formed clouds. Rain fell upon the Earth for many thousands of years, forming the rivers, lakes, and oceans of the world. During this time, large amounts of CO 2 were dissolved in the oceans. Through chemical and Biological processes, much of the CO 2 became locked up in carbonate sedimentary rocks, such as limestone. With much of the water vapor already condensed and the concentration of CO 2 dwindling, the atmosphere gradually became rich in nitrogen (N 2 ), which is usually not chemically active.

4. Chemical Properties of Water The polarity of water allows it to “hook up” with other molecules, including itself. More substances dissolve in water than any other liquid. for this reason, water is often called the Universal Solvent

5. The V-shape of the water molecule is also important because it allows for other configurations of water to be formed. Ice , for instance, has a very ordered lattice structure. Supercooled water ( water below the freezing point ) also has water molecules that are structured in certain way. Snowflakes have yet another shape. The ability of water molecules to quickly break and re-form hydrogen bonds gives it a property called cohesion . Due to this property, water has a high surface tension pH = - Log([H + ]) pH is a measure of the acidity or alkalinity of a substance. Formally, pH is defined as the negative logarithm of the concentration of hydrogen ions in an aqueous solution. For our purposes we need to know that some liquids are acidic ( having more H-ions ) and some are basic ( having more hydroxyls , or OH-ions) [H + ] is the hydrogen concentration, in moles per liter.

8. Water Vapor Pressure The push (force) that the water vapor molecules are exerting against the inside walls of the Parcel. The total air pressure inside the parcel is due to the collision of all the molecules against the walls of the parcel. The total pressure inside the parcel is equal to the sum of the pressures of the individual gases. The air pressure near sea level is the same as inside the parcel, and equals 1000 mb . Since the gases inside include nitrogen (78 %), oxygen (21 %), and water vapor (from 0 to 4 %, but 1 % is more likely) the partial pressures exerted by these gases are: Nitrogen – 780 mb Oxygen – 210 mb Water Vapor – 10 mb An increase in the number of water molecules will increase the total vapor pressure . High actual vapor pressure indicates large numbers of water vapor molecules, whereas low actual vapor pressure indicates small numbers of vapor molecules . Actual vapor pressure indicates that air’s total water vapor content, whereas saturation vapor pressure describes how much water vapor is necessary to make the air saturated at any given temperature.

9. The temperature at which the relative humidity reaches 100 % and Condensation starts is called the Dew Point . When the air becomes saturated with water vapor (that is, the dew point is reached), one of two things happens: either water condenses or , if the temperature is low enough, ice crystals precipitate . In order for a droplet of water or an ice crystal to form, energy is needed. The process is called Nucleation , and energy is required because a new surface is formed. As an unsaturated mass of air rises, it expands and cools at the dry adiabatic rate 10 o C/km. When the air temperature falls to the point where the air is saturated, Condensation commences and latent heat is released

10. Snowflakes and Fractals

11. Climate The average weather patterns for an area over a long period of time (at least 30 years, and above – 1,000,000 years) Average Precipitation Average Temperature Latitude Ocean currents Altitude Where people live? How people live? What they grow and eat? Average It is determined by and Which are influenced by And affects

12. Planetary Biology and Inter-species interactions The Web of Life The appearance of plants and other living forms on Earth constituted one of the most important steps in the development of the current chemical make-up of the atmosphere.

13. PHYSICAL QUANTITIES AND UNITS Observations produce qualitative information about a system Measurements produce quantitative information which is needed in any science that strives for exactness English Units Inch (in) Second (s) Pound (lb) Metric System Meter (m) Second (s) Kilogram (kg) Fundamental Physical Quantities Distance - Time - Mass Scientific Notation Prefix | Abbreviation | Regular Notation | Scientific Notation Tera T 1,000,000,000,000 = 10 12 Giga G 1,000,000,000 = 10 9 Mega M 1,000,000 = 10 6 Kilo k 1,000 = 10 3 Hecto h 1,00 = 10 2 Deca da 10 = 10 1 -------- ---------- 1 = 10 0 Deci d 0.1 = 10 -1 Centi c 0.01 = 10 -2 Milli m 0.001 = 10 -3 Micro μ 0.000,001 = 10 -6 Nano n 0.000,000,001 = 10 -9 Pico p 0.000,000,000,001 = 10 -12 Length : 1 kilometer (km) = 1000 meters (m) = 3281 feet (ft) = 0.62 miles (mi) 1 mile (mi) = 5280 feet (ft) = 1.61 kilometers (km) = 0.87 nautical mile (nm) 1 centimeter (cm) = 0.39 inch (in) 1 inch (in) = 2.54 centimeters (cm) 1 yard (yd) = 3 feet (ft) = 36 inches (in) Time : 1 hour (hr) = 60 minutes (min) = 3600 seconds (s) Mass : 1 kilogram (kg) = 1000 grams (g) = 2.2 pounds (lb) Speed (rate of change of a coordinate in time): 1 knot (kt) = 1 nautical mile per hour (nmph) = 1.15 miles per hour (mph) 1 mile per hour (mph) = 1.61 kilometers per hour (km/hr) = 0.45 m/s

14. THE HYDROSPHERE: WATER ON THE LAND 1. Water and the hydrologic cycle. Streams and their channels. 2. Water in the ground. Porosity and permeability. 3. Glaciers. 4. The Oceans. Ocean circulation 5. El Ni ño/Southern Oscillation 6. Ocean waves and ocean tides Liquid water makes Earth unique in the solar system. Although water has been detected on other bodies of the solar system, it does not appear to be present as a liquid anywhere except on our planet . H 2 O Water’s Chemical Formula 97 % in Oceans Hydrologic Cycle

15. Water temperatures across the world oceans depends on the amount of energy received from the Sun as well as from the deep of a particular area. In this end, Caribbean Sea, Eastern Pacific, Western Pacific and the Indian Oceans, appear as the areas with highest sea surface temperatures. Due to the large capacity of sea water to absorb the sun Energy, oceans are one of the most important regulators of the world climate. If for any reason ocean currents stop flowing around the world, results may be readily catastrophic.

16. Composition of Sea Water, and Fresh Water: Distribution of Elements over Land and Oceans.

17. Hydrologic Cycle over Land in Details Transpiration and Photosynthesis Percolation versus Porosity

18. Which factors determine the groundwater motion, and how far they are responsible for flooding? The answer is in Darcy’s Law. Factors affecting the motion of water under the ground were put together by the famous French scientist Henry Darcy (1803 – 1858). Today, we call this formula, the Darcy’s Law. It may be cast into: In the Darcy’s Laws, we might identify two ratios, the first one Permeability over Porosity , and the second one Rise over Run . Porosity is the proportion of void space in the material – holes or cracks unfilled by solid material, within or between individual grains. Permeability is a measure of how readily fluids pass through the material and is related to the extent to which pores or cracks are interconnected. Compared to the rapid flow of water in surface streams, most ground water moves relative slowly through rocks underground. Because it moves in response to differences in water pressure and elevation, water within the saturated zone tends to move downward following the slope of the water table. The stepper the slope of the water table, the faster ground water moves. Water table slope is controlled largely by topography. How fast ground water flows also depends on permeability of the rock or other materials through which it passes. If rock pores are small and poorly connected, water moves slowly. When openings are large and well connected, the flow of water is more rapid.

19. A simple grid 16 by 16 illustrates the applicability of these two concepts. Cells shaded in red represent a particular grain, while cells in white are voids. Thus, Porosity equals the amount of white cells over the total of cells, which in this case are 110 / 256 = 0.43 or 43 % of porosity. On the other hand this material is highly permeable because it has several interconnected paths. A paved area usually has a surface layer with very low level of both porosity and permeability, such that the first factor is a very small number. This fact is the reason for too low infiltration rate of water into the ground even if the subsurface soil would contain many porous. Clay (45 – 55 %, less than 0.01 m/day) Fine Sand (30 – 52 %, 0.01 – 10 m/day) Gravel (25 – 40 %, 1000 – 10,000 m/day) Sandstone (5 – 30 %, 0.3 – 3 m/day)

21. Planar Figures Perimeter Areas a a a a a b r P = 4 a A = a 2 P = 2a + 2b A = a b P = 3a A = 0.5 a h C = 2 π r A = π r 2 Volumes Surface Areas V = a 3 S = 6a 2 V = π r 2 h S = 2 π r 2 + 2 π r h V = 4 π r 3 / 3 S = 4 π r 2 Solid Bodies Basic Geometrical Formulas

22. Practice Activity about Areas and Fractions 1. Determine the Area of Saint Thomas University main campus by looking at MapQuest Directions 1. Look at the upper right corner. 2. Write down the scale of the map in both units, meters (m) and feet (ft). 3. Determine the length and the width of STU main campus. 4. Find the Area and write your answer in both units (meters), and (feet). 5. Determine the area of the airport of Opa-Locka, following the same method. 6. Determine how many times STU go Into the Opa-Locka airport. 2. In the event that a storm passed over this area of Miami, leaving 1 inch of water. Estimate What would be the volume of water accumulated over STU main campus and the airport of Opa-Locka. How many time the volume over Opa-Locka airport is bigger than the one obtained at STU.

23. Doppler Radar In order to estimate the area affected by the storm we used Mapquest software. By looking at the map scale we can determine the area, which is about 13 km wide by about 40 km long. The total area A = 13 x 40 = 540 km 2 = 540,000,000 m 2 = 5.4 10 7 m 2 . 0.96 0.31 BSO Courthouse 0.72 0.91 Saint Thomas University 1.44 0.45 Broward General Medical Center 0.44 0.88 North Twin Lakes ES 0.88 1.06 Carol City ES 1.00 1.53 BSO Pembroke Park Rain Rate (in/h) Rain (in) Location

24. Estimate the volume of water in a given column of accumulated precipitation and How many rain drops are in the column of precipitation? For the sake of simplicity, let us assume that rainfall occurs over a flat surface and that it is, almost, the same everywhere. In this case the volume of the accumulated water (V w ) may be shown as: Where A is the area covered by the water and d is the column of accumulated water (depth) expressed either in inches, or millimeters. In order to estimate how many rain drops are in the column, we need to find the volume of a single rain drop. We need a model for a rain drop. If we consider it as a sphere of radius R, the rain drop volume V d Therefore, the number of drops

25. What make some areas being more flooded than others in the event of low infiltration? How far a runoff stream may be a treat for your life and properties around you? The answer to the first part is in the topographic slope. In many towns and cities, there are places located higher than other, thus water flows down by gravity. As water cannot infiltrate, it will sink, forming a stream channel like in a river. How much load will enter into this channel again depends on the urban design or connectivity between streets. Of course a fraction of the total volume of water that fell during the storm will contribute to these runoff streams. Let us estimate what may happen? In the assumption that only 1 % of the above volume will impact a particular region, this number is still large enough 1 % of V W = 0.01 x 12 10 5 m 3 = 12 10 3 m 3 ( six millions plastic 2 litters cans of Coca Cola ). Even at moderate velocities ( 5 mi/h = 2.23 m/s ), the kinetic energy associated with the above mass of water is: 0.5 ρV W v 2 ≈ 24 106 J where ρ is the density of water (1000 kg/m 3 ) is too large, and will be enough to move away cars and anything else the stream meets in between.

26. BS in Mathematics PREREQUISITE REQUIRED COURSES : 19 credits MAT 205 Applied Statistics (3 credits) MAT 232 Calculus I (4 credits) MAT 233 Calculus II (4 credits) CHE 101/L General Chemistry I + Laboratory (4 credits) CHE 102/L General Chemistry II + Laboratory (4 credits) MAJOR REQUIREMENTS : 35 credits total Core Mathematics Courses : (13 credits) MAT 234 Calculus III (4 credits) MAT 306 Differential Equations (3 credits) MAT 311 Linear Algebra (3 credits) MAT 316 Complex Variables (3 credits) Mathematics Electives : (6 credits) Take two mathematics courses at the 300 or 400 level. Computing Requirement : (6 credits) Take two courses. CIS 230 Introduction to Java Programming (3 credits) CIS 235 Introduction to C++ Programming (3 credits) CIS 302 Advanced C++ Programming (3 credits) CIS 310 Advanced Java Programming (3 credits) CIS 360 Data Structures (3 credits) CIS 351 Systems Analysis and Design (3 credits) CIS 430 Database Management Systems (3 credits) Physical Science Requirements : (10 credits) PHY 207/L University Physics I + Laboratory (5 credits) PHY 208/L University Physics II + Laboratory (5 credits) Sub-Total Credits: 54 GENERAL EDUCATION REQUIREMENTS : 42 credits (Program requirements will satisfy 9 credits of the GER.) GENERAL ELECTIVES : 24 credits Total credits: 120

27. Mathematics and Atmospheric Sciences Ongoing research project # 1: The effect of Climate and Weather Variability on Hurricane Dynamics

30. The question is… Can we cross the bridge?