The mathematics of rain and flooding: Learning about volumes, ratios, and fractals D. Quesada School of Science, Tec T hnology and Engineering Management St. Thomas University, ty Miami Gardens, FL 33054
Composition of the Atmosphere
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 were dissolved in the 2 oceans. Through chemical and Biological processes, much of the CO became locked up in carbonate sedimentary 2 rocks, such as limestone. With much of the water vapor already condensed and the concentration of CO dwindling, 2 the atmosphere gradually became rich in nitrogen (N ), which is usually not chemically active. 2
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
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 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 concentr pH = - Log([H+]) ation, in moles per liter.
Physical Properties of Water 1. It is colorless, tasteless, and odorless 2. It feels wet 3. It is distinctive in sound when dripping from a faucet or crashing as a wave 4. It dissolves nearly everything 5. It exist in three forms: liquid, solid, and gas 6. It can absorb a great amount of heat (Specific heat of water) 7. It sticks together into beads or drops (Nuclei of Condensation) If we were able to magnify the surface water about a bil ion times, we would see water molecules fairly close together, jiggling, bouncing, and moving about. We would also see Factors affecting Evaporation that molecules are not all moving at the same speed. At the surface, molecules With enough speed would occasionally • Wind enhance evaporation: Some of break away from the liquid surface the vapor molecules already in the and enter into the air above. These air above would be blown away, Molecules, changing from the Liquid creating a difference between actual state into the vapor state, are number of vapor molecules and the Evaporating. While some water total number required for saturation. molecules are leaving the liquid, others • Temperature of the Water: Warm are returning. Those returning are water wil evaporate more readily than Condensing as they are changing cold water. from a vapor state to a liquid state. • Chemical composition of the water The air is said to be Saturated with Solution: Depending on how clean or water vapor when for every molecule pure is the water solution, the ability to that evaporates, one must condense, be warmed changes. A salty water is and no net loss of liquid or vapor more readily warmed that a fresh
Concept of Humidity According to the Laws of Nature, molecules inside this container are in constant motion, thus they have a given Refers to any of a number of ways of amount of kinetic energy. The average value of this kinetic specifying the amount of water vapor energy is what we call Temperature. Therefore, if the Parcel in air. is being warmed, molecules in the interior will move more, and will acquire more kinetic energy, and also exert a large force onto the walls of this container. If we would extract the water vapor content from the parcel, We would specify Humidity in the following ways: 1. We could compare the weight (mass – M) of the water vapor with the Volume – V of air in the parcel and obtain the Water Vapor Density, or Absolute Humidity [g/m3]. M M 3M vapor vapor vapor
3 V 4 3 4 R
parcel R 3 2. We could compare the weight (mass – M) of the water vapor in the parcel with the total weight (mass – M ) of all the air in the T parcel (including vapor) and obtain the Specific Humidity [g/kg]. Volume of air in a thin elastic container
– a Parcel – containing the most 3. We could compare the weight (mass – M) of the water vapor in common gases, Nitrogen, and Oxygen, the parcel with the weight (mass – M’) of the remaining dry air, and also water vapor. and obtain the Mixing Ratio [g/kg].
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 Actual vapor pressure indicates that air’s total water vapor numbers of water vapor molecules, whereas content, whereas saturation vapor pressure describes how low actual vapor pressure indicates small much water vapor is necessary to make the air saturated at numbers of vapor molecules. any given temperature.
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 10o C/km. When the air temperature falls to the point where the air is saturated, Condensation commences and latent heat is released
Snowflakes and Fractals
Climate Average N x i x i 1 The average weather patterns for an area over a long N period of time (at least 30 years, and above – 1,000,000 years) It is determined by Average Precipitation and Average Temperature Which are influenced by Latitude Altitude Ocean currents And affects Where people live? How people live? What they grow and eat?
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.
PHYSICAL QUANTITIES AND UNITS Observations produce qualitative Scientific Notation information about a system Prefix | Abbreviation | Regular Notation | Scientific Notation Tera T 1,000,000,000,000 = 1012 Giga G 1,000,000,000 = 109 Mega M 1,000,000 = 106 Kilo k 1,000 = 103 Hecto h 1,00 = 102 Deca da 10 = 101 -------- ---------- 1 = 100 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 Measurements produce quantitative information which is needed in any Length: 1 kilometer (km) = 1000 meters (m) = 3281 feet (ft) = 0.62 miles (mi) science that strives for exactness 1 mile (mi) = 5280 feet (ft) = 1.61 kilometers (km) = 0.87 nautical mile (nm) 1 centimeter (cm) = 0.39 inch (in) Fundamental Physical Quantities 1 inch (in) = 2.54 centimeters (cm) 1 yard (yd) = 3 feet (ft) = 36 inches (in) Distance - Time - Mass Time: 1 hour (hr) = 60 minutes (min) = 3600 seconds (s) Mass: 1 kilogram (kg) = 1000 grams (g) = 2.2 pounds (lb) English Units Metric System Speed (rate of change of a coordinate in time): Inch (in) Meter (m) 1 knot (kt) = 1 nautical mile per hour (nmph) = 1.15 miles per hour (mph) Second (s) Second (s) 1 mile per hour (mph) = 1.61 kilometers per hour (km/hr) = 0.45 m/s Pound (lb) Kil ogram (kg)
THE HYDROSPHERE: 97 % in Oceans WA W T A ER ON THE LAND Liquid water makes Earth unique in the solar system. Although water has been 1. Water and the hydrologic cycle. Streams and their channels. detected on other bodies of 2. Water in the ground. Porosity and permeability. the solar system, it does not 3. Glaciers. appear to be present as a 4. The Oceans. Ocean circulation liquid anywhere except on 5. El Niño/Southern Oscillation our planet. 6. Ocean waves and ocean tides H O Water’s Chemical Formula Hydrologic Cycle 2
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.
Composition of Sea Water, r and Fresh Water: Distribution of Elements over Land and Oceans.
Hydrologic Cycle over Land in Details Percolation versus Porosity Transpiration and Photosynthe sis
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: groundwater permeabil tiy difference in head
velocity porosity dis tan ce K h
p L In the Darcy’s Laws, we might identify two ratios, the first one Compared to the rapid flow of water in Permeability over Porosity, and the second one Rise over Run. surface streams, most ground water moves Porosity is the proportion of void space in the material – holes or relative slowly through rocks underground. cracks unfilled by solid material, within or between individual Because it moves in response to differences grains. Permeability is a measure of how readily fluids pass in water pressure and elevation, water within through the material and is related to the extent to which pores or the saturated zone tends to move downward cracks are interconnected. 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.
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)
Basic Geometrical Formulas Planar Figures Perimeter Areas Solid Bodies a P = 4 a A = a2 Volumes Surface Areas V = a3 S = 6a2 a P = 2a + 2b A = a b b V = π r2 h S = 2π r2 + 2π r h a a P = 3a A = 0.5 a h a V = 4 π r3 / 3 S = 4 π r2 r C = 2π r A = π r2
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.
Doppler Radar Location Rain (in) Rain Rate (in/h) BSO Pembroke Park 1.53 1.00 Carol City ES 1.06 0.88 In order to estimate the area affected by the North Twin Lakes ES 0.88 0.44 storm we used Mapquest software. Broward General Medical Center 0.45 1.44 By looking at the map scale we can determine the area, which is about 13 km wide by about Saint Thomas University 0.91 0.72 40 km long. The total area A = 13 x 40 = 540 BSO Courthouse 0.31 0.96 km2 = 540,000,000 m2 = 5.4 107 m2.
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 ) may be w shown as: V A d (a b) d w 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 Vd 4 3 V R d Therefore, the number of drops 3 V
w (a b) d 3 (a b) d (a b) d N
2 . 0 4 3 3 V 4 3 4 R R d R
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 12 105 m3 = 12 103 m3 (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 v2 ≈ 24 106 J W where ρ is the density of water (1000 kg/m3) is too large, and will be enough to move away cars and anything else the stream meets in between.
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
Ongoing research project # 1: The effect of Climate and Weather Va V riability on Hurricane Dynamics
Ongoing research project # 2: Asthma – Weather connection Air Quality and Respiratory disorders: Modeling asthma attacks considering the environmental triggers, the mechanics of lung functioning, immune response and genetic factors. Urban Heat Island Effect Asthma Statistics Worldwide: A brief overview # of people diagnosed: more than 150 M Man is likely playing a role in climate Europe: the # of cases has doubled change through urbanization and land USA: the # of cases has increased more than 60% use changes competing with greenhouse India: between 15 and 20 M Gases and cycles of Nature Africa: between 11 and 18% population • In cities, vertical walls, steel and concrete # of deaths yearly: around 180,000 absorb the sun’s heat and are slow to cool Miami Dade County – 7.1% Middle and HS children at night were reported with asthma • Nights may be 10 or more degrees warmer The # of hospitalizations due to asthma has doubled. in and near cities than in rural areas some The # 1 cause of school absences and 35 % of parents nights missed work • Temperatures measured in cities increase as they grow.
Science & Mathematics Fellows Program • Start Date: August 2008 - 30 freshmen & 30 juniors who transfer with an AA. • Qualified students may receive financial aid and academic scholarships. • Research based in state of the art Science & Technology facility.