The mathematics of rain and flooding: Learning The mathematics of rain and flooding: Learning about volumes, ratios, and fractalsabout volumes, ratios, and fractals

D. QuesadaD. QuesadaSchool of Science, Technology and Engineering ManagementSchool of Science, Technology and Engineering Management

St. Thomas University, Miami Gardens, FL 33054St. Thomas University, Miami Gardens, FL 33054

Composition of the AtmosphereComposition of the Atmosphere

Formation of current EarthFormation of current Earth’’s Atmosphere: Outs Atmosphere: Out--gassinggassingAs 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 CO2 were dissolved in the oceans. Through chemical and Biological processes, much of the CO2 became locked up in carbonate sedimentary rocks, such as limestone. With much of the water vapor already condensed and the concentration of CO2 dwindling,the atmosphere gradually became rich in nitrogen (N2), which is usually not chemically active.

Chemical Properties of Water Chemical Properties of Water

The polarity of water allows itto “hook up” with othermolecules, including itself.More substances dissolve inwater than any other liquid.for this reason, water is oftencalled 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. Snowflakeshave 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.

Physical Properties of WaterPhysical Properties of Water1. It is colorless, tasteless, and odorless2. It feels wet3. It is distinctive in sound when dripping from a faucet or crashing

as a wave4. It dissolves nearly everything5. It exist in three forms: liquid, solid, and gas6. 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 thesurface water about a billion times,we would see water moleculesfairly close together, jiggling, bouncing,and moving about. We would also seethat molecules are not all moving at thesame speed. At the surface, molecules With enough speed would occasionally break away from the liquid surface and enter into the air above. TheseMolecules, changing from the Liquid state into the vapor state, are Evaporating. While some watermolecules are leaving the liquid, othersare returning. Those returning are Condensing as they are changingfrom a vapor state to a liquid state.The air is said to be Saturated withwater vapor when for every molecule that evaporates, one must condense, and no net loss of liquid or vapormolecules results.

Factors affecting Evaporation

• Wind enhance evaporation: Some ofthe vapor molecules already in theair above would be blown away,creating a difference between actualnumber of vapor molecules and thetotal number required for saturation. • Temperature of the Water: Warmwater will evaporate more readily thancold water.• Chemical composition of the waterSolution: Depending on how clean or pure is the water solution, the ability tobe warmed changes. A salty water ismore readily warmed that a freshwater.

Concept of Humidity

Refers to any of a number of ways of specifying the amount of water vapor in air.

Volume of air in a thin elastic container – a ParcelParcel – containing the most common gases, Nitrogen, and Oxygen, and also water vapor.

According to the Laws of Nature, molecules inside thiscontainer are in constant motion, thus they have a given amount of kinetic energy. The average value of this kinetic energy is what we call Temperature. Therefore, if the Parcel is being warmed, molecules in the interior will move more, and will acquire more kinetic energy, and also exert a largeforce 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 WaterVapor Density, or Absolute Humidity [g/m3].

2. We could compare the weight (mass – M) of the water vapor in the parcel with the total weight (mass – MT) of all the air in theparcel (including vapor) and obtain the Specific Humidity [g/kg].

3. We could compare the weight (mass – M) of the water vapor in the parcel with the weight (mass – M’) of the remaining dry air, and obtain the Mixing Ratio [g/kg].

33 4

3

34 R

M

R

MVM vaporvapor

parcel

vapor

ππρ ===

Water Vapor PressureThe 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 thewalls of the parcel. The total pressure inside the parcel is equal tothe sum of the pressures of the individualgases.

The air pressure near sea level is the same asinside 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 pressuresexerted by these gases are:Nitrogen – 780 mbOxygen – 210 mbWater Vapor – 10 mbAn increase in the number of water moleculeswill 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 howmuch water vapor is necessary to make the air saturated atany given temperature.

The temperature at which the relative humidity reaches 100 % andCondensation starts is called the Dew Point.

When the air becomes saturated with water vapor (thatis, 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 wateror an ice crystal to form, energyis needed. The process is calledNucleation, and energy is requiredbecause a new surface is formed.

As an unsaturated mass of airrises, it expands and cools at thedry adiabatic rate 10o C/km. Whenthe air temperature falls to thepoint where the air is saturated,Condensation commences andlatent heat is released

Snowflakes and FractalsSnowflakes and Fractals

Climate

The average weather patterns for an area over a longperiod 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

N

xx

N

ii∑

== 1

It is determined by

and

Which are influenced by

And affects

Planetary Biology and InterPlanetary Biology and Inter--species interactionsspecies interactionsThe Web of LifeThe Web of Life

The appearance of plants and otherliving 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 UNITSPHYSICAL QUANTITIES AND UNITSObservations produce qualitative

information about a system

Measurements produce quantitative information which is needed in any

science that strives for exactness

English UnitsInch (in)

Second (s)Pound (lb)

Metric System Meter (m)Second (s)

Kilogram (kg)

Fundamental Physical QuantitiesDistance - Time - Mass

Scientific NotationPrefix | Abbreviation | Regular Notation | Scientific NotationTera 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

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

THE HYDROSPHERE:THE HYDROSPHERE:WATER ON THE LANDWATER 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 circulation5. El Niño/Southern Oscillation6. 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.

HH22O WaterO Water’’s Chemical Formulas Chemical Formula

97 % in Oceans

Hydrologic CycleHydrologic Cycle

Water temperatures across the world oceans depends on the amount of energy received from the Sun as well as fromthe 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 stopflowing around the world, results may be readily catastrophic.

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

Hydrologic Cycle over Land in DetailsHydrologic Cycle over Land in Details

Transpiration and Photosynthesis

Percolation versus Porosity Percolation versus Porosity

Which factors determine the groundwater motion, Which factors determine the groundwater motion, and how far they are responsible for flooding?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 PorosityPermeability over Porosity, and the second one Rise over RunRise over Run. PorosityPorosity is the proportion of void space in the material – holes or cracks unfilled by solid material, within or between individual grains. PermeabilityPermeability is a measure of how readily fluids pass through the material and is related to the extent to which pores or cracks are interconnected.

Lh

pK

cedisheadindifference

porositytypermeabili

velocityrgroundwate

⋅=

=⋅=tan

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.

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.

ClayClay (45 – 55 %, less than 0.01 m/day) Fine SandFine Sand (30 – 52 %, 0.01 – 10 m/day) GravelGravel (25 – 40 %, 1000 – 10,000 m/day) Sandstone Sandstone (5 – 30 %, 0.3 – 3 m/day)

http://geoscape.nrcan.gc.ca/h2o/bowen/images/groundwater_e.jpg

Planar FiguresPerimeter Areas

a

a

a

a

a

b

r

P = 4 a A = a2

P = 2a + 2b A = a b

P = 3a A = 0.5 a h

C = 2π r A = π r2

Volumes Surface Areas

V = a3 S = 6a2

V = π r2 h S = 2π r2 + 2π r h

V = 4 π r3 / 3 S = 4 π r2

Solid Bodies

Basic Geometrical Formulas

Practice Activity about Areas and Fractions1. Determine the Area of Saint Thomas University main campus by looking at MapQuest

Directions1. Look at the upper right corner.2. Write down the scale of the map inboth units, meters (m) and feet (ft).3. Determine the length and the widthof 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 goInto 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 isbigger 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

North Twin Lakes ES 0.88 0.44

Broward General Medical Center 0.45 1.44

Saint Thomas University 0.91 0.72

BSO Courthouse 0.31 0.96

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 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 (Vw) may be shown as:

dbadAVw ⋅⋅=⋅= )(

3

34 RVd ⋅=π

333

)(24.04

)(3

34

)(R

dbaR

dba

R

dbaVV

Nd

w ⋅⋅⋅=

⋅⋅⋅⋅

=⋅

⋅⋅==

ππ

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

Therefore, the number of drops

What make some areas being more flooded than others What make some areas being more flooded than others in the event of low infiltration? How far a runoff stream in the event of low infiltration? How far a runoff stream may be a treat for your life and properties around you?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 willsink, 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?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 VW = 0.01 x 12 105 m3 = 12 103 m3 (six millions plastic 2 litters cans of six millions plastic 2 litters cans of Coca ColaCoca Cola). Even at moderate velocities (5 mi/h = 2.23 5 mi/h = 2.23 m/sm/s), the kinetic energy associated with the above mass of water is:

0.5 ρVW v2 ≈ 24 106 J

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 MathematicsPREREQUISITE REQUIRED COURSES: 19 creditsMAT 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 Ongoing research project # 1: The effect of Climate and Weather Variability Variability on Hurricane Dynamicson Hurricane Dynamics

Ongoing research project # 2: Asthma Ongoing research project # 2: Asthma –– Weather connectionWeather connectionAir Quality and Respiratory disorders: Modeling asthma attacks considering the environmental triggers, the mechanics of lung functioning, immune response and genetic factors.

Asthma Statistics Worldwide: A brief overview# of people diagnosed: more than 150 MEurope: the # of cases has doubledUSA: the # of cases has increased more than 60%India: between 15 and 20 MAfrica: between 11 and 18% population# of deaths yearly: around 180,000Miami Dade County – 7.1% Middle and HS children

were reported with asthmaThe # of hospitalizations due to asthma has doubled.The # 1 cause of school absences and 35 % of parents missed work

Urban Heat Island EffectMan is likely playing a role in climate change through urbanization and land use changes competing with greenhouse Gases and cycles of Nature

• In cities, vertical walls, steel and concrete absorb the sun’s heat and are slow to cool at night

• Nights may be 10 or more degrees warmer in and near cities than in rural areas some nights

• 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.

The question isThe question is……

Can we cross the bridge?Can we cross the bridge?