CE 394K.2 HydrologyAtmospheric Water and Precipitation

• Literary quote for today:

“In Köhln, a town of monks and bones,And pavements fang'd with murderous stonesAnd rags, and hags, and hideous wenches;I counted two and seventy stenches,All well defined, and several stinks!Ye nymphs that reign o'er sewers and sinks,The river Rhine, it is well known,Doth wash your city of Cologne;But tell me, nymphs, what power devineShall henceforth wash the river Rhine?”

Samuel Taylor Coleridge, “The City of Cologne”, 1800Contributed by Eric Hersh

Questions for today

(1)  How is net radiation to the earth’s surface partitioned into latent heat, sensible heat and ground heat flux and how does this partitioning vary with location on the earth?

(2) What are the factors that govern the patterns of atmospheric circulation over the earth?

(3)  What are the key variables that describe atmospheric water vapor and how are they connected?

(4)  What causes precipitation to form and what are the factors that govern the rate of precipitation?

(5)  How is precipitation measured and described?

(Some slides in this presentation were prepared by Venkatesh Merwade)

Questions for today

(1)  How is net radiation to the earth’s surface partitioned into latent heat, sensible heat and ground heat flux and how does this partitioning vary with location on the earth?

(2) What are the factors that govern the patterns of atmospheric circulation over the earth?

(3)  What are the key variables that describe atmospheric water vapor and how are they connected?

(4)  What causes precipitation to form and what are the factors that govern the rate of precipitation?

(5)  How is precipitation measured and described?

(Some slides in this presentation were prepared by Venkatesh Merwade)

Heat energy

• Energy– Potential, Kinetic, Internal (Eu)

• Internal energy– Sensible heat – heat content that can be

measured and is proportional to temperature– Latent heat – “hidden” heat content that is

related to phase changes

fhg

Vyz

g

Vyz

22

22

22

21

11

Energy Units

• In SI units, the basic unit of energy is Joule (J), where 1 J = 1 kg x 1 m/s2

• Energy can also be measured in calories where 1 calorie = heat required to raise 1 gm of water by 1°C and 1 kilocalorie (C) = 1000 calories (1 calorie = 4.19 Joules)

• We will use the SI system of units

Energy fluxes and flows

• Water Volume [L3] (acre-ft, m3)

• Water flow [L3/T] (cfs or m3/s)

• Water flux [L/T] (in/day, mm/day)

• Energy amount [E] (Joules)

• Energy “flow” in Watts [E/T] (1W = 1 J/s)

• Energy flux [E/L2T] in Watts/m2

Energy flow of1 Joule/sec

Area = 1 m2

MegaJoules

• When working with evaporation, its more convenient to use MegaJoules, MJ (J x 106)

• So units are– Energy amount (MJ)– Energy flow (MJ/day, MJ/month)– Energy flux (MJ/m2-day, MJ/m2-month)

Internal Energy of Water

0

1

2

3

4

-40 -20 0 20 40 60 80 100 120 140

Temperature (Deg. C)

Inte

rna

l En

erg

y (

MJ

)

Heat Capacity (J/kg-K) Latent Heat (MJ/kg)Ice 2220 0.33Water 4190 2.5

Ice

Water

Water vapor

Water may evaporate at any temperature in range 0 – 100°CLatent heat of vaporization consumes 7.6 times the latent heat of fusion (melting)

2.5/0.33 = 7.6

Latent heat flux

• Water flux– Evaporation rate, E

(mm/day)

• Energy flux – Latent heat flux

(W/m2), Hl

Area = 1 m2

ElH vl = 1000 kg/m3

lv = 2.5 MJ/kg

)/)(1000/1(*)/)(86400/1(*/1)/(105.2)/(1000/ 632 mmmsdaydaymmkgJmkgmW

28.94 W/m2 = 1 mm/day

Radiation

• Two basic laws– Stefan-Boltzman Law

• R = emitted radiation (W/m2)

= emissivity (0-1) = 5.67x10-8W/m2-K4

• T = absolute temperature (K)

– Wiens Law = wavelength of

emitted radiation (m)

4TR

T

310*90.2

Hot bodies (sun) emit short wave radiationCool bodies (earth) emit long wave radiation

All bodies emit radiation

Net Radiation, Rn

Ri Incoming Radiation

Ro =Ri Reflected radiation

albedo (0 – 1)

Rn Net Radiation

Re

ein RRR )1(

Average value of Rn over the earth and over the year is 105 W/m2

Net Radiation, Rn

Rn Net Radiation

GLEHRn

Average value of Rn over the earth and over the year is 105 W/m2

G – Ground Heat Flux

LE – EvaporationH – Sensible Heat

http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/energy/radiation_balance.html

Energy Balance of Earth

6

4

10070

51

21

26

38

6

20

15

Sensible heat flux 7Latent heat flux 23

19

Energy balance at earth’s surfaceDownward short-wave radiation, Jan 2003

600Z

Energy balance at earth’s surfaceDownward short-wave radiation, Jan 2003

900Z

Energy balance at earth’s surfaceDownward short-wave radiation, Jan 2003

1200Z

Energy balance at earth’s surfaceDownward short-wave radiation, Jan 2003

1500Z

Energy balance at earth’s surfaceDownward short-wave radiation, Jan 2003

1800Z

Energy balance at earth’s surfaceDownward short-wave radiation, Jan 2003

2100Z

Latent heat flux, Jan 2003, 1500z

Questions for today

(1)  How is net radiation to the earth’s surface partitioned into latent heat, sensible heat and ground heat flux and how does this partitioning vary with location on the earth?

(2) What are the factors that govern the patterns of atmospheric circulation over the earth?

(3)  What are the key variables that describe atmospheric water vapor and how are they connected?

(4)  What causes precipitation to form and what are the factors that govern the rate of precipitation?

(5)  How is precipitation measured and described?

(Some slides in this presentation were prepared by Venkatesh Merwade)

Heating of earth surface• Heating of earth

surface is uneven– Solar radiation strikes

perpendicularly near the equator (270 W/m2)

– Solar radiation strikes at an oblique angle near the poles (90 W/m2)

• Emitted radiation is more uniform than incoming radiation

Amount of energy transferred from equator to the poles is approximately 4 x 109 MW

Hadley circulation

Warm air rises, cool air descends creating two huge convective cells.

Coriolis ForceCone is moving southward towards the pole

Camera fixed in the outer space (cone appears moving straight)

Camera fixed on to the globe (looking southward, cone appears deflecting to the right)

the force that deflects the path of the wind on account of earth rotation is called Coriolis force. The path of the wind is deflected to the right in the Northern Hemisphere and the to left in the Southern Hemisphere.

Atmospheric circulation

1. Tropical Easterlies/Trades

2. Westerlies

3. Polar easterlies

1. Intertropical convergence zone (ITCZ)/Doldrums

2. Horse latitudes

3. Subpolar low

4. Polar high

Ferrel Cell

Polar Cell 1. Hadley cell

2. Ferrel Cell

3. Polar cell

Latitudes

Winds

Circulation cells

Effect of land mass distribution

A) Idealized winds generated by pressure gradient and Coriolis Force. B) Actual wind patterns owing to land mass distribution

Uneven distribution of land and ocean, coupled with different thermal properties creates spatial variation in atmospheric circulation

Shifting in Intertropical Convergence Zone (ITCZ)

Owing to the tilt of the Earth's axis in orbit, the ITCZ shifts north and south. 

Southward shift in January

Northward shift in July

Creates wet Summers (Monsoons) and dry winters, especially in India and SE Asia

ITCZ movement

http://iri.ldeo.columbia.edu/%7Ebgordon/ITCZ.html

Questions for today

(1)  How is net radiation to the earth’s surface partitioned into latent heat, sensible heat and ground heat flux and how does this partitioning vary with location on the earth?

(2) What are the factors that govern the patterns of atmospheric circulation over the earth?

(3)  What are the key variables that describe atmospheric water vapor and how are they connected?

(4)  What causes precipitation to form and what are the factors that govern the rate of precipitation?

(5)  How is precipitation measured and described?

(Some slides in this presentation were prepared by Venkatesh Merwade)

Structure of atmosphere

Atmospheric water

• Atmospheric water exists – Mostly as gas or water vapor– Liquid in rainfall and water droplets in clouds– Solid in snowfall and in hail storms

• Accounts for less than 1/100,000 part of total water, but plays a major role in the hydrologic cycle

Water vaporSuppose we have an elementary volume of atmosphere dV and

we want quantify how much water vapor it contains

Atmospheric gases:Nitrogen – 78.1%Oxygen – 20.9%Other gases ~ 1%

http://www.bambooweb.com/articles/e/a/Earth's_atmosphere.html

dV

ma = mass of moist airmv = mass of water vapor

dV

mvv Water vapor density

dV

maa Air density

Specific Humidity, qv

• Specific humidity measures the mass of water vapor per unit mass of moist air

• It is dimensionlessa

vvq

Vapor pressure, e

• Vapor pressure, e, is the pressure that water vapor exerts on a surface

• Air pressure, p, is the total pressure that air makes on a surface

• Ideal gas law relates pressure to absolute temperature T, Rv is the gas constant for water vapor

• 0.622 is ratio of mol. wt. of water vapor to avg mol. wt. of dry air

TRe vv

p

eqv 622.0

Dalton’s Law of Partial PressuresJohn Dalton studied the effect of gases in a mixture. He observed that the Total Pressure of a gas mixture was the sum of the Partial Pressure of each gas.

P total = P1 + P2 + P3 + .......Pn

The Partial Pressure is defined as the pressure of a single gas in the mixture as if that gas alone occupied the container. In other words, Dalton maintained that since there was an enormous amount of space between the gas molecules within the mixture that the gas molecules did not have any influence on the motion of other gas molecules, therefore the pressure of a gas sample would be the same whether it was the only gas in the container or if it were among other gases.

http://members.aol.com/profchm/dalton.html

Avogadro’s lawEqual volumes of gases at the same temperature and pressure contain the same number of molecules regardless of their chemical nature and physical properties. This number (Avogadro's number) is 6.023 X 1023 in 22.41 L for all gases.

Dry airWater vapor

Dry air ( z = x+y molecules)

Moist air (x dry and y water vapor)

d = (x+y) * Md/Volume m = (x* Md + y*Mv)/Volume

m < d, which means moist air is lighter than dry air!

Saturation vapor pressure, es

Saturation vapor pressure occurs when air is holding all the water vaporthat it can at a given air temperature

T

Tes 3.237

27.17exp611

Vapor pressure is measured in Pascals (Pa), where 1 Pa = 1 N/m2

1 kPa = 1000 Pa

Relative humidity, Rh

es

e

sh e

eR Relative humidity measures the percent

of the saturation water content of the airthat it currently holds (0 – 100%)

Dewpoint Temperature, Td

e

Dewpoint temperature is the air temperatureat which the air would be saturated with its current vapor content

TTd

Water vapor in an air column

• We have three equations describing column:– Hydrostatic air pressure,

dp/dz = -ag– Lapse rate of temperature,

dT/dz = - – Ideal gas law, p = aRaT

• Combine them and integrate over column to get pressure variation elevation

Column

Element, dz

aRg

T

Tpp

/

1

212

1

2

Precipitable Water

• In an element dz, the mass of water vapor is dmp

• Integrate over the whole atmospheric column to get precipitable water,mp

• mp/A gives precipitable water per unit area in kg/m2

Column

Element, dz

1

2

Adzqdm avp

Area = A

Precipitable Water, Jan 2003

Precipitable Water, July 2003

January

July

Questions for today

(1)  How is net radiation to the earth’s surface partitioned into latent heat, sensible heat and ground heat flux and how does this partitioning vary with location on the earth?

(2) What are the factors that govern the patterns of atmospheric circulation over the earth?

(3)  What are the key variables that describe atmospheric water vapor and how are they connected?

(4)  What causes precipitation to form and what are the factors that govern the rate of precipitation?

(5)  How is precipitation measured and described?

(Some slides in this presentation were prepared by Venkatesh Merwade)

Precipitation

• Precipitation: water falling from the atmosphere to the earth.– Rainfall– Snowfall– Hail, sleet

• Requires lifting of air mass so that it cools and condenses.

Mechanisms for air lifting

1. Frontal lifting

2. Orographic lifting

3. Convective lifting

Definitions

• Air mass : A large body of air with similar temperature and moisture characteristics over its horizontal extent.

• Front: Boundary between contrasting air masses.

• Cold front: Leading edge of the cold air when it is advancing towards warm air.

• Warm front: leading edge of the warm air when advancing towards cold air.

Frontal Lifting• Boundary between air masses with different properties is

called a front• Cold front occurs when cold air advances towards warm air• Warm front occurs when warm air overrides cold air

Cold front (produces cumulus cloud)

Cold front (produces stratus cloud)

Orographic liftingOrographic uplift occurs when air is forced to rise because of the physical presence of elevated land.

Convective lifting

Hot earth surface

Convective precipitation occurs when the air near the Convective precipitation occurs when the air near the ground is heated by the earth’s warm surface. This warm ground is heated by the earth’s warm surface. This warm air rises, cools and creates precipitation. air rises, cools and creates precipitation.

Condensation

• Condensation is the change of water vapor into a liquid. For condensation to occur, the air must be at or near saturation in the presence of condensation nuclei.

• Condensation nuclei are small particles or aerosol upon which water vapor attaches to initiate condensation. Dust particulates, sea salt, sulfur and nitrogen oxide aerosols serve as common condensation nuclei.

• Size of aerosols range from 10-3 to 10 m.

Precipitation formation• Lifting cools air masses

so moisture condenses• Condensation nuclei

– Aerosols – water molecules

attach• Rising & growing

– 0.5 cm/s sufficient to carry 10 m droplet

– Critical size (~0.1 mm)

– Gravity overcomes and drop falls

Forces acting on rain drop

FdFd

Fb

Fg

D• Three forces acting on rain drop– Gravity force due to weight– Buoyancy force due to

displacement of air– Drag force due to friction

with surrounding air3

6DVolume

2

4DArea

3

6DgF wg

3

6DgF ab

242

22

2 VDC

VACF adadd

Terminal Velocity• Terminal velocity: velocity at which the forces acting on the raindrop

are in equilibrium.• If released from rest, the raindrop will accelerate until it reaches its

terminal velocity

32

23

6246

0

DgV

DCDg

WFFF

wada

DBvert

332

2

6624DgDg

VDC

WFF

wat

ad

BD

1

34

a

w

dt C

gDV

• Raindrops are spherical up to a diameter of 1 mm• For tiny drops up to 0.1 mm diameter, the drag force is specified by

Stokes law

FdFd

Fb

Fg

D

V

Re

24dCa

aVD

Re

At standard atmospheric pressure (101.3 kpa) and temperature (20oC), w = 998 kg/m3 and a = 1.20 kg/m3

Precipitation Variation

• Influenced by – Atmospheric circulation and local factors

• Higher near coastlines

• Seasonal variation – annual oscillations in some places

• Variables in mountainous areas

• Increases in plains areas

• More uniform in Eastern US than in West

Rainfall patterns in the US

Global precipitation pattern

Spatial Representation• Isohyet – contour of constant rainfall• Isohyetal maps are prepared by

interpolating rainfall data at gaged points.

Austin, May 1981 Wellsboro, PA 1889

Texas Rainfall Maps

Temporal Representation

• Rainfall hyetograph – plot of rainfall depth or intensity as a function of time

• Cumulative rainfall hyetograph or rainfall mass curve – plot of summation of rainfall increments as a function of time

• Rainfall intensity – depth of rainfall per unit time

Rainfall Depth and IntensityTime (min) Rainfall (in) Cumulative 30 min 1 h 2 h

Rainfall (in)0 05 0.02 0.0210 0.34 0.3615 0.1 0.4620 0.04 0.525 0.19 0.6930 0.48 1.17 1.1735 0.5 1.67 1.6540 0.5 2.17 1.8145 0.51 2.68 2.2250 0.16 2.84 2.3455 0.31 3.15 2.4660 0.66 3.81 2.64 3.8165 0.36 4.17 2.5 4.1570 0.39 4.56 2.39 4.275 0.36 4.92 2.24 4.4680 0.54 5.46 2.62 4.9685 0.76 6.22 3.07 5.5390 0.51 6.73 2.92 5.5695 0.44 7.17 3 5.5100 0.25 7.42 2.86 5.25105 0.25 7.67 2.75 4.99110 0.22 7.89 2.43 5.05115 0.15 8.04 1.82 4.89120 0.09 8.13 1.4 4.32 8.13125 0.09 8.22 1.05 4.05 8.2130 0.12 8.34 0.92 3.78 7.98135 0.03 8.37 0.7 3.45 7.91140 0.01 8.38 0.49 2.92 7.88145 0.02 8.4 0.36 2.18 7.71150 0.01 8.41 0.28 1.68 7.24Max. Depth 0.76 3.07 5.56 8.2Max. Intensity 9.12364946 6.14 5.56 4.1

Running Totals

Incremental Rainfall

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

Time (min)

Incr

emen

tal

Rai

nfa

ll (

in p

er 5

min

)

Rainfall Hyetograph

Cumulative Rainfall

0

1

2

3

4

5

6

7

8

9

10

0 30 60 90 120 150

Time (min.)

Cu

mu

lati

ve R

ain

fall

(in

.)

30 min

1 hr

2 hr

3.07 in

5.56 in

8.2 in

Rainfall Mass Curve

Arithmetic Mean Method• Simplest method for determining areal average

P1

P2

P3

P1 = 10 mm

P2 = 20 mm

P3 = 30 mm

• Gages must be uniformly distributed• Gage measurements should not vary greatly about the mean

N

iiPN

P1

1

mmP 203

302010

Thiessen polygon method

P1

P2

P3

A1

A2

A3

• Any point in the watershed receives the same amount of rainfall as that at the nearest gage

• Rainfall recorded at a gage can be applied to any point at a distance halfway to the next station in any direction

• Steps in Thiessen polygon method1. Draw lines joining adjacent gages

2. Draw perpendicular bisectors to the lines created in step 1

3. Extend the lines created in step 2 in both directions to form representative areas for gages

4. Compute representative area for each gage

5. Compute the areal average using the following formula

N

iiiPAA

P1

1

P1 = 10 mm, A1 = 12 Km2

P2 = 20 mm, A2 = 15 Km2

P3 = 30 mm, A3 = 20 km2

mmP 7.2047

302020151012

Isohyetal method

P1

P2

P3

10

20

30

• Steps– Construct isohyets (rainfall

contours)– Compute area between each

pair of adjacent isohyets (Ai)– Compute average

precipitation for each pair of adjacent isohyets (pi)

– Compute areal average using the following formula

M

iii pAP

1

A1=5 , p1 = 5

A2=18 , p2 =

15

A3=12 , p3 =

25

A4=12 , p3 = 35

mmP 6.2147

35122512151855

N

iiiPAA

P1

1

Inverse distance weighting

P1=10

P2= 20

P3=30

• Prediction at a point is more influenced by nearby measurements than that by distant measurements

• The prediction at an ungaged point is inversely proportional to the distance to the measurement points

• Steps– Compute distance (di) from ungaged

point to all measurement points.

– Compute the precipitation at the ungaged point using the following formula

N

i i

N

i i

i

d

d

P

P

12

12

1ˆ

d1=25

d2=15

d3=10

mmP 24.25

101

151

251

10

30

15

20

25

10

ˆ

222

222

p

221

22112 yyxxd

Rainfall interpolation in GIS

• Data are generally available as points with precipitation stored in attribute table.

Rainfall maps in GIS

Nearest Neighbor “Thiessen” Polygon Interpolation

Spline Interpolation

NEXRAD

NEXRAD Tower

• NEXt generation RADar: is a doppler radar used for obtaining weather information

• A signal is emitted from the radar which returns after striking a rainfall drop

• Returned signals from the radar are analyzed to compute the rainfall intensity and integrated over time to get the precipitation

Working of NEXRAD

NEXRAD data

• NCDC data (JAVA viewer)– http://www.ncdc.noaa.gov/oa/radar/jnx/

• West Gulf River Forecast Center– http://www.srh.noaa.gov/wgrfc/

• National Weather Service Animation– http://weather.noaa.gov/radar/mosaic.loop/DS.p19r0/ar.us.conus.shtml