CE 394K.2 Hydrology
Atmospheric Water and Precipitation
• Literary quote for today:
“In Köhln, a town of monks and bones,
And pavements fang'd with murderous stones
And 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 devine
Shall henceforth wash the river Rhine?”
Samuel Taylor Coleridge, “The City of Cologne”, 1800
Contributed 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
V12
V22
z1  y1 
 z 2  y2 
 hf
2g
2g
– 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
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 of
1 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
Internal Energy (MJ)
4
Water vapor
3
2
Water
1
Ice
-40
-20
0
0
20
40
60
80
100
120
140
Temperature (Deg. C)
Ice
Water
Heat Capacity (J/kg-K)
2220
4190
Latent Heat (MJ/kg)
0.33
2.5/0.33 = 7.6
2.5
Water may evaporate at any temperature in range 0 – 100°C
Latent heat of vaporization consumes 7.6 times the latent heat of fusion (melting)
Latent heat flux
• Water flux
• Energy flux
– Evaporation rate, E
(mm/day)
 = 1000 kg/m3
lv = 2.5 MJ/kg
– Latent heat flux
(W/m2), Hl
H l  lv E
W / m 2  1000(kg / m3 )  2.5 106 ( J / kg) 1mm / day * (1 / 86400)( day / s) * (1 / 1000)( mm / m)
28.94 W/m2 = 1 mm/day
Area = 1 m2
Radiation
• Two basic laws
– Stefan-Boltzman Law
• R = emitted radiation
(W/m2)
 e = emissivity (0-1)
 s = 5.67x10-8W/m2-K4
• T = absolute
temperature (K)
– Wiens Law
 l = wavelength of
emitted radiation (m)
R  esT
4
All bodies emit radiation
2.90 *10
l
T
3
Hot bodies (sun) emit short wave radiation
Cool bodies (earth) emit long wave radiation
Net Radiation, Rn
Rn  Ri (1  a )  Re
Ri Incoming Radiation
Re
Ro =aRi Reflected radiation
a albedo (0 – 1)
Rn Net Radiation
Average value of Rn over the earth and
over the year is 105 W/m2
Net Radiation, Rn
Rn  H  LE  G
H – Sensible Heat
LE – Evaporation
G – Ground Heat Flux
Rn Net Radiation
Average value of Rn over the earth and
over the year is 105 W/m2
Energy Balance of Earth
6
70
20
100
6
26
4
38
15
19
21
51
Sensible heat flux 7
Latent heat flux 23
http://www.uwsp.edu/geo/faculty/ritter/geog101/textbook/energy/radiation_balance.html
Energy balance at earth’s surface
Downward short-wave radiation, Jan 2003
600Z
Energy balance at earth’s surface
Downward short-wave radiation, Jan 2003
900Z
Energy balance at earth’s surface
Downward short-wave radiation, Jan 2003
1200Z
Energy balance at earth’s surface
Downward short-wave radiation, Jan 2003
1500Z
Energy balance at earth’s surface
Downward short-wave radiation, Jan 2003
1800Z
Energy balance at earth’s surface
Downward 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
Amount of energy transferred from
W/m2)
equator to the poles is approximately
• Emitted radiation is
4 x 109 MW
more uniform than
incoming radiation
Hadley circulation
Warm air rises, cool air descends creating two huge convective cells.
Coriolis Force
Cone 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
Circulation cells
Polar Cell
Ferrel Cell
1.
Hadley cell
2.
Ferrel Cell
3.
Polar cell
Winds
1.
Tropical Easterlies/Trades
2.
Westerlies
3.
Polar easterlies
Latitudes
1.
Intertropical convergence
zone (ITCZ)/Doldrums
2.
Horse latitudes
3.
Subpolar low
4.
Polar high
Effect of land mass distribution
Uneven distribution of land and ocean, coupled with different thermal properties
creates spatial variation in atmospheric circulation
A) Idealized winds generated by pressure gradient and Coriolis Force. B) Actual
wind patterns owing to land mass distribution
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
Creates wet Summers (Monsoons)
and dry winters, especially in India
and SE Asia
Northward shift in July
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 vapor
Suppose we have an elementary volume of atmosphere dV and
we want quantify how much water vapor it contains
Water vapor density
Air density
mv
v 
dV
ma
a 
dV
dV
ma = mass of moist air
mv = mass of water vapor
Atmospheric gases:
Nitrogen – 78.1%
Oxygen – 20.9%
Other gases ~ 1%
http://www.bambooweb.com/articles/e/a/Earth's_atmosphere.html
Specific Humidity, qv
• Specific humidity
measures the mass of
water vapor per unit
mass of moist air
• It is dimensionless
v
qv 
a
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
e  v RvT
e
qv  0.622
p
Dalton’s Law of Partial Pressures
John 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 law
Equal 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 air ( z = x+y molecules)
Moist air (x dry and y water vapor)
Dry air
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 vapor
that it can at a given air temperature
 17.27T 
es  611 exp 

 237.3  T 
Vapor pressure is measured in Pascals (Pa), where 1 Pa = 1 N/m2
1 kPa = 1000 Pa
Relative humidity, Rh
es
e
e
Rh 
es
Relative humidity measures the percent
of the saturation water content of the air
that it currently holds (0 – 100%)
Dewpoint Temperature, Td
e
Td
T
Dewpoint temperature is the air temperature
at which the air would be saturated with its current
vapor content
Water vapor in an air column
• We have three equations
describing column:
2
– Hydrostatic air pressure,
dp/dz = -ag
– Lapse rate of temperature,
dT/dz = - a
– Ideal gas law, p = aRaT
• Combine them and
integrate over column to
get pressure variation
elevation
Column
Element, dz
1
 T2 
p2  p1  
 T1 
g / aRa
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
2
Column
Element, dz
1
Area = A
dm p  qv  a Adz
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 lifting
Orographic uplift occurs when air is forced to rise because of the physical
presence of elevated land.
Convective lifting
Convective precipitation occurs when the air near the ground is heated by the
earth’s warm surface. This warm air rises, cools and creates precipitation.
Hot earth
surface
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 mm.
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 mm droplet
– Critical size (~0.1
mm)
– Gravity overcomes
and drop falls
Forces acting on rain drop
• 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 air
Fg   w g

6
D3
Fb   a g
2
V2
2  V
Fd  Cd  a A
 Cd  a D
2
4 2

6
D3
D
Fb
Fd
Fd
Fg
Volume 
Area 

4

6
D3
D2
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
 Fvert  0  FB  FD  W

D

2

3
2V
  a g D  Cd  a D
  w g D3
6
4
2
6
FD  FB  W
 2 Vt2


Cd  a D
 a g D3   w g D3
4
2
6
6
Vt 
4 gD   w 

 1
3Cd   a

Fb
Fd
At standard atmospheric pressure (101.3 kpa) and temperature (20oC),
w = 998 kg/m3 and a = 1.20 kg/m3
Fd
Fg
V
• 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
Cd 
24
Re
Re 
 aVD
ma
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 Intensity
Time (min)
Rainfall (in)
0
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
Max. Depth
Max. Intensity
0.02
0.34
0.1
0.04
0.19
0.48
0.5
0.5
0.51
0.16
0.31
0.66
0.36
0.39
0.36
0.54
0.76
0.51
0.44
0.25
0.25
0.22
0.15
0.09
0.09
0.12
0.03
0.01
0.02
0.01
0.76
9.12364946
Cumulative
Rainfall (in)
0
0.02
0.36
0.46
0.5
0.69
1.17
1.67
2.17
2.68
2.84
3.15
3.81
4.17
4.56
4.92
5.46
6.22
6.73
7.17
7.42
7.67
7.89
8.04
8.13
8.22
8.34
8.37
8.38
8.4
8.41
Running Totals
30 min
1h
2h
1.17
1.65
1.81
2.22
2.34
2.46
2.64
2.5
2.39
2.24
2.62
3.07
2.92
3
2.86
2.75
2.43
1.82
1.4
1.05
0.92
0.7
0.49
0.36
0.28
3.07
6.14
3.81
4.15
4.2
4.46
4.96
5.53
5.56
5.5
5.25
4.99
5.05
4.89
4.32
4.05
3.78
3.45
2.92
2.18
1.68
5.56
5.56
8.13
8.2
7.98
7.91
7.88
7.71
7.24
8.2
4.1
Incremental Rainfall
0.8
Incremental Rainfall (in per 5 min)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
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)
Rainfall Hyetograph
Cumulative Rainfall
10
9
Cumulative Rainfall (in.)
8
7
6
5
3.07 in
4
8.2 in
30 min
3
5.56 in
2
1 hr
1
2 hr
0
0
30
60
90
Time (min.)
Rainfall Mass Curve
120
150
Arithmetic Mean Method
• Simplest method for determining areal
average
P1 = 10 mm
P1
P2 = 20 mm
P3 = 30 mm
1
P
N
P
N
P
i 1
P2
i
10  20  30
 20 mm
3
P3
• Gages must be uniformly distributed
• Gage measurements should not vary greatly about
the mean
Thiessen polygon method
•
•
•
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 method
1. 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
P
1
Ai Pi

A i 1
P
12 10  15  20  20  30
 20.7 mm
47
P1
A1
P2
A2
P3
A3
P1 = 10 mm, A1 = 12 Km2
P2 = 20 mm, A2 = 15 Km2
P3 = 30 mm, A3 = 20 km2
Isohyetal method
• 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
1M N
PP  
P
Ai pA
i i i
A
i 1 i 1
P
5  5  18 15  12  25  12  35
 21.6 mm
47
10
20
P1
A1=5 , p1 = 5
A2=18 , p2 = 15
P2
A3=12 , p3 = 25
30
P3
A4=12 , p3 = 35
Inverse distance weighting
• 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
P1=10
P2= 20
d2=15
– Compute distance (di) from
ungaged point to all measurement
points.
d12 
d1=25
P3=30
p
d3=10
x1  x2 2   y1  y2 2
N


i 1
 di 
P
 i2 
10 20 30
– Compute the precipitation at the

d 


ungaged point using the following Pˆ  i 1  i  Pˆ  252 152 102  25.24 mm
N 
1 
1
1
1
formula


 2 
2
2
2
25
15
10
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
• 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
NEXRAD Tower
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