solar radiation, earth-sun geometry and climatology

Lecture 7Solar Radiation, Earth-Sun Geometry and , y

Climatology, part III

• Lambert's cosine law• Lambert s cosine law• Earth-Sun geometry

– Earth declination, elevation, zenith, hour and azimuth angles– seasons– sunrise and sunset times

• Radiation Climatology and Functional Relations for gy‘Weather Generation’

ESPM 129 Biometeorology 1

Lambert's cosine law

Definition of zenith and elevation angles and the projection of area normal to incident rays on a flat surface

R i R AA

beam

horizontal

( )

Zenith Angle

horizontal

Elevation angle


R A cos( ) sin( )horizontalR AR C

B

AC

A A

B


Cosine of solar zenith angleCosine of solar zenith angle

1.0

) 0.6

0.7

0.8

0.9

cosin

e(

0 2

0.3

0.4

0.5

0.6

Solar Zenith Angle (degrees)

0 20 40 60 80

0.0

0.1

0.2

Values vary non-linearly between 1 and zero

g ( g )


Values vary non linearly between 1 and zero

S l El i ( ) d Z i h ( ) A lSolar Elevation () and Zenith () Angles

sin( ) sin( )sin( ) cos( )cos( )cos( ) cos( ) h

is latitude, h is the hour angle and is declination angle.

Units: Radians!Units: Radians!


Vectors describing the angle between the sun and a point on EarthVectors describing the angle between the sun and a point on Earth

S n

S

n

n S n n S n ( ) | ||( )|cos


n S n n S n( ) | ||( )|cos

t t2 h t t

224 12

Earth rotates on its axis 2 radians in one day

Hour angle, h, is the fraction of 2 that the earth has turned after local solar noon:

h t t 12 0( )

(radians)12 ( )

t l ec t0 12

lc, longitudinal correction


et, equation of time

lc is the longitudinal correction.lc is the longitudinal correction.

• 4 minutes for each degree of longitude east of the standard meridian

4 i t f h d f l it d t f th• - 4 minutes for each degree of longitude west of the standard meridian


Longitudinal Correction

120105

90

75

60

45

30


Equation of Time hours

L O

Equation of Time, hours

e f f f f f f ft

LNM

OQP

104 7 596 2 2 43 3 12 7 4 4293 2 0 2 193 33600

. sin( ) . sin( ) . sin( ) . sin( ) . cos( ) . cos( ) . cos( )

f d

180279 5 0 9856. . radians

Campbell and Norman, 1998


p ,

Equation of timeEquation of time

urs

) 0.2

0.3

n o

f T

ime

(hou

0.0

0.1

Equati

on

-0.2

-0.1

Day

0 50 100 150 200 250 300 350

-0.3


Values of h and cos(h) for several reference valuesValues of h and cos(h) for several reference values

Solar time (hour) h (radians) Cos(h)6 0 5 06 0.5 0

12 -1

18 1.5 01.5


Geometrical configuration between the Sun and th E th d i th S S l tithe Earth during the Summer Solstice

Equator

Tropic of Cancer


Earth and Sun Geometry during the Spring and Autumnal Equinoxy g p g q

Spring/AutumnalEquinox

Tropic of Cancer

Equator


Configuration between the Sun and Earth during the Winter solstice g g

Winter Solstice

Equator

Tropic ofCapricorn

Equator


Basic Earth/Sun Geometryas c a t /Su Geo et y


Lutgens/Tarbuck, Atmosphere

Solar Declination Angle, radians

2 10( )D

2345180

2 10365

. cos( ( ) )D

D, day of year


Seasonal variation in the declination angle

20

30n

Ang

le 10

Dec

linat

ion

-10

0

-20

ESPM 129 Biometeorology 18Day

0 50 100 150 200 250 300 350-30

Solar elevation and azimuth angles at Sisters, OR (44o N) f d i t l ti d th ifor summer and winter solstices and the equinoxes

90

40

50

60

7060120

10

20

30

4030150

0 10 20 30 40 50 60 700

0102030405060700

10

20

0180

30

40

50

210 330


60

70240

270

300

Solar elevation and azimuth angles at Oak Ridge, TN (36o N) for summer and winter solstices and the equinoxes

Oak Ridge, TN

70 60

90

120 36 oN

30

40

50

60

30150

0 10 20 30 40 50 60 700

10

20

0102030405060700

10

0180

20

30

40

50

210 330

60

70240

270

300


Time of Sunrise/Sunset

sin( )sin( ) cos( )cos( )cos( ) h 0

cos( ) tan tanh

h 180h 180t t hsunrise 12 15

180

t t hsunset 12 15

180


ESPM 129 Biometeorology 22Zachos et al Science 2001

Magnitude and Seasonality in Rg

Oak Ridge, TN199630

35

m-2

d-1

)

20

25

30

daily

sum

(MJ

m

10

15

20

Rg d

0

5

Day

0 50 100 150 200 250 300 350


I CA

Estimating Solar Radiation as f(Tair and VPD)

Ione, CA

25

30

35m

-2 d

-1)

5

10

15

20

35ar R

adia

tion

(MJ

0

5

510

1520

2530

23

45

6

Sola

T air (C

)

0-1

01VPD (kPa)

R MJ m d T VPDg air( ) . . 2 1 0947 1033


FLUXNET database

8000

J m

-2 y

-1) 6000

Rg

(M

2000

4000

0 10 20 30 40 50 60 70 80 900


Latitude


Credit: Youngryel Ryu

How Does Energy Availability Compare with Energy Use?gy y p gy

• US Energy Use: 105 EJ/yeargy y– 1018J per EJ– US Population: 300 106

– 3.5 1011 J/capita/year• US Land Area: 9 8 106 km2=9 8 1012 m2 = 9 8 108 ha• US Land Area: 9.8 106 km2=9.8 1012 m2 = 9.8 108 ha• Energy Use per unit area: 1.07 107 J m-2

• Potential, Incident Solar Energy: 6.47 109 J m-2

– Ione, CAIone, CA• Assuming 20% efficient solar system

– 8.11 1010 m2 of Land Area Needed (8.11 105 km2, the size of South Carolina)


Ivanov et al 2008 WRR

Rn as a f(Rg)

800

600

800

Temperate broadleafRn=-23+0.86Rg

Rn

(W m

-2)

400

0

200

Boreal jack pineRn=-27+0.897Rg

Rg (W m-2)

0 200 400 600 800 1000


Rn = a + b Rg

intercept (W m-2)

slope vegetations source

-73.85 0.787 Zea mays Davies and Idso (1979)

-29 0.842 boreal forest Dubuyah

-27 0.897 Pinus banksiana

ddb + CV

-23 0.860 deciduous forest

ddb +cvforest

-66 0.86 Picea sitchensis

Jarvis et al. (1976)

-110 0.87 Pinus taeda Jarvis et al. (1976)

38 0 91 Pi J i t l (1976)-38 0.91 Pinus sylvestria

Jarvis et al. (1976)

-63 to -124 0.655 to 0.808

wheat Denmead (1976)

-91 0.72 grassland Ripley and Redman (1976)

-60 0.85 Ponderosa pine, OR

Anthoni (unpublished


Ione CAIone, CA

40m

-2 d

-1

30

35

ncom

ing,

MJ

m

25

20

30

Long

wav

e In

rC

20

0

10

01

23

Tair,

VPD, kPa


SummarySummary

• Lamberts cosine law describes the amount of radiation received on the horizontal relative to the sun’s zenith anglethe horizontal relative to the sun s zenith angle.

• The amount of radiation received on an inclined surface is a function of the angle between the sun the that projection normal to the surface.

• The axis of rotation of Earth is tilted 23.5 degrees• The amount of radiation received is a function of its latitude and the

sun’s declination angle.• Seasons arise as the Earth revolves around the sun because the• Seasons arise as the Earth revolves around the sun because the

Earth’s tilted rotation axis causes the shaded and sunlit faces of Earth to change.

• Radiation fluxes can be estimated empirically with weather variables lik t t d h iditlike temperature and humidity


solar radiation, earth-sun geometry and climatology

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