Revie view w for Exam m 1. Electromagnetic radiation exhibits the - - PowerPoint PPT Presentation

Lecture 14. Revie view w for Exam m 1.

Electromagnetic radiation exhibits the dual nature: wave properties and particulate properties Wave nature of radiation: Electromagnetic waves are characterized by wavelength l (or frequency ,or wavenumber n) and speed Relation between l, and : n = /c = 1/l Particulate nature of radiation: can be described in terms of particles of energy, called photons. E photon = h = h c/l = h cn h is Plank’s constant (h = 6.6256x10-34 J s).

n

n ~ n ~ n ~

n ~

Flux (or irradiance) is defined as radiant energy in a given direction per unit time per unit wavelength (or frequency) range per unit area perpendicular to the given direction:

l

l l

dtdAd dE dF 

UNITS: (J sec-1 m-2 mm-1) = (W m-2 mm-1)





  d I F ) cos(

l l

The radiative flux is the integration of normal component of monochromatic intensity over some solid angle.



  

   l l 2

) ( d n I F  

Net radiative flux Monochromatic net flux is the integration of normal component of monochromatic intensity over the all solid angles (over 4):

 m m  m

l  l l

l

d d I F F Fnet

 

  

  

1 1 2

) , (

,

What is the net flux of the isotropic radiative field?

Extinction (scattering +absorption) and emission. Extinction is a process that decreases the radiant intensity, while emission increases it. Absorption is a process that removes the radiant energy from an electromagnetic field and transfers it to other forms of energy. Scattering is a process that does not remove energy from the radiation field, but may redirect it.

Extinction (or attenuation) is due to absorption and scattering.

The fundamental law of extinction is the Beer-Bouguer-Lambert law: the extinction process is linear in the intensity of radiation and amount of matter, provided that the physical state (i.e., T, P, composition) is held constant.

ds I dI

e l l l

 ,  

ds J dI

e l l l

 , 

Extinction: Emission:

where e,l is the volume extinction coefficient (LENGTH-1); Jl is the source function.

l l l

  

, , , s a e

 

ds J ds I dI

e e l l l l l

 

, ,

  

Lecture 3

ds J ds I dI

e e l l l l l

 

, ,

  

Lecture 3 The differential form of radiative transfer equation

l l l l

 J I ds dI

e

  

, l l l l

 J I d dI    

l l l l

 J I d dI  

Using

ds s d

e

) (

,l l

   

We have Elementary solution:

ds J s s s I s I

e s l l l l l l

  

, 1 1 1

1

)) ; ( exp( )) ; ( exp( ) ( ) (



   

See p.12

Solution of the radiative transfer equation in the plane-parallel atmosphere (called the integral form)

  m  m   m m   m  m 

l  l l

          

  



d J I I ) ; ; ( ) exp( 1 ) exp( ) ; ; ( ) ; ; (

  m  m   m m    m   m 

l   l l

        

  



d J I I ) ; ; ( ) exp( 1 ) exp( ) ; ; ( ) ; ; (

*

* *

Lecture 3

Blackbody emission Planck function, Bl(T), gives the intensity (or radiance) emitted by a blackbody having a given temperature.

) 1 ) / (exp( 2 ) (

5 2

  l l

l

T k hc hc T B

B

Stefan-Boltzmann law:

F = sb T4 =  B(T) el = Al

Kirchhoff law: Wien displacement law:

lm = 2898 / T Lecture 4

Molecular Absorption/Emission Spectra

Lorentz profile of a spectral line is used to characterize the pressure broadening and is defined as:

2 2

) ( / ) (  n n   n n    

L

f

n

T T P P T P        ) , (    is the half-width of a line at the half maximum (in cm-1), (often called the line width) Doppler profile is defined in the absence of collision effects (i.e., no pressure broadening) as:

                   

2

exp 1 ) (

D D D

f  n n   n n

D is the Doppler line width

2 / 1

) / 2 ( m T k c

B D

n  

Comparison of the Doppler and Lorentz profiles for equivalent line strengths and widths.

Absorption coefficient is defined by the position, strength, and shape of a spectral line: ka,n = S f(n – n0)



 n

nd

k S

a,

1 ) (  



n n n d f

Dependencies: S depends on T; f(n – n0, ) depends on the line halfwidth  (p, T), which depends

• n pressure and temperature.

Path length (or path) is defined as the amount of an absorber along the path If the amount of an absorber is given in terms of mass density, path length is

ds s u

s s





2 1

) ( 

Homogeneous absorption path: when ka,n does not vary along the path => optical depth is  = ka,n u Inhomogeneous absorption path: ka,n varies along the path du k

u u a





2 1

,n



Absorbing gas (path length u) Absorption coefficient Line intensity (S) cm cm-1 cm-2 g cm-2 cm2 g-1 cm g-1 molecule cm-2 cm2/molecule cm/molecule cm atm (cm atm)-1 cm-2 atm-1

Monochromatic transmittance and absorbance

) exp(

n n

   T ) exp( 1 1

n n n

      T A

Spectral intensity = intensity averaged over a very narrow interval that Bn is almost constant but the interval is large enough to consist

• f several absorption lines.

Narrow-band intensity= intensity averaged over a narrow band which includes a lot of lines; Broad-band intensity= intensity averaged over a broad band (e.g.,

• ver a whole longwave region)

  

  

       

n n n n n n n

n n n  n n  n d u k d d T u T ) exp( 1 ) exp( 1 ) ( 1 ) (

 

 

         

n n n n n n

n n n  n d u k d u T A )) exp( 1 ( 1 )) exp( 1 ( 1 ) ( 1

  m   m m   m  m 

n   n n

        



 

d T B I I )) ( ( ) exp( 1 ) exp( ) ; ( ) ; (

*

* *

  m   m m  m m 

n  n n

         



 

d T B I I )) ( ( ) exp( 1 ) exp( ) ; ( ) ; (

The solutions of the radiative transfer equation for the monochromatic upward and downward intensities in the IR for a plane-parallel atmosphere consisting of absorbing gases (no scattering):

  m   m m    m 

n  n n



        





d B B I ) ( ) exp( 1 ) exp( ) ( ) ; (

*

* *

  m   m m 

n  n

      





d B I ) ( ) exp( 1 ) ; (

For isothermal atmosphere and black body surface

)] * exp( 1 )[ ( ) exp( ) ( ) ; (

* *

m  m   m

n n n

    

 eff

T B B I

For fluxes – see 4.6.2 pp.154-157

Monochromatic net flux (net power per area) at a given height

) ( ) ( ) ( z F z F z F

 

 

n n n

and total net flux

) ( ) ( ) ( z F z F z F

 

 

Introducing the net flux F(z+z) at the level z+z, the net flux divergence for the layer z is

) ( ) ( z F z z F F     

F(z+z) < F(z) (hence F < 0 ) => a layer gains radiative energy => heating F(z+z) > F(z) (hence F > 0 ) => a layer losses radiative energy => cooling The IR radiative heating (or cooling) rate is defined as the rate

• f temperature change of the layer dz due to IR radiative energy

gain (or loss):

dp dF c g dz dF c dt dT

net p net p IR

          1

where cp is the specific heat at the constant pressure (cp = 1004.67 J/kg/K) and  is the air density in a given layer.

Effect of the varying amount of a gas on IR radiation under the same atmospheric condition

1 2 3 4 5 6 7 8 9 10 5 10 15 20 H2O (g/kg) Altitude, km

Consider the standard tropical atmosphere and “dry” tropical atmosphere: same atmospheric characteristics, except the amount of H2O

1 2 3 4 5 6 7 8 9 10 60 160 260 360 460 Flux, W/m2 Altitude, km Fup Fdw Fup, dry Fdw, dry

IR fluxes for tropical (dotted lines) and dry tropical atmospheres (solid lines)

• H2O increases in a layer => increases because more IR radiation emitted in a layer =>

increases

• H2O increases in a layer => decreases because more IR radiation absorbed but

reemitted at the lower temperature => decreases

• Increase of an IR absorbing gas contributes to the greenhouse effect.



F

) (surface F 



F

) (TOA F 

1 2 3 4 5 6 7 8 9 10 60 110 160 210 260 Flux, W/m2 Altitude, km Fnet Fnet, dry

IR net fluxes for tropical (dotted lines) and dry tropical atmospheres (solid lines)

1 2 3 4 5 6 7 8 9 10

• 3
• 2
• 1

Cooling rates, K/day Altitude, km

Cooling rates Cooling rates, dry tropics

IR cooling rates for tropical (dotted lines) and dry tropical atmospheres (solid lines) Heating

IR cooling rates of individual gases:

IR cooling rates in different cloud-free atmospheres: