3D Radiative Transfer in Cloudy Atmospheres: Diffusion
Approximation and Monte Carlo Simulation for Thermal
Emission
K. N. Liou, Y. Chen, and Y. Gu
Department of Atmospheric and Oceanic Sciences, UCLA
3D Clouds in the Atmosphere and in Climate Models
•
3D Radiative Transfer: Diffusion Approximation
•
3D Thermal Radiative Transfer: Monte Carlo Simulation
•
3D Results for Cirrus Clouds: Some Thought on Flux, Heating
Rate, and Absorption

A Global View of Clouds from A GOES Satellite

3-D Cirrus Observations from Lidar
(E. Eloranta, SBIRS ) September 14, 1995
Top View
24 km
North
24 km
Side View
12 km
6 km

Figure 3. Illustrative 2D cirrus cloud images located at the Southern Great Plains (SGP/ARM/DOE) derived from a 35 GHz radar on August 14 (top) and March 22, 2001 (bottom).

Cirrus/Contrail
Aerosol/Dust
Radiative Transfer/A Unified Theory for Light
Scattering by Ice Crystals and Aerosols


Physical Parameterizations
Planetary boundary layer processes: Suarez et al. (1983), Li et al. (1999, 2001)
Cumulus Convection: Prognostic Arakawa-Schubert (Pan and Randall 1998), with downdrafts (Cheng and Arakawa 1997)
Radiation: Harshvardhan et al. (1987) (Control run)
 Fu and Liou (1992, 1993),
Gu et al. (2003)
Prognostic Cloud Water/Ice: Kohler (1999)
+ Fractional clouds/Cloud overlap (Gu et al. 2003)
Gravity Water Drag: Kim and Arakawa (1995)

Dynamics
Horizontal Finite Difference Scheme: Arakawa and Lamb (1981)
Resolution: 5

longitude x 4

latitude
Vertical Finite Difference Scheme: Suarez and Arakawa (1983)
Resolution (top at 1 hPa): 15 layers
Time integration: Leapfrog, Matsuno

Surface Conditions
Prescribed sea surface temperatures (Rayner et al. 1995), albedo, ground wetness, and surface roughness (Dorman and Sellars 1989)

Delta-Diffusion Approximation for 3D Radiative Transfer

The Basic Steady–State RT Equation for Diffuse Intensity:
 1
 e
( s )
( Ω  
) I(s, Ω) = I(s, Ω)
–
J(s, Ω
)

The Source Function is Defined by
J ( s ,

Ω )
( s )
4



( s )
4


4

P ( s ; Ω ,
I ( s , Ω 
) P ( s ;
Ω
0
) F
 exp

 
0 s
0
Ω , Ω 
) d
 
 e
( s

) d s




1

( s ) B [ T ( s )]


Expand Phase Function and Diffuse Intensity in Spherical Harmonics:
P ( s ,

,

)


N
 

0 m

  


Y
 m (

) Y
 m

(

)
I ( s ,
Y
 m (

,

)



N
 

0 m

  
I
 m ( s ) Y
 m (

)
)

(

1 ) ( m
 m ) / 2
( 
( 

 m m
)!
)!
Y
 m

(

,

)

Y

 m (

,

) (

1 ) m
P
 m (cos

) e im



Decomposition of the Basic Equation and Successive Integrations:
1
 e
( s )
(
  
)
N

 
0

 m
  
I
 m ( s ) Y
 m (

)



( s )
4

N 
 
 
0 m
  


I
 m ( s ) Y
 m (

)

N

0 m


 
 


Y
 m (

) Y
 m

(

0
) F
 e
  s

[ 1
 
( s )] B [ T ( s )]

 e
1
( s )
N 
  
4


 
0 m
  
( Ω 
) Y
 m ( Ω ) Y

 
( Ω ) I
 m ( s ) d

  

I


( s )
2

4


1

2

( s )



1
Y

 
(
Ω
0
) F
 e
  s

4 [ 1
 
( s )]

B [ T ( s )]

First-order Approximation (N=1) in Cartesian Coordinates:

I
 z
1
0  1
2



 x
 i

 y

 I
1

1
 1
2



 x
 i

 y

 I
1
1

 e
 e e
( 1

( 1

( 1
  g g ) I
1

1 g ) I
1
0
) I
1
1


3

 
1
1
3
2
3
1
2




I z
0
0

 x




 x
 i
 i

 y
3
4
 g

 y

 I
0
0

0

 I
0
0

3
F
 e
  s
2 4
 g
   e
I
0
0 ( 1

)
( 1
 
0
2 ) 1 / 2 (cos

0


4
 i
 e
F
 e
  s sin

0
) F
 e
  s


3D Nonhomogeneous Diffusion Equation: a 2 nd
order PDE
 
(

I
0
0
 t
 t


F t


 t
)
 e e
( 1
( 1
3

3
 e e

3
 t
  g
I
)
( 1

F
 e

)

0
0

) s

B

4

( T
F t
)
 
0
 
( F t g solar
IR
 t
)

Diffuse Intensity from the Spherical Harmonics Expansion
I ( x , y , z ;
Ω
)

I
0
0


I
0
0

I
1

1 Y
1

1 (
Ω
)
2
3 h j
3


1


I x
0
0 j

I
1
0 Y
1
0
Ω xj

(
Ω
)
9 q

( Ω 
I
1
1 Y
1
1 (
Ω
)
Ω
0
) e
  s
2 h

Delta Function Adjustment for the Phase Function
  e
  e
( 1
 f ),
 
( 1
 f ) ( 1
  f ) , and g
 
( g
 f ) ( 1
 f ).
f =

2
5 .


Three–Dimensional Flux Densities
F
 x i
( x , y , z )
 
2

I ( x , y , z ;

)
 x i d


The Local Rate of Change of Temperature


T t
( x , y ,
F
 i F x z )

 j F y


1 c
 k F z p
 
F

Conventional Definition of the Absorbed Flux in a Cloud Layer:
F abs

[ F

( z t
)

F

( z t
)]

[ F

( z b
)

F

( z b
)]
 t
Absorptance

[ F

( z b
)
 a
F

( z b
)] /

F abs
/ F

( z t
F

( z t
) :
) , reflectance r

F

( z t
) / F

( z t
) , and net transmittance
a = 1 – r – t


Energy Conservation Principle for 3D clouds (Liou and Rao 1996):

A z
F


0 dxdy
 
A x
F

( 1
 
0
2 ) 1 / 2 cos

0 dydz
 
A y
F

( 1
 
0
2 ) 1 / 2 sin

0 dxdz
 
A F z
( z
 
F z

) dxdy
 
A x
( F x
 
F x

) dydz
 
A y
( F y
 
F y

) dxdz
  f
V a dxdydz
where
A x
,
A y ,
and
A z , are areas,
V
is the volume, and f a
is the absorbed flux per volume.

Averaging over the respective areas and volume:
F

[

0
A z

( 1
 
0
2 ) 1 / 2 cos

0
A x

( 1
 
0
2 ) 1 / 2 sin

0
A y
]


F z
 
F z


A z

( F x
 
F x

) A x

( F y
 
F y

) A y
 f a
V

Multigrid Method for Solving Diffusion Equation
Solving the diffusion equation on a series of coarser grids first and subsequently interpolating the coarse grid correction back to the pre-specified fine grid.
Performing the major computational work on the coarse grid.
High frequency errors are relaxed on the fine grid and low frequency errors are relaxed on the coarse grid.

Schematic Overview for 3D Infrared Monte Carlo Radiative Transfer Model

Parameterization of Gaseous and Ice Cloud Emissivities
Cubic Gas Cell Emissivity Cubic Cirrus Cell Emissivity

Monte Carlo “exact” method, delta-diffusion approximation (DDA), and independent pixel-by-pixel approximation (IPA)
Correlated k-distribution method for incorporation of gaseous absorption in multiple scattering atmosphere: 121 wavelength to cover solar (0.2-5
 m) and thermal IR (5-50
 m)
Parameterization of the single-scattering properties in terms of ice/water content and mean effective size/radius

Broadband Infrared Flux and Cooling Rate for Plane-Parallel
Clear Atmosphere: Monte Carlo vs Fu and Liou
Subarctic Winter Tropical
Midlatitude Summer US Standard

Broadband Infrared Flux and Cooling Rate for Plane-Parallel
Cirrus Cloudy Atmosphere: Monte Carlo vs Fu and Liou
Subarctic Winter Tropical
Midlatitude Summer US Standard

June 1, 2002
Combined Satellite and Mm-Wave Radar
3-D Cloud Imaging in a Mesoscale Grid

Input 3D IWC data (Satellite + Mm-wave Radar) for 3D RT Calculation

(a) 3D Cloud Mapping in a Mesoscale Grid (b) Validation
Figure 2 (a) Three-dimensional ice water content (IWC, 0-0.28 g/m 3 ) and mean effective ice crystal size (DE, 0-196 μm) determined from a unification of the optical depth and DE retrieval from the 0.63 and 3.7 μm AVHRR channels aboard the NOAA-14 satellite and the IWC and DE retrieved from the 35 GHz cloud radar over the ARM-SGP CART site at 2023 UTC on April 18, 1997. The 3D IWC and DE results are presented in xy , yz , and xz planes over a 40 km x 40 km x 4.5 km domain. (b) Time series of the flight altitude and the position of the flight track of the North Dakota Citation over the ARM-SGP CART site on April 18, 1997 (upper panel). Comparison of the mean (symbols) and standard deviations (bars) of the retrieved IWC and DE values from the unification approach involving satellite and cloud radar observations derived from in situ measurements taken on board the University of North Dakota Citation, as functions of aircraft leg height (lower panels) (Liou et al.
, 2002).

3D Cloud Mapping in a Mesoscale Grid

3D Radiative Heating Rate in a Mesoscale Grid
Domain-Averaged Values
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5
Solar Heating Rate (k/day)
6
6
5
4
3
8
7
2
1
0
-3.0
-2.0
-1.0
0.0
1.0
IR Heating Rate (k/day)
2.0
Figure 2.
The left panels illustrate three-dimensional solar heating and IR heating/cooling rates in a mesoscale grid, presented in the xy , yz , xz planes computed from the 3D inhomogeneous radiative transfer model developed by Gu and Liou (2001). The input 3D cloud IWC and DE fields were from those presented in Fig. 1a. The solar constant, solar zenith angle, and surface albedo used were 1366 W/m
2
, 60 o , and 0.1 respectively. The standard atmospheric temperature and gaseous profiles were employed in the calculations. The right panels represent the domain-averaged ( xy plane) solar heating and IR heating/cooling rates as functions of cloud height.

Comparison of 3D broadband Solar Heating Rates
3D Monte Carlo 3D DDA relative difference (%) IPA relative difference (%)

Comparison of 3D broadband IR Heating Rates
3D Monte Carlo 3D DDA relative difference (%) IPA relative difference (%)

Domain Average Heating Rate in a Cirrus Cloud
Solar Heating
Thermal IR Heating

3D Radiative Transfer in Cloudy Atmospheres: Diffusion
Approximation and Monte Carlo Simulation for Thermal
Emission
K. N. Liou, Y. Chen, and Y. Gu
Department of Atmospheric and Oceanic Sciences, UCLA
3D Clouds in the Atmosphere and in Climate Models
•
3D Radiative Transfer: Diffusion Approximation
•
3D Thermal Radiative Transfer: Monte Carlo Simulation
•
3D Results for Cirrus Clouds: Some Thought on Flux, Heating
Rate, and Absorption