Transcript Presentation Slides for Chapter 8 of Fundamentals of Atmospheric Modeling 2nd Edition Mark Z. Jacobson Department of Civil & Environmental Engineering Stanford University Stanford, CA 94305-4020 [email protected] March 10, 2005

Presentation Slides
for
Chapter 8
of
Fundamentals of Atmospheric Modeling
2nd Edition
Mark Z. Jacobson
Department of Civil & Environmental Engineering
Stanford University
Stanford, CA 94305-4020
[email protected]
March 10, 2005
Reynolds Stress
Stress
Force per unit area (e.g. N m-2 or kg m-1 s-2)
Reynolds stress
Stress that causes a parcel of air to deform during turbulent
motion of air
Fig. 8.1. Deformation by vertical momentum flux w u 
Stress from vertical transfer of
turbulent u-momentum
(8.1)
 zx   a w u 
zx = stress acting in x-direction,
along a plane (x-y) normal to
the z-direction
Momentum Fluxes
Magnitude of Reynolds stress at ground surface
(8.2)
2 1 2
   

 z  a  w u   w v  


2
Kinematic vertical turbulent momentum flux (m2 s-2)
(8.3)
 zx
w u   
a
w v   
 zy
a
Friction wind speed (m s-1)
(8.8)
Scaling param. for surface-layer vert. flux of horiz. momentum
14
2
2
12


u*   wu  w v    z a s

s
s 
   


Heat and Moisture Fluxes
Vertical turbulent sensible-heat flux (W m-2)
(8.4)
H f  ac p,d w v
Kinematic vert. turbulent sensible-heat flux (m K s-1)
Hf
w v 
acp,d
Vertical turbulent water vapor flux (kg m-2 s-1)
(8.5)
(8.6)
E f  a w q v
Kinematic vert. turbulent moisture flux (m kg s-1 kg-1) (8.7)
w q v 
Ef
a
Surf. Roughness Length for Momentum
Height above surface at which mean wind extrapolates to zero
•
Longer roughness length --> greater turbulence
•
Exactly smooth surface, roughness length = 0
•
Approximately 1/30 the height of the average roughness
element protruding from the surface
Surf. Roughness Length for Momentum
ln z
Method of calculating roughness length
1) Find wind speeds at many heights when wind is strong
2) Plot speeds on ln (height) vs. wind speed diagram
3) Extrapolate wind speed to altitude at which speed equals zero
Fig. 6.5
Roughness Length for Momentum
Over smooth ocean with slow wind
(8.9)
a
a
z0,m  0.11
 0.11
u*
au*
Over rough ocean, fast wind (Charnock relation)
(8.10)
2
u*
z0,m   c
g
Over urban areas containing structures
ho So
z0,m  0.5
Ao
Over a vegetation canopy

z0,m  hc 1  0.91e
0.0075LT
(8.11)

(8.12)
Roughness Length for Momentum
Surface Type
z0,m (m)
0.00001
0.0000150.0015
Ice
0.00001
Snow
0.00005-0.0001
Level desert
0.0003
Short grass
0.03-0.01
Long grass
0.04-0.1
Savannah
0.4
Agricultural crops
0.04-0.2
Orchard
0.5-1.0
Broadleaf evergreen forest
4.8
Broadleaf deciduous trees
2.7
Broad- and needleleaf trees
2.8
Needleleaf-evergreen trees
2.4
Needleleaf deciduous trees
2.4
Short vegetation/C4 grassland
0.12
Broadleaf shrubs w/ bare soil
0.06
Agriculture/C3 grassland
0.12
2500 m2 l ot w/ a b uilding 8-m 0.26
high and 160 m2 silhouette
25,000 m2 lot w/ a building 80-m 5.1
h c (m)
d c (m)
Smooth sea
Rough sea
high and 3200 m2 silhouette
0.02-0.1
0.25-1.0
8
0.4-2
5-10
35
20
20
17
17
1
0.5
1
8
80
4.8
0.27-1.3
3.3-6.7
26.3
15
15
12.8
12.8
0.75
0.38
0.75
Table 8.1
Roughness Length for Energy, Moisture
Surface roughness length for energy
(8.13)
Dh
z0,h 
ku*
Surface roughness length for moisture
(8.13)
Dv
z0,v 
ku*
Molecular thermal diffusion coefficient
(8.14)
a
Dh 
ac p,m
Molecular diffusion coefficient of water vapor
(8.14)
1.94 1013.25 hP a
T



Dv  2.11 105 


273.15 K 
p


a
Turbulence Description
Turbulence
Group of eddies of different size. Eddies range in size from a
couple of millimeters to the size of the boundary layer.
Turbulent kinetic energy (TKE)
Mean kinetic energy per unit mass associated with eddies in
turbulent flow
Dissipation
Conversio
n of turbulence into heat by molecular viscosity
Inertial cascade
Decrease in eddy size from large eddy to small eddy to zero
due to dissipation
Turbulence Models
Kolmogorov scale
14
 3 
k   a 
  d 
(8.15)
Reynolds-averaged models
Resolution greater than a few hundred meters
Do not resolve large or small eddies
Large-eddy simulation models
Resolutio
n between a few meters and a few hundred meters Resolve
large eddies but not small ones
Direct numerical simulation models
Resolutio
n on the order of the Kolmogorov scale
Resolve all eddies
Kinematic Vertical Momentum Flux
Bulk aerodynamic formulae
(8.16-7)
Diffusion coefficient accounts for
Skin drag: drag from molecular diffusion of air at surface
Form drag: drag arising when wind hits large obstacles
Wave drag: drag from momentum transfer due to gravity waves
wu s  CD v hzr u zr  u z0,m
wvs  CD vh zr vzr  vz0,m 
Kinematic Vertical Momentum Flux
Bulk aerodynamic formulae
(8.18)
K-theory
(8.18)
wu s  CD v hzr u zr  u z0,m
u
wu   Km,zx
s
z
Wind speed gradient
 

u u zr   u z0,m

z
zr  z0,m
(8.19)

Eddy diffusion coef. in terms of bulk aero. formulae

Km,zx  Km,zy  CD v hzr  zr  z0,m
(8.20)

Kinematic Vertical Energy Flux
Bulk aerodynamic formulae
(8.21)
K-theory
(8.23)
wv s  CH vh zr v zr  v z0,h
v
wv   Kh,zz
s
z


Potential virtual temperature gradient
(8.24)
 
v v zr  v z0,h

z
zr  z0,h
Eddy diffusion coef. in terms of bulk aero. formulae

Kh,zz  CH v hzr  zr  z0,h

(8.25)
Vertical Turbulent Moisture Flux
Bulk aero. kinematic vertical turbulent moisture flux
(8.26)
wq v s  CE vhzr q v zr   q v z0,v 
CE≈CH --> Kv,zz =Kh,zz
Similarity Theory
• Variables are first combined into a dimensionless group.
• Experiment are conducted to obtain values for each variable in
the group in relation to each other.
• The dimensionless group, as a whole, is then fitted, as a
function of some parameter, with an empirical equation.
• The experiment is repeated. Usually, equations obtained from
later experiments are similar to those from the first experiment.
• The relationship between the dimensionless group and the
empirical equation is a similarity relationship.
• Similarity theory applied to the surface layer is Monin-Obukhov
or surface-layer similarity theory.
Similarity Relationship
Dimensionless wind shear
(8.28)
m
z  vh

k
u* z
Dimensionless wind shear from field data
(8.29)
1  z
m

L


z  1 4
 m  1   m 
L 


1

Integrate (8.28) from z0,m to zr
z
0
L
z
0
L
z
0
L
k v h zr 
u*  z
dz
r
m
z 0,m
z

s tabl e
uns table
neutral
(8.30)
Integral of Dimensionless Wind Shear
Integral of the dimensionless wind shear
zr
dz
z0, m  m z 
m
 zr

zr  z0,m
ln z
L
0,m

14

1
4
z

zr 
0,m 
 
1

1  m   1

  1
m
L 
 
L 

ln
 ln

14
 
zr 
z0,m 1 4

1  m   1
1  m
1
 




L
L 



14
14
z
z


2 tan1 
1   m r 
  2 tan1 1  m 0,m 


L 

L 

ln zr

 z0,m


(8.31)
z
0
L
z
0
L
z
0
L
s tabl e
uns table
neutral
Monin-Obukhov Length
Height proportional to the height above the surface at which
buoyant production of turbulence first equals mechanical (shear)
production of turbulence.
(8.32)
L
u3* v

kg wv
s
u2* v

kg*
Kinematic vertical energy flux
wv s  u**
(8.33)
Potential Temperature Scale
Dimensionless temperature gradient
(8.34)
h
z v

k
* z
Parameterization of *
P r   z
 t h L

 
z 1 2
 h  P rt 1  h 
L 
 

P rt

(8.35)
z
0
L
z
0
L
z
0
L
s tabl e
uns table
neutral
Potential Temperature Scale
Turbulent Prandtl number
P rt 
Km,zx
Kh,zz
Integrate (8.23) from z0,m to zr
h
z v

k
* z
(8.37)

 
k v zr  v z0,h
* 
zr
dz
z0,h h z
Integral of Dimensionless Temp. Grad.
Integral of dimensionless temperature gradient
(8.38)
zr
dz
z0, h  h z 
zr
h


zr  z0,h
P rt ln z
L
0,h


 
12

z


  1   h r   1
1   h
  
L 

P rt ln

ln
12

z
  

1   h r 
  1
1 h

  

L





zr
P
r
ln
 t z
0,h




 1

12

z0,h 
  1
L 


z
0
L
s tabl e
z
0
L
uns table
z
0
L
neutral
12
z0,h 

L 
Equations to Solve Simultaneously
Solution requires iteration
k v h zr 
u*  z
dz
r
m
z 0,m
z
L


3
u* v

kg wv
 
k v zr  v z0,h
* 
zr
dz
z0,h h z
s
2
u* v

kg*
Noniterative Parameterization
Friction wind speed
u* 
k v h zr 

ln zr z0,m
(8.40)

Gm
Potential temperature scale

 
Gh
2
u* P rt ln zr z0,m 
k vh zr   v z r   v z0,h
2
* 
(8.40)
Scale Parameterization
Potential temperature scale
(8.41)
9.4Rib
Gm  1
1
70k
1
Gm ,Gh 
Rib  0
0.5
Rib  0
Rib zr z0,m 
ln 2 zr z0,m 
2
9.4Rib
Gh  1
0.5

50k 2 Rib zr z0,m

ln2 zr z0,m
1
1 4.7Rib 
2


Rib  0
Bulk Richardson Number
Ratio of buoyancy to mechanical shear

(8.39)
 zr  z0,m
Rib 
2
2
v z0,h u zr   v zr  zr  z0,h 
g v zr  v z0,h
2
Gradient Richardson Number
Ri g 
g v
v z
2
(8.42)
2
 u 
 v 
    
 z 
 z 
Table 8.2. Vertical air flow characteristics for different Rib or Rig
Value of
Rib or Rig
Type of
Flow
Level of Turbulence
Due to Buoyancy
Large, negative
Small, negative
Turbulent
Turbulent
Large
Small
Small positive
Large positive
Turbulent
Laminar
None (weak stable)
None (strong stable)
Level of
Turbulence
Due to Shear
Small
Large
Large
Small
Gradient Richardson Number
Ri g 
g v
v z
2
(8.42)
2
 u 
 v 
    
 z 
 z 
Laminar flow becomes turbulent when Rig decreases to less than
the critical Richardson number (Ric) = 0.25
Turbulent flow becomes laminar when Rig increase to greater
than the termination Richardson number (RiT) = 1.0
Similarity Theory Turbulent Fluxes
Friction wind speed
14
2
2


u*   wu  w v  

s
s 
(8.8)
   
Bulk aerodynamic kinematic momentum flux
(8.16)
wu s  CD v hzr u zr  u z0,m
Friction wind speed
(8.43)
u*  v hzr  CD
Rederive momentum flux in terms of similarity theory (8.43)
2
u*
wu   
u zr 
s
v h zr 
 
Eddy Diff. Coef. for Mom. Similarity
K-theory kinematic turbulent momentum fluxes
u
wu   Km,zx
s
z
 
(8.18)
v
wv   Km,zy
s
z
 
Similarity theory kinematic turbulent fluxes
2
u*
wu s   v z  u zr 
h r
(8.44)
2
u*
wvs   v z  v zr 
h r
Combine the two
Km,zx  Km,zy 
(8.46)
2
u*
zr  z0,m 

z 
vh r
Example Problem
---> vh zr 
---> Gm
---> u*
L
--->
---> Km,zx 
z0,m = 0.01
Prt = 0.95
z0,h = 0.0001 m
k = 0.4
u(zr)=10 m s-1
v(zr)= 5 m s-1
v(zr)= 285 K
v(z0,h)= 288 K
= 11.18 m s-1
= 1.046
= 0.662 m s-1
= -169 m
2
u*
--->
--->
Rib
Gh
---> *
zr  z0,m 

z 
= -8.15 x 10-3
= 1.052
= -0.188 K
= 0.39 m2 s-1
vh r
u* *
---> K h,zz 
= 0.41 m2 s-1 ---> K m,zx K h,zz = 0.95
v z
Eddy Diff. Coef. for Mom. Similarity
Dimensionless wind shear
(8.28)
m
z  vh

k
u* z
Wind shear
Km,zx 
2
u*
(8.46)
zr  z0,m 

z 
vh r
Combine expressions above
(8.48)
kzu*
Km,zx  Km,zy 
m
kz = mixing length: average distance an eddy travels before
exchanging momentum with surrounding eddies
Energy Flux from Similarity Theory
Vertical kinematic energy flux
(8.49)
wv s  u**
Surface vertical turbulent sensible heat flux


H f  acp,d wv  acp,du* *
s
(8.53)
Energy, Moisture Fluxes from Similarity
Vertical kinematic water vapor flux
(8.49)
wq v s  u*q *
Surface vertical turbulent water vapor flux


(8.53)
E f  a wq v  au*q*
s
Dimensionless specific humidity gradient
(8.51)
q
z q v

k
q * z
Specific humidity scale

 
k q v zr   q v z0,v
q* 
z r  dz
z
0,v h z
(8.52)
Logarithmic Wind Profile
Dimensionless wind shear
(8.28)
m
z  vh

k
u* z
Rewrite
(8.57)
 vh z u*
u*

m 
1  1 m 
z
kz
kz


Integrate --> surface layer vertical wind speed profile (8.59)
u*
vh z 
k
  z 

ln
   m 


  z0,m 

Logarithmic Wind Profile
Influence function for momentum
(8.61,2)
z
dz
m  

1  m
z 0,m
z
  m
 L z  z0,m

2
1 2

1   m z  1  m z
ln

 
2 
1 2
1   m z0,m
  1   m z0,m





 

1
1
1
1
2 tan m z  2 tan m z0,m

0






 
z
0
L
s tabl e

 
  
z
0
L
z
0
L
uns table
neutral
Logarithmic Wind Profile
Height above surface (m)
Neutral conditions --> logarithmic wind profile
(8.64)
u*
z
vh z  ln
k
z0,m
Height above surface (m)
Logarithmic wind profiles when u* = 1 m s-1.
10
8
6
z
4
0 ,m
=1.0 m
z
0 ,m
=0.1 m
2
0
0
2
4
6
8
-1
Wind speed (m s )
10
12
Fig. 8.3
Potential Virtual Temperature Profile
Dimensionless potential temperature gradient
(8.34)
h
z v

k
* z
Rewrite
(8.58)
v  *
*

h 
1  1 h 
z
kz
kz


Integrate --> potential virtual temperature profile

 *   z 
v z  v z0,h  P rt
ln 
   h 
k 

  z0,h 

 
(8.60)
Potential Virtual Temperature Profile
Influence function for energy
(8.61,3)
z
dz
h  
1  h
z0,h
z
 1  h
 P r L z  z0,h
t



1  h z1
 2ln
1

1 h z0,h


0




 

z
0
L
s tabl e
z
0
L
uns table
z
0
L
neutral
Vertical Profiles in a Canopy
ln z0,m
Relationship among dc, hc, and z0,m
Fig. 8.4
Vertical Profiles in a Canopy
Momentum
u*
vh z 
k
  z  d 

z

d


c   
c 
ln

m
z
 L 

  0,m 

Potential virtual temperature
(8.66)
(8.67)
 *   z  d c 
 z  dc 
v z  v dc  z0,h  P rt
ln

   h 
k 
 L 
  z0,h 



Specific humidity

(8.68)

q*
q v z  q v d c  z0,v  P rt
k
  z  d 

z

d


c   
c 
ln

h
 L 

  z0,v 

Local v. Nonlocal Closure Above Surface
Local closure turbulence scheme
Mixes momentum, energy, chemicals between adjacent layers.
Hybrid
E-l
E-d
Nonlocal closure turbulence scheme
Mixes variables among all layers simultaneously
Free-convective plume scheme
Hybrid Scheme
For momentum for stable/weakly unstable conditions (8.70)
Captures small eddies but not large eddies due to free
convection --> not valid when Rib is large and negative
2
2 Ri  Ri
u
v




c
b
Km,zx  Km,zy  l2e     
 z 
 z 
Ric
Mixing Length
For energy
(8.71)
kz
le 
1  kz lm
K h,zz  K m,zx Prt
E (TKE)-l Scheme
Prognostic equation for TKE
(8.72)
E  
E 
 sqle 2E
  Ps  Pb   d
t z 
z 
Prognositc equation for mixing length
(8.73)
2

2El e   
2El e 
 le  
 sl le 2E
  lee1 Ps  Pb  l e d 1 e2   
t
z 
z 
 kz  



Production rate of shear
 u  2  v 2 
Ps  Km      
 z  

 z 

(8.74)
E-l Scheme
Production rate of buoyancy
(8.75)
g
v
Pb  
Kh
v
z
Dissipation rate of TKE
(8.76)
d 
2 E 3 2
B1l e
Diffusion coefficients
Km  SM l e 2E
(8.77)
Kh  Sh l e 2E
E-d TKE
Prognostic equation for dissipation rate
 d
  Km  d 
d
 
 c 1
Ps  Pb c 2

t
z    z 
E
Eddy diffusion coefficient for momentum
(8.88)
 2d
E
(8.89)
E2
K m  c
d
Diagnostic equation for mixing length
32
34 E
l e  c
d
(8.90)
Heat Conduction Equation
Heat conduction equation
Ts
1   Ts 

 s

t
 g cG z 
z 
(8.91)
Thermal conductivity of soil-water-air mixture
(8.92)


log10  p 2.7
 s  max 418e
, 0.172


Moisture potential
Potential energy required to extract water from capillary and
adhesive forces in the soil
(8.93)
 wg,s  b
 p   p,s 
 w 

 g 
Heat Conduction Equation
Density x specific heat of soil-water-air mixture

(8.94)

gcG  1 wg,s scS  wgwcW
Rate of change of soil water content
(8.95)
wg
   wg

    p

 1  Dg
 Kg 
Kg 
t
z   z
z
 z 

Hydraulic conductivity of soil
Coefficient of permeability of liquid through soil
 wg 2b3
Kg  Kg,s 
 w 

 g,s 
(8.96)
Heat Conduction Equation
Diffusion coefficient of water in soil
(8.97)
b 3
bKg,s  p,s  wg 
bKg,s  p,s
Dg  Kg






wg
wg
wg,s
 wg,s 
 p
 wg  b2

 w 

 g,s 
Heat Conduction Equation
Rate of change of ground surface temperature
(8.98)
Ts
1   Ts


 s
 Fn,g  H f  Le E f 
t
 g cG z 
z

Rate of change of moisture content at the surface
w g


t
z
(8.99)
E f  Pg 
 w g
Dg z  K g  


w 
Surface energy balance equation
Ts
 s,1
 Fn,g  H f  L eE f  0
z
(8.103)
Temp and Moisture in Vegetated Soil
Surface energy balance equation
(8.103)
Ts
 s,1
 Fn,g  H f  L eE f  0
z
Surface irradiance
(8.104)
F n,g  fsFs  Fi   s BTg4
Vertical turbulent sensible heat flux
a c p,d 
H f   fs
p zr 
Ra 

(8.105)
Tg 
a c p,d Taf Tg 
  fv



Pg 
R f 
Pg 

 Pf

Temp and Moisture in Vegetated Soil
Vertical turbulent latent heat flux

 
(8.106)

 
a
a
Le E f   fs Le
 g q v zr  qv,s Tg  fv Le
g q af  q v,s Tg
Ra
Rf
Temperature of air in foliage
(8.109)
Taf  0.3Ta zr  0.6Tf  0.1Tg
Specific humidity of air in foliage
q af  0.3q v zr   0.6q f  0.1q g
(8.110)
Foliage Temperature
Iterative equation for foliage temperature
(8.115)
 v s

4 
F


F



T
v i
B g 
 s
v  s  v s

fv 
  v  2 s   v  s   T 4

v B f
      

 v
s
v s

 Hv  Le Ed  Le Et
Sensible heat flux
Hv  1.1LT
(8.116)
a c p,d
RfPf
Taf  T f 
Foliage Temperature
Direct evaporation

 
ad
Ed  LT
q af  q v,s T f
Rf
Transpiration
(8.117)
(8.118)
a 1   d 
Et   LT
q af  q v,s T f
R f  Rst

Evaporation function
23


W
c



 d  Wc,m ax 


1
 
(8.119)
 
q af  q v,s T f
 
q af  q v,s T f
Foliage Temperature
 
v s


4
F


F



T
v i
B g,t h 
  s

v  s  v s
 f 


v
   v  2 s   v  s

4

3


T
v B f,t,n
 


  v   s   v s





a c p,d

  d


a t 
Taf  Le LT a  
1.1LT

Rf P f


R f R f  Rst 






 

dqv,s T f ,t,n


 q af  q v,s T f ,t,n 
T f ,t,n  

dT
 



 







T f ,t,n1 
a c p,d 
  v  2 s   v  s
3
4


T

1.1L

v B f ,t,n
T







R
P
v
s
v
s
f
f






dq
T
  d a 1   d   v,s f ,t,n


L
L




 e T a  R

R

R

dT
f
f
st










(8.122)
Temperature of Vegetated Soil
 a c p,d 

Tg,t,n1 
 f

p zr  

(8.125)
s R
P

a 
g 





 a c p,d Taf,t Tg,t,n1 




 fv R

P
P

f 
 f
g



  L

a
e
 fs
 g qv zr   q v,s Tg,t,n1 


Ra



L
 fv a e  g q af  q v,s Tg,t,n1


Rf



4
 fs Fs  Fi   B s Tg,t,n1



  s,1

 D T1,t  Tg,t,n1

 1

Tg,t,n  Tg,t,n1 
a c p,d
a c p,d
 s,1
3
fs R P  fv R P  4 s  B Tg,t,n1  D
a g
f g
1








Modeled/Measured Temperatures
Fig. 8.5
Modeled/Measured Temperatures
Fig. 8.5
C)
50
o
Temperature (
Modeled/Measured Temperatures
Predicted
Measured
40
Lodi (LOD)
30
20
10
0
0
24
48
72
Hour after first midnight
96
Fig. 8.5
Road Temperature
(8.128)
a c p,d 
Tg,t,n1 

p zr  

Pg 
 Ra 



a Le

 d q v zr   q v,s Tg,t,n1

 Ra

4
F

F



T
 s
i
B as g,t,n1

  as T
1,c,t  Tg,t,n1

 D1
Tg,t,n  Tg,t,n1 
a c p,d
 as
3
 4 as B Tg,t,n1 
Ra Pg
D1


















Temperatures of Soils and Surfaces
Fig. 8.5
C)
o
Temperature (
Modeled/Measured Temperatures
50
Predicted
Measured
40
Fremont (FRE)
30
20
10
0
0
24
48
72
Hour after first midnight
96
Fig. 8.5
Snow Depth
(8.129)
Ds,t  Ds,t h  hPs

 
 
 


 qv zr   q v,s min Tg,t , Ts,m
 fs
Ra
a 
h

sn 
q af  q v,s min Tg,t , Ts,m
 fv

Rf



a c p,d 
Ts,m 
 

p zr  

  fs
R
P

a 
g 



 



a c p,d Taf ,t Ts,m
 







 fv R f 
P
P


f
g






a Ls


f
q
z

q
T
 s

v  r  v,s s,m
R
a




a Ls
q af  q v,s Ts,m
  fv

R
f




 sn
4
  fs Fs  Fi   B  sn Ts,m 
T1,t  Ts,m 
D

1





h
 sn Lm








Water Temperature
(8.130)
a c p,d 
Tg,t h 

p zr  

Pg 
 Ra 




 a Le
q v zr   q v,s Tg,t  h

 Ra

 F  F    T 4
i
B w g,t h
 s



Tg,t  Tg,t h  h

 sw c p,sw Dl














Sea Ice Temperature
(8.131)
a c p,d 

Tg,t,n1 

p zr  
 
Pg 
 Ra 

 


 a L s q z   q

T
v r
v,s g,t,n1 

R
 a





4
 Fs  Fi   B  i Tg,t,n1






i


Ti, f  Tg,t,n1


Di,t h

Tg,t,n  Tg,t,n1   c
i
a p,d
3
 4 i Tg,t,n1 
Ra Pg
Di,t h





Temperature of Snow Over Sea Ice
Tg,t,n  Tg,t,n1 
(8.134)
a c p,d 

Tg,t,n1 


p zr  

Pg 
 Ra 





 a Ls q z   q



v r
v,s Tg,t,n 1
R
 a



4
Fs  Fi   B  sn Tg,t,n 1



 sn  i

Ti, f  Tg,t,n1 


 sn Di,t h   i Ds,t h

a c p,d
 sn  i
3
 4 sn Tg,t,n1 
Ra Pg
 sn Di,t h   i Ds,t h




