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```									             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
jacobson@stanford.edu
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)
1
2  2

      2
 z  a  w    w   

u         v

Kinematic vertical turbulent momentum flux (m2 s-2)       (8.3)
 zx
w    
u
a
 zy
w    
v
a
Friction wind speed (m s-1)                               (8.8)
Scaling param. for surface-layer vert. flux of horiz. momentum
1
2  4

   2
u*   wu  w      z a s
s
v
s 

12

Heat and Moisture Fluxes
Vertical turbulent sensible-heat flux (W m-2)            (8.4)

v
H f   a c p,d w  

Kinematic vert. turbulent sensible-heat flux (m K s-1)   (8.5)
Hf
v
w   
a c p,d
Vertical turbulent water vapor flux (kg m-2 s-1)         (8.6)
E f  a w  
qv

Kinematic vert. turbulent moisture flux (m kg s-1 kg-1) (8.7)
Ef
w   
qv
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
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
ln z

Fig. 6.5
Roughness Length for Momentum
Over smooth ocean with slow wind                     (8.9)
a         a
z0,m  0.11     0.11
u*         a u*
Over rough ocean, fast wind (Charnock relation)      (8.10)
2
u*
z0,m   c
g
Over urban areas containing structures               (8.11)
ho So
z0,m  0.5
Ao
Over a vegetation canopy                             (8.12)


z0,m  hc 1  0.91e
0.0075LT

Roughness Length for Momentum
Surface Type               z0,m (m)       h c (m)   d c (m)
Smooth sea                       0.00001
Rough sea                        0.000015-
0.0015
Ice                              0.00001
Snow                             0.00005-0.0001
Level desert                     0.0003
Short grass                      0.03-0.01        0.02-0.1
Long grass                       0.04-0.1         0.25-1.0
Savannah                         0.4              8          4.8
Agricultural crops               0.04-0.2         0.4-2      0.27-1.3
Orchard                          0.5-1.0          5-10       3.3-6.7
Broadleaf evergreen forest       4.8              35         26.3
Broadleaf deciduous trees        2.7              20         15
Broad- and needleleaf trees      2.8              20         15
Needleleaf-evergreen trees       2.4              17         12.8
Needleleaf deciduous trees       2.4              17         12.8
Short vegetation/C4 grassland    0.12             1          0.75
Broadleaf shrubs w/ bare soil    0.06             0.5        0.38
Agriculture/C3 grassland         0.12             1          0.75
2500 m2 l ot w/ a b uilding 8-m 0.26              8
high and 160 m2 silhouette
25,000 m2 lot w/ a building 80-m 5.1              80
high and 3200 m2 silhouette
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 
 a c p,m
Molecular diffusion coefficient of water vapor       (8.14)
1.94
5  T          1013.25 hPa 
Dv  2.11 10                             
273.15 K           pa   
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
Decrease in eddy size from large eddy to small eddy to zero
due to dissipation
Turbulence Models
Kolmogorov scale                                    (8.15)
1
3  4
k   a 
 d 
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 s  CD v hzr  zr  u z0,m
u                    u

w s  CD vh zr  zr  vz0,m 
v                    v
Kinematic Vertical Momentum Flux
Bulk aerodynamic formulae                                  (8.18)

w s  CD v hzr  zr  u z0,m
u                    u
K-theory                                                   (8.18)
u
 
w   Km,zx
u
s         z



u u zr   u z0,m   
z     zr  z0,m

Eddy diffusion coef. in terms of bulk aero. formulae       (8.20)


Km,zx  Km,zy  CD v h zr  zr  z0,m      
Kinematic Vertical Energy Flux
Bulk aerodynamic formulae                                 (8.21)

w s  CH vh zr v zr  v z0,h
v
K-theory                                                  (8.23)
 
     
w    Kh,zz v
v
s          z

v  v  r  v 0,h
    z   z       
z        zr  z0,h

Eddy diffusion coef. in terms of bulk aero. formulae      (8.25)


Kh,zz  CH v h zr  zr  z0,h   
Vertical Turbulent Moisture Flux
Bulk aero. kinematic vertical turbulent moisture flux    (8.26)

w s  CE vhzr q v zr   q v z0,v 
qv

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                    z
0     stable
     m
L                 L

       z 1 4         z
 m     m 
1                          0     unst able
       L              L
                          z
1                           0     neut ral
                          L
Integrate (8.28) from z0,m to zr                          (8.30)
k v h zr 
u*  z          dz
r
z0,m  m z
Integral of Dimensionless Wind Shear
Integral of the dimensionless wind shear                         (8.31)
zr          dz
 z0, m  m z 
m
 zr
ln z     
L
          
zr  z0,m
z
L
0   stable
     0,m
                 14                         14
   zr   1                     z0,m 
1 m                      m
1           1
            L                     L 
ln                         ln
               1
zr  4           
1
z0,m  4
   m   1

1
               m
1           1
              L                       L 

2 t an1   
14
zr             
1
z0,m  4   z
1 m   2 t an1   m
                          1                  0   unst able
                   L                      L       L

ln zr                                                     z
0   neut ral
 z0,m
                                                          L
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)

u3 v
*             u2 
* v
L                   

kg w 
v
s
     kg*

Kinematic vertical energy flux                         (8.33)

w s  u**
v
Potential Temperature Scale
h    z 
v

k   * z

Parameterization of *                                    (8.35)

Pr   z                      z
0   stable
 t h L                        L

        z 1 2            z
 h  Prt   h 
1                         0   unst able
        L                 L
                              z
Prt                             0   neut ral
                              L
Potential Temperature Scale

Turbulent Prandtl number
Km,zx
Prt 
Kh,zz

Integrate (8.23) from z0,m to zr                        (8.37)

h

z 
v                     * 

k  zr   z0,h
v         v 
zr      dz
* z
k
z0,h h z
Integral of dimensionless temperature gradient                       (8.38)
zr       dz
z0, h
h
z


h

Prt ln z
zr

L
 zr  z0,h                               z
L
0   stable
         0,h

                                                12    

           zr 1 2                 z0,h 
     1 h   1                 h
1                 1
L                      L           z
Prt 
ln               12     ln                  12         0   unst able
               zr                     z0,h           L
    h   1
     1                          h
1                1
L                      L        
                                                      

         zr                                                  z
Prt ln z                                                       0   neut ral
         0,h                                                 L
Equations to Solve Simultaneously
Solution requires iteration

k v h zr                        3
u* v          2
u* 
u*  z                                                    v
L                     
r
z0,m  m z
dz

kg w 
v
s
     kg*

* 
          
k  zr   z0,h
v         v
zr      dz
z0,h h z
Noniterative Parameterization
Friction wind speed                                  (8.40)
k v h zr 
u*                       Gm

ln zr z0,m     
Potential temperature scale                          (8.40)

* 
            
k vh zr   v z r   v z0,h
2
Gh
u* Prt ln  r z0,m 
2
z
Scale Parameterization
Potential temperature scale                              (8.41)
9.4Ri b
Gm  1                                           Ri b  0

1

70k 2 Ri b zr z0,m           
0.5


ln 2 zr z0,m   
9.4Ri b
Gh  1                                           Ri b  0
Ri b zr z0,m 
2                      0.5
50k
1
ln2 zr z0,m 
1
Gm ,Gh                                           Ri b  0
1 4.7Ri b      2
Bulk Richardson Number
Ratio of buoyancy to mechanical shear                    (8.39)

               zr  z0,m
          2
g  zr   z0,h
v         v
Ri b 

 z0,h u zr   v zr  zr  z0,h 

2         2
v
g 
v                    (8.42)
 z
v
Ri g         2         2
u  v 
    
z  z 
Table 8.2. Vertical air flow characteristics for different Rib or Rig
Value of          Type of      Level of Turbulence      Level of
Rib or Rig         Flow         Due to Buoyancy        Turbulence
Due to Shear
Large, negative     Turbulent             Large              Small
Small, negative     Turbulent             Small              Large
Small positive      Turbulent     None (weak stable)          Large
Large positive      Laminar       None (strong stable)        Small
g 
v
 z                 (8.42)
Ri g         v
2        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                                     (8.8)
1
2  4

   2
u*   wu  w   
       s
v
s 
Bulk aerodynamic kinematic momentum flux                (8.16)

w s  CD v hzr  zr  u z0,m
u                    u
Friction wind speed                                     (8.43)
u*  v h zr  CD
Rederive momentum flux in terms of similarity theory (8.43)
2
u*
 
w    
u
s    v h zr 
u zr 
Eddy Diff. Coef. for Mom. Similarity
K-theory kinematic turbulent momentum fluxes              (8.18)
u                            v
 
w   Km,zx
u
s         z
 
w   Km,zy
v
s         z

Similarity theory kinematic turbulent fluxes              (8.44)
2                                 2
u*                                u*
wu s   v z  u zr      wv s   v z  v zr 
h r                               h r

Combine the two                                           (8.46)
2
u*
K m,zx  K m,zy 
vh zr 
zr  z0,m 
Example Problem
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
---> vh zr    = 11.18 m s-1     ---> Rib            = -8.15 x 10-3
---> Gm         = 1.046           ---> Gh             = 1.052
---> u*         = 0.662 m s-1     ---> *             = -0.188 K
--->   L        = -169 m
2
u*
---> K m,zx 
vh zr 
zr  z0,m      = 0.39 m2 s-1

u* *
---> Kh,zz                 = 0.41 m2 s-1 ---> Km,zx Kh,zz = 0.95
v z
Eddy Diff. Coef. for Mom. Similarity
Dimensionless wind shear                           (8.28)
m    z  vh

k   u* z
Wind shear                                         (8.46)
2
u*
K m,zx 
vh zr 
zr  z0,m 
Combine expressions above                          (8.48)
kzu*
K m,zx  K m,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 s  u**
v

Surface vertical turbulent sensible heat flux      (8.53)

v  
H f  acp,d w   acp,du* *
s
Energy, Moisture Fluxes from Similarity
Vertical kinematic water vapor flux             (8.49)

w s  u*q *
qv
Surface vertical turbulent water vapor flux     (8.53)

     
E f  a w   au*q*
qv
s
q   z q v

k   q * z
Specific humidity scale                         (8.52)

q* 
             
k q v zr   q v z0,v
z r  dz
z 0,v h
z
Logarithmic Wind Profile
Dimensionless wind shear                                     (8.28)
m    z  vh

k   u* z

Rewrite                                                      (8.57)
 vh z 
z

u*
kz
m 
u*
kz

1  1  m       
Integrate --> surface layer vertical wind speed profile      (8.59)

u*     z          
v h z         
ln          m 
k    z0,m 
                 

Logarithmic Wind Profile
Influence function for momentum                            (8.61,2)
z          dz
m  
z 0,m
1   m  z
  m

 L z  z0,m                                         z
L
0   stable


ln      
1   m z 
2

 1   m z    
1 2

                                        1 2
1
        
     m z0,m
   z
2 




1        
m 0,m         



2 t an m z  2 t an m z0,m

1          1          1
           1   z
L
0   un st able

0                                                     z
                                                        0   neut ral
                                                      L
Logarithmic Wind Profile
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
Height above surface (m)

8

6
z 0, m =1.0 m       z 0, m =0.1 m
4

2

0
0         2     4      6     8          10   12     Fig. 8.3
Wind speed (m -1)
s
Potential Virtual Temperature Profile
h    z 
v

k   * z

Rewrite                                                      (8.58)
   *       *
z
v 
kz
h 
kz

1  1 h     
Integrate --> potential virtual temperature profile          (8.60)

 *   z         
 
v z   z0,h  Prt
v                 
ln         h 
k  z0,h        
                
Potential Virtual Temperature Profile
Influence function for energy                          (8.61,3)

z             dz
h  
z0 ,h
1   h  z

 1  h

 Pr L z  z0,h                                 z
L
0   stable
 t

     1   h z1                               z
 2 ln                                               0   un st able
 
 1  h z0,h

1                              L

0                                                z
0   neut ral

                                                 L
Vertical Profiles in a Canopy
Relationship among dc, hc, and z0,m
ln z0,m

Fig. 8.4
Vertical Profiles in a Canopy
Momentum                                                    (8.66)
u*    z  d     z  dc 
v h z         
ln       c   
  m          
k     z0,m 
                L   

Potential virtual temperature                               (8.67)

 *   z  d c       z  dc 
             
v z   dc  z0,h  Prt
v                      
ln
k   z0,h 
  h 
 L 

                               

Specific humidity                                           (8.68)
 z  d               
c   z  dc 
             
q v z  q v d c  z0,v  Prt
q*
k
 
ln        
  z0,v 
h       
 L 
                         
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

u 2 v 2 Ri c  Ri b
Km,zx  K m,zy  l2     
e
 z   z     Ri c
Mixing Length                                       (8.71)
kz
le 
1  kz l m
For energy

Kh,zz  Km,zx Prt
E (TKE)-l Scheme
Prognostic equation for TKE                                   (8.72)
E           E 
 sq l e 2E     Ps  P   d
b
t z         z 

Prognositc equation for mixing length                         (8.73)

2El e             2El e                                 l e  
2
 sl l e 2 E            l ee1 Ps  P   l e d  e2   
b            1
t     z             z                                
     kz  


Production rate of shear                                      (8.74)
 u 2 v 2 


P  K m      
s

 z  z  
                 
E-l Scheme
Production rate of buoyancy                           (8.75)
g    v
P 
b      Kh

v     z

Dissipation rate of TKE                               (8.76)
2E 3 2
d 
B1l e

Diffusion coefficients                                (8.77)

Km  SM le 2E               Kh  Shle 2E
E-d TKE
Prognostic equation for dissipation rate                (8.88)
 d    Km  d        d                   2
  c 1 E P  Pb  c 2
                                          d
s
t   z    z                             E

Eddy diffusion coefficient for momentum                 (8.89)

E2
Km  c
d

Diagnostic equation for mixing length                   (8.90)

34 E3 2
le  c
d
Heat Conduction Equation
Heat conduction equation                                 (8.91)
Ts       1   Ts 
            s 
t     g cG z  z 
Thermal conductivity of soil-water-air mixture           (8.92)
      log 10  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)
b
wg,s 
 w 
 p   p,s      
 g 
Heat Conduction Equation
Density x specific heat of soil-water-air mixture      (8.94)

         
 g cG  1  w g,s s c S  w g  wc W

Rate of change of soil water content                   (8.95)
wg       p       wg       
     g 
K        1  Dg     Kg 
t   z   z     z   z      

Hydraulic conductivity of soil
Coefficient of permeability of liquid through soil   (8.96)
2b3
 wg 
w 
Kg  Kg,s      
 g,s 
Heat Conduction Equation
Diffusion coefficient of water in soil                    (8.97)

b 3                          b2
 p  bKg,s  p,s  wg                bKg,s  p,s  wg 
Dg  Kg                  
w                          
w  
wg       wg        g,s                 wg,s       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   (8.99)
wg         wg        E f  P 
g
      Dg z  Kg         
t   z                   w 

Surface energy balance equation                     (8.103)
Ts
 s,1      F  H f  L eE f  0
n,g
z
Temp and Moisture in Vegetated Soil
Surface energy balance equation                              (8.103)
Ts
 s,1      F  H f  L eE f  0
n,g
z

4
F n,g  fs Fs  F   s  BTg
i

Vertical turbulent sensible heat flux                        (8.105)

 c p,d               Tg       c p,d Taf Tg 
p zr  
a                                  a
H f   fs                              fv                 
Ra                 Pg 
       R f  P f
     Pg 

Temp and Moisture in Vegetated Soil
Vertical turbulent latent heat flux                       (8.106)

Le E f   fs Le


a  q z  q
Ra g v r      v,s        
Rf          
Tg  fv Le a g q af  q v,s Tg      
Temperature of air in foliage                             (8.109)

Taf  0.3Ta zr  0.6Tf  0.1Tg

Specific humidity of air in foliage                       (8.110)

q af  0.3q v zr   0.6q f  0.1q g
Foliage Temperature
Iterative equation for foliage temperature                  (8.115)
                        v s         4 
Fs   v F  
         i                       B Tg 
v  s  v s
fv                                         
  v  2  s   v  s   T 4
                                        
  v   s   v s v B f
                                        


 Hv  Le Ed  Le Et

Sensible heat flux                                          (8.116)
 c p,d
Hv  1.1LT
a
R fPf      Taf  T f 
Foliage Temperature
Direct evaporation                                       (8.117)
 d
Ed  LT a
Rf                  
q af  q v,s T f

Transpiration                                            (8.118)
 1  d 
Et   LT
a
R f  Rst q af  q v,sTf 
Evaporation function                                     (8.119)
        23

 Wc 

 d  Wc,max 
 
q af  q v,s T f



1
                                 
q af  q v,s T f
Foliage Temperature
                           v s                               
   Fs   v Fi                          B Tg,t h 
4
   (8.122)
v  s  v s                  
 fv                                                          
   v  2 s   v  s               4                       
3
                       v  BT f,t,n                     
    v       s     v s


                                                                 

           c p,d                     
 d          t  
a                                         a
1.1LT               T  Le LT                            
           R f P f af                a
R f R f  Rst 
                      
                                                                 
  
  q


dq v,s T f ,t,n           

         

 q af  v,s T f ,t,n              dT
T f ,t,n  
 
                                                    
                                                                 

                                                                 

T f ,t,n1 
  v  2  s   v  s            3                 c p,d 
a
4
                   v BT f ,t,n  1.1LT
      v    s       v s                              R f P f 
                                                              
              d     d   dq v,s T f ,t,n

L e LT          a
1         

            
          a
 R f     R f  Rst           dT              
                                                          
Temperature of Vegetated Soil
  c p,d 
a                    Tg,t,n1       
 f
s R         p zr  
                              (8.125)
         a                 Pg       
                                            
  c p,d Taf ,t Tg,t,n1 
a                                     
 fv                                    
R f  P f
             Pg  
                                            
  L                                       
 fs

a e q z q
Ra

g v  r                  
v,s Tg,t,n1 

                                            


 Le
Rf                 
 fv a  g q af  q v,s Tg,t,n1           

                                            
 fs Fs  F   B s T     4                
i              g,t,n1
                                            
  s,1                                      

  D1
               
T1,t  Tg,t,n1                     

Tg,t,n  Tg,t,n1 
 c p,d
a               c p,d
a                    3         s,1
fs            fv             4 s  B Tg,t,n1 
Ra Pg           R f Pg                           D1
Modeled/Measured Temperatures

Fig. 8.5
Modeled/Measured Temperatures

Fig. 8.5
Modeled/Measured Temperatures
50
Temperature (
C)

40         Predicted      Lodi (LOD)
o

Measured
30
20
10
0
0   24        48          72        96
Hour after first midnight

Fig. 8.5
(8.128)
  c p,d 
a                    Tg,t,n1         
           p zr  
                           
 Ra                   Pg          
                                          
 a                          
  Le  d q v zr   q v,s Tg,t,n1
 Ra
  


                        4                 
F  Fi   B  as Tg,t,n1
 s                                        
                                          
 D    
 as T1,c,t  Tg,t,n1
 1
                


Tg,t,n  Tg,t,n1 
 c p,d
a                      3          as
 4 as  B Tg,t,n1 
Ra Pg                            D1
Temperatures of Soils and Surfaces

Fig. 8.5
Modeled/Measured Temperatures
50
Temperature (
C)

40         Predicted Fremont (FRE)
o

Measured
30
20
10
0
0   24        48          72       96
Hour after first midnight

Fig. 8.5
Snow Depth
Ds,t  Ds,t h  hP                                                                      (8.129)
s


 fs

 q z   q
v r            
v,s min Tg,t , Ts,m    
 
        c p,d 
a                      Ts,m                  
 
a 
Ra                     fs              p zr  
                                 
h                                                       Ra                     
Pg                   
sn 
 f              
q af  q v,s min Tg,t , Ts,m  
     
        c p,d Taf ,t Ts,m 
a


 v
                   Rf                       f
  v

                                 
         R f  P f
            Pg 
                  
                                                      

 fs

 Ls
a
Ra
                 
q v zr   q v,s Ts,m                


                                                      
 fv

 Ls
a
Rf               
q af  q v,s Ts,m                     

                                      sn             
4
 fs Fs  Fi   B  sn Ts,m 
D1
         
T1,t  Ts,m 
                                                      

                                                      

h
 sn Lm
Water Temperature
(8.130)

 c p,d 
a                     Tg,t h      
           p zr  
                       
 Ra                    Pg      
                                       
                                       
 Le

a
 Ra
                 
q v zr   q v,s Tg,t  h   



                                       
F  F    T 4                       
 s     i         B w g,t h            

                                       

Tg,t  Tg,t h  h
 swc p,sw Dl
Sea Ice Temperature
  c p,d 
a                   Tg,t,n1     (8.131)
           p zr 
                    
 Ra                   Pg  

                                     
 Ra


  L s q z  q
a
v r           
v,s         
Tg,t,n1 


                                     
                       4             
F  Fi   B  i Tg,t,n1
s                                  
                                     
                                     
    i
 Di,t h Ti, f  Tg,t,n1          

                                     
Tg,t,n  Tg,t,n1   c
a p,d            3           i
 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)

 c                                            
a p,d              Tg,t,n1 
         p zr                          
 Ra                    Pg               
                                                

a
 Ra

 Ls q z   q T
v r             
v,s g,t,n 1            


                         4                      
Fs  F   B  snTg,t,n 1
i                                         
                                                
          sn  i
 sn Di,t h   i Ds,t h
                 
Ti, f  Tg,t,n1 

                                                
 c p,d
a                  3                       sn i
 4 sn Tg,t,n1 
Ra Pg                           sn Di,t h   i Ds,t h

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