Presentation Slides for
Chapter 8of
Fundamentals of Atmospheric Modeling 2nd Edition
Mark Z. JacobsonDepartment of Civil & Environmental Engineering
Stanford UniversityStanford, CA [email protected]
March 10, 2005
StressForce per unit area (e.g. N m-2 or kg m-1 s-2)
Reynolds Stress
Reynolds stress
Stress that causes a parcel of air to deform during turbulent motion of air
Fig. 8.1. Deformation by vertical momentum flux
Eddymixing
x
yz
Force arisingfrom w'u'
′ w ′ u
Stress from vertical transfer of turbulent u-momentum (8.1)
τzx=−ρa ′ w ′ u
zx = stress acting in x-direction, along a plane (x-y) normal to the z-direction
Magnitude of Reynolds stress at ground surface (8.2)
Momentum Fluxes
Kinematic vertical turbulent momentum flux (m2 s-2) (8.3)
Friction wind speed (m s-1) (8.8)Scaling param. for surface-layer vert. flux of horiz. momentum
′ w ′ u =−τzxρa
′ w ′ v =−τzyρa
τz =ρa ′ w ′ u ( )2
+ ′ w ′ v ( )2⎡
⎣ ⎢ ⎤ ⎦ ⎥
12
u* = ′ w ′ u ( )s2
+ ′ w ′ v ( )s2⎡
⎣ ⎢ ⎤ ⎦ ⎥
14= τz ρa( )s
12
Vertical turbulent sensible-heat flux (W m-2) (8.4)
Heat and Moisture Fluxes
Kinematic vert. turbulent sensible-heat flux (m K s-1) (8.5)
Kinematic vert. turbulent moisture flux (m kg s-1 kg-1) (8.7)
Vertical turbulent water vapor flux (kg m-2 s-1) (8.6)
H f =ρacp,d ′ w ′ θ v
′ w ′ θ v =H f
ρacp,d
Ef =ρa ′ w ′ q v
′ w ′ q v =E fρa
Surf. Roughness Length for MomentumHeight 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 z0,m
|vh|00
ln z
Fig. 6.5
ln z
Over smooth ocean with slow wind (8.9)
Roughness Length for Momentum
Over rough ocean, fast wind (Charnock relation) (8.10)
Over a vegetation canopy (8.12)
Over urban areas containing structures (8.11)
z0,m≈0.11νau*
=0.11ηa
ρau*
z0,m≈αcu*
2
g
z0,m≈0.5hoSoAo
z0,m=hc 1−0.91e−0.0075LT( )
Roughness Length for MomentumSurface Type z0,m (m) h
c (m) d
c (m)
Smooth sea 0.00001Rough sea 0.000015-
0.0015Ice 0.00001Snow 0.00005-0.0001Level desert 0.0003Short grass 0.03-0.01 0.02-0.1Long grass 0.04-0.1 0.25-1.0Savannah 0.4 8 4.8Agricultural crops 0.04-0.2 0.4-2 0.27-1.3Orchard 0.5-1.0 5-10 3.3-6.7Broadleaf evergreen forest 4.8 35 26.3Broadleaf deciduous trees 2.7 20 15Broad- and needleleaf trees 2.8 20 15Needleleaf-evergreen trees 2.4 17 12.8Needleleaf deciduous trees 2.4 17 12.8Short vegetation/C4 grassland 0.12 1 0.75Broadleaf shrubs w/ bare soil 0.06 0.5 0.38Agriculture/C3 grassland 0.12 1 0.75
2500 m2 l ot w/ a b uilding 8-m
high and 160 m2 silhouette
0.26 8
25,000 m2 lot w/ a building 80-m
high and 3200 m2 silhouette
5.1 80 Table 8.1
Surface roughness length for energy (8.13)
Roughness Length for Energy, Moisture
Surface roughness length for moisture (8.13)
Molecular diffusion coefficient of water vapor (8.14)
Molecular thermal diffusion coefficient (8.14)
z0,h =Dhku*
z0,v =Dvku*
Dh =κa
ρacp,m
Dv =2.11×10−5 T273.15 K
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 1.94 1013.25 hPa
pa
⎛
⎝ ⎜
⎞
⎠ ⎟
TurbulenceGroup of eddies of different size. Eddies range in size from a couple of millimeters to the size of the boundary layer.
Turbulence Description
Turbulent kinetic energy (TKE)Mean kinetic energy per unit mass associated with eddies in turbulent flow
Dissipation Conversion of turbulence into heat by molecular viscosity
Inertial cascade
Decrease in eddy size from large eddy to small eddy to zero due to dissipation
Kolmogorov scale (8.15)
Turbulence Models
Reynolds-averaged modelsResolution greater than a few hundred metersDo not resolve large or small eddies
Large-eddy simulation models Resolution between a few meters and a few hundred meters Resolve large eddies but not small ones
Direct numerical simulation models Resolution on the order of the Kolmogorov scaleResolve all eddies
ηk =νa3
εd
⎛
⎝ ⎜
⎞
⎠ ⎟
1 4
Bulk aerodynamic formulae (8.16-7)Diffusion coefficient accounts forSkin drag: drag from molecular diffusion of air at surfaceForm drag: drag arising when wind hits large obstaclesWave drag: drag from momentum transfer due to gravity waves
Kinematic Vertical Momentum Flux
′ w ′ u ( )s=−CD v h zr( ) u zr( )−u z0,m( )[ ]
′ w ′ v ( )s=−CD v h zr( ) v zr( )−v z0,m( )[ ]
Kinematic Vertical Momentum Flux
K-theory (8.18)
′ w ′ u ( )s=−Km,zx
∂u ∂z
Eddy diffusion coef. in terms of bulk aero. formulae (8.20)
Wind speed gradient (8.19)
∂u ∂z
=u zr( ) −u z0,m( )
zr −z0,m
Km,zx=Km,zy≈CD v h zr( ) zr −z0,m( )
′ w ′ u ( )s=−CD v h zr( ) u zr( )−u z0,m( )[ ]
Bulk aerodynamic formulae (8.18)
Kinematic Vertical Energy Flux
K-theory (8.23)
Eddy diffusion coef. in terms of bulk aero. formulae (8.25)
Potential virtual temperature gradient (8.24)
Bulk aerodynamic formulae (8.21)
′ w ′ θ v( )s=−CH v h zr( ) θ v zr( )−θ v z0,h( )[ ]
′ w ′ θ v( )s=−Kh,zz
∂θ v∂z
∂θ v∂z
=θ v zr( )−θ v z0,h( )
zr −z0,h
Kh,zz ≈CH v h zr( ) zr −z0,h( )
Vertical Turbulent Moisture Flux
Bulk aero. kinematic vertical turbulent moisture flux (8.26)
CE≈CH --> Kv,zz =Kh,zz
′ w ′ q v( )s=−CE v h zr( ) q v zr( ) −q v z0,v( )[ ]
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 RelationshipDimensionless wind shear (8.28)
Integrate (8.28) from z0,m to zr (8.30)
Dimensionless wind shear from field data (8.29)
φmk
=zu*
∂ v h∂z
φm=
1+βmzL
zL
>0 stable
1−γmzL
⎛ ⎝ ⎜ ⎞
⎠ ⎟
−14 zL
<0 unstable
1zL
=0 neutral
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
u* =kv h zr( )
φmdzzz0,m
zr∫
Integral of Dimensionless Wind ShearIntegral of the dimensionless wind shear (8.31)
φmdzz
=z0,m
zr∫
lnzr
z0,m+
βmL
zr −z0,m( )zL
>0 stable
ln1−γm
zrL
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 14
−1
1−γmzrL
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 14
+1
−ln
1−γmz0,m
L⎛
⎝ ⎜
⎞
⎠ ⎟ 14
−1
1−γmz0,m
L⎛
⎝ ⎜
⎞
⎠ ⎟ 14
+1
+2tan−1 1−γmzrL
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 14
−2tan−1 1−γmz0,m
L⎛ ⎝ ⎜
⎞ ⎠ ⎟ 14
zL
<0 unstable
lnzr
z0,m
zL
=0 neutral
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Monin-Obukhov LengthHeight proportional to the height above the surface at which buoyant production of turbulence first equals mechanical (shear) production of turbulence. (8.32)
Kinematic vertical energy flux (8.33)
L =−u*3θ v
kg ′ w ′ θ v( )s
=u*2θ v
kgθ*
′ w ′ θ v( )s≈−u*θ*
Potential Temperature ScaleDimensionless temperature gradient (8.34)
Parameterization of * (8.35)
φhk
≈zθ*
∂θ v∂z
φh =
Prt+βhzL
zL
>0 stable
Prt 1−γhzL
⎛ ⎝ ⎜ ⎞
⎠ ⎟ −12 z
L<0 unstable
PrtzL
=0 neutral
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
Potential Temperature Scale
Turbulent Prandtl number
Integrate (8.23) from z0,m to zr (8.37)
Prt =Km,zxKh,zz
θ* =k θ v zr( )−θ v z0,h( )[ ]
φhdzzz0,h
zr∫φhk
≈zθ*
∂θ v∂z
Integral of Dimensionless Temp. Grad.Integral of dimensionless temperature gradient (8.38)
φhdzz
=z0,h
zr∫
Prt lnzr
z0,h+
βhL
zr −z0,h( )zL
>0 stable
Prt ln1−γh
zrL
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 12
−1
1−γhzrL
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 12
+1
−ln
1−γhz0,hL
⎛
⎝ ⎜
⎞
⎠ ⎟ 12
−1
1−γhz0,hL
⎛
⎝ ⎜
⎞
⎠ ⎟ 12
+1
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
zL
<0 unstable
Prt lnzr
z0,h
zL
=0 neutral
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Equations to Solve Simultaneously
Solution requires iteration
θ* =k θ v zr( )−θ v z0,h( )[ ]
φhdzzz0,h
zr∫
L =−u*3θ v
kg ′ w ′ θ v( )s
=u*2θ v
kgθ*u* =
kv h zr( )
φmdzzz0,m
zr∫
Noniterative ParameterizationFriction wind speed (8.40)
Potential temperature scale (8.40)
u* ≈kv h zr( )
ln zr z0,m( )Gm
θ* ≈k2v h zr( ) θv zr( )−θv z0,h( )[ ]
u* Prt ln2 zr z0,m( )Gh
Scale ParameterizationPotential temperature scale (8.41)
Gm=1−9.4Rib
1+70k2 Ribzr z0,m( )
0.5
ln2 zr z0,m( )
Rib ≤0
Gh =1−9.4Rib
1+50k2 Ribzr z0,m( )
0.5
ln2 zr z0,m( )
Rib ≤0
Gm,Gh =1
1+4.7Rib( )2 Rib >0
Bulk Richardson Number
Ratio of buoyancy to mechanical shear (8.39)
Rib =g θ v zr( )−θ v z0,h( )[ ] zr −z0,m( )
2
θ v z0,h( ) u zr( )2
+v zr( )2
[ ] zr −z0,h( )
Gradient Richardson Number(8.42)
Rig =
gθ v
∂θ v∂z
∂u ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2+
∂v ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 2
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
Level of 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
Gradient Richardson Number
(8.42)Rig =
gθ v
∂θ v∂z
∂u ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2+
∂v ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 2
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
′ w ′ u ( )s=−CD v h zr( ) u zr( )−u z0,m( )[ ]
u* = ′ w ′ u ( )s2
+ ′ w ′ v ( )s2⎡
⎣ ⎢ ⎤ ⎦ ⎥
14
′ w ′ u ( )s=−
u*2
v h zr( )u zr( )
u* =v h zr( ) CD
Friction wind speed (8.8)
Bulk aerodynamic kinematic momentum flux (8.16)
Friction wind speed (8.43)
Rederive momentum flux in terms of similarity theory (8.43)
K-theory kinematic turbulent momentum fluxes (8.18)
Eddy Diff. Coef. for Mom. Similarity
Similarity theory kinematic turbulent fluxes (8.44)
Combine the two (8.46)
′ w ′ u ( )s=−Km,zx
∂u ∂z
′ w ′ v ( )s=−Km,zy
∂v ∂z
′ w ′ u ( )s=−
u*2
v h zr( )u zr( ) ′ w ′ v ( )s
=−u*
2
v h zr( )v zr( )
Km,zx=Km,zy≈u*2
v h zr( )zr −z0,m( )
Example Problemz0,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
---> = 0.41 m2 s-1 ---> = 0.95
v h zr( ) RibGm Ghu* θ*L
Km,zx Kh,zz
---> = 0.39 m2 s-1
---> = -169 m
---> = 0.662 m s-1 ---> = -0.188 K
---> = 1.046 ---> = 1.052
---> = 11.18 m s-1 ---> = -8.15 x 10-3
Km,zx=u*2
v h zr( )zr −z0,m( )
Kh,zz =u*θ*
∂θ v ∂z( )
Dimensionless wind shear (8.28)
Eddy Diff. Coef. for Mom. Similarity
Wind shear (8.46)
Combine expressions above (8.48)
φmk
=zu*
∂ v h∂z
Km,zx=Km,zy≈kzu*φm
kz = mixing length: average distance an eddy travels before exchanging momentum with surrounding eddies
Km,zx≈u*2
v h zr( )zr −z0,m( )
Vertical kinematic energy flux (8.49)
Energy Flux from Similarity Theory
Surface vertical turbulent sensible heat flux (8.53)
′ w ′ θ v( )s=−u*θ*
H f ≈ρacp,d ′ w ′ θ v( )s=−ρacp,du*θ*
Surface vertical turbulent water vapor flux (8.53)
Energy, Moisture Fluxes from Similarity
Dimensionless specific humidity gradient (8.51)
Specific humidity scale (8.52)
Ef =ρa ′ w ′ q v( )s=−ρau*q*
φqk
=z
q*
∂q v∂z
q* =k q v zr( ) −q v z0,v( )[ ]
φhdzzz0,v
zr∫
Vertical kinematic water vapor flux (8.49)
′ w ′ q v( )s=−u*q*
Dimensionless wind shear (8.28)
Logarithmic Wind Profile
Rewrite (8.57)
Integrate --> surface layer vertical wind speed profile (8.59)
φmk
=zu*
∂ v h∂z
∂ v h z( )
∂z=
u*kz
φm=u*kz
1− 1−φm( )[ ]
v h z( ) =u*k
lnz
z0,m
⎛
⎝ ⎜
⎞
⎠ ⎟ −ψm
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Influence function for momentum (8.61,2)
Logarithmic Wind Profile
ψm = 1−φm( )z0,m
z∫
dzz
=
−βmL
z−z0,m( )zL
>0 stable
ln1+φm z( )−2[ ]1+φm z( )−1
[ ]2
1+φm z0,m( )−2⎡
⎣ ⎢ ⎤ ⎦ ⎥ 1+φm z0,m( )
−1⎡ ⎣ ⎢
⎤ ⎦ ⎥
2
−2tan−1 φm z( )[ ]−1
+2tan−1 φm z0,m( )[ ]−1 z
L<0 unstable
0zL
=0 neutral
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
Neutral conditions --> logarithmic wind profile (8.64)
Logarithmic Wind Profile
v h z( ) =u*k
lnz
z0,m
0
2
4
6
8
10
0 2 4 6 8 10 12
Height above surface (m)
Wind speed (m s
-1
)
0,
=0.1 m
0,
=1.0 m z
m
z
m
Fig. 8.3
Hei
ght a
bove
sur
face
(m
)
Logarithmic wind profiles when u* = 1 m s-1.
Potential Virtual Temperature Profile
Rewrite (8.58)
Integrate --> potential virtual temperature profile (8.60)
Dimensionless potential temperature gradient (8.34)
φhk
≈zθ*
∂θ v∂z
∂θ v∂z
=θ*kz
φh =θ*kz
1− 1−φh( )[ ]
θ v z( ) =θ v z0,h( )+Prtθ*k
lnz
z0,h
⎛
⎝ ⎜
⎞
⎠ ⎟ −ψh
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Potential Virtual Temperature ProfileInfluence function for energy (8.61,3)
ψh = 1−φh( )z0,h
z∫
dzz
=
−1Prt
βhL
z−z0,h( )zL
>0 stable
2ln1+φh z( )−1
1+φh z0,h( )−1
zL
<0 unstable
0zL
=0 neutral
⎧
⎨
⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪
..
ln(dc+z0,m)
00
|vh|
ln dc
ln hcTop of canopy
ln z
Vertical Profiles in a CanopyRelationship among dc, hc, and z0,m
Fig. 8.4
ln z
0,m
Vertical Profiles in a Canopy
Potential virtual temperature (8.67)
Specific humidity (8.68)
Momentum (8.66)
v h z( ) =u*k
lnz −dcz0,m
⎛
⎝ ⎜
⎞
⎠ ⎟ −ψm
z−dcL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
θ v z( ) =θ v dc +z0,h( )+Prtθ*k
lnz−dcz0,h
⎛
⎝ ⎜
⎞
⎠ ⎟ −ψh
z−dcL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
q v z( ) =q v dc +z0,v( )+Prtq*k
lnz−dcz0,v
⎛
⎝ ⎜
⎞
⎠ ⎟ −ψh
z−dcL
⎛ ⎝ ⎜
⎞ ⎠ ⎟
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Local v. Nonlocal Closure Above SurfaceLocal closure turbulence scheme
Mixes momentum, energy, chemicals between adjacent layers.
HybridE-E-d
Nonlocal closure turbulence scheme Mixes variables among all layers simultaneously
Free-convective plume scheme
Hybrid Scheme
Mixing Length (8.71)
For energy
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
Km,zx≈Km,zy≈λe2 Δu
Δz⎛ ⎝ ⎜ ⎞
⎠ ⎟
2+
Δv Δz
⎛ ⎝ ⎜ ⎞
⎠ ⎟ 2 Ric −Rib
Ric
λe =kz
1+kz λm
Kh,zz ≈Km,zx Prt
E (TKE)- Scheme
Prognositc equation for mixing length (8.73)
Production rate of shear (8.74)
Prognostic equation for TKE (8.72)
∂E∂t
−∂∂z
sqλe 2E∂E∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟ =Ps +Pb −εd
∂ 2Eλe( )∂t
−∂∂z
slλe 2E∂ 2Eλe( )
∂z
⎛
⎝ ⎜
⎞
⎠ ⎟ =λee1 Ps +Pb( )−λeεd 1+e2
λekz
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 2⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Ps =Km∂u ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2+
∂v ∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 2⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
E- Scheme
Dissipation rate of TKE (8.76)
Diffusion coefficients (8.77)
Production rate of buoyancy (8.75)
Pb =−gθ v
Kh∂θ v∂z
εd =2E( )32
B1λe
Km=SMλe 2E Kh =Shλe 2E
E-d TKE
Eddy diffusion coefficient for momentum (8.89)
Diagnostic equation for mixing length (8.90)
Prognostic equation for dissipation rate (8.88)
∂εd∂t
−∂∂z
Kmσε
∂εd∂z
⎛
⎝ ⎜
⎞
⎠ ⎟ =cε1
εdE
Ps +Pb( )−cε2εd2
E
Km=cμE2
εd
λe =cμ34 E3 2
εd
Heat Conduction Equation
Thermal conductivity of soil-water-air mixture (8.92)
Moisture potentialPotential energy required to extract water from capillary and adhesive forces in the soil (8.93)
Heat conduction equation (8.91)∂Ts∂t
=1
ρgcG
∂∂z
κs∂Ts∂z
⎛ ⎝ ⎜
⎞ ⎠ ⎟
κs =max418e−log10ψp −2.7,0.172⎛ ⎝ ⎜
⎞ ⎠ ⎟
ψ p =ψ p,swg,swg
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
b
Heat Conduction Equation
Rate of change of soil water content (8.95)
Density x specific heat of soil-water-air mixture (8.94)
ρgcG = 1−wg,s( )ρscS +wgρwcW
∂wg∂t
=∂∂z
Kg∂ψp∂z
+1⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥ =
∂∂z
Dg∂wg∂z
+Kg⎛
⎝ ⎜
⎞
⎠ ⎟
Hydraulic conductivity of soilCoefficient of permeability of liquid through soil (8.96)
Kg =Kg,swgwg,s
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2b+3
Heat Conduction Equation
Diffusion coefficient of water in soil (8.97)
Dg =Kg∂ψ p∂wg
=−bKg,sψp,s
wg
wgwg,s
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
b+3
=−bKg,sψ p,s
wg,s
wgwg,s
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
b+2
Heat Conduction Equation
Surface energy balance equation (8.103)
Rate of change of moisture content at the surface (8.99)
∂wg∂t
=∂∂z
Dg∂wg∂z
+Kg +Ef −Pg
ρw
⎛
⎝ ⎜
⎞
⎠ ⎟
κs,1∂Ts∂z
+Fn,g−H f −LeE f =0
Rate of change of ground surface temperature (8.98)
∂Ts∂t
=1
ρgcG
∂∂z
κs∂Ts∂z
+Fn,g −H f −LeEf⎛ ⎝ ⎜
⎞ ⎠ ⎟
Temp and Moisture in Vegetated Soil
Surface irradiance (8.104)
Surface energy balance equation (8.103)
Vertical turbulent sensible heat flux (8.105)
κs,1∂Ts∂z
+Fn,g−H f −LeE f =0
Fn,g = fsF s+F i ↓−εsσBT g4
H f =−fsρ acp,d
Raθ p zr( )−
T gP g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥ −fv
ρ acp,dRf
TafP f
−T gP g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Temp and Moisture in Vegetated Soil
Temperature of air in foliage (8.109)
Vertical turbulent latent heat flux (8.106)
Specific humidity of air in foliage (8.110)
LeE f =−fsLeρ aRa
βg q v zr( )−qv,s T g( )[ ]−fvLeρ aRf
βg qaf −qv,s T g( )[ ]
Taf =0.3T a zr( )+0.6Tf +0.1T g
qaf =0.3q v zr( ) +0.6qf +0.1q g
Foliage Temperature
Sensible heat flux (8.116)
Iterative equation for foliage temperature (8.115)
fv
F s+εvF i ↓+εvεs
εv +εs −εvεsσBT g
4
−εv +2εs −εvεsεv +εs −εvεs
εvσBTf4
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
Hv =−1.1LTρ acp,dR fP f
Taf −Tf[ ]
=Hv +LeEd +LeEt
Foliage Temperature
Transpiration (8.118)
Direct evaporation (8.117)
Evaporation function (8.119)
Ed =−LTρ aβdRf
qaf −qv,s Tf( )[ ]
Et =−LTρ a 1−βd( )Rf +Rst
qaf −qv,s Tf( )[ ]
βd =
WcWc,max
⎛
⎝ ⎜
⎞
⎠ ⎟
2 3
qaf <qv,s Tf( )
1 qaf ≥qv,s Tf( )
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
Foliage Temperature
(8.122)
Tf,t,n+1 =
fv
F s +εvF i ↓+εvεs
εv +εs−εvεsσBT g,t−h
4
+3εv +2εs −εvεsεv +εs −εvεs
εvσBTf,t,n4
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
+1.1LTρ acp,dRf P f
Taf +LeLT ρ aβdR f
+ρ aβt
Rf +Rst
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
× qaf − qv,s Tf,t,n( ) −dqv,s Tf,t,n( )
dTTf,t,n
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
4εv +2εs−εvεsεv +εs −εvεs
εvσ BTf,t,n3 +1.1LT
ρ acp,dRfP f
+LeLTρ aβdRf
+ρ a 1−βd( )
Rf +Rst
⎧ ⎨ ⎪
⎩ ⎪
⎫ ⎬ ⎪
⎭ ⎪
dqv,s Tf,t,n( )
dT
⎧
⎨
⎪ ⎪
⎩
⎪ ⎪
⎫
⎬
⎪ ⎪
⎭
⎪ ⎪
Temperature of Vegetated Soil
(8.125)
T g,t,n =T g,t,n−1+
fsρ acp,d
Raθ p zr( ) −
T g,t,n−1
P g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
fvρ acp,d
Rf
Taf,tP f
−T g,t,n−1
P g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
fsρ aLeRa
βg q v zr( )−qv,s T g,t,n−1( )[ ]
fvρ aLeRf
βg qaf −qv,s T g,t,n−1( )[ ]
fsF s+F i ↓−σ BεsT g,t,n−14
κs,1D1
T1,t −T g,t,n−1( )
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
fsρ acp,dRaP g
+fvρ acp,dRf P g
+4εsσBT g,t,n−13 +
κs,1D1
Modeled/Measured Temperatures
Fig. 8.5
10
30
50
70
0 24 48 72 96
Foliage air
Foliage
Ground
Air above canopy
Temperature (
o
C)
Hour after first midnight
Sandy loam
Lodi (LOD)
Modeled/Measured Temperatures
Fig. 8.5
-20
-15
-10
-5
0
5
10
0 24 48 72 96
Foliage air
Foliage
Ground
Δ
(Temperature
o
)C
Hour after first midnight
Sandy loam
minus clay loam
( )Lodi LOD
Modeled/Measured Temperatures
Fig. 8.5
0
10
20
30
40
50
0 24 48 72 96
Predicted
Measured
Temperature (
o
C)
Hour after first midnight
Lodi (LOD)
Road Temperature(8.128)
T g,t,n =T g,t,n−1+
ρ acp,d
Raθ p zr( )−
T g,t,n−1
P g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ρ aLeRa
βd q v zr( ) −qv,s T g,t,n−1( )[ ]
F s +F i ↓−σBεasT g,t,n−14
κasD1
T1,c,t −T g,t,n−1( )
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪
ρ acp,dRaP g
+4εasσBT g,t,n−13 +
κasD1
Temperatures of Soils and Surfaces
Fig. 8.5
10
20
30
40
50
60
70
0 24 48 72 96
Silt loam
Loam
Silt clay loam
Clay
Road
Roof
Temperature (
o
C)
Hour after first midnight
Ground temperature Fremont (FRE)
Modeled/Measured Temperatures
Fig. 8.5
0
10
20
30
40
50
0 24 48 72 96
Predicted
Measured
Temperature (
o
C)
Hour after first midnight
Fremont (FRE)
Snow Depth(8.129)Ds,t =Ds,t−h +hP s
+hρ aρsn
fsq v zr( )−qv,s minTg,t,Ts,m( )( )[ ]
Ra
+fvqaf −qv,s minTg,t,Ts,m( )( )[ ]
Rf
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪
+h
−fsρ acp,d
Raθ p zr( ) −
Ts,mP g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
−fvρ acp,d
Rf
Taf,tP f
−Ts,mP g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
−fsρ aLsRa
q v zr( )−qv,s Ts,m( )[ ]
−fvρ aLsRf
qaf −qv,s Ts,m( )[ ]
−fsF s−F i ↓+σBεsnTs,m4 −
κsnD1
T1,t −Ts,m( )
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
ρsnLm
Water Temperature(8.130)
T g,t =T g,t−h +h
ρ acp,dRa
θ p zr( )−T g,t−h
P g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ρ aLeRa
q v zr( ) −qv,s T g,t−h( )[ ]
F s +F i ↓−σBεwT g,t−h4
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪
ρswcp,swDl
Sea Ice Temperature(8.131)
Tg,t,n =Tg,t,n−1+
ρ acp,dRa
θ p zr( )−T g,t,n−1
P g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ρ aLsRa
q v zr( )−qv,s T g,t,n−1( )[ ]
F s +F i ↓−σBεiT g,t,n−14
κiDi,t−h
Ti, f −T g,t,n−1( )
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪ ⎪
ρ acp,dRaP g
+4εiT g,t,n−13 +
κiDi,t−h
Temperature of Snow Over Sea Ice(8.134)T g,t,n =T g,t,n−1+
ρ acp,dRa
θ p zr( )−T g,t,n−1
P g
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
ρ aLsRa
q v zr( )−qv,s T g,t,n−1( )[ ]
F s +F i ↓−σBεsnT g,t,n−14
κsnκiκsnDi,t−h +κi Ds,t−h
Ti, f −T g,t,n−1( )
⎧
⎨
⎪ ⎪ ⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪ ⎪ ⎪
ρ acp,dRaP g
+4εsnT g,t,n−13 +
κsnκiκsnDi,t−h +κiDs,t−h
