Presentation Slides for Chapter 8 of Fundamentals of Atmospheric Modeling 2 nd Edition

transcript

Presentation Slides for

Chapter 8of

Fundamentals of Atmospheric Modeling 2nd Edition

Mark Z. JacobsonDepartment of Civil & Environmental Engineering

Stanford UniversityStanford, CA 94305-4020jacobson@stanford.edu

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

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⎡

⎣ ⎢ ⎤ ⎦ ⎥

u* = ′ w ′ u ( )s2

+ ′ w ′ v ( )s2⎡

⎣ ⎢ ⎤ ⎦ ⎥

14= τz ρa( )s

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

Fig. 6.5

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

z0,m≈αcu*

z0,m≈0.5hoSoAo

z0,m=hc 1−0.91e−0.0075LT( )

Roughness Length for MomentumSurface Type z0,m (m) h

c (m) d

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

⎝ ⎜

⎠ ⎟

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

⎝ ⎜

⎠ ⎟

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( )

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 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)

∂ v h∂z

1+βmzL

>0 stable

1−γmzL

⎛ ⎝ ⎜ ⎞

⎠ ⎟

−14 zL

<0 unstable

=0 neutral

⎪ ⎪ ⎪

u* =kv h zr( )

φmdzzz0,m

Integral of Dimensionless Wind ShearIntegral of the dimensionless wind shear (8.31)

φmdzz

zr −z0,m( )zL

>0 stable

ln1−γm

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 14

1−γmzrL

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 14

1−γmz0,m

⎝ ⎜

⎠ ⎟ 14

1−γmz0,m

⎝ ⎜

⎠ ⎟ 14

+2tan−1 1−γmzrL

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 14

−2tan−1 1−γmz0,m

L⎛ ⎝ ⎜

⎞ ⎠ ⎟ 14

<0 unstable

=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

′ w ′ θ v( )s≈−u*θ*

Potential Temperature ScaleDimensionless temperature gradient (8.34)

Parameterization of * (8.35)

≈zθ*

∂θ v∂z

Prt+βhzL

>0 stable

Prt 1−γhzL

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −12 z

L<0 unstable

=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

Prt lnzr

zr −z0,h( )zL

>0 stable

Prt ln1−γh

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 12

1−γhzrL

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 12

1−γhz0,hL

⎝ ⎜

⎠ ⎟ 12

1−γhz0,hL

⎝ ⎜

⎠ ⎟ 12

⎢ ⎢ ⎢ ⎢ ⎢

⎥ ⎥ ⎥ ⎥ ⎥

<0 unstable

Prt lnzr

=0 neutral

⎪ ⎪ ⎪ ⎪ ⎪ ⎪

Equations to Solve Simultaneously

Solution requires iteration

θ* =k θ v zr( )−θ v z0,h( )[ ]

φhdzzz0,h

L =−u*3θ v

kg ′ w ′ θ v( )s

=u*2θ v

kgθ*u* =

kv h zr( )

φmdzzz0,m

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( )

ln2 zr z0,m( )

Rib ≤0

Gh =1−9.4Rib

1+50k2 Ribzr z0,m( )

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( )

θ v z0,h( ) u zr( )2

+v zr( )2

[ ] zr −z0,h( )

Gradient Richardson Number(8.42)

∂θ v∂z

∂u ∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟

∂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 =

∂θ v∂z

∂u ∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟

∂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

u* = ′ w ′ u ( )s2

+ ′ w ′ v ( )s2⎡

⎣ ⎢ ⎤ ⎦ ⎥

′ w ′ u ( )s=−

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=−

v h zr( )u zr( ) ′ w ′ v ( )s

=−u*

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

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)

∂ 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

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*

∂q v∂z

q* =k q v zr( ) −q v z0,v( )[ ]

φhdzzz0,v

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)

∂ v h∂z

∂ v h z( )

φm=u*kz

1− 1−φm( )[ ]

v h z( ) =u*k

⎝ ⎜

⎠ ⎟ −ψm

⎣ ⎢ ⎢

⎦ ⎥ ⎥

Influence function for momentum (8.61,2)

ψm = 1−φm( )z0,m

−βmL

z−z0,m( )zL

>0 stable

ln1+φm z( )−2[ ]1+φm z( )−1

1+φm z0,m( )−2⎡

⎣ ⎢ ⎤ ⎦ ⎥ 1+φm z0,m( )

−1⎡ ⎣ ⎢

⎤ ⎦ ⎥

−2tan−1 φm z( )[ ]−1

+2tan−1 φm z0,m( )[ ]−1 z

L<0 unstable

=0 neutral

⎪ ⎪ ⎪ ⎪ ⎪ ⎪

Neutral conditions --> logarithmic wind profile (8.64)

v h z( ) =u*k

0 2 4 6 8 10 12

Height above surface (m)

Wind speed (m s

=0.1 m

=1.0 m z

Fig. 8.3

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)

≈zθ*

∂θ v∂z

=θ*kz

φh =θ*kz

1− 1−φh( )[ ]

θ v z( ) =θ v z0,h( )+Prtθ*k

⎝ ⎜

⎠ ⎟ −ψh

⎣ ⎢ ⎢

⎦ ⎥ ⎥

Potential Virtual Temperature ProfileInfluence function for energy (8.61,3)

ψh = 1−φh( )z0,h

−1Prt

z−z0,h( )zL

>0 stable

2ln1+φh z( )−1

1+φh z0,h( )−1

<0 unstable

=0 neutral

⎪ ⎪ ⎪ ⎪

ln(dc+z0,m)

ln hcTop of canopy

Vertical Profiles in a CanopyRelationship among dc, hc, and z0,m

Fig. 8.4

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⎛ ⎝ ⎜ ⎞

⎠ ⎟

Δv Δz

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 2 Ric −Rib

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( )

⎝ ⎜

⎠ ⎟ =λee1 Ps +Pb( )−λeεd 1+e2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 2⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥

Ps =Km∂u ∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟

∂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

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

Ps +Pb( )−cε2εd2

Km=cμE2

λe =cμ34 E3 2

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

∂∂z

κs∂Ts∂z

⎛ ⎝ ⎜

⎞ ⎠ ⎟

κs =max418e−log10ψp −2.7,0.172⎛ ⎝ ⎜

⎞ ⎠ ⎟

ψ p =ψ p,swg,swg

⎝ ⎜ ⎜

⎠ ⎟ ⎟

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

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥ =

∂∂z

Dg∂wg∂z

+Kg⎛

⎝ ⎜

⎠ ⎟

Hydraulic conductivity of soilCoefficient of permeability of liquid through soil (8.96)

Kg =Kg,swgwg,s

⎝ ⎜ ⎜

⎠ ⎟ ⎟

Diffusion coefficient of water in soil (8.97)

Dg =Kg∂ψ p∂wg

=−bKg,sψp,s

wgwg,s

⎝ ⎜ ⎜

⎠ ⎟ ⎟

=−bKg,sψ p,s

wgwg,s

⎝ ⎜ ⎜

⎠ ⎟ ⎟

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

⎝ ⎜

⎠ ⎟

κs,1∂Ts∂z

+Fn,g−H f −LeE f =0

Rate of change of ground surface temperature (8.98)

∂Ts∂t

∂∂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)

F s+εvF i ↓+εvεs

εv +εs −εvεsσBT g

−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( )[ ]

WcWc,max

⎝ ⎜

⎠ ⎟

qaf <qv,s Tf( )

1 qaf ≥qv,s Tf( )

⎨ ⎪ ⎪

⎩ ⎪ ⎪

Foliage Temperature

(8.122)

Tf,t,n+1 =

F s +εvF i ↓+εvεs

εv +εs−εvεsσBT g,t−h

+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( )

⎪ ⎪

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

⎣ ⎢ ⎢

⎦ ⎥ ⎥

fvρ acp,d

Taf,tP f

−T g,t,n−1

⎣ ⎢ ⎢

⎦ ⎥ ⎥

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

0 24 48 72 96

Foliage air

Foliage

Ground

Air above canopy

Temperature (

Hour after first midnight

Sandy loam

Lodi (LOD)

Fig. 8.5

0 24 48 72 96

Foliage air

Foliage

Ground

(Temperature

Sandy loam

minus clay loam

( )Lodi LOD

Fig. 8.5

0 24 48 72 96

Predicted

Measured

Temperature (

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

⎣ ⎢ ⎢

⎦ ⎥ ⎥

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

0 24 48 72 96

Silt loam

Silt clay loam

Temperature (

Ground temperature Fremont (FRE)

Fig. 8.5

0 24 48 72 96

Predicted

Measured

Temperature (

Fremont (FRE)

Snow Depth(8.129)Ds,t =Ds,t−h +hP s

+hρ aρsn

fsq v zr( )−qv,s minTg,t,Ts,m( )( )[ ]

+fvqaf −qv,s minTg,t,Ts,m( )( )[ ]

⎪ ⎪ ⎪

−fsρ acp,d

Raθ p zr( ) −

Ts,mP g

⎣ ⎢ ⎢

⎦ ⎥ ⎥

−fvρ acp,d

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

⎣ ⎢ ⎢

⎦ ⎥ ⎥

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

⎣ ⎢ ⎢

⎦ ⎥ ⎥

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

⎣ ⎢ ⎢

⎦ ⎥ ⎥

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

Presentation Slides for Chapter 8 of Fundamentals of Atmospheric Modeling 2 nd Edition

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