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Footprint Model Aline Jaimes Kreinovich, Vladik PhD. CYBER-ShARE Meeting Nov ‘08
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Footprint Model
Aline JaimesKreinovich Vladik PhD
CYBER-ShARE MeetingNov lsquo08
CYBER-ShARE meeting
OUTLINE
bull Backgroundbull Footprint definitionbull Model descriptionbull Conclusions
CYBER-ShARE meeting
Air flow in Ecosystem
bull Air flow can be imagined as a horizontal flow of numerous rotting eddiesbull Each eddy has 3D components including a vertical wind componentbull The diagram looks chaotic but components can be measured from the tower
CYBER-ShARE meeting
Eddies at one point
The essence of the method is that vertical flux can be presented as covariance between measurements of vertical velocity the up and down movements and concentration of the entity of interest
CYBER-ShARE meeting
Where is the flux coming from
bullFootprint models and tower flux measurements is a first cut evaluation of the impact of biodiversity on net ecosystem CO2 exchange
Issues related to footprint spatial distribution of vegetation and net ecosystem exchange
1 Representativeness of the integrated tower record2 Appropriateness of conventional gap filling that does not consider the effects of wind
direction on algorithm development
CYBER-ShARE meeting
bullUntil now estimates of annual NEE and its error have not considered the impact of wind direction footprint and upwind vegetation This effect can be a major source of bias error (Kim at al 2006)
bullOne question that needs to be address is whether the temporal sum of net ecosystem CO2 exchange is equal to the net flux if the wind came from the same wind direction and was processed by the same patch
bull To address this question we need to assess NEE for various wind direction sectors
bullBecause wind direction is not uniform gap filling algorithms must be applied
Definition
bull Footprint The contribution per unit surface flux of each unit
element of the upwind surface area to a measured vertical flux
CYBER-ShARE meeting
Figure 1 Plan and profile sketches of the flux footprint function f for a tower-based flux measurement (Horst and Weil 1994)
bullThe measured flux is the integral of the contributions from all upwind surface elements
bullThe flux footprint is the relative weight given each elemental surface flux
x
mm zyxFzyxF )0()( 0
)( dydxzyyxx m
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
OUTLINE
bull Backgroundbull Footprint definitionbull Model descriptionbull Conclusions
CYBER-ShARE meeting
Air flow in Ecosystem
bull Air flow can be imagined as a horizontal flow of numerous rotting eddiesbull Each eddy has 3D components including a vertical wind componentbull The diagram looks chaotic but components can be measured from the tower
CYBER-ShARE meeting
Eddies at one point
The essence of the method is that vertical flux can be presented as covariance between measurements of vertical velocity the up and down movements and concentration of the entity of interest
CYBER-ShARE meeting
Where is the flux coming from
bullFootprint models and tower flux measurements is a first cut evaluation of the impact of biodiversity on net ecosystem CO2 exchange
Issues related to footprint spatial distribution of vegetation and net ecosystem exchange
1 Representativeness of the integrated tower record2 Appropriateness of conventional gap filling that does not consider the effects of wind
direction on algorithm development
CYBER-ShARE meeting
bullUntil now estimates of annual NEE and its error have not considered the impact of wind direction footprint and upwind vegetation This effect can be a major source of bias error (Kim at al 2006)
bullOne question that needs to be address is whether the temporal sum of net ecosystem CO2 exchange is equal to the net flux if the wind came from the same wind direction and was processed by the same patch
bull To address this question we need to assess NEE for various wind direction sectors
bullBecause wind direction is not uniform gap filling algorithms must be applied
Definition
bull Footprint The contribution per unit surface flux of each unit
element of the upwind surface area to a measured vertical flux
CYBER-ShARE meeting
Figure 1 Plan and profile sketches of the flux footprint function f for a tower-based flux measurement (Horst and Weil 1994)
bullThe measured flux is the integral of the contributions from all upwind surface elements
bullThe flux footprint is the relative weight given each elemental surface flux
x
mm zyxFzyxF )0()( 0
)( dydxzyyxx m
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Air flow in Ecosystem
bull Air flow can be imagined as a horizontal flow of numerous rotting eddiesbull Each eddy has 3D components including a vertical wind componentbull The diagram looks chaotic but components can be measured from the tower
CYBER-ShARE meeting
Eddies at one point
The essence of the method is that vertical flux can be presented as covariance between measurements of vertical velocity the up and down movements and concentration of the entity of interest
CYBER-ShARE meeting
Where is the flux coming from
bullFootprint models and tower flux measurements is a first cut evaluation of the impact of biodiversity on net ecosystem CO2 exchange
Issues related to footprint spatial distribution of vegetation and net ecosystem exchange
1 Representativeness of the integrated tower record2 Appropriateness of conventional gap filling that does not consider the effects of wind
direction on algorithm development
CYBER-ShARE meeting
bullUntil now estimates of annual NEE and its error have not considered the impact of wind direction footprint and upwind vegetation This effect can be a major source of bias error (Kim at al 2006)
bullOne question that needs to be address is whether the temporal sum of net ecosystem CO2 exchange is equal to the net flux if the wind came from the same wind direction and was processed by the same patch
bull To address this question we need to assess NEE for various wind direction sectors
bullBecause wind direction is not uniform gap filling algorithms must be applied
Definition
bull Footprint The contribution per unit surface flux of each unit
element of the upwind surface area to a measured vertical flux
CYBER-ShARE meeting
Figure 1 Plan and profile sketches of the flux footprint function f for a tower-based flux measurement (Horst and Weil 1994)
bullThe measured flux is the integral of the contributions from all upwind surface elements
bullThe flux footprint is the relative weight given each elemental surface flux
x
mm zyxFzyxF )0()( 0
)( dydxzyyxx m
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Eddies at one point
The essence of the method is that vertical flux can be presented as covariance between measurements of vertical velocity the up and down movements and concentration of the entity of interest
CYBER-ShARE meeting
Where is the flux coming from
bullFootprint models and tower flux measurements is a first cut evaluation of the impact of biodiversity on net ecosystem CO2 exchange
Issues related to footprint spatial distribution of vegetation and net ecosystem exchange
1 Representativeness of the integrated tower record2 Appropriateness of conventional gap filling that does not consider the effects of wind
direction on algorithm development
CYBER-ShARE meeting
bullUntil now estimates of annual NEE and its error have not considered the impact of wind direction footprint and upwind vegetation This effect can be a major source of bias error (Kim at al 2006)
bullOne question that needs to be address is whether the temporal sum of net ecosystem CO2 exchange is equal to the net flux if the wind came from the same wind direction and was processed by the same patch
bull To address this question we need to assess NEE for various wind direction sectors
bullBecause wind direction is not uniform gap filling algorithms must be applied
Definition
bull Footprint The contribution per unit surface flux of each unit
element of the upwind surface area to a measured vertical flux
CYBER-ShARE meeting
Figure 1 Plan and profile sketches of the flux footprint function f for a tower-based flux measurement (Horst and Weil 1994)
bullThe measured flux is the integral of the contributions from all upwind surface elements
bullThe flux footprint is the relative weight given each elemental surface flux
x
mm zyxFzyxF )0()( 0
)( dydxzyyxx m
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Where is the flux coming from
bullFootprint models and tower flux measurements is a first cut evaluation of the impact of biodiversity on net ecosystem CO2 exchange
Issues related to footprint spatial distribution of vegetation and net ecosystem exchange
1 Representativeness of the integrated tower record2 Appropriateness of conventional gap filling that does not consider the effects of wind
direction on algorithm development
CYBER-ShARE meeting
bullUntil now estimates of annual NEE and its error have not considered the impact of wind direction footprint and upwind vegetation This effect can be a major source of bias error (Kim at al 2006)
bullOne question that needs to be address is whether the temporal sum of net ecosystem CO2 exchange is equal to the net flux if the wind came from the same wind direction and was processed by the same patch
bull To address this question we need to assess NEE for various wind direction sectors
bullBecause wind direction is not uniform gap filling algorithms must be applied
Definition
bull Footprint The contribution per unit surface flux of each unit
element of the upwind surface area to a measured vertical flux
CYBER-ShARE meeting
Figure 1 Plan and profile sketches of the flux footprint function f for a tower-based flux measurement (Horst and Weil 1994)
bullThe measured flux is the integral of the contributions from all upwind surface elements
bullThe flux footprint is the relative weight given each elemental surface flux
x
mm zyxFzyxF )0()( 0
)( dydxzyyxx m
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
bullUntil now estimates of annual NEE and its error have not considered the impact of wind direction footprint and upwind vegetation This effect can be a major source of bias error (Kim at al 2006)
bullOne question that needs to be address is whether the temporal sum of net ecosystem CO2 exchange is equal to the net flux if the wind came from the same wind direction and was processed by the same patch
bull To address this question we need to assess NEE for various wind direction sectors
bullBecause wind direction is not uniform gap filling algorithms must be applied
Definition
bull Footprint The contribution per unit surface flux of each unit
element of the upwind surface area to a measured vertical flux
CYBER-ShARE meeting
Figure 1 Plan and profile sketches of the flux footprint function f for a tower-based flux measurement (Horst and Weil 1994)
bullThe measured flux is the integral of the contributions from all upwind surface elements
bullThe flux footprint is the relative weight given each elemental surface flux
x
mm zyxFzyxF )0()( 0
)( dydxzyyxx m
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
Definition
bull Footprint The contribution per unit surface flux of each unit
element of the upwind surface area to a measured vertical flux
CYBER-ShARE meeting
Figure 1 Plan and profile sketches of the flux footprint function f for a tower-based flux measurement (Horst and Weil 1994)
bullThe measured flux is the integral of the contributions from all upwind surface elements
bullThe flux footprint is the relative weight given each elemental surface flux
x
mm zyxFzyxF )0()( 0
)( dydxzyyxx m
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Objective Estimate flux footprint φ(xyz)
bull Flux portion at (00z) caused by a unit point source at (xy0)
bull In our case z=zm =6m
Figure 1 (a) The crosswind integrated footprint f(xzm) (b) Isopleths of the footprint The solid lines depict the neutral case the dashed and the dotted lines the stable and the instable case respectively
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Parameters describing the model
bull m ndash Describes how horizontal wind velocity depends on height z
bull n ndash Describes how eddy diffusivity depends on the z
bull How to empirically find m
So
mzUzu )(
nzkzK )(m
z
z
zu
zu
2
1
2
1
)(
)(
)log(
))()(log(
21
21
zz
zuzum
k
zK
U
zu
)(
)( - Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =04
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Simplest case (Schuepp et al 1990)
bull Mathematical description
n=1 m=0
bull Wind velocity u(z)=U does not depend on heightbull Diffusivity K(z)=kz linearly increases with height
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
Auxiliary parameters
bull Shape factor r=2+m-n
bull Meaning crosswind integrated concentration c decreases with height z
bull Schuepp case r=1
bull Useful constant
bull Schuepp case micro=1r
m1
Eq 12
CYBER-ShARE meeting
)exp()( rzconstconstzC
c- crosswind integrated concentrationr ndash shape factormicro - (1+m)r ndash constant
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Analytical formula for velocity ū(x) of the plume
ū(x)= rm
rm
UXU
kr
r
2
)1(
)(
Where Г(z) is the gamma function
0
1)( dtetz tz
Is an extension of factorial n=12hellipn to real numbers
Г(n) = (n-1)
Schuepp case ū(x)=U
Eq 18
ū(x) ndash Effective plume velocityГ (x)ndash Gamma functionU ndash Constant of power of law profile wind velocity
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Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Analytical Model
)()( zxfyxDzyx y
bullCrosswind distribution function Dy (xy) described horizontal distribution
bullCrosswind integrated flux footprint f(xz) described vertical distribution
Oslash(xyz) ndash flux footprint or vertical flux per unit point source
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Crosswind distributionFunction Gaussian Model
2
2
)(2exp
)(2
1)(
x
y
xyxDy
where dispersion is
)(x
rmx ū(x)~
)()(
xu
xx v
Since r
m
xx1
~)(
Schuepp case ū(x)~x
Typically Hzv 10σv - Constant crosswind fluctuationσ(x) - Crosswind dispersionū(x)- Effective plume velocityzm ndash Measurement height
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Maximum of a Footprintbull Flux Length scale
bull Crosswind integrated flux footprint or vertical flux per unit point source
f(xz)=
bull Maximum is attained at
bull Maximum footprint value is
x
z
x
z )(exp(
)(
)(
11
1
)(zX
)1exp()()(
)1()(
1
max
zzf
Eq 22
ζ(z) ndash Flux length scaleГ ndash Gamma functionmicro - Vertical profile of the Reynolds-averaged wind velocity
kr
Uzz
r
2)(
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Final Formula
x
z
x
z
x
y
x
zxfyxDzyx y
)(exp
)(
)(
1
)(2exp
)(2
1
)()()(
12
2
σ(x) ndash Crosswind dispersionr ndash shape factormicro - Vertical profile of the Reynolds-averaged wind velocity ζ (z) ndash Flux length scale
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
Towards a more accurate description
bull In analytical model we assumed power law(Eq 11)
bull A more accurate model
bull where
ndash u is friction velocityndash zo Is roughness lengthndash L is Obukhov length where
bull Kinetic energy is equal to the potential energy gz
mUzzu )(
L
z
z
z
k
uzu m
0
ln)(
2
2
1u
CYBER-ShARE meeting
u(z) - Vertical profile of the Reynolds-averaged wind velocity U ndash Constant in power-law profile of the wind velocityΨm(zL) ndash Diabatic integration of the wind profile
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
bull Kormann amp Meixnerrsquos model is a simple approach to estimate footprint models
bull Footprint models will allow estimation of net fluxes when wind direction is not uniform
bull Therefore improving development of gap filling algorithms for JER Station
Conclusions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
CYBER-ShARE meeting
Thank you
bull Questions Comments Suggestions
