land-surface atmosphere interactionnimbus.elte.hu/~acs/pdf/oktatas/fsa_kemia_acs_ferenc.pdf ·...

Land-surface – atmosphere

interaction

Author: Dr. Ferenc Ács

Eötvös Loránd University

Institute of Geography and Earth Sciences

Department of Meteorology

Financed from the financial support ELTE won from the Higher Education Restructuring Fund of the Hungarian

Government


interactionGoals:

TO PROVIDE BASIC PHYSICS KNOWLEDGE, MORE PRECISELY

• knowledge on the phenomenology of radiation transfer above the land-surface,

• knowledge on the phenomenology of heat and water transport processes in the soil,

• knowledge on the phenomenology of the atmospheric transport processes in the vicinity of the land-surface,


interactionGoals:

TO PROVIDE BASIC PHYSICS KNOWLEDGE, MORE PRECISELY

• knowledge in detail about Monin-Obukhov’s similarity theory,

• knowledge on the water transfer processes in the soil-vegetation system,

• knowledge on the energy transfer processes in the soil-vegetation system.

Introduction

(Gaia and the vegetation)

• Characteristics of the soil-vegetation-atmosphere system:

central element: the vegetation

(photosynthesis: the most important and ancient process on the Earth (Gaia)),

Physical, chemical and biological phenomena and processes.

Weather: physical processes.

Climate: phyisical, chemical and biological processes.

Introduction

(Gaia and the vegetation)Monteith et al. (1975)

Introduction

(Gaia and the vegetation)

• Water: Flux

densities and

reservoirs. Soil

is the largest

water reservoir!

Therefore

meteorology

cannot

disregard the

soil.

Monteith et al. (1975)

Introduction

(Gaia and the vegetation)• Resistances: stomatal

resistance is the

largest. Therefore

meteorology cannot

disregard vegetation.

Ψ – potential; rt - soil resistance;

rgy – root resistance; rx – xylem

vessel resistance; rs – stomatal

resistance; rcu – cuticular resistance;

ra – aerodynamic resistances in the

boundary (lower) and turbulent

(upper) atmospheric layers;

légkör = atmosphere; vízkészlet =

water amount in the soil

Rose (1966)

Radiation

• Vegetation canopy:

Radiation features of the leaf (r (reflection), tr

(transmision) and a (absorption) spectra, water

content),

radiation features of the vegetation canopy (r

and tr spectra),

albedo (solar elevation angle),

radiation balance.

Radiation

• Bare soil:

Radiation features of the soil particles (r spectra),

radiation features of the soil types (r spectra,

humus and iron oxides),

albedo (solar elevation angle, soil moisture

content, roughness),

radiation balance.

Radiation - vegetation

Radiation (optical) properties of a "typical" leaf

Jones (1983)

Radiation - vegetation

radiation properties of the leafJones (1983)

Radiation - vegetation canopy

radiation properties of the vegetation canopy, Jones (1983)


radiation properties of the vegetation canopy

Braden (1985)


radiation properties of the vegetation canopy, Braden (1985)

Radiation - vegetated surfaceAlbedo – solar elevation

When the irradiation is

"low" → the albedo is

"high" → and its

changes are great.

When the irradiation is

"high" → the albedo is

"low" → and its

changes are "small". Sellers and Dorman (1987)

Radiation - vegetated surface

• Radiation balance:

4442)1( ccggaa

vvv TTTtrRR

and if trv = 0

4442)1( ccggaa

vv TTTRR

(rough approach and the simplest form)

Radiation - bare soil surface

radiation properties of the soil particles,

Szász and Zilinyi (1994)


radiation properties of the soil types,

Jones (1983)


albedo (solar elevation, soil moisture content,

roughness)

• solar elevation: the same dependence as in the case of vegetation,

• soil moisture content: dry soil → “higher” albedo; moist soil → “lower” albedo; the transition is non-linear,

• roughness: it has the smallest effect of the three parameters.


• Radiation balance:

.)1(44

ggaa

bb TTRR

(rough approach and the simplest form)

Soil – definition

• Soil is a medium consisting of organic and inorganic

materials, where the transfer of matter and energy

occur continuously via physical, chemical and

biological processes. Therefore soil possesses

various horizons, so it has a stratified structure.

• Soil deviates from its bedrock source material by

having such a layered structure. This layered

structure is its important feature, and characterises it.

Soil - profiles

• Soil has a layered structure. The distribution of the horizons according to depth is called the soil profile.

• Each profile is composed of horizons A, B and C.

• The surface horizon A is the most weathered soil layer with the highest humus content.

• The sub-surface horizon B has a lower humus content than the surface horizon A.

• Horizon C is the least weathered soil layer and has the smallest humus content of the soil horizons.

Soil texture

• This notion expresses how large the soil particles are.

• The largest soil particles (50 – 2000 μm) are called sand. Sand’s water conduction is high, consequently its water retention is low. Sandy soils have a very low CEC (Cation Exchange Capacity).

Soil texture

• Medium size soil particles (2 – 50 μm) are called silt. This possesses moderately high (neither good nor bad) water conduction and moderately low water retention. Its ion holding capacity is moderate.

Soil texture

• Soil particles with a diameter smaller than 2 μm are known as clay. Clay possesses low water conduction and a high water retention capacity. Its ion holding capacity is high.

Soil texture

• Soil textural triangle:

schematic diagram for

representing soil

particle composition

(sand, silt and clay

fractions expressed in

per cent). (Remark:

designations in the

triangle represent soil

texture classes)

Stefanovits, Filep, Füleky (1999)

Soil texture: classification according

to soil particle composition

Cosby et al. (1984)

Soil types

• Do not mistake soil type for soil texture!

• Soil type refers to soils formed under

similar environmental conditions, in a

similar state of development, possessing

similar process associations.

Physical properties of soil

• Soil is made up of solids, liquids and gases. It is useful to define severalvariables which describe the physicalcondition of the three-phase soil system.

Mt = total mass, Ms = mass of solids,

Ml = mass of liquid, Mg = mass of gases,

Vt = total volume, Vs = volume of solids,

Vl = volume of liquids, Vg = volume of gases and

Vf = Vl + Vg = volume of fluid (sum of Vl + Vg).

Physical properties of soils

.

,

,

,

s

f

t

f

f

t

sb

s

ss

V

Ve

V

V

V

M

V

M

Particle density

Dry bulk density

Total porosity

Void ratio

Physical properties of soil

.

,

,

Sf

l

l

b

l

b

s

l

b

s

l

l

t

l

s

l

V

VS

wM

M

M

M

V

V

M

Mw

Mass wetness or

mass based water

content

Volume wetness or

volumetric water

content

Degree of saturation

Particle size distribution in soils:

particle size distribution curve

• A particle size distribution curve is a plot of

the number of particles having a given

diameter versus diameter. Particle size

distribution in soil is approximately log-

normal (a plot of number of particles vs.

log diameter would approximate a

Gaussian distribution function).

Soil heat flow

• Heat flow in the soil occures from particle to particle.

• The relationship between heat flux density and temperature is described by the Fourier law (first formulated by Fourier in 1822).

• The highest heat flow is in the vertical direction, since the temperature gradient is the highest in the vertical. Therefore a 1-dimensional treatment is common.

Fourier law

• Fourier law: This is an empirical formula,

i.e. a “parameterization”. The negative sign

“regulates” the direction of fh. λ is thermal

conductivity (Wm-1K-1)

.)(),(z

Tztzfh

Differential equation of heat conduction

• Heat flux density fh is not constant over depth! Where

(divergence)

the temperature has to decrease, and vice versa,

where

(convergence)

the temperature has to increase. Combining the Fourier

law with the continuity equation

0

z

fh

0

z

fh

.t

TC

z

fh

h

Differential equation for heat flow

• The equation can only be physically interpreted

by using a minus sign on the left side of the

equation!

• Namely, in the case of divergence of fhtemperature T has to decrease over time [(δT/δt)

< 0)], while in the case of the convergence of fh T

has to increase over time [(δT/δt) > 0)].

• Ch is the volumetric specific heat of the soil. It is

equal to the product of soil density (kgm-3) and

specific heat (Jkg-1K-1).

Differential equation for heat flow

If λ and Ch are independent of z, the

equation

could be written as

where k=λ/Ch is thermal diffusivity.

t

TzC

z

Tz

zh

)(])([

t

T

z

Tk

2

2

Thermal properties of soil

materials

• The thermal

properties of

soil materials

deviate

markedly.

Campbell (1985)

Parameterization of volumetric

specific heat• The volumetric specific heat of soil is the

weighted sum of the specific heats of all

soil constituents:

• Φ is the volume fraction of the

components (m, w, a and o indicate

mineral, water, air and organic

constituents).

.ooaawmmh CCCCC

Parameterization of volumetric

specific heat

• Since Ca is too small and Φo can be

neglected (2-4% on average), Ch of

mineral soil becomes

.)1( wfmh CCC

Thermal conductivity of soil

• It depends upon

many factors ),,,( oqbf

Campbell (1985)

Thermal conductivity of soil – different

parameterizations

• Thermal

conductivity

change versus

relative soil

moisture content

for fine and coarse

soil textures

(Johansen model) 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.01 0.11 0.21 0.3 0.4 0.5 0.6 0.7 0.79 0.89

Johansen - Coarse

Johansen - Fine

Ács et al. (2012)

Thermal conductivity of soil – different

parameterizations

• Thermal conductivity change versus relative soil moisture content for fine, coarse and very coarse soil textures of mineral soils and of organic soils (Côté Konrad model)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.01 0.11 0.21 0.3 0.4 0.5 0.6 0.7 0.79 0.89

C - vcoarse - minerso

C - coarse - minerso

C - fine - minerso

C - organic - minerso

Ács et al. (2012)

Thermal conductivity of soil –

different parameterizations

• Thermal

conductivity

change versus

relative soil

moisture content

for coarse,

mineral soils

using different

parameterizations

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.01 0.11 0.21 0.3 0.4 0.5 0.6 0.7 0.79 0.89

Relatív talajnedvesség-tartalom

Hő

vezető

kép

esség

(W

m-1

K-1

)

J - coarse

C - coarse

N - coarse

Ács et al. (2012)

Analitical solution to the heat

flow equation• The heat flow equation can be analitically

solved using the boundary conditions asfollows:

• Heat flux density at the soil surface:

At an infinite depth:

.2

),4

sin(),0( 00T

tfftf hhh

.0),( tfh


flow equation• Using former boundary conditions

• At z=dS, the amplitude ΔT is e-1=0.37 times its

value at the surface. This is the so called

damping depth

.22

)sin(),(

C

kdwhere

d

ztTeTtzT S

S

d

z

S

.37,0 Te

TT

Sdz


flow equation• According to the solution

the amplitude of the temperature wave

decreases exponentially over depth,

the phase of the temperature wave is

linearly displaced over depth z.

The shape of fh(z,t)

• Combining the equations T(z,t) and fh(z,t),

fh(z,t) can be written as

• In doing so, we also used the following

equation:

.)4

sin(),(/

0

S

dz

hhd

zteftzf S

.)4

sin(2cossin

xxx

The shape of fh(z,t)

• fh(z,t) can also be written as

• This equation will be used for discussing the so called force-restore method, which serves for predicting soil surface temperature.

.22

2)

2(

,]),(),(1

[),(

02/1 S

S

h

h

dC

dT

fC

whereTtzTt

tzTtzf

Water flow in the soil

• Water flow in the soil is similar to diffusion, leakage. This is caused by the tortuosity of thesoil via the effect of capillary and gravitationalforces.

• Gravitational force is imlicitly directeddownwards. The direction of capillary forces is variable, it is the same with direction of waterpotential gradient. If the water potential gradientis directed upwards and the capillary force is larger than the gravitational force, the waterflows upwards.


• Gravitational force governs water flow in the

macropores where the water is free (not bound to

soil particles). This water is the so-called

gravitational water.

• Capillary force governs water flow in the

micropores where the water is bound by soil

particles. This water is the so called capillary water.

Capillary water is held by cohesion (attraction of

water molecules to each other) and adhesion

(attraction of water molecules to the soil particles).


• Water flux density (fw) is determined by both capillary and gravitational forces. This joint effect could be written as

depending on the units used. Ψ is the water potential and K is the hydraulic conductivity. The formula for flux density fwk is the Darcy law, it is empirically based. Before discussing Ψ and K, let’s get to know their units!

KgorKfandz

Kf

wherefff

wgwk

wgwkw

,

The unit of Ψ

• If the volume of water is considered, Ψ’s

unit is Jm-3, that is Nm-2=Pa.

• Instead of Pa, water column height could

also be used as the unit. The relationship: 1

hPa = 1 cm of water column height.

• If the mass of water is considered, Ψ’s unit

is Jkg-1.

The unit of K

• If the unit of Ψ is water column height and the unit of flux density fw is ms-1 (this comes from m3m-2s-1 because water volume is considered), then the unit of K is also ms-1. In this case, flux density fwg = K.

• If the unit of Ψ is Jkg-1 and the unit of flux density fw is kgm-2s-1, then the unit of K is kg∙s∙ m-3. In this case, flux density fwg = K∙g.

Differential equation for water flow

• Flux density fw is not constant over depth! Where

(divergence)

the soil moisture content (θ) has to decrease and vice

versa, where

(convergence)

the soil moisture content has to increase. Combining the

flux density equation with the continuity equation one can

obtain the so called Richards equation.

0

z

fw

0

z

fw

.tz

fw

w


• If the water flow is mostly governed by capillary forces, that is when fw = fwk (this is the simpler case), then

• Dw is the water diffusivity. C represents the change of soil moisture content for a unit change of Ψ. The most important assumption for this transformation is that Ψ is a function of θ and, vice versa, θ is a function of Ψ. This is true only for capillary and osmotic potentials.

.

],[][

C

KKD

wherez

Dzz

Kzt

w

ww


• In the former equation, the unknown variable is θ. Nevertheless, the equation could also be expressed asa function of Ψ. Then, since Ψ = Ψm

• Ψ is the total water potential, while Ψm is the matric orcapillary potential. The relationship between Ψ and Ψm

as well as their dependence on θ will be discussed later.

.

],[

m

mmw

m

m

w

C

wherez

Kzt

Ct


• If the water flow is governed not only by matric

but also by gravitational potential (this is the

most general case), i.e. when fw = fwk+fwg, then

.][

,

].[

Kgz

Kzt

C

gzSince

zK

ztC

mmw

m

w


• The latter equation approaches reality

closely, since it takes into account both the

capillary and gravitational effects.

• To be able to consider the equation, we

have to know more about Ψ and K. Note

that Ψ is a state variable, while K is a

parameter! Let’s first take a look at Ψ!

Water potential

• Water potential is the potential energy per unit mass (or volume) of water in a system, compared to that of pure, free water.

• According to the convention, the potentialenergy of free water is zero. So, thepotential energy of bound water possessesnegative values.

• The more the water is bound by soilparticles, the more negative Ψ is, i.e. thehigher the absolute value of Ψ is.

Water potential

• In the definition, Ψ is referring to both mass and volume. If it is reffering to volume, Ψ’s unit is Nm-2, that is Pa.

• The negative Ψ can be interpreted as suction, the magnitude of which is equal to the pressure and, implicitly, it is directed opposite to it.

• It was also mentioned that Pa could also be replaced by water column height. 1 hPa = 1 cm water column height. Considering water mass, the unit of Ψ is Jkg-1.

Water potentialCampbell (1985)

Water potential

• Water potential is not only determined by

capillary and gravitational forces.

• In the vicinity of plant roots, water flow is also

influenced by osmotic potential (Ψo). Osmotic

potential is equivalent to the work required to

transport water reversibly and isothermally from a

solution to a reference pool of pure water at the

same elevation.

• If the water column is continous, hydrostatic

pressure could also act as an external force. This

is characterized by a pressure potential Ψp.

Water potential

• Total water potential (Ψ) is the sum of

the water potential components, i.e.

.pogm

Water potential

• Among the water potential components, the matric (the result of the attraction between water and soil particles) and osmotic potentials depend on soil moisture content.

• Ψ is also function of θ via Ψm and Ψo. The Ψm(θ) relationship is of basic importance, it is called the soil moisture characteristic or moisture release curve.

• The Ψm(θ) relationship (in most cases this is the same as Ψ(θ)) is called the pF curve, when Ψ is represented as the logarithm of the water column height expressed in cm (y axis) versus relative soil moisture content (θ/θS) (x axis).

Water potential

S= sand, L= loam, T= clay, WP= wilting point, FK= field capacity

source= internet

Water potential• The function Ψ(θ) can be estimated using

statistical evaluations applied to soil sample data.

• Campbell’s (1974) parameterization is based on the assumtion that the relationship between lnΨ and ln[θ/θS] is linear (this is the simplest approach).

.)( b

S

S

Water potential

• b is the porosity index, ΨS is Ψ at saturation and analogously θS is θ at saturation. Their values were determined by Clapp and Hornberger (Clapp and Hornberger, 1978) using data from USA soil samples.

• Clapp-Hornberger’s data set (Clapp and Hornberger, 1978) is widely used in meteorological models.

.)( b

S

S

Water potentialÁcs (1989)

Water potential

• There are also more complex

parameterizations, van Genuchten’s is one

such parameterization (van Genuchten,

1980).

• This parameterization is widely used in soil

science.

Hydraulic conductivity

• K changes similarly to Ψ in a broad range. In the large pores, where the gravitational effect is dominant, K is a function of ΨS.

• K for saturated soil can be expressed after theoretical considerations as follows:

• where σ is the surface tension of water, ν is the viscosity of water, θS is the saturated soil moisture content, ΨS is the saturated water potential, ρw is the water density and b is the porosity index.

,)22)(12(2

2

22

bbK

Sw

SS


• The former equation can also be written

as

• K is obviously proportional to KS and it is

inversely related to ΨS2. ΨS can be

interpreted as “characteristic microscopic

length”. The characteristic length for a

soil can be taken as the radius of the

largest pores.

.2

constK SS


• Function K(θ) as the function Ψ(θ) could be estimated using statistical evaluations applied to data referring to soil samples. As it was mentioned, one of the simplest relations for K(θ) is obtained by Campbell (Campbell, 1974).

• The values of KS, θS and b are determined by Clapp and Hornberger (Clapp and Hornberger, 1978). Campbell’s parameterization with values of KS, θS

and b obtained for USA are widely used in meteorological applications.

.)( 32 b

S

SKK

Wetness characteristics and soil

texture

• Water flow in the soil is regulated by pores, more precisely by their magnitude and size distribution. These two factors depend indirectly on the features of soil particles (magnitude, form, material composition). So, wetness characteristics as ΨS, KS, θS and b depend indirectly on soil texture.

• How? Is there any rule or relationship?

• Yes, relationships can be observed, in short, they are as follows.

ΨS and the soil texture

• The magnitude of │ΨS│ increases going from coarser (sand) to finer (clay) soil textural classes.

• This increase could be quantified as it is done in the ISBA (Interaction Soil Biospere Atmosphere) biophysical scheme (Meteo France), nevertheless such quantification is not common in meteorological applications.

• The observed increase can be easily explained. At saturation, water retention in smaller pores is higher than water retention in larger pores.

KS and soil texture

• The magnitude of KS decreases going from coarser (sand) to finer (clay) soil textural classes.

• KS is extremely sensitive to the magnitude of the “large” pores since water runs out first from the largest pores when the water content decreases. It is logical but it has to be said: water runs out of the smaller pores only after it has run out of the larger pores.

θS and the soil texture

• Concerning porosity (total pore volume) thebasic question is as follows: How large is theporosity of many small pores with respect to theporosity of much fewer large pores?

• Observations show that porosity increases goingfrom coarser (sand) to finer (clay) soil texturalclasses.

• Since θS is practically equal to porosity, thesame change can also be observed for θS.

b and the soil texture

• b is the slope of the best-fit line between lnΨand ln[θ/θS]. Therefore b represents the changeof lnΨ for a unit change of ln[θ/θS].

• If we construct these straight lines for all soiltextural classes (on the basis of soil sampledata), we shall see that the slope of the linesincreases going from coarser to finer soiltextures.

• More precisely: the lower b is the lower theporosity (light soils) and vice versa, the larger b is the larger the porosity (heavy soils).

Wetness characteristics of different

soil textures for USA and Hungarian

soils

Ács et al. (2010)

Infiltration and redistribution• Water flow in the soil is also determined by soil

surface conditions.

• Precipitation flux density splits into surface run off(liquid water does not enter the soil) and infiltration(liquid water enters the soil).

• This partitioning depends upon relief and soilcharacteristics, primarily upon the soil texture andthe soil moisture conditions.

• Hydrologists are interested in run off, whilemeteorologists and pedologists in infiltration. Let’sfind out more about the most important features of the infiltration!

Infiltration

• Infiltration rate fi(t) depends strongly onsoil moisturecontent. It is higherfor dry and lower formoist soil.

• Infiltration rate is initially high, butdecreases over timeto a constant value.

Campbell (1985)

Infiltration• When water enters soil,

it develops a transmission zone from the soil surface to the “wetting front” (boundary between wet and dry soil).

• This sharp front is a result of the sharp decrease in K. In the transmission zone, K is high because θ is “high”. Below it K is low because θ is much lower than in the transmission zone.

Cambell (1985)

Infiltration

• The observed infiltration rate fi(t) can also be

theoretically deduced.

• Let xf be the depth of the transmission zone. Ψf

and Ψi are the water potential at the wetting front

and at the soil surface. Let [K] be the average

hydraulic conductivity in the transmission zone.

Then, the average infiltration rate is

.][f

if

ix

Kf

Infiltration

• During infiltration the observable “wetting front” moves through the soil with a velocity dxf/dt, threby increasing the water content in the transmission zone by Δθ. Δθ can be expressed as

• where θ0 = soil water content before infiltration,

• θi = soil water content at the inflow and

• θf = soil water content at the wetting front.

,2

0

fi

Infiltration

• On the basis of the continuity equation

• Integration gives xf as a function of time. xf

is directly proportional to the square root of

time.

.][dt

dx

xK

f

f

fi

.)]([2

tKx

fi

f

Infiltration

• Combining xf and fi(t),

• The infiltration rate [fi(t)] is directly proportional

to Δθ1/2 and inversely proportional to t1/2.

.2

)]([)(

t

Ktf

fi

i

Infiltration

• Integrating fi(t) over time one can obtain cumulative infiltration.

• Cumulative infiltration is proportional to t1/2.

.)]([2

,}2

)]([{

,}2

)]([{)(

0

2/12/1

2/1

00

tKI

dttK

I

dtt

KdttfI

fi

tfi

tfi

t

i

Soil water transport equations in

the biophysical scheme SURFMOD

• The movement of water in the soil is

represented in SURFMOD by Richards

equation:

• In this equation, the so called source-sink

term (for instance water uptake by roots) is

not represented. By implementing it one gets

.z

f

t

ww

.SSTz

f

t

ww

Soil water transport equations in

the biophysical scheme SURFMOD

• Integrating the former equation between an

upper level a and a lower level b and

assuming that θ and SST are constant within

the layer thickness Dab, one gets the following

equation:

.

,)(

abab

abwawbabw

zzD

whereSSTDfft

D

Prediction of θ in the top soil layer

• In the SURFMOD, this layer is denoted by D1.

So, Dab = D1 [see Figures 2.3 and 2.5 in Ács et

al. (2000)]

• By substituting these terms, one gets an equationwhich agrees with equation (3.9) in Ács et al. (2000). Now let’s look at Q1!

.

,

1

11

00inf

prunab

Rwb

Rwa

SQSSTD

andQQf

EQPf

Prediction of θ in the top soil layer• Q1 is constituted by both capillary and

gravitational terms. Therefore

• (δΨ/δz) at z1 refers to zb being equal to level D1.

Expressing levels via layer thicknesses and using

finite difference approximation, one can simply

obtain Q1 as

.)1(111 zw

zKQ

,)21(21

2111

DDKQ w

Prediction of θ in the top soil layer

• where D2 is the thickness of the

intermediate soil layer [see Figure 2.3 in

Ács et al. (2000)]

• The obtained Q1 agrees with equation

(5.19) for i=1 in Ács et al. (2000).

Prediction of θ in the intermediate

soil layer

• In the SURFMOD, this layer is denoted by D2. So,

Dab = D2 [see Figures 2.3 and 2.5 in Ács et al.

(2000)]. Furthermore

• By substituting these terms one gets an equation

which agrees with equation (3.12) in Ács et al.

(2000).

.

,

2

2

11

runab

wb

Rwa

QSSTD

andQf

QQf

Prediction of θ in the intermediate

soil layer

• Furthermore

• The obtained Q2 agrees with equation (5.19) for

i=2 in Ács et al. (2000).

• Note that Figures 2.3 and 2.5 in Ács et al. (2000)

can help in understanding how the equations are

obtained.

,)21(32

3222

DDKQ w

Prediction of θ in the bottom soil

layer

• In the SURFMOD, this layer is denoted by D3. So, Dab = D3 [see Figures 2.3 and 2.5 in Ács et al. (2000)]. Furthermore, there are no roots in this layer. So

• By substituting these terms one gets an equation

which agrees with equation (3.13) in Ács et al.

(2000).

.

,

3

3

2

runab

wb

wa

QSSTD

andQf

Qf

Phenomenology of the atmospheric

transport processes in the vicinity of

land-surface

• Structure and

features of the

near surface

atmosphere

(Foken, 2002)


transports in the vicinity of land-

surface

• What is transferred

to where?

• Why and how?

Bonan (2002)


transports in the vicinity of land-surface

• What is the relationship between the flux densities [E (evaporation), H (heat) and τ (momentum)] and state variables [q (specific humidity), T (temperature), u (wind speed)]?

• In common practice, the state variables (q, T, u) are measured (routinely only at one level), while flux densities (except precipitation and radiation) are calculated!

• One important goal in micrometeorological education is to present the most important methods for calculating vertical flux densities, for instance, evapotranspiration.

Flow types

• Laminar flow

(molecular diffusion; feature of the medium; it is

near the surface)

• Turbulent flow

[eddy (diffusion-like transfer) transfer; feature of

the flow; it is far above the surface]

Turbulent flow – domains• Microscale turbulence

ƒ=h/l (l=u∙τ)

h = height above ground

l = horizontal size of the eddy

• viscous subgroup, ƒ >> 1

• inertial subgroup ƒ ≥ 1

• micrometeorological domain

mechanical turbulence 1 > ƒ ≥ 0.3

mechanical and thermal turbulence ƒ ≤ 0.3

Turbulent flow – coefficients

• Eddy diffusivity (K)

• aerodinamic resistance (r)

quantitytheofgradientionconcentrat

quantitytheofdensityfluxK

quantitytheofdensityflux

quantitytheofdifferenceionconcentratr

Turbulent flow – coefficients

• K refers to the level, while r to the layer!

The relationship between them is as

follows:

• This is derived from their definitions!

.)(

12

1

dzzK

r

z

z

Mechanical turbulence – ground

surface• turbulence caused by

wind shear (wind speed change with the height),

• neutral stratification (vertical temperature gradient is equal to zero),

• mass transfer is possible, but heat transfer is not.

• Logarithmic wind profile

)ln()(0

*

z

z

k

uzu


Mechanical turbulence above the

vegetation canopy

• Roughness length (z0)

[wind speed becomes zero not at the surface (this could be called the “geometrical level”), but somewhat above the surface (it could be called the “aerodynamic level”)],

• Zero plane displacement height (d) [there is a shift between “aerodynamic levels” above vegetation and bare soil. Vegetation acts as a protective wall of height dagainst wind, though it is a porous medium.] )ln()(

0

*

z

dz

k

uzu


Mechanical turbulence above the

vegetation canopy

• τ parameterizations, r and K calculations2

*u

aMCzu 2)(

2

*

)(

)(

1

u

zu

Czur

aM

aM

)()( ** dzklaholuldzukKM

Thermal and mechanical

turbulence – ground surface

• Turbulence caused by

both wind shear and

surface heating,

• stable (the vertical

temperature gradient is

positive) and unstable

(the vertical temperature

gradient is negative)

stratifications,

• wind profile: near to the

logarithmic (but not

logarithmic) Bonan (2002)

Thermal and mechanical

turbulence – ground surface

• There is heat transport beside momentum and mass transport.

[all three profiles (wind, humidity, temperature) have to be considered]

• Land-surface: vegetation (d+z0), bare soil (z0).

Aerodynamic method

Let stratification be neutral! Then

,)(* dzku

E

z

q

z

q

EKM

.)(* dzkuz

u

z

uKM

Aerodynamic method

Let stratification be stable or unstable

instead of neutral! Then

,)(*

q

q

MEst dzku

E

K

E

K

E

z

q

,)(*

dzkuc

H

Kc

H

Kc

H

z pMp

Hstp

.)(

*m

m

MMst dzk

u

KKz

u

Aerodynamic method

• According to similarity theory, the functions

φ(ς) are dimensionless so-called universal

functions, where

,monL

z .

3

*

p

mon

c

H

T

gk

uL

• Lmon is that height where the turbulent kinetic

energy generated by wind shear and thermal

stratification is equal.

Aerodynamic method

• We are interested to know the integral form of the equations (Brutsaert, 1982) since the measurements are at discrete levels, so

,)()( 12

*

21

qqku

Eqq

,)()( 12

*

21

pcku

H

.)()( 12*

12 mmk

uuu

Aerodynamic method

where

,)(2

1

dq

q

,)(2

1

d

.)(2

1

dmm

Aerodynamic method

• We are also interested in the relationship between the stable and unstable on the one hand and the neutral stratifications on the other. This could be characterized by introducing the so called stability function (ψ) (Brutsaert, 1982).

)()()ln())(1(11

12

1

2

2

1

d

)()()ln())(1(11

12

1

2

2

1

mmmm d

)()()ln()(1

))(1(11

12

1

2

2

1

2

1

2

1

qq

q

qq dd

d

Aerodynamic method

where

So

.)(1

)(2

1

d

,)()()ln( 12

1

2

*

21

qq

ku

Eqq

,)()()ln( 12

1

2

*

21

pcku

H

.)()()ln( 12

1

2*12

mm

k

uuu

Aerodynamic method

• Stability function

Brutsaert (1982)

Aerodynamic method

according to similarity theory

• The lower level is not in the atmosphere,

instead at the land-surface because of the

lack of the measurements!

.)( 01,2

monL

zheightlayerboundaryplanetaryh

.222111 ,,,0, uuqqanduqq ss

Aerodynamic method

,)(ln0*

q

q

sz

dz

ku

Eqq

,)(ln0*

z

dz

cku

H

p

s

.)(ln0

*

m

mz

dz

k

uu

Aerodynamic method

• In order to integrate ψ we need to know φ. Many

functions of φ are suggested. Here, the functions

suggested by Dyer and Hicks (1970) will be used.

For unstable stratification

,)161( 2/1

.)161( 4/1 m

,)161( 2/1 q

Aerodynamic method

For stable stratification

.16

1051

mq

Aerodynamic method

Universal functions

Brutsaert (1982)

Brutsaert (1982)

Aerodynamic method

Universal functions

Foken (2002)

Businger et al. (1971)

Aerodynamic method

,1

1ln2)(

2

0

2

q

qx

x

,1

1ln2)(

2

0

2

x

x

),(2)(2)1()1(

)1()1(ln)( 02

0

2

0

22

m

mm

m xarctgxarctgxx

xx

.,)161(,)161( 00

4/1

00

4/1

monmon L

zés

L

dzxx

For unstable stratification:

Aerodynamic method

,2

1ln2)(

2

x

,2

1ln2)(

2

xq

.2

)arctan(22

1ln

2

1ln2)(

2

x

xxm

For unstable stratification:

Aerodynamic method

For stable stratification:

.55)()()(mon

mqL

dz

Aerodynamic method

• We could see that flux densities E, H and τ

depend upon Lmon, and, vice versa, Lmon

depends upon E, H and friction velocity (u*).

• When there is such an interdependence

the iterative procedure

has to be applied!

Energy balance of the vegetation

canopy• Beside roughness, the

energy balance (available energy flux density) of the “surface” is also an important factor.

• Let’s take a look at the energy balance of an air column! The air column is within the Prandtl layer. Oke (1978)


canopy

What are flux densities?

• Vertical flux densities:

1. radiation balance at the top of the air column(Rn),

2. heat flux density across the soil surface (G),

3. turbulent heat flux densities (sensible heat flux

density (H) and the latent heat flux density

(λ∙E)) in the air column (we suppose that theyare constant over height)


canopyWhat are flux densities?

• Horizontal flux densities (advection (D)),

• Heat storage:

1. Heat storage in the column of the vegetationcanopy (air, leaves, stems, thin soil surfacelayer) (J),

2. Radiation energy used by photosynthesis(μ∙A). μ is the fixation energy of CO2 (1,15 ∙104

J g-1), A is the assimilation rate (g∙m-2s-1)


canopy

• Adding input and output flux densities referring to

the air column one obtains the energy balance

equation for the vegetation canopy:

The terms D, J and μ∙A could be neglected with

respect to Rn-G, so:

.0 EHAJDGRn

.EHAGR en


canopy

• Ae is the available energy flux density at the

“surface” (note: Ae is energy flux density

(unit: W∙m-2) and not energy (unit: J)).

• Atmosphere gets the Ae (in the form of H+λ∙E),

therefore it is important for us.

• The partitioning of Ae between H and λ∙E is

regulated by the water availability of the “surface”.


canopy

How large are the flux densities? How do

they change during the day?



canopy

.RPA

P= photosynthesis (mg m-2 s-1), Baldocchi (1994)

R= respiration (mg m-2 s-1).

Bowen method

• Input data: air temperature (T), water vapour

pressure (e) at least at two levels and the

available energy flux density at the “surface”

(Ae). Ae is a “new” important term!

• Output quantities: sensible (H) and latent heat

(λ∙E) flux densities.

• There are fewer input data (there is Ae, but there

is no u(z)) as compared to the aerodynamic

method and the energy balance is fulfilled.

Bowen method

.sin,1

,1 1 E

Hce

AH

AE ee

EH

E

H

E

p

Hp

KKbecauseandK

K

e

T

z

eK

cz

TKc

E

H

β is the Bowen ratio. It can be estimated on the

basis of the so called gradient measurements.

.e

T

Bowen method

• The accuracy of β depends on how well the best-fit straight line T(e) is estimated.

Gradient measurement: location - Rimski Sancevi (in Hungarian Római Sáncok), date – 1982, 19th May, local

time - 14 hours, land-surface type – bare soil

Ács (1989)

Bowen method

• Applicability:

The method may be applied well when |Ae|

is large and is less applicable when |Ae| is

small (about zero).

Penman-Monteith’s equation

• Combining the energy balance approach and the aerodynamic treatment one gets Penman-Monteith’s equation. This is possible if “water balance” information is also available and used.

• Input data: air temperature (T), partial water vapor pressure (e) and wind velocity (u) at one level (the levels must not be at the same height), the available energy flux density of the “surface” and information referring to the availability of water on the “surface”.

• Output quantities: sensible (H) and latent heat (λ∙E) flux densities.


• Usually more input data are used than in the

Bowen method since the so called “surface

resistance” of the land-surface also has to be

estimated.

• It takes into account the atmospheric stratification.

The Bowen method does not.

• It is one of the most widespread equations in

environmental meteorology.


• How is it derived? Here are the basic

equations!

• 4 equations, 4 unknowns. The unknowns are:

H, λE, T(0) and e(0).

.)0()]0[(

)()0(

,)()0(

,

E

eTecr

ésE

zeecr

H

zTTcr

EHA

Sp

st

p

aE

paH

e


• Now let’s sum the last and the next to last

equations!

• Herewith e(0) is eliminated.

.)()]0([

E

zeTecrr Sp

staE


• How can T(0) be eliminated?

.)()0(

)()],()0([)]([)]0([

aH

p

e

SSS

rc

EAzTT

andT

TewherezTTzTeTe


• Substituting these

• Redistributing according to λE

.

)(][)]([

E

zerc

EAzTe

crr

aH

p

eS

p

staE

.)]()([)( aHeS

p

aHstaE rAzezTec

ErrrE


• Multiplying by γ and dividing by raH

• Since raH=raE=ra and δe=eS[T(z)]-e(z)

./)(

/)}()]([{

aHstaE

aHSpe

rrr

rzezTecAE

.

)1(

/

a

st

ape

r

r

recAE

Priestley-Taylor’s equation

• Input data: Ae and the air temperaure (T) at one level.

• Output quantities: sensible (H) and latent heat (λE) flux densities.

• Contrary to Penman-Monteith’s equation (PM equation) Priestly-Taylor’s equation does not take into account the stratification effect.

• Priestley-Taylor’s equation (PT equation) is more popular since satellite measurements of radiation became available.


• How to derive it? Let’s start from the PM

equation!

• The PM equation can be divided into two

terms. The first characterises the “surface”

(term ΔAe), while the second the

evaporative demand of the atmosphere

(term δe).


• First supposition: the second term is usually

less than the first. Therefore the second term can

be expressed as a part of the first term.

• Second supposition: the surface is wet,

therefore its surface resistance is small, i.e.

rst→0. If the two suppositions are valid, then

25.1

PTePT whereAE

Synthesis of the methods

• Four methods are presented for estimating H

and λE:

the aerodynamic method,

the Bowen method,

Penman-Monteith’s equation and

Priestly-Taylor’s equation.

• Now let’s compare the methods!

Synthesis of the methods

Aerodynamic method Bowen method

(T2, e2, u2) (T2, e2)

(T1, e1, u1) (T1, e1) and Ae

estimation of no estimation of

stratification stratification

PM equation PT equation

(T1, e1, u1) Ae and Θ T1 and Ae

estimation of no estimation of

stratification of stratification

Evaporation fraction and the

Bowen ratio

• The Bowen ratio has already been introduced.

The evaporation fraction α is defined as

• Both α and β depend on the available water and

energy of the “surface”, so indirectly on weather

and climate. Neverthless, their changes can be

analyzed in a simpler way. How?

.eA

E


Bowen ratio

• Now let’s take the resistances! So far two

resistances were introduced: ra (aerodynamic

resistance) and rst (stomatal resistance). Now

let’s see the so called climatic resistance!

• ri depends both on the state of the “surface” and

on the state of the atmosphere.

.e

p

iA

ecr


Bowen ratio

• α and β can be easily expressed as function of ra,

rst and ri.

• So, they can be also analyzed in terms of

resistances (Jones, 1983)!

.

ia

iast

staa

ia

rr

rrrand

rrr

rr

Soil and vegetation as water

reservoirs

• Water transfer in the soil-vegetation

system will be considered from the

meteorological point of view.

• From the meteorological point of view soil

and vegetation are primarily water

reservoirs. Vegetation stores water not

only in its body, but also on its surface.


reservoirs

• Soil can store the largest amount of water. This

amount is much larger than the amount stored by

vegetation. At the same time the amount of water

storable in the body of vegetation is much larger

than the amount of water storable on its surface.

• The ratio of water storable in the soil, in the body of

vegetation and on the surface of vegetation is

roughly equal to the ratio 100 : 10 : 1.


reservoirs

• It is important to say that soil’s water storage

capacity is comparable to annual flux densities

entering and leaving it.

• It has to be underlined that soil is not only a

great water reservoir but also a great carbon

reservoir. But annual carbon flux densities which

enter and leave it cannot be compared (they are

much less) to its carbon storage capacity.


reservoirs

• Water amount is

changing in both

reservoirs,

nevertheless they are

not independent.

They are connected

via transpiration and

root water uptake.


Water flux densities in the soil-

vegetation system

• Which water flux densities are the largest? Precipitation, evapotranspiration and run-off.

• Evapotranspiration is composed of three components: transpiration, soil evaporation and evaporation of the intercepted water. As we see, the last two terms are both evaporation.

• Meteorologists are interested in precipitation and evapotranspiration, while hydrologist in run-off.


vegetation system

• One important contribution of the

science of the biophysical modeling in

meteorology is that it recognized and

quantified the role and impact of

transpiration in the formation of

weather and climate.


vegetation system

• Which water flux densities are moderately large (not small, not large) but important? The interception and evaporation of the intercepted water. Why? Since this water comes back into the atmosphere without entering the soil. With this, the water cycle becomes faster and the forcing of local convective weather events is stronger.

• This phenomenon is the strongest and therefore the most important in tropical regions.

(Water storage on leaves is called interception.)


vegetation system

• Now let’s take a look at transpiration and root

water uptake! These two water flux densities are

different (root water uptake is always a little bit

greater than transpiration), but they can be

treated as equal from the meteorological point of

view.

• This fact is important since the estimation of

transpiration in meteorological models is based

on this fact. The details related to this topic will

be explained later.

Water storage in the soil-vegetation

system

• Now let’s also consider the characteristics of soiland vegetation as water reservoirs!

• The smallest water reservoir is the vegetationsurface. Its average maximum value inmeteorological models is 0.2 mm/LAI, where LAIis the leaf area index. This means that a maximum of 2 dl water can be kept on a leafsurface of 1 m2 without run-off. So, 1 l water canbe stored on a leaf surface of 5 m2.

• This value is an average value. This variesdepending on vegetation type, but inmeteorological models this is usually not takeninto account.

Vegetation water content

• The ratio between the stored water and

dry mass is an important vegetation

characteristic.

• For herbaceous plants this ratio is 6:1.

This ratio could also be used as a

guideline for other vegetation types.

Vegetation water content

• According to the previous consideration as a first estimation vegetation water content is six times the dry mass contained in a unit of LAI.

• Water content possesses a daily course and it varies during the growing season. Daily maximum is at dawn, while the minimum is during the midday hours. In the growing season, it increases with the increase of biomass.

• At the end of the growing season, the water content of grasses is about 10 mm. This can be interpreted that 10 mm is accumulated over the course of about 100 days, so, the average accumulation rate is 0.1 mm/day.

The accumulation rate and

transpiration• Let’s compare the accumulation rate and

transpiration! We saw that the daily accumulation rate is 0.1 mm/day. The daily sum of transpiration is 1-4 mm/day.

• We can see that the accumulation rate is one order of magnitude or more smaller than transpiration.

• This means that water practically flows through the vegetation, its storage is minimal. Vegetation is simply a channel between the soil and atmosphere. This “flow through the channel” is independent from the stored water.

Soil water content

• What is the maximum storable water in thesoil? When the storage is maximum, thepores are completely full with water, thenθ=θS. For 1 m3 soil this is roughly 0.5 m3,or 500 litres of water.

• Can vegetation gain access to this water? Not completely, only partly. Vegetation cantake up water only from the θf – θw soilmoisture content zone, which is called theplant available water holding capacity.

Soil water content

• Plant available water holding capacity is less than θS and greater than 0. The amount θf – θw also depends upon soiltexture. For sand it is the least (about 0.1 m3, that is 100 litres of water), while forloam it is much greater (about 0.2-0.3 m3, that is 200-300 litres of water).

• These facts also show the reason whyloam is one of the best and sand is theleast appropriate soil texture class for cropproduction.

Soil water content

• The meaning of θf and θw has still not been explained!

• θf is the field capacity soil moisture content, while θw is the wilting point soil moisture content. θf is that minimum soil moisture content for which the force of gravity is still greater than the capillary force for holding the water. Consequently, the soil column is not able to hold the water in it for all cases when θ≥θf.

• θw is that soil moisture content value below which the plant is not able to take up water. In other words, the moisture content of soil after the plants have removed all the water they can.

Soil water holding capacity

• θf and θw values: Clapp and Hornberger (1978)

Soil water content

• We can see that plants are not only able totake up water in extreme dry (θ≤θw) butalso in extreme wet (θ≥θf) conditions.

• This fact shows that plants need some soilair in order to take up water. Of course, this is regulated and solved by plants livingin the water in a certain way (for instance, by building an aerenchyma system).

Soil water content

• There is a certain degree of uncertainity in the definition of θf and θw. Namely, gravitational drainage or the process of wilting cannot be observed unequivocally on the basis of the use of a number of criteria. Consequently, their values are uncertain.

• Different criteria are used to determine their values. Without discussing this issue, let’s underline that in the biophysical modelling of the land-surface their numerical values are uncertain though they are important.

Water transport in the soil-plant

system

• Let’s take a look at water transport in thesoil-plant system! We showed on the diurnalscale that the amount of stored water invegetation (accumulation rate) could be neglected with respect to transpiration.

• Transpiration can be succesfully modelled taking into account the above fact. The useof the aforementioned assumption is widespread in meteorological applications. Therefore, the basic equations of thisapproach will be briefly presented.

Transpiration model: basic

equations• Root water uptake (QR) is the input water flow.

• This water flow will be simulated using an analogy to Ohm’s law (current is the ratio of potential difference and resistance).

• Voltage is the difference between the leaf water potential (Ψleaf) and soil water potential (Ψsoil). Ψleaf refers to the “average” leaf of the canopy which is represented by one “big leaf” located on the level d+z0. Ψsoil reflects an “average” potential reffering to the 1-m deep soil moisture content profile in the root zone.


equations

• Soil puts up a resistance rS, while vegetation a resistance rP to the water flow in the soil-plant system.

• rS is greater the drier the soil is, and vice versa, rS is smaller the wetter the soil is. Note that rS is comparable to stomatal resistance when the soil is dry!

• rP is mainly caused by xylem vessels. It is taken as a constant.

Resistances in the soil-plant

system

Ψ – potential; rt - soil resistance; rgy – root resistance; rx – xylem vessel resistance;

rs – stomatal resistance; rcu – cuticular resistance; ra – aerodynamic resistances

in the boundary (lower) and turbulent (upper) atmospheric layers;

légkör = atmosphere; vízkészlet=water amount in the soil

Rose (1966)


equations

• Root water uptake can be expressed as

)1(.PS

leafsoil

Rrr

Q


equations• Soil water potential and leaf water potential are

given in unit of water column height [m H2O].

• Resistances are given in seconds, though such a resistance unit is very unusual.

• This is true because QR is parameterized after Ohm’s law. Such a parameterization can be done since the water flow in the soil-plant system is almost a steady state (quasi steady-state). The unit in seconds can be interpreted as follows: if the resistance is high, the water flows slowly, consequently the transport needs more time. So, a long time is equivalent to great resistance, and vice versa, a short time corresponds to a low resistance value.


equations

• We have already mentioned that root

water uptake (input water flux, QR) is

practically equal to transpiration (output

water vapor flux, ET), i.e.

)2(.TR EQ


equations

• Transpiration can be calculated either by

Penman-Monteith’s formula or by the

gradient method, as presented below:

)3(.)(

)3(,

)1(

/

brr

eTecEL

a

r

r

recREL

ca

rvgSp

T

a

c

apn

T


equations• One of the most important terms in eq. (3a) is rc.

In the meteorological land-surface modelling community, rc is commonly parameterized byJarvis’ (1976) formula:

• Jarvis’ (1976) formula consists of the product of different environmental functions, often calledstress functions contained in Fad and in Fma.

.min

ma

adstc

FGLFLAI

Frr


equations

• Beside such multiplicative formula, there are

also such formulae where the whole effect is

expressed by the addition of environmental

functions (see, for instance, Federer, 1979).

• Symbols: rstmin is the minimum stomatal

resistance at optimum environmental conditions,

LAI is the leaf area index, GLF is the green leaf

fraction, Fad is the function for representing the

atmospheric demand effect upon stomatal

functioning and Fma is the function for

representing soil moisture availability effect upon

stomatal functioning.


equations

• The function Fma can be expressed via Ψleaf

since it depends upon soil water availability.

Taking these facts into account,

isthatFcrSsoil

crleaf

ma ,,

)4(.)(minleaf

ma

adstc f

FGLFLAI

Frr


equations

• Input data: state variables and fluxes: S, T,

e, U, P; parameters: ρ, cp, γ, L, LAI, GLF,

rstmin, Ψcr, Ψsoil,S.

• Quantities to be calculated: Δ, Rn,δe.

• Parametrizations: rS, rP, Ψsoil, ra, Fad.

• Symbols: see Table 2.1 in Ács et al. (2000,

page 22, 23)

Transpiration model: basic equations

• We have four unknowns in four equations.

The unknowns are: Ψleaf, rc, ET and QR.

• Ψleaf could be expressed by combining

equations. Of course, ET could also be

estimated on the basis of Ψleaf.


• The form of the equation for Ψleafdepends upon how the function Fma is parameterized.

• If the Fma/Ψleaf relationship is linear, the equation for Ψleaf is a quadratic equation. Only the positive signed square root solution is the real, physically based solution.


• Model results:Monteith et al. (1975)

Transpiratiom model: applications

in the SURFMOD

• The derivation of the equation for estimating Ψleaf based on the use of equation (3a) for calculating LET can be found in Ács et al. (2000) on page 59.

• The same, but for equation (3b) can be found in Ács et al. (2000) on page 58.

Vegetation canopy surface

resistance

• As already

mentioned, one of

the most important

parameters in

Penman-Monteith’s

equation is

vegetation canopy

resistance rc.

Rose (1966)

The functioning of stomata

• Since rc is an

important parameter

in calculating

transpiration, the

functioning of stomata

(opening, closing) has

to be described as

fully as possible in

meteorology too.

source: internet

Stomata

• Large area density – small area density

Chaloner (2003) Chaloner (2003)

Stomata

• Low CO2 concentration – large area density

• High CO2 concentration – small area density

Chaloner (2003)


• The value of rc is determined by the functioning

of stomata, which is characterized by the

functions Fad and Fma.

• The function Fad is determined by three

atmospheric factors: global radiation, air

temperature and air humidity.

.ahat

vrad

FF

FF


• Fvr is a function expressing the influence of absorbed visible radiation on stomatal functioning,

• Fat is a function expressing the influence of air temperature on stomatal functioning,

• Fah is a function expressing the influence of air humidity on stomatal functioning.

• The functions vary between 0 and 1, except function Fvr.


• The forms of

environmental

functions which

can also be

described as

impact functions.

Jones (1983)


• The form of impact functions is

determined by experiments

performed in the laboratory.

Consequently they have an empirical

nature.

Parameterization of the impact functions

• There are many parameterizations. One of these is as follows:

• where Krl is a constant, Svis is the absorbed visible radiation.

• Note: this function represents resistance and not relative conductance. The function representing relative conductance is the reciprocal of Fvr, so it is equal to [Svis/(Svis+ Krl)].

,vis

rlvisvr

S

KSF


• The impact of air temperature can be discribed

as follows:

• where To is the optimum temperature of the

canopy (vegetation-specific constant), Tr is the

air temperature at the reference height level

(usually 2 m) and cT is a vegitation-specific

empirical parameter.

,)(1 2

roTat TTcF


• The impact of air humidity can be characterized

by

• where cV is a vegetation-specific empirical

parameter, while [eS(Tr) - er] is the vapor

pressure deficit at the reference height level.

,])([1 rrSVah eTecF


• The impact of soil moisture content can

be expressed by the following equation:

• where θw is the wilting point soil moisture

content and θf is the soil moisture content

at field capacity.

,

0

1

w

fw

wf

w

f

ma

if

if

if

F

The cuticle

• Cuticle is a waxy, protective layer on the leaf

surface. Transpiration through it is minimal,

almost non-existent. Therefore the cuticle-

resistance is large, its value in meteorological

models is about 5000 sm-1 (rcu = 5000 sm-1).

• Leaves which are permanently in sunlight have

a thicker cuticle-layer than leaves which are

permanently in the shade.

Surface resistance of the leaf

• The cuticle and the stomata resistors act

in parallel in regulating vapor exchange

between the leaf and the atmosphere.

• Accordingly, the total resistance of a leaf

can be expressed via cuticle and stomata

resistors as follows

.111

stcul rrr

Surface resistance of the leaf

• In the daytime rcu >> rst. Therefore, then

• Leaf resistors connected in parallel constitute

vegetation canopy resistance (rc). Namely, we

suppose that all leaves are connected to the same

humidity potential gradient between the soil and the

atmosphere. Accordingly

.11

stl rr

.11

i ilc rr

Relationship between rst and rc

• Transpiration of the vegetation canopy is mainly constituted by the transpiration of leaves.

• Nevertheless, the microenvironment (partial vapor pressure, wind, turbulence, insolation, air temperature etc.) of the leaves is different because of the effects of shade. This fact increases the complexity of the system, and it encumbers the estimation.


• As a first guess, we can suppose that there

are no microenvironmental differences in

the vegetation canopy.

• In this case, there are no differences

between rli-values, so rli = rl. The

microenvironment of the leaf possessing

resistance rl represents an “average”

environment in the vegetation canopy.


• Taking these facts into account

• where ρ is the air density, Δq is the specific

humidity difference between the leaf and air, Ai is

the leaf surface, A is the soil surface of 1 m2, i is

the number of the leaves and LAI is the leaf area

index.

.1

,11

i

i

i st

i

i l

i

AA

LAI

r

Aq

Ar

Aq

AE

ii


• Since rst,i = rst, the former equation can be written as:

• Let’s now compare this equation with equation (4)! We can see that we need to know not only rst but also LAI to be able to estimate rc.

.111

cstst rq

LAI

rq

rLAIqE


• The greater the LAI, the smaller the rc is. This is

understandable since evapotranspiration

increases with an increase in surface.

• The procedure presented above is called

upscaling. It illustrates the transition from the

scale referring to rst to the scale referring to rc

[upscaling from the scale of stoma (order of

magnitude: μm) to the scale of vegetation

canopy (order of magnitude: m)] .

The parameter rstmin

• We can see from equation (4) that rc dependsnot only on the impact functions (Fad and Fma) and LAI, but also on rstmin.

• rstmin represents the so called minimum stomatalresistance [unit: s m-1]. It is the reciprocal of themaximum stomatal conductance (gcmax) [unit: m3 m-2 s-1 = m s-1]. The conductance is maximum when the stoma is completely open. In this case, all impact functions in equation (4) are equal to 1.

The parameter rstmin

• LAI is a morphological, whilerstmin is a physiological parameter.

rstmin depends on what?

• The maximum cross section of stomata (gcmax or its reciprocal rstmin) strongly correlates with maximum photosynthesis (Pmax). Pmax

represents the height of the photosynthesis curve at saturation. The relationship was determined by Körner et al. (1979).


• The greater Pmax is, the greater gcmax is and the

smaller rstmin is.Jones (1983)


• Kelliher et al. (1995), rstmin-1 = gsmax

ConductancesKelliher et al. (1995)


• rstmin depends on vegetation types. It

is completely different for cultivated

and natural vegetation. It changes

also by intensity of insolation but also

during the growing season.


• Garratt (1993)


• Cultivated plants possess lower rstmin

values than their corresponding natural

species.

• Sun-loving plants possess lower rstmin

values than shade-loving plants.

• The least rstmin values in the growing

season are in the flowering period. After

flowering, the rstmin values are higher. We

know little about the rate of the changes

during the growing season.

Temperature of the vegetation and

its environment

• Vegetation is able to survive in the

temperature interval ranging from -88 °C

to +58 °C.

• Vegetation is able to grow in the

temperature interval ranging from 0 °C to

+40 °C.

• Growth of the vegetation is most intensive

at the so-called optimum temperature (To).

Temperature change of the

vegetation: physical bases

• Temperature of the vegetation is determined byits energy budget equation. The most generalform of this equation, neglecting thephotosynthetical storage term, is as follows(Jones, 1983):

• The sum of energy fluxes is rarely equal tozero, therefore ST ≠ 0 [ST is the storage term], consequently the vegetation temperature will be changed.

.STELHRn

Equilibrium state

• If ST = 0 [this condition is never fulfilled in reality, only approximately, i.e. ST is close to zero (ST ≈ 0)], we are talking about an equilibrium state.

• If ST ≠ 0, we are talking about a non-equilibrium state. The former case is obviosly simpler than the latter.

Equilibrium state

• In the equilibrium state

• All three terms depend on the

vegetation temperature Tv.

.0 ELHRn

Equilibrium state

• Net radiation flux Rn can be expressed as:

• Let’s express the term εσTv4 using Taylor

series expansion for air temperature Ta! Then

.)1(444

vvabsvvaan TRTTSR

).(4

),(4

3

344

avavnin

avavavabsvvabsn

TTTRR

TTTTRTRR

Equilibrium state

• Rni is the so-called net isothermal

radiation. This term is obtained using the

condition Tv = Ta, which is the same when

we suppose the existence of an

isothermal temperature profile.

• Furthermore

,aH

avp

r

TTcH

Equilibrium state

• and

• By linearization of the term eS(Tv) (water vapor

pressure at saturation at vegetation temperature Tv)

using Taylor series expansion for air temperature Ta

.)(

caE

avSp

rr

eTecEL

),()()()(

)()( avaSavTTS

aSvS TTTeTTT

TeTeTe

a

Equilibrium state

• By substituting terms

.][caE

av

caE

p

rr

TT

rr

ecEL

.0][)(4 3

caE

av

caE

p

aH

avpavavni

rr

TT

rr

ec

r

TTcTTTR

Equilibrium state

• We are interested to know the term Tv-Ta,

accordingly the equation can be written as

follows:

• Introducing the terms rR and rHR,

.]}411

[{)(3

caE

p

ni

p

av

caEaH

pavrr

ecR

c

T

rrrcTT

RaHHRp

av

R rrrand

c

T

r

11141 3

Equilibrium state

• In brief

.]}11

[{)(caE

p

ni

caEHR

pavrr

ecR

rrrcTT

.)(])([

)(

HRcaE

HRni

HRcaEp

caEHRav

rrr

erR

rrrc

rrrTT

Non-equilibrium state

• In this case, ST ≠ 0. The direction of Tv-temperature change is determined by the condition ST → 0. So, for instance, if ST > 0, Tv

increases, consequently the radiation emitted also increases, which diminishes Tv. This is also valid vice versa, if ST < 0, Tv will decrease, consequently the radiation emitted will also decrease, which increases Tv.

Non-equilibrium state

• The rate of temperature change is determined by the heat capacity of vegetation (green vegetation parts possess small heat capacity). The rate of temperature change is larger when the heat capacity is smaller, and vice versa, the rate of temperature change is smaller when the heat capacity is larger.

Non-equilibrium state: a special

case

• Let’s start from the differential equation of heat

transfer in the soil and let’s try to connect it with

the soil surface energy balance equation! The

equations mentioned are as follows:

.

,0

00

1

10

1

10

hn

hhhhhla

h

fGELHR

D

ff

D

ff

z

f

t

TC

isthatz

f

t

TC

Force-restore method• The new terms are as follows: Tla (index la

denotes layer), fh0 which is equal to G0, fh1 and D1. The new terms show that the equation does not refer to level z, rather to the layer of thickness D1. It is obvious that

• fh0 = G0, which is surface heat flux density, fh1 is the ground heat flux density across z1

(obviously z1 = D1). D1 is the thickness of the surface layer, Tla is the mean temperature of the layer.

.t

TCST la

Force-restore method

• By introducing new terms we can start to

consider the so-called force-restore method

(Ács et al., 2000, Appendix A).

• What does the layer D1 amount to? This is a

central question. We can suppose that for z =

D1 the conditions below

are fulfilled.

t

tT

t

tzTandtTtzT la

DzlaDz

)(),()(),(

11


• Obviously D1 is variable since the

previous conditions can be

fulfilled more or less for different

D1 values.

• Term fh0 is known from the energy

balance equation. Term fh1 can be

estimated from fh(z,t).


• Since (see slide 51 “The shape of fh(z,t)”)

• and using the above suppositions

])()(1

[),( 011

11 TtTt

tTftDzf hh

.])()(1

[ 01 TtTt

tTf la

lah


• Substituting term fh1 our starting equation

can be written as

where

])([)(

00 TtTft

tTC lah

lab

,)2

( 1 Cd

DC Sb


• where

• The final equation is

.22

2

2

0

2/1

S

S

h dC

dT

fC

.])([

,)(

01

1

TtTG

whereGELHRt

tTC

la

nla

b


• Let’s discuss the name of the method! The expression “force” refers to the term fh0 = G0, which represents the energy input of the surface coming from the environment. The greater G0 is the greater the rate of increase of Tla.


• The expression “restore” refers to the term G1 because it tends to restitute the previous (“old”) state acting in the opposite direction to G0. So, the greater the temperature Tla is, the greater the term G1 is , i.e. the impact which decreases the value of Tla. The method might be also called the “action-reaction” method or something similar.

Force-restore method: its

application in the SURFMOD

• The force-restore method can also be applied

for the vegetation-covered ground by

expressing the term fh0 for the soil-vegetation

system and expanding the term Cb with the

heat capacity of vegetation. Such an

application is used in the SURFMOD.

.])([

,)1(

,)(

01

1

TtTG

CvegCvegC

whereGELHRt

tTC

vg

bvB

vgvgvg

n

vg

B

Force-restore method: its

application in the SURFMOD

• veg = fraction of the surface covered

by vegetation (in brief vegetation

fraction; for full vegetation cover veg

= 1, for bare ground veg = 0),

• Cv = vegetation heat capcity

Closing remark

• Here, only aspects of heat and water transfer are

considered. These exchange processes are

relevant in weather prediction models.

• Carbon and nitrogen exchange processes are not

discussed at all. Their treatment is important only

in the global climate and/or Earth System models.

In the ELTE curriculum these processes are dealt

with in the framework of Agroclimatology and

Ecological climatology.

References

• Ács, F., Hantel, M., Unegg, J.W., 2000: Climate Diagnostics with the Budapest-Vienna Land-Surface Model SURFMOD. Austrian Contributions to the Global Change Program, Vo. 3, Austrian Academy of Sciences, Vienna, 116 pp.

• http://nimbus.elte.hu/~acs/pdf/OKTATAS/buch_budapest_vienna_2000.pdf

• Ács, F., Horváth, Á., Breuer, H., and Rubel, F., 2010: Effect of soil hydraulic parameters on the local convective precipitation. Meteorol. Z., Vo. 19(2), 143-153.

http://nimbus.elte.hu/~acs/pdf/OKTATAS/buch_budapest_vienna_2000.pdf

References

• Ács, F., Szabó, L., és Jávor, Cs., 2012: A csupasz talaj felszíni hőmérsékletének érzékenysége a talaj sugárzási és termikus tulajdonságainak változásaira. Légkör, 57, 55-60.

• Ács, F., 1989: Prognoza temperature i vlaznosti tla. MSc. Thesis, University of Belgrade, 63 pp.

• Bonan, G., 2002: Ecological Climatology. Conceptsand Applications. Cambridge University Press. Cambridge, 690 pp, ISBN: 0521800323.

• Baldocchi, D., 1994: A comparative study of massand energy exchange rates over a closed C3 (wheat) and an open C4 (corn) crop: II. CO2 exchange and water use efficiency. Agricultural For. Meteorol., 67, 291-321.

References• Braden, H., 1985: Ein Energiehaushalts- und

Verdunstungsmodell für Wasser und Stoffhaushaltsuntersuchungen landwirtschaftlich genutzter Einzugsgebiete. Mitteilgn. Dtsch Bodenkundl. Gesellsch., 42, 294-299.

• Brutsaert, W., 1982: Evaporation into the Atmosphere. Theorz, History and Applications. Reidel Publishing Company. Dordrecht, Boston, London. ISBN: 90-277—1274-6, 299 pp.

• Businger, J.A., Wyngaard, J.C., Izumi, Y., Bradley, E.F., 1971: Flux-profile relationships in the atmospheric surface layer. J. Atm. Sci., 28, 181-189.

• Campbell, G.S., 1974: A simple method for determining unsaturated conductivity from moisture retention data. Soil Sci., 117, 311-314.

References

• Campbell, G.S., 1985: Soil Physics with Basic. Transport models for soil-plant systems. Elsevier, Amsterdam, 15 pp, ISBN 0-444-40882-7.

• Chaloner, W.G., 2003: The role of carbon dioxide inplant evolution. In: Evolution on Planet Earth. The Impact of the Physical Environment, edited byRotschild, L.J. and Lister, A.M., Academic Press, Amsterdam, 65-83.

• Clapp, R.B., and Hornberger, G.M., 1978: EmpiricalEquations for Some Hydraulic Properties. WaterResour. Res., 14, 601-604.

• Cosby, B.J., Hornberger, G.M., Clapp, R.B., and Ginn, T.R., 1984: A Statistical Exploration of theRelationships of Soil Moisture Characteristics to thePhysical Properties of Soils. Water Resour. Res., 20, 682-690.

References

• Dyer, A.J., Hicks, B.B., 1970: Flux-gradientrelationships in the constant flux layer. Quart. J. R. Met. Soc., Vo. 96, 715-721.

• Federer, C.A., 1979: A soil-plant-atmospheremodel for transpiration and availability of soilwater. Water Resour. Res., 15(3), 555-562.

• Foken T., 2002: Angewandte Meteorologie. Mikrometeorologische Methoden. Springer, Berlin, Heidelberg, New York. ISBN: 3-540-00322-3, 289 pp.

• Garratt, 1993: Sensitivity of Climate Simulationsto Land-Surface and Atmospheric Boundary-Layer Treatments – A Review. J. Climate, Vo. 6, 419-449.

References

• Jarvis, P.G., 1976: The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field. Philos. Trans. Roy. Soc. London, Ser. B, 273, 593-610.

• Jones, G.H., 1983: Plants and microclimate. Aquantitative approach to environmental plant physiology. Cambridge University Press, Cambridge, London, New York, Melbourne, Sydney. ISBN: 0 521 27016 2, 309 PP.

• Kelliher, F.M., Leuning, R., Raupach, M.R., and

Schulze, E.-D., 1995: Maximum conductances for

evaporation from global vegetation types. Agricul. For.

Meteorol., Vo. 73, 1-16.

References

• Körner, C., Scheel, J.A., Bauer, H., 1979:

Maximum leaf diffusive conductance in

vascular plants. Photosynthetica 13, 45-82.

• Monteith J.L. et al., 1975: Vegetation and

the Atmosphere, Volume 1., Principles.

Edited by J.L. Monteith, Academic Press,

London, New York, San Francisko, ISBN: 0-

12-505101-8.

• Oke, T.R., 1978: Boundary Layer Climates.

Methuen and Co Ltd., New York. ISBN: 0

416 70530 8.

References

• Rose, C.W., 1966: Agricultural physics.

Pergamon Press, Oxford, London, 226 pp.

• Sellers, P.J., and Dorman, J.L., 1987:

Testung the Simple Biosphere Model (SiB)

Using Point Micrometeorological and

Biophysical Data. J. Clim. Appl. Meteorol.,

26, 622-651.

• Stefanovits, P., Filep, Gy., Füleky, Gy.,

1999: Talajtan, Mezőgazda Kiadó,

Budapest, 470 pp, ISBN 9632860454.

References

• Szász, G., and Zilinyi, V., 1994: The

spectral reflexion of different soils and soil

ingredients. Időjárás, 94, 23-25.

• van Genuchten, M.Th., 1980: A Closed-

Form Equation For Predicting the

Hydraulic Conductivity of Unsaturated

Soils. Soil Sci. Soc. Amer. J., 44, 892-898.

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