Repetition

GEO3020/4020

Lecture 3: Evapotranspiration(free water evaporation)

Flux of water molecules over a surface

2

3

Zveg

Zd

Z0

velocity

22)-(D ln1

0*

z

zzu

kv dm

m

Momentum, sensible heat and water vapour (latent heat) transfer by turbulence (z-direction)

4

Steps in the derivation of LE• Fick’s law of diffusion for matter (transport due to differences in the concentration of water vapour);• Combined with the equation for vertical transport of water vapour due to turbulence (Fick’s law of

diffusion for momentum), gives:

DWV/DM (and DH/DM) = 1 under neutral atmospheric conditions

5

42)-(D )e-(e

ln

622.0ms2

0

2

zzz

vk

PD

DLE

da

maV

M

WV

Lapse rates (stable, neural, unstable)

6

Actual lapse rate

7

Latent heat, LE

Latent heat exchange by turbulent transfer, LE

where

where

a = density of air;

λv = latent heat of vaporization;

P = atmospheric pressure

k = 0.4;

zd = zero plane displacement

height

45)-(D asaLE eevKLE

43)-(D

ln

622.02

0

2

zzz

k

PK

da

aVLE

z0 = surface-roughness height;

za = height above ground surface

at which va & ea are measured;

va = windspeed,

ea = air vapor pressure

es = surface vapor pressure (measured at z0 + zd)

8

Sensible heat, H

Sensible-heat exchange by turbulent transfer, H (derived based on the diffusion equation for energy and momentum):

where

where

a = density of air;

Ca = heat capacity of air;

k = 0.4;

zd = zero plane displacement

height

52)-(D asaH TTvKH

50)-(D

ln

2

0

2

z

zz

kcK

da

aaH

z0 = surface-roughness height;

za = height above ground surface

at which va & Ta are measured;

va = windspeed,

Ta = air temperatures and

Ts = surface temperatures.

Selection of estimation method

• Type of surface• Availability of water• Stored-energy• Water-advected energy

Additional elements to consider:1) Purpose of study

2) Available data

3) Time period of interest

9

10

Estimation of free water evaporation

• Water balance method• Mass-transfer methods

• Energy balance method• Combination (energy +

mass balance) method• Pan evaporation method

Defined by not accounting for stored energy

11

Water balance method• Apply the water balance equation to the water body

of interest over a time period t and solving the equation for evaporation, E

– W: precipitation on the lake– SWin and SWout: inflows and outflows of surface water– GWin and GWout: inflows and outflows of ground water– V change in the amount of stored in the lake during t

But: • Difficult to measure the terms• Large uncertainty in individual terms gives high uncertainty in E• Can however, give a rough estimate, in particular where E and

Δt is relative large

16)-(7 VGWSWGWSWWE outoutinin

12

Water balance methodApply the water balance equation to the water body of

interest over a time period t and solving the equation for evaporation, E

Data needed

Application

16)-(7 VGWSWGWSWWE outoutinin

13

Mass-transfer methodPhysical based equation:

or

Empirical equation:

- Different versions and expressions exist for the empirical constants b0 and b1; mainly depending on wind, va and ea

- If compared with physical based equation; b0=0 and b1=KLE

saaE eevKE saaLE eevKLE

(1802)Dalton ref. )( 10 saa eevbbE

Mass-transfer methodData needed

- va (dependent on measuring height)

- es (from Ts)

- ea (from Ta and Wa)

Application

- gives instantaneous rate of evaporation, but averaging is OK for up to daily values

- requires data for Ts

- KE varies with lake area, atmospheric stability and season

Harbeck (1962) proposed the empirical equation:

where AL is lake area in [km2], KE in [m km-1 kPa-1]

14

19)-(9 1069.1 05.05 LE AK

15

Eddy-correlation approach• The rate of upward movement of water vapor near the surface

is proportional to the time average of the product of the instantaneous fluctuations of vertical air movement, , and of absolute humidity, q’, around their respective mean values,

– Advantages• Requires no assumption about parameter values, the shape of the

velocity profile, or atmospheric stability

– Disadvantages • Requires stringent instrumentation for accurately recording and

integrating high frequency (order of 10 s-1) fluctuations in humidity and vertical velocity

For research application only

'au

21)-(7 '' quE aw

a

16

Energy balance methodSubstitute the different terms into the following equation, the evaporation can

be calculated

where

22)-(7 /

vw

w tQAHGLKE

15)-(7 / tQAHGLKLE w

Latent Heat of Vaporization :v= 2.495 - (2.36 × 10-3) Ta

LE has units [EL-2T-1]

E [LT-1] = LE/ρwλv

17

Bowen ratio

We recognize that the wind profile enters both the expression for LE and H. To eliminate the need of wind data in the energy balance approach, Bowen defined a ratio of sensible heat to latent heat, LE:

where is called the psychrometric constant [kPa K-1]

Needs measurements at two levels.

as

as

asv

asa

ee

TT

ee

TTPc

LE

HB

622.0

v

a Pc

622.0

18

Use of Bowen ratio in energy balance approach

• Original energy balance approach

• Replace sensible heat, H by Bowen ratio, B

• Substitute (7-23) into (7-22)

The advantage of (7-24) over (7-22) is to eliminate H which needs wind profile data

22)-(7 /

vw

w tQAHGLKE

23)-(7 EBLEBH vw

24)-(7 )1(

/

B

tQAGLKE

vw

w

Energy balance method

Data

Data demanding, but in some cases less a problem than in the water balance method (regional estimates can be used)

Application

- gives instantaneous rate of evaporation, but averaging is OK for up to daily values;

- change in energy stored only for periods larger than 7 days (energy is calculated daily and summed to use with weekly or monthly summaries of advection and storage);

- requires data for Ts (Bowen ratio and L);

- most useful in combination with the mass transfer method.

19

20

Penman combination methodPenman (1948) combined the mass-transfer and energy balance

approaches to arrive at an equation that did not require surface temperature data:

I. From original energy balance equation:

Neglecting ground-heat conduction, G, water-advected energy, Aw, and change in energy storage, Q/t, Equation (7-22) becomes

22)-(7 /

vw

w tQAHGLKE

1)-(7B1 vw

HLKE

21

Penman combination methodII. The sensible-heat transfer flux, H, is given by:

• Introduce the slope of saturation-vapor vs. temperature curve:

• Derive an expression for H:

I. + II. gives the Penman equation:

2)-(7B1 asaH TTvKH

3)-(7B1 **

as

as

TT

ee

8)-(7B1 *aa

aH

aE

aH eevK

vK

EvKH

33)-(7

)(

1)( *

vw

aaavwE WevKLKE

22

Penman combination method• Note that the essence of the Penman equation can be

represented as:

• The first term and second term of the equation represents energy (net radiation) and the atmospheric contribution (mass transfer) to evaporation, respectively.

• In many practical application, Ea is simplified as: f(va)(es-ea) and an empirical equations used for f(va).

34)-(7 )(E transfer mass)(Rradiation net an

E

Penman equation – input data

• Net radiation (K+L)

(measured or alternative cloudiness, C or sunshine hours, n/N can be used);

• Temperature, Ta (gives ea*)

• Humidity, e.g. relative humidity, Wa = ea/ea*

(gives ea and thus the saturation deficit, (ea* - ea)

• Wind velocity, va

Measurements are only taken at one height interval and data are available at standard weather stations

23

Penman equation – input data

• Net radiation (K+L)

(measured or alternative cloudiness, C or sunshine hours, n/N can be used);

• Temperature, Ta (gives ea*)

• Humidity, e.g. relative humidity, Wa = ea/ea*

(gives ea and thus the saturation deficit, (ea* - ea)

• Wind velocity, va

Measurements are only taken at one height interval and data are available at standard weather stations

24

GEO3020/4020

Lecture 4: Evapotranspiration- bare soil- transpiration - interception

Lena M. Tallaksen

Chapter 7.4 – 7.8; Dingman

26

Soil Evaporation

• Phase 1: Meteorological controlled

• Phase 2:

Soil controlled

27

Influence of Vegetation

• Albedo• Roughness• Stomata• Root system• LAI • GAI

28

Transpiration

29

Resistance – ConductanceAerodynamic and surface

30

The influence of stomatal aperture on transpiration – leaf scale

31

Modelling transpiration

32

Rearrange to give:

)e-(e

ln

622.0ms2

0

2

zzz

vk

PD

DLE

dm

maV

M

WV

)e-(e

and

)e-(e

ln

622.0

as

as2

0

2

C KE

v

z

zz

k

D

D

P

LEE

atat

m

daM

WV

w

a

wV

Atmospheric conductance, Cat

33

ln25.6

2

0

zzz

vC

dm

mat

Orignal Penman (1948)

Penman (physical based wind function)

Penman (atmospheric conductance)

Penman equation – 3 versions

34

)(

1)( *

vw

aaavwE WevKLKE

)(

1)( *

vw

aaataa WeCcLKE

)(

1)()( *

vw

aa WeufLKE

Estimation of Cleaf

The leaf conductance is a function of:

1. Light intensity

2. CO2 level in the atmosphere

3. Vapour pressure difference (leaf – air)

4. Leaf temperature

5. Leaf water content

where Cleaf* is the maximum value (all stomata full opening; typical values are given in Table 7-5) and f(x) is a proxy used for each variable above.

35

72)-(7 * fTffKfCC aTvpinkleafleaf

Relative leaf conductance [0,1](ref. Fig. 7-13 and Table 7-6)

36

Penman-Monteith

Penman

Penman-Monteith

37

55)-(7

)(

1)( *

vw

aaataa WeCcLKE

where

56)-(7 1

1)( *

CLAIfC

CC

WeCcLKE

leafscan

can

atvw

aaataa

”Big leaf” concept

Evapotranspiration – measuring and modelling

38

• Single leaf or plant • Stand• Mixed vegetation• Regional scale• Seasonal variation in LAI (”big leaf”)

Interception

39

Function of:

i)Vegetation type and age (LAI)

ii)Precipitation intensity, frequency, duration and type

Interception measurements

40

i) Direct measurements

ii) Measurements of throughfall or net precipitation

Interception measurements

41

Measurements of throughfall or net precipitation

Experiemental site in the Huewelerbach catchment, Luxembourg (from TUDelft website)

42

43

Interception modelling

44

• Regression models (empirical equations)e.g. between interception loss (Ei) and

precipitation (R) for a given Δt

• Conceptual based modelse.g. Rutter water balance model which uses the

equation for free water evaporation to estimate interception losses.

- Requires meteorological data and vegetation characteristics.

Regression model to determine the net precipitation rate

45

The Rutter model

46

Regression model to determine S (as the point where the linear line crosses X)

47

48

Forest evapotranspirationExample 7- 8

Thetford forest (UK): 16.5 m, vind speed 3.0 m/sAtmospheric conductance: Cat = 23.2 cm/s

Transpiration rateSoil moisture deficit = 0 cmET=1.8 mm/day

Soil moisture deficit = 7 cmET=1.2 mm/day

Evaporation of intercepted waterET=54 mm/day (1 mm/0.45 hour)

Replacement or addition to transpiration ?

Estimation of potential evapotranspiration

49

Definition: function of vegetation – reference crop

Operational definitions (PET)

1.Temperature based methods (daily, monthly)

2.Radiation based methods (daily)

3.Combination method

4.Pan

50

Actual evapotranspiration

• Two extreme cases– In arid case, P <<PE, water limited

AE = P

– In humid case, P >>PE

AE = PE, energy limited

51

Long-term actual evapotranspiration as presented by Turc-Pike (mid), and Schreiber and Ol’dekop methods.

Estimation of actual evapotranspiration (ET)

• Potential-evapotranspiration approaches– Empirical relationships between P-PET– Monthly water balance– Soil moisture functions– Complementary approach

• Water balance approaches – Lysimeter – Water balance for the soil moisture zone, atmosphere, land

• Turbulent-Transfer/Energy balance approaches– Penman-Monteith– Bowen ratio– Eddy correlation

• Water quality approaches

52

53

Complementary approach

• Based on heuristic arguments of Bouchet (1963)• Simply states that the potential and actual

evapotranspiration are not independent, but form a complementary relationship

Increase of wetness

ETa = 2 ETw - ETp

ETw = wet environment evapotr.ETp = potential evapotr.ETa = actual evapotr.

evap

otra

nspi

ratio

n

The above figure is identical to Fig 7-25 in the book

54

Soil moisture functions (hydrological water balance models)

Daily or monthly time step– General equation

– rel is the relative water content in the soil

– where fc is the field-capacity, pwp is the permanent wilting point

67)-(7 )( PETFET rel

68)-(7 pwpfc

pwprel

55

• Fig 7-24

fccritfc 8.0 fccritfc 8.0

56

Drying of soil moisture by evapotranspiration

GEO3020/4020

Evapotranspiration

• Definition and Controlling factors• Measurements• Physics of evaporation• Estimation of free water evaporation, potential and actual evapotransp.• Processes and estimation methods for bare soil, transpiration,

interception

I. Meteorological Elements

II. Energy Balance

III. Evapotranspiration