Repetition

GEO3020/4020

Lecture 2: I. Energy balance II. Evapotranspiration

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30% 70%

Energy balance equation

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0/ tQAGLEHLK w

where:

K net shortwave radiationL net longwave radiationLE latent heat transferH sensible heat transferG soil fluxAw advective energyΔQ/Δt change in stored energy

Units: [EL-2T-1]

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Calculation of evaporation using 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

[MJkg-1] or 2495 J/g at 0oC

LE has units [EL-2T-1]

E [LT-1] = LE/ρwλv

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Controlling factors of evaporation

I. Meteorological situation• Energy availability• How much water vapour can be received

– Temperature– Vapour pressure deficit– Wind speed and turbulence

Optimal conditions: ?

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Controlling factors of evaporationII. Physiographic and plant characteristics• Characteristics that influence available energy

– albedo– heat capacity

• How easily can water be evaporated– size of the evaporating surface– surroundings– roughness (aerodynamic resistance)– salt content– stomata

• Water supply– free water surface (lake, ponds or intercepted water)– soil evaporation– transpiration

The wind speed immediately above the surface. • The humidity gradient away from the surface.

– The rate and quantity of water vapor entering into the atmosphere both become higher in drier air.

• Water availability. – Evapotranspiration cannot occur if water is not available.

Lena M. Tallaksen(modified from lecture notes Chong-yu Xu, 2008)

Chapter 7.1 – 7.3, Appendix D.6; Dingman

GEO3020/4020

Lecture 3: Evapotranspiration(free water evaporation)

EvapotranspirationMeasurements

Free water evaporation- Pans and tanks- Evaporimeters

Evapotranspiration (includes vegetation)- Lysimeters- Remote sensing

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Definitions• Potential evapotranspiration, PE, is the rate at which

evapotranspiration would occur from a large area completely and uniformly covered with growing vegetation which has access to an unlimited supply of soil water and without advection or heat-storage effects (i.e. the rate is depedent on the vegetation)

• Actual evapotranspiration, ET, is the rate at which evapotranspiration occurs (i.e. describes all the processes by which liquid water at or near the land surface becomes atmospheric water vapor).

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Pan evaporation methods

Pan evaporationEpan = W – [V2-V1]

where

W = precipitation during t

V1 = the storage at the beginning of t

V2 = the storage at the end of t

For American Class-A pan, Kohler et al. (1955) developed an empirical equation to account for energy exchange through sides of a pan, and adjust daily pan evaporation, Epan, to free water evaporation, Efw [mm day-1] (Equations 7-41 and 7-42).

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Pan evaporation methods

Pan coefficient

Elake/Epan = kp

where

k is a coefficient that varies with seasons and lake.

Its annual average over the US is about 0.7

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Pan evaporation methods

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Pan evaporation methods

Example of pan coefficient in the Yangtze River catchment in China

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Lysimeter

One of the most reliable way of measuring potential or actual evapotranspiration is to use large containers (sometimes on the order of several metres across) called lysimeters;

Evapotranspiration is calculated by subtraction considering the different components of the water balance.

A lysimeter is most accurate when vegetation is grown in a large set up which allows the rainfall input and water lost through the soil to be easily calculated from the difference between the weight before and after a given period.

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input (Rainfall R and Additional water A) and output (Percolated water P) collected in the receiver, then PE can be estimated from the equation:

PE = R + A – P

Lysimeter for measuring potential evapotranspiration

R A

P

16Figure. Schematic of a weighable gravitation lysimeter.

Lysimeter for measuring actual evapotranspiration

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Estimation of evapotranspiration by remote sensing

Remote sensing has two potentially very important roles in estimating evapotranspiration (Engman, 1995).

First, remotely sensed measurements offer methods for extending point measurements or empirical relationships to much larger areas, including those areas where measured meteorological data may be sparse.

Secondly, remotely sensed measurements may be used to measure variables in the energy and moisture balance models of ET, such as as radiometric surface temperature, albedo, and vegetation index.

Flux of water molecules over a surface

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19

Zveg

Zd

Z0

velocity

22)-(D ln1

0*

z

zzu

kv dmm

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

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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

21

42)-(D )e-(e

ln

622.0ms2

0

2

zzz

vk

PD

DLE

da

maV

M

WV

Lapse rates (stable, neural, unstable)

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Actual lapse rate

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Physics of Evaporation - Summary

Evaporation is a diffusive process. The rate of evaporation is the rate at which molecules move from the saturated surface layer into the air above, and that rate is proportional to the difference between the vapor pressure of the surface layer, es and the vapor of the overlying air, ea that is

or

where - E is the evaporation rate [L T-1], es and ea have unit of [M L-1 T-1],

va is wind speed [L T-1]- KE is a coefficient that reflects the efficiency of vertical transport

of water vapor by the turbulent eddies of the wind [L T2 M-

1], can be calculated by equation (7-2), or for practical use (7-19)

Equation (D-12) is known as the Dalton’s Law discovered by John Dalton, the English chemist, 1802.

12)-(D as eeE

1)-(7 asaE eevKE

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Latent heat, LE

Latent heat exchange by turbulent transfer, LE

and from equation (D-42)

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)

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Sensible heat, H

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

and from equation (D-49)

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

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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

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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

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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

for example:

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

- Harbeck (1962) found the empirical equation:

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

saaE eevKE saaLE eevKLE

(1802)Dalton ref. )( 10 saa eevbbE

19)-(9 1069.1 05.05 LE AK

18a)-(7 1026.1 3asa eevE

Mass-transfer method

Data 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

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Well-known examples of Mass-transfer equations

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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

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Energy balance methods

0/ tQAGLEHLK w

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Energy balance methodsEnergy balance equationThe general energy balance for an evapotranspiring body during a time period

t can be written as:

where the first six terms represent average energy fluxes (energy per unit area of evaporating surface per unit time [E L-2 T-1],

• Q is the change of energy stored in the body of water• LE - latent heat [E L-2 T-1], • K – net shortwave (solar) radiation input• L – net longwave radiation input• G – net output via conduction to the ground • H – net output of sensible heat exchange with the atmosphere• Aw – net input associated with inflows and outflows of water (water-

advected energy)

tQAGHLKLE w /

38

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

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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]

as

as

asv

asa

ee

TT

ee

TTPc

LE

HB

622.0

v

a Pc

622.0

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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.

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Penman combination method• Penman (1948) was the first to show that mass-transfer and

energy balance approaches could be combined to arrive at an evaporation equation that did not require surface temperature data, Ts

• Derivation of the Penman method starts with the original energy balance equation:

• Neglecting ground-heat exchange, G, water-advected energy, Aw, and change in energy storage, Q/t, Eq. (7-22) becomes:

22)-(7 /

vw

w tQAHGLKE

1)-(7B1 vw

HLKE

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Penman combination method• The sensible-heat transfer flux, H, is given by (7-9)

• The slope of saturation-vapor vs. temperature curve

• then

• and substitute (7B1-4) into (7B1-2),

2)-(7B1 asaH TTvKH

3)-(7B1 **

as

as

TT

ee

4)-(7B1 **

asas

eeTT

5)-(7B1 **as

aH eevK

H

Note: ss ee *

aa ee *

44

Penman combination method• (7B1-5) remains true if ea is added and subtracted from each of

the terms in brackets:

• Use , rearrange eq (7-1) to get,

• Substitute (7B1-7) into (7B1-6) yields

6)-(7B1 **aa

aHas

aH eevK

eevK

H

ss ee * asaE eevKE

8)-(7B1 *aa

aH

aE

aH eevK

vK

EvKH

7)-(7B1 *

aEas vK

Eee

Penman combination method• Substitute (7B1-8) into (7B1-1) :

• and solve for E:

45

*

vw

aE

aHaa

aH

vKEvK

eevK

LK

E

*aa

aH

E

Hvw ee

vKLK

K

EKE

)( *aa

aH

E

Hvw ee

vKLK

K

KE

)( *aaaH

E

Hvw eevKLK

K

KE

46

Penman combination method• Solving for E :

• From definitions of KH (equation (7-10), KE (equation (7-2)), and (equation (7-13), we get

• Substitute (7B1-10) into (7B1-9) and use equation (7-5) ,

yields:

9)-(7B1

*

E

Hvw

aaaH

KK

eevKLKE

10)-(7B1 EvwH KK *aaa eWe

33)-(7

)(

1)( *

vw

aaavwE WevKLKE

aa ee *

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

represented as,

• In practical application, equation (7-33) is simplified as

– Ea = f(u)(es-ea) The first term and second term of the equation represents energy (net radiation) and the atmospheric contribution (mass transfer) to evaporation respectively.

– There are many empirical equations available for f(u), e.g.

34)-(7 transfermassradiationnet

E

an ERE

)54.01(0026.0)( 22 UUf Penman (1948) U in m s-1

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

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49

• Psychrometric Constant ()

where:

= psychrometric constant [kPa C-1],

cp = specific heat of moist air = 1.013 [kJ kg-1 °C-1],

P = atmospheric pressure [kPa],

= ratio molecular weight of water vapour/dry air = 0.622 and

= latent heat of vaporization [MJ kg-1].

• Slope Vapour Pressure Curve ()

PPC p

00163.010 3

6)-(7 3.237

27.17exp

)3.237(

3.2508

)3.237(

)(409822

**

a

a

aa

as

as

as

T

T

TT

Te

TT

ee

can be found by taking the derivative of es*, i.e. des*/dT

Additional material