Repetition GEO3020/4020 Lecture 2:I. Energy balance II. Evapotranspiration.
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Transcript of Repetition GEO3020/4020 Lecture 2:I. Energy balance II. Evapotranspiration.
Energy balance equation
3
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]
4
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
5
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: ?
6
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
8
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).
9
10
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).
11
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
14
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.
15
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
17
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.
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
23
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
24
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)
25
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
26
29
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
30
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
32
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
33
35
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
37
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
39
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
40
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.
41
42
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
43
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 *
47
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
48
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