1. GEO3020/4020 Lecture 3: Evapotranspiration (free water evaporation)
Repetition

2. Flux of water molecules over a surface

3. Zveg Z0 Zd
velocity

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

5. 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
vertical transport of water vapor by the turbulent eddies of the wind

6. Lapse rates (stable, neural, unstable)
Actual lapse rate

7. Latent heat, LE Latent heat exchange by turbulent transfer, LE where
ra = density of air;
λv = latent heat of vaporization;
P = atmospheric pressure
k = 0.4;
zd = zero plane displacement
height
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
ra = density of air;
Ca = heat capacity of air;
k = 0.4;
zd = zero plane displacement
height
z0 = surface-roughness height;
za = height above ground surface
at which va & Ta are measured;
va = windspeed,
Ta = air temperatures and
Ts = surface temperatures.

9. Selection of estimation method
Type of surface
Availability of water
Stored-energy
Water-advected energy
Additional elements to consider:
Purpose of study
Available data
Time period of interest

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 Dt 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
DV change in the amount of stored in the lake during Dt
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

12. Water balance method
Apply the water balance equation to the water body of interest over a time period Dt and solving the equation for evaporation, E
Data needed
Application

13. Mass-transfer method Physical based equation: Empirical 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

14. Mass-transfer method Data needed Application
- 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]

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

16. Energy balance method
Substitute the different terms into the following equation, the evaporation can be calculated
where
LE has units [EL-2T-1]
E [LT-1] = LE/ρwλv
Latent Heat of Vaporization :
lv= (2.36 × 10-3) Ta

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.

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

19. 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.

20. Penman combination method
Penman (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, DQ/Dt, Equation (7-22) becomes

21. Penman combination method
II. 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:

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).

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. 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

25. GEO3020/4020 Lecture 4: Evapotranspiration. - bare soil
GEO3020/4020 Lecture 4: Evapotranspiration - bare soil - transpiration - interception
Lena M. Tallaksen
Chapter 7.4 – 7.8; Dingman

27. Soil Evaporation Phase 1: Meteorological controlled Phase 2:
Soil controlled

28. Influence of Vegetation
Albedo
Roughness
Stomata
Root system
LAI
GAI

29. Transpiration

30. Resistance – Conductance Aerodynamic and surface

31. The influence of stomatal aperture on transpiration – leaf scale

32. Modelling transpiration
Rearrange to give:

33. Atmospheric conductance, Cat

34. Penman equation – 3 versions
Orignal Penman (1948)
Penman (physical based wind function)
Penman (atmospheric conductance)

35. Estimation of Cleaf The leaf conductance is a function of:
Light intensity
CO2 level in the atmosphere
Vapour pressure difference (leaf – air)
Leaf temperature
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.

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

37. Penman-Monteith
Penman Penman-Monteith
”Big leaf” concept

38. Evapotranspiration – measuring and modelling
Single leaf or plant
Stand
Mixed vegetation
Regional scale
Seasonal variation in LAI (”big leaf”)

39. Interception Function of: Vegetation type and age (LAI)
Precipitation intensity, frequency, duration and type

40. Interception measurements
Direct measurements
Measurements of throughfall or net precipitation

41. Interception measurements
Measurements of throughfall or net precipitation
Experiemental site in the Huewelerbach catchment, Luxembourg (from TUDelft website)

44. Interception modelling
Regression models (empirical equations)
e.g. between interception loss (Ei) and precipitation (R) for a given Δt
Conceptual based models
e.g. Rutter water balance model which uses the equation for free water evaporation to estimate interception losses.
- Requires meteorological data and vegetation characteristics.

45. Regression model to determine the net precipitation rate

46. The Rutter model

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

48. Forest evapotranspiration Example 7- 8
Thetford forest (UK): 16.5 m, vind speed 3.0 m/s
Atmospheric conductance: Cat = 23.2 cm/s
Transpiration rate
Soil moisture deficit = 0 cm
ET=1.8 mm/day
Soil moisture deficit = 7 cm
ET=1.2 mm/day
Evaporation of intercepted water
ET=54 mm/day (1 mm/0.45 hour)
Replacement or addition to transpiration ?

49. Estimation of potential evapotranspiration
Definition: function of vegetation – reference crop
Operational definitions (PET)
Temperature based methods (daily, monthly)
Radiation based methods (daily)
Combination method
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.

52. 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

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
ETw = wet environment evapotr.
ETp = potential evapotr.
ETa = actual evapotr.
evapotranspiration
ETa = 2 ETw - ETp
Increase of wetness
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
qrel is the relative water content in the soil
where qfc is the field-capacity, qpwp is the permanent wilting point

55. Fig 7-24
There are different soil type with different dynamic of drying as evapotranspiration continues
From the following slides we see that the linear equation (7-67) or the equation used in HBV model represents only a special case of soil drying dynamics

56. Drying of soil moisture by evapotranspiration

57. GEO3020/4020 Evapotranspiration
Meteorological Elements
Energy Balance
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