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**GEO3020/4020 Lecture 2: I. Energy balance II. Evapotranspiration**

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**Energy balance equation**

where: K net shortwave radiation L net longwave radiation LE latent heat transfer H sensible heat transfer G soil flux Aw advective energy ΔQ/Δt change in stored energy Units: [EL-2T-1]

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**Calculation of evaporation using 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 [MJkg-1] or 2495 J/g at 0oC

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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 evaporation**

II. 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.

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**GEO3020/4020 Lecture 3: Evapotranspiration (free water evaporation)**

Lena M. Tallaksen (modified from lecture notes Chong-yu Xu, 2008) Chapter 7.1 – 7.3, Appendix D.6; Dingman

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**Evapotranspiration Measurements**

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**

Epan = W – [V2-V1] where W = precipitation during Dt V1 = the storage at the beginning of Dt V2 = the storage at the end of Dt 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 13

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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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**Lysimeter for measuring potential evapotranspiration**

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 R A P

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**Lysimeter for measuring actual evapotranspiration**

Figure. Schematic of a weighable gravitation lysimeter.

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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.

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**Flux of water molecules over a surface**

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Zveg Z0 Zd velocity

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**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 vertical transport of water vapor by the turbulent eddies of the wind

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**Lapse rates (stable, neural, unstable)**

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.

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**Latent heat, LE Latent heat exchange by turbulent transfer, LE**

and from equation (D-42) 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)

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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 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.

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**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

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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 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

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**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 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]

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**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

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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

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**Energy balance methods**

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**Energy balance methods**

Energy balance equation The general energy balance for an evapotranspiring body during a time period Dt 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], DQ 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)

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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

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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]

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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

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**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, DQ/Dt, Eq. (7-22) becomes:

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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), Note:

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**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

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**Penman combination method**

Substitute (7B1-8) into (7B1-1) : and solve for E:

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**Penman combination method**

Solving for E : From definitions of KH (equation (7-10), KE (equation (7-2)), and g (equation (7-13), we get Substitute (7B1-10) into (7B1-9) and use equation (7-5) , yields:

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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. Penman (1948) U in m s-1

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**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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**Psychrometric Constant (g)**

Additional material Psychrometric Constant (g) where: g= psychrometric constant [kPa C-1], cp = specific heat of moist air = [kJ kg-1 °C-1], P = atmospheric pressure [kPa], e = ratio molecular weight of water vapour/dry air = and l = latent heat of vaporization [MJ kg-1]. Slope Vapour Pressure Curve (D) can be found by taking the derivative of es*, i.e. des*/dT

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