Hydrology Evaporation

Transcription

1 Hydrology Evaporation Prof. Dr. Christoph Külls, Hydrology and Water Management, Laboratory for Hydrology

2 2 Hydrologie Content 1. Introduction 2. Motivation 3. Basics 4. Measurement of Evaporation 5. Estimation 6. Wind Profile Method 7. Temperature Methods 8. Energy Balance Method 9. Energy and Mass Transfer 10. Combined Equations 11. Actual Evaporation 12. References

3 3 Introduction Objectives To learn about... types of evaporation estimation of evaporation measurement of evaporation Basics to understand evaporation Physical background Hydrological Relevance Application

4 4 Motivation Evaporation Relevance Evaporation is a major component of the water balance accounting for about 60 to 70 % of the annual water balance in mid-latitudes and up to 99 % in arid climates. It is therefore of imminent importance to estimate evaporation for... irrigation projects management of water resources under changing climate management of reservoirs Evaporation only plays a very minor in relation to floods. Only in very long rivers in arid zones evaporation has a significant effect on discharge along the river profile.

5 Basics 5 Physical background

6 6 Basics Atmospheric parameters with ɛ = ρ v ρ d q = ρ v ρ ɛ ρ v ρ d ρ q Mixing Ratio vapour density density of dry air ρ = ρ v + ρ d specific moisture content rf = m m m rf Mixing ratio specific moisture content

7 7 Basics Barometric Pressure vs. Altitude Equation: where P = ( z)5.26 [kpa] 293 P z atmospheric pressure [kpa] elevation above sea level [m]

8 8 Basics Pressure profile Altitude vs. Pressure Pressure [kpa] Altitude

9 9 Basics International Barometric Function Altitude vs. Pressure Equation: P = p 0 [ p 0 = hPa K m h K ] [hpa] where P h K atmospheric pressure [kpa] elevation above sea level [m] temperature in [K]

10 10 Basics Pressure with International Barometric Equation Pressure [hpa] Altitude

11 11 Basics Saturated Vapour Pressure Clasius-Clapeyron e s = (7.48 T /(237+T )) [hpa] (1) e a = rh e s [hpa] (2) (3) e s e a T saturated vapour pressure [hpa] actual vapour pressure [hpa] Temperature in degrees Celsius

12 12 Basics Vapour pressure Magnus-Formula Dampfdruck in hpa Temperatur in Celsius Figure: Vapour pressure calculated with Python

13 13 Basics Programming in Python 1: import numpy as np 2: import matplotlib.pyplot as plt # Create an array of 100 # linearly-spaced points from 0 to 35 3: T = np.linspace(0,35,350) 4: es = 6.11*10^(7.48*T/(237+T)) 5: plt.plot(t,es) 6: plt.savefig( figure/tes.png )

14 14 Basics Saturated Vapour Pressure Clasius-Clapeyron e s = exp(17.27 T /( T )) [kpa] (4) e s T saturated vapour pressure [kpa] Temperature in degrees Celsius

15 15 Basics Programming in R 1: T<-seq(0,35,1.0) 2: es< *exp(17.27*t/(237+t)) 3: plot(t,es,xlab="temperature in Celsius", ylab="vapour pressure in kpa", xlim=c(0, 35), ylim=c(0, 10), pch=1, lty=1, col="blue", axes=false) 4: axis(1, seq(0,35,5)) 5: axis(2, seq(0,10,2)) 6: grid (NULL,NULL, lty = 3, col = "grey") 7: curve(0.6108*exp(17.27*x/(237.3+x)), add = TRUE)

16 16 Basics Vapour pressure Dependency on temperature Dampfdruck in kpa Temperatur in Celsius Figure: Vapour pressure as a function of temperature

17 17 Basics Relative Humidity Relative humidity rf is the percentage of actual humidity e a in relation to potential humidity e s at temperature T : rf = e a /e s [%] The moisture saturation deficit d is defined as: d = (e s e a ) in [hpa]

18 18 Basics Psychrometric Constant γ Psychrometric constant γ is defined as: γ = c p P λ ɛ [kpa] = P[kPa] with γ Psychrometric constant [] c p specific heat at const. pressure [MJ kg 1 C 1 ] λ latent heat of vaporization [MJ kg 1 ] ɛ ratio of molecular weight of vapour dry air [ɛ = 0.622]

19 19 Basics Gradient of saturated vapour pressure curve = 4098 e s T (5) e s T gradient of sat. vapour pressure curve saturated vapour pressure [kpa] Temperature in degrees Celsius

20 Measurement of Evaporation 20 Instruments and Methods

21 21 Measurement of Evaporation Measurement Methods Water Balance Figure: Water Balance-based methods to determine evaporation, source: Shuttleworth (2008)

22 22 Measurement of Evaporation Measurement Methods Vapour Transport Figure: Vapour transport-based methods to determine evaporation, source: Shuttleworth (2008)

23 23 Measurement of Evaporation Measurement Methods Large Scale Figure: Large scale methods to determine evaporation, source: Shuttleworth (2008)

24 24 Measurement of Evaporation Measurement Methods Pans Figure: Pans

25 25 Measurement of Evaporation Measurement of Evaporation Class A pan Wood as support 15 cm Defined diameter cm Water level 5-7 cm below rim Water depth ca. 25 cm Measurement of rainfall Wave protection Protection against animals (grid)

26 26 Measurement of Evaporation Measurement of Evaporation Correction Factors E pot = f c E pan correction factor depends on wind and humidity and surrounding source: FAO, paper 24 65

27 27 Measurement of Evaporation Measurement of Evaporation Example

28 28 Measurement of Evaporation Measurement of Evaporation Sunken pan E pot = f c E pan correction factor source: John Rich,

29 29 Measurement of Evaporation Measurement of Evaporation Sunken pan source: Gangopadhyaya

30 30 Measurement of Evaporation Measurement of Evaporation, Lysimeter Lysimeter, UMS Meter

31 31 Measurement of Evaporation Measurement of Evaporation, Lysimeter Systems Lysimeter, UMS Meter

32 Estimation 32 Methods for the Estimation or Measurement of Evaporation

33 33 Estimation Factors Evaporation depends on... Air exchange, wind, advection Moisture saturation deficit of air Available energy as obtained from the energy balance Key physical parameters are temperature and relative humidity. Wind speed is a parameter for air exchange and advection. Energy balances are only included in physical formulae, most empirical formulae rely on temperature, relative humidity and wind speed, some also on seasonal indices.

34 34 Estimation Estimation Methods - Overview Calculation Dalton-type Empirical formulae Blaney-Criddle Turc Thorntwaite Hargreaves Water balance of basins Energy balance Bowen ratio Penman-Monteith Priestley-Taylor Measurement Class A-pan, evaporimeters Profile method: T, v, e profiles Sap flux measurements Eddy flux correlation Lysimeters Lake studies Basin studies Isotope methods

35 35 Estimation Dalton Formula One of the earliest approaches to estimate evaporation has been proposed by Dalton (1802) [1]. The formula proposed relates evaporation to saturation deficit E s E a and an advection term a. E = f D (e s e a ) with evaporation E in mm, saturated vapour pressure e s and actual vapour pressure e a. The factor f D depends on wind speed.

36 36 Estimation Haude formula Haude [2] proposed an adaption of the Dalton formula to regional conditions in Germany. The same author proposed advection term factors for Egypt. For the application of this formula measurements of temperature and relative humidity at 2 p.m. are needed. E = f H [e s (T 14 ) rf 14 e s (T 14 )] [mm/d] (6) For a green lawn in October f H = 0.2.

37 Estimation 37 Table: Haude Factors Month pasture lawn corn deciduous trees pine trees Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

38 38 Estimation DVWK formula The German Association of Hydrologists recomends the DVWK method: E = B f (v) (e s e a ) [mm/d] (7) The saturated vapour pressure e s and actual vapour pressure e a are given in hpa. B is a factor for surface roughness; B = For rough surfaces with a fetch < 20 m, B can increase to B = The wind function is simply given by f (v) in m/s.

39 Wind Profile Method 39 Boundary Layer Physics

40 40 Wind Profile Method Wind Profiles Surface Roughness and Fetch Figure: Wind profiles over different surfaces

41 41 Wind Profile Method Logarithmic Law Kinney & Merwade, 2007

42 42 Wind Profile Method Derivation of Profile Method Momentum Transfer Kinney & Merwade, 2007

43 43 Wind Profile Method Derivation of Profile Method Merge with Wind Profile Kinney & Merwade 2007

44 44 Wind Profile Method Derivation of Profile Method Solve and Simplify prepared by Kinney & Merwade, 2007

45 45 Wind Profile Method Aerodynamic Method E a = B (e s e a ) [mm/day] (8) e s = (7.48 T /(237+T )) [hpa] (9) e a = rh e s [hpa] (10) B = u 2 2 (11) ln (z 2 /z 0 ) E a e s e a T u 2 z 2 z 0 Evaporation by energy [mm/d] saturated vapour pressure [hpa] actual vapour pressure [hpa] Temperature in degrees Celsius wind speed at 2m height [m/s] height 2m [m] roughness height m [m]

46 Temperature Methods 46 Temperature-based Methods

47 47 Temperature Methods Empirical formulae Blaney-Criddle The Blaney-Criddle formula relies on day light percentage and temperature: with ET = p (0.46 T ) ET T p i potentia evaporation in cm/month, air temperature in degrees Celsius annual daylight percentage of every month

48 48 Temperature Methods Empirical formulae Hamon The Hamon formula relies on day length and temperature: where ( ) e s E p = K N T ( ) T e s = exp T ET potential evaporation in mm/day, e s saturated vapour pressure mbar k empirical coefficient = 1.0 [-] N daylight hours [h] e s saturated vapour pressure in [kpa] T average temperature in degree Celsius (12) (13)

49 49 Temperature Methods Empirical formulae Hargreaves The Hargreaves formula relies on temperature and extraterrestrial radiation: ET = R a (T ) TR 0.50 (14) where ET potential evaporation in mm/day R a extraterrestrial radiation [MJm 2 day 1 ] T average temperature in degree Celsius TR temperature range in degree Celsius

50 Energy Balance Method 50 Energy Balance & Lake Evaporation

51 51 Energy Balance Method Energy Balance of a Lake Q 0 = Q s Q r + Q a Q ar Q bs + Q v Q e Q h Q w where Q 0 energy balance [W /m 2 ] Q s energy from sun (short wave in) [W /m 2 ] Q r energy reflected (short wave out) [W /m 2 ] Q a energy from atmosphere (long wave in) [W /m 2 ] Q ar energy reflected (long wave) [W /m 2 ] Q bs energy emitted from water (long wave) [W /m 2 ] Q e energy transferred by evaporation [W /m 2 ] Q h energy transferred by sensible heat [W /m 2 ] Q v energy advected [W /m 2 ] Q w energy advected by evaporated water [W /m 2 ]

52 52 Energy Balance Method Radiation Balance where R f = Q s Q sr + Q a Q ar Q bs = Q s α s Q s + Q a α l Q a Q bs = Q s (1 α s ) + Q a (1 α l ) Q bs R f radiation balance [W /m 2 ] Q s energy from sun (short wave in) [W /m 2 ] Q sr energy reflected (short wave out) [W /m 2 ] Q a energy from atmosphere (long wave in) [W /m 2 ] Q ar energy reflected (long wave) [W /m 2 ] Q bs energy emitted from water (long wave) [W /m 2 ] α s short wave albedo [-] α l long wave albedo [-]

53 53 Energy Balance Method Long Wave Emission Q bs = 0.97 σ T 4 Q bs energy emitted from water (long wave) [W /m 2 ] σ Stefan-Boltzmann Constant [W m 2 K 4 ] T Temperature in Kelvin [K] The value 0.97 results from the emissivity of water (which is very high for long wave radiation, namely 0.97).

54 54 Energy Balance Method Advected Energy by Evaporation Q w = c p Q e (T e T b ) λ Q w energy advected by evaporation [W /m 2 ] Q e latent energy transfer [W /m 2 ] Q h sensible energy transfer [W /m 2 ] c p specific heat of water [J/kgC] latent energy of evaporation 2260[kJ/kg] T e temperature of water temperature of arbitrary datum T b

55 55 Energy Balance Method Bowen Ratio B = Q h Q e B Q e Q h Bowen ratio latent heat transfer sensible heat transfer

56 56 Energy Balance Method Energy for Evaporation Q e = Q s Q r + Q a Q ar Q bs Q 0 + Q v 1 + B + c p (T e T b )/λ Q e energy transferred by evaporation [W /m 2 ] Q s energy from sun (short wave in) [W /m 2 ] Q r energy reflected (short wave out) [W /m 2 ] Q a energy from atmosphere (long wave in) [W /m 2 ] Q ar energy reflected (long wave) [W /m 2 ] Q bs energy emitted from water (long wave) [W /m 2 ] Q 0 energy balance [W /m 2 ] Q v energy advected [W /m 2 ] B Bowen ratio c p specific heat of water [J/kgC] T e, T b current and reference temperature of water λ latent energy of evaporation 2260[kJ/kg]

57 57 Energy Balance Method Energy for Evaporation E = Q e ρ λ = Q s Q r + Q a Q ar Q bs Q 0 + Q v ρ λ (1 + B) + c p (T e T b ) Q s energy from sun (short wave in) [W /m 2 ] Q r energy reflected (short wave out) [W /m 2 ] Q a energy from atmosphere (long wave in) [W /m 2 ] Q ar energy reflected (long wave) [W /m 2 ] Q bs energy emitted from water (long wave) [W /m 2 ] Q 0 energy balance [W /m 2 ] Q v energy advected [W /m 2 ] B Bowen ratio c p specific heat of water [J/kgC] T e, T b temperature of water ρ density of water in kg/dm 3 λ latent energy of evaporation 2260[kJ/kg]

58 Energy and Mass Transfer 58 Energy, Radiation and Mass Transfer

59 59 Energy and Mass Transfer Energy Method for Terrestrial Surface E r = R n /(λ ρ) (15) λ = T [MJ/kg] = (16) R n = R i (1 α) R e (17) R e = e σ Tp 4 (18) E r Evaporation by energy [mm/d] R n incoming radiation [W /m 2 ] λ latent heat of evaporation in [W ] T Temperature in degrees Celsius ρ density of water in [kg/dm 3 ] R n, R i net and total incoming radiation [W /m 2 ]

60 60 Energy and Mass Transfer Calculation of Incoming Radiation R G = G sc d r π [ω sin(φ) sin(δ) + cos(φ) cos(δ) sin(ω s )] R G extraterrestrial radiation [MJ m 2 day 1 ] G sc solar constant = [MJ m 2 min 1 ] d r inverse relative distance earth-sun ω s sunset hour angle [rad]) φ latitude [rad] δ solar decimation [rad]

61 61 Energy and Mass Transfer Converting decimal degrees to radians rad = π 180 d The variable d represents decimal degrees. For the northern hemisphere degrees in decimal degrees are positive for the southern hemisphere they are entered as negative values.

62 Energy and Mass Transfer 62 Table: Radians degrees rad

63 63 Energy and Mass Transfer Inverse Relative Distance Earth-Sun d r = cos ( ) 2π 365 J J is the Julian day commencing with J=1 on the first of January and ending with J=365 on the 31 st of December.

64 Energy and Mass Transfer 64 Table: Inverse Distance Earth Sun Month Julian Day relative inverse distance earth sun January February March April May June July August September October November Dezember New Year

65 65 Energy and Mass Transfer Inverse Relative Distance Earth-Sun inverse relative distance Julian day Figure: Relative inverse distance earth-sun

66 66 Energy and Mass Transfer Solar Declination ( ) 2 π δ = sin 365 J 1.39 J is the Julian day commencing with J=1 on the first of January and ending with J=365 on the 31 st of December.

67 Energy and Mass Transfer 67 Table: Solar Declination Month Julian Day solar declination January February March April May June July August September October November Dezember New Year

68 68 Energy and Mass Transfer Solar Declination solar declination [rad] Julian day Figure: Solar declination

69 69 Energy and Mass Transfer Sunset Hour Angle ω = arccos [ tan(φ) tan(δ)] (19) ω = π ( ) tan(φ) tan(δ) 2 arctan (20) X N = 24 π ω s (21) N daylength ω sunset hour angle (radians) φ latitude (radians) δ declination (radians) X 1 [tan(φ)] 2 [tan(δ)] 2 N Day length (max. sunshine hours

70 Energy and Mass Transfer 70 Table: Sunshine hours Month Julian Day sunshine hours January February March April May June July August September October November Dezember New Year

71 71 Energy and Mass Transfer Sunshine Hours sunshine hours Julian day Figure: Sunshine hours

72 72 Energy and Mass Transfer Extraterrestrial Radiation R a = 24 (60) G sc d r π [ω s sin(φ) + cos(φ) cos(δ) sin(ω s )] where R a extraterrestrial radiation [MJm 2 day 1 ] G sc solar constant = [MJm 2 min 1 ] d r inverse relative distance earth-sun ω s sunset hour angle (radians) φ latitude (radians) δ declination (radians)

73 73 Energy and Mass Transfer Extraterrestrial Radiation extraterrestrial radiation Julian day Figure: Extraterrestrial radiation

74 74 Energy and Mass Transfer Conversion factors MJ m 2 d Table: convert J m 2 d cal m 2 d W m 2 mm d MJ m 2 d cal m 2 d W m mm d Multiplication with converts m 2 d d which is the equivalent amount of water mm d that can be evaporated by the energy MJ m 2 d. MJ to mm

75 75 Energy and Mass Transfer Short-Wave Radiation on Earth After travelling through atmosphere and clouds radiation on earth mm/day Julian day Figure: Short wave radiation at earth surface in mm/d

76 76 Energy and Mass Transfer Net Long Wave Radiation R l = ( e a ) σ (1.35 (n/n) 0.35) [ ] T 4 max,k Tmin,K 4 2 e a = exp(17.27 T /( T )) R nl Net long wave outgoing radiation [MJm 2 day 1 ] e a actual vapour pressure [kpa] Temperature in Kelvin (min. and max. measured during day) T min,k

77 77 Energy and Mass Transfer Soil Heat Flux [ ] Ti T i 1 G = c s z t G soil heat flux in [MJm 2 day 1 ] c s soil heat capacity [MJm 3 C 1 ] T i air Temperature in Celsius t time step [day] z soil depth [m]

78 Combined Equations 78 Combined Equations

79 79 Combined Equations Seasonal Variation Evaporation

80 80 Combined Equations Combined Method Energy and Aerodynamic Term E c = ( ) ( ) γ E r + E a [mm/d] (22) + γ + γ = 4098 e s T (23) E c E a E r γ T Evaporation from combined method [hpa] Evaporation from aerodynamic method [hpa] Evaporation from energy method gradient of sat. vapour pressure curve psychrometric constant 66.8 Pa/( C) Temperature in degrees Celsius

81 81 Combined Equations Combined Method Energy Term E r = ( n ) S 0 (0.9 n ) N N ( e d ) σ T 4 E r n N e d σ T Evaporation from energy method sunshine hours potential sunshine hours vapour pressure Boltzmann-Constant Temperature in degrees Celsius

82 82 Combined Equations Priestley & Taylor E PT = α ( ) R n + γ R n = R s (1 α) R e R e = e σ Tp 4 E r = R n /(λ ρ) λ = T [MJ/kg] E PT Evaporation by Priestley & Taylor [mm/d] α 1.3 R n net radiation [W /m 2 ] R s total short wave incoming radiation [W /m 2 ] λ latent heat of evaporation in [W ] T Temperature in degrees Celsius ρ density of water in [kg/dm 3 ]

83 83 Combined Equations Makkink ( ET = α + γ R ) s β λ ET Evaporation by Makkink [mm/d] R s solar radiation in [MJm 2 day 1 ] α 0.61 β 0.12 λ latent heat of evaporation in MJkg 1 slope of the vapour pressure curve in [kpa C 1 ] γ psychrometric constant [kpa C 1 ]

84 84 Combined Equations Doorenbos & Pruitt ( ET = a + b + γ R ) s λ a = rh u z rh u z rh uz 2 b = 0.3 ET Evaporation [mm/d] R s solar radiation in [MJm 2 day 1 ] λ latent heat of evaporation in MJkg 1 slope of the vapour pressure curve in [kpa C 1 ] γ psychrometric constant [kpa C 1 ] u z wind speed at 2m in [m/s] rh relative humidity in [%]

85 85 Combined Equations Abtew ET = α ( ) Rs λ ET Evaporation [mm/d] α 0.53 R s solar radiation in [MJm 2 day 1 ] λ latent heat of evaporation in MJkg 1

86 86 Combined Equations Penman-Monteith Method, FAO version E T0 = (R n G) + γ 900 T +273 u 2 (e s e a ) + γ ( u 2 ) (24) E T0 reference evaporation [mm/d] R n net radiation at the crop surface [MJ/m 2 day 1] G soil heat flux [MJ/m 2 day 1] T mean daily temperature at 2m height [ Celsius] u 2 wind speed at 2m height [ms 1 ] e s saturation vapour pressure [kpa] e a actual vapour pressure [kpa] slope of vapour pressure curve [kpa Celsius 1 ] γ psychrometric constant [kpa Celsius 1 ]

87 Actual Evaporation 87 Actual Evaporation and Reduction Functions

88 88 Actual Evaporation Actual Evaporation Reduction by Resistance where as in [3] λ ETo = λ (R n G) + ρ cp λ + γ (es ea) r a ( 1 + rs r a ) (25) ETo reference evaporation [mm/d] R n net radiation at the crop surface [MJ/m 2 day 1] G soil heat flux [MJ/m 2 day 1] T mean daily temperature at 2m height [ Celsius] u 2 wind speed at 2m height [ms 1 ] e s saturation vapour pressure [kpa] e a actual vapour pressure [kpa] slope of vapour pressure curve [kpa Celsius 1 ] γ psychrometric constant [kpa Celsius 1 ]

89 89 Actual Evaporation Actual Evaporation Reduction by Resistance Figure: Reduction of evaporation by resistance

90 90 Actual Evaporation Actual Evaporation Reduction with Crop Coefficient Figure: Reduction of evaporation by crop coefficient

91 91 Actual Evaporation Actual Evaporation Reduction after Infiltration Figure: Reduction of evaporation by crop coefficient

92 92 Actual Evaporation Actual Evaporation Soil Moisture Reduction Figure: Reduction of evaporation by limited soil moisture

93 93 Actual Evaporation Actual Evaporation Soil Function Figure: Reduction of evaporation by limited soil moisture

94 94 Actual Evaporation Actual Evaporation Water Table Reduction Figure: Reduction of evaporation by deep water table

95 95 Actual Evaporation Bagrov-Gugla Method Figure: Reduction of evaporation by (BfG, 2003) [4]

96 96 Actual Evaporation Bagrov-Gugla Soil Moisture

97 Actual Evaporation 97 Runoff in Germany Prof. Dr. C. Ku lls, Labor fu r Hydrologie Lu beck, 2017

98 98 Actual Evaporation Further Topics and Readings... Thorntwaite & soil water balance Jensen-Haise Turc Calculation of r s and r a Wind correction

99 References 98 [1] J. Dalton. Experiments and observations to determine whether the quantity of rain and dew is equal to the quantity of water carried of by the rivers and raised by evaporation; with an enquiry into the origin of springs. Mem. Lit. Philos. Soc. Manchester, V(II): , [2] W. Haude. Zur praktischen Bestimmung der aktuellen und potentiellen Evaporation und Evapotranspiration. Dt. Wetterdienst, [3] R.G. Allen, L.S. Pereira, D. Raes, and M. Smith. Crop evapotranspiration - guidelines for computing crop water requirements - FAO Irrigation and drainage paper 56.

100 References 98 [4] G. Glugla, P. Jankiewicz, C. Rachimow, K. Lojek, Richter K., Fürtig, and Krahe P. Wasserhaushaltsverfahren zur Berechnung vieljähriger Mittelwerte der tatsächlichen Verdunstung und des Gesamtabflusses. BfG, 1342(1342):pp., 2003.