MEC558 Continental hydrology and water resources
Quantitative hydrology
Block 1
Lecturer : Agnès Ducharne
[email protected]
MEC558 is a primer to the concepts and methods, including modelling, that can be mobilized
to address water-related problems in the environment.
The aim of Block 1 is to provide the basis for a quantitative description of the water cycle
and the main governing processes over the continents (from precipitation to river discharge via
evapotranspiration).
Academic year 2013-2014
Table des matières
Textbooks and supporting materials
3
Symbol and acronym denition
3
1 Introduction : water cycles on Earth
7
1.1
The global water cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.2
The terrestrial water balance equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
Residence times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2 Water in the atmosphere
9
2.1
Moist air
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Atmospheric stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.3
Precipitation and moisture convergence
11
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Evapotranspiration
13
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Turbulent uxes in the ABL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.3
From turbulence to aerodynamic resistance
. . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.4
Mass transfer or aerodynamic formulations
. . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.5
The Earth's energy balance
3.6
Formulations related to the surface energy budget
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
13
19
20
4 Water in soils
22
5 Water in rivers
22
6 Water cycles under human inuences
22
2
Textbooks and supporting materials
Brustaert, Hydrology - An intoduction, 2005, Cambridge University Press.
Dingman, Physical Hydrology, 2002, Prentice Hall. (But the publisher may have changed)
Guyot, Climatologie de l'environnement, 1999, Dunod.
Musy & Higy, Hydrologie - 1 Une science de la nature, 2009, PPUR.
Peixoto & Oort, Physics of climate, 1992, AIP.
AMS Glossary of Meteorology :
amsglossary.allenpress.com/glossary
/webworld.unesco.org/water/ihp/db/glossary/
Glossaire International d'Hydrologie :
Symbol and acronym denition
Warning : Many values given in the last column depend on
Symbol
SI Unit
Meaning
as
A
AE
AC
cp
-
Surface albedo
2
Wet section
2
Earth's surface area
m
m
2
m
T
and
p!
Comments
Earth's land surface area
−1
J.kg
−1
.K
Specic heat capacity of moist air for constant
cp ' cpd
of dry air
pressure
−1
.K
−1
.K
cpd
J.kg
cs
d0
e
ea
es
e0s (T )
J.kg
−1
Specic heat capacity of dry air for constant
cpd
= 1005 J.kg
−1
.K
−1
pressure
−1
Specic heat capacity of the Earth's surface layer
-
Zero-plane displacement height
Pa
Partial pressure of water vapor
Pa
Partial pressure of water vapor in the air
Pa
Saturation water vapor pressure
Pa.K
−1
ET
des /dT = ∆
Evapotranspiration
−2
−1
.s
−2 −1
kg.m
.s
−2 −1
kg.m
.s
−2 −1
kg.m
.s
−2 −1
kg.m
.s
−2 −1
kg.m
.s
E
Etr
Ei
Esoil
Esn
EA
kg.m
EP
ETM
ET0
Fv
Fvx
Fvy
Fvz
Fhz
Fmz
g
G
h
h
H
kg.m
−2
−1
.s
−2 −1
.s
kg.m
−2 −1
kg.m
.s
vector
−2
−1
.s
−2 −1
kg.m
.s
−2 −1
kg.m
.s
−2 −1
.s
kg.m
−2 −1
kg.m
.s
−2
m.s
−2
W.m
kg.m
E = Etr + Ei + Esoil + Esn
Evapotranspiration
Transpiration
Interception loss
Soil evaporation
Snow sublimation
Drying power of air (
l'air )
pouvoir évaporant de
Potential ET
Thornthwaite, 1948
Maximum ET
Reference-crop ET
Penman, 1956
Specic mass ux of water vapor
Component of the water vapor ux in the
Component of the water vapor ux in the
Vertical component of the water vapor ux
Vertical component of the sensible heat ux
Vertical component of the horizontal momentum ux
Local gravity acceleration
Heat content of enthalpy
m
Total hydraulic head (
−2
g'
9.8 m.s
Heat ux into the soil
J
W.m
x-direction
y -direction
charge hydraulique )
Sensible heat ux
3
h = z + ψw
−2
Symbol
SI Unit
Meaning
If
k
K
Ks
L
Ls
mm
Md
Mw
-
Leaf area index (
n
p
pa
pd
pv
pw
P (1)
P (2)
Pl
Ps
qa
q0
qs
Q
Q
QC
ra
rs
rc
rg
rst
rsoil
R
Rs
Rb
RE
RC
Rd
Rv
Rm
Rn
Rsd
Rld
Rlu
S (1)
S (2)
Sb
Sf
Sl
Ss
-
indice de surface foliaire )
k'
von Karman constant
−1
m.s
Comments
0.4
Hydraulic conductivity
−1
m.s
Saturated hydraulic conductivity
−1
J.kg
L = 2.501 106 J.kg−1 at STP
Ls = 2.834 J.kg−1 at STP
Latent heat of vaporization
−1
J.kg
Latent heat of sublimation
kg
Mass of soil particles in the volume
Vt
masse molaire ) of dry air Md = 28.966 g.mol−1
(masse molaire ) of water
Mw = 18.016 g.mol−1
−1
Molecular weight (
g.mol
−1
Molecular weight
-
Porosity
Pa
Pressure
g.mol
and water vapor
STP
n = (Vt − Vm )/Vt = 1 − ρb /ρm
pa /ρa = Rd Ta (1 + 0.61 qa ) = Rd Tv
Pa
Atmospheric pressure
Pa
Pressure of dry air
Pa
Pressure of water vapor in the air
Pa
kg.m
pw = γw ψw
Ground water pressure
−2
.s
−1
Precipitation rate
−2
.s
−1
Rainfall rate
m
Wet perimeter
kg.m
−2
−1
.s
−1
kg.kg
−1
kg.kg
−1
kg.kg
Snowfall rate
vector
Vertically integrated horizontal transport of water vapor, or aerial runo kg.m
−3
Specic humidity of the air
Specic humidity of saturated air
−1
.s
−3 −1
m
.s
−1
s.m
−1
s.m
−1
s.m
−1
s.m
−1
s.m
−1
s.m
−2 −1
kg.m
.s
−2 −1
kg.m
.s
−2 −1
kg.m
.s
River discharge
m
Earth's radius
m
−2
kg.m
−1
.s
−1
−1
.K
−1
−1
J.kg
.K
−1
−1
.K
J.kg
−2
W.m
−2
W.m
−2
W.m
−2
W.m
−2
kg.m
3
m
J.kg
-
qa = ρv /ρa
Specic humidity of the air at the surface
qs (Ta ) = es (Ta )/pa
Total discharge from land to oceans
Aerodynamic resistance
Surface resistance
Canopy resistance
Geometric resistance
Stomatal resistance
Soil resistance
Runo (
écoulement total )
ruissellement )
Surface runo (
Subsurface runo
RE
= 6,371 km
Average runo from land to oceans
Specic gas constant for dry air
Specic gas constant for water vapor
Specic gas constant for moist air
Net radiation (
rayonnement net )
Rd = 287 J.kg−1 .K−1
Rv = 461.5 J.kg−1 .K−1
Rm = Rd (1 + 0.61 qa )
Downward short-wave radiation, or global radiation
Downward long-wave radiation, or atmospheric radiation
Upward long-wave radiation
Vertically integrated terrestrial water storage
Total water volume stored in a river reach
River bed slope
-
Friction slope (
pente de frottement )
−2
Vertically integrated terrestrial storage of liquid water
−2
Vertically integrated terrestrial storage of solid water (snow/ice)
kg.m
kg.m
Standard temperature and pressure
4
◦
T=0 C and p=1000 hPa
Symbol
SI Unit
Meaning
T
Ta
Ts
Td
Ts
Tv
Tw
K
Temperature
K
Air temperature at the reference level
K
Surface temperature
K
Dew point temperature
K
Surface temperature
K
Virtual temperature
K
Wet bulb temperature
J
Internal energy
TOA
u
u
u∗
U
v
v
V
Va
Vm
Vt
Vw
w
W
W
W
Wa
Wc
Wt
y
z
zb
z0
z0v
z0h
Comments
es (Td ) = ea
Tv = Ta (1 + 0.61 qa )
Top of atmosphere
−1
m.s
Horizontal wind speed in the
−1
m.s
Friction velocity
-
Relative humidity of the air
vector
x-direction
u∗ = (τ0 /ρa )1/2
U = qa /qs (Ta )
Velocity of the air
−1
Horizontal wind speed in the
−1
Bulk section velocity
m.s
m.s
y -direction
3
Volume of air in
3
Volume of mineral particles in
3
Total volume of soil
3
Volume of water in
m
m
m
m
−1
m.s
Vt
Vt
Vt = Va + Vm + Vw
Vt
Vertical wind speed in the
z -direction
−2
Vertically integrated amount of atmospheric water vapor, or precipitable water
−2
Bulk soil water content, or bulk soil moisture
−2
Bulk available soil water content, or bulk available soil moisture
−2
Vertically integrated amount of condensed atmospheric water (liquid and solid)
(1)
kg.m
(2)
kg.m
(3)
m
Top river width
kg.m
kg.m
−2
kg.m
Vertically integrated amount of total atmospheric water
m
River depth (
m
Elevation above arbitrary datum
m
Elevation of river bed above arbitrary datum
m
Roughness length for momentum
m
Roughness length for water vapor
m
Roughness length for sensible heat
tirant d'eau )
3
α
αd
αv
γ
m .kg
γw
Γ
Γd
Γs
εs
εa
∆
θ
θs
θFC
θWP
λ
N.m
−1
3
−1
m .kg
3
−1
m .kg
−1
Pa.K
α = 1/ρ
Specic volume
Specic volume of dry air
Specic volume of water vapor in the air
Psychrometric constant
γ=
cp pa
L
poids volumique ) of water
−3
Specic weight (
−1
Lapse rate of the air
−1
Dry adiabatic lapse rate of the air
K.m
K.m
−1
K.m
Saturated adiabatic lapse rate of the air
-
= Rd /Rv ' 0.622
-
Surface emissivity (for long-wave)
-
Atmospheric emissivity
Pa.K
−1
−3
.m
−3
.m
−3
.m
−3
.m
m
m
m
m
−3
angle
γ
at p = 1000 hPa et T = 20°C
γw = ρw g
Γ = dT /dz
Γd = −g/cp
dq
Γs = Γd + cLp dz
Equals absorptivity
des /dT = e0s (T )
Volumetric water content (or water content)
−3
Saturated volumetric water content
−3
Volumetric water content at eld capacity
−3
Volumetric water content at wilting point
Longitude
5
−1
= 0.67 hPa.K
θ = Vw /Vt
θs = n
Symbol
SI Unit
ρ
ρa
ρb
ρd
kg.m
ρm
ρv
ρw
σ
τ0
φ
ψm
ψw
ω
kg.m
Meaning
masse volumique )
−3
Density (
−3
Density of moist air
−3
Soil bulk density
−3
Density of dry air
−3
Soil particle density (
kg.m
kg.m
kg.m
−3
kg.m
−3
masse volumique )
Density (masse volumique ) of water vapor
Density (masse volumique ) of water
Comments
ρa = ρv + ρd
ρb = mm /Vt
ρd = (pa − ea )/(Rd Ta )
ρd = 1.2923 kg.m−3 at
ρm = mm /Vm
STP
Water vapor is less dense than dry air
−2
−4
W.m
.K
Stefan-Boltzman constant
ρw = 1000 kg.m−3
σ = 5.6697 108 W.m−2 .K−4
Pa
Shear stress at the surface
(
angle
Latitude
m
Matric head
m
Pressure head
-
Soil wetness, or degree of saturation, or
kg.m
6
contrainte de cisaillement )
ψw = pw /γw
ω = θ/θs
1 Introduction : water cycles on Earth
An important part of hydrology is about quantifying the terms of the water cycles at various scales.
They mostly consist of water uxes, and are deduced from measurements and their interpolation, from
conservation or balance equations, and sometimes modeling.
1.1 The global water cycle
As revealed by the word "cycle", the focus is about uxes. The global water cycle is primarily described
by the mean values of precipitation and evaporation over land and oceans (PC ,
PO , EC , EO ),
and the
land/ocean exchange uxes. There are various estimations of these uxes because of the diculty to
measure them
Ü
Figure 1.1 (Trenberth et al., 2007) + other estimations
q
Overview of measurement/estimation techniques and their uncertainties
PO
and
PC
: in situ measurement, with bias correction (related to rain gauge density, altitude, snowfall
under-catch, etc.) and interpolation ; remote-sensing using passive and active micro-wave sensors
RC
3
: by conversion from river discharge in m .s
−1
to runo in kg.m
−2
.s
−1
; problems related to ungauged
catchments, including coastal streams, to measurement uncertainty, especially at high/low ow (see
section 5) ; to the assumption that subsurface ow is negligible at the outlets
EC
: evaporation is dicult to measure, even at the local scale (see section 3). Some global maps have
recently been produced based on the interpolation of local-scale measurements, or on modeling, with
or without assimilation of remote-sensing data. But large-scale evaporation is most often estimated by
means of water balance (Eq. 3), which allows to deduce the inter-annual mean of terrestrial evaporation
from the measurements of precipitation and river discharge (used as proxy to
R),
in river basins up to
the continental scale.
EO : modeling, possibly using remotely-sensed
divO Q : by a balance approach
q
Link with water resources
surface temperatures, or by a balance approach
Water resources can be dened as the amounts of
fresh water that can be used for either human or
natural needs. Human needs include drinking and domestic use, recreation and tourism, navigation, industrial use including energy production (hydro-power, cooling), and agriculture (rain-fed and irrigated).
The main exploited water resources are river discharge and groundwater, which are uxes (even if groundwater ows are very slow). Thus, we can say that water resources are renewable resources. Yet, it does
cf.
not mean that they cannot be exhausted : this happens if withdrawal rates exceed renewal rates (
residence times in section 1.3).
In water management, terrestrial precipitation,
PC ,
is often split in
green and blue water Ü Figure
1.2, from Falkenmark and Rockström (2006) :
green water is the water inltrating into the soil, taken up by roots, used in photosynthesis and
transpired by natural vegetation and crops ; it is the productive fraction for farming ; white water is
sometimes distinguished as the water intercepted and directly evaporated by the vegetation canopy
and the ground surface ; eventually, green+white water =
EC
blue water is made up from runo to rivers and deep percolation to aquifers that nds its way to rivers
(RC ) ; grey water designates the fraction of blue water that is polluted, as pollution lowers the value
of water as a resource.
1.2 The terrestrial water balance equation
q
Main terrestrial hydrologic processes and relevant scales Ü Figures 1.3 and 1.4
One can distinguish between phase-change processes, mostly vertical ; and transfer processes, or ows,
which are mostly vertical in soils, and horizontal at larger scales (over hills, in ground water and river
systems). Snow processes do not appear on the gure.
7
q
Conservation equation for total terrestrial water
Very generally, a balance equation states that the change in storage results from the balance between
input and output uxes. We focus here on a unit area column of ground (or lithosphere) which extends
below the Earth's surface. The total amount of water
phases,
Sl
and
Ss
S
in this column can be decomposed between two
for liquid and solid (snow and ice) water respectively. The main corresponding input
and output water uxes, usually expressed in volumes per area per unit of time, are :
Sl
outputs : evaporation from water bodies (including intercepted water) and soils, transpiration by
plants, runo (surface and subsurface),
Sl inputs : rainfall, snow/ice melt, dew (negative evaporation),
Ss outputs : snow/ice melt, sublimation
Ss inputs : snowfall, frost deposition by negative sublimation
The resulting conservation equation for
S
is :
∂(Sl + Ss )
∂S
=
=P −E−R
∂t
∂t
(1)
It can be applied to any kind of terrestrial surface, from plots to continents via the river basins, owing
that the suitable spatial averages are applied on the dierent terms
Ü
Figure 1.5 (Dingman p 12).
Note that subsurface runo can be be an input in some instances, but over large enough areas, the net
subsurface runo is usually small compared to the other uxes (it is supposed to be only 6 % of the total
runo to the oceans at the global scale,
q
cf. Oki and Kanae (2006).
Interest of river basins
River basins (or river catchments, or drainage basins, or watersheds) can be dened by the convergence
of surface and most sub-surface runo toward the river network, where the corresponding ows can be
measured
as river discharge :
River basins are usually delineated from topographic basins, assuming that the topographic gradients
Ü Figure 1.5 (Dingman, p11). We will then speak of a topographic basin, which corresarea upstream from an outlet. Ridge lines separate topographic basins from each other
drive the ow
ponds to the
and connect points where the topographic eld is divergent. Therefore, we can assume that there is no
surface runo input into a topographic basin.
In areas where ground water ow is signicant, there can be subsurface runo input or output through the
lateral boundaries of the topographic basin. The subsurface domain characterized by zero-ux boundaries
can be called a
ground water basin.
It is theoretically dened like the topographic basin but using
the piezometric eld (see Block 2) instead of the topography eld. In practice, the piezometric eld
1 is
poorly known (the surveys are local, by wells, while it varies over time and the underground does not
have continuous properties).
q
Long-term equilibrium
Eq. 1 also holds over any time period
∆t,
using the average uxes over the period :
∆S = (P − E − R) ∆t
In most cases, we can neglect
∆S/∆t
(2)
over a long enough period. As the main storage changes occur
on seasonal time scales, this is also often true over a couple of full annual cycles. This leads to the
steady-state, or equilibrium, equation :
P −E 'R
(3)
1.3 Residence times
Another facet of the global water cycle (and other water cycles at smaller scales) is about water stocks
Ü
Figure 1.6, from Jeandel and Mosseri (2011).
1. It is the eld of hydraulic, or piezometric head, which drives undergound ow. Piezometric head can be measured as
the top elevation of water in wells or piezometers.
8
These stocks are not permanent, but the result of the balance between input and output uxes, and
an interesting way to characterize them is by their
residence time, which corresponds to the average
transit time of water in the corresponding reservoirs. Assuming a steady state,
uxes
IR
and
OR
3
−1
(in L .T
i.e. that input and output
) are equal,
tR = SR /IR = SR /OR
where
SR
(4)
is the estimated volume of water stored in the reservoir under the steady state conditions.
Residence time is also called turn-over time, as it is a measure of the time it takes to completely replace
water in the reservoir. Residence time is also largely related to the buering eect of the reservoir,
the way the variability of the outow is reduced compared to the variability of the inow
Ü
i.e.
Figure 1.7,
from Entekhabi et al. (1996).
2 Water in the atmosphere
2.1 Moist air
q
Ideal gas framework
We consider the air as an homogeneous gas, in which the various components obey Dalton's law of partial
pressures and behave as an ideal gas (assuming a unit volume, with
R
the specic gas constant,
i.e., the
universal gas constant divided by the molecular weight of the gas)
pα = RT or p = ρRT
(5)
If we separate dry air and water vapor, we get
ρa =ρd + ρv
(6)
pa =ρa Rm Ta = pd + ea
(7)
pd =ρd Rd Ta
(8)
ea =ρv Rv Ta
(9)
We can dene the apparent molecular weight of dry air,
Md ,
by the weighted means of the molecular
weights of each atmospheric gas but water vapor (mainly N2 and O2
=
Ü
Figure 2.1) and we introduce
Mw
Rd
=
' 0.622
Rv
Md
(10)
It comes that
p a − ea
Rd Ta
ea
ρv =
Rd Ta
pa pa − (1 − )ea
ρa =
Rd Ta
pa
ρd =
q
(11)
(12)
(13)
Water phase change
We focus on the liquid-vapor transition, which controls the fate of atmospheric water, as the latter
increases with evaporation, and decreases with precipitation, requiring condensation (section 2.3).
This transition corresponds to the p-T coexistence curve (
courbe d'équilibre ) between the triple and critical
points on the water phase diagram (Ü Figure 2.2), and can be described by the
Clausius-Clapeyron
relation for phase change
des
L
=
dT
T δα
9
(14)
δα is the variation of specic
αv −αl , the subscripts referring to the specic volumes of the gaseous
and liquid phase. For water vapor in typical atmospheric conditions, we can assume that αl αv , and
es (T ) can be found from Eq. 14 if we know the form of L(T ) Ü Figure 2.3
where
es
is the saturated vapor pressure (corresponding to U=100%), and
volume during the phase change, viz.
Many dierent approximations of
es (T ) exist for the typical range of temperatures on Earth, with mostly
two forms, each of them with man dierent variations, including special cases for the saturation of water
vapor over ice.
es (T ) in hPa and T in
17.62T
es (T ) = 6.112 exp
.
T + 243.12
The WMO Magnus-Tetens approximation, with
◦
C, is :
(15)
Since there is only a weak dependence on temperature in the denominator of the exponential, this equation
shows that saturation water vapor pressure changes approximately exponentially with
T.
es (T ) in hPa and T in K, is valid between -50 to 102
373.16
373.16
− 1 + 5.02808 log10
log10 (p) = − 7.90298
T
T
T
11.344(1− 373.16
−7
)
− 1.3816 × 10
10
−1
373.16
+ 8.1328 × 10−3 10−3.49149( T −1) − 1 + log10 (1013.246) .
The Go-Gratch approximation, with
◦
C :
(16)
Note that this kind of polynomials can be used in nested form to increase computational speed.
q
Variables used to quantify atmospheric moisture
An important variable to characterize the moisture content of air is the
specic humidity, dened as
the mass of water vapor per unit mass of moist air
qa = ρv /ρa =
ea
ea
'
as ea pa
pa − (1 − )ea
pa
We can also quantify the atmospheric vapor content by the
(17)
dew point temperature Td :
es (Td ) = ea .
Finally, the
relative humidity is dened as
U = qa /qs (Ta ) = ea /es (Ta )
(18)
where the s subscript refers to the values at saturation at the pressure and temperature of the air.
q
How can we measure atmospheric moisture ?
Main routine techniques in PC.
2.2 Atmospheric stability
The atmosphere is unstable if a small and rapid vertical displacement leads to further movement in the
same direction, and conversely
buoyancy (
Ü
Figure 2.4. Because of the links between temperature, density and
ottabilité ), atmospheric stability depends on how the vertical rate of temperature decrease,
2
also known as the atmospheric lapse rate ,
Γ = −dT /dz
(19)
compares with the lapse rates of dry and saturated air.
2. The lapse rate is dened as the rate of decrease with height for an atmospheric variable. The variable involved is
temperature unless specied otherwise.
10
We rst need to quantify the change in heat content or enthalpy
dh
during the displacement of an
air parcel. Combining the rst law of thermodynamics (conservation of the internal energy of a closed
system) :
dh = du + p dα
(20)
dp = −ρg dz
(21)
with the hydrostatic law
and the ideal gas equation (equation of state, Eq. 5), and given that the specic heat at constant pressure
is very close for moist and dry air (cp
' cpd ),
we get (see Brutsaert p 29 for the details)
dh = cpd dT + g dz
(22)
For rapid displacements, there is no heat exchange with the surrounding air, what corresponds to an
adiabatic process. Then, the only source of heat change is the internal water phase change.
Thus, in absence of water phase change, we have
rate
dh = 0,
what allows dening the
dry adiabiatic lapse
Γd = g/cpd ' 9.8 K.km−1
(23)
If an ascending unsaturated air parcel cools more than the surrounding air (Γd
density and is brought downward, so that that the air is stable, and conversely
> Γ), it gets a higher
Ü Figure 2.4. Thus, we
get that the following criteria for the stability of unsaturated air :
Γ > Γd : instable
Γ = Γd : neutral
Γ < Γd : stable
If air is saturated, condensation is the only possible source of heat content during the adiabatic displacement ; this can be written as
adiabiatic lapse rate
dh = −L dq ,
which, combined to Eq 22, leads to dening the
Γs = Γd +
Γs
L dq
cpd dz
varies signicantly with pressure and temperature, but it remains lower than
the low layers of the atmosphere is 5
K.km−1 .
saturated
(24)
Γd .
A typical value in
Based on the same considerations as for unsaturated air,
the criteria for the stability of saturated air are
Γ > Γs : instable
Γ = Γs : neutral
Γ < Γs : stable
If air is partly saturated, and the actual lapse rate in the atmosphere is between the dry and saturated
adiabatic lapses rates, that is if
Γs < Γ < Γd ,
we speak of
moist parcel behaves as dry air until saturation (at
zC ),
conditional instability Ü Figure 2.5. The
when it follows the saturated lapse rate
Γs .
The
stability does not just depends on the respective lapses rates, but rather on the respective temperatures
of the parcel and the surrounding air. Under such conditions, the atmosphere is stable in the low levels,
and ascendance needs to be forced to become instable, at the level of free convection
zF .
2.3 Precipitation and moisture convergence
Precipitation is the process of atmospheric water returning to the ground surface, which can be achieved
in dierent forms depending on water phase and grain size and structure : rain, snow, hail (
(
bruine ou brouillard précipitant ), dew (rosée), frost (gelée blanche), etc.
11
grêle ), drizzle
q
Requirements for precipitation
1. Condensation, which is easier if the atmosphere is moist and cold. Thus, condensation is enhanced
by radiative cooling or uplift, which can be caused by orography (we speak of forced convection),
or atmospheric instability (we speak of free convection), or both (in case of conditional instability)
Ü
Figure 2.6
2. Water droplets formation, usually enhanced by condensation nuclei
Ü
Figure 2.7. In above free-
zing temperatures, the air would have to be supersaturated to around 400% before the droplets
could form. Drops grow by further condensation and droplets coalescence. As a rule of thumb, the
transition between precipitation and cloud particles occurs for diameters around 0.1 mm.
q
Balance equation for atmospheric water
We consider a unit area column of air which extends from the Earth's surface to the top of the atmosphere
(TOA). The total amount of water in this column can be separated between the vapor and condensed
phases (liquid and solid) :
Wt = W + Wc ' W as Wc W
The total amount of water vapor in this unit area column of air,
W,
(25)
is called the
precipitable water,
as it corresponds to the amount of liquid water that would be produced if all the vapor condensed :
zTOA
Z
W =
Z
ρq dz =
z0
The changes in total water vapor
W
p0
q
0
dp
g
(26)
result from :
local inputs by evaporation from the Earth's surface,
E,
local outputs by means of condensation, what denes the precipitation
P,
the net ux of water vapor through the lateral boundaries of the air column, which can be dened as
the divergence of
Q,
the vertically integrated horizontal water vapor transport :
Z
p0
Q=
qv
0
dp
= Qλ i + Qφ j
g
(27)
The resulting balance equation for atmospheric water is :
∂W
∂Wt
'
= E − P − divQ
∂t
∂t
Except in the case of severe storms,
a good proxy for
∂W/∂t
is very small compared to the other terms, so that
moisture convergence, even over short time steps :
P − E ' −divQ
(28)
P −E
is
(29)
q Meridional distributions of T , q , W , P , E , and moisture convergence
Ü Figures 2.8 The terms zonal and meridional are used to describe directions over the globe. Zonal means
i.e. following i ; while meridional means "along
i.e. following j. Therefore, meridional distributions result
"along a latitude circle" or "in the westeast direction",
a meridian" or "in the northsouth direction",
from zonal averages.
12
3 Evapotranspiration
3.1 Introduction
Evapotranspiration corresponds to evaporation from land surfaces, an important contribution of which
being transpiration by plants.
As a physical phenomenon, evaporation is the transition of water from the liquid phase to the vapor
phase, with two main requirements
an
Ü
Figure 3.1 :
energy supply to provide water molecules the necessary kinetic energy to escape from the liquid
surface ;
some mechanism to export the escaped molecules from the vicinity of the liquid surface and prevent
them from immediate re-condensation. In the lower part of the atmosphere, this mechanism is linked to
turbulence, which is a very ecient mean of vertical transport for water vapor, but also for sensible
heat (or enthalpy), and momentum.
mass transfer or
aerodynamic formulations (section 3.4), which will be illustrated after an overview of turbulence in the
lower atmosphere (sections 3.2 and 3.3), and the energy budget formulations (section 3.6), preceded by a
This denes the main two classes of methods to describe evaporation, namely the
presentation of the Earth energy balance (section 3.5).
3.2 Turbulent uxes in the ABL
q
The atmospheric boundary layer
Very generally, in a uid in motion along a surface, a boundary layer is the layer of uid in the immediate
vicinity of the surface where the eects of friction are signicant. In the boundary layer, the velocity varies
from zero at the surface, to the velocity of the free uid at the top of the boundary layer. Boundary layers
can be either laminar or turbulent, depending on the viscosity and mean velocity of the uid, and on
cf. Reynolds number).
surface roughness (
It is a turbulent
boundary layer. Its typical structure is comprised of several layers Ü Figure 3.2 :
the atmospheric surface layer (ASL), the lowest 10% or so of the ABL, where the wind direction
The lower part of the atmosphere is called the atmospheric boundary layer (ABL).
remains constant with height, and the vertical turbulent uxes do not change appreciably from their
value at the surface, say less than 10% ;
the lower part of the ASL is the
dynamic layer, where buoyancy (ottabilité ) eects resulting from
temperature and humidity gradients are negligible : this layer can be assumed to have a neutral prole
in the
outer region or defect layer, the ow direction depends on the one in the "free" atmosphere
(Ekman spiraling between the geostrophic wind in the free atmosphere and no wind at the surface).
The depth of outer region typically ranges between 100 m to 2 km, with large diurnal variations under
unstable conditions. As a rule of thumb, it can be taken as 1 km.
Generally speaking, convective transport, here for the water vapor concentration
q
(scalar), is given by
Fv = ρv v = ρa qv
(30)
Note that, in atmospheric science, we often distinguish :
convection
: transport with air movement involving gravity eects, mostly vertical, because of density
stratication (instable proles leading to ascending motions)
advection : transport related to the motion of the uid by large-scale horizontal winds
turbulent uxes : vertical transport by turbulent ow, also called or turbulent diusion,
described by a diusion equation (e.g. Eq. 45)
molecular diusion : negligible when turbulence
as it can be
The rst three kinds of transport correspond to the general sense of convection in uid mechanics.
13
q
Turbulence and its consequences
turbulent
ux. Following turbulence theory, we use Reynolds decomposition into the average and uctuating parts
The evaporation from land and ocean surfaces happens in the ASL and thus corresponds to a
(perturbations) of a quantity
Ü
Figure 3.3
x = x + x0 , with x0 = 0
This decomposition can be applied to the scalar components of
specic humidity, etc. For convenience, we take
so that the
y -components are zero :
(31)
Fv ,
to the wind speed components, to
i as the direction of mean wind velocity near the ground,
Fvx =ρ (q u + qu0 + q 0 u + q 0 u0 )
Fvz =ρ (q w + qw0 + q 0 w + q 0 w0 )
After time-averaging over the suitable period (typical 15 min to 1h), we are left with
Fvx =ρ (q u + q 0 u0 )
Fvz =ρ (q w + q 0 w0 )
The right-hand terms represent the advective transport of water vapor by the mean motion of air for
the rst ones, and by turbulence for the second ones. Statistically speaking, the latter correspond to
covariances.
the horizontal scales of atmospheric ow are larger than
the vertical ones, so that the mean vertical velocities are small in front of the horizontal ones :
In the ABL, and even more in the ASL,
w=0
In addition,
(32)
assuming a uniform source or sink term at the surface, we get that the horizontal
gradients are small compared to the vertical ones, so the mean concentrations change mostly in the
vertical and can be assumed constant in the horizontal direction :
Fvx = 0
q
(33)
Surface turbulent uxes
The above considerations lead the following expression of the mean turbulent transport of moisture in
the ASL
Fvz = ρa w0 q 0
(34)
Similar expressions can be written for other uxes, of horizontal momentum (
horizontal ) and sensible heat 3 , with mean "concentrations" u and cp T
quantité de mouvement
:
Fmz = ρa w0 u0
Fhz = ρa cp
(35)
w0 T 0
(36)
Under steady conditions above a uniform surface, on account of continuity, the inow rate equals the
outow rate, and
the vertical uxes must be constant with
z.
For water vapor and sensible heat,
the surface is a source, via the evaporation and sensible heat uxes at the surface :
Fvz = ρa w0 q 0 = E = ρa w0 q 0 0
(37)
Fhz = ρa cp w0 T 0 = H = ρa cp w0 T 0 0
(38)
where the 0 subscript denotes the value near the surface.
3. The total heat content of the air or enthalpy of air comprises of the sensible heat and the latent heat. The sensible
heat is the heat absorbed or lost during the change in temperature of the air. The latent heat is the heat lost or absorbed
during change in phase of the water vapor.
14
For the horizontal momentum, the surface is a sink by means of shear stress
τ (Ü
Figure 3.4), so that
Fmz = ρa w0 u0 = −τ = −τ0
For convenience, the shear stress at the surface
τ0
(39)
is often expressed as the friction velocity :
u∗ = (τ0 /ρa )1/2
(40)
It has the dimension of a velocity, and depends on the uid viscosity, the surface roughness, and the mean
horizontal velocity (in the sense of Reynolds).
3.3 From turbulence to aerodynamic resistance
The mean conservation equations in turbulent ow (Reynolds-averaged Navier-Stokes equation) are not
linear and introduce more unknowns than equations. This is known as the
turbulence closure problem,
and it requires to introduce additional relationships.
To address the surface turbulent uxes, we rst use the simplications leading to Eqs. 32 and 33, then
we invoke turbulence similarity. This consists in assuming that "universal" relationships, in the sense of
dimensional analysis, exist between the vertical behavior of non-dimensionalized mean ow and turbulence
properties within the ASL :
When the empirical data are plotted on graphs of one dimensionless group versus another,
often data from many disparate meteorological conditions will result in one common curve,
yielding a similarity relationship that may be universal. Dimensional analysis has been used
extensively and successfully in studies of the atmospheric boundary layer, where turbulence
precludes other more precise descriptions of the ow because exact solutions of the equations
of motion are impossible to nd due to the closure problem.
Stull, R. B., 1988 : An Introduction to Boundary Layer Meteorology.
We will develop here the most classical implementation of turbulence similarity, in the case
of neutral conditions, which are often found in the dynamic sub-layer. Dierent formulations exist for
non-neutral conditions, involving additional variables and/or functions describing the eect of stability.
Examples are the Obukov stability length in the Monin-Obukov similarity theory, or the Richardson
number. This is especially relevant for the sensible heat ux, as temperature dierences and sensible heat
ux are relatively small under neutral conditions.
q
Horizontal momentum. In the 1930's, dimensional analysis led to the observation that, under neutral
conditions and in plan-parallel ow, the dimensionless quantity
k=
is a nearly invariant around 0.4 ;
k
u
(41)
is commonly referred to as
height at which the mean velocity gradient
The integration of
u∗
z(du/dz)
with respect to
z
du/dz
von Karman's constant. Here, z is the
is measured.
in Eq 41 leads to the
u2 − u1 =
u∗
ln
k
logarithmic wind prole
z2
z1
,
(42)
where subscripts 1 and 2 refer to two levels within the neutral surface layer. The integration between an
elevation
z
and the surface leads the so-called Prandtl-von Karman universal velocity distribution
u(z) =
u∗
ln
k
z
z0
,
(43)
z0 , corresponding to the height at which u = 0, is called the momentum
roughness length Ü Figures 3.5, 3.6 and 3.7 : graphical interpretation and orders of magnitude
where the integration constant
15
q
Mean specic humidity gradients. Dimensional arguments similarly lead to
k=−
where
z
is the height at which
Firstly, we can express
E,
dq/dz
E
ρa z u∗ (dq/dz)
is measured. This equation can be rearranged in two useful ways.
which is constant along the vertical, as a function of
E = −ρa (k z u∗ )
This equation is a
m2 .s−1
diusion equation,
:
dq
.
dz
(45)
with a diusion coecient
Ke = k z u∗ , which has a unit of
z0 ) and on the height at which
dq/dz .
We can also express
dq/dz
as a linear function of
q0 − q(z) =
z0v
dq/dz
in the SI system, and depends on the surface shear stress (thus on
we measure
being the height at which
than
(44)
z0 .
1/z ,
what leads to a logarithmic prole for
E
ln
ρa ku∗
z
z0v
q(z)
:
(46)
q(z) gets its surface value q0 (Ü Figure 3.8), which is about 10 times smaller
z0v by integrating Eq. 44 between two levels 1 and 2 where q is
Note that we can get rid of
measured (prole method).
Combined with Eq. 43, we get :
E = ρa
k 2 u(z1 )
(q0 − q(z))
ln(z/z0v ) ln(z1 /z0 )
(47)
By analogy with Ohm's law, where E would be equivalent to the electrical current I and q0 − q(z) to
V , we introduce an aerodynamic resistance (in s.m−1 in the SI system), which
the potential dience
characterizes the eect of the turbulent layer between the surface and
rav =
q
z
on evaporation :
ln(z/z0v ) ln(z1 /z0 )
k 2 u(z1 )
Sensible heat ux. The same considerations (Ü Figure 3.9) lead to :
dT
dz
k 2 u(z1 )
H = ρa cp
(T0 − T (z))
ln(z/z0h ) ln(z1 /z0 )
ln(z/z0h ) ln(z1 /z0 )
rah =
k 2 u(z1 )
H = −ρa (k cp z u∗ )
q
(48)
Orders of magnitude of ra over land : Ü Figure 3.10
z0v , and z0h
rah = ra
are theoretically dierent, but rather close, and often assumed to be equal, so that
these values of
z0v = z0h '
ra for water vapor and
z0 (Brutsaert p46), z0
0.1
(49)
(50)
(51)
rav '
sensible heat uxes are often estimated by assuming that
being either experimentally dened (Ü Brutsaert p 45), or
estimated as about 10% of the average height of the vegetation
ra
ra
decreases when
u
increases, what enhances turbulence thus turbulent diusion
is also dependent on atmospheric stability when the conditions are not neutral
in practice,
ra
−1
is mostly found between 10 and 200 s.m
,
from the FAO Report N°56 (Allen et al., 1998), assuming a surface covered by a reference grass of
height
h=0.12 m (cf. section 3.6), and a standardized height for wind speed, temperature and humidity
at 2 m :
z0 = 0.123 h
z0v = z0h = 0.1 z0
this leads to ra = 208/u2m
16
3.4 Mass transfer or aerodynamic formulations
The question now is how to apply the above theoretical developments to measure or estimate
E
from
natural land surfaces.
q
Eddy-covariance
This method is based on the direct measurement of
w0 q 0
(Eq. 37), with Reynolds means over periods of
15 min to 1 h at most, and Reynolds uctuations measured with a frequency of at least 5-10 Hz
Ü Figure
3.11
q
Bulk method
Many similarity formulations have in common that they replace the mean product of temporal uctuations
by the product of the spatial changes of the corresponding mean quantities :
w0 x0 ' −Cx (u2 − u1 )(x4 − x3 )
(52)
where the subscripts 1 to 4 refer to measurement heights above the surface, and
Cx
is a dimensionless
parameter which depends on the levels 1 to 4 and other factors (see below). In Eq. 52, levels 4 and 2 can
be the same as 3 and 1, but level 1 is assumed to be lower than level 2. As
minus sign indicates that
In so-called
for
q
and
bulk methods,
T.
w 0 x0
is against the vertical gradient of
the surface where
This contrasts with the
u=0
u
decreases with height, the
x.
is used as the lowest level for wind speed, and also
mean-prole methods
where the lowest level is higher. Using the
bulk framework, we get the following expressions for the vertical uxes of water vapor, sensible heat
and horizontal momentum (the
∆
operator is the dierence between the quantity at level 2 and at the
surface) :
E =ρa w0 q 0 = −ρa Ce u ∆q
(53)
H =ρa w0 T 0 = −ρa cp Ch u ∆T
(54)
τ0 = − ρa
The
w 0 u0
2
= ρa C d u
(55)
dimensionless transfer coecients, which are constant in time, are called as follows :
Cd : drag coecient (coecient de trainée )
Ce : drag coecient for water vapor or Dalton number
Ch : drag coecient for sensible heat or Stanton number
From Eqs. 43 and 46, we nd that
2
k
Cd =
ln(z/z0 )
1
k2
=
Ce =
ln(z2 /z0 ) ln(z1 /z0v )
rav u(z1 )
(56)
(57)
where the subscripts 1 and 2 refer to the measurement height of wind speed and specic humidity
respectively.
Ce and Cd , either given the values of z0 and z0v , or by
u(z1 ) and q(z2 ). These terms are measured at ux towers (Ü
Figure 3.12) and are further used to deduce E . Given the need to know q0 (in ∆q ), the bulk method
is better suited for estimating of E over oceans, free water and snow/ice.
Bulk methods are based on the determination of
calibration based on the measurements of
q
Resistance models in dierent surface conditions
The principle is here to relate
q0
to the saturated specic humidity
at which the surface water, at temperature
T0 ,
qs (T0 ),
which is the specic humidity
vaporizes. In the following, for simplicity, we note
aerodynamic resistance to the water vapor ux.
17
ra
the
Saturated surfaces : on free water, pounded soils, and water intercepted by leaves (the corresponding
evaporation ux being often called the interception loss), we have
E = ρa
In such cases, the ux
q0 = qs (T0 ),
thus
qs (T0 ) − q(z)
ra
(58)
E is the maximum possible one given the states of the surface and the atmosphere,
potential rate.
Evaporation from soils : the saturation is assumed to be realized somewhere within the soil, at the
and we often speak of
same temperature
T0
as the soil surface. The water vapor ux is all the more reduced as soil moisture
decreases, because of increasing soil water succion (or negative pressure
introduce a
Esoil = ρa
rsoil '
300 s.m
−1
see section
??), so that we
qs (T0 ) − q(z)
ra + rsoil
(59)
A completely dry soil can exert a soil resistance as high as 30,000 s.m
creates
pw ,
soil resistance, which increases when soil moisture decreases :
−2
, so that a dry layer of 1 cm
.
Transpiration from leaves : the saturation is assumed to be realized within the stomatal chamber
(Ü Figure 3.13), at the same temperature T0 as the leaf surface. A stomatal resistance is introduced
to account for the slowing down of water vapor ux across stomates :
Etr = ρa
qs (T0 ) − q(z)
ra + rst
(60)
The stomatal resistance depends on the plant species, and on many environmental factors, which are
not necessarily independent from each other (Ü Figure 3.14) :
soil moisture, which inuences leaf water potential ;
CO2 concentration, radiation, and nutrient availability, by means of photosynthesis, which is closely
linked to transpiration ;
air temperature and vapor pressure decit (VPD =
The minimum
500 s.m
−1
rst
−1
is around 50 to 100 s.m
ea − es (Ta ))
; the order of magnitude in classical conditions can reach
−1
, what remains lower than the resistance of the cuticle (' 2000 to 4000 s.m
).
Complex canopy : we introduce the Leaf Area Index (LAI), If , which is the ratio of total projected
leaf area (one side only) per unit ground area (Ü Figure 3.15). We consider parallel uxes from each
level of leaf, leading to an
ecient resistance, here called re :
If
X 1
1
top
=
< If /rst
i
re
r
i=1 st
The inequality comes from the fact that
i
rst
(61)
increases from the top of the canopy toward the soil because
of reduced visible radiation for photosynthesis. Note also that the direct use of the above equations
with
re
implies that all levels share the same surface temperature
T0 ,
what is not true is reality, but
can be in simplied models.
The structure of a complex canopy can also slow down the vertical water vapor ux (whether interception loss, transpiration, or underlying soil evaporation), what can be described owing to a
resistance :
E = ρa
The values of
rg
qs (T0 ) − q(z)
ra + rg + re
are generally between 0 and 30 s.m
is often between 100 and 500 s.m
−1
−1
18
(62)
, and the overall
, but varies with time, like
geometric
rst .
canopy resistance rc = re + rg
Resistance approach for land surface modeling. This approach is routinely used in climate models
and in some hydrology models. In such models, we cannot measure
T0 ,
and we need to calculate it,
E , and on
H . This ux too depends on T0 , and can be described using the same resistance approach.
The same aerodynamic resistance ra is usually assumed for E and H . A suciently small time step
(ca. 30 min) is needed to correctly describe the diurnal cycle variability. Moreover, E depends on soil
as a result of the surface energy budget (see section 3.5), which depends on the unknown
sensible ux
land surface
models (LSMs) solve jointly the coupled water and energy budgets of land surface units,
moisture, which depends on precipitation but also on soil moisture, like runo. Thus,
usually based on nite-dierence methods. Usually, the inuence of the soil and vegetation properties
are taken into account, what leads to distinguish
Etr , Ei , Esoil ,
and
Esn .
In such a case, we speak of
soil-vegetation-atmosphere transfer (SVAT) models Ü Figure 3.16.
3.5 The Earth's energy balance
q
Top of the atmosphere (TOA)
By means of energy conservation in the Earth system, we have a long-term balance between the radiation
input and output at the TOA
Ü
Figure 3.17.
The input radiation is solar radiation. It mostly covers the ultraviolet to near infrared wavelengths (we
speak of
shortwave
4
the Earth albedo ,
radiation), and on average, the corresponding energy is about
aT OA '
341W.M−2 , lowered by
0.3, on average.
The output radiation is emitted by the Earth surface and atmosphere. Given the mean Earth temperature
ca. 14°C), and Wien displacement law 5 , this radiation mostly belong
speak of longwave radiation). The corresponding energy depends on the
(global mean surface temperature
to the infrared spectrum (we
6
7
temperature of the Earth surface and atmosphere , and on the green-house eect .
Radiation balance at the Earth's surface
We can dene the net radiation Rn from the balance between downward/upward radiation (with s and
q
u
indices) in the shortwave and longwave spectra (with
s
and
l
indices) :
Rn = Rsd (1 − as ) + εs Rld − Rlu
The mean surface albedo,
(63)
as , is about 0.05, but it varies a lot over time and space, as local albedo depends
Ü Figure 3.18).
a lot on the surface type, with values ranging from 0.03 for water to 0.95 for fresh snow
The surface emissivity
εs
is used for the longwave absorptivity of the surface (values between 0.95 and
cf. Brutsaert p64).
Rld depends on radiative transfer, i.e. the complex interplay between absorption and emission of longwave
0.99 for natural surfaces,
radiation in the atmosphere, and
Rlu
depends on
Ts
:
Rlu = εs σTs4
q
(64)
Energy budget of a surface layer
The surface layer may consist of water, soil, plant canopy, snow,
etc., and it can either be innitesimally
thin, or thicker (lake for instance). It is characterized by its depth
by an average temperature
Ts .
∆z ,
its specic heat capacity
cs ,
and
If neglect snow processes, the variation of its heat content (or enthalpy)
per unit surface is :
∂Ts
= Rn − LE − H − G
(65)
∂t
4. Albedo is the ratio of radiation reected by a surface to incident radiation upon it (dimensionless)
5. Wien displacement law says the wavelength of radiation emitted by a black body is inversely proportional to its
absolute temperature T
6. Stefan-Boltzman law states the total energy radiated per unit surface area of a black body of temperature T across
all wavelengths per unit time is σTs4 , where σ is the Stefan-Boltzman constant.
7. Greenhouse eect is a process by which infrared radiation from the Earth surface is absorbed by atmospheric greenhouse gases, then re-radiated in all directions
ρ cs ∆z
19
Eq. 65 makes the link between the surface energy budget and the evolution of the surface temperature.
At steady-state, thus if we reach an equilibrium temperature, or if
∆z
is small enough, it reduces to :
Rn = LE + H + G
The ground heat ux,
G,
(66)
is positive when the surface is warmer than the ground, else negative, so that
it is small when averaged over the diurnal cycle or the annual cycle. In such a case, we nd that the net
radiative ux is balanced by the turbulent uxes, namely the latent and sensible heat uxes :
Rn ' LE + H
(67)
Given the values in Figure 3.17, we get that the latent heat ux allows to dissipate 50% of absorbed solar
radiation, and 80% of net radiation at the surface.
3.6 Formulations related to the surface energy budget
The following formulas have in common to use approximations in the energy budget to get rid of
aerodynamic method. This allows the
T0
in the
calculation of E from standard near-surface meteorological
observations, without the need to solve the coupled energy and water budget at a short time step. They
are thus frequently used in operational hydrology.
q
Case of saturated surfaces
A milestone expression was proposed by Penman (1948). It is often referred to as a
combination equation,
describing the radiative (1) and advective/areodynamic (2) controls of the evaporation ux.
EPenman =
∆ Rn
γ
+
EA
∆+γ L
∆+γ
| {z } | {z }
(1)
(68)
(2)
where
∆ is the derivative of es (T ), des /dT , at Ta
γ is the psychrometric constant : γ = (cp pa )/(L). At 20°C and pa = 1 atm, γ = 0.67 hPa.K−1 , and
many users assume a constant γ
EA is the drying power of the air : EA = f (u)(ρa /pa )(es (Ta ) − ea ), f (u) being a function of the
horizontal wind speed at a certain height. There are dierent expressions of the wind function f (u),
and Penman (1948) calibrated it for free-water and saturated vegetation covers.
q
Important denitions
Potential evaporation : evaporation from a large uniform surface that is suciently wet so that the air
is saturated at the surface (ex : free water, soil or vegetation cover after a rain shower). This quantity
does not depend on the soil/vegetation characteristics, apart from their roughness and albedo, thus
climatic evaporation demand ".
corresponds to the concept of "
Potential ET (Thornthwaite, 1948) : maximum ET from a large area covered completely and uniformly
by an actively growing vegetation with a non-limiting soil moisture supply
ET0
rc = 70 s.m−1
Reference ET,
0.23,
q
: idem, for a reference grass, with specic properties : height = 0.12 m, albedo =
(=
rcmin
as there is no stress)
Reference ET
Météo-France produces time series and maps of
f (u)
ET0
using the Penman equation with a wind function
calibrated for the reference grass :
ET0−Penman−MF =
γ 0.26(1 + 0.4u10 )(es (Ta ) − ea )
∆
Rn +
∆+γ
∆+γ
τ
20
with :
Rn in W.m−2
u10 , the wind speed at 10 m above the surface, in m.s−1
es et ea in hPa ( ! !)
τ = 86400 s
ET0 in mm/s, with daily totals bounded between 0 and 9
mm/d.
The FAO recommends using the Penman-Monteith equation (Monteith, 1965) for unstressed reference
grass, which introduces the minimum stomatal resistance
where
u2
−1
is the wind speed at 2 m in m.s
r0 = = 70 s.m−1 , and a resistance ra = 208/u2 ,
:
ET0−PM =
∆ RLn + γ EA
0
∆ + γ rar+r
a
(69)
A simpler formula is the Priestley-Taylor equation (Priestley and Taylor, 1972), where the aerodynamic
term is replaced by
α>1
:
ET0−P T = α
α = 1.75
α = 1.25
q
∆ Rn − G
∆+γ
Lv
if arid climate (low accuracy)
if humid climate (fair accuracy)
From reference to actual ET
Maximum ET,
ETM ,
is the maximum possible ET at a given time step given the climatic conditions and
vegetation properties. It is thus equivalent to the potential ET for the selected vegetation cover.
For a generic vegetation cover, it is linked to
ET0
by a crop coecient
Kc ,
which accounts for the
dierences in albedo, physiology (rsmin ), height and roughness, with the reference crop :
ETM c = Kc ET0
Kc
(70)
coecient cultural ), and it is time dependent Ü Figure 3.19
is called the crop coecient (
Actual ET (ET
réelle
en français) is the eective ET at a given time. For a given vegetation cover,
is reduced compared to
ETM c
ETc
because of environmental stresses, including soil moisture stress (Ü Figure
3.20) :
ETc = βETM c < ETM c = Kc ET0
β<1
Note that the eect of
Kc
(71)
(72)
and of the environmental stresses on
resistances within the Penman-Monteith equation 69.
21
ETc
can also be described by appropriate
4 Water in soils
5 Water in rivers
6 Water cycles under human inuences
Références
Allen, R., Pereira, L., Raes, D., and Smith, M. (1998). Crop evapotranspiration - guidelines for computing
crop water requirements - fao irrigation and drainage paper no. 56. Technical report, Food and Agriculture
Organization of the United Nations, Rome.
Entekhabi, D., Rodriguez-Iturbe, I., and Castelli, F. (1996). Mutual interaction of soil moisture state and atmospheric processes. Journal of Hydrology, 184(1) :317.
Falkenmark, M. and Rockström, J. (2006). The new blue and green water paradigm : Breaking new ground for
water resources planning and management. Journal of water resources planning and management, 132(3) :129
132.
Jeandel, C. and Mosseri, R. (2011).
Editions.
. CNRS
Le climat à découvert. Outils et méthodes en recherche climatique
Monteith, J. (1965). Evaporation and environment. In 19th
volume 19, pages 205234. University Press, Cambridge.
Symposia of the Society for Experimental Biology
,
Oki, T. and Kanae, S. (2006). Global hydrological cycles and world water resources. science, 313(5790) :10681072.
Penman, H. (1948). Natural evaporation from open water, bare soil and grass. Proceedings
of London. Series A, Mathematical and Physical Sciences, 193(1032) :120145.
of the Royal Society
Priestley, C. and Taylor, R. (1972). On the assessment of surface heat ux and evaporation using large-scale
parameters. Monthly weather review, 100(2) :8192.
Stull, R. (1988).
An Introduction to Boundary Layer Meteorology
. Kluwer Academic Publishers.
Thornthwaite, C. W. (1948). An approach toward a rational classication of climate.
38(1) :5594.
,
Geographical review
Trenberth, K., Smith, L., Qian, T., Dai, A., and Fasullo, J. (2007). Estimates of the global water budget and its
annual cycle using observational and model data. Journal of Hydrometeorology, 8(4) :758769.
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