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1. Evapotranspiration
Evapotranspiration is a collective term for all the processes by which water in liquid or solid phase at or near
the earth’s land surfaces becomes atmospheric water vapor. The term includes both evaporation (from
rivers, lakes, bare soil, and vegetative surfaces) and sublimation (from ice or snow fields). Globally, about 62%
of the precipitation that falls on the continents is evapotranspired. Of this, about 97% is evapotranspiration
from land surfaces and 3% open‐water evaporation. Evapotranspiration exceeds runoff in most of the river
basins on earth.
The following is a mass balance approach for determining the total volume of water that is being evaporated,
vol
E , within a catchment area,
vol in in vol out out
vol vol in in out out
V
W SW GW E SW GW V
E W SW GW SW GW V
I O
where I is the total quantity of water entering the catchment, O is the total quantity of water exiting the
catchment, Δ V is the change in storage,
vol
W is the volume of water entering the catchment through
precipitation,
in
SW is the surface water input,
in
GW is the groundwater input,
out
SW is the surface water
output, and
out
GW is the groundwater output. Evaporation is typically solved for using Eq. 1.1.
Evaporation physics includes the boundary layer mass transport theory, diffusion/dispersive processes, and
Fickian mathematics.
Figure 1.1 : Evaporation over a water surface.
2. Water Vapor
Evaporative processes are affected by the amount of water vapor in the air near the evaporating surface and
there are several ways of expressing that amount. By virtue of their molecular motion and collisions, each
constituent of a mixture of gases exerts a pressure called a partial pressure which is proportional to its
concentration. The partial pressure of water is called the vapor pressure , e. Saturated vapor pressure , e
∗
, is
the maximum vapor pressure that is thermodynamically stable and is given by
2
0.611exp
T F
e
T L
∗
where e
∗
has units of kPa and T is temperature in units of C
D
. A phase diagram is given in Fig. 2.1.
Figure 2.1 : Phase diagram of water.
Absolute humidity (a.k.a. vapor density ),
v
ρ , is the mass concentration of water vapor in a volume of air
v v
ρ = m V. Note that vapor density is lighter, or less dense, than dry air. The water molecule H
2
O weighs
less than a molecule of air N 2
O
2
. The ideal gas law provides the relationship between vapor pressure and
absolute humidity and is expressed as
2
2
M L
PV n R T
T
where P is the absolute pressure of the gas, V is the volume of gas, n is the number of moles of gas, R is
the universal gas constant, and T is the absolute temperature. The ideal gas law for air is written as
a a a
a v v
R e e R T
T ρ ρ
If 0.
a
R = (Eq. 2.3) then Eq. 2.9 may be reduced to
a a
v v
e T e T
where e is in kPa ,
a
T is in K ,
v
ρ is in
3
kg m
−
. Let us now rewrite Eq. 2.9 substituting for
a
R from Eq. 2.3,
a a a v
v a a a a a
e P T P e
q
T P
ρ
ρ ρ ρ ρ
where q is specific humidity , the ratio of water vapor to air (including water vapor and dry air) in a particular
volume. If the temperature and vapor pressure of a parcel of air lie below the curve representing saturation
(Fig. 2.1), the air is unsaturated, and its relative humidity ,
a
W , is the ratio (usually expressed as a percent) of
its actual vapor pressure at air temperature,
a
e , to its saturation vapor pressure:
a
a
a
e
W
e
∗
3. Physics of Evaporation
In Fig. 3.2, dry air with temperature of
a
T lies above a horizontal water surface with a temperature of
s
T. The
molecules at the surface are attracted to those in the body of the liquid by hydrogen bonds, but some of the
surface molecules have sufficient energy to sever the bonds and enter the air. The number of molecules with
this escape energy increases as
s
T increases. The water molecules entering the air move in random motion,
and as these molecules accumulate in the layer of air immediately above the surface, some will re‐enter the
liquid. 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 and
the vapor pressure of the overlying air,
a
e , that is,
s a
E e e
∗
where E is the rate of evaporation transfer ,
a
e is measured at some representative height, and the
proportionality depends on that height and on the factors controlling the diffusion of water vapor in the air.
Figure 3.1 : Schematic diagram of flux of water molecules over a water surface.
Depending on the temperature of the surface and the temperature and humidity of the air, the difference
between the two vapor pressures can be positive, zero, or negative. If
s a
e e
∗
, evaporation is occurring; if
s a
e e
∗
< , there is water condensing on the surface; and if
s a
e e
∗
= , neither condensation or evaporation is
occurring. Evaporation will occur even if the relative humidity equals 100% ( 1)
a a
e e
∗
= , as long as
s a
e e
∗ ∗
However, under these conditions the evaporating water will normally condense in the overlying air to form a
fog or mist.
The latent heat of vaporization ,
v
λ , is the quantity of heat energy that must be absorbed to break the
hydrogen bonds when evaporation takes place. Thus, evaporation is always accompanied by a transfer of heat
out of the water body. Because of this coupling, the rates of latent‐heat and water mass transfer are directly
proportional and given by
v
E M E
LE E
M T T
(3.2a)
where LE is the rate of latent heat transfer. Note that if E is expressed in dimensions of [ L T ], Eq. 3.
becomes
w v 3 2
M E L E
LE E
L M T L T
(3.2b)
4. Physics of Turbulent Transfer Near the Ground
The land surface is the lowest layer of the atmosphere in which the winds are affected by the frictional
resistance of the surface. The frictional resistance produces turbulent eddies, which are irregular and chaotic
motions with vertical components (Fig 4.1). These vertical components are the means by which water vapor
where
a
T is the air temperature, and
b
T is an arbitrary base temperature, usually taken as 0 C
D
. Dividing
both sides of Eq. 4.3 by the air volume,
a
V , gives
( ) [ ]
3 3
a
a a a a b
a a
H m M E E
c T h c T T
V V L M L
where h is the concentration of sensible heat , and
a
ρ is the mass density of air.
The momentum exchange between the atmosphere and the land surface is based on boundary layer theory.
The classical approach to boundary layer theory is a no‐slip boundary condition (zero velocity) at the ground
surface. The wind speed distribution u is dependent on wind speed conditions. For laminar flow conditions
(Fig. 4.2a), slow flow of wind, the horizontal shear stress , τ , due to differences of wind velocity at adjacent
levels may be expressed as
2 2
or
du M M F
dz LT T LT L
where μ is dynamic viscosity, and z is the vertical distance above the ground surface. Note that both τ and
u are functions of z. If flow conditions are turbulent (Fig. 4.2b), higher wind velocities, the dynamic viscosity
within Eq. 4.5 is replaced by an empirical constant known as the eddy viscosity, ε. Note that eddies typically
result in significant deviations in the flow direction.
Figure 4.2 : Particle paths in (a) laminar flow and (b) turbulent flows.
Because of turbulent eddies, the instantaneous horizontal wind velocity at any level,
a
v , fluctuates in time,
and we can separate the instantaneous velocity into a time‐averaged component,
a
v , and a deviation from
that average,
a
v
, caused by the eddies:
a a a
L
v v v
T
Time‐varying vertical air velocity, designated
a
w , are similarly separted into a time‐averaged component,
a
w ,
and an instantaneous deviation form that average,
a
w
a a a
w = w + w ′ (4.7)
The intensity of turbulence can be characterized by a quantity called the friction velocity ,
1 2
a a
u v w
∗
where the minus sign is required because simultaneous values of
a
w
and
a
v
have opposite signs. Dimensional
analysis shows us that at a wall a characteristic velocity (friction velocity), u
∗
, can be defined as
1 2
a
L
u
T
∗
Therefore, the shear stress may be expressed as a function of friction velocity as follows:
2
2
3 2 2 2
or
a
M L M F
u
L T LT L
τ ρ
∗
The Log wind profile is a semi‐empirical relationship used to describe the vertical distribution of horizontal
wind speeds above the ground within the atmospheric surface layer. The relationship is well described in the
planetary boundary layer literature. The logarithmic profile of wind speeds is generally limited to the lowest
100 meters of the atmosphere (i.e., the surface layer of the atmospheric boundary layer). The equation to
estimate the wind speed ( )
a
v at height z above the ground is:
0
0
ln for
d
a d
u z z
v z z z
k z
∗
where k is a dimensionless constant,
d
z is called the zero‐plane displacement ,
0
z is the roughness height
(Fig. 4.3). Eq. 4.11 is known as the Prandtl‐von Karman Universal Velocity‐Distribution for turbulent flows.
Experimental data have shown that k 0.4and that the values of
d
z and
0
z are approximately proportional
to the average height of the vegetation or other roughness elements covering the ground surface.
layer into the air above. Therefore, the evaporation rate is equivalent to the flux of water vapor and Eq. 5.
becomes
v
z v
d
F V E D
dz
Likewise, since the upward latent‐heat transfer rate, LE , is equivalent to the flux of latent heat in the z
direction, ( )
z
F LE , we can write
z
F LE = LE (5.4)
Since latent heat and water vapor are directly coupled (Eq. 3.2a), the diffusion equation for latent heat ( LE )
is
2
4 2
v
z v v z v v
v
z v v
d
F LE E F V D
dz
d L E M E
F LE D
dz T M L L T
Recall from Eq. 2.11 that
v a
v
a a a
e e
P P
and taking the derivative of Eq. 5.6 with respect to z gives,
v a a
a a
d d e de
dz dz P P dz
ρ ρ ρ
Eq. 5.7 can be used to rewrite Eq. 5.5 as
2
a
z v v
a
de E
F LE D
P dz L T
ρ
λ
The upward latent‐heat transfer rate, LE , (Eq. 3.2b) is equivalent to ( )
z
F LE , therefore,
2
a
w v v v
a
de E
LE E D
P dz L T
ρ
ρ λ λ
The diffusion equation for sensible heat , ( H ), is
2
4 2
z H
dh L E E
F H D
dz T L L T
where ( )
z
F H is the flux of sensible heat in the z direction, and
H
D is the diffusivity of sensible heat in air ,
and h is the concentration of sensible heat. Substituting the concentration of sensible heat from Eq. 4.4 into
Eq. 5.10 gives
z H a a a b
d
F H D c T T
dz
where
a
c , is the heat capacity. Assume that the arbitrary base temperature,
b
T , is 0 C
D
and that the density
and heat capacity are essentially constant under prevailing conditions gives
a
z H a a
dT
F H D c
dz
The flux of sensible heat is equivalent to the rate of sensible‐heat transfer, therefore,
a
z H a a
dT
F H H H D c
dz
Recall that the momentum equals mass times velocity, so that the concentration of momentum (momentum
per unit volume) at any level equals the mass density of the air time the velocity,
a a
ρ ⋅ v. The diffusion
equation for momentum , ( M ), is
a a
z M
d v
F M D
dz
where ( )
z
F M is the flux of momentum in the z direction, and
M
D is the diffusivity of momentum in
turbulent air. Assuming constant air density gives,
2
3 2
a
z M a
dv L M M
F M D
dz T L T LT
ρ
The flux of momentum is equivalent to the negative shear stress, τ , therefore,
2
z
M
F M
LT
τ
From Eq. 4.10 shear stress is directly proportional to the square of the friction velocity:
2
z a
F M ρ u
∗
Combining Eq. 5.15 and Eq. 5.17 gives
2
2 2 a a
M a a M M
a
dv dv u
D u D u D
dz dz dv dz
∗
∗ ∗
subsides, it also changes temperature. It warms up, and it is warming up at the dry adiabatic lapse rate.
Make sure you notice that we are talking about moving air (rising or subsiding), not still air. What happens to
the relative humidity (Eq. 2.12),
a a a
W e e
∗
≡ , of a parcel of air when the temperature decreases? The relative
humidity increases because the saturated vapor pressure,
a
e
∗
, decreases for a decrease in temperature (Fig.
2.1). If the air is rising and cooling at a rate of 1 C 100 m
D
, eventually, it's going to cool off enough for the
relative humidity to reach 100%, and condensation can take place. The dew point is the temperature at which
the air becomes saturated and condensation takes place. The lifting condensation level is the altitude at
which condensation begins. You can look up at the windward sides of mountains and see where the lifting
condensation level is, because that is where you will see the bases of clouds that have formed. If
condensation is taking place, latent heat is being released to the surrounding air. So you have two opposing
trends going on at the same time within this parcel of air. It's rising and cooling, but it's also condensing and
being warmed. If water vapor in the air is condensing, the adiabatic rate is less. The air is only cooling off at a
rate of about 0.5 C 100 m
D
. This is called the wet adiabatic lapse rate.
Fig. 6.1 shows unstable, neutral, and stable lapse rates near the ground. The lapse rate is the negative of the
actual change of temperature with altitude for the stationary atmosphere (i.e. the temperature gradient,
a
− dT dz ). When a parcel of air is transported upward in a turbulent eddy, it cools abatically (i.e. without the
loss of heat). Thus if the actual lapse rate is steeper than adiabatic (unstable), the air in the eddy is warmer
and, hence, less dense than the surrounding air and will continue to rise due to buoyancy, enhancing vertical
transport of heat. If the actual lapse rate is less than the adiabatic (stable), the air in the eddy will be cooler
and denser than the surroundings and will sink toward the surface, reducing vertical transport.
Figure 6.1 : Unstable, neutral (adiabatic), and stable lapse rates near the surface.
Under neutral conditions, the diffusivities of water vapor and sensible heat are identical to the diffusivity of
momentum (i.e., 1
V M
D D = and 1
H M
D D = ), because the same turbulent eddies are responsible for the
transport of all three quantities. However, under unstable conditions, the vertical movement of eddies is
enhanced beyond that due to the wind velocity, and there can be significant vertical transport of water vapor
and/or sensible heat but little transport of momentum, so 1
V M
D D > and 1
H M
D D >. These conditions
typically occur when wind speed is low and the surface is strongly heated by the sun. Conversely, when the
lapse rate near the ground is stable, turbulence is suppressed and 1
V M
D D < and 1
H M
D D <. This situation
is typical when warm air overlies a cold surface, such as a snowpack.
Let us first examine latent heat and sensible heat transfer under neutral lapse‐rate conditions,
v H M
D = D = D.
For the rate of latent heat transfer we can replace
V
D in Eq. 5.9 with
M
D from Eq. 5.21 to give
2 2
2
2
a a a
M v d v
a a
a a
d v
a
de dv de
LE D k z z
P dz dz P dz
dv de
LE k z z
P dz dz
2
2 2 1 2 1
2
1
ln
a a
d
d
k
H c v v T T
z z
z z
= − ρ ⋅ ⋅ ⋅ − ⋅ −
Again, we take level 1 as the nominal surface and write
2
2
0
ln
a a a a s
a d
k
H c v T T
z z
z
Let us now relax the assumption of neutral lapse‐rate conditions. To account for the effects of non‐neutral
lapse rates we rely upon stability‐correction factors for momentum, water vapor, and sensible heat,
designated
M
V
Φ , and
H
Φ , respectively. The stability correction factors are incorporated into the general
equations for latent‐ and sensible‐heat transfer. For Latent heat and evaporation (Eq. 6.2):
2
2 2 1 2 1
2
1
ln
a
v
a M V
d
d
k
LE v v e e
P
z z
z z
For Sensible heat (Eq. 6.5):
2
2 1 2 1 2
2
1
ln
a a
M H
d
d
c k
H v v T T
z z
z z
The Φ factors are related to the stability condition of the atmosphere, which is characterized by the
dimensionless Richardson number , Ri , given by
[ ][ ]
[ ]
[ ]
2
2 1 2 1
2 2 2
2 1 2 1
LT L
g z z T T
Ri
L T
T T v v
−
−
where g is the acceleration due to gravity. The Richardson number expresses the ratio of potential to kinetic
energy. Neutral conditions exist when Ri = 0 , stable conditions when Ri < 0 , and unstable conditions when
Ri > 0
. To determine Φ ‐factor values, first calculate
Ri , then use the relations given in Table 6.1.
Table 6.1 : Formulas for stability factors for computing latent‐ and sensible‐heat transfer as functions of Richardson number.
Stability Factor Ri < −0.03 −0.03 ≤ Ri ≤ 0 0 ≤ Ri ≤0.
M
1 4
1 18 Ri
−
1 4
1 18 Ri
−
1
1 5.2 Ri
−
V
H
1 4
1.3 1 18 Ri
−
1 4
1 18 Ri
−
1
1 5.2 Ri
−
7. Evaporation
Knowing that the rates of latent –heat and water mass transfer are directly coupled (Eq. 3.2b) and given the
equation for latent‐heat transfer (Eq. 6.2) allows the evaporation rate, E , to be expressed as
2
2 2 1 2 1
2
1
2
2 1 1 2 2
2
1
ln
ln
a
v
w v w v a M V
d
d
a
a w M V
d
d
LE k
E v v e e
P
z z
z z
k
E v v e e
P
z z
z z
Again, we take level 1 as the nominal surface and write
2
2
0
ln
a
a s a
a w M V
a d
k
E v e e
P
z z
z
