Ground Surface Moisture Flux Boundary Conditions - Unsaturated Soil Mechanics in Engineering Practice - page 303

were taken from 65 weather stations located between 7 ◦
and 50 ◦ North latitude in the United States. The following
relationship between temperature difference during each day

formulation for PE. The Thornthwaite equation depends on
average monthly temperatures whereas the Penman equation
depends on daily temperatures and other weather-related
factors.
Thornthwaite (1948) incorporated the variables of length
of daylight hours, mean monthly temperature, and an empir-
ical constant into the prediction of PE. Penman (1948) uti-
lized daily records of relative humidity, air temperature,
wind speed, and net radiation for the calculation of PE. The
vapor pressure gradient between the water surface and the
air above the water becomes the primary driving mechanism
for evaporation in the Penman equation. Two components
drive evaporation: the net radiation component that char-
acterizes the power of the sun to evaporate water and the
mixing component associated with the power of wind to
remove vapor from the ground surface.
The Penman equation would intuitively appear to have a
distinct advantage over the Thornthwaite equation in pre-
dicting PE. A comparison of the two calculation methods
provides an indication of the reliability of the calculation
methods.
Figures 6.33 and 6.34 present daily PE calculations for
two years (i.e., 2006 and 2007) at a site in northern Canada
(latitude of approximately 50 ◦ ). The daily PE calculations
vary considerably from day to day when using the Penman
(1948) equation. This is as anticipated since temperature,
relative humidity, and wind speed can vary quite widely on
a daily basis.
The Thornthwaite (1948) equation simply uses the aver-
age monthly temperature when calculating PE. Daily PE
is a single value for each month of the year when using
the Thornthwaite equation. The daily PE calculations can
be accumulated over the entire year in order to determine
whether the monthly averaging of temperature is a reason-
able assumption to make when calculating PE.
The Thornthwaite (1948) equation is not defined when the
average monthly temperature is less than 0 ◦ C. The Thorn-
thwaite calculations show that the winter period (i.e., an
inactive period), is between November 1 and April 1 of
each year at this site. The Penman (1948) equation does not
restrict the calculation of PE to the same period. As much as
25% of PE may occur outside the period between November
1 and April 1 in each year.
Figures 6.33 and 6.34 also show that the PE calculations
from the Thornthwaite equation lag behind the Penman cal-
culations for the period from spring to midsummer. The
reverse is true for the period from midsummer to the fall of
each year.
Figures 6.35 and 6.36 show the accumulated PE calcu-
lations for two years (i.e., 2006 and 2007) at the site in
northern Canada. The cumulative precipitation for each year
is also shown. Cumulative precipitation values illustrate the
relationship between PE and precipitation. The cumulative
PE at the end of the year is quite similar for the two methods
of calculation. However, the difference between the results

T max −

T min and the empirical coefficient K t was found to
be related by the equation

0 . 00185 T max − T min 2 0 . 0433 T max

K t = −

T min + − 0 . 4023 (6.47)

6.3.17 Relative Humidity of Air above Water Surface
The vapor pressure in the air above a saturated soil surface
is required in the calculation of the net radiation term and
the mixing term. The relative humidity in the air above a sat-
urated soil surface provides a measure of the vapor pressure
in the air:

u air
v
u air
v 0

h r = (6.48)

or

u air
v

h r u air

= (6.49)

v 0

where:

h r = relative humidity of the air above the saturated

ground (or water) surface,

u air
v 0

= saturated vapor pressure, kPa, and

u air
v

= vapor pressure in the air above ground surface,

kPa.

The above equations use the saturated vapor pressure com-
puted from the measured air temperature while the actual
vapor pressure in the air is computed from the measured
relative humidity of the air above the ground surface.
Other assumptions can be made when solving the Penman
equation for potential evaporation. For example, it is some-
times assumed that the vapor pressure is the same in the
air and at the saturated soil surface (e.g., Priestley-Taylor,
Eq. 6.23). When this assumption is made, the calculated
potential evaporation is typically reduced in drier climate
regimes.
The Penman (1948) equation has been widely used in
geotechnical engineering, and as a result it is important to
understand the role played by each of the variables in the
equation. It has been noted that the Penman equation gen-
erally overestimates the evaporation rate for conditions of
high wind and low humidity.

6.3.18 Comparison of Potential Evaporation
Calculations
The Thornthwaite (1948) equation and the Penman (1948)
equation rely on data collected at a weather station; however,
the Penman formulation is considered to be a more rigorous