Hydrol. Earth Syst. Sci., 15, 267–277, 2011
www.hydrol-earth-syst-sci.net/15/267/2011/
doi:10.5194/hess-15-267-2011
© Author(s) 2011. CC Attribution 3.0 License.
Hydrology and
Earth System
Sciences
Modelling hourly rates of evaporation from small lakes
R. J. Granger and N. Hedstrom
National Water Research Institute, Environment Canada, Saskatoon, Canada
Received: 20 January 2010 – Published in Hydrol. Earth Syst. Sci. Discuss.: 4 May 2010
Revised: 13 December 2010 – Accepted: 21 December 2010 – Published: 21 January 2011
Abstract. The paper presents the results of a field study of
open water evaporation carried out on three small lakes in
Western and Northern Canada. In this case small lakes are
defined as those for which the temperature above the water
surface is governed by the upwind land surface conditions;
that is, a continuous boundary layer exists over the lake, and
large-scale atmospheric effects such as entrainment do not
come into play. Lake evaporation was measured directly using eddy covariance equipment; profiles of wind speed, air
temperature and humidity were also obtained over the water
surfaces. Observations were made as well over the upwind
land surface.
The major factors controlling open water evaporation were
examined. The study showed that for time periods shorter
than daily, the open water evaporation bears no relationship
to the net radiation; the wind speed is the most significant
factor governing the evaporation rates, followed by the landwater temperature contrast and the land-water vapour pressure contrast. The effect of the stability on the wind field
was demonstrated; relationships were developed relating the
land-water wind speed contrast to the land-water temperature contrast. The open water period can be separated into
two distinct evaporative regimes: the warming period in the
Spring, when the land is warmer than the water, the turbulent fluxes over water are suppressed; and the cooling period,
when the water is warmer than the land, the turbulent fluxes
over water are enhanced.
Relationships were developed between the hourly rates of
lake evaporation and the following significant variables and
parameters (wind speed, land-lake temperature and humidity
contrasts, and the downwind distance from shore). The result is a relatively simple versatile model for estimating the
Correspondence to: R. J. Granger
([email protected])
hourly lake evaporation rates. The model was tested using
two independent data sets. Results show that the modelled
evaporation follows the observed values very well; the model
follows the diurnal trends and responds to changes in environmental conditions.
1
Introduction
Evaporation from open water bodies is an important component of the hydrologic cycle for many watersheds. This
is particularly true for boreal and northern regions; for example, in the Western Canadian Boreal region open water
represents from 10% to 15% of the surface area; in portions
of the Canadian Shield open water can represent as much as
20% of the surface area. Open water bodies are distributed
throughout these regions; they appear in sizes ranging from
small ponds to “great lakes”. Many of these lakes act as storage features in complex drainage basins; they can in fact become disconnected and isolated during extended periods of
drought, in which case their hydrology becomes dominated
by the vertical processes of precipitation and evaporation.
The correct representation of the hydrological function of
these water bodies is important to the hydrological modelling
of these watersheds. Also, the effect of these water bodies on
the regional climate needs to be correctly incorporated into
the atmospheric and climate models. Since most hydrological and meteorological models operate with time steps of
the order of an hour, a reliable approach to the calculation of
hourly lake evaporation is necessary to both objectives.
Even though, evaporation from open water remains largely
unmeasured as a course of routine, it is still estimated with
limited confidence. The major source of difficulty is the fact
that the required meteorological variables are rarely measured over the water surfaces, and the thermal lag between
the lake and land surfaces renders the use of land-based
Published by Copernicus Publications on behalf of the European Geosciences Union.
268
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
measurements alone ineffective in the parameterization of
open water evaporation.
Energy budget approaches, such as Morton’s (1983) complementary relationship lake evaporation model have not
proven to be reliable for short-term applications. An appropriate formulation of the transfer processes occurring in the
advective boundary layer is required. This was demonstrated
by Weisman and Brutsaert (1973), who applied analytical
solutions to the advection problem over open water for the
unstable case. Blanken et al. (2000), in a study of evaporation over Great Slave Lake, showed that for daily periods,
open water evaporation is governed by the wind speed and
the vapour gradient over the water. Granger (2000) showed
that lake evaporation is largely uncoupled from (or unsynchronized with) the land surface evapotranspiration. The
land surface processes closely follow the patterns of energy
supply, and the partitioning of the net shortwave radiation is
straightforward; the soil heat flux tends to be relatively small
for most situations and the turbulent fluxes of sensible and latent heat, for the most part, behave in a similar manner. The
partitioning of energy at a lake surface, on the other hand, is
more complex. Because of shortwave radiation penetration,
heat storage effects can be significant. The turbulent fluxes
of sensible and latent heat are not necessarily in phase with
the radiant energy supply, but are governed by the gradients
of temperature and humidity in the boundary layer. These
gradients are controlled both by water surface temperatures,
which are affected by the release of stored energy and intermittent mixing of the water (Oswald and Rouse, 2004) as
well as by the processes occurring over the upwind land surface (heating or cooling of the air and evapotranspiration).
For these reasons, land surface data alone are insufficient to
parameterize the lake evaporation; water surface data are also
required.
Weisman and Brutsaert’s (1973) analysis demonstrated
that in an advective situation, as is the case for open water,
the evaporation rate is not uniform over the lake surface, but
can increase or decrease from the leading edge. Their analytical solution provided for an evaporation rate that is a function of the distance from the upwind shore. Morton (1983)
also provided a method for estimating the change in evaporation as a function of distance from shore for small water
bodies. Mahrer and Assouline (1993), using a meso-scale
model also demonstrated the horizontal variability of open
water evaporation.
Granger (2000), using lake evaporation observations made
over Big Quill Lake, in Saskatchewan, showed that the analytical solutions developed by Weisman and Brutsaert (1973)
can be applied when the boundary layer over the lake is unstable (temperature decreasing with height). For stable conditions, however, these analytical expressions did not work
well; and an analysis of the advection under stable boundary layer conditions, such as those encountered over a lake
during daytime heating of the adjacent land surface, is required. Assouline and Mahrer (1993) and Liu et al. (2009)
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
also showed that the open water evaporation is greatly affected by the stability of the overlying air, which is governed
by the land-lake temperature contrast.
There are limits to the lake size for which an approach such
as that of Weisman and Brutsaert (1973) can be expected to
apply. When a lake is large, meso-scale atmospheric effects
such as the entrainment of higher-level air can become significant. This study is limited to small lakes; that is, those
for which the boundary layer over the water surface remains
intact and essentially governed by the upwind land surface
conditions.
The purpose of the present study was to develop relationships between the short-term (hourly) evaporation rates
from open water and those significant variables: wind speed
over the lake, water surface temperature, the temperature and
vapour pressure gradients over the lake, atmospheric stability
and upwind fetch distance.
2
Study sites
Three lakes, providing a range of fetch distances from 150 m
to 11 000 m, were chosen for the study. The largest of these
is Crean Lake, located within the Prince Albert National Park
in Saskatchewan. A rocky spit extending from a narrow island, near the centre of the lake, provided a stable and secure
platform for the instrumentation tower, with access to nearly
the full spectrum of wind directions and a range of fetch
distances (distance from the upwind shore) from 3600 m to
11 000 m. Observations at Crean Lake were obtained for the
open water seasons from 2005 to 2008.
Landing Lake, near Yellowknife, North West Territories,
with the instrument tower established on a rock outcrop in the
lake, provided fetch distances ranging from 150 m to 900 m.
Observations at Landing Lake were obtained for the period
2007 to 2008.
For the 2009 season, the northern portion of Whiteswan
Lake (known as Whiteswan4), in Northern Saskatchewan
was also instrumented; the instrument tower was set up on a
long, narrow spit extending into the lake. This site provides
fetch distances from 300 m to 4000 m.
Figure 1 shows the shape and size of the study lakes, as
well as the position of the instrumentation on each lake. The
contour lines suggest the surrounding terrain roughness and
provide an indication of the elevation of each lake. Table 1
presents the geographical locations of the lake and land towers, as well as the size characteristics of the study lakes. For
each lake, the instrumentation included a direct measurement
of the latent and sensible heat fluxes using the eddy covariance technique; the equipment used consisted of a Campbell
Scientific CSAT three-axis sonic anemometer, coupled with a
KH2O krypton hygrometer. The eddy covariance instrumentation was typically deployed at a height of 3 m above the
water surface. The incoming and reflected short-wave radiation (Kipp & Zonen Silicon Pyranometers) and net all-wave
www.hydrol-earth-syst-sci.net/15/267/2011/
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
269
Table 1. Characteristics of three study lakes.
Lake
Crean Lake, PANP
Crean Lake, PANP
Landing Lake, NWT
Whiteswan4 Lake, Sask.
Observation period
Location of Flux Tower
2005
2006–2009
2007–2009
2009
54.06353◦ N, 106.19170◦ W
54.06353◦ N, 106.19170◦ W
62.55915◦ N, 114.41365◦ W
54.17640◦ N, 105.1642◦ W
1 1
1
2 2
2
Range of Fetch Dist.
Location of Land Tower
3600 m–11 000 m
3600 m–11 000 m
150 m–900 m
300 m–4000 m
53.8926◦ N, 106.12066◦ W
54.00806◦ N, 106.20035◦ W
62.59559◦ N, 114.43857◦ W
53.98711◦ N, 105.11773◦ W
Distance to Land Tower
19.2 km
6.2 km
4.2 km
21.3 km
Instrumented
showing
location
of
instrument
towers;
a) of
Crean
Lake,
3 3 Figure
1. 1.
Instrumented
Lakes
location
of
instrument
towers;
a)
Crean
Lake,
Princea) Crean(b)
Figure
Instrumented
showing
location
instrument
towers;
Lake,
Prince
3 Lakes
Figure
1.showing
Fig. 1. Instrumented
Lakes
showing
location
of
instrument
towers;
(a) Lakes
Crean
Lake, Prince
Albert
National
Park,Prince
Saskatchewan;
Landing
Lake, North West
Territory;
(c)
Whiteswan4
Lake,
Saskatchewan.
National
Park,
b) b)
Landing
Lake,
North
West
Territory;
Whiteswan4
4 4 Albert
Albert
National
Park,
Saskatchewan;
Landing
Lake,
North
West
Territory;
c)
Whiteswan4
Albert
National
Park,
Saskatchewan;
b)
Landing
Lake,c)North
West Territory; c) Whiteswan4
4 Saskatchewan;
Saskatchewan.
5 5 Lake,
Lake,
Saskatchewan.
5 Lake, Saskatchewan.
radiation (REBS Q7 radiometers) were also measured. Profiles were obtained consisting of at least two levels of air
temperature and humidity (calibrated and shielded HMP45
Vaisala RH and Temperature Probes) and wind speed (calibrated NRG cup anemometers). The lower levels of the
temperature/RH and wind speed sensors were established at
1.5 and 2.0 m, respectively; the height of the second level
varied, depending on the type and height of tower in use.
Wind direction and the infrared water surface temperature
were also measured. For each lake a water temperature profile was established near the centre of the lake. Near each
lake, an instrumented tower over the land surface provided
the shortwave and net all-wave radiation fluxes as well as the
wind speed, wind direction, air temperature and humidity.
All data were recorded on half-hour intervals.
www.hydrol-earth-syst-sci.net/15/267/2011/
2.1
Data control and analysis
The data from all sites were systematically examined for
quality and completeness. Particular attention was paid to
the turbulent fluxes of latent and sensible heat; the following corrections were applied to the eddy covariance measurements: coordinate rotation (Kaimal and Finnigan, 1994) the
WPL adjustment (Webb et al., 1980), adjustments for sonic
path length, high frequency attenuation and sensor separation
(Massman, 2000; Horst, 1997) and oxygen extinction for the
krypton hygrometer.
The Thorthwaite and Holzman (1939) aerodynamic equations with stability corrections were applied to the profile
1818
measurements of wind speed, air temperature
and vapour 18
pressure, providing a second, independent determination of
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
270
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
the latent and sensible heat fluxes. Granger (1991) showed
that these equations provide reliable estimates of latent heat
flux rates. These profile calculations allowed for additional
quality control and gap-filling of the direct eddy flux measurements of evaporation. Gaps in the eddy covariance flux
measurements, generally caused by rain events, represented
from 9 to 13% of the observations at the various sites. The
gap-filling thus allowed for the determination of seasonal totals; however, only the valid eddy covariance measurements
were used in the model development.
Wind speed, air temperature and humidity data from the
land tower provided the contrast between the land and lake
surfaces.
2.2
Controls on evaporation
Most evaporation models are based on the representation and
parameterization of one or more of the conditions required
for evaporation to occur; these are: the supply of water at the
surface; the supply of energy required for the phase change
from liquid to vapour; and the transport mechanism which
carries the water vapour away from the surface.
Understanding how these conditions are controlled in various situations is necessary for the correct parameterization of
the evaporation process. For land surface evaporation, most
models in fact represent all three conditions; the moisture
availability is parameterized using soil moisture or a “stomatal resistance”, the net radiation absorbed at the surface is
the available energy, and a vapour transfer function based on
wind speed is applied. For lakes, however, the supply of water at the surface is non-varying, and as such is not a useful
parameter. Also, since the short-wave radiation penetrates
the lake surface, the radiant energy is absorbed at depth and
is not immediately available for the phase change at the surface; there can be very little relationship between the available energy and the turbulent exchanges of heat and water
vapour over the lake surface for sub-daily time periods.
This leaves essentially the vapour transport mechanism to
work with in the development of a model for the hourly estimate of open water evaporation. The transport mechanism
is governed by the vapour pressure gradient above the surface, and the efficiency of the exchange is controlled by the
wind speed and the atmospheric stability; stability, in turn, is
a function of the temperature gradient above the surface.
However, the air temperature and humidity are rarely measured above the lake water surface. Meteorological stations
are invariably placed on land. For small lakes, however, since
the boundary layer generally remains intact and undisturbed
by large-scale atmospheric effects, the conditions over the
lake will be governed by the upwind land surface conditions.
To test this hypothesis, the land-lake temperature contrast
(air temperature over land minus water surface temperature)
was compared to the temperature gradient over the lake. At
Crean Lake and Landing Lake, these were of the same sign
for 95% and 92% of the hourly periods, respectively. These
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
values are quite large when one considers the thermal lag between the land and lake surfaces. They do suggest that the
land-lake temperature contrast can be used as a proxy for the
temperature gradient over water in the determination of the
atmospheric stability conditions over the lake. This is consistent with Badrinath et al. (2004) who suggested that for
small water bodies the land-lake temperature contrast exercises control over the turbulent fluxes over the water.
In order to determine which are the most significant variables affecting lake evaporation, the relationships between
the hourly evaporation rates over water and several meteorological variables were examined. Figure 2 presents these relationships for (a) net radiation at the lake surface, (b) wind
speed over the lake, (c) land-lake temperature contrast, and
(d) lake-land vapour pressure contrast. The figure shows the
results for the 2006 Crean Lake data only; the relationships
for the other sites and other field seasons are similar. Figure 2a shows that, unlike evaporation from land surfaces, the
hourly open water evaporation rate bears little or no relationship to the net radiation. The hourly open water evaporation rate is most strongly affected by the wind speed over the
lake. Although the vapour transfer function is defined by the
vapour pressure gradient, the land-lake temperature gradient
actually shows a stronger relationship: this is likely due to
the strong effect of atmospheric stability on the evaporation
rates.
Since many of the current evaporation models, such as
that of Morton (1983) use net radiation as one of the controlling parameters, it is useful to demonstrate how the relationship between evaporation and net radiation is affected
by the time scale and by water depth. Crean Lake is a relatively deep lake; the portion of the Lake that was studied
is typically 21 to 25 m deep. Landing Lake is a relatively
shallow lake, with an average depth of 3 to 4 m. For each
lake the correlation coefficients were determined for the relationship between evaporation and net radiation for hourly,
daily, weekly and monthly periods. For Crean Lake, the relationship showed R 2 values of 0.009, 0.003, 0.004 and 0.023,
respectively. Thus, for a deep lake, even at the monthly time
scale, one sees very little relationship between evaporation
and net radiation. This suggests that modelling approaches
such as the Priestley and Taylor (1972) technique, which uses
only net radiation and a correction factor, will be of little use
in estimating evaporation from open water bodies such as
Crean Lake. For the shallower Landing Lake, the respective
R 2 values were 0.212, 0.354, 0.731 and 0.927. Thus, for a
shallow lake, the energy input is reflected more quickly in the
surface temperature, and there may be a definable relationship between evaporation and net radiation only for periods
greater than weekly; but for hourly periods, the relationship
remains weak.
The open water evaporation is most closely controlled by
the wind speed over the lake (Fig. 2b). Although wind speed
is a standard, routinely observed parameter over land surfaces, it is not commonly measured over lake surfaces. The
www.hydrol-earth-syst-sci.net/15/267/2011/
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
b) 400
a) 400
4.5
4
350
y = 0.02x + 90.68
2
Lake-Land Wind Speed
Contrast, m/s
350
2
R = 0.01
300
300
250
250
3.5
y = 0.57x + 19.16x
2
L E , W /m ²
L E , W /m ²
R = 0.66
200
150
200
150
100
100
50
-200
0
200
350
350
250
400
c)
2
600
800
1000
350
y = 0.02x + 90.68
2
R = 0.01
300
250
d) 400
2
L E , W /m
L E² , W /mL²E , W /m ²
R = 0.01
400
Net Radiation, W/m²
L E , W /m ²
y = 0.02x + 90.68
0
400 0
b)
a) 4000
b) 400
300
3
2.5
y = 0.94x + 2.23
2
1.5
1
0.5
50
1
271
300
200
350
250
300
150
200
250
100
150
200
50
100
150
0
50-200
100
0
400
50 0
y = 0.57x + 19.16x
2
y = 0.62x - 4.18x + 81.82
2
R = 0.66
0
-1.5
2
1
4
-1
6
-0.5
0
0.5
1
1.5
8
10
12 Contrast, °C
Lake-Land
Temperature
Wind Speed, m/s
Figure
3. The
observed relationship
relationship between
the 5-daythe
average
lake-land
wind speed
2y = 0.57x
Fig.
3. 2 +The
observed
between
5-day
average
lake-contrast
19.16x
2
3 Rand
the
lake-land
temperature
contrast,
for
Crean
Lake,
2006.
land
wind
speed
contrast
and
the
lake-land
temperature
contrast,
for
= 0.66
Crean Lake, 2006.
200
350
2
R = 0.20
water surface is smoother than the land surface, and so the
wind
speeds will tend to be greater over the water surfaces.
2
y = -82.53x + 193.46x
To
examine
how the wind speed over a lake may be affected
2
50
200
R = 0.11
by atmospheric stability, the lake-land wind speed contrast
1500
was
compared
to6 the lake-land
temperature
contrast for 5-day
0
2
4
8
10
12
0
200
400
600
800
1000
periods at Crean Lake in 2006. These are presented in Fig. 3.
100
Wind Speed, m/s
Net Radiation, W/m²
The figure shows that for neutral conditions, the wind speed
b)
50
1
2
4
6
8
10
12
600
800
1000
over Crean Lake tends, on average, to be 2.23 m/s greater
3500
Wind Speed, m/s
y = 0.02x + 90.68
0
m²
than 0.5the wind speed
over1.5land. Figure
3 also shows that the
2
2
400
400
d)
R = 0.01
0
1
-15
0
5
10
2
c)
y = -10
0.57x + 19.16x -5
300
2
wind
speed
over
water
is
affected
by
the
atmospheric stabil2
y
=
0.62x
4.18x
+
81.82
= 0.66
R Land-Lake
Temperature Contrast, °C
Lake-Land Vapour Pressure Contrast, kPA
350
350
2
ity
over
the
lake;
the
wind
speed
contrast
increases in un250
R
=
0.20
400
d)
2
300
300
2
stable
situations
(lake-land
temperature
>
0).
This wind iny = 0.62x - 4.18x + 81.82
200
350
2
crease
over water
was
alsoLake,
documented
by Derecki (1981)
250
3 Figure
2. The observed relationship between the 250
hourly
evaporation
over
Crean
2006,
R = 0.20
2
y = -82.53x + 193.46x
150
300
who
reported monthly wind ratios (Ulake /Uland ) for Lake Su2
200
R = 0.11
4 and200
(a)
net
radiation,
(b)
wind
speed,
(c)
land-lake
temperature
and the
(d)smallest
lake-land
100
250
perior.
Hiscontrast,
data showed
ratio for the months of
2
y = -82.53x + 193.46x
150
150
2
April to June, the warming portion of the season being char5 vapour
pressure
50
200
R = 0.11 contrast.
100
100
acterised by stable conditions; the largest ratios occurred in
0
150
November and December when conditions are unstable and
0
2
4
6
8
10
12
50
50
600
800
1000
100
the contrast between the land and lake temperatures is greatWind
Speed,
m/s
m²
0
0
est (cf.
5, below).
However,
this relationship is
50-15
0
0.5 Figs. 4 and
1
1.5
-10
-5
0
5
10
2
alsoLake-Land
likelyVapour
affected
thekPA
distance from the upwind shore.
Land-Lake Temperature Contrast, °C
Pressureby
Contrast,
0
0.5
1
1.5
5
2
To test this, the data for both lakes and for all seasons were
2 10 d) 400 0
2
y
=
0.62x
4.18x
+
81.82
separated into stable and unstable cases, and sorted accordContrast, °C
Lake-Land Vapour Pressure Contrast, kPA
350
2
R = 0.20
3 Figure 2. The observed relationship between the hourlying
evaporation
overfetch
Crean
Lake, the
2006,
to the upwind
distance;
following relationships
300
were developed to relate the wind speed over the lake to that
4 and
(a) net evaporation
radiation,
(b)
wind
speed,
(c)2006,
land-lake temperature contrast, and (d) lake-land
ionship between
the 250hourly
over
Crean
Lake,
2
over land:
y = -82.53x + 193.46x
L E , W /m ²
100
250
L E , W /m ²
L E² , W /mL ²E , W /m ²
L E , W /m
L E , W /m ²
150
300
2
5 land-lake
vapour
pressure
200
R = 0.11 contrast.
ind speed, (c)
temperature
contrast, and (d) lake-land
ontrast, °C
50
0
10
(1)
Tland is the air temperature over the adjacent land surface and
Tlake is the surface temperature of the water. The coefficients,
b and c, are related to the fetch distance, X (m):
For Stable conditions:
100
5
Ulake = Uland × (b + c × (Tland − Tlake ))
150
0
0.5
1
1.5
2
Lake-Land Vapour Pressure Contrast, kPA
Fig. 2. The observed relationship between the hourly evapora-
onship between
theCrean
hourly
evaporation
over
Lake,
2006,
tion over
Lake,
2006, and (a)
net Crean
radiation,
(b) wind
speed,
(c) land-lake temperature contrast, and (d) lake-land vapour pres-
nd speed, (c)
land-lake temperature contrast, and (d) lake-land
sure contrast.
www.hydrol-earth-syst-sci.net/15/267/2011/
b = 1.0 + 0.0001247 × X
19
c = − 0.0125 − 4.87 × 10−6 × X
(2)
For Unstable conditions:
b = 1.0 + 0.0001247 × X
c = − 0.0125 − 2.3 × 10−5 × X
(3)
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
20
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
25
800
400
Crean Lake
20
600
500
300
20
200
15
100
10
15
400
10
300
200
Temperature, C
Cumulative Flux, mm eq.
25
Landing Lake
700
5
100
0
0
Temperature, C
272
5
0
Evap
-100
140
1
2
H
190
Rn
Tsf (lake)
240
Julian Day, 2007
Ta (land)
290
Evap
-5
-100
160
340
H
190
Rn
Tsf (lake)
220
250
Ta (land)
0
280
Julian Day, 2007
Figure 4. The cumulative evaporation, sensible heat and net radiation for Crean Lake, 2007;
Fig.
4. The cumulative evaporation, sensible heat and net radiation
3 showing 5-day average water surface temperature and air temperature over land.
for Crean Lake, 2007; showing 5-day average water surface temperature and air temperature over land.
Equations (1)–(3) can be used to estimate the wind speed
over small lakes when no measurements are available.
The effect of the land-lake temperature contrast, or atmospheric stability, can also be demonstrated by examining
the seasonal trends. Figure 4 shows the cumulative seasonal
evaporation, net radiation and turbulent heat exchange rates
21
as well as the temperature contrast between the lake surface
and adjacent land surface at Crean Lake for the 2007 open
water period. The energy terms are expressed in mm equivalent of evaporation. For the sake of clarity, the temperature
values are five-day running averages. The figure shows that
for the warming period, the land temperature is greater than
the lake water temperature, resulting in stable conditions over
the lake; whereas for the cooling period the inverse is true.
The trends shown in Fig. 4 are similar to those for the other
years. The figure shows that the lake evaporation is strongly
affected by the thermal contrast between the lake and the adjacent land surface. In the Spring, the land surface warms
more rapidly than the water surface does; this results in a
predominantly stable boundary layer over the lake. In stable conditions, turbulence is suppressed, and so are the exchanges of water vapour and heat. Figure 4 shows that, although evaporation occurs during this period, there is very
little turbulent heat exchange between the lake and the atmosphere. At Crean Lake, for the three study years, the average
evaporation rate during the warming period was 1.85 mm/d.
The land surface reaches its maximum seasonal temperature
generally a few days before the lake surface, and it cools
more rapidly than the water does. After this point, an unstable boundary layer (temperature decreasing with height) develops over the lake: the instability enhances turbulence and
the exchanges of water vapour and heat. For the three study
seasons, the average evaporation rate during the cooling period was 3.05 mm/d. This pattern of temperatures was also
demonstrated by Bussières and Granger (2007), who derived
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
1
2
3
Figure 5. The cumulative evaporation, sensible heat and net radiation for Landing Lake,
Fig. 5. The cumulative evaporation, sensible heat and net radia2007; showing 5-day average water surface temperature and air temperature over land.
tion for Landing Lake, 2007; showing 5-day average water surface
temperature and air temperature over land.
seasonal curves of water temperature for large lakes and suggested that these will have an effect on the evaporation rates.
Figure 5 shows the cumulative seasonal evaporation, net
radiation and turbulent heat exchange rates, and the temperature contrast between the lake and adjacent land surfaces
at Landing Lake for the 2007 open water period. Unlike
Crean Lake, the warming period at Landing Lake was not
characterised by a consistent stable regime; it showed al22
ternating periods of stable and unstable conditions. Landing Lake is located north of Great Slave Lake, such that the
thermal regime over the area is significantly affected by the
proximity of the large lake when southerly flows are occurring. Nonetheless, Fig. 5 does show that the turbulent fluxes
are significantly modified when the regime changes from unstable to stable. For example, during the first 9-day period,
shown in Fig. 5, conditions were unstable; the average evaporation rate observed was 3.86 mm/d. During the next 8day period, conditions were stable and the average evaporation rate dropped to 2.04 mm/d. During the cooling period,
Landing Lake behaved the same as did Crean Lake, with
a consistent unstable regime and enhanced turbulent fluxes
(3.3 mm/d). Figure 4 shows that evaporation from Crean
Lake represents approximately 2.5 times more energy transfer than does the turbulent heat exchange, while Fig. 5 shows
that for Landing Lake the latent heat is 3.8 times the turbulent sensible heat. This suggests that, for regions such as the
boreal plain and arctic shield where open water represents a
significant proportion of the surface area, correct knowledge
of the lake evaporation rates are important in the regional energy balance, and indeed for the correct assessment of the
meteorology of these regions.
www.hydrol-earth-syst-sci.net/15/267/2011/
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from
small lakes
0
Modelling evaporation rates
-0.50
The approach used in the model development was one of successive regression; that is, the most significant parameter was
identified, and a relationship developed between it and the
evaporation rate. The effects of the other parameters on the
relationship were then incorporated in order of decreasing
importance. The relative importance of the wind speed, the
lake-land humidity contrast, the land-lake temperature contrast, and the net radiation on the hourly evaporation rates
was shown if Fig. 2.
1
The Fig. 2a shows that, in fact, unlike for land surfaces,
there is very little relationship between the available radiant1
energy and the hourly evaporation from open water; therefore, the net radiation was excluded from the analysis. The
wind speed shows by far the strongest relationship with evaporation. Although the vapour transfer function is defined by
the vapour pressure gradient, the temperature gradient (or
land-lake temperature contrast) actually shows a stronger relationship. Hence, the model development proceeded from
the relationship between wind speed and evaporation; this relationship was then modified to include in turn the effects of
the land-lake temperature and lake-land humidity contrasts.
Weisman and Brutsaert (1973) showed lake evaporation
2
includes advection and that fetch distance, or the distance
from the upwind shore is also significant; therefore the effect
32
of the fetch distance was also included in each step of the de4
velopment. Most lakes are not uniform in shape, so the fetch3
distance was determined as a function of the wind direction.54
The data from Crean Lake (2006, 2007, 2008), Landing Lake (2007, 2008) and from Whiteswan4 Lake (2009)5
were used for the model development. The data from the
combined open water periods were separated into stable and
unstable categories; stable cases were defined as those for
which the land-based air temperature was greater than the
lake surface temperature. Data for wind directions affected
directly by the small island on Crean Lake and by the land
point on Whiteswan4 Lake were eliminated from the analysis. The resultant data sets included approximately 10 000
and 18 000 values respectively for the stable and unstable categories. These were then sorted into distinct ranges of fetch
distance for the analysis.
The relationship between the wind speed over the lake surface, u, and the evaporation rate is the basis for the model.
Although Fig. 2b shows a non-linear over-all trend, for any
specific set of conditions, such as a narrow range of land-lake
temperature difference, the relationship is linear:
coefficient
coefficient
"m" "m"
Unstable
y = 0.0904Ln(x) - 1.758
-1
-0.5
Unstable
y = 0.0904Ln(x) - 1.758
-1
-1.5
Stable
y = 0.420Ln(x) - 4.584
-1.5
-2
Stable
y = 0.420Ln(x) - 4.584
-2.5
-2
-2.5
-3
100
1000
a)
10000
Fetch, m
-3
100
1000
10000
(a)
Fetch, m
a)
35
30
35
Unstable
y = -0.0008x + 26.525
25
30
coefficient
"n"
coefficient
"n"
2.3
273
Unstable
y = -0.0008x + 26.525
20
25
15
20
Stable
y = -0.0011x + 20.256
10
15
Stable
y = -0.0011x + 20.256
5
10
0
5
100
1000
10000
(b)
b)
1000
10000
m
Fig. 6. The coefficients m andFetch,
n plotted
against fetch distance, for
b)stable 6.
Figure
coefficients
m and n plotted against fetch distance, for stab
andThe
unstable
conditions.
conditions.
Figure 6. The coefficients m and n plotted against fetch distance, for stab
Fetch, m
0
100
(lake-land contrast) vapour pressure (e), and of the fetch disconditions.
tance over the open water.
a = f (1 T ,1 e,X)
1 T = Ta(land) − Tsf(lake) , [◦ C]
1 e = esf(lake) − ea(land) , [kPa]
X = fetch · over · water, [m]
The coefficient, a, takes the form:
a = b + m · 1T + n · 1e
(5)
Figure 6 shows the effect of fetch distance on the coefficients,
m and n, respectively; the relationships for both stable and
unstable conditions are shown.
For stable conditions over the lake, i.e. Ta(land) > Tsf(lake) :
b = 3.395 + 0.0008 · X
m = − 4.584 + 0.420 · ln (X)
n = 20.256 − 0.0011 · X
(6)
For unstable conditions over the lake, i.e. Ta(land) < Tsf(lake) :
E = a ·u
(4)
where u is the 2 m wind speed over water [m/s]; E is expressed as the latent energy flux [W/m2 ].
The coefficient, a, was determined as a function of the horizontal gradients (land-lake contrast) of temperature (T ) and
www.hydrol-earth-syst-sci.net/15/267/2011/
b = 2.373 + 0.0002 · X
m = − 1.758 + 0.0904 · ln (X)
n = 26.525 − 0.0008 · X
(7)
Equations (4–7) represent a relatively simple series of expressions which can be applied to the calculation of hourly
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
274
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
1
Aug. 13 '05
200
200
150
100
Evaporation, W/m²
Model verification and discussion
250
250
Evaporation, W/m²
evaporation rates from open water; they require as inputs,
the wind speed, the surface temperature of the lake, the air
temperature and humidity above the adjacent land; the wind
direction and the relationship between lake fetch and wind
direction are also required.
150
100
6
Evaporation, W/m² Evaporation, W/m²
W/m²
Evaporation,
W/m² Evaporation,
Evaporation,
W/m²
Evaporation,
W/m²
Evaporation,
W/m²Evaporation,
W/m²
Evaporation, W/m²
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
Evaporation, W/m²
Evaporation, W/m²
Evaporation, W/m² Evaporation, W/m²
Evaporation, W/m²
Two data sets were available for the model verification. The
1
50
50
first data set included the evaporation data collected on Crean
LE, m
LE, meas
Lake in 2005; this data set was not used in the model develLE, m
LE, mod
1
0
0
250
250
opment since there was no land-based tower deployed adja0
0
600
1200
1800Aug. 13 '05 2400
cent to the lake during this first observation season. However,
Time, CST
200
200
2
land-based data from 250
the “Mixedwood” forest site, south of
250
Aug. 14 '05
Aug. 13 '05
Waskesiu Lake, were still available, and were used in the
150
150
verification calculations.
200
200 Although the Mixedwood tower is
250
250
Aug.
15
'05
LE, me
not adjacent to Crean Lake, the study, for which the tower
LE, mo
100
100
had been deployed, showed
that the boreal forest is relatively
150
150
200
200
uniform in that the above-canopy temperature and humid50
50
ity does not vary greatly
over the region (Pomeroy et al.,
100
100
150
150
LE, m
LE, meas
1997). A second data set involved the data collected over
LE, m
LE, mod
1
0
0
Quill Lake, Saskatchewan
in 1993 (HEATMEX Experiment,
50
50
100
100
0
0
1200
1800
2400
LE, meas600
LE, meas
unpublished).
Time, CST
LE, mod
LE, mod
2
Equations (4–7) were
0
250
50
50
2500 applied to the half-hourly observa13 '05 2400
0
1200
1800Aug. 14 '05 2400
0
600
1200
LE, meas600
tions from Crean Lake and
the Mixedwood
tower for 1800
theAug.
peLE, mod
Time, CST
Time, CST
riod May to 2
September
200
0
0
200 2005. Figure 7 shows the compar250
250
0
0
600
1200
1800Aug. 15 '05 2400
LE, me
ison between measured and calculated half-hourly evaporaLE, mo
Time, CST
tion rates (expressed 150
in energy units of W/m2 ) over Crean
150
200
3
200
250
250
Aug. 16 '05
Aug. 15 '05
Lake for 4 consecutive days. The figures show that the modLE, meas
LE, mod
elled evaporation follows
the
observed
values
very
well;
the
4
Figure
7.
The
measured
and
calculated
half-hourly150evaporat
100
100
150
200
200
model follows the diurnal trends and responds correctly to
5 Crean Lake (August 13 – 16, 2005).
changes in environmental
conditions (13, 14 August).
50
50
100
100
150
150
LE, meas
LE, meas
Figure 8a shows the comparison
between measured and
LE, mod
LE, mod
6
calculated evaporation100rates
(W/m2 ) for all the time periods
0
50
500
100
0
1200
1800
2400
600the comparison
1200
1800
2400
LE, meas600
in the 2005 data set; Fig.0 8b shows
between
LE, mod
CST
Time, CST
the calculated
cumulativeTime,
seasonal
evapora0
0
2 and measured
50
50
0
0
600
1200
1800
2400
tion, expressed in mm of water.
The Fig. 8a shows excellent
LE, meas
2
LE,
mod
Time,
CST
agreement, with an R value
of 0.86, for the 2250 data points.
3
0
250
2500
Figure 8b shows that the0 seasonal
evapora0
1200
1800Aug. 16 '05 2400
600 total modelled
1200
1800Aug. 15 '05 2400
LE, meas600
LE, mod
Time,
CST
Time,
CST
tion is in very close agreement with the observed evapora4 Figure
7. The measured and calculated half-hourly evapora
200
3
200
tion, with a total divergence between the two of 7 mm over
5 Crean Lake (August 13 – 16, 2005).
84 days.
4 Figure
7. The measured and calculated half-hourly150evaporation rates for 4 consecutive days on
150
The model (Eqs. 4–7) was also applied to the hourly obser6
Crean
(August
13 – 16, in2005).
vations from 5
the Quill
LakeLake
HEATMEX
Experiment
1993.
100
100
The “measured” lake evaporation values were obtained using
6 a tower located near the centre of the lake.
profile data from
50
50
For the Quill Lake test, the LE,
land-based
data were obtained
meas
LE, modon the western shore of the
from an instrument tower located
0
0
lake. In this case, all calculations
were made
0
600
1200
1800
2400
0
600
1200for hourly
1800 val2400
ues.
Time, CST
Time, CST
3
Whereas Crean and Whiteswan4 Lakes are surrounded by
Fig. 7. The measured and calculated half-hourly evaporation rates
boreal forest4and Figure
Landing7.Lake
is surrounded
by half-hourly
The (NWT)
measured
and calculated
evaporation rates for 4 consecutive days on
for 4 consecutive days on Crean Lake (13–16 August 2005).
forest and rock, Quill Lake is a classic Prairie lake, sur5 fields
Crean
(August
13this
– 16,
2005).
rounded by open
andLake
pasture
land. For
reason
also,
www.hydrol-earth-syst-sci.net/15/267/2011/
2
200
150
100
50
1
0
0
100
150
200
250
300
Aug 29, 1993
LE, mod
Evaporation, W/m²
100
1
100
50
2400
LE, mod
Aug 30, 1993
LE, mod
LE, meas
Aug 31, 1993
50
0
400
LE, meas
800
1200
1600 Aug 2000
29, 1993
0
100
50
2400
0
230
250
150
250
270
290
3
310
meas
JulianLE,
Day,
2005
Aug 31, 1993
LE, mod
200
100
Fig. 8. Comparison between the measured and calculated evapon the measured
and calculated evaporation from Crean Lake,
ration from Crean Lake, 2005;
(a) half-hourly evaporation rates,
150
50
2 ; (b) cumulative evaporation, mm.
W/m
W/m²; b) cumulative evaporation, mm .
400
800
1200
1600
2000
Evaporation, W/m²
Quill Lake represents a good test for the transferability
of the
Time, CST
50
2
model.
Figure 9 shows the comparison
between measured and cal0
250
2 ) over Quill Lake for
culated hourly evaporation
(W/m
0rates
400
800
1200
1600 Aug 2000
31, 1993
LE, meas
LE,
mod
Time,that
CST the mod4 consecutive days
in
1993.
The
figures
show
3
200
elled evaporation follows the “measured” values very well; it
also responds correctly
to the9.diurnal
variations as
well
as to
4 Figure
The measured
and
calculated
150
changes in environmental conditions (1 September).
5 Quill
Lake (Aug.
29 – Sep.1,
1993).
Figure 10a shows
the comparison
between
measured
and
100
calculated evaporation rates (W/m2 ) for all the time periods
in the 1993 data set; Fig. 10b shows the comparison between
50
the calculated and measured cumulative seasonal evaporation, expressed in mm of water.
Although the comparison for
0
the 1800 data points produces
a
smaller
R 2 value
0
400
800
1200 (0.75)
1600 than
2000
Time,
CSTgood. The
for the Crean Lake
data,
the
agreement
is
still
very
3
seasonal total modelled evaporation is in very close agreement with the observed
evaporation,
a totaland
divergence
4 Figure
9. The with
measured
calculated
of 8 mm over 75 days.
5
1600
2000
2400
800
400
LE, meas
1200
1200
LE, meas
LE, mod
200
100
150
50
1000
0
1600 Aug 2000
30, 1993
2400
1600
2400
0
250
Aug 2000
31, 1993
0
Time, CST
LE, mod
LE, 400
meas
LE, mod
250
200
150
200
Sep 1, 1993
LE, meas
mod
Figure
9.LE,The
measured and calculated hourly evaporation
r
200
150
150
100
5
Quill Lake (Aug. 29 – Sep.1, 1993).
2400
150
100
50
100
500
50
0
400
800
0
400
800
3
4
1200
1600
2000
2400
1600
2000
2400
1600
Sep2000
1, 1993
2400
Time, CST
50
0
1200
0
400
0
LE, 400
meas
800
1200
mod
Time, CST
Figure
9.LE,The
measured
and calculated hourly evaporation
200
5 Quill
Lake (Aug.
– Sep.1,
1993). days on
hourly
evaporation
rates29for
4 consecutive
150
100
50
0
2400
0
Time, CST
0
250
2400
100
0
400
800
1200
Time, CST
1600
25
2000
2400
Fig.
9. The
measured and
calculated
hourly evaporation
hourly
evaporation
rates
for 4 consecutive
daysrates
on for
4 consecutive days on Quill Lake (29 August–1 September 1993).
Quill Lake (Aug. 29 – Sep.1, 1993).
www.hydrol-earth-syst-sci.net/15/267/2011/
400
50
Time, CST
800
250
150
4
1000
0
400
LE, meas
LE, mod
0
250
200
200
1200
Time, CST
0
210
800
0
50
250
Time, CST
LE, mod
50
100
150
400
LE, 400
meas
200
LE, meas
0
0
Time, CST
Julian Day, 2005
200
250
2
0
250
m²
on rates,
29, 1993
310 1600 Aug 2000
Evaporation, W/m²
Modeled
150
0
300
1200
200
LE, mod
2005;
a) half-hourly
evaporation rates, W/m²; b) cumulative evaporation,
mm .
150
100
200
Observed 150
2
250
800290
LE, mod
Evaporation, W/m²
4
100
0
250
LE, 400
meas
270
200
150
200
250
Figure 8. Comparison
between the measured and calculated evaporation
from Crean Lake,
C umulative Evaporation, mm
3
150
50
0
250
230
2
LE, meas
250
100
2500
210
250
150
50
0
50
LE (meas), W/m²
b)
1
Evaporation,
W/m²
Evaporation,
W/m²
Evaporation,
W/m²
2
100
LE, mod
200
200
Modeled
150
LE, meas
LE, mod
Observed
50
Aug 29, 1993
LE, meas
Evaporation,
W/m²
Evaporation,
W/m²
Evaporation,
W/m²
LE (Mod), W/m²
R = 0.86
250
250
200
Evaporation, W/m²
y = 0.99x
1
Evaporation, W/m²
Evaporation, W/m²
C umulative Evaporation, mm
250
275
250
Evaporation, W/m²
b)
a) 300
Evaporation, W/m²
Evaporation, W/m²
200
1 J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
R.
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
1276
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
b) 250
coefficient. Equation (5) then shows that the atmospheric
Observedis characterised by the land-lake temperature constability
600
Modeled
trast,
and the vertical vapour gradient by the land-lake vapour
200
pressure
contrast. This is not inconsistent with the analyt500
ical development of Weisman and Brutsaert who, in trans150
400
forming the boundary layer equations into nondimensional
form, scaled the vertical gradients of temperature and humid300
100
ity with the land-lake contrasts for these variables. And, as
with the Weisman and Brusaert (1973) analytical solutions,
200
the effect of distance from the upwind shore was significant
50
in the development of Eqs. (4–7).
100
The lake evaporation model presented here may be limited
0
0
to small lakes; that is to say, lakes over which a consistent in0
100
200
300
400
500
600
700
210
230
250
270
290
ternal boundary layer can develop and be maintained. The
LE (Meas), W/m²
Julian Day 1993
data presented
here suggest that this is the case at least for
b) 250
lakes
with
a
fetch
distance up to 10 000 m. For larger lakes,
Observed
Figure 10. Comparison
between the measured and calculated
evaporation
Lake,
mesoscale
effects, suchfrom
as airQuill
mass entrainment
and local cirModeled
200
culations, may break down the internal boundary layer and
1993; a) hourly evaporation rates, W/m²; b) cumulative evaporation, mm .
the coupling between the land and lake surfaces; this would
150
seriously reduce the applicability of this approach.
a) 700
2
3
4
5
Cumulative Evaporation, mm
LE (mod), W/m²
Cumulative Evaporation, mm
y = 1.00x
R2 = 0.75
3
100
50
0
500
600
700
210
230
m²
250
270
290
Julian Day 1993
Fig. 10. Comparison between the measured and calculated evapoeen the measured
and calculated evaporation from Quill Lake, 2
ration from Quill Lake, 1993; (a) hourly evaporation rates, W/m ;
evaporation,
mm. mm .
ates, W/m²; (b)
b)cumulative
cumulative
evaporation,
Development of the hourly lake evapoartion model
(Eqs. 4–7) has demonstrated that the lake evaporation process is governed by the wind speed over the lake (this is consistent with the findings of Blanken et al., 2000) and by the
horizontal advection of air from the adjacent land surface (as
has been shown by Weisman and Brutsaert, 1973 and by Liu
et al., 2009).
It is intersting and useful to compare the form of the
model (Eqs. 4–7) to the well-known bulk aerodynamic transfer equation:
E = LCE u (es − ea )
(8)
Where, L is the latent heat of vaporization, CE is a bulk
transfer coefficient, u is the wind speed and es and ea are the
vapour pressures at the water surface and in the overlying air,
respectively; E is given in energy units [W/m2 ]. The Bulk
transfer coefficient, CE , is generally formulated from the
Monin-Obukhov similarity theory with corrections for stability. The coefficient a in Eq. (4) is then related to CE (es −ea ),
that is to the vertical vapour gradient and to a stability-related
Hydrol. Earth Syst. Sci., 15, 267–277, 2011
Summary and conclusions
Three successful field campaigns were carried out for measuring and modelling evaporation from small to mediumsized lakes: Crean Lake and Whiteswan Lake in the boreal
plain of Saskatchewan, and Landing Lake in the Canadian
Shield, north of Great Slave Lake, North West Territory. The
data collected allowed for the development of a model capable of calculating the lake evaporation rates for hourly time
periods. The model validation, on Crean Lake for 2005 and
on Quill Lake for 1993, showed that the model does provide
accurate and reliable results. The modelled evaporation follows the observed values very well; it also responds correctly
to the diurnal fluctuations as well as to changes in environmental conditions.
Obtaining reliable hourly operational estimates of evaporation from individual lakes, using this method, is feasible
if lake water surface temperature measurements are made
available. Further study will be required to extend the limits
of the model to lake fetches less than 150 m and exceeding
10 000 m. The application of this type of lake evaporation
model within meteorological or climate models will require
the development of a technique for characterizing lake size
and shape.
Acknowledgements. The authors wish to express their appreciation
for the support provided by the staff of the Prince Albert National
Park, for the use of Park facilities and equipment. The authors
also acknowledge the financial support provided
by Environment
27
Canada, and the IP3 and IPY research programs. The provision of additional data from colleagues at Environment Canada
(Chris Spence, Garth van der Kamp and Randy Schmidt), is also
gratefully acknowledged.
Edited by: J. Pomeroy
www.hydrol-earth-syst-sci.net/15/267/2011/
R. J. Granger and N. Hedstrom: Modelling hourly rates of evaporation from small lakes
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Blanken, P. D., Rouse, W. R., Culf, A. D., Spence, C., Boudreau,
L. D., Jasper, J. N., Kochtubajda, B., Schertzer, W. M., Marsh,
P., and Verseghy, D.: Eddy covariance measurements of evaporation from Great Slave Lake, Northwest Territories, Canada,
Water Resour. Res., 36(4), 1069–1077, 2000.
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