Bowen-Ratio Energy-Budget

Bowen (1926) completed a pioneering study of evaporation that empirically defined the ratio of sensible to latent-heat flux (the Bowen ratio) as a function of temperature and vapor-pressure gradients between the atmosphere and any water surface. Lake evaporation rates ($E$), in meters per second, were computed as follows:

where

$Q_e$ is computed using a rearranged form of the daily energy budget for the lake (eq. 2):

where All terms in equation 2 were defined in units of watts per square meter. Averaging was performed using all available instantaneous 15- or 30-minute measurements during a given day.

Daily net radiation ($Q_{nr}^{ }$) was estimated using the CNR1 four-component net radiometer (fig. 2). Daily advected energy from surface flows ($Q_{as}^{ }$) was assumed to be negligible given the sparse water temperature data and measured daily inflows and outflows that never exceeded 1 percent of the total lake volume, based on daily flow data published by the SFWMD and available on the DBHYDRO website (South Florida Water Management District, 2023). Daily advected energy from groundwater ($Q_{ag}^{ }$) also was assumed to be negligible in the energy budget, because measurements of groundwater seepage and the temperature of groundwater beneath the lake were not available. Advected energy from precipitation ($Q_{ap}$) was computed as

where

Rainfall was collected by the SFWMD at the site and used to assign values for $q_i$. Precipitation temperature was assumed equal to the dew point temperature, computed as a function of air temperature and relative humidity. $T_b$ was assumed equal to the average water temperature at the LZ40 station during the study period, as measured by vertical temperature profiles (fig. 2). ${\rho }_w$ and $c_w$ are assumed to have constant values of 1,000 kilograms per cubic meter and 4,184 Joules per kilogram degree Celsius, respectively.

Sensible heat flux ($Q_h$) was derived from the Bowen (Bowen, 1926) ratio ($B$) using the function

with Bowen’s ratio equal to where Vapor pressure was computed as a function of air temperature and relative humidity (Buck, 1981).

Water surface temperatures from October 7, 2013 to February 14, 2014, were obtained from two sources: the DBHYDRO database (South Florida Water Management District, 2023) LZ40 surface-water temperature (DBKEY= IY025), and thermistors mounted on two buoys at the water surface (fig. 2). DBHYDRO is SFWMD’s environmental database that stores hydrologic data. The DBHYDRO data were used only when surface thermistor data from the buoy system were not available. For example, the USGS buoy system failed on October 7, 2013, causing thermistors to remain static in the water column about 2 m below the surface. The USGS buoy system to measure surface-water temperature was repaired on February 14, 2014.

Changes in heat energy stored within Lake Okeechobee were assumed to be represented by a single temperature profile at the station location; that is, areal variations in the temperature profile were not considered because of the cost and logistical issues associated with complete and repeated thermal surveys of the entire lake. The change in stored heat within the one-dimensional (vertical) lake control volume (Q_s) for a given day is given by

where

${\overline{T}}_{i-1}$ and ${\overline{T}}_i$ were computed using a backward (1-day) moving average of submerged thermistors to minimize spurious values created by local thermal eddies in the vicinity of the LZ40 station. Combining equations 1–6 results in an equation to compute daily evaporation ($E$), in millimeters per day (mm/d):

where

$T_b$ (Anderson, 1954) was assumed equal to the average water temperature at the LZ40 station during the study period, as measured by vertical temperature profiles. Unrealistically large evaporation rates occurred when daily Bowen ratios were between −0.65 and −1.3 (German, 2000). These unrealistic values (4 percent of the evaporation dataset) were removed and replaced by assuming complete conversion of net radiation to evaporation. Negative values for $E$ were set to zero because negative values may represent dew formation, rather than lake evaporation. For reference, the sum of negative evaporation was −36 millimeters over the period of record (4 years) for this study.

Alternative Methods for Computation of Lake Evaporation

Five equations for lake evaporation were selected for comparison to the BREB monthly and annual results. The equations were chosen with the intention of reducing data requirements for computing evaporation in the interior of Lake Okeechobee. Prior knowledge of the sensitivity of various data types (Abtew, 1996a, b; German, 2000; Shoemaker and others, 2005; Shoemaker and Sumner, 2006; Shoemaker and others, 2011) guided selection of the equations. For example, the Simple, Priestley-Taylor, Penman, and Turc equations are based on components of the surface-energy budget that are known to explain much of the variability in evapotranspiration in the region (Abtew, 1996a, b; German, 2000), in particular, solar radiation, net radiation and (or) available energy. The Mass-Transfer equation was chosen because it is based on two variables to which computed evaporation showed sensitivity during preliminary analysis of the BREB results; specifically, wind speed and vapor pressure deficits.

The first equation is the “Simple” equation (Abtew 1996a, b), which is based on the concept that most of the variation in evaporation is explained by solar radiation. The Simple equation takes the form

where

The value of $K_1$ is generally set to 0.53 (open water conditions); however, this variable was estimated by regression (tables 1 and 2) to minimize residuals from daily and monthly BREB estimates of evaporation.

The Priestley-Taylor (PT) equation (Priestley and Taylor, 1972) estimates evaporation from an extensive wet surface under conditions of minimum advection. The PT equation takes the form

where

For this analysis, α was estimated by regression to minimize residuals from daily and monthly BREB estimates of evaporation (tables 1 and 2). Evaporation, in millimeters per day, was computed by dividing $\lambda E$ by $\lambda {\rho }_w$.

The Penman (1948) combination equation is derived for evaporation from an open-water surface. This equation accounts for both energy budget and aerodynamic principles, and takes the form

where

Evaporation, in millimeters per day, was computed by dividing $\lambda E$ by the product of ${\rho }_w$ and $\lambda$ . Aerodynamic resistance for the Penman equation was estimated using an equation published by Shuttleworth (1993):

where

The variable $d$ (Brutsaert, 1982) was estimated as 0.67 times an estimate of average wave height (0.45 m) visually observed during field work at the station. The variable $z_o$ was approximated as 0.0035 for the water surface. The variable $z_{ov}$ was approximated as 0.1 $z_o$ (Jensen and others, 1990). The variables e_s and e were estimated as described by Allen and others (1998) using daily minimum and maximum air temperatures and relative humidity. The Penman equation was calibrated to daily and monthly BREB results using correction coefficients, as outlined in Shoemaker and Sumner (2006). A linear correction coefficient formulated as a function of relative humidity was effective for debiasing the Penman equation.

Turc (1961) simplified earlier versions of his equation (Turc, 1954) for climatic conditions in western Europe. Trajkovic and Kolakovic (2009) presented the adjusted Turc similar to the following equation:

where The values of $C_u$ and $C_s$ were defined by regression (tables 1 and 2) to minimize the sum-of square residuals between daily (table 1) and monthly (table 2) BREB evaporation rates and values estimated by equation 13.

The Mass-Transfer method (Anderson and others, 1950; Marciano and Harbeck, 1954; and Harbeck, 1962) is based on the assumption that evaporation is proportional to the product of the measured water-to-air vapor pressure differential and wind speed as

where

Wind speed was measured from 3.6 to 5.1 m above the water surface, depending upon water levels in Lake Okeechobee. Wind speeds at 2 and 5 m above the water surface were calculated to be within 10 percent of each other during an assessment of error in evaporation introduced by not assuming a log profile in the vertical distribution of horizontal wind speed at heights >3.6 m above the lake surface. Furthermore, $N$ was estimated by regression between daily and monthly BREB values of evaporation and values estimated by equation 14. The value of $N$ is site-specific and represents a variety of effects, including the wind profile with height, roughness of the water surface, atmospheric stability, and averaging period during which the variables in the equation are measured (Harbeck, 1962; Jobson, 1972).

Lake Evaporation at LZ40

Using the BREB method, annual evaporation averaged 1,825 millimeters per year (mm/ yr) (averaging 5.0 mm/d) at the LZ40 platform in Lake Okeechobee during the 4-year study. Net radiation can be a helpful upper-limit check on the BREB evaporation rates by assuming net radiation converts solely into latent heat, the energy equivalent of evaporation. For example, annual evaporation would be 1,903 mm/yr (averaging 5.22 mm/d) if net radiation was converted solely into latent heat rather than sensible heat. BREB rates for calendar years 2013, 2014, 2015, and 2016 were 1,760, 1,840, 1,805, and 1,880 mm/yr, respectively. These evaporation rates are among the highest rates computed in Florida and are due to frequent clear-sky conditions over the interior of the lake. For comparison, Abtew (2001) estimated Lake Okeechobee evaporation as 1,320 mm/yr using evaporation pans adjacent to the perimeter of the lake, and calibrated empirical equations that were driven by solar radiation and air temperature. Cloud cover surrounding the perimeter of the lake likely reduced evaporation rates and calibration parameters ($K_1$) documented in the Abtew (2001) pan evaporation study. German (2000) estimated evapotranspiration rates in the Florida Everglades ranging from about 1,000 to 1,400 mm/yr using the BREB method. Price and others (2007) estimated a mean annual evaporation rate of about 1,700 mm/yr for Florida Bay during a 33-year period using vapor-flux and energy-budget based methods. Sumner (2001) estimated an annual evapotranspiration of about 900 and 1,000 mm in 1998 and 1999, respectively, over a cypress and pine forest in east-central Florida subjected to logging and natural fires. Swancar and others (2000) estimated evaporation rates of 1,450 and 1,422 mm/yr at Lake Starr in central Florida.

The high evaporation rates observed at LZ40 are explained by frequent clear skies over the interior of Lake Okeechobee (fig. 3) (Segal and others, 1997) that enhance evaporation by allowing more solar radiation to reach the lake’s water-surface than surrounding areas. Surface water has a greater heat capacity than land, which maintains cooler temperatures in the interior of the lake that inhibit convective cloud formation. In contrast, surrounding land areas heat more rapidly during the day and promote convective cloud formation. Frequent clear skies are also apparent over other large lakes in Florida, such as Lakes Apopka and George, as well as in the marine environment (fig. 3). These evaporation results and maps of cloud cover have implications for the design of water storage reservoirs. Specifically, storage reservoirs could benefit from smaller scales that do not suppress convective cloud formation and enhance evaporation of water in the interior of a reservoir caused by frequent exposure to relatively higher levels of solar radiation.

Seasonality was clearly observed in Lake Okeechobee evaporation rates, which increased in the summer with increasing vapor pressure gradients, air temperature, and solar and net radiation (fig. 4A–E). In general, evaporation declined in the winter as solar and net radiation decreased. Exceptions to this pattern included the occurrence of relatively cold air temperature events in the winter (fig. 4E), which occasionally created very large evaporation rates (fig. 4C; January 2013, 2014, 2015). These cold events documented in data releases by Wacker (2017, 2020) were associated with (1) increased wind speeds, (2) decreased humidity that increased vapor pressure deficits, and (3) abrupt changes in air temperature that initiated the release of heat energy stored in the lake (fig. 4C) for evaporation. In fact, similar to results from Shoemaker and others (2005) for the Everglades, available energy for evaporation increased considerably from changes in stored heat-energy in Lake Okeechobee, overwhelming low values of solar and net radiation.

Comparison of Methods for Computing Evaporation at LZ40

The BREB method requires a large amount of input data to account for many variables that explain open-water evaporation and provide reliable results (German, 2000). This 4-year dataset for Lake Okeechobee provided an opportunity to compare and calibrate several other approaches for estimating evaporation from the lake. The goal of this analysis was to identify methods that could reproduce the BREB results reasonably well, while reducing input data, instrumentation, and operational expenses.

Calibration of all five methods to daily BREB evaporation rates represented annual evaporation reasonably well (table 1). Annual totals were all within ±10 percent of the BREB results. Calibration was performed by estimating single parameter values for the period of record that minimized sum-of-squared residuals from daily BREB evaporation rates. Notably, the Mass-Transfer method underestimated annual evaporation by about 8 percent (table 1). The Penman and Mass-Transfer methods accurately estimated relatively large evaporation rates that occurred when cold fronts passed through the study area in the winter (fig. 4C), because these methods accounted for large wind speeds and vapor pressure deficits that occurred during the passage of winter cold fronts. For operational implementation, the Simple, Mass-Transfer, and Turc methods have the benefit of simplicity and limited data requirements (table 1) for accurately computing annual evaporation totals.

The PT equation provided a comparable prediction of annual Lake Okeechobee evaporation relative to the BREB method. Regression estimated an α value of 1.25 (table 1), which is within 1 percent to the theoretical value of 1.26 established by Priestley and Taylor (1972). Annual evaporation totals were underestimated by 2 percent, and the standard error of daily residuals was 0.76 mm, a low value relative to those obtained using the other equations (table 1). Large evaporation rates of ~10 mm/d (fig. 4C) during cold fronts in January 2013, 2014, 2015, and 2016 were well represented because changes in stored heat-energy are accounted for in the PT equation (eq. 10). A limitation of the PT equation (eq. 10) is the data requirements, which include net radiation and changes in heat-energy stored in the lake. This study confirms findings in prior studies, such as Douglas and others (2009), that demonstrate the impressive performance of the PT equation compared to alternate formulations.

In practice, a single set of monthly regression coefficients for all five methods could be tractable and useful in future water-budgeting studies of Lake Okeechobee, as long as reasonable accuracy is maintained in the evaporation results. Thus, a single set of monthly coefficients was identified with regression for each alternate evaporation equation (table 2) by minimizing sum-of-squared residuals from monthly BREB evaporation totals. Accuracy statistics for evaporation are presented here to provide information on potential uncertainty (table 2).

Monthly evaporation rates were more problematic for fitting in the fall, winter, and early spring for every equation compared to late spring and summer (fig. 5, table 2). The Mass-Transfer method occasionally underestimated monthly evaporation by as much as −28 percent. Likewise, errors for the Penman and Priestley-Taylor equations were >10 percent in the winter. The Turc and Simple methods performed well, given the simplicity and limited data requirements for computing monthly evaporation totals, with errors ranging from −2 to 10 percent and −5 to 11 percent, respectively. The Turc equation has two calibration parameters (C_u and C_s) that were identified by regression for computing monthly evaporation at the LZ40 station. Future hydrologic studies could use these optimized parameters with the Turc and Simple equations to accurately compute evaporation from Lake Okeechobee using only air temperature and solar radiation for input data. The Simple equation requires only solar radiation for input data. As expected, there are errors for each method of computing evaporation (fig. 5). Nevertheless, the errors are likely acceptable for many analyses of water resources, including investigations of long-term seasonal trends, droughts that extend over several months, or predictions of declines in lake water levels.