Calculation of Open Water Evaporation as a Climate Parameter

Calculation of open water evaporation is important for hydrology, industry, agriculture, environment, and other fields. The available methods of calculating evaporation are based on field or laboratory experiments and should not be used for scale-up to open water evaporation for similitude relationships cannot be correctly obtained. The methods are thus unjustified scientifically. In addition, surface evaporation is not a local phenomenon that is a function of independent meteorological parameters. These are in fact dependent parameters, and the solar energy exchanged with the surface of the earth is the only independent variable for open water evaporation. Contrary to the existing methods, meteorological records and measurements are therefore not required. Many parts of the world do not have full or partial records available. For these, the available methods are likely not to be useful. In addition, future meteorological records or measurements cannot be made available for evaporation projection in a warming world. This may well place a limit on using the existing methods. The work presented in this manuscript reveals a new understanding of evaporation as a climate parameter instead and can be calculated as such. Minimal to no meteorological records or measurements may be required. The advantages of the proposed method are scientific justification, simplicity, accuracy, versatility, low to virtually no cost, and can be used to map present and future evaporation in a short period of time.


Introduction
Evaporation applications by solar energy are characterized by low cost, and solar evaporation pans have been used to produce salt, minerals, and chemicals for a long time [1] [2].Waste streams are collected and evaporated in ponds for zero discharge applications [3] [4].They are increasingly desirable to meet environmental requirements.Pond and reservoir utilization is not limited to industry; they are used for irrigation, farming, recreational, and other purposes.Artificial lakes and lagoons are created following dam construction, and ponds are built for fish farms, waste water treatment, and farm drainage collection [5].Open water evaporation is a key design parameter for most of these applications and therefore merits consideration.In addition, open water evaporation can be used to calculate evapotranspiration and crop irrigation [6].
The importance of determining present and future evaporation and the time and costs required for preparing meteorological records motivated this submission.Open water evaporation is not the same as class A or test pan evaporation.
While these vary with local ambient conditions, open water evaporation is a climate parameter.It varies with the solar energy exchanged with the surface of the earth as this work indicates.Its variability during the year is caused mostly by variation in the earth's axial tilt with the motion of the earth around the sun.
This motion is an established knowledge and can be used as basis for evaporation calculation.Because world average surface temperature and evaporation are measured, they can be used as pilot data.The latitude at which the world average values manifest can be determined.Similitude relationships can then be established between the latitudes, and open water evaporation can be scaled-up and obtained.Minimal to no meteorological variables are necessary.Scale-up from laboratory and pilot data are scientific methodologies widely used in the chemical and other industries.
The objective of this manuscript is to derive the equation of open water evaporation utilizing the physics of the earth and scale-up methodologies, calculate evaporation for sample locations, and compare the calculated evaporation with observations.The calculated evaporation is found to be in agreement with observations.This offers an inexpensive way of mapping present and future evaporation cost effectively in a short period of time, which has merit at the societal level.

Background Information
Reference [6] presented Food and Agriculture Organization (FAO) paper No.

The paper addresses among other subjects calculation of evapotranspiration
and evaporation from open water.These are important parameters for crop irrigation, hydrology, and environmental considerations.Empirical equations and pan evaporation methods are explained.The hurdles and uncertainties with the procedures are highlighted in the paper.Reliable meteorological records are necessary, which require time and considerable expenditure.Publication [7] derived a formula based on energy and dynamic considerations, which is widely used for calculating local evaporation.A similar approach was adopted by [8] for larger water bodies.Numerous papers have been written to calculate evaporation Clearly, the existing methods assume that evaporation as a mathematical N. H. Swedan function and the meteorological parameters as independent parameters of the function.This may be true for cooling towers and other industrial applications but not for surface evaporation of the earth.Evaporation is a heat transfer phenomenon, and in the absence of variation in the heat exchanged with the surface, surface evaporation remains unchanged regardless of the values of wind speed, relative humidity, or surface temperature.In reality, evaporation and the meteorological parameters are dependent parameters; they depend on the heat exchanged with the surface, which is the only independent parameter.This heat exchanged varies with the motion of the earth around the sun.Because the motion of the earth is known, meteorological record is not required for calculating surface evaporation.
Unlike the existing methods, open water evaporation is calculated by knowing the physics of the earth and the available data relative to world average surface temperature and evaporation.These can be used as pilot data.The scale-up from laboratory and pilot data using similitude is a scientific method that is widely used for complex applications.Most of the commercial complexes presently operating in the world have been designed based on scale-up procedures.Chemical engineering reference [16] present detailed explanations of the different methods

Data
Evaporation data for validation are gathered to subject the proposed procedure to a vigorous test.Sample locations of the world have been selected such that they have considerably different geographic and climatic conditions.The Dead Sea, 31˚N, Jordan, is the lowest location on the surface.The related evaporation is obtained from an engineering study conducted by [12].

Methods
The main concept is that annual average evaporation at the surface of the earth is measured; it is equal to annual precipitation.Therefore, if a mathematical equation correlating evaporation at a given location and annual average evapora-

Thermodynamics
The earth's subsystems that exchange solar radiation include atmosphere, surface water, and land.Land has a small thermal capacity and can be neglected.
While the solar energy exchanged with surface water is thermal, or enthalpy, in nature, the energy exchanged with the atmosphere is thermal and potential energy.When the atmosphere absorbs solar heat, it rises against gravity and expands into the surrounding outer space that has negligible mass and pressure.
Air expansion ceases at equilibrium.Seasonal variations affect this equilibrium because the distance between the earth and the sun varies.The axial tilt of the earth alters the energy exchanged with the geographic northern and southern hemispheres.Therefore, thermodynamic transformations must result from the motion of the earth around the sun, and the atmosphere and surface water must N. H. Swedan exchange energy with seasonal variations.Surface evaporation varies as a consequence.
Because the surrounding outer space has negligible mass, the potential energy and enthalpy of the atmosphere cannot be exchanged with outer space.Only radiation may be exchanged with outer space.Therefore, variation in the energy of the atmosphere can only be exchanged with the surface and the following must be valid: ) where a E = Energy of the atmosphere (enthalpy and potential energy), J.
s Q = Surface energy (enthalpy or heat), J.
e E = Energy of the atmosphere and surface combined, J.
1 C = Constant of integration, J.
The solar energy absorbed by the atmosphere and surface, e E , is equal to the latent heat of surface evaporation based on observations: At the conclusion of a full revolution of the earth around the sun, variation in the energy of the surface and atmosphere are negligible for it is a repeatable cycle.The only observed change is water evaporation and its subsequent condensation as precipitation. Therefore where e E = Annual rate of solar heat absorbed by the atmosphere and surface, J•yr −1 .E = Annual rate of surface evaporation, which is equal to precipitation, kg•yr −1 .
The solar energy absorbed, e E , can be calculated.It is approximately equal to the latent heat of condensing 2.61 × 365 = 953 mm of rain annually [18].This is equivalent to where 5.1 × 10 14 is the total surface area of the earth measured in m 2 .

Beer-Lambert Representation
The solar heat absorbed by the earth raises the atmosphere (air and clouds) to its current position and maintains present average surface temperature.The energy absorbed by the atmosphere is enthalpy and potential energy, whereas the surface gains solar energy as heat.Because the temperature of the sun is considerably greater than the temperature of the earth, radiation from the earth to the sun may be neglected.The net incident solar radiation may thus be assumed to be absorbed by the side of the earth's sphere facing the sun.The other side radiates N. H. Swedan heat to outer space.Using Beer-Lambert Law equation, the radiative energy absorbed by the atmosphere and surface are ( ) where ar E = Rate of radiative solar energy absorbed by the atmosphere, W.
s Q = Rate of radiative solar energy absorbed by the surface, W.
f = A factor that accounts for the reflected solar energy by the earth, dimen- sionless.
c A = Earth's circle area as viewed from the sun, ( ) The radiative energy absorbed by the atmosphere, ar E , can be determined from data available.It is approximately equal to one third of the total solar energy absorbed by the atmosphere and surface, e E , based on [19].The radia- tive energy exchanged with the atmosphere is thus equal to ured value of the factor, f , is approximately equal to 0.30 [20].From Equation (5), the average value of the optical depth of the atmosphere is equal to τ = 0.107.Also, this value of optical depth can be calculated by trial and error solution of Equations ( 4), ( 5), and (6) for the entire surface at a surface temperature T s = 286.70K.
Figure 1 illustrates the earth viewed from the sun as a disc, and Figure 2 is a cross section of the earth with a perpendicular plane through the noon line A-A.
In Figure 3, a cross section of the earth with a perpendicular plane through the line B-B at arbitrary latitude is presented.As Figure 1 reveals, the distance traveled by sunrays along any diameter is the same, and the average distance traveled is equal to that at noon. Figure 2 illustrates incident sunrays at noon for arbitrary latitude θ.The average distance traveled, t Z , is equal to that calculated at noon, Figure 2, as follows: ( ) ( ) N. H. Swedan  where θ is arbitrary latitude and ( ) n Z θ is the distance traveled by sunrays in the atmosphere at noon for the arbitrary latitude where Z is the height of the atmosphere, approximately equal to the average height of the mesopause, 96 km [21].From Figure 2 and Figure 3, the average distance traveled by sunrays, ( ) t Z θ , at the arbitrary latitude, θ, is equal to where N. H. Swedan ( In Table 1, the average distance traveled by sunrays in the atmosphere, ( ) t Z θ , is tabulated.The table reveals that the average distance traveled at noon, ( ) t Z θ , for θ = 0 is 223.2 km.Based on Equation ( 7), the average coefficient of solar energy absorption by the atmosphere a = 0.107/223.2= 0.00048 km −1 , which can be reasonably used for all other latitudes.The optical depth of the atmosphere for other latitudes is then determined by multiplying ( ) t Z θ by 0.00048, they are presented in Table 1.

Energy Exchange
From Equations ( 5) and ( 6) ) Not considered in Equation ( 12) is variation in the radiation term ( 4 2 6).The reason is that, unlike solid surfaces, surface water has negligible thermal conductivity, and the value of this term is controlled by convention heat transfer of surface water.The convection heat transfer coefficient does not vary tangibly with the observed surface temperature variation, and variation in this radiation term may be neglected.Equation (12) indicates that variation in the optical depth of the atmosphere induces a thermodynamic transformation where energy is exchanged between the atmosphere and surface.For the scenario where energy is transferred from the cold atmosphere to the warm surface, the external energy required is available; it is equal to the variation in the potential energy of the atmosphere and the laws of thermodynamics are thus satisfied.For a given latitude, θ, the solar heat exchanged with the surface follows based on Equation (12): Table 1.Latitude optical depth and solar energy.The solar energy absorbed by the atmosphere and the incident solar energy on the surface are for a plane perpendicular to the incident solar radiation.
Seasonal variation occurs infinitesimally with time and the height of the atmosphere varies infinitesimally as well.The distance traveled by solar radiation is variable and the optical depth of the atmosphere varies as a consequence.Because the diameter of the atmosphere is large, its circumference can be assumed to be a straight line in a small area.Similarities between triangles following small N. H. Swedan variation in the height of the atmosphere, Figure 2 and Figure 3, reveal that Both sides of this last equality can be multiplied by (1/π) 2 and double integrated with respect to θ and ф, in a similar fashion to equations ( 8) and (10).Keeping in mind that the functions and variables are separable, the result of the double integration is where E is a known surface water evaporation at a known latitude, and the measured world average precipitation of the entire surface of the earth will be used for E; ΔT s is variation in the world average surface temperature, K; and W s is the world average water vapor mixing ratio, kg water per kg dry air.These world average meteorological parameters are measured or available and can thus be used as pilot data for scale-up.Therefore evaporation at any latitude can be determined by measuring or calculating latitude surface temperature, T s (θ).The rest of the variables are known from the motion of the earth around the sun and Table 1.

Results
The solution of Equation ( 18) requires obtaining the latitude at which world average surface temperature and evaporation occur.This latitude can be determined by computing the average optical depth for a hemisphere h τ ( ) ( ) where h A is surface area of half hemisphere (  The upper limit of the summation, f θ , is the latitude where surface water exists.It is approximately equal to 70 degrees.Thereafter, the surface is assumed to be covered with Arctic and Antarctic ice.Integration of Equation ( 20) yields to h τ = 0.128 based on Table 1.From this table, the world average meteoro- logical parameters are equal to those measured at approximate latitude of 40˚.
Therefore, the latitude of 40˚ can be used for evaporation scale-up, as pilot latitude at which world average surface temperature and evaporation are known and available.

Calculation steps of open water evaporation follow:
Step one: given day of the year, d , where January 1 st is day 1, calculate the declination angle, δ , in de- Step two: calculate the instantaneous latitude θ = geographic latitude-δ.In the southern hemisphere the geographic latitude is negative.
Step three: read from used.Dimensionless groups of parameters are derived that apply for the pilot and the commercial units based on the laws of conservation of mass and energy at steady state operation.Mathematical expressions are formulated in accordance with the laws of physics governing the phenomenon being considered.For open water evaporation, the measured precipitation and surface temperature by researchers over the years are pilot data and can be used for scale-up from one latitude to another.A scale-up relationship for latitude evaporation with respect to the world average evaporation is then derived.The advantage of this scale-up methodology is that it is representative because it is between earth subsystems and similarity exists.The procedure thus adheres to the scientific methodologies.This is not the case for the existing methods that have been developed based on small laboratory or field experiments.The earth cannot be tested in laboratory or engineering setting, and similitude between these experiments and the earth does not exist.The existing procedures are thus incorrect, and using the current methods of calculating evaporation in climate models or other evaporation calculation products is not supported by science.The proposed method in this work on the other hand is scientifically justified and yields good values of open water evaporation.
tion can be established, then meteorological records and measurements are unnecessary.The mathematical correlation is summarized in equation (21).The equation is derived based on the physics of the earth and representative scale-up of the measured world average surface temperature and evaporation.Validation of the equation is presented under Results.Derivation of the equation is divided into three stages.Stage one covers basic thermodynamics of the earth, stage two utilizes Beer-Lambert Law equation to model radiative energy exchanged with the atmosphere, and the third and last stage applies energy exchange relationships between the atmosphere and the surface of the earth.

Figure 1 .
Figure 1.The earth's circle area, A c , as viewed from the sun.

Figure 2 .
Figure 2. Cross section, A-A, viewed from the right showing the earth's circle at noon and the incident sunrays on an arbitrary latitude θ.

Figure 3 .
Figure 3. Cross section, B-B, viewed from the top showing the earth's longitude circle and the incident sunrays on an arbitrary longitude ф.
[17] of the test evaporation pans are not adjusted for rainfall.Therefore, rainfall data are added to obtain actual evaporation.Data for Lake IJssel, Netherland, are part of a study report prepared by[17]for the Royal Netherlands Meteorological Institute.This lake is next to the Atlantic Ocean at about 52˚N.For Melbourne, The measured read-N.H. Swedan Journal of Water Resource and Protection gives latitude surface evaporation, E(θ), if the angle θ and surface temperature change, ΔT s (θ), of the latitude in consideration are known.On the other hand, the world average values of E, ΔT s , and W s are a result of solar energy exchanged.If the solar radiation is imagined to cease, surface air temperature approaches zero absolute.At steady state when surface evaporation, E, is equal to the observed average value, variation of surface air temperature, ΔT s , measures average surface temperature rise with respect to zero absolute.The ratio, ΔT s /ΔT s (θ), can thus be replaced by T s /T s (θ), where the values of the temperature are in degrees Kelvin.The ratio W s (θ)/W s is reasonably equal to the ratio of absolute humidity at saturation.This ratio can be determined if T s (θ) is known, for the average surface temperature T s is available.The world average

Table 1
Lake Okanagan, Canada, average evaporation for the month of October: d = 288.5;number of days = 31; latitude = 49.5˚N;Ts(40)=14.9˚C;declinationangle = −9.8˚;θ=49.5+9.8 = 59.3˚N;T s (θ) = 10.2˚C,Kelowna; from Table1τ(θ) = 0.171, τ(40) = 0.129; at saturation W s (θ)/W s (40) = 0.71; E(θ) = 1.6 mm•day −1 ; E(θ) = 1.6 × 31 = 50 mm for October.Observed open water evaporation is unavailable, observed class A pan evaporation is 70 mm; estimated pan coefficient = 50/70 = 0.71.Similarly, brine evaporation can be calculated by knowing the total dissolved solids.These reduce vapor pressure, which can be obtained and W s (θ) determined.Evaporation from brine can be calculated following the same procedure.Calculation of monthly evaporation for sample locations is presented in Table2.Evaporation calculation and projection in a warming world are important at the societal level.Presently, there are no low cost and accurate methods at the same time for calculating evaporation.Background Information and Data sections present examples where calculation of evaporation with accuracy is determined.It is complex and requires substantial resources.The inconveniences and limitations of the present methods are inherent to considering surface evaporation as a function of ambient conditions.Meteorological data and measurements are thus required.The work presented in this manuscript reveals that evaporation is a function of the solar heat exchanged with the surface, and can be addressed as a nual evaporation multiply average daily evaporation by 365.Justification for using the pilot data for average surface temperature, T s (40), is explained under the Discussion section.Example: N. H. Swedan

Table 2 .
Calculated and actual observed open water evaporation, mm•day −1 .