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Net radiation (Rnet) was calculated using equations II.8 to II.17. For the five-minute interval calculations, incoming shortwave radiation was measured rather than calculated. Therefore, calculation of net radiation only required the use of equations II.15 to II.17. Furthermore, in equation II.15, the measured temperature for each interval was used rather than the minimum and maximum temperature, similar as for equation II.4.
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Table II.1: Explanation and value of symbols used in equations II.1 to II.17. This table is only valid for this appendix. Note that some symbols are similar as used in the main report, but this table only applies to this appendix. Table continues on next page.
Symbol Unit Description Value
(if constant)
Used in equations a as - Empirical constant; fraction of extraterrestrial
radiation reaching the earth on clear-sky days
0.25 13, 14
aw - Activity of water 0.9 1
bs - Empirical constant; fraction of extraterrestrial radiation reaching the earth on clear-sky days
0.50 13, 14
cp MJ kg-1 °C -1 Specific heat at constant pressure 1.013 x 10-3 1, 7
dr - Inverse relative distance Earth-Sun 8, 9
DOY - Day of year 9, 10
e0 kPa Vapor pressure 3, 4, 5, 6
ea kPa Actual vapor pressure 1, 5, 16
es kPa Saturation vapor pressure 1, 4
E mm d-1 Evaporation 1
Gsc MJ m-2 d-1 Solar constant 118.08 8
n Hours d-1 Sunshine duration per day 8 14
N Hours d-1 Maximum possible sunshine duration 12, 14
pair kPa Atmospheric pressure 7
ra s m-1 Aerodynamic resistance 1, 2
Ra MJ m-2 d-1 Extraterrestrial solar radiation 8, 13, 14
Rnet MJ m-2 d-1 Net radiation 1
Rnl MJ m-2 d-1 Net longwave radiation 16, 17
Rns MJ m-2 d-1 Net shortwave radiation 15, 17
Rs MJ m-2 d-1 Incoming solar radiation 14, 15, 16
Rso MJ m-2 d-1 Clear-sky solar radiation 13, 14, 16
Tdew °C Dewpoint temperature 5
Tmax °C Daily maximum temperature 4
Tmin °C Daily minimum temperature 4
Tmax,K K Daily maximum temperature 16
Tmin,K K Daily minimum temperature 16
u2 m s-1 Wind speed at 2 meters height 2
α - Albedo of water 0.05 15
γ kPa °C-1 Psychrometric constant 1, 7
δ rad Solar declination 8, 10, 11
Δ kPa °C-1 Slope of saturation vapor pressure deficit curve 1, 6 ε - Ratio molecular weight of water vapor / dry air 0.622 7
λ MJ kg-1 Latent heat of vaporization 2.45 1, 7
rad Latitude 0.213 8, 11
ρair kg m-3 Mean air density at constant pressure 1.15 1
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Symbol Unit Description Value
(if constant)
Used in equations a σ MJ K-4
m-2 d-1
Stefan-Boltzmann constant 4.903 x 10-9 16
ωs rad Sunset hour angle 8, 11, 12
a All equation numbers have prefix II. as they refer to equations used in this appendix.
Dam conductance
The simplest dam model, given by equation 3.3a, assumed one constant conductance over the whole dam (1-layer dam model). This is illustrated in figure II.1 A by the dashed line. The solid line in this figure is the total flux through the dam as function of (the average of sea and lake) level. This equation originated from the Darcy equation for flow through porous media, which states (in one dimension):
where: Q is the flux through a porous medium [m3 d-1] k is the permeability of the porous material [m d-1] A is the area perpendicular to the flow direction [m2]
ΔH is the head difference between the two sides of the porous medium [m]
L is the distance between the two sides of the porous medium (thickness) [m].
In equation 3.3a, ΔH was the difference in water level between Lac Goto and the sea. Equation II.18 was translated into a one dimensional flux q, by dividing through the area of Lac Goto (AGoto). Furthermore, rewriting the area of the porous medium perpendicular to the flow direction, which is the area of the dam, in width (W) times height ( , where it is assumed that the dam is cubic, yields:
Introducing the dam conductance, c, as the combination of permeability, dam width and – thickness and the area of Lac Goto, finally yields equation II.19, which is similar as equation 3.3a:
where: q is the flux through the dam [md-1]
c is the conductance of the dam [m-1 d-1], calculated as (k * W) / (AGoto * L)
is the height of the dam through which water flows [m], defined as ddam + (HGoto + Hsea) / 2 ddam is the lowest point of the dam, fixed at -10 [m +BRL]
HGoto is the height of the water level of Lac Goto [m +BRL]
Hsea is the height of the water level of the sea [m +BRL].
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Figure II.1: Dam conductance (dotted line) and total flux through the dam (solid line) as function of the average water height of sea and lake, using c = c1 = cmin = 1 m-1d-1, c2 = cmax = 2 m-1d-1, ΔH = 0.1 m and γ = 2 d-1. Panel A shows the situation for the 1-layer dam model (with a constant conductance), panel B for the 2-layer dam model (where a transition of the layers is located at -4 m +BRL) and panel C for the n-layer dam model, where the conductance is higher at the top of the dam.
The 2-layer dam model, using two layers with a different conductance, is shown in panel B in figure II.1.
The equation describing this function calculated whether the water level at a certain moment in time was high enough to permit flow through both layers (which was the case when the average water level of the sea and Lac Goto was above a threshold, htrans). If water could only flow through the lower layer, an equation similar as for the 1-layer model was used. However, if flow through both layers occurred, the flow was calculated as the sum of fluxes through both layers, shown in equation III.20.
where: c1 is the conductance of the lower layer of the dam [m-1 d-1]
c2 is the conductance of the upper layer of the dam [m-1 d-1]
htrans is the height of transition between the first and second layer above the bottom of the dam [m].
The n-layer dam model was defined similarly as equation II.19, albeit that here the conductance was a function of the average level of Lac Goto and the sea ( ):
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Conductance as function of height was calculated using equation 3.4. This equation was derived from equation II.22, which calculates the conductance at a certain height of the dam over a smooth continuous function. For h=0, the conductance equals cmin, for h=hmax, the conductance equals cmax. Note that, even if water levels rise above hmax, this function still produces results. An example of this function is shown in figure II.1 C.
where: cmin is the conductance at the bottom of the dam [m-1 d-1]
cmax is the conductance at the top of the dam [m-1 d-1]
γ is the shape factor of the curve, defining how gradual the transition from cmin to cmax is [d-1] h is the height of the water level, above the bottom of the dam [m]
hmax is the maximum height above the bottom of the dam for which this function is defined [m].
For this function to be of use for the model, an average of the conductance between h=0 and h= was required, as given by equation II.23.
To arrive at equation 3.4, the integral was calculated
resulting in equation III.24, which is similar as equation 3.4.
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