Hydrological Model J2000g

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<math>P_K = P_M + P_M \cdot WK_{s,r} + BV_{Som,Win} </math>
 
<math>P_K = P_M + P_M \cdot WK_{s,r} + BV_{Som,Win} </math>
  
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==Strahlungsberechnung==
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==Radiation Calculation==
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Für die Verdunstungsberechnung nach Penman-Monteith wird die Nettostrahlung als Eingangswert benötigt. Diese kann aus der Globalstrahlung berechnet werden. Liegt die Globalstrahlung nicht als Messwert vor kann sie näherungsweise aus der Sonnenscheindauer bestimmt werden. Hierzu sind eine Anzahl von Teilberechnungen notwendig. Die folgenden Berechnungen gehen von einer täglichen Modellierung aus. Wird das Modell mit monatlichen Zeitschritten betrieben erfolgen die unten aufgeführten Berechnungen für den 15. Tag jeden Monats. Die resultierenden Terme werden dann anhand der Tage pro Monat aufsummiert.
+
For the evaporation calculation according to Penman-Monteith, the net radiation is needed as initial value and can be calculated on the basis of the global radiation. If the global radiation is not available, it can be defined approximately on the basis of the sunshine duration. For this purpose, a number of intermediate calculations are necessary. The following calculations act on the assumption of a daily modeling. When the model runs in monthly time steps, the calculations listed below are carried out on the 15th of each month. The resulting terms are then accumulated on the basis of the days per month.  
  
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===Extraterrestrische Strahlung===
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===Extraterrestrial Radiation===
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[[Bild:radiation.jpg|thumb|Jahresgang der extraterrestrischen Strahlung (Ra), Globalstrahlung (Rg) und Nettostrahlung (Rn) für einen Standort im Thüringer Wald]]
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[[Bild:radiation.jpg|thumb|Annual course of extraterrestrial radiation (Ra), global radiation (Rg) and net radiation (Rn) for a position in the Thuringian Forrest]]
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Die extraterrestrische Strahlung (Ra) ist der kurzwellige Strahlungsenergiefluss der Sonne an der Obergrenze der Atmosphäre. Ra berechnet sich für einen spezifischen Ort in Abhängigkeit von seiner geographischen Breite (lat in radians), der Deklination der Sonne (decl in radians), der Solarkonstante (Gsc in MJ / m<sup>2</sup>min), dem Stundenwinkel bei Sonnenuntergang (ws in radians) und der inversen relativen Distanz zwischen Erde und Sonne (dr in radians) nach:
+
The extraterrestrial radiation (Ra) is the short-wave radiation energy flux of the sun at the upper boarder of the atmosphere. Ra is calculated for a specific place in dependence of its latitude (lat in radians), the declination of the sun (decl in radians), the solar constant (Gsc in MJ / m<sup>2</sup>min), the hour angle at sundown (ws in radians) and the inverse relative distance between earth and sun (dr in radians) according to:  
  
 
<math> Ra = \frac{24\cdot60}{\pi} \cdot Gsc \cdot dr \cdot \left(ws \cdot \sin(lat) \cdot \sin(decl) + \cos(lat) \cdot \cos(decl) \cdot \sin(ws)\right)</math>
 
<math> Ra = \frac{24\cdot60}{\pi} \cdot Gsc \cdot dr \cdot \left(ws \cdot \sin(lat) \cdot \sin(decl) + \cos(lat) \cdot \cos(decl) \cdot \sin(ws)\right)</math>
  
−
Die '''Solarkonstante''' (Gsc in MJ / m<sup>2</sup>min) ergibt sich aus der julianischen Tageszahl (jD [1... 365,366]) nach:
+
The '''solar constant''' (Gsc in MJ / m<sup>2</sup>min) results from the Julian Date (jD [1... 365,366]) as follows:  
  
 
<math>Gsc = 0.08139 + 0.00291 \cdot \cos\left(\frac{\pi}{180} \cdot (jD - 15)\right)</math> [MJ / m<sup>2</sup>min]
 
<math>Gsc = 0.08139 + 0.00291 \cdot \cos\left(\frac{\pi}{180} \cdot (jD - 15)\right)</math> [MJ / m<sup>2</sup>min]
  
−
Die '''relative Distanz Erde Sonne''' (dr in radians) ergibt sich aus der julianischen Tageszahl (jD [1... 365,366]) nach:
+
The '''relative distance between earth and sun''' (dr in radians) results from the Julian Date (jD [1... 365,366]) as follows:  
  
 
<math>dr = 1 + 0.033 \cdot \cos\left(\frac{2\pi}{365} \cdot jD\right)</math> [rad.]
 
<math>dr = 1 + 0.033 \cdot \cos\left(\frac{2\pi}{365} \cdot jD\right)</math> [rad.]
  
−
Die '''Deklination der Sonne''' (decl in radians) ergibt sich aus der julianischen Tageszahl (jD [1... 365,366]) nach:
+
The '''declination of the sun''' (decl in radians) results from the Julian Date (jD [1... 365,366]) as follows:  
  
 
<math> decl = 0.40954 \cdot \sin(0.0172 \cdot (jD - 79.35)) </math> [rad.]
 
<math> decl = 0.40954 \cdot \sin(0.0172 \cdot (jD - 79.35)) </math> [rad.]
  
−
Der '''Stundenwinkel bei Sonnenuntergang''' (ws in radians) ergibt sich aus der geographischen Breite (lat in radians) und der Deklination (decl in radians) nach:
+
The '''hour angle at sundown''' (ws in radians) results from the latitude (lat in radians) and the declination (decl in radians) as follows:  
  
 
<math> ws = \arccos(-1 \cdot \tan(lat) \cdot \tan(decl)) </math> [rad.]
 
<math> ws = \arccos(-1 \cdot \tan(lat) \cdot \tan(decl)) </math> [rad.]
  
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===Globalstrahlung===
+
===Global Radiation===
−
Die Globalstrahlung (Rg) berechnet sich aus der extraterrestrischen Strahlung (Ra in MJ/m²d) und dem Bewölkungsgrad. Der Bewölkungsgrad wird hierbei aus dem Verhältnis der gemessenen Sonnenscheindauer (S in h/d) zur astronomisch möglichen Sonnenscheindauer (S<sub>0</sub> in h/d) unter Zuhilfenahme der Angström Formel approximiert. Rg berechnet sich damit nach:
+
The global radiation (Rg) is calculated on the basis of the extraterrestrial radiation (Ra in MJ/m²d) and the degree of cloudiness. At this, the degree of cloudiness is approximated on the basis of the relation of the measured sunshine duration (D in h/d) to the astronomic possible sunshine duration (S<sub>0</sub> in h/d) with the help of the Angström formula. Thus, Rg is calculated as follows:
  
 
<math> Rg = Ra \cdot \left(a + b \cdot \frac{S}{S_0}\right) </math> [MJ/m²d]
 
<math> Rg = Ra \cdot \left(a + b \cdot \frac{S}{S_0}\right) </math> [MJ/m²d]
  
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Die Koeffizienten a und b müssen für den Standort geschätzt werden. Vielfach wird für a = 0.25 und für b = 0.50 eingesetzt.
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The coefficients a and b need to be estimated for the position. Often 0.25 is used for a and 0.50 is used for b.  
  
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Die maximale '''astronomisch mögliche Sonnenscheindauer''' (S<sub>0</sub> in h) ergibt sich aus dem Stundenwinkel bei Sonnenuntergang (ws in radians) nach:
+
The '''maximum astronomic possible sunshine duration''' (S<sub>0</sub> in h) is calculated on the basis of the hour angle at sundown (ws in radians) as follows:
  
 
<math> S_0 = \frac{24}{\pi \cdot ws} </math> [h/d]
 
<math> S_0 = \frac{24}{\pi \cdot ws} </math> [h/d]
  
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===Nettostrahlung===
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===Net Radiation===
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Als resultierende der einzelnen Strahlungskomponenten ergibt sich die Nettostrahlung (Rn in MJ/m²d), die die Energie für die Verdunstung bereitstellt. Die Nettostrahlung ergibt sich aus der Differenz von Globalstrahlung (Rg in MJ/m²d) und der effektiven langwelligen Ausstrahlung (Rl in MJ/m²d) wobei die Globalstrahlung noch durch die Albedo (alpha) der jeweiligen Bodenbedeckung abgemindert wird, als:
+
The net radiation (Rn in MJ/m²d) results from the single radiation components and provides the energy for the evaporation. The net radiation is calculated on the basis of the difference of the global radiation (Rg in MJ/m²d) and the effective long-wave radiation (Rl in MJ/m²d). The global radiation is reduced by the albedo (alpha) of the referring land cover.
  
 
<math> Rn = (1 - \alpha) \cdot Rg - Rl </math>
 
<math> Rn = (1 - \alpha) \cdot Rg - Rl </math>
  
−
Die '''effektive langwellige Ausstrahlung''' (Rl in MJ/m²d) berechnet sich aus der Bolzmann Konstanten (Bk  = 4.903E-9 MJ/K<sup>4</sup>m²d), der absoluten Lufttemperatur (Tabs in K), dem aktuellen Dampfdruck der Luft (ea in kPa), der aktuellen Globalstrahlung (Rg in MJ/m²d) und der maximalen Globalstrahlung bei unbedecktem Himmel (Rg0 in MJ/m²d) nach:
+
The '''effective long-wave radiation''' (Rl in MJ/m²d) is calculated on the basis of the Bolzmann constant (Bk  = 4.903E-9 MJ/K<sup>4</sup>m²d), the absolute air temperature (Tabs in K), the actual vapor pressure of the air (ea in kPa), the actual global radiation (Rg in MJ/m²d) and the maximum global radiation for unclouded sky (Rg0 in MJ/m²d):
 +
 
  
 
<math> Rl = Bk \cdot Tabs^4 \cdot (0.34 - 0.14 \cdot \sqrt{ea}) \cdot \left(1.35 \cdot \frac{Rg}{Rg0} - 0.35 \right)</math> [MJ/m²d]
 
<math> Rl = Bk \cdot Tabs^4 \cdot (0.34 - 0.14 \cdot \sqrt{ea}) \cdot \left(1.35 \cdot \frac{Rg}{Rg0} - 0.35 \right)</math> [MJ/m²d]
  
−
Der '''aktuelle Dampfdruck der Luft''' (ea in kPa) berechnet sich aus dem Sättigungsdampfdruck (es in kPa) und der relativen Feuchte (U in %) nach:
+
The '''actual vapor pressure of the air''' (ea in kPa) is calculated on the basis of the saturation vapor pressure (es in kPa) and the relative humidity (U in %) according to the following equation:  
  
 
<math> ea = es \cdot \frac{U}{100} </math> [kPa]
 
<math> ea = es \cdot \frac{U}{100} </math> [kPa]
  
−
Der '''Sättigungsdampfdruck der Luft''' (es in kPa) ergibt sich aus der Lufttemperatur (T in °C) nach:
+
The '''saturation vapor pressure of the air''' (es in kPa) results from the air temperature (T in °C) according to:  
  
 
<math> es = 0.6108 \cdot e^{\frac{17.27 \cdot T}{237.3 + T}} </math> [kPa]
 
<math> es = 0.6108 \cdot e^{\frac{17.27 \cdot T}{237.3 + T}} </math> [kPa]
  
−
Die '''maximale Globalstrahlung bei unbedecktem Himmel''' (Rg0 in MJ/m²d) ergibt sich aus der extraterrestrischen Strahlung (Ra in MJ/m²d) und der Geländehöhe (h in m ü.N.N.) nach:
+
The '''maximum global radiation for uncovered sky''' (Rg0 in MJ/m²d) results from the extraterrestrial radiation (Ra in MJ/m²d) and the terrain height (h in m above N.N.) as follows:  
  
 
<math> Rg0 = (0.75 + 2\cdot10^{-5} \cdot h) \cdot Ra </math> [MJ/m²d]
 
<math> Rg0 = (0.75 + 2\cdot10^{-5} \cdot h) \cdot Ra </math> [MJ/m²d]

Revision as of 16:37, 12 February 2010

Contents

Abstract

The Model J2000g was developed as a simplified hydrological model to calculate temporally aggregated, spatially allocated hydrological target sizes. The representation and calculation of the hydrological processes is carried out one-dimensionally for an arbitrary number of points in the space. These model points enable the use of different distribution concepts (Response Units, grid cells, catchment areas) without further model adjustments.

The temporal discretization of the modeling can be carried out in daily or monthly steps. During the modeling the following processes are calculated for each time step: regionalization of punctual existing climate data to the referring model units, calculation of global and net radiation as access for the evaporation calculation, calculation of the land-cover-specific potential evaporation according to Penman-Monteith, snow accumulation and snowmelt, soil water budget, groundwater recharge, retardation in runoff (translation and retention). The individual processes are described in more detail below.

Distribution and Attribute Tables

The model J2000g is not bound to any specific distribution concept because the processes are calculated on the basis of points in the space that are independent of each other (1D-model concept). These points can represent different space units, e.g. single station positions, polygons, grid cells, catchment areas or subcatchments but also rather administrative units such as field unit or administrative districts. In the following text the term “model unit” is used for these points.


Model Unit Attribute Table

Each model unit needs to be described with specific attributes for the modeling. These are: thumb|Sample for a model unit-parameter-table

  • ID - a clear numeric ID
  • X - the easting of the centre (centroid) as Gauss-Krüger coordinate
  • Y - the northing of the centre (centroid) as Gauss-Krüger coordinateder
  • area - the area of the model unit in m²
  • elevation - the mean elevation of the model unit in m above N.N.
  • slope - the mean slope of the model unit in degree
  • aspect - the aspect of the model unit in degree from North clockwise
  • soilID - a clear numeric ID for the soil type (serves as allocation to the soil attribute table)
  • landuseID - a clear numeric ID for the land use/land cover type (serves as allocation to the land use table)
  • hgeoID - a clear numeric ID for the hydrological unit of the model unit (serves as allocation to the hydrogeology table)

The model unit attribute table contains the attribute names in the first row. These must not be changed because they are used as variable name for the attribute generation during the model parameterization. The second row contains the minimal possible value of the referring attribute, whereas the third one contains the maximum possible value. In the fourth row the physical unit of the attribute needs to be entered. Then, as many rows as needed are given that contain the attribute values of the single model units. The table needs to be completed by a comment row which starts with the comment symbol (#). The tab (\t) has to be used to separate the rows.


Soil Attribute Table

thumb|Sample for a soil attribute table The soil attribute table contains soil-physical characteristics for each soil unit that occurs in the area. In the current model version only the usable field capacity for each decimeter is needed. On the basis of the field capacity the maximum storage capacity of the soil water storage is calculated in dependence of the effective root depth of the vegetation on the model unit during the model parameterization.

The format of the soil attribute table is very similar to the model units attribute table. The table can be started with an arbitrary number of comment rows that need to be started with the comment symbol (#). In the first interpreted row the attribute names need to be given. The spelling is very important here. The following attributes need to be included:

  • SID - clear numeric ID with which the connection to the model unit table is generated
  • depth - depth of the soil in cm
  • fc_sum - entire usable field capacity of the soil in mm
  • fc_1 bis fc_n - usable field capacity for each decimeter in mm/dm

To complete the table, a comment row must be given (introduced with #). The tab (\t) has to be used to separate the rows.

Land Use Attribute Table

thumb|sample for a land use attribute table The land use attribute table contains vegetation parameters that are nearly exclusively needed for the evaporation calculation according to Penman-Monteith. In the table the following attributes need to be given for each land use unit and land cover unit that occurs in the area:

  • LID - a clear numeric ID with which the connection to the model unit table is generated.
  • description - a description as text
  • albedo - albedo [0 ... 1]
  • RSC0_1 bis RSC0_12 - the stomatal resistances for good water availability for the months January (RSC0_1) to December (RSC0_12) in s/m
  • LAI_d1 bis LAI_d4 - the area leaf index in m²/m² for the Julian Days 110 (d1), 150 (d2), 250 (d3) and 280 (d4) for a terrain height of 400 m above N.N.
  • effHeight_d1 bis effHeight_d4 - effective vegetation height in meter for the Julian Days 110 (d1), 150 (d2), 250 (d3) und 280 (d4) for a terrain height of 400 m above N.N.
  • rootDepth - the effective root depth in dm

To complete the table, a comment row must be given (introduced with #). The tab (\t) has to be used to separate the rows.

Hydrogeology Attribute Table

thumb|sample for a hydrogeology attribute table In the hydrogeology attribute table the maximum possible ground water recharge rates per time unit are set up for each hydrogeologic unit. It only contains the following two attributes:

  • GID - a clear numeric ID with which the connection to the model unit table is generated.
  • mxPerc - the maximum possible percolation rate (ground water recharge rate) per time interval in mm per time unit

To complete the table, a comment row must be given (introduced with #). The tab (\t) has to be used to separate the rows.

Input Data - Model Driver

thumb|sample for an input data table For the model run climate time series of an arbitrary number of climate and precipitation stations are needed. The time series have to be preprocessed in the referring formatted data files. The files have the following format:

Rows 1 to 13 contain meta information about the data. They are arranged in blocks which are started with descriptions which in turn are started with the AT-symbol (@). Multiple entries per row are separated via the tab (\t).


  • @dataSetAttribs (attribute of the dataset)
    • missingDataVal - value that marks missing data
    • dataStart - start date of the dataset (DD.MM.JJJJ HH:MM)
    • dataEnd - end date of the dataset (DD.MM.JJJJ HH:MM)
    • tres - termporal resolution (days "d", months "m")
  • @statAttribVal (attributes of the climate stations)
    • name - the names of the climate stations
    • ID - a numeric, clear ID
    • elevation - the elevation of the station in m above N.N.
    • x - easting as Gauss-Krüger-coordinate
    • y - northing as Gauss-Krüger-coordinate
    • dataColumn - describes the column in which the data for the referring station can be found

In the following rows the actual data for each time step is listed – starting with:

  • @dataVal

The format is Date, Time followed by Data that is separated by the tab.

Installation and Start

Model Initialization

Regionalization

The regionalization module is used for the transfer of punctual values to the model units. The procedure was taken from the hydrological Model J2000 without any changes and is arranged in the following steps:

  1. Calculation of a linear regression between the station values and the station heights for each time step. Thereby, the coefficient of determination (r2) and the slope of the regression line (bH) is calculated.
  2. Definition of the n gaging stations which are nearest to the particular model unit. The number n which needs to be given during the parameterization is dependent on the density of the station net as well as on the position of the individual stations.
  3. Via an Inverse-Distance-Weighted Method (IDW) the weightings of the n stations are defined dependently on their distances for each model unit. Via the IDW-method the horizontal variability of the station data is taken into account according to its spatial position.
  4. Calculation of the data value for each model unit with the weightings from point 3 and an optional elevation correction for the consideration of the vertical variability. (The elevation correction is only carried out when the coefficient of determination –calculated under point 1 – goes beyond the threshold of 0.7.)

The calculation of the data value for each model unit (DWU) without elevation correction is carried out with the weightings (W(i)) and the values (MW(i)) of each n gaging station according to:

DW_{U} = \sum_{i=1}^{n}MW(i) \cdot W(i)

For the calculation with elevation correction the elevation difference (HD(i)) between the gaging station and the model unit as well as the slope of the regression line (bH) are taken into account. Thus, the data value for the model unit (DWU) is calculated according to:

DW_U = \sum_{i=1}^{n}\left(\left(HD(i) \cdot b_H + MW(i)\right) \cdot W(i)\right)

Precipitation Correction

Precipitation values partly show a clear systematic error in measurement which is caused instrument-determined or due to the selection of the instrument position. This error in measurement has two causes: (1) the moistening error and evaporation error, which each depends on the type of the instrument and (2) the wind error which emerges due to the drifting of the precipitations. Both errors in measurement are strongly dependent on the type (rain or snow) of the precipitation amount.

For the correction of both errors a correction method according to Richter (1995) is used which is applied in the same way in the Model J2000. The procedure for the evaporation correction differs with regard to the temporal resolution of the precipitation data.


Monthly Precipitation Correction

thumb|Monthly percental precipitation errors in measurement for slightly sheltered stations, concentrated regionally (Richter 1995)

The corrected precipitation (Pkorr) is calculated for monthly time steps on the basis of the decrease of the measured precipitation (Pm) and the percental monthly error in measurement (MFt) (see adjacent table):

P_{korr} = P_m + P_m \cdot \frac{MF_t}{100}

Daily Precipitation Correction

For the daily precipitation correction the two error terms are calculated explicitly. For this purpose, correction functions were derived on the basis of the tabled error values according to Richter (1995).

Moistening Loss and Evaporation Loss

thumb|Moistening loss and evaporation losses by Hellmann precipitation gauges separated for the summer half year and winter half year in dependence of the daily value of the precipitation amount (according to Richter) and the adjusted correction function (Krause 2001)

For a continual error correction of the moistening loss and evaporation loss continual correction functions were adjusted to the tabled values by Richter (1995). They are shown in the adjacent figure. The correction functions were derived each separately for the winter half year (November till April) and summer half year (May till October). With these functions the moistening loss and evaporation loss is calculated in mm for precipitations <= 9 mm according to:

BV_{Som} = 0.08 \cdot \ln(N) + 0.225

BV_{Win} = 0.05 \cdot \ln(N) + 0.13

If the amount of precipitation is greater than 9 mm, a constant error of 0.47 mm in the summer half year and of 0.3 mm in the winter half year is assumed.

Wind Error

thumb|Characteristic of the relative correction function for the wind error based on the tabled values by Richter (1995) (Krause 2001) The calculation of the wind error of the precipitation measurement with the Hellman precipitation gauge is also carried out according to the tabled error values from Richter (1995) which was adjusted to the referring continual correction functions. For the correction it is differentiated whether the precipitation was rain or snow. The way of internal differentiation is described further below. The relative correction value is calculated as follows:

WK_s = 0.5424 \cdot N^{-0.211} for snow

WK_r = 0.1421 \cdot N^{-0.519} for rain.

The corrected precipitation value (PK) is finally calculated on the basis of the value (PM), the relative correction value for the wind error (WKs, WKr) as well as the moistening loss and evaporation loss (BVSom, BVWin) as follows:

P_K = P_M + P_M \cdot WK_{s,r} + BV_{Som,Win}

Radiation Calculation

For the evaporation calculation according to Penman-Monteith, the net radiation is needed as initial value and can be calculated on the basis of the global radiation. If the global radiation is not available, it can be defined approximately on the basis of the sunshine duration. For this purpose, a number of intermediate calculations are necessary. The following calculations act on the assumption of a daily modeling. When the model runs in monthly time steps, the calculations listed below are carried out on the 15th of each month. The resulting terms are then accumulated on the basis of the days per month.

Extraterrestrial Radiation

thumb|Annual course of extraterrestrial radiation (Ra), global radiation (Rg) and net radiation (Rn) for a position in the Thuringian Forrest The extraterrestrial radiation (Ra) is the short-wave radiation energy flux of the sun at the upper boarder of the atmosphere. Ra is calculated for a specific place in dependence of its latitude (lat in radians), the declination of the sun (decl in radians), the solar constant (Gsc in MJ / m2min), the hour angle at sundown (ws in radians) and the inverse relative distance between earth and sun (dr in radians) according to:

Ra = \frac{24\cdot60}{\pi} \cdot Gsc \cdot dr \cdot \left(ws \cdot \sin(lat) \cdot \sin(decl) + \cos(lat) \cdot \cos(decl) \cdot \sin(ws)\right)

The solar constant (Gsc in MJ / m2min) results from the Julian Date (jD [1... 365,366]) as follows:

Gsc = 0.08139 + 0.00291 \cdot \cos\left(\frac{\pi}{180} \cdot (jD - 15)\right) [MJ / m2min]

The relative distance between earth and sun (dr in radians) results from the Julian Date (jD [1... 365,366]) as follows:

dr = 1 + 0.033 \cdot \cos\left(\frac{2\pi}{365} \cdot jD\right) [rad.]

The declination of the sun (decl in radians) results from the Julian Date (jD [1... 365,366]) as follows:

decl = 0.40954 \cdot \sin(0.0172 \cdot (jD - 79.35)) [rad.]

The hour angle at sundown (ws in radians) results from the latitude (lat in radians) and the declination (decl in radians) as follows:

ws = \arccos(-1 \cdot \tan(lat) \cdot \tan(decl)) [rad.]

Global Radiation

The global radiation (Rg) is calculated on the basis of the extraterrestrial radiation (Ra in MJ/m²d) and the degree of cloudiness. At this, the degree of cloudiness is approximated on the basis of the relation of the measured sunshine duration (D in h/d) to the astronomic possible sunshine duration (S0 in h/d) with the help of the Angström formula. Thus, Rg is calculated as follows:

Rg = Ra \cdot \left(a + b \cdot \frac{S}{S_0}\right) [MJ/m²d]

The coefficients a and b need to be estimated for the position. Often 0.25 is used for a and 0.50 is used for b.

The maximum astronomic possible sunshine duration (S0 in h) is calculated on the basis of the hour angle at sundown (ws in radians) as follows:

S_0 = \frac{24}{\pi \cdot ws} [h/d]

Net Radiation

The net radiation (Rn in MJ/m²d) results from the single radiation components and provides the energy for the evaporation. The net radiation is calculated on the basis of the difference of the global radiation (Rg in MJ/m²d) and the effective long-wave radiation (Rl in MJ/m²d). The global radiation is reduced by the albedo (alpha) of the referring land cover.

Rn = (1 - \alpha) \cdot Rg - Rl

The effective long-wave radiation (Rl in MJ/m²d) is calculated on the basis of the Bolzmann constant (Bk = 4.903E-9 MJ/K4m²d), the absolute air temperature (Tabs in K), the actual vapor pressure of the air (ea in kPa), the actual global radiation (Rg in MJ/m²d) and the maximum global radiation for unclouded sky (Rg0 in MJ/m²d):


Rl = Bk \cdot Tabs^4 \cdot (0.34 - 0.14 \cdot \sqrt{ea}) \cdot \left(1.35 \cdot \frac{Rg}{Rg0} - 0.35 \right) [MJ/m²d]

The actual vapor pressure of the air (ea in kPa) is calculated on the basis of the saturation vapor pressure (es in kPa) and the relative humidity (U in %) according to the following equation:

ea = es \cdot \frac{U}{100} [kPa]

The saturation vapor pressure of the air (es in kPa) results from the air temperature (T in °C) according to:

es = 0.6108 \cdot e^{\frac{17.27 \cdot T}{237.3 + T}} [kPa]

The maximum global radiation for uncovered sky (Rg0 in MJ/m²d) results from the extraterrestrial radiation (Ra in MJ/m²d) and the terrain height (h in m above N.N.) as follows:

Rg0 = (0.75 + 2\cdot10^{-5} \cdot h) \cdot Ra [MJ/m²d]

Verdunstungsberechnung

Die Berechnung der potentiellen Verdunstung kann wahlweise nach Penman-Monteith oder Haude erfolgen. Der Vorteil der Berechnung nach Penman-Monteith ist die höhere physikalische Basiertheit allerdings werden auch deutlich mehr Eingangsdaten benötigt. Für die Berechnung nach Haude sind lediglich Lufttemperatur und relative Feuchte notwendig.

Potentielle Verdunstung nach Penman-Monteith

Die Berechnung der Bestandesverdunstung nach Penman-Monteith (PET in mm/d) erfolgt nach:

PET = \frac{1}{L_d} \cdot \frac{s \cdot (Rn - G) + \rho \cdot c_P \cdot \frac{es - ea}{r_a}}{s + \gamma \cdot \left(1 + \frac{r_s}{r_a}\right)} [mm/d]

mit:

Ld : Latente Verdunstungswärme [MJ/kg]

s : Steigung der Dampfdruckkurve [kPa / K]

Rn : Nettostrahlung [MJ/m²d]

G : Bodenwärmestrom [MJ/m²d]

ρ : Dichte der Luft [kg/m³]

cP : Spezifische Wärmekapazität der Luft (=1.031E-3 MJ/kg K]

es : Sättigungsdampfdruck der Luft [kPa]

ea : aktueller Dampfdruck der Luft [kPa]

γ : Psychrometerkonstante [kPa / K]

ra : aerodynamischer Widerstand der Bodenbedeckung [s/m]

rs : Oberflächenwiderstand der Bodenbedeckung [s/m]


Die latente Verdunstungswärme (Ld in MJ/kg) ergibt sich aus der Lufttemperatur nach:

L_d = \frac{2501 - 2.361 \cdot T}{1000} [MJ/kg]

Die Steigung der Sättigungsdampfdruckkurve (s in kPa/K) berechnet sich aus der Lufttemperatur nach:

s = \frac{4098 \cdot 0.6018 \cdot e^{\frac{17.27 \cdot T}{T + 237.3}}}{(T+237.3)^2} [kPa/K]

Der Bodenwärmestrom (G in MJ/m²d) ergibt sich aus der Nettostrahlung (Rn in MJ/m²d) nach:

G = 0.1 \cdot Rn [MJ/m²d]

Die Dichte der Luft (ρ in kg/m³) ergibt sich aus dem Luftdruck (p in kPa) und der virtuellen Temperatur (vT in K) nach:

\rho = 3.486 \cdot \frac{p}{vT} [kg/m³]

Liegt der Luftdruck (p in kPa) nicht als Messwert vor kann er näherungsweise aus Geländehöhe (h in m ü.NN) und der absoluten Lufttemperatur (Tabs in K) wie folgt berechnet werden:

p = \frac{1013 \cdot e^{\left(-1\cdot\frac{9.811}{8314.3 \cdot Tabs} \cdot h\right)}}{10} [kPa]

Die virtuelle Temperatur (vT in K) ergibt sich aus dem Luftdruck (p in kPa), der absoluten Lufttemperatur (Tabs in K) und dem aktuellen Dampfdruck der Luft (ea in kPa) nach:

vT = \frac{Tabs}{1 - 0.378 \cdot \frac{ea}{p}} [K]

Die Psychrometerkonstante (γ in kPa / K) berechnet sich aus der spezifischen Wärmekapazität der Luft (=1.031E-3 MJ/kg K), dem Luftdruck (p in kPa), dem Verhältnis der Molgewichte von trockener und feuchter Luft (VM = 0.622) und der latenten Verdunstungswärme (Ld in MJ/kg) nach:

\gamma = \frac{c_P \cdot p}{VM \cdot L_d}

Der Oberflächenwiderstand der Bodenbedeckung (rs in s/m) berechnet sich aus dem Blattflächenindex (Leaf Area Index LAI), dem Stomatawiderstand zum gegebenen Zeitpunkt (rsc in s/m) und der Oberflächenwiderstand einer unbewachsenen Oberfläche (rss = 150 s/m) nach:

r_s = \frac{1}{\frac{1 - 0.7^{LAI}}{rsc}+\frac{0.7^{LAI}}{rss}}

Die aerodynamische Rauhigkeit der Bodenbedeckung (ra in s/m) berechnet sich aus der Windgeschwindkeit (v in m/s) und der effektiven Wuchshöhe der Vegetation (eH in m)

r_a = \frac{9.5}{v} \cdot \ln\left(\frac{2}{0.1 \cdot eH}\right)^2 [s/m] für Bestände mit eH < 10 m

r_a = \frac{20}{0.41^2 \cdot v} [s/m] für Bestände mit eH >= 10 m

Potentielle Verdunstung nach Haude

Die potentielle Verdunstung nach Haude berechnet sich aus dem Sättigungsdefizit der Luft und einem empirischen, dimensionslosen Haude-Faktor (hF). Das Sättigungsdefizit ergibt sich aus dem Sättigungsdampfdruck (es in kPa) und der relativen Luftfeuchte (U in %). Der Haude-Faktor muss für unterschiedliche Vegetationsarten bestimmt werden. Die Berechnung der potentiellen Verdunstung (PET in mm) erfolgt nach:

PET = es \cdot \left(1 - \frac{U}{100}\right) \cdot hF

Der Sättigungsdampfdruck der Luft (es in kPa) ergibt sich aus der Lufttemperatur (T in °C) nach:

es = 0.6108 \cdot e^{\frac{17.27 \cdot T}{237.3 + T}} [kPa]

Schneedeckenberechnung

Die Schneedeckenberechnung ist als einfacher Akkumulations- und Schmelzansatz implementiert. Das Verfahren entscheidet anhand der Lufttemperatur ob Wasser als Schnee auf einer Modelleinheit gespeichert wird oder ob potentiell vorhandener Schnee schmilzt und Schneeschmelzabfluss produziert. Hierzu werden zunächst zwei Temperaturen aus Minimum- (Tmin), Durchschnitts- (Tavg) und Maximumtemperatur (Tmax) des jeweiligen Zeitschrittes berechnet:

Die Akkumulationstemperatur als: T_{acc} = \frac{T_{min}+T_{avg}}{2} [°C] und

die Schmelztemperatur als: T_{melt} = \frac{T_{avg}+T_{max}}{2} [°C]

Liegt die Akkumulationstemperatur (Tacc) gleich oder unterhalb eines vom Anwender anzugebenden Grenzwertes (Tbase) wird davon ausgegangen, dass eventuell auftretender Niederschlag als Schnee fällt. Dieser wird dann auf der Modelleinheit zwischengespeichert.


Wenn die Schmelztemperatur den Temperaturgrenzwert Tbase übersteigt wird Schneeschmelze mit Hilfe eines einfachen Schneeschmelzfaktors (TMF) berechnet. Hierzu wird eine potentielle Schneeschmelzrate aus TMF (in mm/d K), dem Temperaturgrenzwert und der Schmelztemperatur berechnet:

SM_p = TMF \cdot (T_{melt} - Tbase) [mm/d]

Diese potentielle Schmelzrate wird dann gegen das aktuell gespeicherte Schneewasseräquivalent verglichen, das dann teilweise oder vollständig geschmolzen wird. Das resultierende Schneeschmelzwasser wird an das folgende Modul als Input weitergegeben.

Bodenwasserhaushalt

Das Bodenwasserhaushaltsmodul dient als Verteiler des Inputs (Niederschlag und Schneeschmelze) auf die Outputpfade (Verdunstung, Direktabfluss, Grundwasserneubildung). Zentrales Element ist der Bodenspeicher der durch die nutzbare Feldkapazität des durchwurzelten Bodenbereiches dimensioniert ist. Das maximale Füllvolumen kann durch einen multiplikativen, anwendergesteuerten Eichparameter (FCA) angepasst werden.

Der Input (Niederschlag und Schneeschmelze) wird zunächst der Verdunstung zugeführt bis die potentielle Verdunstung erreicht ist. Der Überschuss (Inflow) wird dann in Direktabfluss (SQ) und Infiltration (INF), abhängig von der relativen Bodenwassersättigung (Θ) und einem Eichparameter (DFB) aufgeteilt. Die Berechnung erfolgt nach:

SQ = \Theta^{DFB} \cdot Inflow [mm/d]

INF = (1 - \Theta^{DFB}) \cdot Inflow [mm/d]

Die Infiltration (INF) wird dem Bodenspeicher zugeschlagen bis dieser vollkommen gesättigt ist. Entsteht dann Überschuss (excess water EW) wird dieses in Abhängigkeit von der Hangneigung (α) und einem Eichparameter (LVD) auf die beiden Pfade Interflow (SSQ) und Grundwasserneubildung (GWR) aufgeteilt. Die Berechnung erfolgt nach:

SSQ = \tan\alpha \cdot LVD \cdot EW [mm/d]

GWR = (1 - \tan\alpha) \cdot LVD \cdot EW [mm/d]

Die Grundwasserneubildung wird im Folgenden als Quelle für den Basisabfluss (BQ) betrachtet.

Abflusskonzentration und Abflussverzögerung

Die Abflusskonzentration und Abflussverzögerung wird als einfache Funktion des Einzugsgebietes für die drei Abflusskomponenten (Direktabfluss SQ, Zwischenabfluss SSQ und Basisabfluss BQ). Hierzu werden die Bildungsraten der drei Komponenten flächengewichtet aufsummiert und durch eine jeweils eigene lineare Speicherkaskade (Nash-Kaskade) geleitet. Für jede dieser drei Kaskaden muss der Anwender die beiden Parameter Anzahl der Speicher (n) und Retentionskoeffizient (k) bestimmen.

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