Budyko water- and energy-balance model

The Budyko water- and energy-balance model is based on work conducted by Mikhail Ivanovich Budyko over a series of decades (Budyko 1974, 1980).  To run Budyko’s model, you must calculate the radiation balance at the surface of the Earth by the following formula:



       (1)        B = Qa - Ql


where B is the radiation balance (ly/day W/m2); Qa is the absorbed shortwave radiation (ly/day or W/m2) at the surface; and Ql is the net longwave radiation at the surface (ly/day or W/m2).


Absorbed shortwave radiation (Qa) is calculated by the following:


       (2)        Qa = Q(1-á)


where Q is the total monthly solar radiation at the surface (ly/day or W/m2); á is the mean monthly albedo of the surface.  Albedo data are not available in Müller (1982).  Therefore ClimVegWin uses a default value of 0.15, which appears to be about average for most vegetated surfaces (Budyko 1974, p. 54, Table 6; Budyko 1986, p. 22, Table 2.3).  These are the albedo values listed by Budyko:


Vegetation type

Albedo Range

Rye and wheat fields

Potato fields

Cotton fields

Meadows

Dry steppe

Tundra

Coniferous forest

Deciduous forest

0.10-0.25

0.15-0.25

0.20-0.25

0.15-0.25

0.20-0.30

0.15-0.20

0.10-0.15

0.15-0.20


Total monthly solar radiation at the surface is either obtained directly from measured values or else estimated using the radiation estimation routine.  Estimated values are adjusted to take into account the cloudiness (fraction of sky obscured by clouds) of the atmosphere.  This is done using the following formula:


       (3)        Q = Qo(1-an-bn2)


Where Qo is the estimated monthly solar radiation at the surface; n is mean monthly cloudiness (default is 0.5, but program allows user to select values ranging from 0.1 to 1.0); and a and b are empirically derived coefficients.  The coefficient b is relatively stable;  Budyko (1974, p. 48) recommends the mean value of 0.38 and that is what is used in ClimVegWin.  The coefficient a varies with latitude.  The value used in ClimVegWin is interpolated from the values given in Budyko (1974, p. 48, Table 4).


Latitude (N or S)        

a

85

80

75

70

65

60

55

50

45

40

35

30

25

20

15

10

5

0

0.14

0.15

0.16

0.18

0.25

0.36

0.41

0.40

0.38

0.38

0.38

0.36

0.35

0.37

0.39

0.40

0.40

0.38


Budyko (1974) and Monserud et al. (1993) use a different technique to estimate Qo, but the differences between what they use and my estimation routine appear (on a quick glance) to produce similar results.


Net longwave radiation (Ql) is calculated by the following formula (Monserud 1993, p. 62):



       (4)        Ql = εóT4(11.7-0.23E)(1-cnd)


Where ε is the coefficient of emissivity (0.95; Budyko 1974, p. 58), ó is the Stefan-Boltzman constant (1.17x10-7 ly/day-K4), T is the mean monthly temperature (°K), E is the mean monthly vapor pressure (mb), c is an empirically derived coefficient (Budyko 1974) and d describes the relationship between cloudiness and net longwave radiation.


The first step in calculating the vapor pressure, E, is to calculate the saturation vapor pressure estimated using the formula given in Murray (1967):


       (5)        es = 6.1078*exp((asvp(T-273.16))/(T-bsvp))


where es is the saturation vapor pressure (mb); T is the mean monthly temperature (°K); and asvp and bsvp are coefficients.  Murray recommends the following values for asvp: 21.8745584 over ice and 17.2693882 over water.  For bsvp, he recommends: 7.66 over ice and 35.86 over water.  ClimVegWin uses Murray’s recommendations for ice when temperatures are less than or equal to 0°C.  For temperatures greater than 0°C, ClimVegWin uses Murray’s recommendations for water.


The values for c are interpolated from values given in Budyko (1974, p. 49, Table 9).  They are:


Latitude (N or S)

c

75

70

65

60

55

50

45

40

35

30

25

20

15

10

5

0

0.82

0.80

0.78

0.76

0.74

0.72

0.70

0.68

0.65

0.63

0.61

0.59

0.57

0.55

0.52

0.50


Once the radiation balance, B, is obtained, the two key components of the Budyko classification scheme can be calculated.  The first is potential evaporation (PE) which is calculated using the following formula:


       (6)        PE = B/L


Where L is the latent heat of vaporization of water (585 cal/g).  Finally Budyko’s dryness index can be calculated:


       (7) DI = PE/MAP


Where MAP is mean annual precipitation (cm).


Vegetation types are classified using Budyko’s original method, as described in Budyko (1986, p. 94, Table 3.8), as well as the modified method of Monserud et al. (1993, p. 61, Table 1).

Copyright © 2003-2011, David M. Lawrence

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