Appendix B. Water Balance Model (WBM) Description (Vörösmarty 1998)

 

The WBM simulates soil moisture variations, evapotranspiration, and runoff on single grid cells using biophysical data sets that include climatic drivers, vegetation, and soil parameters. The state variables are determined by interactions among time-varying precipitation, potential evaporation, and soil water content. The original model has been described in detail by Vörösmarty and coworkers (Vörösmarty et al. 1989, Vörösmarty et al. 1996, Vörösmarty and Moore 1991). The governing equations can be summarized as follows:

 

Equation variables:

dWs/dt              change in soil moisture

p                      probability that a day has a wetting event

Pa                     available precipitation (rainfall and snowmelt) [mm day-1]

g(Ws)                unitless soil drying function

Ep                     potential evaporation [mm day-1]

Dws                    soil moisture deficit equal to amount of water required within a time step to fill soil to its water holding capacity and simultaneously satisfy potential evaporation [mm day-1]

Es                     estimated actual evapotranspiration [mm day-1]

Xr                      excess rainfall (available for runoff and runoff detention pool) [mm day-1]

Pr                     rainfall

Xs                     excess snowfall (available for runoff and runoff detention pool) [mm day-1]

Ms                    snowmelt

Ws                    soil moisture [mm]

 

Change in soil moisture

dWs/dt = pPa{g(Ws) + [1-g(Ws)]e-(Ep/Pa) – e-(Dws/Pa)} – Epg(Ws)           Pa < Ep

(original equation = -g(Ws)(Ep – Pa))

            = Pa – Ep                       Ep < Pa < Dws

            = Dws – Ep                      Dws < Pa

 

Estimated actual evapotranspiration

Es         = Pa – dWs/dt                Pa < Ep

            = Ep                              Ep < Pa

 

Excess rainfall available for runoff and recharge of runoff detention pools

Xr          = 0                               Pa < Dws

            = Pr –Dws                       Dws < Pa

 

Excess snowfall available for runoff and recharge of runoff detention pools

Xs         = 0                               Ms < Dws

            = Ms – Dws                     Dws < Ms

 

The unitless drying function

Equation variables:

a                      empirical constant

Ws                    soil moisture [mm]

Wc                    soil and vegetation-dependent available water capacity [mm]

 

g(Ws)    = 1 – e (-aWs/Wc) / 1 – e-a

a = 5.0 so that the drying curve would resemble that of Pierce (1958) when g(Ws) = Es/Ep is plotted as a function of Ws/Wc during periods of no precipitation.

 

Wetting Event Probability

Equation variables:

p                      probability that a day has a wetting event

u = 0.005          best fit for broad geographic range

Prm                    monthly rain

 

p = 1 – e-u(Prm)        

Assumes that there is an exponential distribution of rainfall amounts during those days that are rainy within a month.

 

Precipitation available for soil recharge as rainfall

Equation variables:

Pr                     rainfall

Prm                    monthly rain

nd number of days in a month

 

Pr = Prm/ndp

During periods of snowmelt recharge we add Ms directoly to Prm, increasing the probability of soil wetting. We have assumed that the within-month variability in potential evaporation is much less than for rainfall and snowmelt recharge and can be approximated by Ep = Epm/nd, where Epm is computed potential evaporation.

 

Rainfall-derived Runoff

Equation variables:

Dr                                          rainfall runoff detention pool

Rr                     rainfall-derived runoff emerging from the grid cell

Xr                      excess rainfall (available for runoff and runoff detention pool) [mm day-1]

b and g              empirical constants set to 0.0167 [day-1] and 0.5 respectively

 

dDr/dt = ( 1 – g) Xr – bDr

Rr = gXr +bDr

 

Snowpack

Snowpack accumulates when monthly temperatures are below –1.0 oC. Snowmelt is a prescribed

function of temperature and elevation. Snowmelt occurs at or above –1.0 oC.

 

Equation variables:

Ps                     snowfall [mm day –1]

Ks                     accumulated snowpack [mm water equivalent]

Ms                    snowmelt excess [mm day –1]

Ep                     potential evaporation [mm day-1]

T                      monthly temperature

 

At or below 500 m elevation, Ms = Ks in the first month where T >= -1.0 oC.

dKs/dt = Ps – Ep             T < -1.0 oC

dKs/dt = -Ms –Ep             T >= -1.0 oC

 

Above 500 m elevation, snowmelt proceeds over 2 months with 0.5 of Ks lost within each month.

 

Snowmelt detention pool

A detention pool is tracked which is used to generate runoff associated with snowmelt.

 

Rs                     snowmelt-derived runoff [mm month-1]

Ds                     snowmelt-derived detention pool

Xs                     excess snowfall (available for runoff and runoff detention pool) [mm day-1]

 

 

dDs/dt = Xs – Rs

Rt = Rs +Rr

 

At or below 500 m:

Rs = 0.1 Ds        month = 1 of T >+ -1/0 oC

Rs = 0.5 Ds        month > 1 of T >= -1.0 oC

 

Above 500 m:

Rs = 0.1 Ds        month = 1 of T >= -1.0 oC

Rs = 0.25 Ds      month = 2 of T >= -1.0 oC

Rs = 0.5 Ds        month > 2 of T >= -1.0 oC

 

The expected monthly changes in the pools are calculated as the average daily change multiplied by nd, the number of days in each month. Likewise, the associated water fluxes computed by the WBM are initially expressed as a daily average for the duration of each month. These also are multiplied by nd to obtain corresponding monthly values. The time varying changes in Ws, Ks, Dr and Ds are solved using a fifth-order Runge-Kutta integration technique (International Mathematical and Statistical Libraries, Houston TX).