In the ‘Catchments as simple dynamical systems’-approach, the main assumption is that discharge solely depends on the amount of water stored in a catchment, and that the function describing this relation is unique. A motivation for this approach is that the physical processes taking place in catchments are highly complex and heterogeneous. In this approach, these processes are considered in a lumped fashion at the catchment scale (Kirchner, 2009). This approach is derived from the water balance:
dS
dt =P−E−Q, (2.1)
with dSbeing the change in storage in the catchment over time dt,Pis precipitation,E
evaporation andQdischarge. It is assumed that there is a functionfdescribing the re-
lation between discharge and storage, and that this relation is invertible (no hysteresis):
Q=f(S), S=f−1(Q). (2.2)
The change in discharge over time can be related to the water balance: dQ dt = dQ dS dS dt = dQ dS(P−E−Q), (2.3) in whichdQ
dS is the sensitivity of discharge to changes in storage. This is the derivative of the functionfdescribing the relation between storage and discharge. Generally, this
measurable. Discharge is, and since it is assumed that the relation between storage and discharge is invertible, the derivative can be expressed as a function of discharge:
dQ
dS =f
0(S) =f0(f−1(Q)) =g(Q). (2.4)
Thefunctiondescribingthederivativeofdischargetostorageistypicallyreferredtoasthe
sensitivity functiong(Q). Whileg(Q)is not restricted to a particular form, the common
power-law representation was chosen (Brutsaert and Nieber, 1977;Kirchner, 2009;Troch et al., 2013) in Chapter 3. Applying a power-law results in a model with only two parame-
ters:
g(Q) = dQ
dS =αQ
β. (2.5)
The change in discharge (dQ) due to a change in storage (dS) is described by the param-
etersαandβand the dischargeQ. Note that the dimension ofαdepends on the value
ofβ. There are fundamental differences in the behaviour of the system forβ <1,β = 1
andβ >1(Kirchner, 2009). For example,β <1implies that there is a residual storage
S0which remains in the catchment if discharge reduces to zero, forβ = 1there will be
discharge at all storage values, which implies that storage can decline indefinitely, and forβ >1S0is no longer the lower but the upper bound of storage in the catchment. For
a comprehensive discussion on the effect ofβ, we refer toKirchner(2009).
The ‘Catchments as simple dynamical systems’-approach has been explored in the Riet- holzbach catchment. The Rietholzbach experiences intermittent snow cover in winter, and therefore snowmelt had to be accounted for. SnowmeltM(mm h-1) is assumed to
be dependent on radiation and temperature, following the Restricted Degree-Day Radi- ation Balance approach byKustas et al.(1994):
M =F1(T2m−T0) +F2Rg. (2.6)
F1(mm h-1◦C-1) andF2(mm h-1(W m-2)-1) are parameters controlling the melt rate,T2m
(◦C) is the measured temperature at 2 m height,T
0(◦C) is a threshold temperature, and Rg(W m-2) is, in contrast to the net radiation whichKustas et al.(1994) proposed, the global radiation.Teuling et al.(2010) argued that for small catchments with partial snow
cover, like the Rietholzbach, net radiation is highly variable and therefore global radia- tion was used, which is independent of local surface conditions.
When the snowmelt parameters, and the parametersαandβof the sensitivity function g(Q)are identified, the discharge can be simulated. Due to the strong non-linearity
ofg(Q), numerical stability improves if changes in log(Q) are simulated, rather than
changes inQ(Kirchner, 2009). This leads to the following differential equation:
d(log(Q)) dt = 1 Q dQ dt =g(Q)· P+M −ET Q −1 , (2.7)
wherePis rainfall (mm h-1),M snowmelt (mm h-1) andETevapotranspiration (mm
h-1). Equation 2.7 is solved with a fourth-order Runge-Kutta scheme with variable time
step as proposed byKirchner(2009) and adopted byTeuling et al.(2010) andBrauer et al.
2.2. Model descriptions
The SAC-SMA model
RAIN
UPPER Z
ONE
LOWER Z
ONE
tension water storage free water storage
tension water storage primary free water storage tension water storage supple- mentary free water storage impervious and direct runoff surface runoff interflow baseflow
Figure 2.7: Overview of the processes incorporated in the Sacramento Soil Moisture Accounting model (SAC-SMA). Adapted from National Weather Service (2002).
The Sacramento Soil Moisture Accounting model (SAC-SMA,Burnash et al., 1973) was de-
veloped by the US National Weather Service, with the goal to provide relatively short- term discharge predictions. The following description of the SAC-SMA model (from now onreferredto simply asSAC)isbasedon thereport oftheNational Weather Service(2002).
The two basic components of SAC are tension water, water present in the soil but due to absorption to soil particles only removable through evaporation and transpiration, and free water, water that is available for percolation and drainage. Furthermore, SAC divides the soil into an upper and a lower zone. The tension water in the upper zone (represented in the UZTWM parameter) represents the amount of water which can be absorbed by the soil before drainage takes place. The free water in the upper zone (represented in the UZFWM parameter) either moves laterally through the soil as interflow, or drains vertically into the deeper soil. In the lower zone, the tension water (LZTWM) represents the amount of moisture necessary to satisfy the moisture requirements of the soil due to molecular attraction. The free water in the lower zone provides the reservoirs which gen- eratebaseflow. Freewaterinthelowerzoneisdividedintoaprimarytype(LZFPM),which represents slow drainage, and a secondary type (LZFSM), which represents fast drainage after a recent rainfall event. The fast drainage can supplement the slow drainage. Direct runoff is generated from impervious areas (which can be parameterized with the PCTIM parameter) and when rainfall intensity exceeds the infiltration rate of the soil or when the soil is saturated. An overview of the processes described in the SAC model is given in Figure 2.7.