WATER RESOURCES RESEARCH, VOL. 38, NO. 2, 10.1029/2001WR000482, 2002

2. Flood Peak Estimation Using a Simple Storage Model

[9]   To throw light on the differences between the two definitions of nonlinearity, we introduce a simple, lumped, and linear model of catchment response. Catchment response consists of two independent component processes: runoff generation on hillslopes and flow routing in the channel network. For simplicity, these are represented through two linear stores, arranged in series. The hillslope store receives rainfall input at rate i [L/T] over a storm duration t_r and delivers runoff to the network store, located downstream of it, at a rate q_h[L/T]: the latter is assumed to be a linear function of the volume of water stored in the hillslope, denoted by S_h[L]. The outflow from the network store, q[L/T], which is the same as the outflow from the catchment at its outlet, is assumed to be a linear function of the water stored in the network, denoted by S_n[L]. Figure 1 provides a schematic of the simple linear model. The water balance equations for these two parts of the model are expressed as

where t_h is the mean hillslope response time and t_n is the mean channel response time, both of which are assumed to be constants for the catchment in question. All of the quantities in (5a), (5b) – (6a), (6b) are normalized by catchment area and are expressed in units of [L] or [L/T]. We also define an unscaled discharge by converting q(t) to Q(t) [L³/T] by multiplying by the catchment area A[L²], i.e., Q(t) = Aq(t).

[10]   For simplicity, we look at the case where the rainfall input to the catchment falls at a constant (in time and space) rate iand where the catchment is dry initially, meaning the stores S_h and S_n are empty at the beginning of the storm. The effects of antecedent conditions have been studied previously in detail within the same context by Robinson and Sivapalan [1997b] and will not be repeated here. The outflow hydrograph for the simple case can be derived in a straightforward manner. Here we present only the final expressions for the peak of the resulting hydrograph, denoted by q_p[L/T] and Q_p[L³/T]:

The unscaled peak discharge Q_pis simply given by

[11]   Equations (7a), (7b), and (7c) are used to compute the magnitudes of the dimensionless flood peaks, q_p/i, for different values of the timescales t_r, t_h, and t_n. The results are presented in Figure 2a as families of curves relating q_p/i to t_n for different values of t_r and t_h. They show that the flood peaks, q_p/i [L/T], remain constant for small values of t_n (scaling exponent with respect to t_n is zero), while for larger values of t_n they decrease linearly with increasing values of t_n (scaling exponent is -1). The effect of t_h is to smooth and thus reduce the magnitude of the flood peaks, without changing the scaling exponents with respect to t_n. This can easily be seen by asymptotic analysis of (7a), (7b), (7c) for t_n 0 and t_n .

[12]   Since we are interested in the relationship of the flood peaks q_por Q_pto catchment area A, we assume, for simplicity, that (1) t_h is independent of catchment area (a reasonable assumption used often; see Robinson et al. [1995]) and (2) t_n is a scaling function of catchment size A [Robinson and Sivapalan, 1997b] of the form

where t_n is in hours and A is in km². In this case, for a start, we assume that = 0.28, and = 0.5. The results shown in Figure 2a can now be converted to a relationship of q_p/i and of Q_p (using equations (7a), (7b), (7c) and (8) with an assumed value of i = 50 mm/h), against catchment area A. The results are presented in Figures 2b and 2c. In the case of Figure 2b the results are similar to those shown in Figure 2a, except that the scaling exponents with respect to catchment area Aare 0.0 and -0.5, i.e., -.

[13]   Focusing on Figure 2c, we can clearly see that the scaling exponent of Q_pwith respect to catchment area A changes from 1.0 (unity) for small catchments to 0.5 for large catchments, with a transition zone in the middle with variable exponents. This is similar in character to the “nonlinear” relationship, in the sense of (3), exhibited in previous studies, including that at Walnut Gulch [Goodrich et al., 1997]. It should be noted here that for general values of the exponent for our simple case changes from 1 for small catchments to 1 – for large catchments.

[14]   Note that while the catchment response, in the sense of (1) or (2), remains linear for all catchment sizes, the observed change in scaling exponent is caused by a change of runoff process, as shown by Robinson and Sivapalan [1997a] and Gupta and Waymire [1998]. In small catchments, storm duration is long compared to the catchment's residence time, and consequently the catchment reaches steady state, with the whole catchment area contributing to the flood peak. As long as this remains true, the flood peak increases linearly with catchment area, and thus the scaling exponent remains 1 (unity). On the other hand, in “large catchments,” storm duration is smaller than the catchment's residence time. The fraction of the catchment area contributing to the flood peak is proportional to the ratio of storm duration to catchment residence time. This ratio, following (9), decreases at the rate of A with an increase of catchment area A. Thus the partial area contributing to flood peak increases only at the rate of A, leading to the exponent 1 - , which is less than unity.

[15]   The above demonstration has been based on the response to a single storm. We have not chosen to present results on the scaling behavior of the mean annual flood or of the flood peak with a specified return period, in the true sense of (3). This would require much more detailed statistical analyses or random simulations, involving the derived flood frequency method. Such extensive analyses with more complex models have indeed been carried out and do support the conclusions reached here [Gupta and Waymire, 1998; Blöschl and Sivapalan, 1997; Robinson and Sivapalan, 1997a, 1997b]. For example, for the case corresponding to t_h = 0, Robinson and Sivapalan [1997a] have shown that the scaling exponents of mean annual flood changed from 1.0 to 1 - , as it was for the simple case presented here earlier.

[16]   In summary, what we have demonstrated is that a simple, linear model in the sense of (2) can still lead to the “nonlinear” scaling type of behavior in the sense of (3) and (4). While the scaling exponent changed from 1 to 1 - with increasing area A, the dynamical catchment rainfall-runoff response has remained linear. Indeed, our model confirms the (partial) explanation given by Gupta and Waymire [1998], Robinson and Sivapalan [1997a], and Blöschl and Sivapalan [1997] for the observed decrease of scaling exponents with increasing catchment area, namely the interaction between mean storm duration and catchment residence time.

Citation: Sivapalan, M., C. Jothityangkoon, and M. Menabde, Linearity and nonlinearity of basin response as a function of scale: Discussion of alternative definitions, Water Resour. Res., 38(2), 10.1029/2001WR000482, 2002.

Copyright 2002 by the American Geophysical Union.