Figure 3 should be here
By comparing (a) and (b) of Fig. 3, it could be seen that q was in general positively correlated with the flood wave, but the positive and negative extremes (i.e.q +max, |q -max|) appeared before the peak and the initial value of the stream stage, respectively. This finding was consistent with previous studies (Chen and Chen, 2003; McCallum et al., 2010; Doble et al., 2012; Gu et al., 2012; Siergieiev et al., 2015; McCallum and Shanafield, 2016; Shuai et al., 2017); however, previous explanations to the mechanism of this phenomenon are insufficient. Therefore, the response mechanism ofq to a flood wave was further investigated by theoretical derivation in this study.
It could be seen from Formula (8) that q has a linear relationship with the polynomial “ωh + v ”. When the flood wave first rose and then fell in a sinusoidal manner, vfirst decreased and then increased in a cosine manner. The superposition of the two processes could well explain whyq +max and |q -max| appeared before the peak and initial value of the flood wave, respectively. In other words, although the flood wave had not yet reached its peak, the value of the polynomial “ωh + v”had reached its maximum, which made q reachq +max. The same was hold for why |q -max| occurred prior to the initial water level. In addition, the occurrence time of q +max and |q -max| were close to that of the peak or initial flood wave value, because the influence of h on q was much larger than that of v : the flood wave of the base case in Fig. 3 (i.e. T = 3d) causedω (= 2pi/T) to be greater than 1.
According to Formula (8), we randomly selected a flood wave as an example and showed the variations of the flood wave and its change rate as well as their polynomial ”ωh + v ” through Fig. 4. The moments of t1 and t2 corresponded to the two inflection points of the polynomial ”ωh + v ”. It could be seen that they appeared before the peak and initial value of the flood wave, respectively, which further verified the findings from the numerical simulation (Fig. 3).