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).