COMPARISON BETWEENKINEMATIC AND DYNAMIC HYDRAULIC DIFFUSIVITIES
In diffusion wave modeling of catchment dynamics,
the hydraulic diffusivity may be expressed in two alternative ways:
(1) kinematic; or (2) dynamic. The kinematic hydraulic diffusivity
is due to Hayami (1951), who pioneered the diffusion wave approach
to surface flow. An extension of the concept
of hydraulic diffusivity to the domain of dynamic waves
followed, notably first by Dooge and his collaborators
( 1991a;
1991b).
The kinematic hydraulic diffusivity is
(
^{_______}2 S_{o}
(1) |
in which S = bottom slope.
_{o}
^{_______} ( 1 - ^{_______} )2 S_{o} 4
(2) |
in which
Subsequenty,
β of
the discharge-area rating curve, wherein Q = α A^{ β}.
The enhanced formulation is the following:
^{_______} [ 1 - (β - 1)^{2} F^{2} ]2 S_{o}
(3) |
Ponce
(
^{_______} (1 - V^{ 2})2 S_{o}
(4) | in which the Vedernikov number is:
While the
This paper sheds light on the role of the Vedernikov number in catchment dynamics. The specific
question to be answered is the following:
How important is the
Overland flow is surface runoff that flows across the land
immediately following a rainfall event.
The diffusion wave model is an improvement over the conventional
kinematic wave model
because it is
A reference test problem was chosen for use together
with script
- Row 11 (overland flow area
*A*), - Row 13 (slope of left plane
*S*)._{lp} - Row 15 (rating exponent for left plane
*β*), and_{lp} - Row 20 (slope of channel
*S*)._{ch}
These parameters are selected to perform sensitivity analysis. The objective is to determine the difference in catchment response (i.e., the outflow hydrograph), between the two types of hydraulic diffusivities considered here: (1) kinematic (Eq. 1); and (2) dynamic (Eq. 4). The results of the sensitivity analysis are presented in Sections 4 to 7.
In this section we describe the response of the diffusion
wave model of catchment dynamics to the variation of drainage area, i.e.,
the overland flow area
P/t = 24 cm / 12 hr = 2 cm/hr.
_{r}
i _{e}A =
2 cm/hr × 18 ha = 1 m^{3}/s.
Figure 2 shows the hydrographs calculated by
We note that while hydrographs a, b, c, and d are
Figure 3 shows the calculated
Vedernikov number as a function of drainage area, using the dynamic hydraulic
diffusivity (Eq. 4). In order to accomodate the outflow for the
larger drainage areas, the design channel depth in the reference test problem
(Table 1, line 23) was increased to 1.2 m.
In the planes, the Vedernikov numbers are quite low,
increasing with an increase in drainage area
In this section we describe the response of the diffusion wave model
of catchment dynamics to the variation of plane slopes, i.e.,
the slope of the left plane S (Table 1, row 16).
In all cases, it is assumed that _{rp}S
= _{rp}S (reference test problem).
_{lp}
As with Section 4, the response
of the reference test problem is a superconcentrated
catchment flow hydrograph, wherein the rainfall duration (12 hr)
exceeds the time of concentration (
P/t = 24 cm / 12 hr = 2 cm/hr.
_{r}
i _{e}A =
2 cm/hr × 18 ha = 1 m^{3}/s.
Figure 4 shows the hydrographs calculated by
As expected, hydrographs a and b are
Figure 5 shows the calculated
Vedernikov number, as a function of plane slope, using the dynamic hydraulic
diffusivity (Eq. 4).
In the planes, Vedernikov numbers are quite low,
increasing with an increase in slope, from V =
0.05 for a slope S = 0.01.
In the channel,
the Vedernikov number is high (_{p}V = 0.58) and remains constant
(with the variation in plane slope),
since the relatively high channel slope
(S = 0.01) is constant and
corresponds to the _{ch}reference test problem (Table 1, row 20).
In this section we describe the response of the
diffusion wave model of catchment dynamics to the variation in rating exponent
Figure 6 shows the hydrographs calculated by
Note that all three hydrographs, a, b, and c,
are
Figure 7 shows the calculated
Vedernikov number as a function of rating exponent reference test problem (Table 1, row 20).
Figure 8 shows the hydrographs calculated by
As expected, hydrographs a and b are
Figure 9 shows the calculated
Vedernikov number as a function of channel slope,
using the dynamic hydraulic
diffusivity (Eq. 4).
In the planes, Vedernikov numbers remain constant and equal to
S = 0.001,
corresponding to the _{rp}reference test problem (Table 1, rows 13 and 16, respectively).
As expected, in the channel
the Vedernikov numbers increase sharply, from V = 0.008S = 0.00001),
to _{ch}V = 0.55 (near critical flow)
for a relatively steep slope S = 0.01)._{ch}
The results of Sections 4 to 7 have clearly shown that the difference
in outflow hydrographs between the two formulations for hydraulic
diffusivity (kinematic, Eq. 1; and dynamic, Eq. 4), is
negligible for the tested flow conditions listed in Table 1.
The reason for this
behavoir strikes at the core of shallow wave propagation mechanics
(
The wave tested in the reference test problem (Table 1) is a typical wave
found in hydrologic practice; therefore, it is very likely to be kinematic.
To prove this assertion, we use the kinematic
wave applicability criterion developed by
T S (_{o}u /_{o}d) ≥ 171._{o}T = 12 hr, and the channel slope is
S = 0.01._{o}u
and flow depth _{o}d,
based on the channel
dimensions (Table 1, rows 22 and 24) are the following: _{o}u = 2 m/s, and
_{o}d = 0.2 m. This leads to: _{o}T S (_{o}u /_{o}d)
= 4320 ≥ 171_{o}
We reckon that diffusion and mixed kinematic-dynamic waves
are more likely to be present in unusual situations,
appearing to be the exception rather than the rule
(
Two existing formulations for the coefficient of hydraulic diffusivity in
diffusion wave modeling of catchment dynamics (Eqs. 1 and 4)
are evaluated and compared.
While the kinematic hydraulic diffusivity (Eq. 1)
is independent of the Vedernikov number,
the dynamic hydraulic diffusivity (Eq. 4)
is dependent on the Vedernikov number.
We use the script
Testing of the model varying the drainage area S,
rating exponent _{rp}β, and channel slope S, show the model's consistency in simulating
catchment response under a variety of flow conditions.
Calculated outflow hydrographs correctly
show the type of catchment response, showing
either superconcentrated, concentrated, or subconcentrated catchment flow,
depending on the input data.
_{ch}
The
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Dooge, J. C. I. 1973.
Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkoswki. 1982.
Hayami, I. 1951.
Lighthill, M. J., and G. B. Whitham. 1955.
Ponce, V. M. 1978.
Ponce, V. M. 1986.
Ponce, V. M. 1991a.
Ponce, V. M. 1991b.
Ponce, V. M. 2014a.
Ponce, V., M. 2014b.
Ponce, V. M. 2014c.
Ponce, V., M, and V. Vuppalapati. 2016.
Ponce, V., M. 2019.
Wooding, R. A. 1965. A hydraulic model for the catchment-stream problem.
I. Kinematic wave theory. |

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