ABSTRACT
The concept of hydraulic diffusivity and its extensions to the
dynamic regime are examined herein. Hayami (1951) originated the concept
of hydraulic diffusivity in connection with the propagation of flood waves.
Dooge (1973) extended Hayami's hydraulic diffusivity to the realm of
dynamic waves, specifically for the case of
Chezy friction in hydraulically wide channels.
Dooge's formulation amounts to
a dynamic hydraulic diffusivity.
Subsequently, Dooge
et al. (1982) expressed the dynamic hydraulic diffusivity
in terms of the exponent β of the dischargearea rating Q =
αA^{β}.
Lastly, Ponce (1991a, 1991b) expressed the
dynamic hydraulic diffusivity
in terms of the Vedernikov number, further clarifying the mechanics of flood wave propagation.

1. INTRODUCTION
The concept of hydraulic diffusivity is due to Hayami (1951), who combined the equations of continuity and momentum of
unsteady openchannel
flow to derive a convectiondiffusion equation, i.e., a partial differential equation containing a term of
convection (first order) and a term of diffusion (second order).
The modeling of flood wave propagation in terms of the convectiondiffusion equation has been referred to
as Hayami's diffusion analogy (Ponce, 1989).
The coefficient of the second order term of the convectiondiffusion equation
is the channel hydraulic diffusivity, or Hayami's hydraulic diffusivity.
Since he neglected inertia in his formulation, his hydraulic diffusivity is properly a kinematic hydraulic diffusivity.
Dooge (1973) extended Hayami's hydraulic diffusivity to the realm of dynamic waves,
specifically for the case of Chezy friction in hydraulically wide channels.
Dooge's formulation amounts to
a dynamic hydraulic diffusivity.
Subsequently, Dooge et al. (1982) expressed the dynamic hydraulic diffusivity
in terms of the exponent β of the dischargearea rating Q = αA^{β}.
Later, Ponce (1991a, 1991b) expressed the
dynamic hydraulic diffusivity
in terms of the Vedernikov number, further clarifying the mechanics of flood wave propagation.
These propositions are now explained in detail.
2. THE HYDRAULIC DIFFUSIVITY
Hayami's hydraulic diffusivity, or kinematic hydraulic diffusivity, is (Hayami, 1951):
q_{o}
ν_{k} = ^{_______}
2 S_{o}
 (1) 
in which q_{o} = unitwidth discharge, and S_{o} = bottom slope.
Dooge's dynamic hydraulic diffusivity, applicable for Chezy friction in hydraulically wide channels, is
(Dooge, 1973: Eq. 33b):
q_{o} F_{o}^{ 2}
ν_{d} = ^{_______} (1  ^{_____} )
2 S_{o} 4
 (2) 
in which F_{o} = normalflow Froude number.
Dooge et al.'s dynamic hydraulic diffusivity is (Dooge et al., 1982):
q_{o}
ν_{d} = ^{_______} [1  (β  1)^{2} F_{o}^{ 2}]
2 S_{o}
 (3) 
The variable β in Eq. 3 is the exponent of the rating Q = αA^{β}. For Chezy friction
in hydraulically wide channels, β = 3/2, and Eq. 3 reduces
to Eq. 2.
Ponce's dynamic hydraulic diffusivity is (Ponce, 1991b):
q_{o}
ν_{d} = ^{_______} (1  V_{o}^{ 2})
2 S_{o}
 (4) 
in which V_{o} is the Vedernikov number, defined as follows:
V_{o} = (β  1) F_{o}
 (5) 
As the Froude number F_{o} → 0, the Vedernikov number V_{o} → 0 (Eq. 5),
and the dynamic hydraulic diffusivity (Eq. 4) reduces to the kinematic hydraulic diffusivity (Eq. 1),
a finite value which is independent of the Froude or Vedernikov numbers.
On the other hand, when the Vedernikov number V_{o} → 1, the dynamic hydraulic diffusivity
→ 0. The condition V_{o} = 1 is the threshold of neutral stability, where roll waves tend to develop (Fig. 1).
Fig. 1 Roll waves in a steep irrigation canal, CabanaMañazo, Puno, Peru.
Under normal openchannel flow conditions, the Froude number F_{o} > 0; therefore,
the Vedernikov number V_{o} > 0. Thus, the dynamic hydraulic diffusivity ν_{d} (Eq. 4) is always smaller
than the kinematic hydraulic diffusivity ν_{k} (Eq. 1). In practice, the use of ν_{k}
in lieu of ν_{d} will always exaggerate the amount of wave diffusion.
In the limit, for V_{o} = 1, no diffusion is possible, and kinematic and dynamic waves
travel at the same speed and are not subject to dissipation,
thus leading to the development of roll waves such as those shown
in Fig. 1 (Ponce and Simons, 1977).
3. THE DYNAMIC HYDRAULIC DIFFUSIVITY
In deriving the hydrodynamic Muskingum parameters including inertia,
Dooge et al. (1982) expressed X in terms of hydraulic variables as follows (op. cit., Eq. 37b):
1 M R
X = ^{___} + ^{____}  ^{____}
2 N P
 (6) 
in which:
g y_{o}
M = ^{______} (1  F_{o}^{2})
L
 (7) 
∂S_{f}
N = g A_{o} ^{_____}
∂A
 (8) 
∂S_{f}
P = g A_{o} ^{_____}
∂Q
 (9) 
2 Q_{o} 1 (∂S_{f} / ∂A)
R = ^{______} + ^{___ ___________}
A_{o} L L (∂S_{f} / ∂Q)
 (10) 
in which
y_{o} = flow depth,
A_{o} = flow area,
Q_{o} = flow discharge,
S_{f} = friction slope,
g = gravitational acceleration,
and
L = reach length Δx.
The ratio M /N is:
M 1 y_{o}
^{______} = ^{__________} ^{______} (1  F_{o}^{2})
N ∂S_{f} /∂A A L
 (11) 
The ratio R /P is:
R 1 2 Q_{o} ∂S_{f} /∂A 1
^{______} = ^{__________} ^{___________} + ^{____________} ^{_________}
P ∂S_{f} /∂Q g A_{o}^{2} L
(∂S_{f} /∂Q)^{2} g A_{o} L
 (12) 
Dooge et al. (1982) defined the variable m:
∂S_{f} / ∂A A_{o}
m =  ^{___________} ^{_____}
∂S_{f} / ∂Q Q_{o}
 (13) 
This variable (m) is the same as the exponent β of the dischargearea rating Q = αA^{β} (Ponce, 1989: Eq. 913).
Therefore:
∂S_{f} / ∂A A_{o}
∂S_{f} / ∂Q =  ^{___________} ^{_____}
m Q_{o}
 (14) 
With Eq. 14, the ratio R /P reduces to:
R m 2 Q_{o}^{2} m^{2} Q_{o}^{2}
^{______} =  ^{__________} ^{___________} + ^{___________} ^{_________}
P ∂S_{f} /∂A g A_{o}^{3} L
∂S_{f} /∂A g A_{o}^{3} L
 (15) 
R 1 Q_{o}^{2}
^{______} =  ^{__________} ^{___________} (2 m  m^{ 2})
P ∂S_{f} /∂A g A_{o}^{3} L
 (16) 
From the definition of the Froude number, with T_{o} = top width:
Q_{o}^{2} T_{o}
F_{o}^{2} = ^{_________}
g A_{o}^{3}
 (17) 
R 1 F_{o}^{2}
^{______} =  ^{__________} ^{________} (2 m  m^{ 2})
P ∂S_{f} /∂A T_{o} L
 (18) 
R 1 y_{o} F_{o}^{2}
^{______} =  ^{__________} ^{__________} (2 m  m^{ 2})
P ∂S_{f} /∂A A_{o} L
 (19) 
Replacing Eqs. 11 and 19 into Eq. 6:
1 1 y_{o}
X = ^{_____} + ^{__________ ______} [1  (m  1)^{2} F_{o}^{ 2}]
2 ∂S_{f} / ∂A A_{o} L
 (20) 
From Eq. 14:
Q_{o}
∂S_{f} / ∂A =  m (∂S_{f} / ∂Q) ^{_______}
A_{o}
 (21) 
Since:
Q_{o}^{2}
S_{f} = ^{_______}
K^{2}
 (22) 
in which K = conveyance, then:
2 S_{f}
∂S_{f} / ∂Q = ^{_______}
Q_{o}
 (23) 
Replacing Eq. 23 into Eq. 21:
S_{f}
∂S_{f} / ∂A =  2 m ^{_______}
A_{o}
 (24) 
Replacing Eq. 24 into Eq. 20:
1 1 y_{o}
X = ^{_____}  ^{__________ _____} [1  (m  1)^{2} F_{o}^{ 2}]
2 2 m S_{f} L
 (25) 
Or:
1 y_{o}
X = ^{_____} { 1  ^{__________ } [1  (m  1)^{2} F_{o}^{ 2}] }
2 m S_{f} L
 (26) 
Replacing L with Δx, m with β,
and S_{f} with S_{o} :
1 q_{o}
X = ^{_____} { 1  ^{____________ } [1  (β  1)^{2} F_{o}^{ 2}] }
2 S_{o} c_{o} Δx
 (27) 
in which c_{o} is the flood wave celerity, c_{o} = m u_{o} =
β u_{o}.
Replacing Eq. 5 in Eq. 27:
1 q_{o}
X = ^{_____} [ 1  ^{____________ } (1  V_{o}^{ 2}) ]
2 S_{o} c_{o} Δx
 (28) 
The kinematic hydraulic diffusivity ν_{k} is given by Eq. 1.
The grid diffusivity is ν_{g} = c_{o} (Δx /2) (Ponce, 1989). Thus, from Eq. 28, it follows that the dynamic hydraulic diffusivity is:
q_{o}
ν_{d} = ^{_______} (1  V_{o}^{ 2})
2 S_{o}
 (29) 
Equation 29 is the same as Eq. 4. It is confirmed that the expression within parenthesis in
the dynamic hydraulic diffusivity is a function not of the Froude number, as presented by Dooge (1973),
but of the Vedernikov number, as presented by Ponce (1991b).
The script ONLINE DYNAMIC HYDRAULIC DIFFUSIVITY
calculates the kinematic and dynamic hydraulic diffusivities, given a set of
hydraulic variables consisting of (mean) velocity u, flow depth y,
bottom slope S_{o}, and exponent β of the rating.
4. SUMMARY
The Dooge et al. equation for dynamic hydraulic diffusivity, Eq. 3, is shown to be a function of the Froude number F_{o} and the exponent β of the dischargearea rating Q = α A^{β}. The Ponce formulation for dynamic hydraulic diffusivity, Eq. 4, is shown to be a function of the Vedernikov number V_{o}. In view of the relation between Froude and Vedernikov numbers,
Eq. 5, the formulations of Dooge et al. (1982) and Ponce (1991) are equivalent.
The Vedernikovnumber dependent formulation for hydraulic diffusivity is recommended
for increased modeling accuracy in the following applications:
(1) channel routing using the MuskingumCunge method
(Ponce and Yevjevich, 1978); and
(2) overland flow routing using the diffusion wave model of catchment dynamics
(Ponce, 1986). An online calculator
is provided to round up the experience.
REFERENCES
Dooge, J. C. I. (1973). Linear theory of hydrologic systems, Technical Bulletin No. 1468, Agricultural Research Service, U.S. Department of Agriculture,
Washington, D.C.
Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkowski. (1982). Hydrodynamic derivation of storage parameters
of the Muskingum model, Journal of Hydrology, Vol. 54, 371387.
Hayami, S. (1951). On the propagation of flood waves. Bulletin No. 1, Disaster Prevention Research Institute,
Kyoto University, Kyoto, Japan, December.
Ponce, V. M., and D. B. Simons. (1977). Shallow wave propagation in open channel flow.
Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, pages 14611476, December.
Ponce, V. M., and V. Yevjevich. (1978).
MuskingumCunge method with variable parameters.
Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY12, pages 16631667, December.
Ponce, V. M. (1986). Diffusion wave modeling of catchment dynamics.
Journal of Hydraulic Engineering, ASCE, Vol. 112, No. 8, pages 716727, August.
Ponce, V. M. (1989). Engineering hydrology:
Principles and practices, Prentice Hall, Englewood Cliffs, New Jersey.
Ponce, V. M. (1991a). The kinematic wave controversy.
Journal of Hydraulic Engineering, ASCE, Vol. 117, No. 4, pages 511525, April.
Ponce, V. M. (1991b). New perspective on the Vedernikov number. Water Resources Research, Vol. 27, No. 7, pages 17771779, July.
