Rock Creek reservoir, on the Feather river, Plumas County, California.
 


ABSTRACT
The concept of runoff diffusion is reexamined in order
to clarify its role in connection with unsteady freesurface flow routing in reservoirs,
channels, and catchments.
Diffusion is the spreading of the hydrograph in time and, therefore, in space.
Diffusion is inherent to reservoirs and it is always produced in flow through reservoirs.
In channel flow, diffusion is produced in the absence
of kinematic wave conditions, i.e., under diffusion or mixed kinematicdynamic wave conditions,
provided the Vedernikov number is less than 1.
In catchment runoff, diffusion is produced: (1) for all wave types, when the time of
concentration exceeds the effective rainfall duration, i.e., for subconcentrated catchment flow,
a condition which is usually associated with midsize and
large basins, or (2) for all effective
rainfall durations,
when the wave is a diffusion wave (or mixed wave type), which
is usually associated with a sufficiently mild catchment slope.

1. INTRODUCTION
Runoff diffusion is characterized by the spreading of the hydrograph in time and space.
Flow through a reservoir always produces runoff diffusion.
Flow in stream channels may or may not produce runoff diffusion,
depending on the relative scale of the flood wave, provided the Vedernikov number is less
than 1. The relative scale of the flood wave relates to whether the wave is: (a) kinematic,
(b) diffusion, or (c) mixed kinematicdynamic.
In catchment runoff, diffusion is produced: (1) for all wave types,
when the
time of concentration exceeds the effective rainfall duration, or (2) for all effective
rainfall
durations,
when the wave is a diffusion wave.
These propositions are now examined in detail.
2. DIFFUSION IN RESERVOIRS
Reservoirs are natural or artificial
surfacewater hydraulic features that provide runoff diffusion.
Runoff diffusion is depicted by the attenuation of the inflow hydrograph, as shown in Fig. 1.
Figure 1 Hydrograph attenuation in reservoir routing.

Flow through a reservoir is governed by the onedimensional
equation of storage (Fig. 2):
dS
I  O = ^{_____}
dt
 (1) 
in which S = inflow, O = outflow, and dS/dt = rate of change of storage,
expressed in L^{3} T^{ 1} units.
Figure 2 Inflow, outflow, and rate of change of storage in a control volume.

A commonly used relationship between outflow and storage is the following:
in which K = storage coefficient and m = exponent.
For m = 1, Eq. 2 reduces to the linear form:
in which K is a proportionality constant or linear storage coefficient,
which has the units of time (T). When combined with Eq. 1,
either Eq. 2 or Eq. 3 will provide runoff diffusion
(Ponce, 1989).
3. DIFFUSION IN OPENCHANNEL FLOW
Channels are surfacewater hydraulic features which may or may not provide runoff diffusion,
depending on the relative scale of the
flood wave. The amount of wave diffusion is characterized by the
dimensionless wavenumber σ, as shown in Fig. 3.
The dimensionless wavenumber is defined as:
2 π
σ = ^{_______} L_{o}
L
 (4) 
in which L = wavelength of the disturbance, and L_{o} = the length of channel in which
the equilibrium flow drops a head equal to its depth (Lighthill and Whitham, 1955):
d_{o}
L_{o} = ^{______}
S_{o}
 (5) 
Four types of waves are identified:

Kinematic waves,
 Diffusion waves,
 Mixed kinematicdynamic waves, and
 Dynamic waves.
Kinematic waves lie on the left side of the wavenumber spectrum, featuring constant dimensionless relative wave celerity and zero attenuation.
Dynamic waves lie on the right side, featuring constant dimensionless relative wave celerity and zero attenuation.
Mixed kinematicdynamic waves lie in the middle of the spectrum, featuring variable dimensionless relative wave celerity and medium to high attenuation.
Diffusion waves are intermediate between kinematic and mixed kinematicdynamic waves, featuring mild attenuation. [Note that in hydraulic engineering practice,
dynamic waves are commonly referred to as Lagrange or "short" waves, while the mixed kinematicdynamic waves are commonly referred to as "dynamic waves,"
fueling a semantic confusion].
For flood routing computations, the governing equations of continuity and motion, commonly referred to as the
Saint Venant equations,
may be linearized and combined into a convectiondiffusion equation
with discharge Q as the dependent variable
(Hayami, 1951;
Dooge, 1973;
Dooge et al., 1982;
Ponce, 1991a ;
Ponce, 1991b):
∂Q dQ ∂Q Q_{o} ∂^{2}Q
^{______} + ( ^{______} ) ^{______} = [ ( ^{________} ) ( 1  V ^{2} ) ] ^{_______}
∂t dA ∂x 2 T S_{o} ∂x^{2}
 (6) 
in which V = Vedernikov number,
defined as the ratio of relative kinematic wave celerity to relative dynamic
wave celerity (Ponce, 1991b):
(β  1) V_{o}
V = ^{______________}
(g d_{o})^{1/2}
 (7) 
in which β = exponent of the dischargeflow area rating Q = A^{β}, V_{o} = mean
flow velocity, d_{o} = mean flow depth,
and g = gravitational acceleration.
In Eq. 6, for V = 0, the coefficient of the secondorder term reduces
to the kinematic hydraulic diffusivity, originally due to
Hayami (1951).
On the other hand, for V = 1, the coefficient of the secondorder term reduces to zero,
and the diffusion term vanishes. Under this latter flow condition, all waves, regardless of scale,
travel with the same speed, fostering the development of roll waves (Fig. 4).
Fig. 4 Roll waves in a steep irrigation canal.


4. DIFFUSION IN CATCHMENT DYNAMICS
Surface runoff in catchments may be one of three types
(Ponce, 1989a; 2014a):
Concentrated flow, when the effective rainfall duration is equal to the time of concentration,
Superconcentrated flow, when the effective rainfall duration is longer than the time of concentration, and
Subconcentrated flow, when the effective rainfall duration is shorter than the time of concentration.
Figure 5 shows a typical openbook schematization for
overland flow modeling. Input is effective rainfall on two planes adjacent ot a channel.
Output is the outflow hydrograph at the catchment outlet.
Fig. 5 Openbook catchment schematization.


Figure 6 shows dimensionless catchment outflow hydrographs for the three cases described above
(Ponce and Klabunde, 1999).
The maximum possible peak outflow is: Q_{p} = I_{e} A,
in which I_{e} = effective rainfall intensity, and A = catchment area.
By definition, the maximum possible peak outflow is reached for
superconcentrated and concentrated flow.
However, in the case of subconcentrated flow, the peak outflow fails to reach the maximum possible value. Effectively, this
amounts to runoff diffusion, because the flow has actually been spread in time (and space).
Thus, runoff diffusion is produced for all waves
when the time of concentration exceeds the effective rainfall duration.
This is typically the case of midsize and large basins, for which the
catchment slope (along the hydraulic length) is sufficiently mild (small).
The time of concentration is directly related to catchment hydraulic length and bottom friction,
and inversely related to bottom slope and
effective rainfall intensity (Ponce, 1989b;
2014b).
Figure 6 shows dimensionless rising overland flow hydrographs for a kinematic wave model (labeled KW)
and for several storageconcept models,
for the rating exponent m (Eq. 2) ranging from m = 1, corresponding to a
linear reservoir, to m = 3, corresponding to laminar flow
(Ponce et al., 1997). The kinematic wave timetoequilibrium,
akin to the time of concentration, is theoretically equal to onehalf
of the time of concentration of the storagebased models
(Ponce, 1989;
2014).
It is seen that the storage models spread the hydrograph and, consequently, produce diffusion, while the
kinematic wave model lacks runoff diffusion altogether. The kinematic timetoequilibrium is the shortest possible value
of time of concentration, resulting, in the aggregate, in the largest peak flows.
Thus, under pure kinematic flow, runoff diffusion vanishes.
In practice, a numerical kinematic wave model may not be entirely devoid of diffusion, due to the appearance
of numerical diffusion (Ponce, 1991a). In fact, firstorder schemes of the kinematic wave equation produce numerical diffusion.
This diffusion, however, is uncontrolled, not based on physical parameters and, therefore, unrelated to the true
diffusion of the physical problem.
5. DIFFUSION IN ONLINE COMPUTATIONS
Examples of runoff diffusion can be calculated with the following online scripts:
6. SUMMARY
The concept of runoff diffusion is reexamined in order
to clarify its role in connection with unsteady freesurface flow routing in reservoirs,
channels, and catchments.
All flood routing calculations are effectively computing the amount of wave diffusion.
Diffusion is the spreading of the hydrograph in time and, therefore, in space.
Diffusion is inherent to reservoirs and it is always produced in flow through reservoirs.
In channel flow, diffusion is produced in the absence
of kinematic wave conditions, i.e., under diffusion or mixed kinematicdynamic wave conditions,
provided the Vedernikov number is less than 1.
At the threshold V = 1, all diffusion disappears and this flow condition
promotes the development of roll waves.
In catchment runoff, diffusion is produced: (1) for all wave types, when the time of
concentration exceeds the effective rainfall duration, i.e., for subconcentrated catchment flow,
a condition which is usually associated with midsize and
large basins, or (2) for all effective
rainfall durations,
when the wave is a diffusion wave (or mixed wave type), which
is usually associated with a sufficiently mild catchment slope.
In conclusion, in catchment runoff, diffusion
is produced in: (1) midsize and large basins, with long time of concentration, and/or (2)
mild catchment slopes, which typically feature diffusion waves.
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