Encyclopedia of Environmental Science and Engineering, Volume I and II

482 HYDROLOGY

can be considerably greater and can occur over much shorter

time periods. These contrasting areal distribution and time

distribution effects are partly self-compensating. Snowmelt

is greatest at low elevations which tends to make hydrograph

response more peaked, but the lower intensity of snowmelt

tends to produce a fl atter response. Rain, on the other hand,

is of greater intensity, producing a higher peak outfl ow, but is

greatest in the more remote regions of the watershed, which

has a tendency to reduce the peak fl ow. There seems to be

little or no conclusive data to test this contrast between rain

and snow response. This lack of conclusive data may be that

the difference between rain and snow events are small, or

because the data base is not suffi ciently detailed and accurate

to demonstrate any difference.

Returning to the question of the watershed routing, some

simple theoretical models will be examined to explore the

representativeness of linear routing models.

The linear routing process is usually assumed to repre-

sent a relationship between the outfl ow, QO, and storage S,

of a linear reservoir, with a storage constant K,

S 
 K ( QO ). (51)

Cascades of such linear reservoirs are used to produce a range

of unit hydrograph responses to inputs of rain or snowmelt.

A linear Weir is not a structure which can easily be imagined as

part of a watershed system, because it is much more realistic to

imagine fl ow through various interstices and fl ow paths in the

soil layers and to assume such fl ow to be friction controlled.

Such a conceptualisation leads to examination of the ground-

water fl ow equations, for example, the Darcy equation,

QKA

y

x

d

d

. (52)

The area of fl ow, A, can be represented by a width, B, and a

saturated depth of fl ow, y. Assuming B to be constant, then

y is a linear measure of storage. The equation can be linear-

ized if it is assumed that d y /d x is constant for all fl ows, an

assumption which might be reasonable for steep mountain

catchments where the ground slope dominates the fl ow pro-

cess. Eq. (6) might therefore be reduced to the linear fl ow-

storage equation, like Eq. (5).

A single storage-fl ow relationship produces an instan-

taneous unit hydrograph response which is a simple expo-

nential decay, as given by the Nash single reservoir result.

Even such a simple model, when convoluted with the time

distributed input of effective rainfall, will produce a typical

time distribution of outfl ow, modeling the rising and falling

limbs of a unit hydrograph. It is not even strictly necessary

to resort to a cascading of linear storages, although, from a

modeling viewpoint, a cascade of storages offers a fl exible

method of controlling hydrograph shape.

Other frictionless and friction-controlled equations can

also be shown to imply relationships between storage and fl ow,

but many of the relationships are non-linear, for example, a

Weir formula and the Manning equation,

Weir: Q 
 KBH n^ (53)

Manning: Q

A

n

 RSh^23  012. (54)

In these equations H and R h can be considered as expressions

of storage.

A more interesting relationship can be derived from con-

sideration of kinematic wave behavior. Following Lighthill

and Whitham’s work and later work by Henderson and

Wooding, many authors have proposed that kinematic wave

behavior is very representative of hydrologic run-off pro-

cesses, Lighthill and Whitham,^27 Quick and Pipes,^48 either

from watersheds or in channels. The starting point of the

kinematic approach is the continuity equation

∂

∂

∂

∂

Q

x

B

y

t


0. (55)

Re-writing and comparing with the total differential of y,

∂

∂

∂

∂

∂

∂

Q

y

y

x

B

y

t


 0 (56)

and

d

d

d

d

y

t

x

t

y

x

y

t



∂

∂

∂

∂

0. (57)

These equations are equivalent if,

1

B

Q

y

x

t

C

∂

∂

d

d

.

(58)

So far these results only have the restriction that Q should be

a function of y only. Consider the special case when Q is a

linear function y, as previously argued for the Darcy ground-

water equation,

i.e. ,

Q 
 ky. (59)

Then, solving for C,

C

B

Q

y

k

B

1d

d

constant. (60)

The wave travel time, T, for a catchment dimension or chan-

nel length, L, is

T

L

C

LB

k

 
constant. (61)

Therefore we have a system in which any and all discharges,

Q, travel through the catchment or channel in exactly the

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