Geoscience Reference
In-Depth Information
1 Introduction: Surface Water Floods in the Earth System
Flood waves moving along river systems are both a key determinant of globally important
biogeochemical and ecological processes and, at particular times and particular places, a
major environmental hazard. The time and length scales of such waves vary depending on
river basin area, basin shape, basin slope, geology, vegetation and land use. In the very
smallest urban catchments, rivers respond near instantaneously to rainfall, whilst in the
world's largest river systems, there may a single annual flood pulse and river flood waves
may span whole continents. Such waves therefore vary in length from perhaps 1 to 1,000 s
of km and in duration from a few minutes to a whole year. Compared to their length, river
flood waves are extremely low amplitude: for example, the Amazon flood wave in the
middle reach of the river has a maximum amplitude of *12 m for a wave thousands of
kilometres in length. In most other basins, even catastrophic flash floods have amplitudes
much less than this, and typically, flood waves are just a few metres in height. Flood waves
are therefore shallow water phenomenon where typical horizontal length scales far exceed
those in the vertical. Hydraulically, most flood waves are gradually varying sub-critical
flows (Froude number \1) where the influence of downstream water level controls can
propagate upstream (the so-called backwater effect). Sub-critical hydrodynamics occur
because most river longitudinal slopes are low (typical river slopes are in the range
1-100 cm km
-1
) and change only gradually. Flood waves are translated with speed or
celerity, c, and attenuated by frictional losses such that in downstream sections, the hyd-
rograph is flattened out. Wave speeds vary with discharge (see NERC
1975
) such that
maximum wave speed occurs at approximately two-thirds bankfull capacity (Knight and
Shiono
1996
). Typical observed values for c reported by NERC (
1975
) and Bates et al.
(
1998
) for UK rivers are in the ranges 0.5-1.8 and 0.3-0.67 ms
21
, respectively.
Shallow water waves are described, in one dimension, by the Saint-Venant equation:
0
@
1
A
ou
ot
þ
u
o
u
ox
oh
ox
u
þ
g
þ
S
f
S
o
¼
0
ð
1
Þ
|{z}
ð
iv
Þ
|{z}
ð
v
Þ
|{z}
ð
i
Þ
|{z}
ð
ii
Þ
|{z}
ð
iii
Þ
where wave propagation is controlled by the balance of the various forces in Eq.
1
. Here,
(i) represents the local inertia (or acceleration), (ii) represents the advective inertia, (iii)
represents the pressure differential, and (iv) and (v) account for the friction and bed slope,
respectively. The relative magnitude of these terms for different types of shallow water
flow is discussed in detail by Hunter et al. (
2007
), but in general, for sub-critical flow, the
advective inertia term (ii) can be disregarded, and the pressure differential, friction slope
and bed slope terms (iii, iv and v, respectively) are significantly more important than local
inertia. For super-critical flow however, where shocks, hydraulic jumps and bores may
exist, term (i) assumes much greater importance, and term (ii) cannot be disregarded.
This one-dimensional description is reasonable when flood waves are contained within
defined river channels; however, when bankfull height is exceeded and water is transferred
to floodplains and wetlands adjacent to the main channel, this description is insufficient.
Here, water flow paths cannot be predicted a priori and such flows are clearly two-
dimensional phenomena where flow spreads according to the hydraulic gradient and
floodplain topography, which may be exceedingly complex (see, for example, Nicholas
and Mitchell
2003
). Floodplains and wetlands act as additional routes for flow conveyance
or areas of water storage. Even when floodplains convey flow, the typically higher friction
