be on the order of 21.9 3.9C (7.5 7F) and that
characterize the intense cold periods simply on the
a period of 10 consecutive days of cold weather
basis of air temperature. A complete analysis of air
averaging 14.1 4.3C (6.6 7.7F) is the norm.
temperature was performed in the previous sec-
It should be noted that negative values of cu-
tion of this report.
mulative freezing-degree-days FDDn indicate that
During the periods of intense cold, the water cools
rapidly. When the water temperature cools to 0C
the average air temperature during the corre-
(32F), ice begins to form in the river. Observations
sponding n consecutive days was above freezing.
show that this ice forms as frazil or skim ice that
moves in the downstream direction. The moving ice
collects into large pans as it travels. As long as the ice
STATISTICAL ANALYSIS OF THE
is in motion, it will have a negligible impact on the
ICE-IMPACTED DISCHARGE DATA
discharge. At some point, the ice may bridge or arch
During periods of intense cold weather, the
across the river and its motion will be arrested. This
production of ice on the Missouri River can cause
may occur in bends, at islands, in slow-moving
reductions in discharge in the river that can have
reaches, or at other points. Moving ice collects at the
tremendous negative impacts, primarily by expos-
upstream edge of the stationary ice, and the station-
ing important water intakes along the river. In this
ary cover progresses upstream. The presence of a
section we will investigate the occurrence of the
stationary cover changes the hydraulic conditions of
ice-impacted discharge periods. We will do this
the channel dramatically from those of an open
statistically, by describing the exceedance prob-
channel. By presenting an additional stationary
ability of the maximum discharge deficits caused
boundary, the ice cover makes a portion of the chan-
by ice. We can determine the exceedance prob-
nel unavailable for flow, changes the channel wetted
abilities on an annual basis or on a half-month
perimeter and hydraulic radius, and adds additional
basis through the winter months of December,
roughness. The change in the hydraulic radius is
January, and February. We can also describe the
quite significant. For wide channels, the hydraulic
exceedance probability on the basis of other peri-
radius is essentially reduced by half (Wuebben 1986).
ods of time, in this case defined by time required to
The net effect is that the relation between the ice cover
accumulate a determined amount of freezing-de-
depth, DI, and the open water depth, Do, becomes
gree-days. The half-month basis is valuable for
0.6
planning future discharges. If the probability of a
DI = 1.32 Do N I
(3)
certain deficit is known, the risk of maintaining
No
low discharge levels can be assessed and the
amount of water required carefully assigned. The
where NI is the effective Manning's roughness of
accumulated freezing-degree-day (AFDD) ap-
the ice-covered channel, and No is the Manning's
proach is valuable for providing guidance during
roughness of the open channel. Carey's (1966)
the course of winter. During the winter period, the
calculations indicate 0.73 < NI / No< 1.37 such that:
number of accumulated freezing-degree-days can
1.09 < DI < 1.59
be tracked, and the probabilities of a particular
(4)
discharge deficit being exceeded can be deter-
Do
mined on an updated basis.
Equations 3 and 4 are valid only for the case of
constant discharge. In the case of the Missouri
Background
River, we can imagine the ice cover progressing
In outline, the following sequence of events
upstream in short lengths, ∆X. The depth of each
leads to the occurrence of discharge deficits.
section is initially Do, the open water elevation.
First, there is an intense cold period. This causes
The water level of each section must rise to the new
a large heat transfer rate from the water surface to
elevation, DI. By mass conservation we can state
the atmosphere. While the heat transfer rate from
the water surface is a function of many parameters,
QIN QOUT = B ∆ d ∆X
(5)
including wind speed, relative humidity, long and
∆t
shortwave radiation, as well as air temperature, it
where QIN = flow into the section
has been found that, during the winter period, heat
QOUT = flow out of the section
transfer from the water surface is very well corre-
B = river width
lated with the difference between the air and water
∆d = rise in water level over time ∆t.
temperature (Ashton 1988). Therefore, we can
10
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