in turn affects the calculation of ice roughness. In
14 Russian rivers meeting the following criteria
practice, then, roughness is generally estimated
was used to determine ni for rough ice covers: the
during calibration of numerical models, where it
ice cover was either rough or had frazil deposits,
essentially becomes a lumped variable represent-
sufficient measurements were made each year to
ing other unknown jam components.
determine time-related effects, and slope was
The most commonly used roughness coefficient
measured. The following equation obtained from
is the Manning's n, followed by the Darcy friction
the Manning formula, assuming Ri = yi/2, was
used to calculate nc:
factor f. The Chezy C and the equivalent sand
roughness k are occasionally used as well, but less
so in recent years, and they are not included in
2
yi
Sf
3
the most frequently used hydraulic models. The
nc = 0.63
(21)
vi
following discussion will be limited to use of the
various Manning's n values for the underside of
and ni was then back-calculated using eq 13.
the ice, ni, and the composite roughness, nc. Nu-
Based on his results, Nezhikhovskiy concluded
merous methods of determining nc are given in
that the roughest ice covers were observed dur-
Uzuner (1975). No attempt has been made to con-
ing the first three days of ice cover and that ni tends
vert roughness estimates reported in the form of
to decrease over the course of the winter. Loose
f, C, or k to n. Table 1 and Figure 6 summarize re-
frazil deposits tended to smooth more rapidly than
ported values of ni and nc.
dense deposits. He also noted that the bottom of
frazil deposits tended to be wavy, and that thicker
Roughness calculated from discharge
deposits had larger amplitude ridges; both thick
measurements or estimates
and thin deposits smoothed during the winter. By
Based on about 500 observations between 1936
categorizing ice covers into three categories (ice
and 1959, Nezhikhovskiy reports that ni for
covers formed primarily from loose frazil, dense
smooth ice covers (i.e., generally formed through
frazil, and sheet ice), Nezhikhovskiy developed
heat transfer processes as opposed to accumula-
the curves shown in Figure 7 to describe ni at the
tion) ranges from 0.010 to 0.012 during initial ice
time of initial ice cover formation in terms of thick-
cover formation and from 0.008 to 0.010 thereaf-
ness. In the case of frazil accumulations or depos-
ter. A total of 368 measurements at 19 stations on
Table 1. Values of ice roughness coefficient (ni) and composite roughness coefficient (nc)
calculated from discharge measurements.
(ni)
(nc)
Comment
Reference
0.0100.012
Sheet ice, early winter*
Nezhikhovskiy (1964)
0.0080.010
Sheet ice, late winter
Nezhikhovskiy (1964)
0.0100.06†
Ice cover formed from loose frazil*
Nezhikhovskiy (1964)
0.0130.09†
Ice cover formed from dense frazil*
Nezhikhovskiy (1964)
0.0150.10†
Ice cover formed from sheet ice*
Nezhikhovskiy (1964)
0.0100.028**
0.0180.027
Sheet ice
Carey (1966)
0.0040.013**
0.0150.022
Sheet ice
Carey (1967)
0.10
0.0900.109
Breakup jams
Beltaos (1978)
0.0570.065,
0.0410.046
Breakup jam
Andres (1980)
ni = 0.060
0.0100.015
Breakup jam
Knowles and Hodgins (1980)
0.0530.142
Breakup jams
Michel (1980)
0.0130.040
Freezeup jam*
Beltaos (1981)
0.0330.041††
Freezeup jam*
Beltaos (1983)
0.072
Breakup jams
Andres and Doyle (1984)
0.020-0.15
Freezeup jam, frazil deposits
Majewski and Grzes (1986)
*Within three days of formation.
†Higher values for thicker accumulations.
**Lower values earlier in the winter.
††Higher values for thinner accumulations.
9
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