If the upstream Froude number
Ci = cohesion factor for ice (can range from
zero for breakup jams to 20 lb/ft2 for
V
freezeup jams)
τ = shear force on underside of accumu-
gH
lation, approximated by ρg (yi/2)S,
is less than or equal to the right-hand side of eq
where yi = under-ice depth.
1, a solution by juxtaposition is assumed possible.
The accumulation's leading edge thickness h,
Ice thinning
which satisfies eq 1, is then found by trial and
The water velocity beneath an ice cover may
error.
be high enough to erode ice pieces and thin the
The second method, derived by Ashton (1974),
accumulation in a manner analogous to sediment
is based on the particle Froude number Fp:
transport. The user inputs a threshold velocity
(VEROS), above which erosion or thinning of the
h
2 1 -
ice cover takes place. The "thinned" ice cover
H
Vc
Fp =
≤
thickness is estimated by the following form of the
ρ
2
.
(3)
h
continuity equation:
gh 1 i
5 - 3 1 -
ρ
H
V
1
H - (H - Si h)
ht =
Again, ICETHK checks to determine whether a
(5)
Vc
Si
solution is possible, i.e., left-hand side less than
or equal to right-hand side for h/y = 1/3, then
solves for h, the leading edge thickness.
where ht = thickness of "thinned" ice cover
H = open water depth
Shoving
V = average water-current velocity
Thickening due to shoving is calculated by eq
Vc = maximum non-eroding velocity (input
4. The underlying theory was developed through
by user) (VEROS)
the work of many, including Kennedy (1958),
h = ice thickness before thinning
Pariset and Hausser (1961), Michel (1965), Pariset
Si = specific gravity of ice, assumed to be
et al. (1966), Uzuner and Kennedy (1976), and
0.916.
Beltaos (1978).
As stated earlier, the equations describing thick-
Typical values for VEROS range from 3 to 5 ft/s
ness by shoving apply only to the equilibrium
for freezeup-type jams and 4 to 8 ft/s for break-
reach of the jam (see Fig. 1). Assumptions include
up. If the HEC-2 calculated water-current velocity
uniform flow, constant ice thickness, and the trans-
is greater than VEROS, eq 5 will reduce the ice
fer of all downstream forces on the ice accumula-
thickness, but not to a value less than the thick-
tion to the banks. It should be restated that diffi-
ness of the initial (pre-jam) ice cover (ITHICK).
culties arise when using ICETHK to model the
head and toe of the jam, where conditions of var-
Roughness of the ice accumulation
ied flow and changing ice thickness exist.
Of the four roughness options available in
Under steady-state conditions, the uniform
ICETHK, two involve direct assignment of Man-
section of an ice accumulation, compressed by
ning's n values and two involve calculation of
shoving, can be described as
roughness. Ice roughness can be calculated as a func-
tion of ice thickness or as a function of ice piece size.
ρ
1 i gh2 - ( gρi SB - 2 Ci ) h - τB = 0
Existing field data show that thick jams are typ-
(4)
ρ
ically made up of larger ice pieces and are hydrau-
lically rougher than thin jams. Relationships in the
ICETHK model, based on Nezhikovskiy's (1964)
where h = thickness of the ice accumulation
= coefficient related to the internal
data, relate Manning's n values for the ice cover
to the ice accumulation thickness. The relation-
strength of the accumulation, ranging
ships take the form of a similar equation by Beltaos
from 0.8 to 1.3
ρ, ρi = densities of ice and water
(1983). Nezhikovskiy's data were measured in
wide canals 6.6 to 9.8 ft deep for ice floes, dense
slush, and loose slush. For breakup situations with
S = energy slope
ice accumulations greater than 1.5 ft thick,
B = channel width at bottom of ice cover
3
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