The Uzuner and Kennedy formulation of the force balance associated with gradu-
ally varied, unsteady water flow was too complex for a general solution. They did
show that, for a condition of quasi-steady jam formation in which the jam progresses
upstream at a constant rate, the unsteady water flow equations are constant with
time. For the section of the jam considered to be in equilibrium, eq 19 can easily be
solved for ηo using the quadratic formula by setting the term on the left-hand side
to zero.
The formulations proposed by Pariset et al. and Uzuner and Kennedy compose
the basis for most subsequent analyses of static ice jams. Beltaos (1983), most nota-
bly, adapted the formulations for wide-river jams and expressed flow depth
beneath a jam h as
1
-
2
4 gSo
3
h=
(24)
q3
f
o
where fo is a composite value of the Darcy-Weisbach resistance coefficients for the
bed and the jam underside. Solution of eq 10 for jam thickness, assuming that cohe-
sion is negligible, and that f3 is given by eq 5, yields
1
1 2
q2 3
1
BSo (2 fo ) 3 (1 - si ) fi gSo
η=
1 + 1 +
f BS
2(1 - si )
o
si
o
(25)
where fi is a Darcy-Weisbach resistance factor for flow along the jam underside.
Beltaos also presented field data, consisting of thickness measurements for several
ice jams that had refrozen in place. Using eq 25, he back-calculated values of and
found them to range from 0.6 to 3.5, with these upper and lower limits obtained for
conditions of considerable uncertainty. If the two extreme values are excluded, his
data show consistently that = 0.8 to 1.3. Beltaos found, on average, that ≅ 1.2,
which is in good agreement with the value of 1.28 suggested by Pariset et al.
Numerical modeling
Several numerical models of jams have been developed. They assume that a
balance exists between forces acting on the jam, predict equilibrium jam thickness,
and estimate jam effects on water levels. Existing open-water models for steady
and unsteady flow simulations have been adapted by the use of equations similar
to eq 25 to provide estimates of ice jam conditions. Other models have been devel-
oped using steady or unsteady water flow and equilibrium (uniform) or
nonequilibrium thickness.
HEC-2, the step-backwater program developed by the U.S. Army Corps of Engi-
neers, was modified to include an ice cover (HEC 1979). In its initial version, the
cover, or jam, was treated simply as a boundary, floating at hydrostatic pressure,
that provides an additional resistance to flow at the water surface. The cover is
taken as being static, with the user of the program inputting values of cover thick-
ness, roughness, location, specific gravity, and a value of . The program calculates
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