Therefore, a minimum of 13 variables is needed to describe the discharge of a
continuous layer of ice moving in a channel. The influences of gravity, which moti-
vates the discharge of water and ice, relate to the discharge relationships for water
and ice, and in the relationship between layer thickness H and ice discharge rate G.
To describe ice discharge in two channels that differ only in channel geometry
and the discharges of water and ice, the number of variables increases to 20, add-
ing, for the second channel, Q, Y, b, k, ki, H, and η. The material properties of water
and ice, layer porosity, and friction are taken to be the same for all channels. To
describe the merging of ice flow from two confluent channels, further variables
are needed. The number of variables increases with the addition of variables de-
scribing the orientation of the outflow channel relative to the confluent channels
(α and β) and the hydraulic characteristics of the outflow channel (Qc, bc, Yc, k and
ki). The total number of variables is now 27.
The number of variables can be reduced to 23 if the roughness heights k and ki
are assumed the same for each channel.
For the simple case of no ice jamming (i.e., no significant channel storage of
water and ice in the confluence), continuity of water, and ice discharge through
the confluence without jam formation, gives respectively
Q1 + Q2 = Qc
(16)
and, for ice discharge,
G1 + G2 = Gc
(17)
or
Gc = ηcQc (1p)1 = η1Q1 (1p)1 + η2Q2 (1p)1
(18)
with ice discharge expressed as a volumetric proportion η of water discharge.
However, for the limiting condition of incipient jamming, eq 11 pertains.
The pertinent variables may be assembled in the following functional relation-
ship, for merging ice layers comprising ice a given size (small compared to chan-
nel width), with ice outflow as the dependent variable:
η = fL (Q1 , Q2 , b1 , b2 , bc , Y1 , Y2 , Yc , kb , ki , η1 , η2 , H1 , H2 , p, ρi , ρ, g, , φ, α, θ) . (19)
The 23 variables in eq 19 reduce to 20 nondimensional parameters, given three
basic dimensions (mass, length, and time) involved with ice discharge through a
confluence. If the dimensional analysis is carried out using bc, Q2, and ρ as the
repeating variables, the following functional relationship emerges for the limiting
condition of a single layer of free-drifting ice discharging through a confluence:
Q
Y
Q2
b1 b2
η = ϕL 1 ,
,
,
,
Q2 Db gD b2 b3c
bc 1,2,3
2
ρi
kb ki
H1 H2
, , , η1 , η2 ,
, p, , φ, , α, θ .
,
(20)
ρ
b3 b3
b3 b3
The foregoing analysis, though simplifying the actual processes, nonetheless leads
to useful sets of nondimensional parameters for describing the general charac-
21
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