respectively for water,
Q1 + Q2 = Q3 + ∆∀/∆t
(7)
and for ice,
G1+ G2 = G3 + ∆∀i/∆t.
(8)
In eq 7 and 8, ∆∀ and ∆∀i are changes in water volume and ice volume stored in
the confluence reach during time period ∆t, respectively. Up to the condition of
incipient jamming in a confluence of river channels, ∆∀ = 0 and ∆∀i = 0 may be
assumed. Once jamming initiates, ice inflow begins to exceed ice outflow from the
confluence, ∆∀i ≥ 0, and water inflow may exceed outflow, ∆∀ ≥ 0.
Up to incipient jamming, eq 8 is simply
G1 + G2 = G3
(9)
or
(C[h/Y]Q)1 + (C[h/Y]Q)2 = (C[h/Y]Q)3 .
(10)
and Q3 ≈ 0, such that
Q1 = ∆∀/∆t
(11)
and G2 = 0, and G3 ≈ 0, such that
G1 = ∆∀i/∆t.
(12)
The present analysis considers incipient ice jamming at a confluence of rivers and
at a river discharging into a reservoir or lake. Under the assumption that ice piece
dimensions, D and h, are the same for all channels, and that the channels have
the same roughness, k, the number of variables reduces to 27 4 = 23. If it is fur-
ther assumed that the flows are subcritical, the effects of gravity g are treated by
use of an open-channel discharge relationship for Q, and by specifying that ice
floats. The influence of water viscosity, ν, can be neglected if it is assumed that
flow in the channels and around ice pieces is fully rough. For ice pieces in actual
rivers, surface tension σ is negligible. The number of variables finally reduces to
23 3 = 20.
The ice pieces are assumed here to move through the confluence as a single
layer of ice pieces of a given size. The following functional relationship may be
written for the areal concentration of ice discharge at the narrowest cross section
of flow in the confluence outflow channel Cc as the dependent variable of interest:
Cc = fd (Q1 , Q2 , b1 , b2 , bc , Y1 , Y2 , Yc , k, D, h, C1 , C2 , α, θ, , ρ, ρi , g) .
(13)
Equation 13 assumes that, for a confluence of rivers, Q3 = Q1 + Q2, and for a conflu-
ence of river and reservoir or lake, Q2, G2 = 0, and eq 11 and 12 pertain. The 20
variables in eq 13 reduce to 16 nondimensional parameters, given three basic di-
mensions (length, mass, and time) involved with the volumetric discharge of ice
through a confluence. If a dimensional analysis is carried out using D, Q2, and ρ as
the repeating variables, the following functional relationship emerges for the limit-
ing condition of a single layer of free-drifting ice discharging through a confluence:
19
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