to be the same for all channels. To describe the
Q1
Q2
merging of ice flow from two channels confluent
into a single outflow channel, additional variables
Y1
Y2
are needed to describe the orientation of the out-
flow channel relative to the confluent channels, α
b1
b2
and β, and the hydraulic characteristics of the out-
a
flow channel (Qc, bc, Yc, and k). The total number
C2
C1
of variables is now 26.
For the simple case of no ice jamming, continu-
ity of water and ice discharges through the
confluence gives, respectively, for water
D
Q1 + Q2 = Qc + ∆∀/∆t
(1)
and for ice
G1 + G2 = Gc + ∆∀i/∆t .
(2)
Cc
b
In eq 1 and 2, ∆∀ and ∆∀i are changes in water
and ice volume stored in the confluence reach dur-
Drifting
Ice
ing time period ∆t. Up to the condition of incipi-
ent jamming in a confluence of river channels, it
k
may be assumed that ∆∀ = 0 and ∆∀i = 0. Once
jamming takes place, ∆∀ > 0 and ∆∀I > 0, ice in-
bc
Yc
flow begins to exceed ice outflow from the
confluence, and water inflow may exceed outflow.
Up to incipient jamming, eq 2 may be written
Qc
simply as
Figure 3. Variables influenc-
ing ice free-drift through a
G1 + G2 = Gc
(3)
confluence.
or
k) instead of channel slope S. The present analysis
(C[h/Y]Q)1 + (C[h/Y]Q)2 = (C[h/Y]Q)c .
(4)
uses Q, as it gives more meaningful parameters for
describing confluent flows than do q, V, or S. The
For water and ice conveyed by a river discharg-
ing into a reservoir or lake, Q2 = 0 and Qc ≈ 0, such
fluid properties of concern are kinematic viscosity
ν, density ρ, and surface tension strength σ. The ice
that
pieces, taken to be of uniform size, are described
Q1 = ∆∀/∆t
(5)
using a characteristic plan dimension D, thickness
among ice pieces and with the channel banks . The
and G2 = 0 and Gc ≈ 0, such that
flow is driven by gravitational acceleration g. The
G1 = ∆∀i/∆t .
discharge of free-drifting ice pieces moving at nearly
(6)
the surface water velocity in a single channel can be
described in terms of areal concentration C; ice dis-
The present analysis considers incipient ice jam-
charge G ≈ C(hb)(Q/bY) = C(h/Y)Q.
ming at a confluence of rivers and at a river dis-
A total of 13 variables are needed to describe
charging into a reservoir or lake. By assuming that
the discharge of free-drifting ice in a channel. To
ice piece dimensions, D and h, are the same for all
describe ice discharge in two channels, which dif-
channels, and that the channels have the same
fer only in geometry and discharges of water and
roughness k, the number of variables reduces to
ice, the number of variables increases to 20; added
22. If it is further assumed that the flows are sub-
are Q, Y, b, k, C, D, and h for the second channel.
critical, the effects of gravity g are taken care of by
The material properties of water and ice are taken
use of an open-channel discharge relationship for
4
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