the following functional relationship, for merging
commensurately with the porosity of the layer p,
i.e., G = ηQ(1 p). The material behavior of the
ice layers comprising ice of a given size (small com-
pared to channel width), with ice outflow as the
layer can be defined using its thickness H, angle
of internal resistance φ, porosity p, the density of
dependent variable:
η = fL (Q1 , Q2 , b1 , b2 , bc , Y1 , Y2 , Yc , kb , ki , η
explained above, the effects of water viscosity and
η1 , η2 , H1 , H2 , p, ρi , ρ, , φ, α, β) .
surface tension can be neglected for the scale of
(13)
flows in rivers. Water density ρ must be retained
for use with ρi as a variable. Therefore, a mini-
The 22 variables in eq 13 are reducible to 19
mum of 12 variables is needed to describe the dis-
nondimensional parameters, given three basic
charge of a continuous layer of ice moving in a
dimensions (mass, length, and time) involved with
channel. The influences of gravity, which motivate
ice discharge through a confluence. If the dimen-
the discharge of water and ice, are accounted for
sional analysis is carried out using bc, Q1 + Q2
(= Qc), and ρ as the repeating variables, the fol-
in the discharge relationships for water and ice,
and in the relationship between layer thickness H
lowing functional relationship emerges for the lim-
and ice discharge rate G.
iting condition of a contiguous layer of accumu-
To describe ice discharge in two channels that
lated ice discharging through a confluence:
differ only in geometry and the discharges of
Q
Y
water and ice, the number of variables increases
k k
b b
η = ϕL
, b, i,
1
, 1, 2
,
Q1 + Q2 bc bc
to 19, adding, for the second channel, Q, Y, b, k, ki,
bc 1,2,c bc bc
H, and η. The material properties of water and ice,
ρi
layer porosity, and friction are taken to be the same
H1 H2
, p, , φ, , α, β .
η1 , η2 ,
,
for all channels. To describe the merging of ice flow
(14)
ρ
bc bc
from two channels confluent into a single outflow
channel, the number of variables increases with
An awkward aspect of the foregoing dimen-
the addition of variables describing the orienta-
sional analysis is the precise definition of incipi-
tion of the outflow channel relative to the confluent
ent jamming. The analysis assumes that incipient
channels, α and β, and the hydraulic characteris-
jamming happens when the water and ice inflows
tics of the outflow channel (Qc, bc, Yc, k, and ki).
to a confluence begin to exceed outflows of water
The total number of variables is now 26. The num-
and ice. Actually, there probably are shades or
ber of variables can be reduced to 22 if the
degrees of incipient jamming; outflows of water
roughnesses k and ki are assumed the same for each
and ice may be less than inflows and yet a jam
channel.
may not have formed. Jam formation may be
attended by unsteady, interactive processes in
For the simple case of no ice jamming (i.e., no
which water and ice flows adjust in accordance
significant channel storage of water and ice in the
with, for example, changes in ice concentration
confluence), continuity of water and ice discharge
and layer thickness. The foregoing analysis,
through the confluence, without jam formation,
though simplifying the actual processes, nonethe-
gives respectively
less leads to useful sets of nondimensional param-
Q1 + Q2 = Qc
(10)
eters for describing the general characteristics of
ice movement through confluences.
and, for ice discharge
G1 + G2 = Gc
(11)
DISCUSSION
or
Having identified categories for classifying dif-
ferent situations of ice discharge through
G = ηcQc(1 p) = η1Q1(1 p)
confluences and having established sets of
+ η2Q2(1 p)
(12)
nondimensional parameters for discussing, in gen-
eral terms, ice discharge through confluences, it is
with ice discharge expressed as a proportion η of
interesting to evaluate which categories actually
water discharge. However, for the limiting condi-
seem to be the most problematic. The categories
tion of incipient jamming, eq 4 pertains.
indicating a higher incidence of problems should
The pertinent variables may be assembled in
be given a higher priority for further investiga-
6
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