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*.

and the discharges of water and ice, the number of variables increases to 20, add-

ing, for the second channel, *Q*, *Y*, *b*, *k*, *k*i, *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 (*Q*c, *b*c, *Y*c, *k *and

The number of variables can be reduced to 23 if the roughness heights *k *and *k*i

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

(16)

and, for ice discharge,

(17)

or

(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 = *f*L (Q1 , *Q*2 , *b*1 , *b*2 , *b*c , *Y*1 , *Y*2 , *Y*c , *k*b , *k*i , η1 , η2 , *H*1 , *H*2 , *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 *b*c, *Q*2, 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*

η = ϕL 1 ,

,

,

,

*Q*2 Db gD b2 b3*c*

*b*c 1,2,3

2

ρi

, , , η1 , η2 ,

, *p*, , φ, , α, θ .

,

(20)

ρ

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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