where *R*i is the hydraulic radius of the ice-covered

transport is assumed to occur at locations where a

channel, and τ is the shear stress on the underside

solution to eq 15 is obtained; if no solution is pos-

sible (i.e., Φn < Φc), deposition is assumed to oc-

of the ice cover.

Because the width of a river channel is often

cur. An iterative process is required to obtain an

much wider than its depth, the open-water hy-

draulic radius *R *(the ratio of cross-sectional area

hydraulic model is thickened at the locations

to wetted perimeter) of an open-water reach is of-

where deposition is identified, beginning with the

ten approximated as *H*, the water depth. In the

most upstream location. The ice-covered hydrau-

ice-covered case, the wetted perimeter is effec-

lic model is rerun, providing new values of *R *and

tively doubled, so *R*i can be approximated by *y*i/

2, where *y*i is the under-ice depth, often calculated

cross section, thus identifying locations of trans-

from Manning's formula as

port and deposition under the updated hydraulic

conditions. The process is repeated until no fur-

ther deposition is possible or until the volume of

3

*cqn *

5

the deposited frazil ice nears the estimated vol-

(12)

ume produced. The variables required in this

*S*f 2

method are ρ, ρi, *d*n, *C*f, *S*f, and the variables nec-

essary to calculate *R: q, n*i, *n*b, and *n*c.

where *c *is a coefficient (1.5875 in SI units of s/m1/3

and 1.0654 in English units of s/ft1/3), *q *is the dis-

charge per unit width, and *n*c is the composite

Manning's *n *for the channel. Nezhikhovskiy

When the forces exerted by incoming ice in-

(1964) reviewed methods of determining compos-

crease to the point that they exceed the internal

ite channel roughness and suggested the use of

strength of ice accumulations formed by juxtapo-

the Belokon-Sabaneev formula:

sition or thickened by deposition, the ice accumu-

lation will fail. The failure may be large scale, re-

2

*n*3 2 + *n*3 2

3

sulting in the transport of ice for large distances

b

(13)

until the accumulation stops again, or it may be

2

smaller in scale, resulting in a local shoving and

thickening of the accumulation. The shoving and

in which *n*i is the Manning's *n *for underside of ice

thickening process is an inherently unsteady pro-

and *n*b is the Manning's *n *for the bed. Other for-

cess that may eventually cause the formation of

mulations are presented in Uzuner (1975). The

what is often termed an equilibrium jam; that is, a

roughness ratio can be used to modify the ice-

jam in which a uniform section of ice has reached

affected hydraulic radius as follows:

a thickness where the external and internal forces

3

are in equilibrium (Fig. 4).

*n *

2

Zufelt and Ettema (1997) developed an un-

.

(14)

*n*c

2

steady model of the shoving and thickening pro-

cess, and a stationary ice cover subroutine has been

used in conjunction with the one-dimensional

White and Acone (1998) modeled frazil depo-

unsteady flow model UNET developed by the

sition beneath an ice cover at a river-reservoir

Hydrologic Engineering Center (HEC) (USACE

confluence utilizing both ICETHK and the sim-

1995). The use of a dynamic ice cover in UNET is

plified version of Shen and Wang's functional re-

under development (McGilvary et al. 1995, Daly

lationship in eq 6 to 10:

et al. 1997). However, most currently used mod-

els assume steady state conditions, including the

Φn = 5.487(Θn Θc )

1.5

(15)

HEC's River Analysis System (HEC-RAS) (USACE

1997), the ICETHK option of the USACE step-

where Θc = 0.041. The following process was used:

backwater computer program HEC-2 (USACE

1990), and Canadian models RIVJAM and ICESIM

1) a hydraulic model with stable ice cover must

(Healy et al. 1997). This assumption is actually con-

be developed for the reach of river in question; 2)

servative except in the downstream transition

variables *R *and *S*f calculated by the hydraulic

model are used to estimate Θn using eq 10 and 11,

zone, and hence has been deemed acceptable.

from which Φn is calculated using eq 15; 3) frazil

Zufelt (1999) presents a test that can be used to

6