= [-2.26(

) +

0.7 m/s (2.25 ft/s) and the channel Froude number

ρi

(V

)

ρ)

+ 0.015]- 2

1

(40)

2.14

does not exceed 0.08. Ashton (1974) analyzed the

surface stability of floating ice blocks and obtained the

where *V*c = flow velocity below the cover and *L *= length

following submergence condition:

of ice floe. The experimental study of Kawai et al.

2(1 -

(1997) showed that the critical Froude number depends

)

=

not only on *t*b /L. For different floe sizes, their

(39)

ρ

experimental results showed that the *F*e,c = *f*(*t*b/*L*)

relationships are different. This shows the complexity

where

of the problem, and the difficulty in describing the

mechanism by a simple formula. For typical Missouri

River ice floes of thickness *t*b = 0.15 m, and size *L *= 1.5

~ 6 m, eq 40 gives *V*e,c = 0.76 ~ 1.4 m/s. In a 1:25

physical model test with natural ice scaled to the

This formula compared well with experimental data.

observed ice size distribution of the Missouri River ice,

Larsen (1975) studied the stability of thin blocks and

Tuthill and Gooch (1998) measured an erosion velocity

showed that *F*c is much larger than 1.4 when

to be about 1.5 m/s (5 ft/s).

these studies. More recent field studies showed that,

under certain conditions, specially designed booms

The present model assumes that each span of the

can perform successfully at water velocities as

ice boom is designed to submerge and to allow ice to

high as 0.76 m/s (2.5 ft/s) and Froude numbers

overtop it when a critical value of cable tension is

above 0.12 (White 1992). Results of a recent 1:25

exceeded (Shen et al. 1997). When the load is reduced

scale physical model study by Tuthill and Gooch

and the weight of the ice above the boom is smaller

(1998) support existing ice entrainment criteria, find-

than the net buoyancy of the boom, the boom will rise

ing that 0.3-m-thick-ice blocks submerge in a

again to prevent ice from passing through. The value

4.5-m-deep channel at a velocity of about 0.76 m/s (2.5

ft/s) (all units prototype), and a channel Froude number

and size of booms, has to be specified for the model

of 0.1.

simulation.

When an ice jam forms, its thickness may be limited

When the boom stops the ice movement, mechanical

by the stability of ice particles on its underside. As the

thickening will occur and the internal stress will build

jam thickens near its downstream end, the water flow

up quickly in the ice rubble. The load on the boom and

area decreases, increasing water velocity and shear

the span cable tension can be calculated from the ice

on the ice underside. The shear may become large

rubble stresses.

enough to erode ice pieces from the jam's underside

Consider a triangular differential element of ice

rubble *abc *(Fig. 5) in contact with the boom, which has

and transport them downstream. Thinning the

a finite thickness *t*i. The total force acting in the *x*-

downstream end of the jam by erosion lowers the water

level at the upstream end, increasing the velocity and

direction is

the tendency for ice entrainment. This process of under-

∑ *F*x = -σ xx Ntidy - σ yx Ntidx + *XNt*idl +

ice erosion near the toe and entrainment at the upstream

1 τ *Ndxdy *+ 1 τ * Ndxdy *+ 1 ρ *gNt dxdy *η

end of the jam may be sufficient to halt upstream

.

2 sx

2 wx

2 i

i

progression.

The stability of ice floes on the underside of an ice

(41)

The total force acting in the *y*-direction is

cover or ice jam has been investigated by Ashton (1974),

Uzuner (1977), Tatinclaux and Gogus (1981), and Daly

∑ *F*y = -σ yy Ntidx - σ xy Ntidy + *YNt*idl +

and Axelson (1990). Tatinclaux and Gogus (1981)

η

recommended an empirical stability criterion in terms

+ 1 τsy Ndxdy + 1 τ wy Ndxdy + 1 ρi gNtidxdy

2

2

2

of the critical Froude number *F*e,c and the floe aspect

ratio *t*b /L

(42)