Wake and Rumer 1983, Shen et al. 1993). The present

along the boundary are set to be zero. Since the

study uses the Mohr-Coulomb yield criterion (Gutfraind

hydrodynamic model is a depth-averaged model, it is

and Savage 1997), i.e.

assumed that the floating ice boom does not affect the

flow condition. However, if the boom configuration can

(

)

(35)

σ l σ2 = σ l + σ2 sin φ

significantly affect the flow, an internal boundary

condition may be added.

where σ1 and σ2 are the principal stresses and φ is the

in terms of a nonlinear viscosity , as follows:

Initial conditions for ice dynamic simulations are

ice concentration, ice velocity, and ice thickness in the

(36)

σ ij = 2 ε ij - ε kk δ ij - *P*δ ij

channel. Ice thickness, ice concentration, and ice

where εij is the strain rate tensor. Assuming that the

velocity are required upstream boundary conditions.

principal axes of stress and strain rate coincide, we can

External and internal forces acting on the ice govern

express the viscosity as

the boundary ice velocity. However, when the ice

concentration is not very high, the internal ice resistance

is not significant, which is usually the case at the

*P *sin φ

(37)

= min

, max

upstream boundary, and the ice velocity at the upstream

ε1 - ε2

boundary may be approximated by the water velocity.

where ε1 and ε2 are the principal components of the

strain rate tensor and max, the maximum value of the

viscosity, determines whether the stress state is inside

When surface ice arrives at the boom from upstream,

or on the yield envelope. When = max, the ice will

it may stop behind the boom to accumulate into an ice

flow as a viscous fluid, whereas it will flow in a plastic

cover. However, if the current velocity exceeds a critical

manner when is smaller than max. The following

entrainment velocity, the surface ice will submerge and

expression obtained by extending the constitutive

be transported downstream. If the flow condition

relationship for static ice jams (Shen et al. 1990) is used

permits the accumulation of the ice rubble behind the

to determine the pressure *P*:

boom, the upstream progression is limited by the

possible entrainment of surface ice at the upstream edge

πφ

1

of the ice accumulation. In addition, an increase in the

2

42

current velocity beneath the ice accumulation, caused

(38)

ρ ρ gt

either by the thickening of the cover or an increase in

(1 - i ) i i (

)

ρ

water discharge, can erode ice particles from the

2 N max

underside of the cover and limit its progression. Neither

in which *j *= 15, an empirical constant, and the + and

surface ice entrainment nor undercover erosion was

signs are for convergent and divergent states,

considered in the lake ice boom simulation model of

respectively.

Shen et al. (1997), owing to the low current velocity in

lakes. The critical condition for ice accumulation behind

a lake ice boom is the spillover of ice rubble when the

boom load exceeds a critical value. This condition

Initial values of unit discharges *q*x,*q*y, and water level

should also be considered for river ice booms. However,

η at every finite-element node have to be specified for

owing to the increased effect of bank resistance, the ice

the hydrodynamic simulation. A steady-state condition

load on river ice booms rapidly approaches a limiting

that corresponds to the specified boundary conditions

value as the ice cover progresses a few channel widths

at *t = *0 is generated with the model and used as the

upstream (Latyshenkov 1946, Tuthill and Gooch 1998).

initial conditions for *q*x,*q*y, and η. Boundary conditions

In this section, the limiting conditions for ice

used are the water level at the downstream boundary

accumulation behind boom will be discussed.

and discharge at the upstream boundary. The unit normal

discharge distribution across the upstream boundary is

estimated by the stream-tube method. The calculated

unit-discharge distribution is adjusted to ensure that the

When water velocity is high, surface ice arriving at

water level across the width of the boundary is constant.

the boom will submerge and pass under it. Past field

Along land boundaries, such as shorelines and dikes,

experience suggests that ice retention is possible at river

the unit normal discharges at the finite-element nodes

locations where the surface water velocity is at or below