coupled numerical model of ice jam formation. Such a model includes both ice

velocity and the effects of ice momentum on the force balance, utilizing the full

conservation of mass and momentum equations for the ice.

Formulated here are the one-dimensional, unsteady flow equations for water

and ice. The equations, derived in integral form, are based on the conservation of

mass and momentum for water and ice flow during jam formation and breakup.

The integral equations are then discretized as finite-difference equations, approxi-

mating the conservation laws in their integral form. The equations are expressed in

terms of four dependent variables that fully describe the flow, as shown in Figure

23, namely the velocities of the ice cover and under-ice water flow (υ and *u, *respec-

tively), the ice cover thickness (η), and the under-ice water depth (*d*). All four vari-

ables are functions of time and space. The equations are derived first in a general

form, and then simplified in accordance with the assumptions listed below. Addi-

tional simplifications for certain flow conditions are discussed subsequently.

The assumptions made in developing the equations for water and ice flow

include the usual St. Venant assumptions for one-dimensional flows (e.g., Cunge et

al. 1980) are as follows:

Flow is one-dimensional, with velocity uniform across each cross section, and

water level horizontal at each cross section.

pressure distribution is hydrostatic.

The effects of turbulence and boundary friction can be accounted for through

resistance laws identical to those used for steady-state flow.

Average channel-bed slope is small so that the cosine of the angle it makes

with the horizontal may be taken as unity. (This assumption is valid for bed

slopes to about 0.01.)

Ice

η(x,t)

υ(x,t)

Water

u(x,t)

d(x,t)

Bed

29