→

→

+ *M *i∇ ⋅ *V *= 0 .

(32)

→

→

→

Since the ice mass per unit area, *M*i, is determined by

the ice concentration and the ice layer thickness, one

(σ xx Nti ) +

(σ xy Nti )

(23)

more conservation equation is needed. The equation of

conservation of ice area within an elemental area can

be obtained by considering the ice area flux into and

(σ yy Nti ) +

(σ yx Nti )

(24)

out of the control area and mechanical redistribution.

→

in which, σxx, σyy = normal stress components, and

+ *N*∇ ⋅ *V *+ *R*a = 0

(33)

σxy = σyx = shear stress components. These stresses

in which *R*a = rate of change of ice area attributable to

can be determined by the constitutive relationship.

mechanical redistribution.

→

A Lagrangian discrete-parcel method (DPM) (Shen

and Chen 1992, Shen et al. 1993) is used to simulate

v

→

→

→

the dynamics of the ice transport. The basic concept of

the discrete-parcel method is that the ice, considered as

→

τax = ρa ca | *W *| *W*x

(25)

a continuum, can be represented by a sufficiently large

number of individual parcels. Each parcel has well-

→

τay = ρaca | *W *| *W*y

(26)

defined properties, such as mass, concentration,

thickness, and velocity, and is deformable in shape. Ice

in which

properties at parcel locations or finite-element nodes

can be interpolated from the properties of parcels within

→

→

→

the close vicinity. The theoretical background

the water surface

underlying this method is the smoothed particle

ρ = density of air

hydrodynamics (SPH) developed by Lucy (1977) and

Gingold and Monaghan (1977).

Unlike the original smoothed particle hydro-

→

dynamics, the present discrete-parcel model deals with

ice movement in a bounded domain, such as a river or

→

→

→

a lake. A natural boundary condition at a stationary

boundary is a partial-slip boundary condition with zero

normal ice flux. As an ice parcel moves along a solid

→

→

τ wx = -ρ*c*w V w - *V *i (*u *- *V*wx )

boundary, it is subjected to a frictional force. The

(27)

method of images is used in this model for such a

boundary condition.

This model applies a dynamic Mohr-Coulomb yield

→

→

τ wy = -ρ*c*w V w - *V *i (*v *- *V*wy )

criterion to calculate the boundary frictional force as

(28)

follows

(34)

→

in which

→

→

→

boundary

η

(29)

η

(30)

The ice mass conservation equation is

In order to calculate the internal ice resistance, a

+

+

=0.

(31)

constitutive law relating stresses with the motion of ice

is required. The most widely used constitutive law for

This equation can also be written as

ice dynamics is the viscousplastic law (Hibler 1979,