→

→

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

i

Similarly, the *y*-component of the momentum

(17)

s

equation is

2

in which

2

2

+

+

)+

+

)=

(

(

→

→

t

t

= water current velocity

1

(τ iy + τsy - τ by )

(11)

→

→

→

ρ

η 1 * T*xy

+ (

+

- *g*(*H * u + *H *′)

) .

varies with ice concentration and ice

→

The flow model solves for components of q t and the

This coefficient can be related to Manning's coefficient

water depth *H *using eq 3, 9, and 11. A finite-element

of the underside of the cover as

model with the lumping technique and leapfrog time

2

integration (Connor and Brebbia 1978, Wake and Xiao

.

(18)

1989) is used. The bed shear stresses can be expressed

1

[(1 - α h )*H *′]

3

as

1

2

2

For a partially ice-infested water surface, the surface

2

τ bx = *c*f ρ

(12)

shear stress is assumed to be a linear combination of

2

τ(a-w) and τ(i-w)

s

s

1

2

2

2

τ by = *c*f ρ

→

→

→

(13)

τ s = (1 - *N *) τ s (a-w ) + *N *τ s (i-w ) .

2

(19)

The drag of the seepage flow on ice is

terms of Manning's coefficients of the bed and the shear

stress distribution coefficient αh as

τix = -ρ*gH*u

(20)

2

2

τiy = -ρ*gH*u

.

1

α hH ′

.

(21)

3

2

On the open water surface, the surface shear stress

attributable to wind effect can be expressed as

The momentum equation of the surface ice can be

τ(a-w) = ρa γ 2W 2 cos θa

(14)

sx

written in the Lagrangian form as (Shen et al. 1990)

→

τ(a-w)

→

= ρa γ W sin θa

2

2

(15)

= *R*+ *F *a + *F *w + *G*

sy

(22)

in which

in which

→

γ2 = wind drag coefficient (Wu 1973)

= acceleration of ice

= ρiNti = ice mass per unit area

→i

surface

= internal ice resistance

→

ρa = density of air

= wind drag

→

θa = angle between the wind direction and the

= water drag

→

ρi = density of ice

For a fully ice-covered water surface, the surface

shear stress components on the water can be written as

→

→

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

Force terms in the momentum equation can be

i

(16)

s

expressed in two-dimensional forms as follows.