The conservation of ice mass equation becomes

η

η

υ

+υ

+η

= 0.

(113)

*t*

*x*

*x*

Expanding the terms in eq 110, combining with eq 112, and dividing by *d*, the con-

servation of water momentum equation becomes

η

*d*

*u*

*u*

+*u*

+*g*

+ *gs*i

- *gS*o

*x*

*x*

*x*

*t*

*fbu*2 (B + 2*d*)

*u *- υ

2

*f*i

*B*

(114)

1 +

+

= 0.

*f*b (B + 2*d*) *u *

8*Bd*

Similarly, after expansion of the partials in eq 111, combining with eq 113, and

dividing by η, the conservation of ice momentum equation becomes

υ

υ

η

η

+ *g*(1 - *s*i )Kp (1 - *p*)

*d*

+υ

+ *gs*i

+*g*

*t*

*x*

*x*

*x*

*x*

*g*(1 - *s*i )

(115)

*f*

*k*0λ*K*p (1 - *p*)η - *gS*o - i (u - υ) = 0.

2

+

8*s*i η

*B*

Figure 30 shows the computational grid used for the numerical simulations in

this study. The discretization of the governing equations assumes that the values of

the dependent variables and functions of the variables between the computational

grid points can be expressed in terms of the values at the grid points. A function

*f(x,t) *within the space interval *j *and (*j *+ 1) is replaced by the following weighted

average

*f *(x, *t*n ) = Ψ*f*j*n*1 + (1 - Ψ) fj*n*

(116a)

+

and

*f *(x, *t*n+1) = Ψ*f*j*n*+1 + (1 - Ψ) fj*n*+1

(116b)

+1

t

∆x

t n+1

D(j, n+1)

C(j+1, n+1)

∆t

tn

A(j,n)

B(j+1, n)

t n1

x

xj

x j+1

x j1

*Figure 30. Computational grid used for the numerical simulations.*

44