where xj ≤ *x *≤ *x*j+1. Similarly, for the time interval tn ≤ *t *≤ *t*n +1

()

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

(117a)

and

(

)

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

(117b)

+1

+

where 0 ≤ θ ≤ 1 and 0 ≤ Ψ ≤ 1 are weighting factors. Subscripts denote the *x*-

location while superscripts denote time level. By use of these relations, the finite

difference equations that approximate the conservation laws expressed by the dif-

ferential eq 112, 113, 114, and 115 can be developed. When Ψ = 0.5, the resulting

equations represent the Preissmann four-point scheme of finite differences. In the

discretization, the derivatives of a function *f(x,t) *are estimated as

(

)(

)

*f*j*n*+1 + *f*j*n*+1 - *f*j*n*1 + *f*j*n*

+1

+

*f*

=

(118a)

2∆*t*

*t*

(

)

(

)

and

+ (1 - θ) fj*n*1 - *f*j*n*

*n*+1

*n*+1

*f * θ *f*j+1 - *f*j

+

=

(118b)

∆*x*

*x*

and the values of the variables, or the functions themselves, are estimated as

(

)

(

).

θ *f*j*n*+1 + *f*j*n*+1 + (1 - θ) fj*n *+ *f*j*n*1

+1

+

*f *(x, *t*) =

(119)

2

The above rules of discretization can be used to recast the conservation of water

mass equation as

(d

) - (d

) + *u * θ(d

) + (1 - θ)(d

)

*n*+1

*n*+1

*n*+1

*n*+1

*n*

*n*

*n*

*n*

j+1 - *d*j

j+1 + *d*j

j+1 + *d*j

j+1 - *d*j

2∆*t*

∆*x*

(

)

(

) = 0

θ *u*n+1 - *u*n+1 + (1 - θ) un - *u*n

(120)

j+1

j+1

+*d *

j

j

∆*x*

where

(

)

(

)

θ *u*j*n*+1 + *u*j*n*+1 + (1 - θ) uj*n *1 + *u*j*n*

+1

+

*u*=

(121)

2

and

(

)

(

).

θ *d*j*n*+1 + *d*j*n*+1 + (1 - θ) dj*n *1 + *d*j*n*

+1

+

*d*=

(122)

2

Similarly, the conservation of ice mass equation becomes

(η

) - (η

) + υ θ (η

) + (1 - θ )(η

)

*n*+1

*n*+1

*n*+1

*n*+1

*n*

*n*

*n*

*n*

j+1 - ηj

j+1 + ηj

j+1 + ηj

j+1 - ηj

i

i

2∆*t*

∆*x*

(

)

(

) = 0

θ υn+1 - υn+1 + (1 - θ ) υn - υn

(123)

i j+1

j+1

+ η

j

i

j

∆*x*

45