This solution mode is the same technique used by several models where the

unsteady water flow is solved, followed by a calculation of the ice thickness. In this

mode, the water coefficient matrix is solved and then the ice thickness calculated

by an abbreviated form of the conservation of ice momentum equation, in which

ice velocity is always assumed to be zero. The resulting calculation of ice thickness

may violate ice mass conservation, since the ice velocity is neglected and is an

instantaneous (not time-dependent) reassessment of ice thickness. Equation 129 is

modified to

(

) + *g* (d

)

ηn+1 - ηn+1

- *d*j*n*+1

(

)

j+1

j+1

- *gS*

j

o

∆*x*

∆*x*

(150)

- i u2 = 0

+

8*s*i η

where the average (bar) terms are given as simple *x*-dependent averages. Previous

formulations for jam thickness by a relation similar in form to eq 150 indicate that

the integration can move either in an upstream or downstream direction (Beltaos

1993). This study found, however, well behaved solutions only for integration in

the downstream direction, given a thickness relation for the upstream boundary.

An equilibrium thickness condition is specified at the upstream boundary by set-

ting the / *x *terms of eq 150 to zero and solving the resulting relation for η. In

recognizing that the thickness does not decrease upon a reduction in the forces

acting on the cover, the newly calculated thickness is compared to the previous

thickness and the greater value adopted as the new thickness. This solution mode

is the least computationally intensive but neglects the effects of ice momentum. It

is included in this study only for comparison to the loosely coupled and fully coupled

modes.

Boundary condition equations are necessary for the closure of the system of equa-

tions for each solution technique described above. In general terms, upstream and

downstream boundary conditions are specified in each case. For the fully coupled

solution mode, ice and water boundary conditions are necessary at the upstream

and downstream boundaries. Similar to open-water modeling, the water bound-

ary conditions are typically a specified water discharge at the upstream and a speci-

fied depth relation at the downstream boundaries. The ice boundary conditions

include ice thickness at the upstream boundary and ice velocity at the downstream

boundary. As mentioned in the foregoing sections describing the various solu-

tion techniques, various relations can be used to describe these specified boundary

conditions.

54

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