Static-unsteady thickness solution
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
- djn+1
n+1
(
)
j+1
j+1
g(1 - si )Kp (1 - p) + gsi
- gS
j
o
∆x
∆x
g(1 - si )k0λKp (1 - p)η
(150)
f
- i u2 = 0
+
8si η
B
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 conditions
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.
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