x

x

x

x

x

x

x

x

a

b

c

d

e

f

g

h

a′

b′ c′ d′ e′

f′

g′ h′

a′′

b′′ c′′ d′′ e′′

f′′

g′′ h′′

a′′′

b′′′ c′′′ d′′′ e′′′

f′′′

g′′′ h′′′

a

b

c

d

e

f

g

h

a′

b′

c′ d′ e′

f′

g′

h′

a′′ b′′ c′′ d′′ e′′

f′′

g′′ h′′

a′′′ b′′′ c′′′ d′′′ e′′′

f′′′

g′′′ h′′′

[A]=

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

h

a

b

c

d

e

f

g

a′ b′

c′

d′ e′

f′

g′

h′

h′′

a′′ b′′

c′′

d′′ e′′

f′′

g′′

h′′′

a′′′ b′′′

c′′′

d′′′ e′′′

f′′′

g′′′

x

x

x

x

x

x

x

x

*Figure 31. Coefficient matrix for the Newton-Raphson iteration procedure.*

*m*+1 *n*+1 *m n*+1

*d*j = *d*j + ∆*d*j .

(146)

The values of the variables are updated in this way for the `*m *+ 1' iteration and

solved again until a specified tolerance is met. The tolerance is expressed in a least

squares form

(

)

*N *-1

*N*

2

*Tol *= ∑ * m*+1*F*-*mF * = ∑ (∆' *s*) .

2

(147)

*j*=1

*j*=1

When the tolerance is met, the final `*m *+ 1' iteration values are used as the values of

the dependent variables for time `*n *+ 1'. The time is then incremented and the

iteration procedure begins again. For the first iteration at a time step (`*m *= 0'), the

initial values of the variables are taken to be equal to the final value of the variables

at the previous time step.

**Ice cover stability, solution methods, and boundary conditions**

The conservation of ice momentum, as expressed in eq 102 or 115, is actually an

equation expressing a unique balance of ice momentum and external forces acting

on an ice jam element. Through assumption of the passive-pressure failure criteria

and adopting the Rankine equation for *K*p (the passive pressure coefficient), the

equation is valid for a static element of an ice jam at its limit of stability. The equa-

tion is also valid for an ice jam element failing and in motion. In the former case, if

the forces on a static element exceed the passive pressure limit, application of the

equation (in concert with the other conservation equations) results in downstream

ice movement. For an ice element in motion, application of the equation for a change

in the forces may result in changes in ice velocity or thickness. For many cases, the

49