stress on the ice cover. Figure 26 identifies the two layers. Note that α refers to the

area of each respective layer, *P *is the wetted perimeter, and the subscripts *b *and *i*

designate the bed- and ice-affected variables. The dashed line indicates the line of

nominal zero shear or the boundary between the two layers. The total shear force

per unit length of flow area is

τ*P *= τ bPb + τi Pi .

(55)

If the "two-layer approach" is valid for any value of ice velocity υ, such as depicted

in Figure 27, then the shear stress at each boundary is expressible as

ρ*f*bu u

τb =

(56)

8

and

ρ*f*i (u - υ) u - υ

τi =

.

(57)

8

The absolute value sign captures directional shear. It can be dropped provided that

stress direction is preserved in the momentum equation.

The "two-layer approach" assumes that each layer can be adequately described

using the Darcy-Weisbach relationship for flow resistance and thus can be related

to the friction slope of the water flow, i.e.

2

= i

.

(58)

8*R*b g

8*R*i g

0<υ<u

τi

τ=0

u

τb

a)

υ=u

τi = 0

u

τb

b)

35

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