→
→
DM i
+ M i∇ ⋅ V = 0 .
Internal ice resistance (R)
(32)
Dt
→
→
→
R = Rx i + Ry j
Since the ice mass per unit area, Mi, is determined by
the ice concentration and the ice layer thickness, one
Rx =
(σ xx Nti ) +
(σ xy Nti )
(23)
more conservation equation is needed. The equation of
x
y
conservation of ice area within an elemental area can
be obtained by considering the ice area flux into and
Ry =
(σ yy Nti ) +
(σ yx Nti )
(24)
out of the control area and mechanical redistribution.
y
x
→
DN
in which, σxx, σyy = normal stress components, and
+ N∇ ⋅ V + Ra = 0
(33)
Dt
σxy = σyx = shear stress components. These stresses
in which Ra = rate of change of ice area attributable to
can be determined by the constitutive relationship.
mechanical redistribution.
→
Wind drag at the airice interface ( F a )
A Lagrangian discrete-parcel method (DPM) (Shen
and Chen 1992, Shen et al. 1993) is used to simulate
v
→
→
→
F a = (τax N ) i + (τay N ) j
the dynamics of the ice transport. The basic concept of
the discrete-parcel method is that the ice, considered as
→
τax = ρa ca | W | Wx
(25)
a continuum, can be represented by a sufficiently large
number of individual parcels. Each parcel has well-
→
τay = ρaca | W | Wy
(26)
defined properties, such as mass, concentration,
thickness, and velocity, and is deformable in shape. Ice
in which
properties at parcel locations or finite-element nodes
can be interpolated from the properties of parcels within
→
→
→
W = Wx i + Wy j = wind velocity at 10 m above
the close vicinity. The theoretical background
the water surface
underlying this method is the smoothed particle
ρ = density of air
hydrodynamics (SPH) developed by Lucy (1977) and
ca = wind drag coefficient.
Gingold and Monaghan (1977).
Unlike the original smoothed particle hydro-
→
Water drag at the icewater interface ( F w)
dynamics, the present discrete-parcel model deals with
ice movement in a bounded domain, such as a river or
→
→
→
F w = (τwx N ) i + ( τ wy N ) j
a lake. A natural boundary condition at a stationary
boundary is a partial-slip boundary condition with zero
normal ice flux. As an ice parcel moves along a solid
→
→
τ wx = -ρcw V w - V i (u - Vwx )
boundary, it is subjected to a frictional force. The
(27)
method of images is used in this model for such a
boundary condition.
This model applies a dynamic Mohr-Coulomb yield
→
→
τ wy = -ρcw V w - V i (v - Vwy )
criterion to calculate the boundary frictional force as
(28)
follows
Ff = Fc + FN tan φ B
Gravitational force ascribable to the water
(34)
→
surface slope (G)
in which
→
→
→
G = Gx i + Gy j
Ff = frictional force between ice and the solid
boundary
η
Gx = - Mi g
Fc = ice cohesive force, assumed to be zero
(29)
x
FN = normal ice force against the boundary
η
Gy = - Mi g .
(30)
y
Constitutive law
The ice mass conservation equation is
In order to calculate the internal ice resistance, a
Mi
Miu
Mi v
+
+
=0.
(31)
constitutive law relating stresses with the motion of ice
t
x
y
is required. The most widely used constitutive law for
This equation can also be written as
ice dynamics is the viscousplastic law (Hibler 1979,
7
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