λ m = λρλ3

(5)

L

= λT

(2)

scale is

and

λF

λ

λa =

,

(6)

= λV = L .

λm

(3)

λT

or from velocity and time,

Ice modeling usually requires consideration of

λL

λa =

(7)

.

water movement, ice movement, and ice deforma-

λ2

T

tion and failure. The present discussion begins

with a brief review of the criteria for dynamic si-

The force scale can then be expressed as

militude of water flow and ice-piece transport,

λ4 λρ

then reviews the similitude criteria for ice-sheet

L

λF =

(8)

.

deformation and failure and the criteria for the

λ2

T

deformation and failure of ice rubble. The criteria

for ice-sheet, or rubble, deformation are less well

All dynamic quantities can be expressed in terms

developed or generally accepted than are the cri-

of the length, time, and density scales.

teria for water flow and ice-piece transport.

Most hydraulic processes are governed by

Modeling complexity and the constraints on

momentum balances involving inertial, gravita-

model ice selection increase markedly with mod-

tional, and viscous forces. Surface tension forces

eling situations requiring that both hydraulics and

usually are insignificant, except when considering

ice deformation criteria be met. Some modeling

very shallow flows or the movement of small ice

situations require consideration only of the simili-

pieces. The relative influences of inertial and grav-

tude criteria associated with water flow and ice-

itational forces can be expressed nondimensional-

piece transport. For those situations, an unbreak-

ly as a Froude number,

able model ice of appropriate buoyancy suffices.

Other modeling situations involve ice deforma-

(9)

tion in static water and require satisfaction of the

criteria for ice deformations and failure as well as

for ice-piece movement. Those situations require a

or as a densimetric Froude number,

model ice that deforms and breaks appropriately.

Modeling becomes complicated when the mod-

(10)

eled situation requires simulation of water flow as

well as the failure and transport of ice. When ther-

modynamic processes are important, thermody-

in which *V *is a representative velocity, *y *is flow

depth or an alternate length of interest, and ρs and

namic similitude criteria guide the modeling, and

ρ are solid and fluid densities, respectively. The

the skill of the modeler becomes vital. Generally,

relative influences of inertial and viscous forces

the more criteria to be met, the less accurate is the

can be expressed nondimensionally as a Reynolds

modeling, and greater reliance must be placed on

number,

the experience and interpretive abilities of the

modeler.

For more detailed coverage and discussion of

(11)

ν

the similitude criteria see Ashton (1986), ITTC

(1987, 1990, 1993), and Ettema et al. (1992).

in which ν is the kinematic viscosity of the fluid

(usually water). The relative magnitudes of inertial

and surface-tension forces can be expressed

Dynamic similarity relates prototype and model

nondimensionally as a Weber number,

forces by a force scale; i.e.,

(12)

(4)

= λF .

As the scale for mass is

in which *k *is the surface-tension strength of water.

3