ness coefficient assignment as a function of thickness based on the data of
Nezhihovskiy (1964). The program then automatically updates the HEC-2 input
file to reflect these new values of ice thickness and roughness. Iterations continue
until a specified tolerance is met. Considerable judgment is necessary in the jam-
toe area, where ice thickness conditions cannot be expressed adequately by the
equilibrium thickness as provided by eq 25.
Lal and Shen (1991) developed the jam model RICE, which is intended to simu-
late unsteady conditions of water flow, water temperature, ice concentration, and
thermal growth and decay of ice. In their model, ice travels downstream at the
water velocity until it reaches some location where a jam forms, by either ice-piece
juxtaposition or the narrow-jam or wide-jam accumulation modes. The wide-jam
mode is taken to be governed by the ice force balance equation proposed by Pariset
et al. Lal and Shen did recognize that as progression (by shoving) is taking place,
there is a simultaneous change in the flow hydraulics. They take care of flow changes
by solving the equilibrium thickness and step-backwater equations simultaneously
in the reach where the jam is thickening. The RICE model has been used success-
fully in simulations of ice conditions on the St. Lawrence, Niagara, Ohio, and Yel-
low rivers, though, like other models, it requires significant calibration to match
field data.
Tsai et al. (1988) developed a jam model to investigate ice transport in rivers and
ice jam initiation. They used a one-dimensional numerical scheme for solving the
ice transport equations, i.e., conservation of ice momentum, ice mass, and ice area.
The equations are solved in a Lagrangian form, where the trajectories of ice ele-
ments at fixed Eulerian grid points at the beginning of a time step are traced on the
x t plane. Values of ice variables are then interpolated back to the grid points at
the end of the time step. The de Saint Venant equations for unsteady water flow are
solved using a four-point implicit finite-difference scheme. The ice transport and
water flow equations are loosely coupled by first solving the water flow equations
and then the ice transport equations based on the new values of the water flow
variables.
Shen et al. (1990) elaborated further aspects of this model, examining the vari-
ous plausible constitutive relationships possible for describing the internal stresses
and bank shear. For example, they describe a rapid flow regime as one in which the
ice concentration is low and interaction between ice particles is minimal. Commen-
surately, they characterize a slow flow regime as one in which higher (multi-layer)
ice concentrations typically form, and where internal resistance is attributable to
prolonged interaction of contacting particles. Their expressions for the streamwise
stress σx and the stress normal to the bank τxy are equivalent to those for passive
pressure resistance, as described by Pariset et al. The authors state the model
appears to adequately describe the time and location of jam initiation in river chan-
nels, but that more research is necessary to improve the constitutive laws.
Summary
While considerable progress has been made in modeling the unsteady flow
associated with stationary ice jams, the unsteady aspects of ice movement have
not been adequately addressed. Most models treat shoving and thickening as an
instantaneous phenomenon, with no consideration for the effects of ice momen-
tum on the resulting jam thickness and profile. Physics and field observations sug-
gest that ice momentum should substantially affect jam thickness.
The following sections describe laboratory and numerical experiments aimed at
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