Modeling Wide-River Ice Jams

siThe wide river ice jam is probably the most common type of river ice jam. In this type, all stresses acting on the jam are ultimately transmitted to the channel banks. The stresses are estimated using the ice jam force balance equation:

1)	$\begin{array}{l}\displaystyle \displaystyle \frac{d (\overline{\sigma}_x t)}{dx} + \frac{2 \tau _b t}{B} = \rho ' gS_w t+ \tau _i\end{array}$

Symbol	Description	Units
$\begin{array}{l}\overline{\sigma} _x\end{array}$	the longitudinal stress (along stream direction)
$\begin{array}{l}t\end{array}$	the accumulation thickness
$\begin{array}{l}\tau _b\end{array}$	the shear resistance of the banks
$\begin{array}{l}B\end{array}$	the accumulation width
$\begin{array}{l}\rho '\end{array}$	the ice density
$\begin{array}{l}g\end{array}$	the acceleration of gravity
$\begin{array}{l}S_w\end{array}$	the water surface slope
$\begin{array}{l}\tau _i\end{array}$	the shear stress applied to the underside of the ice by the flowing water

This equation balances changes in the longitudinal stress in the ice cover and the stress acting on the banks with the two external forces acting on the jam: the gravitational force attributable to the slope of the water surface and the shear stress of the flowing water on the jam underside.

Two assumptions are implicit in this force balance equation: that $\begin{array}{l}\overline{\sigma} _x\end{array}$ , $\begin{array}{l}t\end{array}$ , and $\begin{array}{l}\tau _i\end{array}$ are constant across the width, and that none of the longitudinal stress is transferred to the channel banks through changes in stream width, or horizontal bends in the plan form of the river. In addition, the stresses acting on the jam can be related to the mean vertical stress using the passive pressure concept from soil mechanics, and the mean vertical stress results only from the hydrostatics forces acting in the vertical direction. In the present case, we also assume that there is no cohesion between individual pieces of ice (reasonable assumption for ice jams formed during river ice breakup). A complete discussion of the granular approximation can be found elsewhere (Beltaos 1996).

In this light, the vertical stress, $\begin{array}{l}\overline{\sigma} _x\end{array}$ , is:

2)	$\begin{array}{l}\displaystyle \overline{\sigma} _x = \gamma _e t\end{array}$

Where:

3)	$\begin{array}{l}\displaystyle \gamma _e = 0.5 \rho ' g (1-s)(1-e)\end{array}$

Symbol	Description	Units
$\begin{array}{l}e\end{array}$	the ice jam porosity (assumed to be the same above and below the water surface)
$\begin{array}{l}s\end{array}$	the specific gravity of ice

The longitudinal stress is then:

4)	$\begin{array}{l}\displaystyle \overline{\sigma} _x = k_x \overline{\sigma} _z\end{array}$

Where:

5)	$\begin{array}{l}\displaystyle \displaystyle k_x = tan^2 \left( 45 + \frac{\varphi}{2} \right)\end{array}$

Symbol	Description	Units
$\begin{array}{l}\varphi\end{array}$	the angle of internal friction of the ice jam

The lateral stress perpendicular to the banks can also be related to the longitudinal stress as

6)	$\begin{array}{l}\displaystyle \displaystyle \overline{\sigma} _y = k_1 \overline{\sigma} _x\end{array}$

Symbol	Description	Units
$\begin{array}{l}k_1\end{array}$	the coefficient of lateral thrust

Finally, the shear stress acting on the bank can be related to the lateral stress:

7)	$\begin{array}{l}\displaystyle \tau _b = k_0 \overline{\sigma} _y\end{array}$

Where:

8)	$\begin{array}{l}\displaystyle k_0 = tan \varphi\end{array}$

Using the above expressions, we can restate the ice jam force balance as:

9)	$\begin{array}{l}\displaystyle \displaystyle \frac{dt}{dx} = \frac{1}{2k_x \gamma _e} \left[ \rho ' gS_w + \frac{\tau _i}{t} \right] - \frac{k_0 k_1 t}{B} =F\end{array}$

Symbol	Description	Units
$\begin{array}{l}F\end{array}$	a shorthand description of the force balance equation

To evaluate the force balance equation, the under-ice shear stress must be estimated. The under-ice shear stress is:

10)	$\begin{array}{l}\displaystyle \tau _i = \rho g R_{ic} S_f\end{array}$

Symbol	Description	Units
$\begin{array}{l}R_{ic}\end{array}$	the hydraulic radius associated with the ice cover
$\begin{array}{l}S_f\end{array}$	the friction slope of the flow

$\begin{array}{l}R_{ic}\end{array}$ can be estimated as:

11)	$\begin{array}{l}\displaystyle \displaystyle R_{ic} = \frac{n_i}{n_c} ^{1.5} R_i\end{array}$

The hydraulic roughness of an ice jam can be estimated using the empirical relationships derived from the data of Nezhikovsky (1964). For ice accumulations found in wide river ice jams that are greater than 1.5 ft thick, Manning's n value can be estimated as:

12)	$\begin{array}{l}\displaystyle n_i = 0.069 H^{-0.23} t^{0.40}_i\end{array}$

and for accumulations less than 1.5 ft thick

13)	$\begin{array}{l}\displaystyle n_i = 0.0593 H^{-0.23} t^{0.77}_i\end{array}$

Symbol	Description	Units
$\begin{array}{l}H\end{array}$	the total water depth
$\begin{array}{l}t_i\end{array}$	the accumulation thickness