Modeling Ice Covers with Known Geometry

Ice covers are common on rivers during the cold winter months and they form in a variety of ways. The actual ways in which an ice cover forms depend on the channel flow conditions and the amount and type of ice generated. In most cases, river ice covers float in hydrostatic equilibrium because they react both elastically and plastically (the plastic response is termed creep) to changes in water level. The thickness and roughness of ice covers can vary significantly along the channel and even across the channel. A stationary, floating ice cover creates an additional fixed boundary with an associated hydraulic roughness. An ice cover also makes a portion of the channel cross sectional area unavailable for flow. The net result is generally to reduce the channel conveyance, largely by increasing the wetted perimeter and reducing the hydraulic radius of a channel, but also by modifying the effective channel roughness and reducing the channel flow area.

The conveyance of a channel or any subdivision of an ice-covered channel, K_i, can be estimated using Manning's equation:

1)	$\begin{array}{l}\displaystyle \displaystyle K_i = \frac{1.486}{n_c} A_i R^{2/3}_i\end{array}$

Symbol	Description	Units
$\begin{array}{l}n_c\end{array}$	the composite roughness
$\begin{array}{l}A_i\end{array}$	the flow area beneath the ice cover
$\begin{array}{l}R_i\end{array}$	the hydraulic roughness modified to account for the presence of ice

The composite roughness of an ice-covered river channel can be estimated using the Belokon-Sabaneev formula as:

2)	$\begin{array}{l}\displaystyle \displaystyle n_c = \left( \frac{n^{3/2}_b +n^{3/2}_i}{2} \right) ^{2/3}\end{array}$

Symbol	Description	Units
$\begin{array}{l}n_b\end{array}$	the bed Manning's roughness value
$\begin{array}{l}n_i\end{array}$	the ice Manning's roughness value

The hydraulic radius of an ice-covered channel is found as:

3)	$\begin{array}{l}\displaystyle \displaystyle R_i = \frac{A_i}{P_b + B_i}\end{array}$

Symbol	Description	Units
$\begin{array}{l}P_b\end{array}$	the wetted perimeter associated with the channel bottom and side slopes
$\begin{array}{l}B_i\end{array}$	the width of the underside of the ice cover

It is interesting to estimate the influence that an ice cover can have on the channel conveyance. For example, if a channel is roughly rectangular in shape and much wider than it is deep, then its hydraulic radius will be cut approximately in half by the presence of an ice cover. Assuming the flow area remains constant, we see that the addition of an ice cover, whose roughness is equivalent to the beds, results in a reduction of conveyance of 37%.

Separate ice thickness and roughness can be entered for the main channel and each overbank, providing the user with the ability to have three separate ice thicknesses and ice roughness at each cross section. The ice thickness in the main channel and each overbank can also be set to zero. The ice cover geometry can change from section to section along the channel. The suggested range of Manning's n values for river ice covers is listed in Table 11- 1.

The amount of a floating ice cover that is beneath the water surface is determined by the relative densities of ice and water. The ratio of the two densities is called the specific gravity of the ice. In general, the density of fresh water ice is about 1.78 slugs per cubic foot (the density of water is about 1.94 slugs per cubic foot), which corresponds to a specific gravity of 0.916. The actual density of a river ice cover will vary, depending on the amount of unfrozen water and the number and size of air bubbles incorporated into the ice. Accurate measurements of ice density are tedious, although possible. They generally tell us that the density of freshwater ice does not vary significantly from its nominal value of 0.916. In any case the user can specify a different density if necessary.

Table 11-1 Suggested Range of Manning's n Values for Ice Covered Rivers
The suggested range of Manning's n values for a single layer of ice

Type of Ice	Condition	Manning's n value
Sheet ice	Smooth	0.008 to 0.012
	Rippled ice	0.01 to 0.03
	Fragmented single layer	0.015 to 0.025
Frazil ice	New 1 to 3 ft thick	0.01 to 0.03
	3 to 5 ft thick	0.03 to 0.06
	Aged	0.01 to 0.02

The suggested range of Manning's n values for ice jams

Thickness	Manning's n values
ft	Loose frazil	Frozen frazil	Sheet ice
0.3	-	-	0.015
1.0	0.01	0.013	0.04
1.7	0.01	0.02	0.05
2.3	0.02	0.03	0.06
3.3	0.03	0.04	0.08
5.0	0.03	0.06	0.09
6.5	0.04	0.07	0.09
10.0	0.05	0.08	0.10
16.5	0.06	0.09	-