Solution Procedure

The ice jam force balance equation is solved using an approach analogous to the standard step method. In this, the ice thickness at each cross section is found, starting from a known ice thickness at the upstream end of the ice jam. The ice thickness at the next downstream section is assumed and the value of F found. The ice jam thickness at this downstream cross section, $//<![CDATA[ \begin{array}{l}t_{ds}\end{array} //]]>$, is then computed as:

1)	$$//<![CDATA[ \begin{array}{l}\displaystyle t_{ds} = t_{us} + \overline{F} L\end{array} //]]>$$

And

2)	$$//<![CDATA[ \begin{array}{l}\displaystyle \displaystyle \overline{F} = \frac{F_{us} + F_{ds}}{2}\end{array} //]]>$$

The assumed value and computed value of $//<![CDATA[ \begin{array}{l}t_{ds}\end{array} //]]>$ are then compared. The new assumed value of the downstream ice jam thickness set equal to the old assumed value plus 33% of the difference between the assumed and computed value. This "local relaxation" is necessary to ensure that the ice jam calculations converge smoothly to a fixed value at each cross section. A maximum of 25 iterations is allowed for convergence. The above steps are repeated until the values converge to within 0.1 ft (0.03 m) or to a user defined tolerance.

After the ice thickness is calculated at a section, the following tests are made:

The ice thickness cannot completely block the river cross section. At least 1.0 ft must remain between the bottom of the ice and the minimum elevation in the channel available for flow.
The water velocity beneath the ice cover must be less than 5 fps (1.5 m/s) or a user defined maximum velocity. If the flow velocity beneath the ice jam at a section is greater than this, the ice thickness is reduced to produce a flow velocity of approximately 5 fps or the user defined maximum water velocity.

The ice jam thickness cannot be less than the thickness supplied by the user. If the calculated ice thickness is less than this value, it is set equal to the user supplied thickness.

It is necessary to solve the force balance equation and the energy equation

 simultaneously for the wide river ice jam. However, difficulties arise because the energy equation is solved using the standard step method, starting from the downstream end of the channel and proceeding upstream, while the force balance equation is solved starting from the upstream end and proceeding downstream. The energy equation can only be solved in the upstream direction because ice covers and wide river jams exist only under conditions of subcritical flow. To overcome this incompatibility and to solve both the energy and the ice jam force balance equations, the following solution scheme was adopted.

A first guess of the ice jam thickness is provided by the user to start this scheme. The energy equation is then solved using the standard step method starting at the downstream end. Next, the ice jam force balance equation is solved from the upstream to the downstream end of the channel. The energy equation and ice jam force balance equation are solved alternately until the ice jam thickness and water surface elevations converge to fixed values at each cross section. This is "global convergence."

Global convergence occurs when the water surface elevation at any cross section changes less than 0.06 ft, or a user supplied tolerance, and the ice jam thickness at any section changes less than 0.1 ft, or a user supplied tolerance, between successive solutions of the ice jam force balance equation. A total of 50 iterations (or a user defined maximum number) are allowed for convergence. Between iterations of the energy equation, the ice jam thickness at each section is allowed to vary by only 25% of the calculated change. This "global relaxation" is necessary to ensure that the entire water surface profile converges smoothly to a final profile.