Effective cross-section

To model the process of the mouth silting up I will now perform some jiggery-pokery which works in a qualitative sense even if there are some pretty fundamental shortcomings. The whole ``virgin state'' mouth model is admittedly a weak point of the salinity model. I make no claim of accuracy, rather I claim to get something that ultimately behaves qualitatively in the way the experts say the mouth behaved. In fact the model is fairly insensitive to the exact behaviour of the mouth, so these approximations are good for most purposes.

Consider a simple form of the gravity driven linear reservoir as a model for the estuary.

Where l is the lake level, C(l) is the cross-sectional area depending on level, is the sea level, R is the relaxation time and V is the volume of water flowing out of the lake.

But we know from Huizenga's work that

Where a, b and c are constants. Therefore for the purposes of emulating the full complexity of the long ``narrows'' part of the lake mouth, we can declare the mouth to have a cross-section that fits the modeled outflow. Note that this is NOT modeling the mouth physically, this is just a mathematical convenience to integrate the estuary model component with the lake model. We cannot really regard the estuary mouth as having such a cross-section.

So...

Thus ...

In a linear reservoir model, the sea level is the level of zero flow, thus...

Which implies that...

Or, as shown in figure 4...

  
Figure 4: Effective Crossection area.

At sea level we have a 0/0 situation which we can resolve by L'Hospitals rule...

Now we can regard the width as being the derivative of the cross-sectional area with respect to level. Thus...

Or as shown in figure 5...

Where is the channel bottom level.

  
Figure 5: Effective channel width

Now for expedience, I cross the gulf of mathematical reality from convenience back to physical modeling, and consider the channel silting up.

Yes, I know I argued initially that this was just a trick for mathematical convenience, yet I need an extra trick to get some grip on the silting up of the mouth. What I have here is an ``effective'' width/area relationship which I'm using to ``effectively'' create a siltation model. Sigh! Lousy physics, but good expedience.

There are a couple of approaches of various degrees of tractability that one can take here...

• Assume the silt deposition is constant depth t on the walls of the channel.

  Then...
  
  

  Let l' = l-t, then as shown in figure 6...
  
  

    
  Figure 6: Outflow function modified by siltation
  

  The glitch with this approach is you never actually shut off the mouth so sea water still keeps coming in and the silt level keeps on going up...

• Assume a thickness t of silt on the walls of the channel. The bitch here is to invert W(l) to find the ``bottom'' of the channel where . This would have to be done numerically and would take just too long.
• Assume the siltation parameter is how much the channel area is decreased by. This last approach is the one I took. In fact the siltation parameter is the silt level, which is then used to get the effective cross-section after siltation.