Constant in/out flow, linear level dependent estuary.

If we have a constant river+rainfall+evaporation inflow, one square sided cell and a linearly level dependent estuary, (See figure 15)

  
Figure 15: Single lake cell the level dependent estuary and time dependent inflow

Opening or closing Potter's channel would correspond to changing the value of k.

Rearranging slightly so we may more easily see the homogeneous part.

Then the solution is...

Integrating this gives us...
 

Note one small detail. is not the volume of water in the cell at time t=0. is merely a constant of integration. Now for interests sake let us put this in terms of , the volume of water at time t=0.

Which implies...

So the equation (51) for the volume becomes...

Volume as a function of time.

 

A pair of terms of the form is the mathematics way of taking the system from the initial condition A at time t=0 to the final condition B at time .

The final steady state condition is made up of two terms...

1. Term one is a head difference due to in flow I. The magnitude of the head difference due to I is governed by the estuary flow constant k.
2. The last term is just the effect of the sea level, which 'pins' the differential equation to move about sea level. Without a level dependent estuary the volumes can wander to arbitrarily large or small values.

Denote the salt load by S. Now the load calculation, assuming no salt in the inflow, is...

 
Where C is the salt concentration of sea water () when V/A > L and is S/V when the flow is from lake to sea.