“An estuary is a partly enclosed coastal body of brackish water with one
or more rivers or streams flowing into it, and with a free connection to the
open sea” [Wikipedia].
The salt concentration in an estuary depends on the sweet water inflow
from the river as well as the tide from the sea that brings in saltwater.
Write a model that gives the temporal variation of the salt concentration
according to the following assumptions:
(a) The estuary is modeled as a compartment of volume V = V (t). We
assume that the sweet water and saltwater mix immediately to form
the brackish water.
(b) The river brings in sweet water with a constant rate Jriver > 0. The
sweet water replaces the same volume of brackish water that flows
into the sea.
(c) The tidal wave is periodic, causing a flow Jtide = Jtide(t), e.g.,
Jtide(t) = Jmax cos ωt.
(d) When Jtide(t) > 0, the tide brings in saltwater, while when Jtide(t) <
0, the outflow is brackish water.
Denoting by V1(t) and V2(t) the volumes of the sweet water and salt water,
respectively, write differential equations for the partial volumes. Write a
differential equation for the total volume V , and check that the total volume
depends on the tidal flow. What are the solutions (V (t), V1(t), V2(t))
if the tidal flow is set to zero? How does the solution behave in this case
asymptotically as t → ∞?
Assuming that the sea water has a salt concentration c0, using your differential
equations for V1 and V2, write a differential equation for the salt
concentration c = c(t) of the brackish water in