Where S: salinity (or any other constituent such as chlorophyll or dissolved oxygen), (e.g. kg m-3); t: time (s); A: cross-sectional area (m2); Q: river flow (m3 s-1); x: length of box (m); E: dispersion coefficient (m2 s-1).
For a given length delta x, Adx = V, the box volume. For a set value of Q, the equation becomes:
VdS/dt = QdS - (d(EA)/dx) dS (Eq. 2)
EA/x, i.e. (m2 X m2) / (m s) = E(b), the bulk dispersion coefficient, units in m3 s-1, i.e. a flow, equivalent to Q
At steady state, dS/dt = 0, therefore we can rewrite Eq. 2 for one estuarine box as:
Q(Sr-Se)=E(b)r,e(Sr-Se)-E(b)e,s(Se-Ss) (Eq. 3)
Where Sr: river salinity (=0), Se: mean estuary salinity; Ss: mean ocean salinity
E(b)r,e: dispersion coefficient between river and estuary, and E(b)e,s: dispersion coefficient between the estuary and ocean.
By definition the value of E(b)r,e is zero, otherwise we are not at the head (upstream limit of salt intrusion) of the estuary. Likewise Sr is zero, otherwise we're not in the river. Therefore:
QSe=E(b)e,s(Se-Ss) (Eq. 4)
At steady state
E(b)e,s = QSe/(Se-Ss) (Eq 5)
The longitudinal dispersion simulates the turbulent mixiing of water in the estuary during flood and ebb, which supplies salt water to the estuary on the flood tide, and make the sea a little more brackish on the ebb.
You can use the upper slider to turn off dispersion (set to zero), and see that if the tidal wave did not mix with the estuary water due to turbulence, the estuary would quickly become a freshwater system.
The lower slider allows you to simulate a variable river flow, and understand how dispersion compensates for changes in freshwater input.