This model represents the four-dimensional version of the advection-dispersion equation (1) for an estuary.
dS/dt = (1/A)d(QS)/dx - (1/A)d(EA)/dx(dS/dx) (1)
Where S: salinity (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 (2)
EA/x, i.e. (m2 X m2) / (m s) = E(b), the bulk dispersion coefficient (m3 s-1, i.e. a flow, equivalent to Q)
We can rewrite (2) for the first estuarine box as:
Where Sr: river salinity (=0), S1: mean estuary salinity for box 1; S2: mean estuary salinity for box 2; E(b)r,1: dispersion coefficient between river and estuary box 1; and E(b)1,2: dispersion coefficient between the estuary boxes 1 and 2.
Because we're at the head of the estuary, E(b)r,1 is zero, wich means: no salt enters the river. Sr is also zero, because the river salinity is zero. Therefore: