[ This model represents the four-dimensional version of the advection-dispersion equation (1) for an estuary.
dN/dt = (1/A)d(QN)/dx - (1/A)d(EA)/dx(dN/dx) (1)
Where N: Nitrates (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:
VdN/dt = QdN - (d(EA)/dx) dN (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 Sn: river nitrates(=5), N1: mean estuary Nitrates for box 1; N2: mean estuary nitrates 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.
At steady state
E(b)1,2 = QN1/(N1-N2) (5) ]