Estuarine Nitrates 4 boxes model
[ 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:

Q(Nr-N1)=E(b)r,1(Nr-N1)-E(b)1,2(N1-N2) (3)

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. 

 Therefore:

QN1=E(b)1,2(N1-N2) (4)

At steady state

E(b)1,2 = QN1/(N1-N2) (5) ]

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