Clone of Estuary Salinity 4 Boxes Model

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.

dS/dt = (1/A)d(QS)/dx - (1/A)d(EA)/dx(dS/dx)

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:

Q(Sr-S1)=E(b)r,1(Sr-S1)-E(b)1,2(S1-S2)** (3)**

VdS/dt = QdS - (d(EA)/dx) dS

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

Q(Sr-S1)=E(b)r,1(Sr-S1)-E(b)1,2(S1-S2)

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:

QS1=E(b)1,2(S1-S2)**(4)**

At steady state

E(b)1,2 = QS1/(S1-S2)**(5)**

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:

QS1=E(b)1,2(S1-S2)

At steady state

E(b)1,2 = QS1/(S1-S2)