B.3 Scalar Conservation Equations
Analogous conservation equations are solved for the transport of scalar constituents in the water column.
The conserved scalar variable is:
\[\begin{equation} \mathbf{U} = \begin{bmatrix} hC \end{bmatrix} \tag{B.7} \end{equation}\]
where:
- \(C\) is the constituent concentration
The flux components of the scalar conservation equation are:
\[\begin{equation} \mathbf{F}_x^I = \begin{bmatrix} huC \end{bmatrix}, \quad \mathbf{F}_x^V \approx \begin{bmatrix} -h(D_{xx} \frac{\partial C}{\partial x} + D_{xy} \frac{\partial C}{\partial y}) \end{bmatrix} \end{equation}\]
\[\begin{equation} \mathbf{F}_y^I = \begin{bmatrix} hvC \end{bmatrix}, \quad \mathbf{F}_y^V \approx \begin{bmatrix} -h(D_{yx} \frac{\partial C}{\partial x} + D_{yy} \frac{\partial C}{\partial y}) \end{bmatrix} \tag{B.8} \end{equation}\]
\[\begin{equation} \mathbf{F}_z^I = \begin{bmatrix} hwC \end{bmatrix}, \quad \mathbf{F}_z^V \approx \begin{bmatrix} -h\nu_t' \frac{\partial C}{\partial z} \end{bmatrix} \end{equation}\]
The source components may include scalar decay and settling:
\[\begin{equation} \mathbf{S} = \begin{bmatrix} - K_d hC - w_s C \end{bmatrix} \tag{B.9} \end{equation}\]
where:
- \(K_d\) is a scalar decay-rate coefficient
- \(w_s\) is a scalar settling velocity