Many water bodies have problems with anoxia ie low levels of dissolved oxygen $($DO$)$ in the water column

due to phytoplankton respiration. Lake Monger in Western Australia is one such water body. A colleague

asks you to predict DO profiles in the lake to determine whether anoxia will occur. Preliminary

investigations reveal that there are negligible horizontal gradients of DO across the lake. However, the

competing effects of oxygen consumption by phytoplankton and reaeration from the surface cause

significant vertical gradients of $DO.$

$($a$)$ Assuming steady$-$state conditions, derive the governing equation for the vertical $($z$)$ profile of dissolved

oxygen from the Advection$-$Diffusion equation. Also assume that the phytoplankton represent a first$-$order

oxygen sink $($i$.$e$.$ the rate of consumption of dissolved oxygen is proportional to its concentration in the

water, with constant of proportionality $=k).$ Vertical flows in the lake can also be assumed to be

insignificant.

$($b$)$ Find the solution to this equation, subject to the following boundary conditions:

$-$reaeration at the surface $(z=0)$ keeps the water at the surface saturated with dissolved oxygen at a

concentration of $8m\frac{g}{L}.$

$-$the sediments $(z=-10m)$ are sufficiently impermeable and abiotic to represent a no$-$flux boundary for

dissolved oxygen.

Use typical values of $D=0\mathrm{.}1c\frac{m^2}{s}$ and $k=0\mathrm{.}05da{y}^{-1}($which corresponds to high phytoplankton levels$).$

$($c$)$ Where in the water column would you expect the lowest concentration of DO$?$

$($d$)$ Your colleague tells you that DO concentrations below $1m\frac{g}{L}$ are considered dangerous. Determine

whether Lake Monger will have dangerously low levels of dissolved oxygen. What if eutrophication

increases the phytoplankton population, such that $k=0\mathrm{.}1$ day ${?}^{-1}?$