Ch 2.6 Case Study: Lake Burley Griffin
Introduction
- In this section, we apply the lake pollution model to Lake Burley Griffen in Canberra, Australia's capital.
- This lake was named after Walter Burley Griffin, an American architect who won the competition to design Canberra.
- Lake Burley Griffin was created artificially in 1962 and serves recreational and aesthetic purposes.
Introduction
- We will use the analytical results from Ch2.5 to address specific questions regarding pollution levels identified below.
- We will also implement numerical solutions for varying flow rates \( F(t) \) and incoming concentrations \( c_{in}(t) \), for which it might prove difficult to determine solutions analytically.
Humor
Who invented the round table?
Sir Cumference
Background
- Lake pollutants can come in a variety of chemical and biological forms.
- Our focus will be on fecal (thermotolerant) coliform bacteria.
- These bacteria are considered primary indicators of fecal contamination and disease bacteria in water.
Background: Fecal Coliform Bacteria
Background: Lake Burley Griffen
- Pollution levels measured in 1970s.
- It was determined that while upstream sewage plants in Queanbeyan contributed to the problem, rural and urban runoff were factors as well.
- Canberra and Queanbeyan are in two separate districts, governed by different safety standards.
- Queanbeyan River flood, March 2021 (image below left).
Background: Lake Burley Griffen
- In 1974 the mean concentration of bacteria fecal coliform count was \( 10^7 \) bacteria per \( \mathrm{m}^3 \) where river feeds into lake.
- The safety threshold for contact recreational sports requires that no more than 10% of total samples over a 30-day period should exceed \( 4 \times 10^6 \) bacteria per \( \mathrm{m}^3 \).
Problem Statement
Use our Ch2.5 lake pollution model to address the following:
- If sewage management were improved, would pollution levels would drop below the safety threshold?
- Investigate numerical results when flow rate \( F(t) \) and incoming pollutant concentration \( c_{in}(t) \) vary with time.
Identify Variables and Parameters
IVP and Analytic Solution
From Ch2.5, the IVP and exact solution are given by
\[ \begin{aligned} \frac{dC}{dt} & = \frac{F}{V}\left[c_{in} - C(t) \right] , \,\, C(0)= c_0 \\ \\ C(t) & = c_{in} - (c_{in} - c_0) e^{-Ft/V } \end{aligned} \]
Unit Check for Differential Equation
Model Predictions: Case 1, Analytical
Model Predictions: Case 1, Numerical
#Perform Rk4 on Lake Burley Griffen Pollution IVP
F1 <- 4; c1 <- 0; V <- 28
f <- function(x, C) { F1/V*(c1 - C) }
LakeModel(f,10,8)
Model Predictions: Case 2, Analytical
Model Predictions: Case 2, Numerical
#Perform Rk4 on Lake Burley Griffen Pollution IVP
F1 <- 4; c1 <- 3; V <- 28
f <- function(x, C) { F1/V*(c1 - C) }
LakeModel(f,10,20)
Model Predictions: Case 3
- Flow rate \( F(t) \) and \( c_{in}(t) \) typically vary seasonally.
- Using data from 1970s, rough estimates are:
\[ \small{ \begin{aligned} F(t) & = 10^6 \left[ 10 + 6 \sin(2 \pi t) \right] \mathrm{m}^3/ \mathrm{year} \\ c_{in}(t) & = 10^6 \left[ 10 + 10 \cos(2 \pi t) \right] \mathrm{bacteria}/ \mathrm{m}^3 \end{aligned} } \]
- Desmos graph (left) and Oregon watershed study (right).
Model Predictions: Case 3, Analytical
Model Predictions: Case 3, Numerical
#Perform Rk4 on Lake Burley Griffen Pollution IVP
F1 <- function(x) {(10 + 6*sin(2*pi*x))}
c1 <- function(x) {(10 + 10*cos(2*pi*x))}
f <- function(x, C) { F1(x)/V*(c1(x) - C) }
V <- 28; LakeModel(f,0,20)
Model Predictions: Case 3, Continued
- Adapt parameters to match Case 2 conditions:
\[ \small{ \begin{aligned} F(t) & = 10^6 \left[ 4 + 2 \sin(2 \pi t) \right] \\ c_{in}(t) & = 10^6 \left[ 3 + 2\cos(2 \pi t) \right] \end{aligned} } \]
- See graph on next slide, starting with \( c_0 = 10 \) (1974 levels).
- Roughly 14 months required for pollution to drop below \( C_L \).
Model Predictions: Case 3, Continued
#Perform Rk4 on Lake Burley Griffen Pollution IVP
F1 <- function(x) {(4 + 2*sin(2*pi*x))}
c1 <- function(x) {(3 + 2*cos(2*pi*x))}
f <- function(x, C) { F1(x)/V*(c1(x) - C) }
V <- 28; LakeModel(f,10,20)
Discussion of Results
- The results of each of the three model prediction cases are discussed above, along with each successive refinement in assumptions used.
- Models are simplistic in assuming a well-mixed body of water.
- If concentration decreased from river entry to river outflow, with general form \( C(x,y,z,t) \) instead of \( C(t) \), then flushing time could take much longer.
Discussion of Results
- Most lakes have main channel of flow that flushes regularly (advection), with adjacent areas of trapped water.
- Margin areas flush gradually via diffusion at microscopic level.
- Various pockets of lake may have a much higher (or lower) pollution concentration than others, including protected bays where swimming is most likely to take place.
- A PDE advection-diffusion model is introduced in Ch12.4, and is a starting point for more advanced models.
