1. Explain the changes in the predicted Cu concentration in Torch L. that result from using different time steps in the numerical model.

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1 1. Explain the changes in the predicted Cu concentration in Torch L. that result from using different time steps in the numerical model. ANSER: Time steps longer than 2 years cause the model to overpredict the peak concentration, and time steps of 5 years result in a sawtooth pattern of overshooting and then undershooting the predictions with shorter time steps. All of these time steps are too long relative to the rates of the processes involved. The concentration would change substantially within the duration of the time step such that the model is unable to accurately respond to changes in rates that result from changes in concentrations. 2. Consider a lake with a eutrophication problem. You wish to predict the future phosphorus concentration in the lake. You are given the following information: current concentration: 0 /L Loading (g/): 40,000 Outflow (m /): 2x10 6 Area (m 2 ) 100,000 Mean depth (m) 15 Settling velocity (m/) 1 a. hat will be the steady-state concentration of phosphorus in this lake under these conditions? = 0 = QoC vsac C = = Qo vsa 210 x 1 10m 5 2 g = = 19 m L b. Use a first-order Runge-Kutta with a timestep of 1 to predict the concentration one year from now.

2 b C Q v A o s g 4x10 2x10 1 g m m g = = 15 m L C C = t Δt = = 15 L L L H c. Use Heun's Method with a timestep of 1 to predict the concentration one year from now. e can use a lot of the information above within Heun's method. Heun's method is: C=C o slope (time interval) where the slope is the average of the slopes at t = t o and t = t o Δt. n Part b we calculated the slope at t o, so now we must merely calculate the slope at t o Δt. b C Q v A o s g 210 x 1 g m m H g = = 6 m L C = Ct 05. bslope1 slope2g Δt = = 255. L H L L L d. Use a fourth-order Runge-Kutta with a timestep of 1 year to predict the concentration one year from now. n addition to the two slopes calculated above, we have to estimate two additional slopes for the 4th order method. The first of these slopes is the one at t=t o 0.5Δt and at C=C o 0.5ΔC = 0 0.5(-15) = 22.5 /L The slope at this concentration is:

3 b C Q v A o s g 210 x 1 g m m H g = = 4. 6 m L The final slope estimate is calculated for t=t o 0.5Δt assuming that concentrations are changing at the slope just calculated. f the previous slope existed for 1/2 the time step, then the concentration after 1/2 the timestep would be: C = C o 0.5ΔC = *( ) = The final slope estimate is then calculated as: b C Q v A o s g 210 x 1 g m m H g = = 12 m L Now we are in a position to average these estimates of the slope; for the 4th order Runge- Kutta, the last two slope estimates are weighted twice as heavily as the first two. Thus: Slope = 1/6*(S 1 S 2 ) 1/*(S S 4 ) = 1/6*( ) 1/*( ) = -7.1 /L-. The concentration after one year would be C = C o slope(time interval) = 22.9 /L. e. Use the analytical solution to the mass balance equation to predict the concentration one year from now. The analytical solution is obtained by integrating the mass balance equation. The integrated equation is: λt λt C = Coe c1 e h λ where λ equals (Q/ v s /H) = Substituting the appropriate values into this equation yields a predicted concentration of 21.7 /L.. Consider Torch Lake with its narrow constriction between the south and the central basin. rite an equation for horizontal mixing that occurs between these two basins.

4 K x ' = ΔC ( wh ) = vx ΔC ( wh ) = Kx ΔC Δx where K x is the horizontal dispersion coefficient (m 2 /d), Δx is the horizontal distance over which concentration gradients are observed, ΔC is the concentration difference between the two basins, w is the wih of the connecting channel, and h is the depth of the connecting channel (the product wh equals the cross-sectional area of the connecting channel), v x is the dispersion velocity (m/d), and K x is the volumetric dispersion coefficient (m /d). 4. rite the equation for the flux of Cu from the sediment porewaters to the hypolimnion. D = z ΔC A Δz This time, D z is the effective vertical diffusion coefficient, Δz is the vertical distance, and A is the surface area of sediments. 5. rite the equation for the exchange of phosphorus between the epilimnion and hypolimnion of a lake. K = z ΔC A Δz n this equation, a vertical dispersion coefficient, K z, has replaced the diffusion coefficient, Δz is the thickness of the thermocline, and A is the area of the thermocline.

5 6. Predict the shape of the concentration vs. time curve for PCBs in Lake Superior assuming that the historical exponential increase is reversed and becomes an exponential decrease. C time 7. Predict the effect of building a wastewater treatment plant and piping the effluent into a lake with a one year residence time vs. a lake with a 5 year residence time. Draw the shapes of the concentration vs. time curves for both lakes on one graph assuming a step increase in the loading due to the construction of the wastewater treatment plant. Conc. or Loading Loading Time () This graph assumes that the pollutant residence times are 1 and 5 years. 8. List three considerations that would influence your decision as to whether to use a numerical or analytical solution for determining future concentrations of a pollutant in a lake. ANSER 1. Are hydrologic flows and inputs constant, or is seasonal or annual variation significant?

6 2. Can the temporal variations in flows and inputs be described with simple algebraic expressions?. s the desired result an accurate estimation of concentrations over time or simply an approximate picture of how concentrations will change in the future. 9. rite an equation for the flux (mass area -1 time -1 ) resulting from the following processes. Be certain to define the terms that you use in your equations. a. ertical mixing between the epilimnion and hypolimnion of a lake K = z ΔC A Δz This is identical to question 5. n this equation, a vertical dispersion coefficient, K z, has replaced the diffusion coefficient, Δz is the thickness of the thermocline, and A is the area of the thermocline. b. as transfer across the surface of a lake * = Kaw ( C Cw ) A n this equation, K aw, is the air-water mass transfer coefficient (m/d), C * is the aqueous concentration that would exist in equilibrium with the concentration in the air, and A is the lake area. 10. The magnitude of the horizontal dispersion coefficient calculated for transfer of Cu from the nearshore to the offshore zone in Lake Superior was 10 4 cm 2 sec -1 or 8.6 x 10 4 m 2 d -1. The mean water depth at the interface between near- and offshore zones is 50 m, the horizontal thickness of the mixing zone is about 500 m, and this mixing zone occurs about 5 km offshore over a length of coastline of about 150 km. a. Explain what parameter is needed to convert the dispersion coefficent to a dispersion velocity (m d -1 ) and calculate this velocity. (Just for comparison's sake, the velocity of the Keweenaw Current is about 5 km/d, and the wave velocity on the lake yesterday was about 6000 km/d.) b. State what parameters you need to convert the dispersion coefficient to a volumetric dispersion coefficient (m s -1 ). Do the calculation. (or comparison's sake, the average flow of the Ontonagon River is 4 m s -1, the average outflow of L. Superior at the St. Mary's River is 250 m s -1.)