Dr. Martin T. Auer MTU Department of Civil & Environmental Engineering CE5504 Surface Water Quality Modeling Lab 5. One-Dimensional Models Thermal Stratification, Vertical Mass Transport and Interactions with Trophic State

Temperature and the Density of Water Density (g∙cm 3 ) Temperature (°C) Maximum Density 3.94 °C

Thermal Stratification spring summer Temperature (°C)

Thermal Stratification

Seasonality in Stratification and Mixing  = °C

Dollar Bay - Temperature Depth (m) Temperature (°C) Depth (m) M A M J J A S O N D

Productivity and Oxygen

Trophic State and Oxygen Profiles Oligotrophic Lake Eutrophic Lake Temperature (°C)

Dollar Bay - Temperature Temperature (°C) Depth (m) M A M J J A S O N D

Dollar Bay – Dissolved Oxygen Depth (m) Dissolved Oxygen (mg∙L -1 ) M A M J J A S O N D

Onondaga Lake – Dissolved Oxygen

Oneida Lake – Dissolved Oxygen

Lake Erie – Dissolved Oxygen

Gulf of Mexico – Dissolved Oxygen

8 Lake Superior – 4/28/00 Completely-Mixed Lake

8 8 Lake Superior – 8/25/00 Vertical Resolution - Stratification

Advection and Diffusion

Diffusion

Scales of Diffusion

Mathematics of Diffusion I

Fick’s 1 st Law X-direction E C Mathematics of Diffusion II

l Mathematics of Diffusion III

Mathematics of Diffusion IV

E, turbulent diffusion coefficient, m 2 /d E’, bulk diffusion coefficient, m 3 /d Mathematics of Diffusion V

Vertical Mass Transport

Application to Vertical Mass Transport turbulent diffusion coefficient thermocline heat transfer coefficient

Calculation of the Thermocline Heat Transfer Coefficient Epilimnion Heat Balance change in heat tributary heat in discharge heat out surface heat flux thermocline heat exchange

Calculation of the Thermocline Heat Transfer Coefficient Hypolimnion Heat Balance change in heat thermocline heat exchange assumes no direct heating of hypolimnion or sediments upper mixed layer (~epilimnion) lower mixed layer (~hypolimnion) thermoclinev t ∙A s

Calculation of the Thermocline Heat Transfer Coefficient Canceling the  and C p terms, dividing by V h and expanding … And defining … Yields … uml lml

Calculation of the Thermocline Heat Transfer Coefficient and re-arranging, yields … Which, for T h at the beginning of the calculation = Th i, yields … further re-arranging, yields … where a plot of t s versus term yields 1/v t as the slope term

Application to Dollar Bay

Dollar Bay - Temperature Temperature (°C) Depth (m) M A M J J A S O N D

Dollar Bay – Dissolved Oxygen Depth (m) Dissolved Oxygen (mg∙L -1 ) M A M J J A S O N D

For diffusion calculation, need to choose an interval where T epi is stable and T hyp is warming. Dollar Bay – Temperature Calculations Hypolimnion temperature mass balance.

Sediment oxygen demand reflects the diffusion of oxygen into the sediment to meet the metabolic needs of microorganisms participating in the aerobic decomposition of organic matter. SOD is also exerted through the aerobic oxidation of reduced species byproducts of anaerobic respiration. The mass balance, for SOD alone is … Sediment Oxygen Demand uml lml

For the SOD calculation, we need to choose the interval of oxygen depletion. Dollar Bay – Oxygen Calculations (the question of the cones)

Time-Variable vt Values for Onondaga Lake (Doerr et al. 1996)