In the process of retaining minimal level of nutrients in lakes, rivers, and streams to support lakes productivity, the nutrients accumulate overtime causing eutrophication within the water column [1]. By definition, eutrophication in lakes and reservoirs is caused by over enrichment of lakes and reservoirs with high level of nutrients (nitrogen and phosphorus). In most cases, excess levels of nutrients lead to accumulations of high levels of algal biomass including harmful algae blooms that alters the aquatic environment [2]. Some major effects of eutrophication include nuisance levels of algae, taste and odor problems with water supply, occurrence of harmful algae blooms, increased bacterial production, and depletion of DO within the hypolimnion of lakes and reservoirs [3] [4] [5]. In the United States, impairment of approximately 60% of lakes, rivers, and reservoirs is a result of nutrient enrichment and eutrophication [3]. For example, freshwater impairments due to eutrophication have been documented as an important factor affecting the economy of counties, states, and other sectors in United States [6]. As an example, costs related to harmful algae blooms in eutrophic lakes and reservoirs that provide drinking water services and other recreational activities have risen for water treatment costs to over a million United States (US) dollars per algal bloom event, while assessment and monitoring costs in impaired waterbodies have exceeded $50 million US dollars annually [3].

In 2010, 2012, and 2014 the lower segment of Grand Lake was listed in Oklahoma’s Integrated Report of the 303(d) list of impaired waterbodies for the state. The causes of impairment shown on the 303(d) list included low dissolved oxygen (DO) levels. Major contributing factors for low DO were summer stratification and the effects of algal blooms [7]. On July 4, 2011, Grand Lake, the third largest lake in Oklahoma experienced a significant harmful algal bloom of blue-green algae. The Grand River Dam Authority (GRDA) prohibited swimming in the lake and huge tourism-related financial losses were incurred by the regional economy over the July 4^th holiday.

Hypolimnetic depletion of DO is a common occurrence in lakes and reservoirs during the summer months in the Central Plains because of seasonal thermal stratification. The formation of a thermocline reduces the resupply of DO from the upper mixed layer to reduce DO levels in the hypolimnion to less than 2 mg/L. Under the Clean Water Act (CWA), water quality management planning efforts by the State of Oklahoma to develop Total Maximum Daily Loads (TMDLs) for lakes and reservoirs are based on models that describe the cause-effect relationships between watershed loading and the seasonal occurrence of hypolimnetic DO depletion.

DO modeling in natural waters has evolved from simple one-dimensional, steady-state “Streeter-Phelps BOD-DO” stream models that accounted for atmospheric reaeration and organic matter mineralization [8] [9] to advanced three-dimensional, time variable biogeochemical models that account for physical transport processes and eutrophication related interactions of nutrients, organic matter, sediment diagenesis, and phytoplankton production with DO [10] and [11] Although advanced models provide the highest resolution to characterize horizontal, temporal, and vertical DO distributions in a waterbody, the level of effort and the field data required to calibrate an advanced lake model requires a commitment of substantial resources and time.

As horizontal gradients of DO in a lake or reservoir are much smaller than vertical gradients, seasonal stratification is a major physical process that controls hypolimnetic DO depletion. One-dimensional (vertical) models, developed to represent horizontally averaged, depth-dependent distributions of DO, can provide insight for assessments of the relative importance of physical and biogeochemical processes that control the formation of seasonal DO profiles in a reservoir or lake. Initial progress for a water quality planning study can be made with a vertical DO model and modest resources to understand the key processes that control hypolimnetic DO depletion in a lake or reservoir.

Models have been developed to represent vertical profiles of DO above and below the thermocline using approaches based on mixed layer box models and continuous diffusive mixing in the water column. Bell [12] developed a time dependent two-layer DO model of Bassenthwaite Lake in England and O’Connor and Mancini [8] developed a steady-state two-layer model of DO in the New York Bight. Steady-state three-layer DO models have been developed for Chesapeake Bay [13] and Tenkiller Ferry Lake [14]. Mixed layer model input parameters were calibrated to account for stratification and components of the DO balance based on kinetic processes and observed data sets. Diffusive mixing models, developed to account for a stack of multiple layers of the water column, have been coupled with physical models of water temperature to represent stratification. Input parameters for DO components have been based on model calibration, kinetic processes, and observed data sets [15] and [16] and coupling of the DO sub-model with physical and biogeochemical models [17].

The objective of this study is to develop a simplified three-layer, steady state vertical DO model for representative station locations in Grand Lake, Oklahoma. Model input parameters developed to represent thickness of the three layers, DO sources from reaeration and phytoplankton production and DO sinks from sediment oxygen demand and DO consumption are derived from observed data and calibration of kinetic coefficients. In addition, derived model input rates and kinetic processes are used to determine the relative importance of DO source sink terms on DO depletion within the hypolimnion of Grand Lake under summer-stratified condition.

With a drainage area of 10,298 square miles, Grand Lake O’The Cherokees stretches through Delaware, Ottawa, and Mayes counties in the northeast area of Oklahoma. The lake is a multi-use reservoir formed from the construction of the Pensacola dam in 1941. A hydrographic survey by OWRB (2009) showed that the surface area and storage volume of the lake has decreased by about 10% from the 1940 design specifications. The physical characteristics of the lake include surface area of 41,779 acres (16,907 ha), mean depth of 36.3 ft (11.1 m), maximum depth of 133 ft (40.5 m), and cumulative storage volume of the lake of 1,515,415 acre-feet [18] and [19]. The lake provides flood control, hydroelectric power supply, drinking water, recreational activities, fish, and wildlife propagation for the state of Oklahoma. The primary land use is agriculture [20] ; the three major tributaries to the lake are the Neosho River, Elk River, and Spring River as shown in Figure 1 below.

In the development of the three-layer steady state vertical DO model for Grand Lake, observed water quality data were used to calculate the Trophic State Index (TSI) [24] for Grand Lake to help us understand trophic state condition of the lake within the riverine, transition, lacustrine zones, and the site close to the dam. Table 3 and Table 4 summarize the observed data used to determine the TSI lake classification as mesotrophic, eutrophic, and hypereutrophic.

In 2013, the TSI results show that Grand Lake was eutrophic in all the zones except for the riverine zone which was classified as hypereutrophic. On the other hand, the TSI results for 2015 indicate that Grand Lake was hypereutrophic in the riverine and transition zone, while the lacustrine zone and the site close to the dam were classified as eutrophic. Based on the results obtained from the TSI analysis for both 2013 and 2015, Grand Lake can be characterized as a reservoir with high productivity for algae.

The plots in Figure 4 illustrate typical water temperature profiles (Figure 4(a) and Figure 4(c)) and the results of DO model simulations (Figure 4(b) and Figure 4(d)) for 2013 and 2015 survey dates. Water depth is plotted on the horizontal axis of the vertical profiles. The vertical dotted and solid lines in each

plot mark the breaks in water temperature used to determine depths and thickness of each layer. As can be seen in Figure 4(b) and Figure 4(d), the DO model shows very good agreement with the observed profile data for DO.

The plots in Figure 5 illustrate a water temperature profiles and results of DO model simulations for the P-dam station in Grand Lake. Figure 5(a) and Figure 5(c) water temperature profiles indicate significant stratification conditions in Grand Lake at the dam during summer of 2013 and 2015 respectively. The vertical dotted and solid lines in each plot mark the breaks in water temperature used to determine depths and thickness of each layer. Figure 5(b) and Figure 5(d) show a comparison of the vertical distribution of modeled DO to the observed DO data collected at P-dam Station on Grand Lake for the 2013 and 2015 survey dates. The DO model predictions indicate reasonable agreement with the observations. The exception is shown in Figure 5(b) for the July 2015 survey where the surface results of the model did not match the observed higher levels of DO. This discrepancy in the model prediction for the surface was due to the following reasons: 1). higher light available at the surface of the photic zone, 2) supersaturated conditions for DO of 140% obtained on that survey date indicate very high rates of primary production in the 0 - 2 m of the surface layer that the depth-averaged model over the euphotic zone layer was unable to match.

The graphs in Figure 6(a) and Figure 6(b) illustrate the results of the sensitivity analysis using SOD and the vertical mixing rate as the variables tested to

observe the effects on the model results for Tree Station survey date of July 17, 2013. In Figure 6(a) the circular blue marks are the observed dissolved oxygen, the red line is the base value of SOD = 1.0 g O₂/m²-day, the green line is the low value of SOD (0.5 g O₂/m²-day) and the purple line is the high value of SOD 2.0 g O₂/m²-day). From the graph, the SOD of 1.0 g O₂/m²-day shows good agreement with the observed DO throughout the three layers of the water column. The sensitivity tests of the model using the lower (0.5 g O₂/m²-day) and higher (2.0 g O₂/m²-day) SOD rates showed that neither the low or high SOD rates achieved good agreement with the observed DO. The sensitivity results demonstrate that SOD has an effect on the oxygen profile in all three layers.

In addition to the sensitivity analysis runs using low and high values of SOD (Figure 6(a)), the effect of low and high values of the vertical mixing coefficient (E₂) were also tested using parameter values ranging from E₂ = 0.045 cm²/sec (low), 0.45 cm²/sec (calibration baseline) to 0.75 cm²/sec (high) respectively. The dramatic effect of the low and high values for the sensitivity analysis of vertical mixing clearly show that the modeled oxygen results for the thermocline layer and the hypolimnion layer are extremely sensitive to the vertical mixing rate assigned for model input. Vertical mixing rates used for development of the vertical DO profile model (see Table 2) were determined by calibration to the observed DO profiles for each station and survey date.

A three-layer steady-state model of the vertical structure of DO in a stratified lake has been successfully developed based on observed water chemistry data, physical data such as water depth, water temperature, and kinetic coefficients used to derive input data to the vertical dissolved oxygen model. The model was developed using water quality data collected during the summer months of 2013 and 2015 for stations within the characteristic riverine, transition, lacustrine zones and at the dam in Grand Lake. Application of the model to a set of water quality data obtained for 2013 and 2015 gives an understanding of reservoir eutrophication processes and how physical, chemical and biological processes are connected to DO production and DO consumption in the epiliminion, thermocline, and hypolimnion layers of the lake.

Results obtained from the model simulations at each station and zone give reasonable agreement between the predicted versus observed dissolved oxygen profiles within the water column. Based on the pre-processing analysis used to derive rate estimates for model input, the vertical DO model successfully accounted for both oxygen production terms and oxygen consumption terms. Based on the percentage analysis for each source and sink term, the pre-processing analysis clearly identified the relative importance of each component term for oxygen production and oxygen consumption.

In summary, the relative effects on oxygen production in the photic zone were based on phytoplankton biomass, high light limitation, and nutrient limitation. The relative effects on oxygen consumption were based on the phytoplankton biomass and the levels of organic matter and ammonia in the water column. In a study of Lake Erie, Clevinger [25] found that SOD contributed 19.2 percent to the total oxygen demand in the hypolimnion, while nitrification accounted for 32.6 percent in the hypolimnion and 28 percent of oxygen demand in the epiliminion layer. In contrast to other studies [26] where sediment oxygen demand accounted for a large component of hypolimnetic oxygen demand, Clevinger’s results from Lake Erie are consistent with the results derived from this analysis of the contributions of nitrification and sediment oxygen demand to oxygen consumption in Grand Lake. Key findings from this study show that high light limitation at the surface layer, phytoplankton growth rate, and phosphorus were the most critical factors for oxygen production within the epiliminion layer while nitrogen availability was the least critical factor. Phosphorus was the limiting factor to phytoplankton production in most cases, except for the P-dam site on July 17, 2013 where it was found that nitrogen was the limiting factor for phytoplankton production due to supersaturation conditions of 140 percent at the 0 - 2 m depth and high light limitation observed at the surface.

Analysis of sink terms for oxygen (phytoplankton respiration, nitrification, organic carbon decomposition, and sediment oxygen demand) showed that the most critical factor for oxygen consumption within the entire water column was nitrification and the least critical factor was sediment oxygen demand. The effect of stratification on the oxygen profile was demonstrated with the sensitivity analyses simulation results for the vertical mixing rate. The analysis clearly showed that a smaller mixing coefficient will result in stronger stratification, while a larger mixing coefficient will result in weaker stratification. The model proved to be very useful to provide insight into the relative contributions of the different physical and kinetic processes for formation of the oxygen profile in a stratified lake. The model was able to quantify the relative importance of the different source and sink terms on the seasonal occurrence of hypoxia and anoxia in the hypolimnion.

First of all, I thank God for this great opportunity. This study was funded by theEnvironmental Science Graduate Program at Oklahoma State University. The authors ofthis work are grateful for the opportunity and part of it goes to Grand River DamAuthority (GRDA) management for providing data for this work. I would like to thank my advisor Dr. Scott Stoodley, and committee members Dr. Daniel E. Storm, Dr. Andrew R. Dzialowski, and Dr. Andrew Stoddard for their meaningful contributions throughout this study. Finally, I dedicate this work to Borsuah, Mansaray, and Kamara family.

The authors declare no conflicts of interest regarding the publication of this paper.

Borsuah, J.F., Stoodley, S., Storm, D., Dzialowski, A. and Stoddard, A. (2018) Development of a Three-Layer Steady State Vertical Dissolved Oxygen Model in Grand Lake, Oklahoma. Natural Resources, 9, 448-467. https://doi.org/10.4236/nr.2018.912028

A one-dimensional model has been developed to simulate the vertical profile of dissolved oxygen (DO) under summer stratified conditions. The model, based on a conceptual representation of the vertical structure of water temperature in a lake or reservoir as three mixed layers in a stratified water column, represents the epiliminion, metalimnnion, and hypolimnion as shown in Figure A1 of Appendix-A. The model, based on mass balance principles, is developed from analytical solutions of the three layer set of coupled differential equations presented in [13]. Kinetic relationships used to derive model input values from observed data for model parameters listed in this appendix are presented in Table 1 of this paper. Model parameter values used to develop the model are summarized in Table 2.

The differential equation for the surface euphotic zone layer (n = 1) is given by;

0 = d d z ( E 1 d D d z ) + R 1 − P a

the equation for the surface layer, D is the oxygen deficit; E₁ is the rate of vertical mixing within the surface layer; and R₁ is water column consumption of oxygen from algal respiration, nitrification and decomposition of detrital organic carbon; whereas P_a is the depth integrated algal photosynthetic production of oxygen within the euphotic zone [21].

The differential equation for the thermocline layer (n = 2) and the hypolimnion layer (n = 3) is depicted by;

0 = d d z ( E n d D n d Z ) + R n

The differential equations for each layer were integrated to give solutions for each layer (n = 1, n = 2, and n = 3). The water column oxygen consumption computed for each layer from nitrification, algal biomass respiration and detrital organic carbon decomposition is depth-integrated over the entire water column to yield a depth-averaged rate of oxygen consumption applied as a constant (R) to each layer. The surface layer depth-averaged photosynthetic oxygen production rate (P_a) is computed from chlorophyll-a (algal biomass), carbon: chlorophyll and oxygen: carbon stoichiometric ratios for algae, and the algal growth rate computed as a function of water clarity, light availability, ambient nutrient concentrations and water temperature [21].

At the air-water boundary of the surface layer (n = 1) the dissolved oxygen deficit is computed from:

D o = S K L + R H K L + P a H 1 K L

where,

D_o = is the oxygen deficit (mg/L)

S = Sediment oxygen demand (g O₂ m⁻² day^-1)

R = Depth averaged water column oxygen consumption (g O₂/m³-day)

P_a = Euphotic zone algal oxygen production (g O₂/m³-day)

H = Water column depth t the sediment bed-water interface (m)

H₁ = Depth at the bottom of the surface euphotic layer (m)

Systematic representation of oxygen deficit in the surface layer (n = 1) was computed from:

D 1 ( z ) = S K L ( 1 + K L Z E 1 ) + R H K L [ 1 + K L Z E 1 ( 1 − Z 2 H ) ] − P a H 1 K L [ 1 + K L Z E 1 ( 1 − Z 2 H 1 ) ]

Definition of terms and units for surface layer (n = 1) solutions are:

D₁(z) = dissolved oxygen deficit at depth, z with the layer (mg/L)

Z = depth in layer (Z = 0 is air-water interface) (m)

S = sediment oxygen demand (g O₂/m²-day)

R = depth averaged water column oxygen consumption (g O₂/m³-day)

P_a = euphotic zone algal oxygen production (g O₂/m³-day)

H = depth at sediment bed-water interface (m)

H₁ = depth of the surface layer (n = 1) at interface with thermocline layer (n = 2) (m)

K_L = air-water transfer coefficient for oxygen (m/day)

E₁ = vertical mixing rate for surface layer (n = 1) (m²/day)

The oxygen deficit D_n(z) for each sub-surface layer (n = 2 and n = 3) is computed from:

D n ( z ) = S K L [ 1 + K L Z E n + ∑ i = 2 n K L H i − 1 E 1 ( 1 − H i − 1 2 H ) ( E 1 E i − 1 − 1 ) ]     + R H K L [ 1 + K L Z E n ( 1 − Z 2 H ) + ∑ i = 2 n K L H i − 1 E 1 ( 1 − H i − 1 2 H ) ( E i E i − 1 − 1 ) ]     − P a H 1 K L [ 1 + K L H 1 2 E 1 ] (A1)

where,

D_n(z) = Oxygen deficit at depth, z within layer n = 2 and n = 3 (mg/L)

Z = depth in layer n = 2 and n = 3 (Z = 0 is air-water interface; Z = H is bottom depth (m)

H₁ = depth of euphotic zone layer (n = 1) at interface with the thermocline layer (n = 2) (m)

H₂ = depth of thermocline layer (n = 2) at interface with hypolimnion layer (n = 3) (m)

H₃ = depth of hypolimnion layer (n = 3) at interface with sediment bed (n = 3) (m)

H = depth at sediment bed-water interface (i.e., bottom depth) (m)

E₂ = vertical mixing rate for thermocline layer (n = 2) (m²/day)

E₃ = vertical mixing rate for hypolimnion layer (n = 3) (m²/day)