#### Abstract

Dissolved oxygen (DO) is essential for an aquatic ecosystem since it controls the biological productivity. Here, we propose a unidimensional dynamic model for DO by incorporating biological (photosynthesis, respiration, and mineralization), physical (atmospheric reaeration) and chemical (nitrification) processes so characteristic of shallow coastal water bodies. The analytical study of the proposed model is focussed on supersaturation and undersaturation of oxygen in the water body. The controllability of the ecosystem health has also been investigated. Model results indicate that, while undersaturation of oxygen is largely governed by nitrification and Net Ecosystem Metabolism (NEM), the supersaturation is controlled by photosynthetic activity. The model results are corroborated with observed data collected from Chilka lagoon, India. Subsequently, a biogeochemical model to study the DO variations in Chilka lagoon has been constructed. The model is properly calibrated and validated with observed data. Two independent sets of data (2004-2005 and 2005-2006) were used for model calibration and validation and Chi-square tests supported its robustness ( and 0.987; ). The model was used to evaluate independently the influence of individual taxa (diatoms, microphytobenthos, and cyanophyceans) on DO variations. Simulations indicate the vital role of microphytobenthos in lagoon DO dynamics and the overall wellbeing.

#### 1. Introduction

Dissolved oxygen (DO) is an essential component which determines the water quality and trophodynamics of an aquatic system [1–3]. A fluctuation of DO near its saturation indicates relatively healthy waters [4]. In a healthy aquatic lagoon ecosystem the concentration of oxygen must always be above the chronic criterion for growth (4.8 mg L^{−1}) developed by the U.S. EPA [5, 6]. A fall of oxygen below this chronic criterion can cause adverse effects on the health of the aquatic system. The system can suffer with hypoxia/anoxia conditions, resulting in depletion of fish stocks and other important aquatic animals [7, 8]. These losses can have significant damaging effects on ecological health, economic health, and stability of aquatic systems [8, 9].

DO dynamics are complex in nature and are affected by many physical, chemical, and biological processes. The important factors that affect DO dynamics in an aquatic environment are atmosphere-water surface exchange, photosynthesis, respiration, and mineralisation [5, 10, 11]. The other processes that affect the oxygen dynamics in the water column include nitrification, sediment-water exchange. The availability of the oxygen in the water column also depends on topography of the system, the nature of the soil, and so forth ([4] and references there in).

Many experimental studies are carried out to study the conditions that are responsible for hypoxia/anoxia in costal ecosystems [3, 12–14]. These studies pointed out that factors such as excess of nutrient load (eutrophic condition) and nitrification are the primary reasons for the hypoxic/anoxic conditions. To support these conclusions, many theoretical and empirical models (ranging from simple one-dimensional to three-dimensional) are constructed [10, 15–18]. All these studies concentrate on specific ecosystems (such as Cartagena Bay, Chesapeake Bay, Gulf of Mexico, Lake Zurich, and Virginia Estuaries, etc.) and to our knowledge no general theory exists which presents the influence of ecological and environmental factors on the qualitative behaviour of DO. Such a study would be useful not only to understand the strength of the external factors on the dynamics of DO but also to design control strategies to elevate the DO levels in the ecosystems.

In the present paper we construct a universal model to represent the DO dynamics that takes into consideration the essential factors that influence the concentration of DO in aquatic systems. This work presents a novel approach to evaluate the conditions under which an aquatic system experiences super/under saturated conditions. This analysis also highlights the influence of Net Ecosystem Metabolism (a vital concept related to studies on aquatic ecosystems which is introduced by Odum [19]) on the saturation characteristics of the aquatic systems. The theoretical study indicates that the system will always be driven to undersaturated state whenever Net Ecosystem Metabolism (NEM) is negative. On the other hand, when NEM is positive the system can exist in any one of the saturated states depending on the strengths of other processes (such as mineralisation, nitrification, and reaeration). The model analysis also brings forward control strategies to steer the system from undersaturated state to supersaturated state. To our knowledge these types of analysis and results have not been presented in the literature so far.

The main objective of the current study is to develop a universal model for DO dynamics and to bridge the theoretical results obtained from the abstract model with that of simulation modelling studies for better understanding of the ecosystem dynamics. This approach has been successfully applied to examine the causes and identify the crucial processes that are responsible for the saturation states observed in Chilka Lagoon (the largest brackish water lagoon in Asia, situated on the east-coast of India). By this study, it also has been possible to identify the key biogeochemical processes taking place in the Chilka Lagoon, based on which it is hypothesised that MPB are playing a key role in maintaining the health of the Chilka Lagoon.

The paper is structured as follows. An abstract model representing the oxygen dynamics in a water column is formulated in next section. This is followed by a section that contains detailed analysis of the model which highlights conditions in terms of the system parameters that ensure the presence of oxygen in water column. In Section 4 the conditions for the system to be super-/undersaturated are inferred and the results pertaining to system controllability are presented. The consequence of variations in the involved parameters on the health of the ecosystem is studied in Section 5. The theoretical findings are corroborated with the observations made for Chilka Lake in Section 6. In Section 7 appropriate forms have been attributed to the coefficient functions involved in the general model (by taking into consideration various biogeochemical events taking place in Chilka Lagoon), enabling simulation of DO trends observed in Chilka Lagoon and evaluation of vital parameters for the considered aquatic system. Finally, discussions and conclusions are presented in Section 8.

#### 2. Model Description and Formulation

In this section we construct an abstract model to represent the dynamics of DO in an aquatic system by taking into consideration various important processes that influence DO in the (system) water column. These processes include atmosphere-water surface exchange, photosynthesis, nitrification, mineralisation of detritus, respiration by plant (phytoplankton, macrophytes, and microphytobenthos (MPB)) and animal (zooplankton, fish, etc.) community, and loss due to other processes. Figure 1 presents a schematic diagram that describes the effect of these processes on DO in the water column.

There are a few other significant pathways which can also influence the oxygen dynamics in aquatic systems. These include ground water incursion and entrainment of low DO from the lake sediments. These processes are hard to measure in times, in particular, the DO contribution through the ground water. Moreover, the addition of entrainment of DO from sediments increases model dimension from one to two which makes the system relatively complex and hard to track. To avoid this complexity, we have added a loss/gain term to our model and calibrated this term to fit the model results. This seems to be a convenient approach as it gives a rough estimate on how much of DO is being lost/gained due to other processes which are hard to estimate using available tools. The other important aspect which influences the DO is stratification of water column in certain periods of time. In our model we have neglected this effect due to the fact that Chilka is a shallow lagoon (mean depth 1.5 m), which is seen through absence of any appreciable column variability in water temperature and/or salinity. Often, the strong surface winds overpower vertical gradients, if any, in the lake.

If stands for DO concentration in the water column, then the following equation represents its associated dynamics:

The mathematical formulation of the various processes involved in the above equation and the description of parameters used in the model formulation are presented in Tables 1 and 2, respectively.

Incorporating the above mathematical expressions presented in Table 1 into (1), we obtain the following one-dimensional model representing the dynamics of DO in the water column:

The above equation can be conveniently rearranged to obtain the following simpler representation: where Here the expression represents efflux of DO due to mineralisation of dead organic matter, exchange with atmosphere, and other processes, represents the influx of DO by photosynthetic activity, and represents the efflux of DO due to respiration by the plant and animal communities.

#### 3. Model Analysis

The general model formulated in the previous section is analysed to study the existence and nonexistence of equilibrium solution for the considered system. This study enables the identification conditions under which the system moves towards oxic/anoxic state.

To minimise the complexity in the analysis we first reduce the number of parameters in the model (3) by the process of nonimpersonalisation using the following transformations: These transformations reduce the model (3) to the form where

From the model description presented in the previous section and the formulation presented in (9), the positive parameter is directly proportional to the maximum rate of nitrification and ammonium concentration in the system. The cumulative effect of loss due to mineralisation process, exchange with atmosphere, and other processes is represented by . It can be easily observed that this parameter is always greater than unity (cf. (4), (10)). The parameter is given to be the sum of two terms given by and (cf. (11)). While the latter term is always positive, the former term represents the net ecosystem metabolism (NEM) [20] and can take either sign. Here NEM represents the net effect of production (due to photosynthesis) and consumption (due to respiration by all communities).

If the net gain due to photosynthetic activity and diffusion from atmosphere is greater than the loss due to respiration, then we have to be positive else it is negative. Therefore positivity of NEM () implies positivity of and negativity of implies negativity of NEM () as the term is always positive.

The model analysis gives us valuable information regarding the cause and relations of various events that take place in water bodies. It also helps in deriving solutions for vital problems such as avoidance of anoxic conditions to prevent ecological and economic losses in the ecosystem. As a part of this analysis we shall first study the existence of equilibrium solutions for the considered model and investigate their qualitative behaviour. The considered model admits at most one positive equilibrium solution given by

The stability of the equilibrium depends on the sign of the derivative of (cf. RHS of (8)) evaluated at the equilibrium. This equilibrium will be stable if this sign is nonpositive and unstable otherwise [21]. Further this equilibrium is asymptotically stable if the derivative sign is negative. Clearly, the existence and stability nature of the equilibrium solution depend on the involved parameters , , and . We know that is always positive, is greater than unity, and can take any sign. Thus for a given , it is enough to study the qualitative behaviour of (8) in the first and fourth quadrants of -space.

Figure 2 presents the relationship between these parameters and the existence of equilibrium solution. Note that the curves and divide the first and fourth quadrants of -space into three significant regions given by

It is easy to observe that (8) admits a unique globally asymptotically stable equilibrium solution when the parameters belong to Regions I and II and it does not admit any equilibrium solution in Region III. We have the following result.

Theorem 1. *For a given , (8) admits a unique globally asymptotically stable equilibrium if the parameters and belong to the first quadrant of the -space and it admits no equilibrium solution if they belong to the fourth quadrant including .*

Theorem 1 is a significant result in the sense that it gives the essential conditions under which the water column supports DO. This result along with the expression for the equilibrium solution (12) and Figure 2 helps us to derive conditions for improving DO levels in the water column.

In Region III we have to be negative indicating negativity of NEM for the aquatic system. In this case the system loses oxygen and gets driven to anoxic conditions as cumulative gain due to the fact that photosynthetic and atmospheric diffusion processes are dominated by cumulative loss due to respiration, nitrification, and mineralisation.

In Regions I and II we have to be positive representing the dominance of photosynthetic and atmospheric diffusion processes over the process of respiration. In this case the system supports DO (reflected by the existence of asymptotically stable equilibrium) but its magnitude depends upon strength of relative to .

If the system parameters belong to Region I, then the characteristic of the aquatic system is such that the activities of photosynthesis and atmospheric diffusion dominate not only the respiration process (i.e., ) but also the activities of nitrification and mineralisation (i.e., ). In this case the aquatic system retains higher levels of oxygen (owing to higher magnitude of the equilibrium).

In Region II although the photosynthetic and atmospheric diffusion dominate the respiration process (i.e., ), the cumulative effect of respiration, nitrification, and mineralisation dominates the cumulative effect of photosynthesis and atmospheric diffusion processes (i.e., ). Thus the system could tend to hypoxic conditions (reflected by the smaller magnitude of the equilibrium).

Here, it is important to note that the level of DO continues to decrease as the value of decreases. The system gets driven to hypoxia followed by anoxia in the vicinity of as crosses the -axis. Further decrease in from this level () makes the system free from DO. These findings are in accordance with the experimental studies carried out on hypoxia/anoxia [8, 13, 14].

#### 4. Regions of Supersaturation, Undersaturation

In this section a novel approach is presented to derive the conditions under which the equilibrium state of DO will be in two possible states given by supersaturation or undersaturation. By definition, oxygen in a water body is said to be super- (under)saturated if DO concentration is greater (lesser) than the saturating oxygen in the water body. To study the conditions under which the DO will be super- (under)saturated, we first identify the ranges on the parameters that lead the system to the respective saturation states. For this we proceed as follows.

For a given saturating oxygen level O_{ss} of the water body, the equilibrium state (under the assumption that it exists) of the system (8) satisfies

Here is nothing but , a proportion of the DO concentration and represents the NEM of the system. Treating O_{ss} as a variable and defining let us consider the following equations:
in the positive quadrant of the -space.

The curve (15) is a representative of the set of all admissible equilibrium solutions of (8). Henceforth we shall label this curve (15) as admissible equilibrium curve. Note that the equilibrium of the system for a given saturating oxygen O_{ss} is given by .

Observe that the line (16) divides the positive quadrant into two regions such that one of them (with ) represents region of undersaturation and the other (with ) represents region of supersaturation. Figures 3 and 4 present the curves (15), (16) in the positive quadrant of the -space for the cases and , respectively. These figures give further information on the saturation characteristics of the equilibrium state and associated controllability aspects.

Note that the curves (15) and (16) intersect in the interior of the positive quadrant only if (Figure 3). On the contrary, if is nonpositive, then there does not exist any such intersection in the positive quadrant and the curve (15) lies completely above (16) (cf. Figure 4). Observe that the admissible equilibrium curve (15) is a hyperbola and intersects -axis at having a positive slope . This curve tends to infinity as . The slope of this curve highlights the vital role the mineralisation and nitrification processes play in improving the equilibrium state of the water body. This aspect will be discussed in the next section.

In order to study the relationship between the eventual state of oxygen with the saturating oxygen level, let us consider the intersection point of the curves (15) and (16) in the positive quadrant of -space. The component of this intersection point is given by the positive root () of the quadratic equation: where

Let be the intersection of the curves (15) and (16) which is represented in Figure 3. Thus for positive NEM (i.e, ), the admissible equilibriums curve can be divided into two significant segments given by

If is the saturation level of the water body with and is the corresponding equilibrium of (8), then Segment 1 (Segment 2) implies that the water body will be in super- (under)saturation state eventually.

For the case of negative NEM (i.e., ) the system always equilibrates in undersaturated state as the curve (23) lies completely in the region of undersaturation (cf. Figure 4). From these observations, we conclude that while a water body may equilibrate in a supersaturated state or undersaturated state when the NEM is positive, it always equilibrates in undersaturated state if NEM is negative.

#### 5. Ecosystem Health and System Controllability

In this section we study the consequences of variations in the involved parameters (, , and ) on the health of the aquatic system. We also find strategies to drive the equilibrium state of the system from undersaturation to supersaturation state. From the graph of (15) (Figures 3 and 4), observe that the DO equilibrium level decreases as the saturating oxygen decreases. Also from the qualitative properties of this curve (in particular slope) it is easy to see that for a given saturating oxygen concentration the corresponding equilibrium DO concentration increases with decrease in either or or both. This clearly indicates that the DO concentration in the system decreases with an increase in the ammonium concentration in the system (which increases ) or increase in the mineralisation activity (which increases ) or both.

From the analysis presented in the previous section, it is clear that an equilibrium state of the considered system cannot be driven to supersaturation state when NEM is negative. Thus the possibility of transition from undersaturation to supersaturation exists only when NEM is positive. Observe that, for a given representative of saturating oxygen () if the corresponding DO equilibrium () satisfies , then the system would be under- (super)saturated at the equilibrium.

Thus given that the system is undersaturated at a given saturation level , the way to drive the system from undersaturated state to supersaturated state is to alter the function in such a way that the corresponding DO equilibrium state, denoted by , satisfies the supersaturation condition (). The monotonicity property of ensures such a possibility through a sufficient reduction in its slope (which is ). This can be achieved by reducing the coefficients and/or sufficiently. It can be easily observed that reduction in decreases the slope of faster when compared to reduction in .

For the purpose of illustration, we assume the following hypothetical values (but, these values can also be found using field experiments) for the nondimensional parameters , , and of the system (8): The admissible equilibrium curve (15) with the foresaid parameter values is presented in Figures 5 and 6 with dashed lines. Taking the saturation level () of the system to be 7, we observe that the DO in the system equilibrates in undersaturated state with a magnitude of 4.85784 (cf. Figures 5 and 6).

Following the theory, the system can be driven to a supersaturated state (since ), by varying the parameter or or both. Fixing the values of and it can be easily inferred from (18) that the system will equilibrate in supersaturated state if is made to stay below 2.6857. Similarly, fixing and , the system can be made to equilibrate in supersaturated state by ensuring that is less than 1.3357.

Suppose we wish that system should equilibrate in supersaturated state with , (18) guarantees such eventual state for the choices of parameters or (8.0,1.155,9.35). The simulations demonstrating the move from undersaturation state to supersaturation state for both the above said choices are presented in Figures 5 and 6. The required reductions in the parameters and to reach a state with clearly demonstrate the stronger effect of reducing over in driving the state to supersaturation.

In the case, where (i.e., system with negative NEM), the system can be driven from anoxic condition to hypoxic and further to conditions of fairly high DO concentration by reducing and sufficiently. However, it would not be possible to drive the system to supersaturated state by altering and .

Thus, irrespective of the sign of NEM, reduction in ammonium concentration and mineralisation activity enhances the health of the ecosystem and enriches the waters with oxygen. At this juncture, it is worth noting that, for the case it is also possible to drive the system to supersaturation state by reducing the saturating oxygen level to a state below . One of the ways to reduce the saturating oxygen level in the water body is to increase the salinity of the system; other possibilities are increase in temperature and decrease in partial pressure [5]. Thus increasing the salinity of the water body is yet another solution to increase the health of the water bodies.

From (9)–(11), it can be noted that the values of the parameters , , and are dependent on the factors such as reaeration coefficient, ammonium concentration, nitrification rate constant, half-saturation constant for nitrification, plankton biomass, mortality rates of plankton groups, and losses due to other processes, and so forth. Having knowledge of all these parameter (either through field surveys or experimental studies) for a real-life system under consideration, it is possible to determine the values of , , and , from which we can derive the management strategies to drive the system to supersaturation conditions. Please refer to Section 7.3 and Figure 12 for evaluation of these coefficient values and their seasonal dependent variations for the case of Chilka Lagoon.

#### 6. A Few Lessons in the Light of the Analysis

Analysis of the abstract model presented in the previous sections can be used to answer several questions pertaining to water bodies such as Chilka Lake, a brackish water lagoon on the east-coast of India. In this section the events that are responsible for the observed DO levels in Chilka Lagoon during 2004-2005 are interpreted using the analysis presented in previous sections.

##### 6.1. Study Area

Chilka Lake (19°19°N 85°85°E) (Figure 7) is situated on the eastern seaboard of India some 350 km south of Kolkata. The lagoon is pear shaped and covers a total area of about 1000 km^{2} during wet season (August–October), but the volume is reduced by nearly 60% during summer months (April-May). Topographically, Chilka Lake can be divided into two major regions—the main area and the outer channel. The main area, which is about 65 km long and 20 km at its greatest breadth, constitutes the lagoon proper and can be arbitrarily divided into south, central, and north sectors. In general, the lagoon is shallow (1.5 m) particularly in the north sector where considerable silting has taken place. Overall, the topography is not even since large variations in depth occur as a result of heavy deposition of terrigenous materials from catchments and river discharge. In the outer channel, which connects the lagoon proper to the sea (Bay of Bengal), depth variations are even marked owing to erosion and accretion processes. There are many small and large islands scattered throughout Chilka Lagoon. The largest is the Nalaban (35 km^{2}), which is covered by considerable marsh vegetation. During monsoon when the lagoon receives enormous quantities of freshwater from the catchments and surrounding rivers, most of these islands are inundated. However, during summer when freshwater discharge is negligible and evaporation exceeds precipitation, the lagoon recedes to a considerable distance exposing a vast expanse of wetland area. Chilka Lagoon is unique not only from the point of its shear extent but also hydrologically as it is nature's most intriguing habitat supporting a wealth of marine, brackish, and freshwater life. In the year 1981 Chilka Lake was designated as a Ramsar Site—a wetland of international importance, because of its rich biodiversity and socioeconomic value.

##### 6.2. Data Collection Procedures

###### 6.2.1. Depth Measurements

Time series observation on the bathymetry of the lagoon has been obtained from 36 GPS fixed locations (cf. Figure 7) at frequent intervals of time (every 15-day interval) during the period of study May 3, 2004 to September 22, 2006 using the conventional depth measurement technique (weighted line and a graduated bamboo pole) described by [22].

###### 6.2.2. Salinity Measurements and Local Weather Data

Salinity was determined according to Knudsen’s method [23] at all the 36 GPS locations during the study period on a monthly basis. A complex combination of freshwater discharge, evaporation, wind condition, and tidal inflow of seawater govern the spatial and temporal changes in salinity of the lagoon. In October the entire lagoon is occupied with freshwater. Salinities between 1.8 to 4.6 PSU in the main area and between 7 to 8 PSU in the outer channel (cf. Figure 7) were noticed. In February salinities rise (5–20 PSU) due to decrease in freshwater flow and they are maximum (16.5–34.7 PSU) in May. Meteorological data (rainfall, wind speed, atmospheric temperature, and relative humidity), recorded at Gopalpur station (some50 km south of Chilka), were obtained from Indian Meteorological Department, Bhubaneswar. The annual maximum temperature (34.8°C) is recorded in May and minimum temperature (15.6°C) in December. The annual rainfall in the region is 1123 mm. Rainfall is very high (151–298 mm) during June–September and the least (0.4–4.9 mm) during January–March. The lagoon experiences Southwest and Northeast monsoons during June–September and November-December, respectively. Wind speeds are high (6.5–9m/s) during monsoon months (March–August) and low (4.8–6.4 m/s) during winter months (January-February). Wind direction is mostly N and NE during postmonsoon months (October-November) and SW during monsoon months. During the study period the mean monthly atmospheric minimum temperature was 21.51°C and maximum temperature was 30.29°C; mean total rainfall is amounted to 1514.1 mm the bulk of which 72.2% was recorded during the months July–September of every year corresponding to the southwest monsoon.

##### 6.3. Model Application to Chilka

Figure 8 presents the Lake-wide mean DO (along with computed DO saturation), ammonium concentration, and phytoplankton biomass for the period July 19, 2004 to July 18, 2005. The DO saturation for Chilka is computed using the formula [5]
where O_{ss} is the saturating oxygen concentration in salt water at 1 atm [mg L^{−1}], O_{sf} is the saturating oxygen concentration in fresh water at 1 atm [mg L^{−1}], is absolute temperature [K], and is the salinity [ppt] of the lagoon.

The period September–November, 2004, in Figure 8 represents the active river flow period and the period February–April, 2005, represents the lean flow conditions. Here, active flow period represents the period during which the lake receives enormous amount of freshwater from the catchments.

During the period of active flow, it is observed that the lake experiences high turbidity (mean 34.25 NTU) and low salinity (mean 9.44 PSU) conditions. In this period the findings of Gupta et al. [24] reported that the NEM of the Chilka Lagoon is negative, which is attributed to reduced activity of phytoplankton and MPB. According to the theory developed in Sections 3 and 4, the lake is supposed to experience undersaturation conditions during active flow period, where the NEM is negative. The observations on DO presented in Figure 8 confirm this fact. The fall in ammonium concentrations during this period indicates the activity of nitrification, which further reduces the oxygen concentration.

During lean flow period, Gupta et al. [24] reported that the NEM of the Chilka is positive, which is due to high photosynthetic activity by autotrophic organisms. Again, according to the theory the lake can experience either of the saturation states during lean flow periods depending on the ambient conditions. Here undersaturated conditions prevail if the processes of nitrification and mineralisation dominate NEM. However supersaturation conditions can be observed if the photosynthetic activity by autotrophic organisms dominates the processes of nitrification and mineralisation.

From Figure 8 it may be seen that in the first half of the lean flow period the phytoplankton activity dominates nitrification and mineralisation processes. Thus the system experiences supersaturated conditions, validating the concepts presented in the model. In the latter half of the lean flow period, increase in ammonium concentrations may be attributed to mineralisation activity. This observation may be substantiated by the decrease in phytoplankton concentration following the bloom. Thus the oxygen concentrations during this period almost reach the level of saturation, which is also in accordance with the theory developed in Sections 3 and 4.

Coming to intermittent conditions (particularly during the periods August 2004, November 15, 2004 to January 14, 2005), we notice that the system is in supersaturated state even though the phytoplankton activity is low. From the theory, supersaturated state is possible only when NEM is positive. So it would appear that, during the said periods, the low phytoplankton activity is compensated by the activities of other autotrophic organisms such as seagrasses and MPB which seem to be driving NEM to a positive state. This is an important observation from the perspective of ecosystem health. This indicates that the seagrasses, MPB, and other plant community (other than phytoplankton) also play a vital role in maintaining the health of the Chilka Lagoon.

#### 7. Development of Biogeochemical Model for DO in Lake Chilka

In this section, the general model discussed so far is applied to Chilka Lagoon by attributing appropriate forms to the involved coefficient functions. These functions are derived by taking into consideration various biogeochemical events taking place in the lagoon. Throughout this modelling exercise carbon is taken to be the currency to express plant and animal biomass.

The reaeration coefficient () in the atmosphere-water surface exchange process is calculated using formula [5, 25] where is the mean current velocity in m/sec, is the wind speed measured 10 m above the water surface in m/sec, and is the mean depth in meters.

Saturating oxygen levels (O_{ss}) for Chilka Lagoon are computed using formulae (21).

Diatoms, MPB, and cyanophyceans are found to be dominating groups of primary producers in Chilka Lagoon during the period of study. It is well known that the diatoms and MPB utilise the available ammonia [5, 26, 27] in the water column for their growth during the photosynthetic activity and the amount of oxygen liberated in this process is proportional to their growth rates. Thus, the rate of oxygen liberation due to photosynthetic activity by diatoms and MPB can be given by [5] where and are the growth rates of diatoms and MPB, respectively. It is well known that cyanophyceans prefer nitrate and convert nitrate to ammonium, which is subsequently used for photosynthesis activity [5]. Thus, the rate of oxygen liberation due to photosynthesis activity by cyanophyceans is given by [5] where is nitrogen to carbon conversion factor and is the growth rate of cyanophyceans.

The nitrification process can be represented by the expression [5] where is the water temperature in °C, is the nitrification rate is the half-saturation constant for oxygen limitation during the nitrification activity, and is the ammonium concentration in the water column.

The respiration process is the reverse process of photosynthesis and is modelled by multiplying the concentrations of phytoplankton and zooplankton by their respective respiration rates. Thus this process is mathematically represented as where , , , and are the respiration rate constants for diatoms, cyanophyceans, MPB, and zooplankton, respectively.

The mineralisation process in aquatic system is governed by particulate organic matter mineralisation. During this process, one molecule of is consumed per micromole of particulate matter. The particulate matter depends on the mortality rates of phytoplankton and zooplankton. The mineralisation process is temperature dependent and this effect is modelled as Arrhenius type expressions [5, 17]. Thus the mineralisation process can be represented by the following expression: where is the mineralisation temperature correction coefficient. , , , and are the mortality rates of diatoms, MPB, cyanophyceans, and zooplankton, respectively.

Finally, the loss due to other processes is modelled as a linear function of available DO with a constant loss rate, . Since, quantification of this loss is difficult, calibration has been done to determine this rate constant.

Thus the dynamics of DO in the water column of Chilka Lagoon can be represented by

The monthly average values of salinity, biomass of plant and animal communities, water temperature, and ammonia obtained from the periodical field surveys are interpolated with Hermite interpolation to obtain their daily estimates and are presented in Figure 9. These estimates are used as inputs for the model. The description of the parameters and their units along with the assumed values are presented in the Table 3.

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##### 7.1. Sensitivity Analysis

Sensitivity analysis attempts to provide a measure of the sensitivity of either parameters, or forcing functions to the state variables of greatest interest in the model [17, 28]. Sensitivity analysis is performed using the following formula: where is sensitivity, is state variable (here ), and is parameter. and are change of initial values of state variable, parameters, and forcing functions, respectively, at ±10% level and ±20% levels. Sensitivity analysis results are summarized in Table 4. Parameters, which are not possible to determine from field survey experiments are calibrated using first set data and further validated using second set data.

##### 7.2. Model Calibration and Validation

Model calibration is done by adjusting the selected parameters such as growth rates, loss rates in the model to obtain a best fit between the model calculations and the monthly average field data (Set number 1) collected during first year (June 18, 2004 to June 27, 2005). The monthly average field data (Set number 2) collected during second year (July 31, 2005 to 22, September 2006) is used to validate the model.

##### 7.3. Results

The simulated values of DO for Chilka are presented in Figure 10. Clearly the results of simulation reflect the trends observed with the field data indicating that the chosen parameters are closer to reality.

The goodness of fit for statistical significance between the field data and modelled results is performed using Chi-square test with the help of SPSS Software. The performance of the model is quantified from the RMSE value of the Chi-square test. The RMSE value for the model output is found to be 0.3028. Further, for the first and second years of data collection the Chi-square values are found to be 0.982 and 0.987, respectively, () and for the entire simulation period the Chi-square value is found to be 0.993 (). The model validation results using regression analysis are presented in Figure 11.

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The nondimensional parameters , , , and for the model (8) have been computed for the period of simulation and are presented in Figure 12. From Figure 12(a) we observe that the parameter which is representative of nitrification activity belongs to the range , and the parameter that represents the cumulative effect of mineralisation, surface exchange, and loss due to other processes ranges between 1.0815 and 2.7773 (cf. Figure 12(b)). From Figure 12(c) we observe that the parameter which is representative of NEM varies between 0.20091 and 9.891 and is low during monsoon and high during recovery period (December–April) which coincides with the occurrence of massive cyanophyceans bloom. Figure 12(d) presents the variations in the parameter which is the cumulative effect of NEM and atmospheric diffusion. The range for this parameter during the period of study is found to be (3.5685, 13.543). We also observe that the term is negative all through the period of simulation ranging from −10.872 to −2.4444 (cf. Figure 12(e)) and it is inversely related to the oxygen present in the system (as predicted by the theory).

##### 7.4. Influence of Individual Plankton Groups on the DO Dynamics

The biogeochemical model developed to simulate the DO trends in Chilka is used to assess the strength of each of the individual plankton groups in oxygenating the waters of Chilka. The expected scenarios for DO in Chilka Lagoon in the presence of only one of the plankton groups among diatoms, cyanophyceans and MPB are presented in Figure 13.

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From Figure 13 we clearly observe that the system would have been undersaturated all through the period if either diatoms or MPB alone were present in the system. Even in the case of cyanophyceans (Figure 9(b)) the system would be undersaturated during the simulation period except for a short duration (January 14 to April 14, 2005) during the massive bloom formation.

Figure 14 presents the expected DO trends if two out of the three plankton groups were present in the system. These numerical simulations clearly indicate that the supersaturation conditions observed are due to the presence of both diatoms and MPB (cf. Figure 13).

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Observe that the system cannot be driven to supersaturation in these periods in the absence of either diatoms or MPB (cf. Figures 12 and 14). The pronounced supersaturation observed during January–March 2006 is due to the cyanophyceans bloom (cf. Figure 10). From Figures 10 and 13 it can be inferred that the near saturation state of DO after the massive cyanophyceans bloom is due to the presence of MPB and diatoms in the system else the system would have experienced hypoxic conditions during June 2006. Thus the health of Chilka Lagoon appears to be maintained by the presence of ammonium consuming plant community, that is, diatoms and MPB.

#### 8. Discussions and Conclusions

Dissolved oxygen (DO) is known to be a vital component in determining the health of aquatic systems. Much work was carried out on the events that are responsible for hypoxic/anoxic conditions in the aquatic systems [20, 29–33], characterised by differing topographical conditions, physical and chemical attributes. Excess nutrient inflows, nitrification, and denitrification events are the main causes for the observed conditions. A few key research concerns raised in this regard include “When does large-scale hypoxia occur?” “What amount of nutrient load reductions is needed to reduce the hypoxia?” [7, 34]. We show that mathematical models coupled with biology, physics, and chemistry can be used to address the above issues.

In this paper a theoretical model of fairly general nature has been constructed to understand the dynamics of DO in the aquatic systems. Forcing factors for the model include atmosphere-water surface exchange, photosynthesis, respiration, nitrification, and mineralisation. In the proposed model the concentrations of plant and animal community in the water body are taken as parameters. This gives the advantage of studying the responses of the system with respect to variations in these concentrations and enhances the predictive potential of the model under various scenarios involving conditions on the concentrations of nutrient, plant, and animal community. The main interest of the study is to bridge the theoretical results obtained from the abstract model with that of simulation modelling studies for better understanding of the ecosystem dynamics. This type of approach has proved to be of great help in understanding system dynamics of Chilka Lagoon, India.

The analysis of the theoretical model revealed several interesting phenomena that are responsible for decrease and increase in DO of aquatic systems. It is found that Net Ecosystem Metabolism (NEM), nitrification and/or mineralisation of detritus could play a key role in the eventual state of the system. The analysis indicates that the effect of mineralisation is stronger than the effect of nitrification in reducing the oxygen concentration in the system. For a system where in the processes of photosynthesis and surface exchange dominate the process of respiration, it is found that the saturation characteristics depend on the magnitude of nitrification and/or mineralisation rates. Supersaturation conditions prevail if the said rates are small enough. On the other hand, the system can get driven to hypoxic conditions at higher rates of nitrification and/or mineralisation. This observation is in conformity with studies done on hypoxia [13, 14].

Further, the analysis reveals that an aquatic system can get driven to anoxic conditions even if respiration exceeds net contribution by photosynthesis and atmosphere-water surface exchange. In such a case the analysis indicates that the health of the system can only be improved (DO can be enhanced) by external supply of oxygen to water column as observed in the studies [35, 36]. Apart from the above findings, this study provides a few management strategies to drive the system away from hypoxic conditions towards higher DO concentrations. These strategies can be used to steer the system from undersaturation to supersaturation whenever NEM is positive. Such a transition is not possible for a system with negative NEM although the DO concentration can be enhanced.

After a thorough analysis of the general model an effort is made to corroborate the theoretical findings with the field observations made for Chilka Lagoon, situated in East-Coast of India. Insights drawn from the theoretical analysis are used to explain the oscillations observed in DO for Chilka. From these investigations it is learnt that the macrophytes, submerged vegetation play a vital role in maintaining the health of the Chilka Lagoon. It is found out that during the active flow period (September–November) of 2005, the Chilka experienced undersaturation conditions due to NEM being less (in fact close to zero) and high nitrification activity. During the lean flow conditions (February–April) of 2005, Chilka experienced supersaturation conditions due to enhanced NEM (as a result of high photosynthetic activity of plant community) and reduced nitrification activity.

Having learnt the importance of the plant species in Chilka Lagoon other than phytoplankton, a biogeochemical model has been developed to simulate the dynamics of DO in Chilka Lagoon. The construction of this model is done based on insights gained from the general model proposed for DO dynamics. Biogeochemical events taking place in the lagoon are taken into consideration when assigning forms to various coefficients in the model. Values for some of the involved parameters are drawn from the literature and a few have been calibrated. The results of the simulation are validated with that of the observed trends of DO in Chilka. Statistical tests such as Chi-square test for goodness of fit, regression analysis, and RMSE are conducted for establishing the ability of the proposed biogeochemical model to imitate the observed trends. From these simulations it has become possible to not only to estimate vital parameters present in the proposed model but also to capture the seasonal variations associated with these parameters. The proposed biogeochemical model has enabled evaluation of strength of each of the plant community in oxygenating water of Chilka. As a consequence, it is hypothesised that diatoms and MPB are mainly responsible for maintaining the DO levels of Chilka around saturation during the period of study.

Finally, we wish to conclude that inclusion of macrophyte dynamics into the considered biogeochemical model would take the study much closer to reality. This calls for a model of higher dimension since the nutritional requirements of macrophytes are different from that of phytoplankton and MPB. Due to lack of data related to macrophytes for the case of Chilka, the simulation study is confined only to a model that involves explicit roles of phytoplankton and MPB. It is noteworthy that the considered model itself is able to replicate the system dynamics sufficiently close to reality enabling us to assess the roles of phytoplankton and MPB in the DO dynamics associated with Chilka Lagoon.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author B. S. R. V. Prasad acknowledges the support from the University Grants Commission, Govt. of India, New Delhi, for financial support in the form of Dr. D. S. Kothari Post-Doctoral fellowship. Authors wish to express their sincere gratitude to the Ministry of Earth Sciences (formerly Department of Ocean Development), New Delhi for funding the Project on Chilka Lake (no. DOD/ICMAM-PD/9/2001). The authors are thankful to Dr. B. R. Subramanian, Project Director, ICMAM, Chennai, and the Indian Navy at INS Chilka for several courtesies. They thank all those who helped in field work. This work was carried out at Andhra University and they are grateful to the authorities for the facilities provided. Authors B. S. R. V. Prasad and S. Ray acknowledge the support of the Department of Zoology, Visva-Bharati University.