**2.2 Modeling in Lake Mangueira**

For Lake Mangueira, we applied a dynamic ecological model describing phytoplankton growth, called IPH-TRIM3D-PCLake (Fragoso Jr. et al., 2009). Although this model can represent the three-dimensional flows and the entire trophic structure dynamically, in this study case we used a simplified version of the model consisting of three modules: (a) a detailed horizontal 2-D hydrodynamic module for shallow water, which deals with winddriven quantitative flows and water levels; (b) a nutrient module, which deals with nutrient transport mechanisms and some conversion processes; and (c) a biological module, which describes phytoplankton growth in a simple way. An overview of the modeling processes is given in Fig. 3.

The hydrodynamic model is based on the shallow-water equations derived from Navier-Stokes, which describe dynamically a horizontal two-dimensional flow:

$$\frac{\partial \mathfrak{\partial} \eta}{\partial t} + \frac{\partial \left[ \left( h + \eta \right) \mu \right]}{\partial \mathfrak{x}} + \frac{\partial \left[ \left( h + \eta \right) \upsilon \right]}{\partial y} = 0 \tag{1}$$

$$\frac{\partial \mathbf{u}}{\partial t} + \mu \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \upsilon \frac{\partial \mathbf{u}}{\partial y} = -\mathbf{g} \frac{\partial \eta}{\partial \mathbf{x}} - \gamma \boldsymbol{u} + \boldsymbol{\tau}\_{\boldsymbol{x}} + A\_{h} \nabla^{2} \boldsymbol{u} + f \boldsymbol{\upsilon} \tag{2}$$

$$\frac{\partial \mathbf{v}}{\partial t} + \mu \frac{\partial \mathbf{v}}{\partial \mathbf{x}} + \mathbf{v} \frac{\partial \mathbf{v}}{\partial y} = -\mathbf{g} \frac{\partial \eta}{\partial y} - \mathbf{\gamma} \mathbf{v} + \mathbf{r}\_y + A\_h \nabla^2 \mathbf{v} - f \mu \tag{3}$$

Hydrodynamic Control of Plankton Spatial and

Cattani, 1994).

zooplankton:

Temporal Heterogeneity in Subtropical Shallow Lakes 33

10 m above the water surface; and 2 2 *W WW <sup>x</sup> <sup>y</sup>* is the norm of the wind velocity vector. We solved the partial differential equations numerically by applying an efficient semi-implicit finite differences method to a regular grid, which was used in order to assure stability, convergence and accuracy (Casulli, 1990; Casulli & Cheng, 1990; Casulli &

The nutrient module includes the advection and diffusion of each substance, inlet and outlet

is the horizontal scalar diffusivity assumed as 0.1 m2 day-1 (Chapra, 1997). Equation 6 was applied to model total phosphorus, total nitrogen and phytoplankton. All these equations are solved dynamically, using a simple numerical semi-implicit central finite differences scheme (Gross et al., 1999a; 1999b) (Fig. 2). Thus, the mass balances involving

*h h*

+ source or sink (6)

+ inlet/outlet (7)

+ inlet/outlet (8)

(9)

+ h is the total depth; and *Kh*

*K K*

*eff h h*

*Ha K K*

*na eff h h*

*pa eff phos h h*

 

*P L TNIa a* , , – (10)

*a Ha K K*

 

*Hp uHp vHp Hp Hp a Ha k p K <sup>K</sup> tx y x xy y*

where *a*, *n* and *p* are the chlorophyll-a, total nitrogen and total phosphorus concentrations, respectively; *ana* is the N/Chla ratio equal to 8 mg N mg Chla-1; *apa* is the P/Chla ratio equal to 1.5 mg N mg Chla-1, inlet/outlet represents the balance between all inlets and outlets in a

The effective growth rate itself is not a simple constant, but varies in response to environmental factors such as temperature, nutrients, respiration, excretion and grazing by

where *μP(T, N, I)* is the primary production rate as a function of temperature (*T*), nutrients (*N*), and light (*I*); *μL* is the loss rate due to respiration, excretion, and grazing by

The hydrodynamic module was calibrated by tuning the model parameters within their observed ranges taken from the literature (Table 1). Nonetheless, the hydraulic resistance caused by the presence of emerged macrophytes in the Taim Wetland was represented by a

inlet /outlet

loading, sedimentation, and resuspension through the following equation:

*HC uCH vCH HC HC*

*t x yx x y y* 

 

*t x y x xy y* 

 

*tx y x xy y* 

*ef* 

control volume *∂x ∂y ∂z*; and kphos is the settling coefficient.

zooplankton; and *a* is the chlorophyll-a concentration.

*Hn uHn vHn Hn Hn*

*Ha uHa vHa Ha Ha*

where *C* is the mean concentration in the water column; H=

phytoplankton and nutrients can be written as:

Fig. 3. Simplified representation of the interactions involving the state variables (double circle), and the processes (rectangle) used for the modeling of Lake Mangueira.

where *u(x,y,t)* and *v(x,y,t)* are the water velocity components in the horizontal *x* and *y* directions; *t* is time; (x,y,t) is the water surface elevation relative to the undisturbed water surface; *g* is the gravitational acceleration; *h(x,y)* is the water depth measured from the undisturbed water surface; *f* is the Coriolis force; *<sup>x</sup>* and *<sup>y</sup>* are the wind stresses in the *x* and *y* directions; *xi x <sup>j</sup>* is a vector operator in the *x-y* plane (known as nabla operator, or del operator); *Ah* is the coefficient of horizontal eddy viscosity; and 2 2 *z g u v C* (Daily & Harlerman, 1966) where *Cz* is the Chezy friction coefficient.

Usually, the wind stresses in the *x* and *y* directions are written as a function of wind velocity (Wu, 1982):

$$
\tau\_x = \mathbf{C}\_D \cdot \mathcal{W}\_\mathbf{x} \cdot \left\| \mathcal{W} \right\| \tag{4}
$$

$$
\tau\_y = \mathbf{C}\_D \cdot \mathbf{V} \mathbf{V}\_y \cdot \left\| \mathbf{V} \right\| \tag{5}
$$

where *CD* is the wind friction coefficient; and *Wx* and *Wy* are the wind velocity components (m.s-1) in the *x* and *y* directions, respectively. Wind velocity is measured at

Fig. 3. Simplified representation of the interactions involving the state variables (double

where *u(x,y,t)* and *v(x,y,t)* are the water velocity components in the horizontal *x* and *y*

surface; *g* is the gravitational acceleration; *h(x,y)* is the water depth measured from the

operator, or del operator); *Ah* is the coefficient of horizontal eddy viscosity; and

Usually, the wind stresses in the *x* and *y* directions are written as a function of wind velocity

where *CD* is the wind friction coefficient; and *Wx* and *Wy* are the wind velocity components (m.s-1) in the *x* and *y* directions, respectively. Wind velocity is measured at

(Daily & Harlerman, 1966) where *Cz* is the Chezy friction coefficient.

(x,y,t) is the water surface elevation relative to the undisturbed water

 and *<sup>y</sup>* 

is a vector operator in the *x-y* plane (known as nabla

*x Dx CWW* (4)

*y Dy CW W* (5)

are the wind stresses in the *x*

circle), and the processes (rectangle) used for the modeling of Lake Mangueira.

directions; *t* is time;

2 2

*z g u v C*

(Wu, 1982):

and *y* directions; *xi x <sup>j</sup>*

undisturbed water surface; *f* is the Coriolis force; *<sup>x</sup>*

10 m above the water surface; and 2 2 *W WW <sup>x</sup> <sup>y</sup>* is the norm of the wind velocity vector. We solved the partial differential equations numerically by applying an efficient semi-implicit finite differences method to a regular grid, which was used in order to assure stability, convergence and accuracy (Casulli, 1990; Casulli & Cheng, 1990; Casulli & Cattani, 1994).

The nutrient module includes the advection and diffusion of each substance, inlet and outlet loading, sedimentation, and resuspension through the following equation:

$$\frac{\partial \left( HC\right)}{\partial t} + \frac{\partial \left( uCH \right)}{\partial \mathbf{x}} + \frac{\partial \left( vCH \right)}{\partial y} = \frac{\partial}{\partial \mathbf{x}} \left( K\_h \frac{\partial \left( HC \right)}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial y} \left( K\_h \frac{\partial \left( HC \right)}{\partial y} \right) + \text{source or sink} \tag{6}$$

where *C* is the mean concentration in the water column; H= + h is the total depth; and *Kh* is the horizontal scalar diffusivity assumed as 0.1 m2 day-1 (Chapra, 1997). Equation 6 was applied to model total phosphorus, total nitrogen and phytoplankton. All these equations are solved dynamically, using a simple numerical semi-implicit central finite differences scheme (Gross et al., 1999a; 1999b) (Fig. 2). Thus, the mass balances involving phytoplankton and nutrients can be written as:

$$\frac{\partial \left(Ha\right)}{\partial t} + \frac{\partial \left(uHa\right)}{\partial \mathbf{x}} + \frac{\partial \left(vHa\right)}{\partial y} = \mu\_{\text{eff}} \cdot \text{Ha} + \frac{\partial}{\partial \mathbf{x}} \left(K\_h \frac{\partial \left(Ha\right)}{\partial \mathbf{x}}\right) + \frac{\partial}{\partial y} \left(K\_h \frac{\partial \left(Ha\right)}{\partial y}\right) + \text{inlet/outlet} \tag{7}$$

$$\frac{\partial \left(H\mathbf{n}\right)}{\partial t} + \frac{\partial \left(uH\mathbf{n}\right)}{\partial \mathbf{x}} + \frac{\partial \left(vH\mathbf{n}\right)}{\partial y} = -a\_{nu}\mu\_{\text{eff}}H\mathbf{a} + \frac{\partial}{\partial \mathbf{x}}\left(K\_{h}\frac{\partial \left(H\mathbf{n}\right)}{\partial \mathbf{x}}\right) + \frac{\partial}{\partial y}\left(K\_{h}\frac{\partial \left(H\mathbf{n}\right)}{\partial y}\right) + \text{inlet/outlet (8)}$$

$$\frac{\partial \left(Hp\right)}{\partial t} + \frac{\partial \left(uHp\right)}{\partial \mathbf{x}} + \frac{\partial \left(vHp\right)}{\partial y} = -a\_{pu}\mu\_{vg}Ha - k\_{phs}p + \frac{\partial}{\partial \mathbf{x}} \left(K\_h \frac{\partial \left(Hp\right)}{\partial \mathbf{x}}\right) + \frac{\partial}{\partial y} \left(K\_h \frac{\partial \left(Hp\right)}{\partial y}\right) + \tag{9}$$
 
$$\text{ + inlet / outlet}$$

where *a*, *n* and *p* are the chlorophyll-a, total nitrogen and total phosphorus concentrations, respectively; *ana* is the N/Chla ratio equal to 8 mg N mg Chla-1; *apa* is the P/Chla ratio equal to 1.5 mg N mg Chla-1, inlet/outlet represents the balance between all inlets and outlets in a control volume *∂x ∂y ∂z*; and kphos is the settling coefficient.

The effective growth rate itself is not a simple constant, but varies in response to environmental factors such as temperature, nutrients, respiration, excretion and grazing by zooplankton:

$$
\mu\_{\preccurlyeq} = \mu\_{\mathbb{P}}\left(T, \ N, I\right)a \ -\mu\_{\mathbb{L}}a \tag{10}
$$

where *μP(T, N, I)* is the primary production rate as a function of temperature (*T*), nutrients (*N*), and light (*I*); *μL* is the loss rate due to respiration, excretion, and grazing by zooplankton; and *a* is the chlorophyll-a concentration.

The hydrodynamic module was calibrated by tuning the model parameters within their observed ranges taken from the literature (Table 1). Nonetheless, the hydraulic resistance caused by the presence of emerged macrophytes in the Taim Wetland was represented by a

Hydrodynamic Control of Plankton Spatial and

**3. Modeling results** 

for daily and seasonal periods.

Três Forquilhas

River

**3.1 Itapeva Lake** 

Temporal Heterogeneity in Subtropical Shallow Lakes 35

In Itapeva Lake, wind action generated oscillations of the water level between the North and South parts of the lake. The meteorological and hydrological variables were characterized

Simulations using a mathematical bidimensional horizontal hydrodynamic model succeeded in reproducing this phenomenon, and helped to calculate the synthesis of velocity and direction of the water. It was possible to confirm the complexity of the circulation in the lake and to distinguish different behaviors among the South, Center and North. Besides the similarities in morphometry between the Center and South parts, the flow from the Três Forquilhas River enters the center part of the lake and the prevailing N-

0 5.5 11.0 cm/s

Fig. 4. Numerical simulation of the vertically averaged velocity field in Itapeva Lake during the combination of high-flow condition in the Três Forquilhas River and low wind speed.

The hydrological variables analyzed and modeled showed a quite characteristic seasonal behavior at each sampling point in Itapeva Lake, closely related to the velocity and direction of the wind. The water level responded to wind action in a very direct manner, since NE winds displaced water from north to south, along the main lake axis. Winds from the SW

The environmental sources selected for the analysis were suspended solids and turbidity, due to their influence on many physical, chemical and biological factors. The results for hydrodynamics, such as water column and water velocity generated by the model for the sampling points, were used as the basis for the study of the environmental variations. The waves generated by wind action were the third source used to explain the variations of

Red arrows indicate the direction of the prevailing currents.

suspended solids and turbidity in the lake (Figs 6 and Table 2).

quadrant produced the opposite effect.

NE winds move water toward the South part of the lake (Figs 4 and 5).

smaller Chezy's resistance factor than was used in other lake areas (Wu et al., 1999). Calibration and validation of the hydrodynamic parameters were done using two different time-series of water level and wind produced for two locations in Lake Mangueira (North and South).


Sources: 1 White (1974); 2 Wu (1982); 3 Chow (1959); 4 Jørgensen (1994); 5 Schladow & Hamilton (1997); 6 Eppley (1972); 7 Lucas (1997); 8 Chapra (1997)

Table 1. Hydrodynamic and biological parameters description and its values range.

For the parameters of the phytoplankton module, we used the mean values for the literature range given in Table 1. To evaluate its performance, we simulated another period of 86 days, starting 12/22/2002 at 00:00 hs (summer). Solar radiation and water temperature data were taken from the TAMAN meteorological station, situated in the northern part of Lake Mangueira. Photosynthetically active radiation (PAR) at the Taim wetland was assigned as 20% of the total radiation, in order to represent the indirect effect of the emergent macrophytes on the phytoplankton growth rate according to experimental studies of emergent vegetation stands in situ. For the lake areas, we assumed that the percentage of PAR was 50% of the total solar radiation (Janse, 2005).

The resulting phytoplankton patterns were compared with satellite images from MODIS, which provides improved chlorophyll-a measurement capabilities over previous satellite sensors. For instance, MODIS can better measure the concentration of chlorophyll-a associated with a given phytoplankton bloom. Unfortunately, there were no detailed chlorophyll-a and nutrient data available for the same period. Therefore, we compared only the median simulated values with field data from another period (2001 and 2002).
