# Algal growth (PhD Thesis, O. Fovet, 2010, p.101)

### Definition

The biomass growth is calculated at time t, in each section x as follows :

$\frac{\partial B}{\partial t}(x,t) = \mu(x,t) B(x,t) F_{lim}(B’(x,t))$

where :

• $F_{lim}(B’(x,t)) = \left ( 1 - \frac{B’(x,t)}{B_{Max}} \right )$
• $\mu(x,t) = \mu_{0} \theta^{T(t)-T_{0}} \frac{I(x,t)}{I_{opt}} e^{1- \frac{I(x,t)}{I_{opt}}} \textup{min} \left ( \frac{N_{i}(x,t)}{N_{i}(x,t) + K_{N_{I}}} \right )$
• $I(x,t) = I_{s}(x,t) e^{-k_{ext} h(x,t)}$
• $I_{s}(x,t) = (1-C_m(x))(1-a)R_N(t)$

### Variables and parameters

• $B(x,t)$ : fix biomass modified by the law (kg m-1)
• $B’(x,t)$ : fix biomass parameter in the law (kg m-1)
• $B_{Max}$ : maximum value of fix biomass (kg m-1)
• $\mu_{0}$ : reference growth rate (s -1)
• $\theta$ : growth coefficient
• $T(x,t)$ : water temperature (°C)
• $T_{0}$ : reference temperature (°C)
• $I(x,t)$ : light intensity at the section’s bottom (W m-2)
• $I_{s}(x,t)$ : solar light intensity (W m-2)
• $C_{m}(x)$ : mask coefficient (cf. Temperature simulation)
• $a$ : albedo (cf. Temperature simulation)
• $R_{N}(t)$ solar radiation (W m-2) (cf. Temperature simulation)
• $k_{ext}$ : extinction coefficient (due to turbidity)
• $h(x,t)$ : mean water level (m)
• $I_{opt}(x,t)$ : optimum light intensity (W m-2)
• $N_{i}(x,t)$ : concentration of nutrient i (kg m-3)
• $K_{N_{i}}$ : half-saturation constant of nutrient i (kg m-3)

### Limiting nutrient

This law can take into account up to 3 nutrients for algae growth. If less than three nutrients are used, $K_{N_{i}}$ for unused nutrients should be set to 0.

### Specifications

• Law’s ID : 301
• Number of acting classes: 6
• Number of meteo parameters : 3
• Number of parameters : 9

Acting classes :

1. $B(x,t)$ :the class modified by the law
2. $B’(x,t)$ : the parameter class of the law
3. $T(x,t)$ :Water temperature
4. $N_{1}(x,t)$ : Nutrient 1
5. $N_{2}(x,t)$ : Nutrient 2
6. $N_{3}(x,t)$ : Nutrient 3

Meteo parameters :

• $C_{m}(x)$
• $a$
• $R_{N}(t)$

Parameters

1. $B_{Max}$ : maximum value of fix biomass
2. $\mu_{0}$ : reference growth rate
3. $\theta$ : growth coefficient
4. $T_{0}$ : reference temperature
5. $k_{ext}$ : extinction coefficient
6. $I_{opt}$ : optimum light intensity
7. $K_{N_{1}}$ : half-saturation constant of nutrient 1
8. $K_{N_{2}}$ : half-saturation constant of nutrient 2
9. $K_{N_{3}}$ : half-saturation constant of nutrient 3