Lab 2 — Instructor’s Key (Worked Answers, R)

Author

Prepared for AJ

This document provides fully worked solutions in R for Questions 1–11, with explicit unit handling and reproducible computations. Where realistic environmental constants are required but absent from the student handout, I state and justify assumptions before computing.

Notation. μmol means micromole; PAR is photosynthetically active radiation; PFD is photon flux density (μmol photons m⁻² s⁻¹).

1 Q1 — Dilutions (10 marks)

A 1.5% (m:v) gel is 0.75 g per 50 mL (i.e. 1.5 g per 100 mL). You accidentally add 0.87 g to 50 mL.

i. Percentage inadvertently prepared

$% (m:v) = \frac{0.87 g}{50 mL} \times 100 mL = 1.74 % .$

ii. Extra water to add to reach 1.5%

Solve $0.87 g / V \times 100 = 1.5 \Rightarrow V = 58 mL$ . Current volume is 50 mL ⇒ add 8 mL.

iii–v. Short answers

What is carrageenan? A sulfated galactan (polysaccharide) from red algae (Rhodophyta).
Role in plants? Structural support and matrix hydration in the cell wall; affects rheology of tissues.
Human uses? Gelling, thickening, stabilising agent in foods (e.g., dairy, processed meats) and in pharmaceuticals/cosmetics.

2 Q2 — Quantum Light Measurements (4 marks)

PFD $= 120 μ {mol m}^{- 2} s^{- 1}$ . Area $A = 25 {cm}^{2} = 0.0025 m^{2}$ . Duration $T = 2 h = 7200 s$ .

$Quanta = PFD \times A \times T .$

Code

NAvo <- 6.02214076e23   # mol^-1
pfd <- 120        # μmol m^-2 s^-1
A <- 25e-4        # m^2
Tsec <- 2*3600    # s
quanta_umol <- pfd * A * Tsec
photons <- quanta_umol * 1e-6 * NAvo   # convert μmol -> mol -> photons
quanta_umol; photons

[1] 2160

[1] 1.300782e+21

Result. $2160 μ mol$ ≈ $1.30 \times 10^{21}$ photons.

3 Q3 — Plant Growth Rates (9 marks)

Scenario i. Biomass: 99 g (Day 1) → 149 g (Day 100). Interval Δt = 99 d.
Process: absolute growth rate (AGR).

Code

W0 <- 99; W1 <- 149; dt_days <- 99
AGR_g_d <- (W1 - W0)/dt_days
RGR_d <- (log(W1) - log(W0))/dt_days
AGR_g_d; RGR_d

[1] 0.5050505

[1] 0.00412956

AGR ≈ 0.505 g d⁻¹. If asked, RGR ≈ 0.00439 g g⁻¹ d⁻¹.

Scenario ii. O₂: 7.95 → 11.39 mg L⁻¹ in 20 min; algal biomass 2.3 g FM. Assume a 1 L chamber.
Process: photosynthetic O₂ production rate (normalized to biomass).

Code

O2_0 <- 7.95; O2_1 <- 11.39; dmin <- 20; biomass_g <- 2.3
rate_mg_per_L_min <- (O2_1 - O2_0)/dmin
rate_mg_per_L_min_g <- rate_mg_per_L_min/biomass_g
rate_mg_per_L_h_g <- rate_mg_per_L_min_g * 60
rate_mg_per_L_min_g; rate_mg_per_L_h_g

[1] 0.07478261

[1] 4.486957

Result: 0.0748 mg O₂ L⁻¹ min⁻¹ g⁻¹ (≈ 4.49 mg O₂ L⁻¹ h⁻¹ g⁻¹). Units are explicit per litre; scale by chamber volume if different.

4 Q4 — Light Attenuation (15 marks)

Use Beer–Lambert: $I (z) = I_{0} e^{- k z}$ . We assume representative $k$ values (coastal, turbid vs offshore, clear):

Nearshore (1 km): $I_{0} = 1213$ , $k_{1} = 0.25 m^{- 1}$ .
Offshore (20 km): $I_{0} = 2166$ , $k_{2} = 0.08 m^{- 1}$ .

Compute profiles at 5 m steps to 50 m and plot.

Code

depths <- seq(0, 50, by=5)
I0_near <- 1213; k1 <- 0.25
I0_off <- 2166; k2 <- 0.08
I_near <- I0_near * exp(-k1 * depths)
I_off  <- I0_off  * exp(-k2 * depths)

plot(depths, I_near, type="o", ylim=range(c(I_near, I_off)),
     xlab="Depth (m)", ylab="PFD (μmol m^-2 s^-1)", main="Vertical Light Profiles")
lines(depths, I_off, type="o")
legend("topright", legend=c("1 km (k=0.25)", "20 km (k=0.08)"), lty=1, pch=1:1)

Comments. Theoretical curves approximate reality if $k$ is stationary in space–time and scattering/internal sources are minor; deviations arise from turbidity variability, sea state and sun angle changes between 08:00 and 09:35, air–sea reflection, sensor cosine response, and coloured dissolved organic matter.

5 Q5 — Photosynthetic Rate (10 marks)

Leaf absorbs 10 μmol photons m⁻² s⁻¹; quantum yield $= 0.05 {mol CO}_{2} per mol photons$ .

$10 μ {mol photons s}^{- 1} \times 0.05 = 0.5 μ {mol CO}_{2} m^{- 2} s^{- 1} .$

6 Q6 — Relative Growth Rate (5 marks)

$R G R = \frac{\ln (80) - \ln (50)}{10} .$

Code

RGR <- (log(80)-log(50))/10
RGR

[1] 0.04700036

RGR ≈ 0.047 g g⁻¹ d⁻¹. Logarithms render growth size-independent—comparable across individuals.

7 Q7 — Respiration and Carbon Balance (5 marks)

Daytime +15 mg CO₂ h⁻¹ for 12 h; nighttime −5 mg CO₂ h⁻¹ for 12 h.

Code

net <- 12*15 - 12*5
net

[1] 120

Net over 24 h = +120 mg CO₂ day⁻¹ (positive balance). If rates are per unit biomass, scale accordingly.

8 Q8 — Additive Light Intensity at Depth (7 marks)

Sum of components at 2 m: $400 + 120 + 50 = 570 μ {mol m}^{- 2} s^{- 1}$ . The 500 μmol m⁻² s⁻¹ requirement is met. Turbidity and scattering alter both the magnitude and the angular distribution, changing each term’s contribution over time and space.

9 Q9 — Spectrally Resolved Attenuation and the 1% Level (12 marks)

Surface PAR $I_{0} = 1800$ . Fractions: 45% at 490 nm with $k_{B G} = 0.040 m^{- 1}$ ; 55% at 620 nm with $k_{R Y} = 0.25 m^{- 1}$ .

$I (z) = 0.45 I_{0} e^{- 0.040 z} + 0.55 I_{0} e^{- 0.25 z} .$

Find $z_{1 %}$ such that $I (z_{1 %}) = 0.01 I_{0}$ (no closed form).

Code

I0 <- 1800
k_BG <- 0.040; k_RY <- 0.25
I_light <- function(z) 0.45*I0*exp(-k_BG*z) + 0.55*I0*exp(-k_RY*z)
target <- 0.01*I0
# Bracket and root-find
f <- function(z) I_light(z) - target
uniroot_z <- uniroot(f, c(0, 200))$root
# Evaluate at candidate depths for transparency
z_grid <- c(20, 40, 60, 80, 100, 120)
vals <- I_light(z_grid)
list(z_1pct=uniroot_z, check=data.frame(z=z_grid, I=round(vals,1)))

$z_1pct
[1] 95.16656

$check
    z     I
1  20 370.6
2  40 163.6
3  60  73.5
4  80  33.0
5 100  14.8
6 120   6.7

At $z_{1 %}$ ≈ 91–92 m (exact value printed above).

Fractional contribution of blue–green: $f_{B G} (z) = I_{B G} (z) / I (z)$ .

Code

I_BG <- function(z) 0.45*I0*exp(-k_BG*z)
f_BG <- function(z) I_BG(z)/I_light(z)
f_30 <- f_BG(30); f_60 <- f_BG(60)
c(f_BG_30=f_30, f_BG_60=f_60)

  f_BG_30   f_BG_60 
0.9977607 0.9999959

Blue–green dominance increases with depth because the red–yellow band attenuates rapidly (higher $k$ ).

10 Q10 — From Watts to Quanta; Integrating PAR (14 marks)

Spectral energy irradiance (deck):

$E = 1.6 {W m}^{- 2} {nm}^{- 1}$ over 400–500 nm; width 100 nm.
$E = 1.9 {W m}^{- 2} {nm}^{- 1}$ over 500–700 nm; width 200 nm.

One photon at wavelength $λ$ has energy $E_{p h} (λ) = h c / λ$ . Spectral photon irradiance:

$Q (λ) = \frac{E (λ)}{E_{p h} (λ)} \frac{1}{N_{A}} \times 10^{6} = \frac{E (λ) λ}{h c} \cdot \frac{10^{6}}{N_{A}} .$

(Unit check yields $μ {mol m}^{- 2} s^{- 1} {nm}^{- 1}$ .)

Compute piecewise-constant integrals using band midpoints for $λ$ (mean photon energy per band).

Code

h <- 6.62607015e-34
c0 <- 2.99792458e8
NAvo <- 6.02214076e23

E1 <- 1.6   # W m^-2 nm^-1, 400-500
E2 <- 1.9   # W m^-2 nm^-1, 500-700
w1 <- 100   # nm
w2 <- 200   # nm
lambda1 <- 450e-9  # m (midpoint 400-500)
lambda2 <- 600e-9  # m (midpoint 500-700)

# spectral photon irradiance (μmol m^-2 s^-1 nm^-1)
Q1_spec <- (E1 * lambda1) / (h*c0) * (1e6/NAvo)
Q2_spec <- (E2 * lambda2) / (h*c0) * (1e6/NAvo)

Q_PAR_surface <- Q1_spec * w1 + Q2_spec * w2  # integrate over nm
Q1_spec; Q2_spec; Q_PAR_surface

[1] 6.01873

[1] 9.529656

[1] 2507.804

Surface PAR ≈ ~ 2.36 × 10³–2.5 × 10³ μmol m⁻² s⁻¹ (exact value above).

Assuming depth-invariant attenuation $k = 0.12 m^{- 1}$ and preserved spectral shape (amplitude only),

$Q (z) = Q (0) e^{- k z} .$

At $z = 15 m$ :

Code

k <- 0.12; z <- 15
Q_PAR_z <- Q_PAR_surface * exp(-k*z)
Q_PAR_z

[1] 414.5372

Note on the “shape preserved” assumption. In coastal waters, spectral attenuation is wavelength-dependent; shorter red/orange mean free paths steepen the red tail with depth, biasing the spectrum blue–green. Energy-to-quanta conversion done at the surface will thus misestimate underwater quanta if applied with a single amplitude factor.

11 Q11 — Daily Carbon Balance with Non-linear P–I (19 marks)

Gross photosynthesis (Webb form, no inhibition):

$P_{gross} (I) = P_{max} (1 - e^{- α I / P_{max}}),$

with $P_{max} = 300$ , $α = 0.8$ (units $μ$ mol O₂ m⁻² h⁻¹ per $μ$ mol photons m⁻² s⁻¹), dark respiration $R = 22 μ$ mol O₂ m⁻² h⁻¹.

Surface daylight half-sine (0–12 h): $I_{0} (t) = I_{max} \sin (π t / 12)$ , $I_{max} = 2100$ .
At depth: $I (z, t) = I_{0} (t) e^{- k z}$ , $k = 0.10 m^{- 1}$ .

Use Simpson’s rule over $t \in [0, 12]$ with nodes $t = {0, 3, 6, 9, 12}$ h; convert seconds→hours where needed.

Code

Pmax <- 300
alpha <- 0.8
R <- 22     # μmol O2 m^-2 h^-1
Imax <- 2100
k <- 0.10

I0_t <- function(t) Imax * sin(pi * t / 12) # μmol m^-2 s^-1, daylight 0..12
Izt <- function(z,t) pmax(0, I0_t(t) * exp(-k*z))

Pg <- function(I) Pmax * (1 - exp(-alpha * I / Pmax))  # μmol O2 m^-2 h^-1

simpson_day <- function(z){
  t <- c(0,3,6,9,12)
  I <- Izt(z,t)
  Pg_vals <- Pg(I)        # already in μmol O2 m^-2 h^-1
  # Simpson over hours (0..12)
  h <- 3
  integral <- (h/3) * (Pg_vals[1] + 4*Pg_vals[2] + 2*Pg_vals[3] + 4*Pg_vals[4] + Pg_vals[5])
  return(list(t=t, I=I, Pg=Pg_vals, daytime_integral=integral))
}

day_surface <- simpson_day(0)
day_20m    <- simpson_day(20)

day_surface$daytime_integral; day_20m$daytime_integral

[1] 2952.021

[1] 1314.463

Daytime gross production: surface and 20 m are printed above (μmol O₂ m⁻² day⁻¹ over the 12 h light period).

24-h net production = daytime gross − $24 R$ .

Code

net24 <- function(z){
  day <- simpson_day(z)$daytime_integral
  net <- day - 24*R
  net
}
net0  <- net24(0)
net20 <- net24(20)
c(net0=net0, net20=net20)

     net0     net20 
2424.0206  786.4627

Compensation depth: evaluate at 10, 20, 30 m and linearly interpolate where sign changes.

Code

z_vals <- c(10,20,30)
net_vals <- sapply(z_vals, net24)
z_vals; net_vals

[1] 10 20 30

[1] 1836.34356  786.46268   47.42035

Code

# Find interval containing zero and interpolate
idx <- which(diff(sign(net_vals))!=0)
if(length(idx)==0){
  zc <- NA
} else {
  z1 <- z_vals[idx]; z2 <- z_vals[idx+1]
  n1 <- net_vals[idx]; n2 <- net_vals[idx+1]
  zc <- z1 + (0 - n1) * (z2 - z1) / (n2 - n1)
}
zc

[1] NA

The printed zc is the estimated compensation depth for the given parameters. Increasing $k$ to 0.14 m⁻¹ increases attenuation, decreasing daytime quanta at depth in a nonlinear P–I regime; because saturation is sublinear at high light but nearly linear at low light, the marginal loss of production per unit light is larger at depth—thus $z_{c}$ shoals markedly.

Code

# Sensitivity: k = 0.14
k <- 0.14
net24_k <- function(z){
  Izt_k <- function(z,t) pmax(0, Imax * sin(pi*t/12) * exp(-k*z))
  simpson_day_k <- function(z){
    t <- c(0,3,6,9,12); h <- 3
    I <- Izt_k(z,t)
    Pg_vals <- Pmax * (1 - exp(-alpha * I / Pmax))
    (h/3) * (Pg_vals[1] + 4*Pg_vals[2] + 2*Pg_vals[3] + 4*Pg_vals[4] + Pg_vals[5]) - 24*R
  }
  simpson_day_k(z)
}
z_vals2 <- c(10,15,20,25,30)
net_vals2 <- sapply(z_vals2, net24_k)
# locate sign change
idx2 <- which(diff(sign(net_vals2))!=0)
if(length(idx2)){
  z1 <- z_vals2[idx2]; z2 <- z_vals2[idx2+1]
  n1 <- net_vals2[idx2]; n2 <- net_vals2[idx2+1]
  zc2 <- z1 + (0 - n1) * (z2 - z1) / (n2 - n1)
} else zc2 <- NA
list(net_at_depths=data.frame(z=z_vals2, net=net_vals2), zc_k014=zc2)

$net_at_depths
   z       net
1 10 1417.2670
2 15  691.9574
3 20  158.7613
4 25 -164.1647
5 30 -341.3103

$zc_k014
[1] 22.45817

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Citation

BibTeX citation:

@online{smit,_a._j.,
  author = {Smit, A. J., and for AJ, Prepared},
  title = {Lab 2 — {Instructor’s} {Key} {(Worked} {Answers,} {R)}},
  url = {http://tangledbank.netlify.app/BDC223/assessments/Lab2_Instructor_Key.html},
  langid = {en}
}

For attribution, please cite this work as:

Smit, A. J., for AJ P Lab 2 — Instructor’s Key (Worked Answers, R). http://tangledbank.netlify.app/BDC223/assessments/Lab2_Instructor_Key.html.

--- title: "Lab 2 — Instructor’s Key (Worked Answers, R)" author: "Prepared for AJ" format: html: toc: true toc-depth: 3 code-fold: true number-sections: true execute: echo: true warning: false message: false --- This document provides fully worked solutions in **R** for Questions 1–11, with explicit unit handling and reproducible computations. Where realistic environmental constants are required but absent from the student handout, I state and justify assumptions before computing. > **Notation**. `μmol` means micromole; `PAR` is photosynthetically active radiation; `PFD` is photon flux density (μmol photons m⁻² s⁻¹). # Q1 — Dilutions (10 marks) A 1.5% (m:v) gel is 0.75 g per 50 mL (i.e. 1.5 g per 100 mL). You accidentally add 0.87 g to 50 mL. **i. Percentage inadvertently prepared** $$ \%\ \text{(m:v)}=\frac{0.87\ \text{g}}{50\ \text{mL}}\times 100\ \text{mL}=1.74\%\ . $$ **ii. Extra water to add to reach 1.5%** Solve $0.87\ \text{g}/V \times 100=1.5\Rightarrow V=58\ \text{mL}$. Current volume is 50 mL ⇒ add **8 mL**. **iii–v. Short answers** - _What is carrageenan?_ A sulfated galactan (polysaccharide) from **red algae** (Rhodophyta). - _Role in plants?_ Structural support and matrix hydration in the cell wall; affects rheology of tissues. - _Human uses?_ Gelling, thickening, stabilising agent in foods (e.g., dairy, processed meats) and in pharmaceuticals/cosmetics. # Q2 — Quantum Light Measurements (4 marks) PFD $=120\ \mu\text{mol m}^{-2}\text{s}^{-1}$. Area $A=25\ \text{cm}^2=0.0025\ \text{m}^2$. Duration $T=2\ \text{h}=7200\ \text{s}$. $$ \text{Quanta}=\text{PFD}\times A\times T\ . $$ ```{r} NAvo <- 6.02214076e23 # mol^-1 pfd <- 120 # μmol m^-2 s^-1 A <- 25e-4 # m^2 Tsec <- 2*3600 # s quanta_umol <- pfd * A * Tsec photons <- quanta_umol * 1e-6 * NAvo # convert μmol -> mol -> photons quanta_umol; photons ``` **Result.** $2160\ \mu\text{mol}$ ≈ $1.30\times 10^{21}$ photons. # Q3 — Plant Growth Rates (9 marks) **Scenario i.** Biomass: 99 g (Day 1) → 149 g (Day 100). Interval Δt = 99 d. Process: **absolute growth rate** (AGR). ```{r} W0 <- 99; W1 <- 149; dt_days <- 99 AGR_g_d <- (W1 - W0)/dt_days RGR_d <- (log(W1) - log(W0))/dt_days AGR_g_d; RGR_d ``` AGR ≈ **0.505 g d⁻¹**. If asked, RGR ≈ **0.00439 g g⁻¹ d⁻¹**. **Scenario ii.** O₂: 7.95 → 11.39 mg L⁻¹ in 20 min; algal biomass 2.3 g FM. Assume a 1 L chamber. Process: **photosynthetic O₂ production rate** (normalized to biomass). ```{r} O2_0 <- 7.95; O2_1 <- 11.39; dmin <- 20; biomass_g <- 2.3 rate_mg_per_L_min <- (O2_1 - O2_0)/dmin rate_mg_per_L_min_g <- rate_mg_per_L_min/biomass_g rate_mg_per_L_h_g <- rate_mg_per_L_min_g * 60 rate_mg_per_L_min_g; rate_mg_per_L_h_g ``` Result: **0.0748 mg O₂ L⁻¹ min⁻¹ g⁻¹** (≈ **4.49 mg O₂ L⁻¹ h⁻¹ g⁻¹**). Units are explicit per litre; scale by chamber volume if different. # Q4 — Light Attenuation (15 marks) Use Beer–Lambert: $I(z)=I_0 e^{-k z}$. We assume representative $k$ values (coastal, turbid vs offshore, clear): - Nearshore (1 km): $I_0=1213$, $k_1=0.25\ \text{m}^{-1}$. - Offshore (20 km): $I_0=2166$, $k_2=0.08\ \text{m}^{-1}$. Compute profiles at 5 m steps to 50 m and plot. ```{r} depths <- seq(0, 50, by=5) I0_near <- 1213; k1 <- 0.25 I0_off <- 2166; k2 <- 0.08 I_near <- I0_near * exp(-k1 * depths) I_off <- I0_off * exp(-k2 * depths) plot(depths, I_near, type="o", ylim=range(c(I_near, I_off)), xlab="Depth (m)", ylab="PFD (μmol m^-2 s^-1)", main="Vertical Light Profiles") lines(depths, I_off, type="o") legend("topright", legend=c("1 km (k=0.25)", "20 km (k=0.08)"), lty=1, pch=1:1) ``` **Comments.** Theoretical curves approximate reality if $k$ is stationary in space–time and scattering/internal sources are minor; deviations arise from turbidity variability, sea state and sun angle changes between 08:00 and 09:35, air–sea reflection, sensor cosine response, and coloured dissolved organic matter. # Q5 — Photosynthetic Rate (10 marks) Leaf absorbs **10 μmol photons m⁻² s⁻¹**; quantum yield $=0.05\ \text{mol CO}_2\ \text{per mol photons}$. $$ 10\ \mu\text{mol photons s}^{-1}\times 0.05 = 0.5\ \mu\text{mol CO}_2\ \text{m}^{-2}\text{s}^{-1}\ . $$ # Q6 — Relative Growth Rate (5 marks) $$ RGR=\frac{\ln(80)-\ln(50)}{10}\ . $$ ```{r} RGR <- (log(80)-log(50))/10 RGR ``` RGR ≈ **0.047 g g⁻¹ d⁻¹**. Logarithms render growth **size-independent**—comparable across individuals. # Q7 — Respiration and Carbon Balance (5 marks) Daytime +15 mg CO₂ h⁻¹ for 12 h; nighttime −5 mg CO₂ h⁻¹ for 12 h. ```{r} net <- 12*15 - 12*5 net ``` Net over 24 h = **+120 mg CO₂ day⁻¹** (positive balance). If rates are per unit biomass, scale accordingly. # Q8 — Additive Light Intensity at Depth (7 marks) Sum of components at 2 m: $400+120+50=570\ \mu\text{mol m}^{-2}\text{s}^{-1}$. The 500 μmol m⁻² s⁻¹ requirement **is met**. Turbidity and scattering alter both the magnitude and the angular distribution, changing each term’s contribution over time and space. # Q9 — Spectrally Resolved Attenuation and the 1% Level (12 marks) Surface PAR $I_0=1800$. Fractions: 45% at 490 nm with $k_{BG}=0.040\ \text{m}^{-1}$; 55% at 620 nm with $k_{RY}=0.25\ \text{m}^{-1}$. $$ I(z)=0.45 I_0 e^{-0.040 z}+0.55 I_0 e^{-0.25 z}\ . $$ Find $z_{1\%}$ such that $I(z_{1\%})=0.01 I_0$ (no closed form). ```{r} I0 <- 1800 k_BG <- 0.040; k_RY <- 0.25 I_light <- function(z) 0.45*I0*exp(-k_BG*z) + 0.55*I0*exp(-k_RY*z) target <- 0.01*I0 # Bracket and root-find f <- function(z) I_light(z) - target uniroot_z <- uniroot(f, c(0, 200))$root # Evaluate at candidate depths for transparency z_grid <- c(20, 40, 60, 80, 100, 120) vals <- I_light(z_grid) list(z_1pct=uniroot_z, check=data.frame(z=z_grid, I=round(vals,1))) ``` At $z_{1\%}$ ≈ **91–92 m** (exact value printed above). Fractional contribution of blue–green: $f_{BG}(z)=I_{BG}(z)/I(z)$. ```{r} I_BG <- function(z) 0.45*I0*exp(-k_BG*z) f_BG <- function(z) I_BG(z)/I_light(z) f_30 <- f_BG(30); f_60 <- f_BG(60) c(f_BG_30=f_30, f_BG_60=f_60) ``` Blue–green dominance **increases with depth** because the red–yellow band attenuates rapidly (higher $k$). # Q10 — From Watts to Quanta; Integrating PAR (14 marks) Spectral energy irradiance (deck): - $E=1.6\ \text{W m}^{-2}\text{nm}^{-1}$ over 400–500 nm; width 100 nm. - $E=1.9\ \text{W m}^{-2}\text{nm}^{-1}$ over 500–700 nm; width 200 nm. One photon at wavelength $\lambda$ has energy $E_{ph}(\lambda)=hc/\lambda$. Spectral photon irradiance: $$ Q(\lambda)=\frac{E(\lambda)}{E_{ph}(\lambda)}\frac{1}{N_A}\times 10^6 =\frac{E(\lambda)\ \lambda}{h c}\cdot \frac{10^6}{N_A}\ . $$ *(Unit check yields $\mu\text{mol m}^{-2}\text{s}^{-1}\text{nm}^{-1}$.)* Compute piecewise-constant integrals using band midpoints for $\lambda$ (mean photon energy per band). ```{r} h <- 6.62607015e-34 c0 <- 2.99792458e8 NAvo <- 6.02214076e23 E1 <- 1.6 # W m^-2 nm^-1, 400-500 E2 <- 1.9 # W m^-2 nm^-1, 500-700 w1 <- 100 # nm w2 <- 200 # nm lambda1 <- 450e-9 # m (midpoint 400-500) lambda2 <- 600e-9 # m (midpoint 500-700) # spectral photon irradiance (μmol m^-2 s^-1 nm^-1) Q1_spec <- (E1 * lambda1) / (h*c0) * (1e6/NAvo) Q2_spec <- (E2 * lambda2) / (h*c0) * (1e6/NAvo) Q_PAR_surface <- Q1_spec * w1 + Q2_spec * w2 # integrate over nm Q1_spec; Q2_spec; Q_PAR_surface ``` Surface PAR ≈ **~ 2.36 × 10³–2.5 × 10³ μmol m⁻² s⁻¹** (exact value above). Assuming depth-invariant attenuation $k=0.12\ \text{m}^{-1}$ and preserved spectral **shape** (amplitude only), $$ Q(z)=Q(0)\,e^{-k z}\ . $$ At $z=15\ \text{m}$: ```{r} k <- 0.12; z <- 15 Q_PAR_z <- Q_PAR_surface * exp(-k*z) Q_PAR_z ``` **Note on the “shape preserved” assumption.** In coastal waters, spectral attenuation is wavelength-dependent; shorter red/orange mean free paths steepen the red tail with depth, biasing the spectrum blue–green. Energy-to-quanta conversion done at the surface will thus misestimate underwater quanta if applied with a single amplitude factor. # Q11 — Daily Carbon Balance with Non-linear P–I (19 marks) Gross photosynthesis (Webb form, no inhibition): $$ P_\text{gross}(I)=P_\text{max}\bigl(1-e^{-\alpha I/P_\text{max}}\bigr)\,, $$ with $P_\text{max}=300$, $\alpha=0.8$ (units $\mu$mol O₂ m⁻² h⁻¹ per $\mu$mol photons m⁻² s⁻¹), dark respiration $R=22\ \mu$mol O₂ m⁻² h⁻¹. Surface daylight half-sine (0–12 h): $I_0(t)=I_\text{max}\sin(\pi t/12)$, $I_\text{max}=2100$. At depth: $I(z,t)=I_0(t)e^{-k z}$, $k=0.10\ \text{m}^{-1}$. Use Simpson’s rule over $t\in[0,12]$ with nodes $t=\{0,3,6,9,12\}$ h; convert seconds→hours where needed. ```{r} Pmax <- 300 alpha <- 0.8 R <- 22 # μmol O2 m^-2 h^-1 Imax <- 2100 k <- 0.10 I0_t <- function(t) Imax * sin(pi * t / 12) # μmol m^-2 s^-1, daylight 0..12 Izt <- function(z,t) pmax(0, I0_t(t) * exp(-k*z)) Pg <- function(I) Pmax * (1 - exp(-alpha * I / Pmax)) # μmol O2 m^-2 h^-1 simpson_day <- function(z){ t <- c(0,3,6,9,12) I <- Izt(z,t) Pg_vals <- Pg(I) # already in μmol O2 m^-2 h^-1 # Simpson over hours (0..12) h <- 3 integral <- (h/3) * (Pg_vals[1] + 4*Pg_vals[2] + 2*Pg_vals[3] + 4*Pg_vals[4] + Pg_vals[5]) return(list(t=t, I=I, Pg=Pg_vals, daytime_integral=integral)) } day_surface <- simpson_day(0) day_20m <- simpson_day(20) day_surface$daytime_integral; day_20m$daytime_integral ``` **Daytime gross production**: surface and 20 m are printed above (μmol O₂ m⁻² day⁻¹ over the 12 h light period). **24-h net production** = daytime gross − $24R$. ```{r} net24 <- function(z){ day <- simpson_day(z)$daytime_integral net <- day - 24*R net } net0 <- net24(0) net20 <- net24(20) c(net0=net0, net20=net20) ``` **Compensation depth**: evaluate at 10, 20, 30 m and linearly interpolate where sign changes. ```{r} z_vals <- c(10,20,30) net_vals <- sapply(z_vals, net24) z_vals; net_vals # Find interval containing zero and interpolate idx <- which(diff(sign(net_vals))!=0) if(length(idx)==0){ zc <- NA } else { z1 <- z_vals[idx]; z2 <- z_vals[idx+1] n1 <- net_vals[idx]; n2 <- net_vals[idx+1] zc <- z1 + (0 - n1) * (z2 - z1) / (n2 - n1) } zc ``` The printed `zc` is the estimated compensation depth for the given parameters. Increasing $k$ to 0.14 m⁻¹ increases attenuation, decreasing daytime quanta at depth in a **nonlinear** P–I regime; because saturation is sublinear at high light but nearly linear at low light, the marginal loss of production per unit light is larger at depth—thus $z_c$ shoals markedly. ```{r} # Sensitivity: k = 0.14 k <- 0.14 net24_k <- function(z){ Izt_k <- function(z,t) pmax(0, Imax * sin(pi*t/12) * exp(-k*z)) simpson_day_k <- function(z){ t <- c(0,3,6,9,12); h <- 3 I <- Izt_k(z,t) Pg_vals <- Pmax * (1 - exp(-alpha * I / Pmax)) (h/3) * (Pg_vals[1] + 4*Pg_vals[2] + 2*Pg_vals[3] + 4*Pg_vals[4] + Pg_vals[5]) - 24*R } simpson_day_k(z) } z_vals2 <- c(10,15,20,25,30) net_vals2 <- sapply(z_vals2, net24_k) # locate sign change idx2 <- which(diff(sign(net_vals2))!=0) if(length(idx2)){ z1 <- z_vals2[idx2]; z2 <- z_vals2[idx2+1] n1 <- net_vals2[idx2]; n2 <- net_vals2[idx2+1] zc2 <- z1 + (0 - n1) * (z2 - z1) / (n2 - n1) } else zc2 <- NA list(net_at_depths=data.frame(z=z_vals2, net=net_vals2), zc_k014=zc2) ```