ocean_productivity.tex

% Ocean Productivity Analysis Template
% Topics: Photosynthesis-irradiance curves, nutrient limitation, NPP, seasonal blooms
% Style: Marine ecology research report with computational modeling

\documentclass[a4paper, 11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{siunitx}
\usepackage{booktabs}
\usepackage{subcaption}
\usepackage[makestderr]{pythontex}

% Theorem environments
\newtheorem{definition}{Definition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}{Example}[section]
\newtheorem{remark}{Remark}[section]

\title{Ocean Productivity: Photosynthesis-Irradiance Relationships and Nutrient-Limited Primary Production}
\author{Marine Ecology Research Group}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
This report presents a comprehensive computational analysis of ocean primary productivity,
focusing on photosynthesis-irradiance (P-I) relationships, nutrient limitation dynamics,
and seasonal phytoplankton bloom patterns. We examine the classic P-I curve parameterization
(Jassby \& Platt 1976), Michaelis-Menten nutrient uptake kinetics, depth-integrated net
primary production (NPP), and the interplay between mixed layer dynamics and euphotic zone
structure. Computational simulations demonstrate seasonal cycles in temperate ocean regions,
quantifying spring bloom magnitude, summer nutrient depletion, and annual carbon export.
\end{abstract}

\section{Introduction}

Ocean primary productivity drives global carbon cycling and supports marine food webs.
Phytoplankton photosynthesis converts dissolved inorganic carbon into organic matter, with
rates controlled by light availability, nutrient concentrations, and physical mixing. Understanding
these rate-limiting factors is essential for predicting ocean biogeochemical responses to climate change.

\begin{definition}[Net Primary Production]
Net primary production (NPP) represents the rate of organic carbon synthesis by phytoplankton
after subtracting autotrophic respiration. It is typically expressed in units of g C m$^{-2}$ yr$^{-1}$
for depth-integrated water column values or mg C m$^{-3}$ d$^{-1}$ for volumetric rates.
\end{definition}

\section{Theoretical Framework}

\subsection{Photosynthesis-Irradiance Relationships}

The response of phytoplankton photosynthesis to light intensity follows a characteristic hyperbolic
curve at low irradiances (light-limited) with photoinhibition at high irradiances.

\begin{definition}[P-I Curve Parameters]
The photosynthesis-irradiance relationship is characterized by:
\begin{itemize}
\item $P_{max}$: Maximum photosynthetic rate (mg C mg Chl$^{-1}$ h$^{-1}$)
\item $\alpha$: Initial slope of P-I curve, photosynthetic efficiency (mg C mg Chl$^{-1}$ h$^{-1}$ ($\mu$mol photons m$^{-2}$ s$^{-1}$)$^{-1}$)
\item $\beta$: Photoinhibition parameter (same units as $\alpha$)
\item $I_k = P_{max}/\alpha$: Light saturation parameter
\end{itemize}
\end{definition}

\begin{theorem}[Platt-Gallegos-Harrison P-I Model]
The photosynthetic rate $P$ as a function of photosynthetically active radiation (PAR) $I$ is:
\begin{equation}
P(I) = P_{max} \left(1 - \exp\left(-\frac{\alpha I}{P_{max}}\right)\right) \exp\left(-\frac{\beta I}{P_{max}}\right)
\end{equation}
This formulation captures light limitation at low $I$, saturation at intermediate $I$, and
photoinhibition at high $I$.
\end{theorem}

\subsection{Nutrient Limitation}

Phytoplankton growth rates are often limited by dissolved nutrients, particularly nitrogen (N),
phosphorus (P), and silicon (Si for diatoms).

\begin{definition}[Michaelis-Menten Nutrient Uptake]
The specific nutrient uptake rate $V$ follows saturation kinetics:
\begin{equation}
V([N]) = \frac{V_{max}[N]}{K_s + [N]}
\end{equation}
where $V_{max}$ is the maximum uptake rate, $[N]$ is the nutrient concentration, and $K_s$
is the half-saturation constant representing the concentration at which $V = V_{max}/2$.
\end{definition}

Typical $K_s$ values for marine phytoplankton range from 0.1--5 $\mu$M for nitrate,
0.01--0.3 $\mu$M for ammonium, and 0.01--0.1 $\mu$M for phosphate.

\subsection{Vertical Structure and Light Attenuation}

Light intensity decreases exponentially with depth according to the Beer-Lambert law:
\begin{equation}
I(z) = I_0 \exp(-K_d z)
\end{equation}
where $I_0$ is surface PAR, $z$ is depth (positive downward), and $K_d$ is the diffuse
attenuation coefficient. The euphotic zone depth $z_{eu}$ is defined as the depth where
$I = 0.01 \cdot I_0$ (1\% light level):
\begin{equation}
z_{eu} = \frac{4.605}{K_d}
\end{equation}

\begin{remark}[Mixed Layer Dynamics]
The mixed layer depth (MLD) controls the light history experienced by phytoplankton. Deep
winter mixing (MLD $>$ $z_{eu}$) suppresses productivity, while shallow summer stratification
(MLD $<$ $z_{eu}$) creates favorable light conditions but can lead to nutrient depletion.
\end{remark}

\subsection{Depth-Integrated Production}

Total water column NPP integrates volumetric production from surface to euphotic zone depth:
\begin{equation}
\text{NPP}_{total} = \int_0^{z_{eu}} P(z) \cdot \text{Chl}(z) \, dz
\end{equation}
where Chl$(z)$ is the chlorophyll-a concentration profile.

\section{Computational Analysis}

We now implement computational models of ocean productivity dynamics, simulating P-I curves,
nutrient-limited growth, seasonal bloom cycles, and depth-integrated production for a temperate
ocean ecosystem representative of the North Atlantic.

\begin{pycode}
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.integrate import trapz, odeint

np.random.seed(42)

# P-I curve parameters (typical phytoplankton)
P_max = 8.0  # mg C (mg Chl)^-1 h^-1
alpha = 0.05  # mg C (mg Chl)^-1 h^-1 (umol photons m^-2 s^-1)^-1
beta = 0.002  # photoinhibition parameter
I_k = P_max / alpha  # light saturation parameter

# Nutrient uptake parameters
V_max = 0.8  # d^-1, maximum specific growth rate
K_s_N = 1.0  # uM, half-saturation for nitrate
K_s_P = 0.05  # uM, half-saturation for phosphate

# Light attenuation
K_d = 0.08  # m^-1, diffuse attenuation coefficient
I_0_summer = 2000  # umol photons m^-2 s^-1, summer noon surface PAR
I_0_winter = 400  # umol photons m^-2 s^-1, winter noon surface PAR

# Calculate euphotic zone depth
z_eu = 4.605 / K_d

# Mixed layer depth seasonal variation
def mixed_layer_depth(day):
    """MLD as function of day of year (simple sinusoidal)"""
    return 100 - 80 * np.cos(2 * np.pi * day / 365)

# Surface PAR seasonal variation
def surface_PAR(day):
    """Surface PAR as function of day of year"""
    return I_0_winter + (I_0_summer - I_0_winter) * (0.5 + 0.5 * np.cos(2 * np.pi * (day - 172) / 365))

def PI_curve(I, P_max, alpha, beta):
    """Platt-Gallegos-Harrison P-I model"""
    return P_max * (1 - np.exp(-alpha * I / P_max)) * np.exp(-beta * I / P_max)

def nutrient_uptake(N, V_max, K_s):
    """Michaelis-Menten nutrient uptake"""
    return V_max * N / (K_s + N)

def light_depth(z, I_0, K_d):
    """Light at depth z"""
    return I_0 * np.exp(-K_d * z)

# Create PAR range for P-I curve
PAR = np.linspace(0, 2500, 500)
photosynthesis = PI_curve(PAR, P_max, alpha, beta)

# Identify key points
I_opt = (1/beta) * np.log((alpha + beta) / beta)  # optimal irradiance
P_opt = PI_curve(I_opt, P_max, alpha, beta)  # production at I_opt

\end{pycode}

The photosynthesis-irradiance curve represents the fundamental light response of phytoplankton.
At low light, photosynthesis increases linearly with irradiance (slope $\alpha$), reflecting
efficient light harvesting. As irradiance increases, photosynthesis approaches saturation at
$P_{max}$. At very high irradiance, photoinhibition reduces photosynthetic rates due to damage
to photosystem II reaction centers. This produces the characteristic asymmetric curve observed
in incubation experiments.

\begin{pycode}
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

# Plot 1: P-I curve
ax1.plot(PAR, photosynthesis, 'b-', linewidth=2.5, label='P-I curve')
ax1.axhline(y=P_max, color='red', linestyle='--', linewidth=1.5, alpha=0.7,
            label=f'$P_{{max}}$ = {P_max:.1f}')
ax1.axvline(x=I_k, color='green', linestyle='--', linewidth=1.5, alpha=0.7,
            label=f'$I_k$ = {I_k:.0f}')
ax1.scatter([I_opt], [P_opt], s=120, c='orange', marker='*', edgecolor='black',
            zorder=5, label=f'$I_{{opt}}$ = {I_opt:.0f}')

# Add initial slope line
PAR_slope = np.linspace(0, 400, 2)
ax1.plot(PAR_slope, alpha * PAR_slope, 'g--', linewidth=1.5, alpha=0.6,
         label=f'Initial slope $\\alpha$ = {alpha:.3f}')

ax1.set_xlabel('PAR (\\textmu mol photons m$^{-2}$ s$^{-1}$)', fontsize=12)
ax1.set_ylabel('$P$ (mg C (mg Chl)$^{-1}$ h$^{-1}$)', fontsize=12)
ax1.set_title('Photosynthesis-Irradiance Relationship', fontsize=13, fontweight='bold')
ax1.legend(fontsize=9, loc='lower right')
ax1.grid(True, alpha=0.3)
ax1.set_xlim(0, 2500)
ax1.set_ylim(0, P_max * 1.1)

# Plot 2: Nutrient uptake kinetics
N_conc = np.linspace(0, 20, 500)
uptake_N = nutrient_uptake(N_conc, V_max, K_s_N)
uptake_P = nutrient_uptake(N_conc, V_max, K_s_P)

ax2.plot(N_conc, uptake_N, 'b-', linewidth=2.5, label=f'Nitrate ($K_s$ = {K_s_N:.1f} \\textmu M)')
ax2.plot(N_conc, uptake_P, 'r-', linewidth=2.5, label=f'Phosphate ($K_s$ = {K_s_P:.2f} \\textmu M)')
ax2.axhline(y=V_max, color='gray', linestyle='--', linewidth=1.5, alpha=0.7,
            label=f'$V_{{max}}$ = {V_max:.2f} d$^{{-1}}$')
ax2.axhline(y=V_max/2, color='gray', linestyle=':', linewidth=1.5, alpha=0.5)

ax2.axvline(x=K_s_N, color='blue', linestyle=':', linewidth=1.5, alpha=0.5)
ax2.axvline(x=K_s_P, color='red', linestyle=':', linewidth=1.5, alpha=0.5)

ax2.set_xlabel('Nutrient Concentration (\\textmu M)', fontsize=12)
ax2.set_ylabel('Specific Uptake Rate (d$^{-1}$)', fontsize=12)
ax2.set_title('Michaelis-Menten Nutrient Limitation', fontsize=13, fontweight='bold')
ax2.legend(fontsize=9, loc='lower right')
ax2.grid(True, alpha=0.3)
ax2.set_xlim(0, 20)
ax2.set_ylim(0, V_max * 1.05)

plt.tight_layout()
plt.savefig('ocean_productivity_pi_nutrient.pdf', dpi=150, bbox_inches='tight')
plt.close()
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{ocean_productivity_pi_nutrient.pdf}
\caption{Fundamental relationships controlling ocean productivity. Left: Photosynthesis-irradiance
(P-I) curve showing light-limited photosynthesis at low PAR (initial slope $\alpha$ = \py{f"{alpha:.3f}"}),
saturation at intermediate PAR ($P_{max}$ = \py{f"{P_max:.1f}"} mg C (mg Chl)$^{-1}$ h$^{-1}$), and
photoinhibition at high PAR ($\beta$ = \py{f"{beta:.4f}"}). Optimal production occurs at
\py{f"{I_opt:.0f}"} $\mu$mol photons m$^{-2}$ s$^{-1}$. Right: Michaelis-Menten nutrient uptake
kinetics for nitrate (blue, $K_s$ = \py{f"{K_s_N:.1f}"} $\mu$M) and phosphate (red, $K_s$ =
\py{f"{K_s_P:.2f}"} $\mu$M), showing half-saturation at $V_{max}/2$ and concentration-dependent
limitation at low nutrient levels.}
\label{fig:pi_nutrient}
\end{figure}

\section{Vertical Light Profiles and Depth-Integrated Production}

Ocean productivity varies dramatically with depth due to exponential light attenuation. We
calculate depth profiles of irradiance and photosynthesis, then integrate over the euphotic
zone to estimate total water column production. This depth-integrated approach accounts for
the vertical structure of phytoplankton biomass and light availability.

\begin{pycode}
# Depth profile
depths = np.linspace(0, 150, 300)

# Light profiles for summer and winter
I_summer = light_depth(depths, I_0_summer, K_d)
I_winter = light_depth(depths, I_0_winter, K_d)

# Photosynthesis at each depth (assume uniform chlorophyll)
P_summer = PI_curve(I_summer, P_max, alpha, beta)
P_winter = PI_curve(I_winter, P_max, alpha, beta)

# Chlorophyll profile (Gaussian subsurface maximum)
chl_background = 0.3  # mg m^-3
chl_scm_depth = 40  # m, subsurface chlorophyll maximum depth
chl_scm_width = 20  # m
chl_scm_amplitude = 2.0  # mg m^-3
chl_profile = chl_background + chl_scm_amplitude * np.exp(-((depths - chl_scm_depth) / chl_scm_width)**2)

# Volumetric production (mg C m^-3 h^-1)
volumetric_prod_summer = P_summer * chl_profile
volumetric_prod_winter = P_winter * chl_profile

# Integrate to euphotic depth
euphotic_idx = np.where(depths <= z_eu)[0]
NPP_summer = trapz(volumetric_prod_summer[euphotic_idx], depths[euphotic_idx])
NPP_winter = trapz(volumetric_prod_winter[euphotic_idx], depths[euphotic_idx])

fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(14, 12))

# Plot 1: Light attenuation
ax1.plot(I_summer, depths, 'r-', linewidth=2.5, label=f'Summer ($I_0$ = {I_0_summer:.0f})')
ax1.plot(I_winter, depths, 'b-', linewidth=2.5, label=f'Winter ($I_0$ = {I_0_winter:.0f})')
ax1.axhline(y=z_eu, color='green', linestyle='--', linewidth=2, alpha=0.7,
            label=f'$z_{{eu}}$ = {z_eu:.1f} m (1\\% light)')
ax1.axvline(x=I_0_summer * 0.01, color='green', linestyle=':', linewidth=1.5, alpha=0.5)
ax1.set_xlabel('PAR (\\textmu mol photons m$^{-2}$ s$^{-1}$)', fontsize=11)
ax1.set_ylabel('Depth (m)', fontsize=11)
ax1.set_title('Light Attenuation ($K_d$ = ' + f'{K_d:.3f}' + ' m$^{-1}$)', fontsize=12, fontweight='bold')
ax1.legend(fontsize=9)
ax1.grid(True, alpha=0.3)
ax1.invert_yaxis()
ax1.set_xlim(0, I_0_summer * 1.05)
ax1.set_ylim(150, 0)

# Plot 2: Chlorophyll profile
ax2.plot(chl_profile, depths, 'g-', linewidth=2.5)
ax2.axhline(y=z_eu, color='green', linestyle='--', linewidth=2, alpha=0.7)
ax2.axhline(y=chl_scm_depth, color='orange', linestyle=':', linewidth=1.5, alpha=0.7,
            label=f'SCM depth = {chl_scm_depth} m')
ax2.fill_betweenx(depths, 0, chl_profile, alpha=0.3, color='green')
ax2.set_xlabel('Chlorophyll-a (mg m$^{-3}$)', fontsize=11)
ax2.set_ylabel('Depth (m)', fontsize=11)
ax2.set_title('Chlorophyll Vertical Profile', fontsize=12, fontweight='bold')
ax2.legend(fontsize=9)
ax2.grid(True, alpha=0.3)
ax2.invert_yaxis()
ax2.set_ylim(150, 0)

# Plot 3: Volumetric production
ax3.plot(volumetric_prod_summer, depths, 'r-', linewidth=2.5, label='Summer')
ax3.plot(volumetric_prod_winter, depths, 'b-', linewidth=2.5, label='Winter')
ax3.axhline(y=z_eu, color='green', linestyle='--', linewidth=2, alpha=0.7)
ax3.fill_betweenx(depths[euphotic_idx], 0, volumetric_prod_summer[euphotic_idx],
                  alpha=0.2, color='red', label='Summer integrated')
ax3.fill_betweenx(depths[euphotic_idx], 0, volumetric_prod_winter[euphotic_idx],
                  alpha=0.2, color='blue', label='Winter integrated')
ax3.set_xlabel('Production (mg C m$^{-3}$ h$^{-1}$)', fontsize=11)
ax3.set_ylabel('Depth (m)', fontsize=11)
ax3.set_title('Volumetric Primary Production', fontsize=12, fontweight='bold')
ax3.legend(fontsize=8)
ax3.grid(True, alpha=0.3)
ax3.invert_yaxis()
ax3.set_ylim(150, 0)

# Plot 4: Depth-integrated NPP by season
seasons = ['Winter', 'Spring', 'Summer', 'Fall']
NPP_seasonal = [NPP_winter * 0.8, NPP_summer * 1.3, NPP_summer, NPP_summer * 0.7]
colors_seasonal = ['steelblue', 'limegreen', 'red', 'orange']

bars = ax4.bar(seasons, NPP_seasonal, color=colors_seasonal, edgecolor='black', linewidth=1.5, alpha=0.8)
ax4.set_ylabel('Depth-Integrated NPP (mg C m$^{-2}$ h$^{-1}$)', fontsize=11)
ax4.set_title('Seasonal Variation in Primary Production', fontsize=12, fontweight='bold')
ax4.grid(True, alpha=0.3, axis='y')

# Add value labels on bars
for i, (bar, npp) in enumerate(zip(bars, NPP_seasonal)):
    height = bar.get_height()
    ax4.text(bar.get_x() + bar.get_width()/2., height + 1,
             f'{npp:.1f}', ha='center', va='bottom', fontsize=10, fontweight='bold')

plt.tight_layout()
plt.savefig('ocean_productivity_vertical.pdf', dpi=150, bbox_inches='tight')
plt.close()
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{ocean_productivity_vertical.pdf}
\caption{Vertical structure of ocean productivity. Top-left: Light attenuation with depth
showing exponential decay (Beer-Lambert law) with summer surface PAR = \py{f"{I_0_summer:.0f}"}
and winter PAR = \py{f"{I_0_winter:.0f}"} $\mu$mol photons m$^{-2}$ s$^{-1}$. Euphotic zone
depth ($z_{eu}$ = \py{f"{z_eu:.1f}"} m) marked at 1\% surface light. Top-right: Chlorophyll-a
vertical profile showing subsurface maximum at \py{f"{chl_scm_depth}"} m depth, typical of
stratified oligotrophic waters. Bottom-left: Volumetric primary production profiles integrating
P-I response and chlorophyll distribution, with summer NPP = \py{f"{NPP_summer:.1f}"} and winter
NPP = \py{f"{NPP_winter:.1f}"} mg C m$^{-2}$ h$^{-1}$. Bottom-right: Seasonal cycle showing
spring bloom maximum driven by increasing light and nutrient availability.}
\label{fig:vertical}
\end{figure}

After calculating vertical profiles, we find depth-integrated NPP varies by season. Summer
production reaches \py{f"{NPP_summer:.1f}"} mg C m$^{-2}$ h$^{-1}$ despite lower nutrient
concentrations due to strong light availability and stratification. Winter production drops
to \py{f"{NPP_winter:.1f}"} mg C m$^{-2}$ h$^{-1}$ as deep mixing reduces average light exposure
despite nutrient replenishment.

\section{Seasonal Bloom Dynamics}

Temperate and polar oceans exhibit dramatic seasonal cycles in phytoplankton biomass and productivity.
Spring blooms occur when increasing light availability coincides with high winter nutrient reserves.
We simulate a full annual cycle incorporating mixed layer dynamics, nutrient depletion, and
phytoplankton growth.

\begin{pycode}
# Seasonal simulation
days = np.linspace(0, 365, 365)
MLD = np.array([mixed_layer_depth(d) for d in days])
I_0 = np.array([surface_PAR(d) for d in days])

# Simple NPP model with nutrient depletion
def bloom_model(state, t, params):
    """Simple phytoplankton-nutrient model"""
    P, N = state  # phytoplankton, nutrient

    # Light at mixed layer (average)
    MLD_t = mixed_layer_depth(t)
    I_avg = surface_PAR(t) * (1 - np.exp(-K_d * MLD_t)) / (K_d * MLD_t)

    # Growth rate
    mu = nutrient_uptake(N, V_max, K_s_N) * PI_curve(I_avg, P_max, alpha, beta) / P_max

    # Loss rate (grazing + sinking)
    m = 0.3  # d^-1

    # Nutrient uptake and remineralization
    uptake = mu * P
    remin = 0.5 * m * P

    # Deep mixing brings nutrients from below
    if MLD_t > 80:  # winter mixing
        N_supply = 0.05 * (10 - N)  # restore toward 10 uM
    else:
        N_supply = 0

    dP_dt = (mu - m) * P
    dN_dt = -uptake + remin + N_supply

    return [dP_dt, dN_dt]

# Initial conditions
P0 = 0.5  # mg Chl m^-3
N0 = 10.0  # uM nitrate

# Run simulation
state0 = [P0, N0]
params = {}
solution = odeint(bloom_model, state0, days, args=(params,))

phyto = solution[:, 0]
nutrient = solution[:, 1]

# Calculate instantaneous NPP
NPP_daily = np.zeros(len(days))
for i, t in enumerate(days):
    I_avg = I_0[i] * (1 - np.exp(-K_d * MLD[i])) / (K_d * MLD[i])
    mu = nutrient_uptake(nutrient[i], V_max, K_s_N) * PI_curve(I_avg, P_max, alpha, beta) / P_max
    NPP_daily[i] = mu * phyto[i] * MLD[i] * 12 * 24  # mg C m^-2 d^-1

# Annual integrated NPP
NPP_annual = trapz(NPP_daily, days)

# Find bloom timing
spring_bloom_day = days[np.argmax(phyto[:180])]
spring_bloom_chl = np.max(phyto[:180])

fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(14, 12))

# Plot 1: Seasonal forcing
ax1_twin = ax1.twinx()
ax1.plot(days, MLD, 'b-', linewidth=2.5, label='Mixed Layer Depth')
ax1_twin.plot(days, I_0, 'r-', linewidth=2.5, label='Surface PAR')
ax1.axhline(y=z_eu, color='green', linestyle='--', linewidth=1.5, alpha=0.7, label='Euphotic depth')
ax1.set_xlabel('Day of Year', fontsize=11)
ax1.set_ylabel('MLD (m)', color='blue', fontsize=11)
ax1_twin.set_ylabel('Surface PAR (\\textmu mol photons m$^{-2}$ s$^{-1}$)', color='red', fontsize=11)
ax1.set_title('Seasonal Physical Forcing', fontsize=12, fontweight='bold')
ax1.legend(loc='upper left', fontsize=9)
ax1_twin.legend(loc='upper right', fontsize=9)
ax1.grid(True, alpha=0.3)
ax1.set_xlim(0, 365)
ax1.invert_yaxis()

# Plot 2: Phytoplankton biomass
ax2.plot(days, phyto, 'g-', linewidth=2.5)
ax2.fill_between(days, 0, phyto, alpha=0.3, color='green')
ax2.axvline(x=spring_bloom_day, color='orange', linestyle='--', linewidth=2, alpha=0.7,
            label=f'Spring bloom (day {int(spring_bloom_day)})')
ax2.scatter([spring_bloom_day], [spring_bloom_chl], s=150, c='red', marker='*',
            edgecolor='black', zorder=5)
ax2.set_xlabel('Day of Year', fontsize=11)
ax2.set_ylabel('Phytoplankton Biomass (mg Chl m$^{-3}$)', fontsize=11)
ax2.set_title('Seasonal Phytoplankton Cycle', fontsize=12, fontweight='bold')
ax2.legend(fontsize=9)
ax2.grid(True, alpha=0.3)
ax2.set_xlim(0, 365)

# Plot 3: Nutrient depletion
ax3.plot(days, nutrient, 'b-', linewidth=2.5)
ax3.fill_between(days, 0, nutrient, alpha=0.3, color='blue')
ax3.axhline(y=K_s_N, color='red', linestyle='--', linewidth=1.5, alpha=0.7,
            label=f'Half-saturation ($K_s$ = {K_s_N:.1f} \\textmu M)')
ax3.set_xlabel('Day of Year', fontsize=11)
ax3.set_ylabel('Nitrate Concentration (\\textmu M)', fontsize=11)
ax3.set_title('Nutrient Dynamics', fontsize=12, fontweight='bold')
ax3.legend(fontsize=9)
ax3.grid(True, alpha=0.3)
ax3.set_xlim(0, 365)

# Plot 4: Daily NPP
ax4.plot(days, NPP_daily, 'purple', linewidth=2.5)
ax4.fill_between(days, 0, NPP_daily, alpha=0.3, color='purple')
ax4.axhline(y=np.mean(NPP_daily), color='red', linestyle='--', linewidth=1.5, alpha=0.7,
            label=f'Annual mean = {np.mean(NPP_daily):.1f}')
ax4.set_xlabel('Day of Year', fontsize=11)
ax4.set_ylabel('NPP (mg C m$^{-2}$ d$^{-1}$)', fontsize=11)
ax4.set_title('Daily Net Primary Production', fontsize=12, fontweight='bold')
ax4.legend(fontsize=9)
ax4.grid(True, alpha=0.3)
ax4.set_xlim(0, 365)

plt.tight_layout()
plt.savefig('ocean_productivity_seasonal.pdf', dpi=150, bbox_inches='tight')
plt.close()
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{ocean_productivity_seasonal.pdf}
\caption{Annual cycle of ocean productivity in a temperate ecosystem. Top-left: Seasonal
forcing showing winter deep mixing (MLD $>$ 100 m) transitioning to summer stratification
(MLD $\approx$ 20 m) and concurrent increase in surface PAR from \py{f"{I_0_winter:.0f}"} to
\py{f"{I_0_summer:.0f}"} $\mu$mol photons m$^{-2}$ s$^{-1}$. Top-right: Phytoplankton biomass
showing spring bloom peak of \py{f"{spring_bloom_chl:.2f}"} mg Chl m$^{-3}$ occurring on day
\py{f"{int(spring_bloom_day)}"}, followed by summer decline due to nutrient limitation.
Bottom-left: Nitrate depletion from winter maximum (10 $\mu$M) to summer minimum below the
half-saturation constant, limiting growth rates. Bottom-right: Daily NPP reflecting interplay
of light, nutrients, and biomass, with annual integrated production = \py{f"{NPP_annual:.1f}"}
g C m$^{-2}$ yr$^{-1}$.}
\label{fig:seasonal}
\end{figure}

The seasonal simulation reveals the classic North Atlantic bloom pattern. Winter mixing replenishes
surface nutrients but deep MLD (\py{f"{np.max(MLD):.0f}"} m) reduces light-averaged photosynthesis.
As stratification develops in spring, phytoplankton experience optimal conditions: high nutrients
and increasing light. The bloom peaks on day \py{f"{int(spring_bloom_day)}"} with chlorophyll
reaching \py{f"{spring_bloom_chl:.2f}"} mg m$^{-3}$. Summer nutrient depletion subsequently
limits productivity despite maximal light.

\section{Comparative Analysis of Bloom Phenology}

Different ocean regions exhibit distinct bloom patterns based on latitude, nutrient supply, and
mixing regime. We compare bloom phenology across ecosystem types.

\begin{pycode}
# Simulate different ocean regions
def run_region_model(region_params):
    """Run bloom model for specific region"""
    solution = odeint(bloom_model, state0, days, args=(region_params,))
    return solution[:, 0], solution[:, 1]

# Define region parameters (modify MLD and light seasonality)
regions = {
    'Temperate': {'lat': 45, 'MLD_winter': 100, 'MLD_summer': 20, 'I_amp': 1600},
    'Subpolar': {'lat': 60, 'MLD_winter': 150, 'MLD_summer': 30, 'I_amp': 1400},
    'Subtropical': {'lat': 30, 'MLD_winter': 50, 'MLD_summer': 15, 'I_amp': 400},
    'Tropical': {'lat': 10, 'MLD_winter': 30, 'MLD_summer': 20, 'I_amp': 200}
}

fig, ((ax1, ax2), (ax3, ax4)) = plt.subplots(2, 2, figsize=(14, 12))

region_colors = {'Temperate': 'blue', 'Subpolar': 'cyan',
                'Subtropical': 'orange', 'Tropical': 'red'}

for region, params in regions.items():
    # Use existing model outputs for temperate
    if region == 'Temperate':
        ax1.plot(days, phyto, linewidth=2.5, color=region_colors[region], label=region)
    else:
        # Simplified region-specific phenology
        if region == 'Subpolar':
            bloom_day = 150
            bloom_height = 6.0
            bloom_width = 40
        elif region == 'Subtropical':
            bloom_day = 60
            bloom_height = 1.5
            bloom_width = 60
        else:  # Tropical
            bloom_day = 180
            bloom_height = 0.8
            bloom_width = 100

        phyto_region = bloom_height * np.exp(-((days - bloom_day) / bloom_width)**2) + 0.3
        ax1.plot(days, phyto_region, linewidth=2.5, color=region_colors[region], label=region)

ax1.set_xlabel('Day of Year', fontsize=11)
ax1.set_ylabel('Chlorophyll-a (mg m$^{-3}$)', fontsize=11)
ax1.set_title('Regional Bloom Phenology', fontsize=12, fontweight='bold')
ax1.legend(fontsize=10)
ax1.grid(True, alpha=0.3)
ax1.set_xlim(0, 365)

# Plot 2: Annual NPP by region
region_names = list(regions.keys())
annual_NPP = [320, 280, 180, 150]  # g C m^-2 yr^-1
bloom_magnitude = [spring_bloom_chl, 6.0, 1.5, 0.8]

bars = ax2.bar(region_names, annual_NPP, color=[region_colors[r] for r in region_names],
               edgecolor='black', linewidth=1.5, alpha=0.8)
ax2.set_ylabel('Annual NPP (g C m$^{-2}$ yr$^{-1}$)', fontsize=11)
ax2.set_title('Annual Production by Region', fontsize=12, fontweight='bold')
ax2.grid(True, alpha=0.3, axis='y')

for i, (bar, npp) in enumerate(zip(bars, annual_NPP)):
    height = bar.get_height()
    ax2.text(bar.get_x() + bar.get_width()/2., height + 5,
             f'{npp}', ha='center', va='bottom', fontsize=10, fontweight='bold')

# Plot 3: NPP vs SST (temperature dependence)
SST = np.linspace(0, 30, 100)  # deg C
Q10 = 2.0  # temperature sensitivity
NPP_temp = 100 * Q10**((SST - 15) / 10)

ax3.plot(SST, NPP_temp, 'r-', linewidth=2.5)
ax3.axvline(x=15, color='blue', linestyle='--', linewidth=1.5, alpha=0.7, label='Reference T')
ax3.set_xlabel('Sea Surface Temperature (\\textdegree C)', fontsize=11)
ax3.set_ylabel('Relative NPP (\\%)', fontsize=11)
ax3.set_title('Temperature Dependence ($Q_{10}$ = ' + f'{Q10:.1f}' + ')', fontsize=12, fontweight='bold')
ax3.legend(fontsize=9)
ax3.grid(True, alpha=0.3)

# Plot 4: Nutrient vs light colimitation
N_range = np.linspace(0.1, 10, 50)
I_range = np.linspace(50, 2000, 50)
N_grid, I_grid = np.meshgrid(N_range, I_range)

# Growth rate as function of N and I
mu_grid = nutrient_uptake(N_grid, V_max, K_s_N) * PI_curve(I_grid, P_max, alpha, beta) / P_max

cs = ax4.contourf(N_grid, I_grid, mu_grid, levels=20, cmap='YlGnBu')
cbar = plt.colorbar(cs, ax=ax4)
cbar.set_label('Growth Rate (d$^{-1}$)', fontsize=10)
ax4.contour(N_grid, I_grid, mu_grid, levels=10, colors='black', linewidths=0.5, alpha=0.3)
ax4.axvline(x=K_s_N, color='red', linestyle='--', linewidth=2, alpha=0.7, label=f'$K_s$ = {K_s_N} \\textmu M')
ax4.axhline(y=I_k, color='orange', linestyle='--', linewidth=2, alpha=0.7, label=f'$I_k$ = {I_k:.0f}')
ax4.set_xlabel('Nitrate Concentration (\\textmu M)', fontsize=11)
ax4.set_ylabel('PAR (\\textmu mol photons m$^{-2}$ s$^{-1}$)', fontsize=11)
ax4.set_title('Nutrient-Light Colimitation', fontsize=12, fontweight='bold')
ax4.legend(fontsize=9, loc='lower right')

plt.tight_layout()
plt.savefig('ocean_productivity_regional.pdf', dpi=150, bbox_inches='tight')
plt.close()
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{ocean_productivity_regional.pdf}
\caption{Regional and environmental controls on ocean productivity. Top-left: Bloom phenology
across latitudinal zones showing pronounced spring blooms in temperate and subpolar regions
versus weak seasonality in tropical oligotrophic waters. Top-right: Annual depth-integrated
NPP by region, with highest productivity (\py{f"{annual_NPP[0]}"} g C m$^{-2}$ yr$^{-1}$) in
temperate zones and lowest (\py{f"{annual_NPP[3]}"} g C m$^{-2}$ yr$^{-1}$) in tropical gyres.
Bottom-left: Temperature dependence of NPP showing $Q_{10}$ relationship where metabolic rates
double per 10°C increase. Bottom-right: Two-dimensional colimitation surface illustrating how
growth rate depends simultaneously on nutrient concentration and light availability, with
transition from light limitation (low PAR) to nutrient limitation (low [N]) marked by
half-saturation thresholds.}
\label{fig:regional}
\end{figure}

Regional comparisons demonstrate that temperate and subpolar oceans achieve highest annual
productivity (\py{f"{annual_NPP[0]}"} g C m$^{-2}$ yr$^{-1}$) despite seasonal extremes,
while permanently stratified tropical regions remain nutrient-limited year-round with lower
productivity (\py{f"{annual_NPP[3]}"} g C m$^{-2}$ yr$^{-1}$).

\section{Results Summary}

\begin{pycode}
print(r"\begin{table}[htbp]")
print(r"\centering")
print(r"\caption{Summary of Ocean Productivity Parameters}")
print(r"\begin{tabular}{lcc}")
print(r"\toprule")
print(r"Parameter & Value & Units \\")
print(r"\midrule")
print(f"$P_{{max}}$ & {P_max:.2f} & mg C (mg Chl)$^{{-1}}$ h$^{{-1}}$ \\\\")
print(f"$\\alpha$ & {alpha:.4f} & (\\textmu mol photons m$^{{-2}}$ s$^{{-1}}$)$^{{-1}}$ h$^{{-1}}$ \\\\")
print(f"$\\beta$ & {beta:.4f} & (\\textmu mol photons m$^{{-2}}$ s$^{{-1}}$)$^{{-1}}$ h$^{{-1}}$ \\\\")
print(f"$I_k$ & {I_k:.1f} & \\textmu mol photons m$^{{-2}}$ s$^{{-1}}$ \\\\")
print(f"$I_{{opt}}$ & {I_opt:.1f} & \\textmu mol photons m$^{{-2}}$ s$^{{-1}}$ \\\\")
print(f"$V_{{max}}$ & {V_max:.2f} & d$^{{-1}}$ \\\\")
print(f"$K_s$ (nitrate) & {K_s_N:.2f} & \\textmu M \\\\")
print(f"$K_d$ & {K_d:.3f} & m$^{{-1}}$ \\\\")
print(f"$z_{{eu}}$ & {z_eu:.1f} & m \\\\")
print(r"\midrule")
print(f"Spring bloom peak & {spring_bloom_chl:.2f} & mg Chl m$^{{-3}}$ \\\\")
print(f"Bloom day & {int(spring_bloom_day)} & day of year \\\\")
print(f"Annual NPP & {NPP_annual:.1f} & g C m$^{{-2}}$ yr$^{{-1}}$ \\\\")
print(f"Mean daily NPP & {np.mean(NPP_daily):.1f} & mg C m$^{{-2}}$ d$^{{-1}}$ \\\\")
print(r"\bottomrule")
print(r"\end{tabular}")
print(r"\label{tab:summary}")
print(r"\end{table}")
\end{pycode}

\section{Discussion}

\begin{example}[Critical Depth Hypothesis]
Sverdrup (1953) proposed that spring blooms occur when mixed layer depth becomes shallower
than a critical depth where integrated photosynthesis exceeds integrated respiration. Our
simulations support this framework: bloom initiation on day \py{f"{int(spring_bloom_day)}"}
coincides with MLD shoaling to \py{f"{mixed_layer_depth(spring_bloom_day):.1f}"} m, well
above euphotic depth (\py{f"{z_eu:.1f}"} m).
\end{example}

\begin{remark}[Photoinhibition]
High-latitude spring blooms often develop under ice cover or at high latitudes where surface
PAR rarely exceeds photoinhibition thresholds. Our optimal irradiance $I_{opt}$ =
\py{f"{I_opt:.0f}"} $\mu$mol photons m$^{-2}$ s$^{-1}$ represents the balance between
light-saturated photosynthesis and photoinhibitory damage, typically occurring at 40--60\%
of full sunlight.
\end{remark}

\begin{remark}[Nutrient Regeneration]
The ratio of nutrient uptake to remineralization controls whether ecosystems are net carbon
sources or sinks. Our model includes 50\% remineralization efficiency, consistent with
observed f-ratios (new production / total production) of 0.3--0.6 in temperate oceans. Higher
f-ratios indicate stronger biological carbon pumps.
\end{remark}

\section{Conclusions}

This computational analysis quantifies the fundamental controls on ocean primary productivity:

\begin{enumerate}
\item Photosynthesis-irradiance relationships follow the Platt-Gallegos-Harrison model with
$P_{max}$ = \py{f"{P_max:.1f}"} mg C (mg Chl)$^{-1}$ h$^{-1}$, initial slope $\alpha$ =
\py{f"{alpha:.4f}"}, and photoinhibition parameter $\beta$ = \py{f"{beta:.4f}"}

\item Nutrient limitation follows Michaelis-Menten kinetics with half-saturation constants
$K_s$ = \py{f"{K_s_N:.1f}"} $\mu$M for nitrate, controlling growth rates when nutrient
concentrations drop below this threshold

\item Seasonal bloom dynamics in temperate oceans produce spring chlorophyll maxima of
\py{f"{spring_bloom_chl:.2f}"} mg m$^{-3}$ on day \py{f"{int(spring_bloom_day)}"}, driven
by the transition from deep winter mixing to shallow stratification

\item Annual depth-integrated NPP reaches \py{f"{NPP_annual:.1f}"} g C m$^{-2}$ yr$^{-1}$ in
temperate regions, with mean daily production of \py{f"{np.mean(NPP_daily):.1f}"} mg C
m$^{-2}$ d$^{-1}$ modulated by seasonal light and nutrient availability

\item Regional productivity patterns reflect the interplay of mixing regime, light climate,
and nutrient supply, with highest annual NPP in temperate and subpolar regions where seasonal
mixing replenishes nutrients
\end{enumerate}

These results demonstrate how computational models integrating P-I curves, nutrient kinetics,
and vertical mixing can quantitatively predict ocean productivity patterns and their response
to environmental forcing.

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\end{document}