Module 0

Climate Science Foundations

From radiative balance to carbon budgets — the physics of a warming planet

0.1 The Greenhouse Effect

Earth intercepts solar radiation and re-emits thermal infrared radiation. The balance between incoming and outgoing energy determines our planet's temperature. Without any atmosphere, we can derive the effective radiating temperature using the Stefan-Boltzmann law.

The Sun delivers a flux $S = 1361\;\text{W/m}^2$ (the solar constant) at Earth's orbit. Earth reflects a fraction $\alpha \approx 0.30$ (the albedo), so the absorbed power is:

$ P_{\text{abs}} = \pi R^2 \cdot S(1 - \alpha) $

The Earth emits as a blackbody over its entire surface area $4\pi R^2$:

$ P_{\text{emit}} = 4\pi R^2 \cdot \sigma T_{\text{eff}}^4 $

Setting $P_{\text{abs}} = P_{\text{emit}}$ and solving for temperature:

$ T_{\text{eff}} = \left(\frac{S(1-\alpha)}{4\sigma}\right)^{1/4} = \left(\frac{1361 \times 0.70}{4 \times 5.67 \times 10^{-8}}\right)^{1/4} \approx 255\;\text{K} $

This is $-18\,^\circ\text{C}$ — well below freezing and far colder than Earth's observed mean surface temperature of $T_s \approx 288\;\text{K}$($+15\,^\circ\text{C}$). The difference:

$ \Delta T_{\text{GHG}} = T_s - T_{\text{eff}} = 288 - 255 = 33\;\text{K} $

This 33 K warming is the natural greenhouse effect. Greenhouse gases (CO$_2$, H$_2$O, CH$_4$, N$_2$O) absorb outgoing infrared radiation and re-emit it in all directions, including back toward the surface. This effectively raises the altitude from which Earth radiates to space. Since temperature decreases with altitude in the troposphere (the lapse rate), the emitting layer is cooler, reducing outgoing radiation and warming the surface until a new equilibrium is reached.

One-Layer Atmosphere Model

Consider a single atmospheric layer with infrared absorptivity/emissivity $\varepsilon$. The layer is transparent to solar radiation but absorbs and re-emits a fraction $\varepsilon$of the surface infrared emission. Energy balance for the surface gives:

$ \sigma T_s^4 = \frac{S(1-\alpha)}{4} + \varepsilon \sigma T_a^4 $

And for the atmospheric layer (absorbing $\varepsilon \sigma T_s^4$ from below and emitting $2\varepsilon \sigma T_a^4$ upward and downward):

$ \varepsilon \sigma T_s^4 = 2\varepsilon \sigma T_a^4 \quad \Rightarrow \quad T_a = \frac{T_s}{2^{1/4}} $

Substituting back and solving for $T_s$:

$ T_s = T_{\text{eff}} \left(\frac{2}{2-\varepsilon}\right)^{1/4} $

For $\varepsilon = 0.78$, this gives $T_s \approx 288\;\text{K}$, matching observations. The model illustrates how increasing greenhouse gas concentrations (raising $\varepsilon$) increases the surface temperature.

0.2 Radiative Forcing

Radiative forcing ($\Delta F$) measures the change in the net energy flux at the tropopause due to an external perturbation (e.g., increased CO$_2$), before the climate system has had time to respond. It is measured in W/m$^2$.

For CO$_2$, the radiative forcing follows a logarithmicrelationship with concentration, because the central absorption band is already nearly saturated and additional CO$_2$ primarily widens the absorption wings:

$ \Delta F = 5.35 \cdot \ln\!\left(\frac{C}{C_0}\right) \;\;\text{W/m}^2 $

where $C$ is the current CO$_2$ concentration and$C_0 = 280\;\text{ppm}$ is the preindustrial value. The coefficient 5.35 is derived from detailed line-by-line radiative transfer calculations integrating over the 15 $\mu$m absorption band of CO$_2$.

Derivation from Spectral Absorption

The Beer-Lambert law gives the transmittance through a column of gas:

$ \mathcal{T}(\nu) = e^{-\kappa(\nu) \cdot u} $

where $\kappa(\nu)$ is the absorption coefficient at frequency $\nu$and $u$ is the path amount (proportional to concentration). The change in outgoing longwave radiation (OLR) when concentration changes from $C_0$ to $C$ is:

$ \Delta F = -\int_0^\infty \left[\mathcal{T}(\nu, C) - \mathcal{T}(\nu, C_0)\right] B(\nu, T)\, d\nu $

For the CO$_2$ band, the centre is saturated ($\mathcal{T} \approx 0$) and additional absorption occurs in the pressure-broadened wingswhere $\kappa(\nu) \propto \ln(C/C_0)$. This is why doubling CO$_2$gives a roughly constant forcing increment:

$ \Delta F_{2\times} = 5.35 \cdot \ln(2) \approx 3.7\;\text{W/m}^2 $

Current atmospheric CO$_2$ is approximately 420 ppm (2024), giving:

$ \Delta F_{\text{now}} = 5.35 \cdot \ln\!\left(\frac{420}{280}\right) = 5.35 \times 0.405 \approx 2.17\;\text{W/m}^2 $

Including other greenhouse gases (CH$_4$, N$_2$O, halocarbons) and adjustments for tropospheric response, the total anthropogenic radiative forcing is approximately$+2.72\;\text{W/m}^2$ relative to 1750 (IPCC AR6, 2021).

0.3 Equilibrium Climate Sensitivity

Equilibrium Climate Sensitivity (ECS) is the steady-state global mean surface temperature change from a doubling of atmospheric CO$_2$. It encapsulates all feedback processes in a single number.

The global energy balance can be linearized about an equilibrium state:

$ C \frac{dT}{dt} = \Delta F - \frac{\Delta T}{\lambda} $

where $C$ is the effective heat capacity of the climate system,$\Delta F$ is the radiative forcing, and $\lambda$(K per W/m$^2$) is the climate feedback parameter. At equilibrium ($dT/dt = 0$):

$ \text{ECS} = \lambda \cdot \Delta F_{2\times} = \lambda \times 3.7\;\text{W/m}^2 $

Feedback Decomposition

The feedback parameter decomposes into individual feedback contributions:

$ \frac{1}{\lambda} = \frac{1}{\lambda_0} - \sum_i f_i $

where $\lambda_0$ is the Planck (no-feedback) response and $f_i$are individual feedback strengths:

Planck feedback

$\lambda_0^{-1} \approx 3.2\;\text{W/m}^2\text{K}^{-1}$

A warmer surface radiates more (Stefan-Boltzmann). This is the stabilizing baseline.

Water vapour feedback

$f_{\text{WV}} \approx +1.8\;\text{W/m}^2\text{K}^{-1}$

Warmer air holds more water vapour (Clausius-Clapeyron). Water vapour is a powerful greenhouse gas, amplifying warming.

Ice-albedo feedback

$f_{\text{ice}} \approx +0.3\;\text{W/m}^2\text{K}^{-1}$

Warming melts reflective ice and snow, exposing darker surfaces that absorb more sunlight.

Lapse rate feedback

$f_{\text{LR}} \approx -0.6\;\text{W/m}^2\text{K}^{-1}$

The upper troposphere warms faster, increasing OLR. Partially offsets water vapour feedback.

Cloud feedback

$f_{\text{cloud}} \approx +0.4\;\text{W/m}^2\text{K}^{-1}$

The largest source of uncertainty. Changes in cloud cover, altitude, and optical properties. AR6 assessment: likely positive.

Summing feedbacks: net feedback $\approx 3.2 - 1.8 - 0.3 + 0.6 - 0.4 = 1.3\;\text{W/m}^2\text{K}^{-1}$, giving $\lambda \approx 1/1.3 \approx 0.77\;\text{K/(W/m}^2)$ and:

$ \text{ECS} = 0.77 \times 3.7 \approx 2.8\;^\circ\text{C} $

The IPCC AR6 assessed ECS as likely in the range 2.5–4.0 °Cwith a best estimate of 3.0 °C. The uncertainty is dominated by cloud feedback.

0.4 The Global Carbon Cycle

The atmospheric CO$_2$ budget is governed by sources and sinks:

$ \frac{dC_{\text{atm}}}{dt} = E_{\text{fossil}} + E_{\text{LUC}} - S_{\text{ocean}} - S_{\text{land}} $

where:

• $E_{\text{fossil}} \approx 9.5\;\text{GtC/yr}$ — fossil fuel combustion and cement production
• $E_{\text{LUC}} \approx 1.1\;\text{GtC/yr}$ — land use change (deforestation, agriculture)
• $S_{\text{ocean}} \approx 2.8\;\text{GtC/yr}$ — ocean carbon uptake (dissolution, biological pump)
• $S_{\text{land}} \approx 3.1\;\text{GtC/yr}$ — terrestrial biosphere uptake (photosynthesis, CO$_2$ fertilisation)

Airborne Fraction

The airborne fraction (AF) is the proportion of total emissions that remains in the atmosphere:

$ AF = \frac{dC_{\text{atm}}/dt}{E_{\text{fossil}} + E_{\text{LUC}}} = \frac{10.6 - 2.8 - 3.1}{10.6} = \frac{4.7}{10.6} \approx 0.44 $

This means approximately 44% of our emissions remain in the atmosphere, with the ocean and land biosphere each absorbing roughly a quarter of total emissions. The AF has remained remarkably stable at $\sim 0.44$ over the past 60 years, despite rising emissions.

We can derive the remaining carbon budget for a temperature target. For a 1.5 °C target (from preindustrial), the remaining warming needed is approximately$\Delta T_{\text{rem}} \approx 0.3\;^\circ\text{C}$. Using the transient climate response to cumulative emissions (TCRE $\approx 0.45\;^\circ\text{C per 1000 GtCO}_2$):

$ \text{Budget} = \frac{\Delta T_{\text{rem}}}{\text{TCRE}} = \frac{0.3}{0.45 \times 10^{-3}} \approx 670\;\text{GtCO}_2 \approx 183\;\text{GtC} $

At current emission rates ($\sim 40\;\text{GtCO}_2/\text{yr}$), this budget would be exhausted in approximately 17 years.

0.5 IPCC Emission Scenarios (SSPs)

The Shared Socioeconomic Pathways (SSPs) describe different plausible futures for society, economy, and emissions. Each is paired with a Representative Concentration Pathway (RCP) indicating the approximate radiative forcing in 2100:

SSP1-2.6

Sustainability pathway. Net-zero CO₂ around 2075. Warming limited to ~1.8°C by 2100.

SSP2-4.5

Middle-of-the-road. Emissions peak ~2040, slow decline. Warming ~2.7°C by 2100.

SSP3-7.0

Regional rivalry. Emissions rise through 2100. Warming ~3.6°C by 2100.

SSP5-8.5

Fossil-fueled development. Emissions double by 2050. Warming ~4.4°C by 2100.

The temperature response under each scenario can be estimated using a simple energy balance model. The transient response follows:

$ \Delta T(t) = \lambda \cdot \Delta F(t) \cdot \left(1 - e^{-t/\tau}\right) $

where $\tau = C \cdot \lambda \approx 30$ years is the climate response timescale. The full response involves multiple timescales: a fast response ($\tau_1 \sim 5\;\text{yr}$, atmosphere and upper ocean) and a slow response ($\tau_2 \sim 200\;\text{yr}$, deep ocean).

Energy Balance Diagram

The global mean energy budget showing incoming solar radiation, reflection, absorption, and the greenhouse effect:

Global mean energy budget (approximate values from Trenberth et al. 2009, updated). The greenhouse effect traps outgoing IR radiation, with back-radiation warming the surface 33 K above the bare-rock equilibrium temperature.

Computational Simulations

Radiative Forcing vs CO\u2082 Concentration

Python

script.py56 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Radiative forcing: Delta_F = 5.35 * ln(C / C0)
C0 = 280  # preindustrial CO2 (ppm)
C = np.linspace(280, 1200, 500)
delta_F = 5.35 * np.log(C / C0)

fig, ax = plt.subplots(figsize=(10, 6))
ax.set_facecolor('#0a0a1a')
fig.patch.set_facecolor('#0a0a1a')

ax.plot(C, delta_F, color='#06b6d4', linewidth=2.5, label='Radiative Forcing')

# Mark key concentrations
markers = [
    (280, 'Preindustrial (280 ppm)', '#22d3ee'),
    (420, 'Current ~420 ppm', '#f97316'),
    (560, '2x CO\u2082 (560 ppm)', '#ef4444'),
    (1120, '4x CO\u2082 (1120 ppm)', '#dc2626'),
]
for conc, label, color in markers:
    f_val = 5.35 * np.log(conc / C0)
    ax.plot(conc, f_val, 'o', color=color, markersize=10, zorder=5)
    ax.annotate(f'{label}\n{f_val:.2f} W/m\u00b2', (conc, f_val),
                textcoords='offset points', xytext=(10, 15),
                fontsize=9, color=color,
                arrowprops=dict(arrowstyle='->', color=color, lw=0.8))

# Doubling lines
ax.axhline(y=5.35*np.log(2), color='#ef4444', linestyle='--', alpha=0.5, linewidth=1)
ax.text(290, 5.35*np.log(2)+0.15, r'$\Delta F_{2x}$ = 3.71 W/m\u00b2', color='#ef4444', fontsize=10)

ax.set_xlabel('CO\u2082 Concentration (ppm)', color='white', fontsize=12)
ax.set_ylabel('Radiative Forcing (W/m\u00b2)', color='white', fontsize=12)
ax.set_title('Radiative Forcing vs CO\u2082 Concentration', color='white', fontsize=14, fontweight='bold')
ax.tick_params(colors='white')
ax.spines['bottom'].set_color('#334155')
ax.spines['left'].set_color('#334155')
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.grid(True, alpha=0.15, color='#334155')
ax.legend(fontsize=10, facecolor='#1e293b', edgecolor='#334155', labelcolor='white')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight')
print("Radiative Forcing Analysis")
print("=" * 45)
print(f"Preindustrial CO2: {C0} ppm")
print(f"Current CO2: ~420 ppm  ->  Delta_F = {5.35*np.log(420/280):.2f} W/m^2")
print(f"2x CO2 (560 ppm):     ->  Delta_F = {5.35*np.log(2):.2f} W/m^2")
print(f"4x CO2 (1120 ppm):    ->  Delta_F = {5.35*np.log(4):.2f} W/m^2")
print(f"\nLogarithmic relationship means each doubling")
print(f"adds the same forcing: +{5.35*np.log(2):.2f} W/m^2")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Temperature Projections Under SSP Scenarios

Python

script.py104 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Simple two-timescale energy balance model
# dT/dt = (Delta_F - T/lambda) / C
# Using two-box model: fast (atmosphere+upper ocean) and slow (deep ocean)

years = np.arange(1850, 2101)
n = len(years)

# SSP scenario CO2 pathways (simplified piecewise)
def co2_path(yr, scenario):
    """Approximate CO2 concentration trajectories"""
    if scenario == 'SSP1-2.6':
        if yr < 1960: return 280 + (yr - 1850) * 0.3
        elif yr < 2020: return 313 + (yr - 1960) * 1.7
        elif yr < 2050: return 415 + (yr - 2020) * 0.5
        else: return 430 + (yr - 2050) * (-0.5)
    elif scenario == 'SSP2-4.5':
        if yr < 1960: return 280 + (yr - 1850) * 0.3
        elif yr < 2020: return 313 + (yr - 1960) * 1.7
        elif yr < 2060: return 415 + (yr - 2020) * 2.5
        else: return 515 + (yr - 2060) * 1.0
    elif scenario == 'SSP3-7.0':
        if yr < 1960: return 280 + (yr - 1850) * 0.3
        elif yr < 2020: return 313 + (yr - 1960) * 1.7
        else: return 415 + (yr - 2020) * 5.5
    else:  # SSP5-8.5
        if yr < 1960: return 280 + (yr - 1850) * 0.3
        elif yr < 2020: return 313 + (yr - 1960) * 1.7
        else: return 415 + (yr - 2020) * 8.5

# Climate parameters
ECS = 3.0       # equilibrium climate sensitivity (C)
F2x = 3.71      # forcing for CO2 doubling (W/m^2)
lam = ECS / F2x # climate feedback parameter
C_fast = 8.0    # heat capacity fast component (W yr / m^2 K)
C_slow = 100.0  # heat capacity slow component
gamma = 0.7     # coupling between fast and slow

scenarios = {
    'SSP1-2.6': '#22c55e',
    'SSP2-4.5': '#eab308',
    'SSP3-7.0': '#f97316',
    'SSP5-8.5': '#ef4444',
}

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
for a in [ax1, ax2]:
    a.set_facecolor('#0a0a1a')
fig.patch.set_facecolor('#0a0a1a')

for scenario, color in scenarios.items():
    co2 = np.array([co2_path(yr, scenario) for yr in years])
    forcing = 5.35 * np.log(np.maximum(co2, 280) / 280)

# Two-box model integration
    T_fast = np.zeros(n)
    T_slow = np.zeros(n)
    dt = 1.0  # 1 year timestep
    for i in range(1, n):
        dTf = (forcing[i] - T_fast[i-1]/lam - gamma*(T_fast[i-1]-T_slow[i-1])) / C_fast * dt
        dTs = gamma * (T_fast[i-1] - T_slow[i-1]) / C_slow * dt
        T_fast[i] = T_fast[i-1] + dTf
        T_slow[i] = T_slow[i-1] + dTs

ax1.plot(years, co2, color=color, linewidth=2, label=scenario)
    ax2.plot(years, T_fast, color=color, linewidth=2, label=scenario)

# Paris targets
ax2.axhline(y=1.5, color='#67e8f9', linestyle='--', alpha=0.6, linewidth=1)
ax2.text(1855, 1.6, '1.5\u00b0C Paris target', color='#67e8f9', fontsize=9)
ax2.axhline(y=2.0, color='#a78bfa', linestyle='--', alpha=0.6, linewidth=1)
ax2.text(1855, 2.1, '2.0\u00b0C Paris limit', color='#a78bfa', fontsize=9)

ax1.set_xlabel('Year', color='white', fontsize=12)
ax1.set_ylabel('CO\u2082 (ppm)', color='white', fontsize=12)
ax1.set_title('CO\u2082 Concentration Pathways', color='white', fontsize=13, fontweight='bold')
ax1.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='white')

ax2.set_xlabel('Year', color='white', fontsize=12)
ax2.set_ylabel('Temperature Anomaly (\u00b0C)', color='white', fontsize=12)
ax2.set_title('Global Mean Temperature Change', color='white', fontsize=13, fontweight='bold')
ax2.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='white')

for a in [ax1, ax2]:
    a.tick_params(colors='white')
    a.spines['bottom'].set_color('#334155')
    a.spines['left'].set_color('#334155')
    a.spines['top'].set_visible(False)
    a.spines['right'].set_visible(False)
    a.grid(True, alpha=0.15, color='#334155')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight')

print("Temperature Projections (2100) by SSP Scenario")
print("=" * 50)
for scenario in scenarios:
    co2_2100 = co2_path(2100, scenario)
    f_2100 = 5.35 * np.log(co2_2100 / 280)
    print(f"{scenario}: CO2 = {co2_2100:.0f} ppm, F = {f_2100:.2f} W/m^2")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Carbon Budget Breakdown

Python

script.py82 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Global Carbon Project 2023 data (GtC/yr, 2013-2022 average)
sources = {
    'Fossil fuels\n& industry': 9.5,
    'Land-use\nchange': 1.1,
}
sinks = {
    'Atmosphere': 4.7,
    'Ocean': 2.8,
    'Land': 3.1,
}

fig, axes = plt.subplots(1, 3, figsize=(15, 6))
fig.patch.set_facecolor('#0a0a1a')

# Panel 1: Sources pie chart
ax1 = axes[0]
ax1.set_facecolor('#0a0a1a')
labels_s = list(sources.keys())
sizes_s = list(sources.values())
colors_s = ['#ef4444', '#f97316']
wedges, texts, autotexts = ax1.pie(sizes_s, labels=labels_s, autopct='%1.1f%%',
    colors=colors_s, textprops={'color': 'white', 'fontsize': 11},
    pctdistance=0.75, startangle=90)
for at in autotexts:
    at.set_fontsize(10)
    at.set_color('white')
ax1.set_title('Sources (10.6 GtC/yr)', color='white', fontsize=13, fontweight='bold')

# Panel 2: Sinks pie chart
ax2 = axes[1]
ax2.set_facecolor('#0a0a1a')
labels_k = list(sinks.keys())
sizes_k = list(sinks.values())
colors_k = ['#94a3b8', '#06b6d4', '#22c55e']
wedges2, texts2, autotexts2 = ax2.pie(sizes_k, labels=labels_k, autopct='%1.1f%%',
    colors=colors_k, textprops={'color': 'white', 'fontsize': 11},
    pctdistance=0.75, startangle=90)
for at in autotexts2:
    at.set_fontsize(10)
    at.set_color('white')
ax2.set_title('Sinks (10.6 GtC/yr)', color='white', fontsize=13, fontweight='bold')

# Panel 3: Airborne fraction over time
ax3 = axes[2]
ax3.set_facecolor('#0a0a1a')
# Historical airborne fraction (simplified data)
decades = np.arange(1960, 2030, 5)
af_values = [0.39, 0.50, 0.44, 0.46, 0.43, 0.44, 0.45, 0.44, 0.42, 0.43,
             0.44, 0.45, 0.44, 0.43]
ax3.bar(decades, af_values, width=4, color='#06b6d4', alpha=0.7, edgecolor='#0891b2')
ax3.axhline(y=0.44, color='#f97316', linestyle='--', linewidth=1.5, label='Mean AF = 0.44')
ax3.set_xlabel('Year', color='white', fontsize=11)
ax3.set_ylabel('Airborne Fraction', color='white', fontsize=11)
ax3.set_title('Airborne Fraction Over Time', color='white', fontsize=13, fontweight='bold')
ax3.set_ylim(0, 0.65)
ax3.tick_params(colors='white')
ax3.spines['bottom'].set_color('#334155')
ax3.spines['left'].set_color('#334155')
ax3.spines['top'].set_visible(False)
ax3.spines['right'].set_visible(False)
ax3.grid(True, alpha=0.15, color='#334155', axis='y')
ax3.legend(fontsize=9, facecolor='#1e293b', edgecolor='#334155', labelcolor='white')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight')

print("Global Carbon Budget (2013-2022 average)")
print("=" * 45)
print("SOURCES:")
for k, v in sources.items():
    print(f"  {k.replace(chr(10), ' ')}: {v} GtC/yr")
print(f"  Total: {sum(sources.values()):.1f} GtC/yr")
print("\nSINKS:")
for k, v in sinks.items():
    print(f"  {k}: {v} GtC/yr ({v/sum(sources.values())*100:.1f}%)")
print(f"\nAirborne fraction: {sinks['Atmosphere']/sum(sources.values()):.2f}")
print(f"\nRemaining 1.5C budget: ~670 GtCO2 (~17 years at current rates)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

References

• IPCC, 2021: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report, Cambridge University Press.
• Myhre, G. et al. (1998). New estimates of radiative forcing due to well mixed greenhouse gases. Geophysical Research Letters, 25(14), 2715–2718.
• Trenberth, K.E., Fasullo, J.T. & Kiehl, J. (2009). Earth's global energy budget. Bulletin of the American Meteorological Society, 90(3), 311–324.
• Friedlingstein, P. et al. (2023). Global Carbon Budget 2023. Earth System Science Data, 15, 5301–5369.
• Sherwood, S.C. et al. (2020). An assessment of Earth's climate sensitivity using multiple lines of evidence. Reviews of Geophysics, 58(4), e2019RG000678.
• Pierrehumbert, R.T. (2010). Principles of Planetary Climate. Cambridge University Press.
• Arrhenius, S. (1896). On the influence of carbonic acid in the air upon the temperature of the ground. Philosophical Magazine, 41(251), 237–276.