Sources and References

Primary textbook — No required textbook. Supplementary texts — Pierrehumbert, R. T. (2010). Principles of Planetary Climate. Cambridge University Press; Strogatz, S. H. (2018). Nonlinear Dynamics and Chaos, 2nd ed. CRC Press (for dynamical systems sections); Northrop, M., & Hufbauer, G. C. (Eds.) (2019). Playing It Cool: High-value Strategies for Reducing Greenhouse Gas Emissions. (Economics.) Online resources — IPCC Assessment Reports (ipcc.ch); NOAA Climate.gov; NASA GISS Surface Temperature Analysis (data.giss.nasa.gov); NCAR Climate Data Guide (climatedataguide.ucar.edu); Fourier Analysis MIT OCW 18.03 notes.

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Chapter 1: Overview — Mathematics, Science, and Impacts of Climate Change

1.1 Why Mathematics for Climate?

Climate is a complex physical system governed by conservation laws (energy, mass, momentum) and the interactions among atmosphere, ocean, land surface, and biosphere. Mathematical models provide the only rigorous framework for:

• Understanding mechanisms and feedback loops.
• Predicting future states under different emission scenarios.
• Quantifying uncertainty in projections.
• Evaluating policy options.

Reductionism in mathematics: a hallmark of applied mathematics is deliberately simplifying complex systems to extract essential behaviour. We build a hierarchy of models — from zero-dimensional energy balance to coupled GCMs (General Circulation Models) — each capturing different aspects of climate at different levels of complexity.

1.2 Climate vs. Weather

The distinction matters mathematically: weather is chaotic (sensitive to initial conditions; predictability horizon ~2 weeks), while climate statistics can be much more predictable because they are governed by slower physical processes (Earth’s energy balance, ocean heat uptake, ice feedbacks).

1.3 Key Climate Observations

• Global mean surface temperature (GMST) has increased ~1.1°C above pre-industrial levels (IPCC AR6, 2021).
• Atmospheric CO₂: pre-industrial ~280 ppm; current ~420 ppm (Keeling Curve, Mauna Loa Observatory, 1958–present).
• Sea level rise: ~20 cm since 1900; accelerating.
• Arctic sea ice: summer minimum area declined ~13%/decade since 1979.
• Extreme events: increased frequency and intensity of heat waves, heavy precipitation, tropical cyclones.

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Chapter 2: Energy Balance Models

2.1 The Earth as a Blackbody

The simplest climate model treats Earth as a sphere in radiative equilibrium with the Sun.

Solar constant: the solar flux at Earth’s mean orbital distance is \( S_0 \approx 1361 \, \text{W m}^{-2} \).

\[ Q_{in} = \frac{S_0(1-\alpha)}{4} \approx 238 \, \text{W m}^{-2} \]\[ Q_{out} = \epsilon\sigma T_s^4 \]

where \( \epsilon \approx 1 \) for the surface but greenhouse gases reduce the effective emissivity to space.

2.2 Zero-Dimensional Energy Balance Model (EBM)

\[ \frac{S_0(1-\alpha)}{4} = \sigma T_e^4 \]\[ T_e = \left(\frac{S_0(1-\alpha)}{4\sigma}\right)^{1/4} \approx 255 \, \text{K} = -18°\text{C} \]

The observed mean surface temperature is ~288 K = +15°C. The difference (~33 K) is the greenhouse effect: greenhouse gases (CO₂, H₂O, CH₄, N₂O) absorb outgoing longwave radiation and re-emit it, warming the surface.

2.3 The Greenhouse Effect and Forcing

\[ \Delta F_{2\times CO_2} \approx 3.7 \, \text{W m}^{-2} \]\[ \Delta T_0 = \frac{\Delta F}{4\sigma T_e^3} \approx \frac{3.7}{3.3} \approx 1.1 \, \text{K} \]

2.4 Climate Feedbacks

Feedback parameter \( \lambda \) (W m\(^{-2}\) K\(^{-1}\)): total feedback is \( \lambda = \lambda_0 + \sum \lambda_i \), where \( \lambda_0 = -4\sigma T_e^3 \approx -3.3 \, \text{W m}^{-2}\text{K}^{-1} \) is the Planck (blackbody) feedback (stabilizing).

\[ \text{ECS} = -\frac{\Delta F_{2\times CO_2}}{\lambda} = \frac{3.7}{3.3 - \sum \lambda_i^{pos}} \]

Feedback	\( \lambda_i \) (W m\(^{-2}\)K\(^{-1}\))	Sign	Mechanism
Water vapour	~+1.6	Positive	Warmer atmosphere holds more H₂O (strong GHG)
Lapse rate	~−0.6	Negative	Warming alters temperature gradient; tropical troposphere warms more than surface
Surface albedo (ice-albedo)	~+0.4	Positive	Melting ice → less reflection → more absorption
Cloud (SW)	~+0.4	Positive (uncertain)	High clouds decrease, reducing SW reflection
Cloud (LW)	~−0.6	Negative (uncertain)	High clouds decrease, reducing LW trapping

Net feedback is positive (amplifying), giving ECS ~2.5–4.0 K (IPCC likely range: 2.5–4.0°C per doubling CO₂).

2.5 Ice-Albedo Feedback and Bifurcations

\[ C\frac{dT}{dt} = \frac{S_0(1-\alpha(T))}{4} - \sigma T^4 \]

where \( C \) is the ocean heat capacity.

Snowball Earth: for sufficiently low solar forcing or high albedo, the system can bifurcate to a fully ice-covered state (snowball Earth). The system exhibits hysteresis — different equilibria are reached depending on initial conditions, and recovering from a snowball state requires dramatically higher solar forcing than caused it.

Mathematical structure: The equilibrium condition \( dT/dt = 0 \) may have up to three fixed points (cold glaciated, ice-free, and an unstable intermediate). Stability analysis reveals which fixed points are attractors.

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Chapter 3: Climate Data and Fourier Analysis

3.1 Time Series Analysis of Climate Data

Climate datasets (temperature, precipitation, sea level, CO₂) are time series: sequences of measurements \( \{x_1, x_2, \ldots, x_N\} \) at times \( \{t_1, t_2, \ldots, t_N\} \). Key tasks:

• Identify trends (long-term changes).
• Identify periodic cycles (seasonal, annual, decadal).
• Identify anomalies (El Niño, volcanic eruptions).
• Quantify noise and variability.

3.2 The Discrete Fourier Transform

Physical interpretation: \( |X_k|^2 \) is proportional to the power spectral density at frequency \( f_k = k/(N\Delta t) \) (Hz), where \( \Delta t \) is the sampling interval. The Fourier transform decomposes the signal into sinusoidal components at different frequencies.

Sampling theorem (Nyquist): to detect cycles at frequency \( f \), the sampling rate must be at least \( 2f \) (Nyquist frequency \( f_N = 1/(2\Delta t) \)).

3.3 Climate Signals in the Frequency Domain

Seasonal cycle: dominant annual periodicity (\( f = 1 \, \text{yr}^{-1} \)) in virtually all surface climate records.

Milanković cycles: periodic changes in Earth’s orbital parameters:

• Eccentricity: ~100 kyr cycle (shape of orbit).
• Obliquity: ~41 kyr cycle (tilt of spin axis, 22.1°–24.5°).
• Precession: ~26 kyr cycle (orientation of spin axis).

These cycles are visible as spectral peaks in paleoclimate proxy records (ice cores, deep-sea sediments) and are the pacemaker of the ice ages.

Trend removal: climate records exhibit superimposed trends (anthropogenic warming) plus variability. Linear detrending isolates the variability; or compute anomalies by subtracting the climatological mean for each calendar month.

ENSO spectral signature: El Niño events have a characteristic period of 2–7 years, visible in power spectra of tropical Pacific SST anomalies.

3.4 Moving Averages and Filters

A low-pass filter retains low-frequency (slowly varying) components and removes high-frequency noise:

• Running mean over \( M \) time steps: \( \bar{x}_n = \frac{1}{M}\sum_{m=0}^{M-1} x_{n-m} \).
• In Fourier space, multiplication by a window function.

A high-pass filter retains variability; a band-pass filter selects a range of frequencies (e.g., to isolate ENSO signal: ~2–7 year band).

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Chapter 4: Spatial Data and Linear Algebra — EOF Analysis

4.1 Climate Fields as Matrices

\[ \mathbf{X} \in \mathbb{R}^{M\times N} \]

Each row is a spatial location’s time series; each column is a global spatial snapshot.

4.2 Empirical Orthogonal Functions (EOF Analysis / PCA)

Mathematical procedure:

1. Form the anomaly matrix by subtracting the temporal mean: \( \mathbf{X}' = \mathbf{X} - \bar{\mathbf{X}} \).
2. Compute the spatial covariance matrix: \( \mathbf{C} = \frac{1}{N}\mathbf{X}'\mathbf{X}'^T \in \mathbb{R}^{M\times M} \).
3. Eigendecompose: \( \mathbf{C} = \mathbf{E}\mathbf{\Lambda}\mathbf{E}^T \) where columns of \( \mathbf{E} \) are the EOFs (orthogonal) and \( \mathbf{\Lambda} = \text{diag}(\lambda_1, \ldots, \lambda_M) \) with \( \lambda_1 \geq \lambda_2 \geq \ldots \).
4. Principal components: \( \mathbf{PC}_k = \mathbf{e}_k^T \mathbf{X}' \) (time series of amplitude of EOF \( k \)).

Interpretation: EOF 1 (associated with the largest eigenvalue \( \lambda_1 \)) explains the largest fraction of total variance \( \lambda_1 / \sum_k \lambda_k \). Each EOF is a spatial pattern; its PC is how that pattern varies in time.

4.3 SVD Connection

\[ \mathbf{X}' = \mathbf{U}\mathbf{\Sigma}\mathbf{V}^T \]

The columns of \( \mathbf{U} \) are the EOFs; the columns of \( \mathbf{V} \) are the normalized PCs; the diagonal of \( \mathbf{\Sigma} \) contains the singular values (related to the square roots of eigenvalues). The SVD is computationally more efficient and numerically stable than eigendecomposition of the covariance matrix.

4.4 Physical Application: EOF of SST → ENSO

EOF 1 of tropical Pacific SST anomalies is the ENSO mode: a large warm or cool anomaly centered in the eastern tropical Pacific. PC 1 is the Niño 3.4 index (time series of east-central Pacific SST anomaly, a standard ENSO index).

Higher EOFs capture other modes of variability (e.g., Pacific Decadal Oscillation, Indian Ocean Dipole).

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Chapter 5: El Niño–Southern Oscillation (ENSO)

5.1 What is ENSO?

5.2 Walker Circulation and Normal Conditions

Under normal (non-ENSO) conditions, trade winds blow westward across the tropical Pacific, piling up warm water in the western Pacific (warm pool). Cold water upwells in the eastern Pacific (La Niña-like conditions). This sustains the Walker Circulation: rising air over the warm west Pacific, east-west pressure gradient.

5.3 ENSO Dynamics — Bjerknes Feedback

Bjerknes positive feedback (1969): the key amplifying mechanism:

• Anomalous warming in eastern Pacific → weakens trade winds.
• Weaker trades → less cold upwelling → further warming.
• Self-reinforcing loop creates El Niño.

\[ \frac{dT}{dt} = \mu_a b T - \alpha T + \mu_a h \]\[ \frac{dh}{dt} = -r h - \mu_b T \]

This linear system oscillates with a period of ~3–4 years for physically realistic parameters. ENSO arises from the interaction between fast SST dynamics and slow thermocline recharge/discharge.

5.4 Global Teleconnections

El Niño events affect weather worldwide through atmospheric teleconnections:

• North America: warmer, drier winters in the Pacific Northwest; wetter winters in the southeastern US.
• Australia/Indonesia: drought, bushfire risk.
• East Africa: above-normal rainfall.
• Peru/Ecuador: heavy rainfall, flooding.
• India: weakened monsoon (correlation not deterministic).

ENSO is the single most important source of global interannual climate variability.

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Chapter 6: The Role of Oceans and Earth’s Rotation

6.1 The Coriolis Effect

\[ \mathbf{F}_{Cor} = -2m\mathbf{\Omega}\times\mathbf{v} \]

The Coriolis parameter \( f = 2\Omega\sin\phi \) (s\(^{-1}\)), where \( \phi \) is latitude.

Consequences:

• Geostrophic balance: large-scale atmospheric and oceanic flows balance pressure gradient against Coriolis force → winds/currents flow along, not across, isobars.
• Cyclones/anticyclones: low-pressure systems spin counterclockwise in NH, clockwise in SH.
• Ocean gyres: large-scale circulation cells driven by wind stress; western boundary currents (Gulf Stream, Kuroshio) are intense and narrow due to Sverdrup balance.
• Ekman transport: surface ocean layer driven perpendicular to wind direction (90° to the right in NH), causing coastal upwelling/downwelling.

6.2 Ocean Heat Uptake and the Thermohaline Circulation

Oceans as heat buffers: the ocean has ~1000× the heat capacity of the atmosphere; it takes up ~90% of the excess heat from the enhanced greenhouse effect, slowing surface warming.

Thermohaline circulation (THC) / Atlantic Meridional Overturning Circulation (AMOC):

• Surface heat loss in the North Atlantic makes water cold and dense → sinks.
• Deep water flows south; surface water flows north (Gulf Stream extension).
• Returns heat to Europe; maintains European climate significantly warmer than similar latitudes elsewhere.

AMOC weakening: freshwater input from Greenland ice sheet melt can reduce deep water formation → weaken AMOC → potential tipping point in the climate system.

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Chapter 7: Natural Climate Variability (Extra Topics)

7.1 Paleoclimate and Ice Ages

Ice cores (Vostok, EPICA Dome C, NEEM): provide records of temperature (from δ¹⁸O and δD), CO₂, CH₄, dust, and other quantities over hundreds of thousands of years. Key findings:

• CO₂ and temperature are tightly correlated over glacial-interglacial cycles.
• Pre-industrial CO₂ never exceeded ~300 ppm in the past 800,000 years; current ~420 ppm is unprecedented.
• Glacial-interglacial temperature changes: ~5°C global average, ~10°C at the poles.

Milanković theory: glacial cycles are paced by orbital forcing; however, the orbital changes are small (< 1 W m⁻²) and amplified by feedbacks (ice-albedo, CO₂, methane).

7.2 Volcanic Forcing

Large volcanic eruptions (Pinatubo 1991, Tambora 1815) inject sulfur dioxide into the stratosphere → sulfate aerosols → increased albedo → temporary surface cooling (~0.3–0.5°C for 1–3 years). Detected as negative spikes in temperature records and in ice core sulfate layers.

7.3 Solar Variability

The Sun’s luminosity varies by ~0.1% over the ~11-year sunspot cycle — a small forcing (~0.1 W m⁻²). Solar activity has been flat or slightly declining since 1980, ruling it out as a cause of recent warming.

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Chapter 8: Population Dynamics and Climate Change

8.1 Malthus and Logistic Growth

\[ \frac{dN}{dt} = rN \implies N(t) = N_0 e^{rt} \]

where \( r \) is the intrinsic growth rate.

\[ \frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right) \]

Equilibria: \( N^* = 0 \) (unstable) and \( N^* = K \) (stable).

\[ N(t) = \frac{K}{1 + \left(\frac{K}{N_0} - 1\right)e^{-rt}} \]

S-shaped (sigmoidal) growth; approaches \( K \) asymptotically.

8.2 Effect of Climate Change on Carrying Capacity

Climate change affects the carrying capacity \( K(t) \) through:

• Agricultural productivity: higher temperatures and altered precipitation patterns reduce crop yields in many regions; CO₂ fertilization partially offsets this.
• Water availability: shifting precipitation patterns, glacial retreat, increased drought frequency.
• Extreme events: floods, heat waves, tropical cyclones reduce food production.
• Ecosystem services: biodiversity loss, ocean acidification (reduced fisheries), deforestation.

\[ \frac{dN}{dt} = rN\left(1 - \frac{N}{K(T(t))}\right) \]

where \( T(t) \) is the time-evolving global mean temperature and \( K(T) \) is a decreasing function of temperature above an optimum.

8.3 Multi-Species Models (Lotka-Volterra)

\[ \frac{dN}{dt} = \alpha N - \beta NP \quad \text{(prey)} \]\[ \frac{dP}{dt} = \delta NP - \gamma P \quad \text{(predator)} \]

Climate change affects prey and predator parameters differently, potentially desynchronizing oscillations (phenological mismatch) and causing species collapse.

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Chapter 9: The Economics of Climate Change

9.1 The Social Cost of Carbon

Estimating the SCC requires:

1. An emissions scenario: how much CO₂ is emitted over time.
2. A climate model: relating CO₂ to temperature change.
3. A damage function: relating temperature change to economic losses.
4. A discount rate: how to weigh future losses against present costs.

Discount rate controversy: choosing a 5% vs. 1% discount rate dramatically changes the SCC. Nordhaus (2018 Nobel Prize) uses ~4%: SCC ~$40/tonne. Stern Review (2006) uses ~1.4%: SCC ~$85–$350/tonne. The choice reflects ethical judgements about intergenerational equity.

9.2 Integrated Assessment Models (IAMs)

IAMs combine economic and climate models:

• DICE model (Nordhaus): a single-region, aggregate model. Economy emits CO₂; CO₂ warms climate; damage function reduces GDP; optimal policy minimizes total cost (mitigation + damages) over time.

\[ Y(t) = A(t) K(t)^\gamma L(t)^{1-\gamma} \quad \text{(production)} \]\[ C(t) = Y(t)\left[1 - \Omega(T)\right] - I(t) \quad \text{(consumption, net of climate damage)} \]

where \( \Omega(T) = 1 - (1 + \psi_1 T + \psi_2 T^2)^{-1} \) is the damage function.

9.3 Policy Instruments

Instrument	Description	Mathematical form
Carbon tax	Price placed on each tonne of CO₂ emitted	Sets \( p_{CO_2} = \text{SCC} \); emitters reduce until marginal abatement cost = \( p_{CO_2} \)
Cap-and-trade	Total emissions cap; permits traded	Market price = shadow price of cap; efficient allocation via equilibrium
Regulation	Technology or performance standards	Minimum efficiency or maximum emission rate
Subsidy	Support for clean technologies	Reduces cost of low-carbon alternatives

Efficiency theorem (Pigou): a Pigouvian tax equal to the externality (SCC) achieves the socially optimal level of emissions without command-and-control mandates.

Political economy: even efficient policies face distributional concerns (carbon taxes can be regressive unless revenues are recycled) and political barriers (concentrated fossil fuel interests; short political time horizons relative to climate time horizons).

9.4 Tipping Points and Non-Linearity

The economic analysis is complicated by tipping elements — large, potentially irreversible changes at climate thresholds:

• West Antarctic and Greenland ice sheet collapse.
• Amazon rainforest dieback.
• Permafrost carbon release.
• AMOC collapse.

These introduce non-linearity, fat-tailed risk distributions, and the possibility of climate tipping cascades — one tipping point triggering others, potentially precluding stabilization.

Weitzman’s Dismal Theorem: if there is significant probability of catastrophic, arbitrarily large damages (fat-tailed distribution), standard cost-benefit analysis may assign infinite weight to mitigation — making the discount rate irrelevant. The key policy question becomes: how much insurance is worth buying against low-probability catastrophic outcomes?