Theoretical Environmental Analysis

Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Prerequisites and Corequisites

The course requires elementary physical reasoning, and therefore 8.01T Physics I is a prerequisite. The course also requires mathematical expertise at the level of 18.02SC Multivariable Calculus. Finally, 18.03SC Differential Equations is a corequisite.

Course Description

This course analyzes cooperative processes that shape the natural environment, now and in the geologic past. It emphasizes the development of theoretical models that relate the physical and biological worlds, the comparison of theory to observational data, and associated mathematical methods. Topics include carbon cycle dynamics; ecosystem structure, stability and complexity; mass extinctions; biosphere-geosphere coevolution; and climate change. Employs techniques such as stability analysis; scaling; null model construction; time series and network analysis.

Objectives

The principal objectives of this course are twofold. First, it provides students with an understanding of the mechanisms that underly the organization of the natural environment; i.e., how nature works. Second, it introduces students to methods of quantitative analysis that are useful for investigating how any system works. The course teaches students how to identify fundamental phenomena, how to formulate theoretical models, and how to quantitatively test models by comparison to observations. Students are provided with real datasets so that they can engage in these processes independently and creatively.

A secondary but no less important objective is to provide students with a unified view of environmental science. The unification is made possible by emphasizing aspects of earth, atmospheric, and planetary sciences that collectively act to create the natural environment, both physical and biological. We feature several of the great advances of 20^th-century science (e.g., plate tectonics, climate cycles, and chaos theory) and introduce modern mathematical models of complex phenomena that remain to be understood. In so doing, we teach methods of analysis that are applicable throughout science and engineering.

Overview

The first part of the course (topics 1–3) is loosely organized around Earth’s carbon cycle: The injection of CO₂ into the atmosphere and oceans by volcanos and other tectonic processes, the exchange of CO₂ between the atmosphere and oceans, and the runoff of dissolved carbon from rivers into the oceans. In each case, we emphasize the role of diffusion, perhaps the simplest and most important mode of transport in the natural environment.

In the second part of the course (topics 4–5) we focus on climate cycles, their foundation in orbital dynamics, and methods for the analysis of periodic phenomena in general. We learn how to compute and interpret power spectra, one of the most important tools used in the analysis of any system that evolves with time.

In the final part of the course (topics 6–7) we discuss the physical basis of ecological organization, the dynamics of ecological communities, and nonlinear dynamics in general. Here we meet concepts of scaling, stability, and the geometry of natural networks. The course ends with an introduction to the greatest intellectual achievement arising from the study of environmental dynamics: the theory of chaos.

Topics

1. Introduction
   1. Themes, objectives, and expectations
   2. The biological and geological carbon cycles
   3. Climate cycles
   4. Ecological organization and dynamics
2. Plate Tectonics: The Volcanic Source
   1. Volcanism as a long-term CO₂ source
   2. Thermal convection within the Earth; the Rayleigh number
   3. Seafloor heat flux, topography, and thermal diffusion
   4. Diffusive scaling
3. Short-term Evolution Atmospheric CO₂
   1. The Keeling curve
   2. The radiocarbon bomb spike as an impulse response
   3. Microscopic (random-walk) model of molecular diffusion
   4. Diffusive exchange with the oceans
4. Scaling Laws for Rivers and Runoff
   1. Fluvial transport as a sink for CO₂
   2. The geometry of river basins
   3. Power laws, fractals, allometry, and scale invariance
   4. Random-walk model, null models, and universality
5. Natural Climate Change: Glacial Cycles
   1. Ice-core records of climate change
   2. Milankovitch cycles
   3. Precession, obliquity, eccentricity, and insolation
   4. Enigmatic significance of the eccentricity time scale
   5. Enigmatic correlation of climate and CO₂
6. Quantitative Analysis of Periodic Phenomena
   1. Discrete Fourier transform
   2. Power spectrum and autocorrelation function
   3. Power spectra of periodic signals and white noise
   4. Identification of spectral peaks
7. Ecological Organization
   1. Energetic limits on the length of food chains
   2. Food webs and scale-free networks
   3. Body size, temperature, and metabolic scaling
   4. Ecological equipartition
8. Ecosystem Stability and Chaos
   1. Elements of population dynamics
   2. Linear stability analysis
   3. Predator-prey cycles: The Lotka-Volterra equation
   4. Stability vs. complexity; the fossil record of biodiversity and mass extinctions
   5. Discrete logistic model and period doubling
   6. Chaos

Requirements

There are quasi-weekly problem sets, a take-home midterm exam, and a take-home final exam due on the last day of classes. Students are welcome to collaborate on weekly problem sets but, if you do so, we ask that you list the names of the students with whom you have collaborated along with your own name. The take-home midterm and final exam must, however, be individual work. Students are not permitted to consult old solution sets.

Grading

ACTIVITIES	PERCENTAGES
Problem Sets	50%
Midterm Exam	10%
Final Exam	30%
Class Participation	10%

Late problem sets will be accepted until the graded problems sets are returned. However, late problem sets will be penalized by 10% each day. Specifically, if the grade would have been s₀ had the problem set been turned in on time, the effective grade S_n for a problem set handed in n days late is s_n= s₀ x 0.9ⁿ