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Contents

Syllabus

This graduate-level course based on the book “Predictability of Weather and Climate”[PH06] will help you walk through some fundamental ideas in chaos (a.k.a. Butterfly effect), predictability and information theory. The goal is to learn how to (1) qualitatively and quantatively describe “predictability”, (2) physically interpret dynamical systems and (3) find the optimal patterns favoring model error growth in various systems. Prerequisites for this course including: applied math (ODE, PDE and linear algebra), numerical analysis (Python or Julia), statistics and atmospheric dynamics.

Note

This is the v1.0 handout of the course “Chaos and Predictability” taught in the Department of Atmospheric Science, National Taiwan University. There might be tons of typos. If you find any error, feel free to DM me (email Dr. Kai-Chih Tseng: kaichiht@princeton.edu)

Course Outlines

Part I: Lecturing

• Week 1: Predictability of Weather and Climate

  • Climatological distribution vs forecast distribution

  • State-dependent predictability and the mathematical assumptions

  • Where the uncertainty comes from?

  • Introduction to ensemble forecast in weather and climate

  • HW1: Lorenz 63 model (testing state-dependent predictability) (10%)

• Week 2: Predictability source in atmosphere

  • The origins of predictability (Scale Separation) and balanced dynamics

  • Atmospheric Blocking

  • Teleconnections

  • HW2: Baroclinic instability and Predictability (15%)

• Week 3: The minimalist models for studying chaos and predictability I: Lorenz 63

  • The origins of the story

  • Rayleigh-Barnard Convection

  • Lorenz 63 model I (solutions)

  • Lorenz 63 model II (stability analysis)

• Week 4: The minimalist models for studying chaos and predictability II: Lorenz 96

  • The physical and mathematical background

  • Single Scale

  • Multi-scale interations

  • HW3: Lorenz 96 model (calculating the Layapynov exponent) (10%)

• Week 5: The connection between statistical mechanics and ensemble forecast I: Theory

  • Introduction to Liouville equation

  • Jacobian Matrix and Determinant

  • Connections between Liouville equation and ensemble forecasts

• Week 6: The connection between statistical mechanics and ensemble forecast II: Applications

  • Liouville equation as a function of initial state vs current state

  • Solution to Liouville equation

  • Liouville equation in 1st-order ODE case

  • Liouville equation in Lorenz 63 model

  • Challengs of applying Liouville Equation to weather forecast

  • Liouville equation and Singular Value Decomposition

  • HW4: Derivation of Liouville equation + LE in 1st-order ODE and Lorenz 84 model (20%)

• Week 7: The stability theory

  • Direct Method

  • von Neumann method

  • Energy Method

  • Stability in Geophysical Fluid Dynamics (Shear instability and Eady problem)

• Week 8: The generalized stability theory

  • Generalized stability theorem

  • Modal vs non-modal systems

  • Reduced-dimension problems

• Week 9: Predictability through the lens of Information Theory

  • Perfect model experiments

  • Shannon Entropy and Average Predictability Time

  • Predictable Components

  • HW5: Identifying the predictable components in full-physics operational forecast systems (20%)

• Week 10: Future projection problems (Suppelmentary information: Nobel Prize Problem)

  • Initial value problems vs boundary value problems

  • Generalized stability and Predictable Components in a non-linear/boundary value problem

  • Hasselmann (1993) Optimal fingerprints for the detection of time-dependent climate change

  • Thompson et al. (2015) Quantifying the Role of Internal Climate Variability in Future Climate Trends

Part II: Literature Reviews (25%)

• Week 11: The predictability of a flow which possesses many scales of motions (Lorenz 1969)

  • Key Lecture: Week 8

• Week 12: Stochastic Forcing of ENSO by the Intraseasonal Oscillation (Moore and Kleeman 1999)

  • Key Lecture: Week 9

• Week 13: Ensemble-based sensitivity analysis (Hakim and Torn 2008)

  • Key Lecture: Week 9

• Week 14: The Critical Role of Non-Normality in Partitioning Tropical and Extratropical Contributions to PNA Growth (Henderson et al. 2020)

  • Key Lecture: Week 9

• Week 15: Physically Interpretable Neural Networks for the Geosciences: Applications to Earth System Variability (Toms et al. 2020)

  • Key Lecture: Week 10

Grading

HW X 5 (10%, 15%, 15%, 15% 20%) Literature Review (25%)

Book

PH06

Tim Palmer and Renate Hagedorn. Predictability of weather and climate. Cambridge University Press, 2006.

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Week 1: Predictability of Weather and Climate