ATMO 430

Computational Methods in Atmospheric Sciences

Introduction to computational methods used in solving problems in the Atmospheric Sciences. Emphasis is on numerical techniques used in developing numerical weather prediction and climate models and in radiative transfer. Knowledge of FORTRAN is required. Also includes an introduction to statistical analysis of observational data and statistical prediction.

Prequisites: Math 254, ENGR 170, ATMO 300a,b or consent of the instructor.

Required Textbook:

Handbook of Mathematical Formulas and Integrals, Alan Jeffrey

Supplemental Textbooks:

An Introduction to Numerical Weather Prediction Techniques, T. Krishnamurti and L. Bounoua

Numerical Prediction and Dynamic Meteorology, G. Haltiner and R.T. Williams

Grading Policy:

There will be weekly homework problems and a series of laboratory exercises. The final grade will be based on the homework, labs, a midterm exam and a final exam as follows:

Homework 1/5

Lab Exercises 1/5

Midterm 1/5

Final 2/5

 

Homeworks and lab exercises must be handed in on time in order to receive full credit. A penalty of 10% per day for each day late will be incurred.

Topics

Fundamentals (Week 1)

Complex and real numbers

Logarithms and Exponentials

Trigonometric and Hyperbolic Functions

Differential Calculus (Weeks 2-3)

Derivatives, product, quotient and chain rules.

Derivatives of elementary functions

Taylor Series representation of functions

Partial Derivatives

Finite Difference Formulae for Evaluating Total and Partial Derivatives

COMPUTER EXERCISE: Numerical Differentiation

Scalars and Vectors

Gradient of a Scalar

Divergence and Curl of Vector Fields

Vector Identities

COMPUTER EXERCISE: Newton's Method for Solving Transcendental Equations

Integral Calculus (Weeks 4-5)

Basic Techniques of Integration

Use of Tables of Integrals

Fourier Series and Integrals

COMPUTER EXERCISE: Numerical Integration using the Trapezoid and Simpsons's rule

Differential Equations (Weeks 6 - 12)

Fundamentals of Second Order Linear Differential Equations.

Series Solutions of Second Order ODEs

Introduction to Special Functions

Numerical Solutions of ODEs

COMPUTER EXERCISE: Numerical Solution of ODE's by Euler's and Runge-Kutta Methods

Fundamentals

Laplace and Poisson's Equations

Numerical Solutions of Second Order, Linear Partial Differential Equations

COMPUTER EXERCISE: SOR Solution of Poisson's Equation

Solutions of the Advection Equation

Euler and Leapfrog Schemes

CFL Stability Criteria

Explicit, Implicit and Semi-Implicit Schemes

Spectral Techniques

COMPUTER EXERCISE: Passive Scalar Advection

Statistics (Weeks 13 -16)

Basic definitions

Theoretical probability distributions

Hypothesis testing

Time series analysis

Curve Fitting, square regression and correlation coefficients

Probabilistic field (ensemble) forecasts

COMPUTER EXERCISE: Hypothesis testing of meteorological data

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