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    • Tropical Cyclones
      • Tropical cyclone climatology and overview
      • Tropical cyclone life cycle and motion
      • Tropical cyclone structure
      • Hurricane forecasting learning material
      • Hurricane forecasting tools
      • Storm surge
      • TUTTs and LA/MS flood event of 2016
    • Ocean applications
      • Waves
      • Tides
      • Miscellaneous ocean products
    • Streamlines
    • Regression, MOS, and NBM
    • Forecasting baroclinic systems and other products
      • Analysis
      • Model guidance
      • Model Output Statistics and National Blend of Models
      • Useful forecast products
  • Synoptic class
    • Syllabus
    • Exam Information
    • Homework
    • Stability
      • The Basics
      • The SkewT and related diagnostic tools
      • POP, air mass thunderstorms, sea breeze thunderstorms
      • Severe Weather
    • Map analysis
      • Upper-level synoptic charts
      • Contouring
      • Surface analysis and fronts
      • Vertical structure
      • Jet Streaks
    • Dynamics
      • Review of dynamics
      • Dynamics applications
      • QG Theory and the Omega equation
      • Cyclogenesis and baroclinic instability
    • Modeling
    • Fog
    • Winter Weather
  • Intro Dynamics
    • Syllabus
    • Homework
    • Sample exam questions
    • Introduction (Chap. 1, HH)
    • Basic equations of meteorology (Chap. 2, HH)
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 2 material
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 3 material
    • Planetary boundary layer
    • Vorticity (Chap.4, HH)
    • Pertubation method and atmospheric waves (Chap 5, HH)
  • Numerical methods
    • Syllabus
    • Homework
    • Number Series
    • Interpolation
    • Basic matrix math
    • Filters And Fourier Analysis
    • Numerical derivatives
    • Numerical integration, random numbers, and Monte Carlo
    • Numerical solutions of differential equations and atmospheric modeling
    • Parameterization, data assimilation, and overview on WRF model
    • Final computer exercises
  • Dashboard
 Basic differential equation formulations
  • Classifications of partial differential equations (PDEs)
  • ODEs versus PDEs
Solving Dx/Dt=f(x,t)
  • Examples of time differencing schemes
  • Overview on Euler, Runge-Kutta, and multistep methods
  • A comparison of RK versus Leapfrog. Note the errors in phase speed and amplitude. Also note Leapfrog has a (fictional) computation mode. This is why WRF uses a high-order Runge-Kutta.
  • Centered-in-time, centered-in-space scheme (Leapfrog scheme) example for advection equation
  • Semi-Lagrangian schemes
  • Supplementary material:
          1)  Swathmore College lecture on numerical solutions to DEs
          2)  CK12.org material on Euler's Method and Runge-Kutta
          3)  Saylor Academy notes on Runge-Kutta
Trajectories 
  • Pertinent equations and basics
  • Example applications for computing trajectories
          1) a. Run NOAA's HYSPLIT model! b. Recent BAMS paper on HYSPLIT and its applications
          2) Dr. Fitz's simulation of the Deepwater Horizon oil spill
          3) Inflow trajectories into hurricanes
          4) a. Cross-ocean balloon flight planning b. More on the first Transpacific balloon flight c. More on Steve Fossett d. Article on Steve Fossett
Modeling overview
  • Overview on weather modeling process (dated document on modeling, but conceptually correct)
  • Stull's overview
Spectral method (used in global models and climate models)
  • Pseudo-spectral example
  • Advantage and disadvantage of the spectral method
  • Comparison of grid point and spectral methods
  • Example with spherical harmonics. What are spherical harmonics?: 1) The classic paper on spherical harmonics by Baer. 2) Advanced graduate school material from CSU professor David Randall.
Example solutions
  • Sample solutions
Spreadsheet examples
  • Euler compared to 2nd-order Runge-Kutta
  • 3rd-order Runge-Kutta
  • 4th-order Runge-Kutta
  • Euler compared to improved Euler with VBA function
  • Euler compared to 4th-order Runge-Kutta with VBA function
FORTRAN basics
  • What is FORTRAN?
  • Compiling and running FORTRAN code
  • Supplementary material - Dr. Paul Tackley's FORTRAN notes and additional tutorials
  • Supplementary material - website on FORTRAN
FORTRAN examples (to be used as part of final exam)
  • Basic Leapfrog example
  • Basic Leapfrog sensitivity to Courant number. What happens if the Courant number>1, =1, and <1?
  • Forward-in-time, backward-in-space (FTBS). How well does this actually work?  
  • 4th-order Runge-Kutta example
Other computer algorithm examples
  • Matlab example with ODE45. More information on the ODE45 algorithm.
  • More MATLAB examples. Accompanying code is below:
          1) EulerODE
          2) HeunODE
          3) Function RK4
          4) Adams-Bashforth
          5) Adams-Bashforth-Moulton predictor-corrrector
  • FORTRAN, First order ODE -  Euler, Predictor-Corrector, Runge-Kutta 4th order
  • FORTRAN, Second order single ODE - Euler, Predictor-Corrector, Runge-Kutta 4th order
  • Mathematica ODE solver
"Simple" model codes
  • Dr. Fitz's barotropic model
  • Kerry Emanuel's balanced vortex model. The README.
  • A barotropic/ reduced gravity 1.5 layers QG model. Courtesy of Dr. Tziperman
  • ENSO delay oscillator model. Courtesy of Dr. Tziperman. Documented in this paper. 
Advanced material on stability, amplitude errors, phase speed errors, and computational mode
  • Dr. Roger Smith's notes
  • Dr. Fitz's detailed analysis of FTCS scheme
  • Dr. Fitz's detailed analysis of Leapfrog scheme
  • Dr. Fitz's detailed analysis of implicit scheme
  • Dr. Fitz's von Neumann analysis on a filter scheme
  • Dr. Roger Pielke's analysis plots
  • Wicker and Skamarock (2002) paper on Runka-Kutta
  • Dr. Eugenia Kalnay's' summary analysis of time-differencing schemes
  • Dr. Eugenia Kalnay's detailed lectures on the subject. Part 1. Part 2.
  • Positive definite advection schemes
  • Review on positive definite advection schemes
  • ECMWF lecture on Semi-Lagrangian schemes
Recorded lectures
  • Time-stepping schemes, Part 1. Audio only.
  • Time-stepping schemes, Part 2. Audio only.
  • The spectral method. Audio only.
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  • Home
  • Contact information
  • Synoptic lab
    • Syllabus
    • Take-home exams
    • Homework
    • Tropical Cyclones
      • Tropical cyclone climatology and overview
      • Tropical cyclone life cycle and motion
      • Tropical cyclone structure
      • Hurricane forecasting learning material
      • Hurricane forecasting tools
      • Storm surge
      • TUTTs and LA/MS flood event of 2016
    • Ocean applications
      • Waves
      • Tides
      • Miscellaneous ocean products
    • Streamlines
    • Regression, MOS, and NBM
    • Forecasting baroclinic systems and other products
      • Analysis
      • Model guidance
      • Model Output Statistics and National Blend of Models
      • Useful forecast products
  • Synoptic class
    • Syllabus
    • Exam Information
    • Homework
    • Stability
      • The Basics
      • The SkewT and related diagnostic tools
      • POP, air mass thunderstorms, sea breeze thunderstorms
      • Severe Weather
    • Map analysis
      • Upper-level synoptic charts
      • Contouring
      • Surface analysis and fronts
      • Vertical structure
      • Jet Streaks
    • Dynamics
      • Review of dynamics
      • Dynamics applications
      • QG Theory and the Omega equation
      • Cyclogenesis and baroclinic instability
    • Modeling
    • Fog
    • Winter Weather
  • Intro Dynamics
    • Syllabus
    • Homework
    • Sample exam questions
    • Introduction (Chap. 1, HH)
    • Basic equations of meteorology (Chap. 2, HH)
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 2 material
    • Imbedded processes in equations of meteorology (Chap. 3, HH). Exam 3 material
    • Planetary boundary layer
    • Vorticity (Chap.4, HH)
    • Pertubation method and atmospheric waves (Chap 5, HH)
  • Numerical methods
    • Syllabus
    • Homework
    • Number Series
    • Interpolation
    • Basic matrix math
    • Filters And Fourier Analysis
    • Numerical derivatives
    • Numerical integration, random numbers, and Monte Carlo
    • Numerical solutions of differential equations and atmospheric modeling
    • Parameterization, data assimilation, and overview on WRF model
    • Final computer exercises
  • Dashboard