96843 - Numerical Laboratory of the Atmosphere and Ocean

Learning outcomes

Understanding the formulation of general circulation models and climate models. Knowledge of historical development and applications of General Circulation Models for the ocean and the atmosphere. Basic knowledge of the numerical algorithms used to solve the primitive equations. Ability to write numerical codes to integrate hyperbolic conservation laws in one and multiple dimensions. Basic knowledge of a UNIX-like shell and job submission in a HPC environment. Compiling and running a model, design and implementation of a numerical simulation. Interpretation and use of a self-describing, array-oriented data format. Expertise with analysis and post-processing of model outputs and simple strategies to handle large datasets. Interpretation of model results in view of governing equations.

Course contents

MODULE I

-Introduction
Overview of the course, History and development of Atmospheric General Circulation Models, global and mesoscale numerical weather predictions, climate models.

-Numerical Methods I
Classification of PDEs, consistency and stability analysis,
semi-lagrangian schemes, spectral methods, grids

-Hands-on Session: Numerical Integration of the Lorenz 63 system
Deriving the Lorenz system, the coding environment, writing the code for the integration, exercises on fundamentals of chaotic systems, conceptual example of an ensemble forecast.

-Hands-on Session: Numerical integration of the barotropic vorticity equation
Implementation of a code to solve the barotropic vorticity equation, filtering approximations, recreation of the
pioneering numerical weather prediction made by Charney, Fjörtoff and von Neumann.

-Hands-on Session: Simulations with an atmospheric General Circulation Model
Vertical coordinates in AGCMs, subgrid-scale processes, overview of model dynamical core and parameterizations. Computation of the model climatology, a sensitivity simulation.

-A Numerical Weather Prediction with the ECMWF Integrated Forecast System
What is a weather forecast system, the rationale of data assimilation, forecast and discussion of a case study, post-processing of the output and forecast products.

 

MODULE II

-Introduction
Overview of the course, Ocean General Circulation Models, Governing Equations.

-Numerical methods for Hyperbolic Equations
Hyperbolic Conservation Laws, Method of characteristics, Linear and Non-linear hyperbolic equations, Finite Difference and Finite Volume Methods, First-order linear schemes (e.g. Godunov Upwind, Lax-Friedrichs), Second-order linear schemes (e.g. Lax-Wendroff, Warming and Beam, Fromm), Monotone schemes, Godunov’s theorem, Non-linear schemes, Total Variation Diminishing (TVD) Methods.

-Hands-on Session: Numerical Integration of the 1D non-linear Inviscid Burgers’ equation
Coding environment, implementation of the program to solve the 1D inviscid Burger’s equation, comparison of the numerical solutions for several numerical schemes.

-Hands-on Session: Numerical Integration of the 1D shallow water equations
Derivation of the shallow water equations, linearization and analytic solution of the shallow water equations, implementation of the program to solve the Shallow water equations.

-Numerical methods for Parabolic Equations
Diffusion-Reaction Equations, Explicit and Implicit schemes. Crank-Nicolson method.

-Hands-on Session: Numerical Integration of the 1D diffusion equation
Implementation of the program to solve the 1D diffusion equation, comparison of explicit and Implicit Finite Difference schemes.

-Analysis and Applications of the NEMO ocean modelling code.
Analysis of the NEMO ocean modelling code, description of the simulation frameworks, Bathymetry, Numerical Grids, Surface and Lateral Boundary Conditions, Parametrization of Subgrid Scale Processes, Vertical mixing processes, Dynamical downscaling techniques.

-Hands-on Session: Implementation of 1D model of water column

Implementation of a 1D model of water column with NEMO-1D, comparison of the effects of different ocean vertical mixing schemes on the vertical structure of the water column.

-Hands-on Session: Implementation of a high-resolution limited area, nested ocean model
Spatial interpolation methods, the SURF-NEMO Relocatable Platform, implementation of a high-resolution nested ocean model, visualization, and analysis of the model results.

 

Readings/Bibliography

 

Atmospheric Modeling, Data Assimilation and Predictability, E. Kalnay, Cambridge university press

Riemann solver and Numerical Methods for Fluid Dynamics: A Practical Introduction, Eleuterio Toro, Springer.

Fundamental of ocean climate models, Stephen M. Griffies, 2003, Princeton University Press.

Numerical Methods for Fluid Dynamic with Applications to Geophysics, Dale R. Durran, Springer

Lecture slides

Teaching methods

Classroom lectures and hands-on sessions with numerical simulations

Assessment methods

Reports developed during the course are evaluated with a qualitative mark. The final mark is determined with an oral examination focused on the theoretical part of the lectures and on the discussion of the reports. 

The duration of the exam is about 30 minutes. 

Teaching tools

Slide projector and laboratory.

Office hours

See the website of Paolo Ruggieri

See the website of Francesco Trotta