# Lecture notes

## Introductory lectures for NWP Chaos and weather prediction

by Roberto Buizza

### Contents

1. Introduction
2. The Lorenz system
3. Numerical weather prediction
4. Sources of forecast error
5. The ECMWF Ensemble Prediction System
6. Targeted observations
7. Summary and future developments
8. Conclusion
9. Acknowledgements
10. References Atmospheric waves

by Martin Miller

### Contents

1. Introduction
2. Basic equations
3. Exact solutions of the linearized equations
4. Simplified solutions to the linearized equations - filtering approximations
5. Surface gravity waves
6. Equatorial waves
7. Summary diagram and modelling implications
8. References Properties of the equations in motion

by Mike Cullen

### Contents

1. Introduction
2. Observed behaviour
3. Toy problems
4. Shallow water equations
5. Three dimensional equations
6. Averaged equations
7. Summary
8. References Data assimilation concepts and methods

by F Bouttier and P Courtier

### Contents

1. Basic concepts of data assimilation
2. The state vector, control space and observations
3. The modelling of errors
4. Statistical interpolation with least-squares estimation
5. A simple scalar illustration of least-squares estimation
6. Models of error covariances
7. Optimal interpolation (OI) analysis
8. Three-dimensional variational analysis (3D-Var)
9. 1D-Var and other variational analysis systems
10. Four-dimensional variational assimilation (4D-Var)
11. Estimating the quality of the analyses Implementation techniques
12. Dual formulation of 3D/4D-Var (PSAS)
13. The extended Kalman filter (EKF)
14. Conclusion
15. Appendix A. A primer on linear algebra
16. Appendix B. Practical adjoint coding
17. Appendix C. Exercises
18. Appendix D. Main symbols
19. References The general problem of parametrization

by M Tiedtke

### Contents

1. Introduction
2. The spectrum of atmospheric motions
3. The non-parametrized equations
4. Principles of parametrization
5. References

## Data Assimilation Data assimilation concepts and methods

By F Bouttier and P Courtier

### Contents

1. Basic concepts of data assimilation
2. The state vector, control space and observations
3. The modelling of errors
4. Statistical interpolation with least-squares estimation
5. A simple scalar illustration of least-squares estimation
6. Models of error covariances
7. Optimal interpolation (OI) analysis
8. Three-dimensional variational analysis (3D-Var)
9. 1D-Var and other variational analysis systems
10. Four-dimensional variational assimilation (4D-Var)
11. Estimating the quality of the analyses Implementation techniques
12. Dual formulation of 3D/4D-Var (PSAS)
13. The extended Kalman filter (EKF)
14. Conclusion
15. Appendix
• A. A primer on linear algebra
• C. Exercises
• D. Main symbols
16. References Assimilation Algorithms

By EV Holm

### Contents

1. Basic concepts
2. Variational data assimilation
3. Common assimilation algorithms
4. References Assimilation techniques (3): 3dVar

By M Fisher

### Contents

1. Introduction
2. The incremental method
3. The background cost function
4. References Assimilation techniques (4): 4dVar

By M Fisher

### Contents

1. Introduction
2. Comparison between the ECMWF 3dVar and 4dVar systems
3. The current operational configuration of 4dVar
4. Increments from a single observation
5. A cautionary example
6. References Assimilation techniques (5): Approximate Kalman Filters and Singular Vectors

By M Fisher

### Contents

1. Introduction
2. Why is the Kalman filter impractical for very large systems?
3. The ensemble Kalman filter
4. Subspaces, projections and Hessian singular vectors
5. The ECMWF reduced-rank Kalman filter
6. Examples
7. References A data assimilation tutorial based on the Lorenz-95 system

By M Leutbecher

### Contents

1. Introduction
2. Data assimilation experiments
3. Outlook
4. References Adaptive observations, the Hessian metric and singular vectors

By M Leutbecher

### Contents

1. Introduction
2. Adaptive observations in the Lorenz 95 system - Methodology
3. Adaptive observations in the Lorenz 95 system - Results
4. Reduced rank prediction of forecast error variance reductions in an operational NWP context
5. Discussion
6. Conclusions
7. References Observation Impact on the Short Range Forecast

By C Cardinali

### Contents

1. Introduction
2. Observational Impact on the Forecast
3. Results
4. Conclusion
5. References Observation influence diagnostic of a data assimilation system

By C Cardinali

### Contents

1. Introduction
2. Classical Statistical Definitions of Influence Matrix and Self-Sensitivity
3. Observational Influence and Self-Sensitivity for a DA Scheme
4. Results
5. Conclusions
6. Appendix:
• A. Influence Matrix Calculation in Weighted Regression Data Assim. Scheme
• B. Approximate calculation of self-sensitivity in a large variational analysis system
7. References Observations and diagnostic tools for data assimilation

By H Järvinen

### Contents

1. Observational preprocessing
2. The observation screening
3. Use of feedback information
4. Diagnostic tools for an assimilation system
5. References The control of gravity waves in data assimilation

By A Simmons

### Contents

1. Introduction
2. Non-linear normal-mode initialization
3. Control of gravity waves in the ECMWF variational data assimilation
4. Digital filtering
5. Appendix
• A. Definition of operators for the ECMWF vertical finite-difference scheme
• B. The Lamb wave
• C. The non-recursive implementation of the recursive filter
6. References Principles of remote sensing of atmospheric parameters from space

By R Rizzi (revised by R Saunders)

### Contents

1. Introduction
2. Absorption and transmission of monochromatic radiation
4. Emissivity, Kirchoff law and local thermodynamical equilibrium
5. The equation for radiative transfer
6. Spectral distribution of radiance leaving the atmosphere
7. Modelling the interaction
8. Line shapes and the absorption coefficient
9. Continuum absorption
10. Integration over frequency
11. The direct problem
12. References Inversion methods for satellite sounding data

By J Eyre

### Contents

1. Basic ideas
2. Temperature profile inversion methods
3. Constituent profile inversion
4. Clouds
5. Satellite sounding in numerical weather prediction
6. References Introduction to microwave radiative transfer

By P Bauer

### Contents

3. Atmospheric absorption
4. Surface emission / reflection
5. Hydrometeors
6. Fast models
7. Outstanding issues
8. References Microwave radiative transfer modeling in clouds and precipitation - Part I

By P Bauer

### Contents

1. Introduction
4. Optical properties
5. References Microwave radiative transfer modeling in clouds and precipitation - Part II

By E Moreau, P Bauer and F Chevallier

### Contents

1. Introduction
3. Comparison with other models
4. References Land surface assimilation

By J-F Mahmouf and P Viterbo

### Contents

1. Introduction
2. Design of land surface parametrizations
3. Introduction to land surface assimilation
4. Simple land surface initialisation methods
5. Soil moisture initialisation using SYNOP observations
6. Other techniques to initial soil moisture
7. Initialisation of other land surface quantities
8. Conclusions
9. References

## Numerical Methods Atmospheric Waves

By M Miller

### Contents

1. Introduction
2. Basic equations
3. Exact solutions of the linearized equations
4. Simplified solutions to the linearized equations - filtering approximations
5. Surface gravity waves
6. Equatorial waves
7. Summary diagram and modelling implications
8. References Properties of the equations of motion

By M Cullen

### Contents

1. Introduction
2. Observed behaviour
3. Toy problems
4. Shallow water equations
5. Three dimensional equations
6. Averaged equations
7. Summary
8. References Numerical methods

By RW Riddaway and M Hortal

### Contents

1. Some introductory ideas
2. Finite differences
4. Towards the primitive equations
5. The semi-Lagrangian technique
6. The spectral method
7. The finite-element technique
8. Solving the algebraic equations
9. References Adiabatic formulation of large-scale models of the atmosphere: Finite schemes for the horizontal discretization

By AJ Simmons

### Contents

1. Introduction
2. Distribution of grid points
3. Staggering of variables
4. Conservation and Eulerian advection schemes
5. Treatment of the poles
6. References Adiabatic formulation of models

By T Davies

### Contents

1. Introduction
2. Governing equations
3. Horizontal discretization
4. Vertical coordinates
5. Vertical discretization
6. References The wave model

By P Janssen

### Contents

1. Derivation of the energy balance equation
2. The WAM model
3. Benefits for atmospheric modelling
4. References

## Parametrization The general problem of parametrization

By M Tiedtke

### Contents

1. Introduction
2. The spectrum of atmospheric motions
3. The non-parametrized equations
4. Principles of parametrization
5. References Parametrization of non-convective condensation processes

By M Tiedtke

### Contents

1. Thermodynamics of moist air
2. Cloud physical processes
3. Parametrization of non-convective processes and precipitation processes in numerical models
4. References Atmospheric moist convection

By P Bechtold

### Contents

1. The nature of moist convection
2. Parameterization of convection
3. The IFS convection parameterization
4. Forecasting - case studies
5. Appendix
• Simple wave types and quasi-geostrophic adjustment
6. References Atmospheric thermodynamics

By P Bechtold

### Contents

1. Ideal gas law
2. First law of thermodynamics
3. Second law of thermodynamics
4. Enthalpy
5. State functions and Maxwell relations
6. Humidity variables
7. Virtual temperature
8. Reversible adiabatic transformation without phase change
9. Reversible adiabatic transformation with phase change
10. Clausius Clapeyron equation
11. Ways of reaching saturation
12. Energy diagrams - Tephigram
13. Saturation adjustment - numerical procedure
14. References Improved middle atmosphere climate and forecasts in the ECMWF model through a non-orographic gravity wave drag parametrization

By A Orr, P Bechtold, J Scinocca, M Ern and M Janiskova

### Contents

1. Introduction
2. The S03 gravity wave scheme
3. Climate experiments, parameter settings, and validating data
4. The extratropical temperature structure and circulation
5. Planetary waves
6. Tropical oscillations
7. Parametrized and resolved momentum flux
8. High resolution middle atmosphere forecast error and analysis
9. Summary and discussion
10. References The mass-flux approach to the parametrization of deep convection

By D Gregory

### Contents

1. Introduction
2. The need for parametrization
3. Mass-flux theory of convection
4. Evaluation of mass-flux theory
5. Evaluation of convective schemes using cloud-resolving models
7. References Sensitivity of general circulation model performance to convective parametrization

By D Gregory

### Contents

1. Introduction
2. The importance of shallow convection
3. Sensitivity of climate simulations to the parametrization of deep convection
4. Summary
5. References The parametrization of cloud cover

By AM Tompkins

### Contents

1. Introduction
2. Background
3. Relative humidity schemes
4. Statistical schemes
5. The ECMWF prognostic cloud cover scheme
6. Summary
7. References The parametrization of the planetary boundary layer

By A Beljaars

### Contents

1. Introduction
2. Similarity theory and surface fluxes
3. PBL schemes for atmospheric models
4. List of symbols
5. References A review of parametrization schemes for land surface processes

By P Viterbo

### Contents

1. Introduction
2. General remarks
3. Soil energy and water budget
4. Examples of parametrization schemes
5. Subgrid-scale heterogeneity
6. Validation and intercomparison
7. Initial values
8. Snow modelling
9. Conclusions
10. References The role of the land surface in the climate system

By P Viterbo

### Contents

1. Introduction
2. Surface energy and water budget
3. Time scales and the role of soil moisture
4. Impact of land surface on weather: A brief literature survey
5. Examples from ECMWF recent experience
6. Conclusions
7. References Theory of linear gravity waves

By T Palmer

### Contents

1. Simple properties of internal gravity waves
2. Gravity-wave drag
3. References Subgrid-scale orographic drag

By MJ Miller

### Contents

1. General principles
2. Description of the scheme
3. Specification of subgrid-scale orography
4. Appendix
• A. List of symbols Radiation transfer

By J-J Morcrette

### Contents

1. Introduction: An historical perspective
2. The Earth's radiative balance and its implications
3. The theory of radiation transfer
4. Radiation schemes in use at ECMWF
5. Comparisons with observations
6. Conclusions and perspective
7. References

## Predictability Chaos and weather prediction

By R Buizza

### Contents

1. Introduction
2. The Lorenz system
3. Numerical weather prediction
4. Sources of forecast error
5. The ECMWF Ensemble Prediction System
6. Targeted observations
7. Summary and future developments
8. Conclusion
9. Acknowledgements
10. References Predicting uncertainty in forecasts of weather and climate

By T Palmer

### Contents

1. Introduction
2. The Liouville equation
3. The probability density function of initial error
4. Representing uncertainty in model formulation
5. Error growth in the linear and nonlinear phase
6. Applications of singular vectors
7. Forecasting uncertainty by ensemble predictions
8. Verifying forecasts of uncertainty
9. The economic value of predicting uncertainty
10. Concluding remarks