Dear Haftamu,

Prediction of weather is connected with solving partial differential equations (Navier -Stokes equations). If you need to study this subject you should be good at solving numerically PDE (or ODE in simple model).

Nonlinear dynamics and mathematical fluid theory

go with Rui

Dear Haftamu,

I have my own experience of using mathematics for calculating and building air parcel trajectories. I have attached several articles related to applying trajectories for weather forecasting and prediction needs.

I can even tell you a story about my research concerning mathematics and weather prediction. In 2010 I met a man (a leading programmer from our capital Kiev) who had developed our national Ukrainian forecaster workstation. It was when his participating some scientific conference in Odessa. I simply asked him for the project source codes of the forecaster workstation (current operational version), and he gave me (at that time I was an unknown postgraduate student) a full C++Builder project (about 20 Mb of plain text files *.h, *.cpp and forms for C++Builder 5). He just laughed and said to me that I could not redistribute his soft in the Internet. He even joked (using words: play if you want) that I could compile and make my own versions of the forecaster workstation. This year I really compiled and used his powerful project for maintaining backward trajectory calculations (with wind velocity components taken from GRIB data of the American model NAVGEM) based on iterative algorithms of the famous Petterssen scheme and I made many visualizations on the built in maps. The results obtained had become an important part of the master’s thesis for a student whose thesis advisor was me. By this story I just wanted to say that unusual technical goodwill assistance of another (programming) specialist from the distant city had played a great role for my own small research. Now I have an outstanding opportunity to make various numeric (programming) experiments involving our national Ukrainian forecaster workstation and a full set of meteorological data from national hydrometeorological service ftp-server.

Unfortunately, I cannot attach many papers due to poor Internet connection. Some items are mentioned below.

1. Innocentini V. A successive substitution method for the evaluation of trajectories approximating the parcel path by a linear function of space and time // Mon. Wea. Rev. – 1999. – Vol. 127, No. 7. – P. 1639–1650.

2. Kuo Y.H., Skumanich M., Haagenson P.L., Chang J.S. The accuracy of trajectory models as revealed by the observing system simulation experiments // Mon. Wea. Rev. – 1985. – Vol. 113, No. 11. – P. 1852–1867.

3. Methods of Interpreting Numerical Weather Prediction Output for Aeronautical Meteorology: Technical Note No. 195 / developed by the CAeM Working Group on Advanced Techniques Applied to Aeronautical Meteorology. – Second Edition. – Geneva: Secretariat of the World Meteorological Organization, 1999. – 123 p.: ill., charts, maps. – (WMO–No. 770). – ISBN 92-63-12770-2.

4. Reap R.M. An operational three-dimensional trajectory model // J. Appl. Met. – 1972. – Vol. 11, No. 12. – P. 1193–1202.

5. Seibert P. Convergence and accuracy of numerical methods for trajectory calculations // J. Appl. Met. – 1993. – Vol. 32, No. 3. – P. 558–566.

Dear Haftamu,

I have attached a paper (Chapter 4 "Grids in Numerical Weather and Climate Models"), i.e. a part of the book, which may contain useful information on gridding techniques used in numerical weather prediction. This paper has a huge description of mathematical applications for numerical weather prediction tools. It also contains a brief history of the model grids. It can be interesting for a mathematician because there are some geometric (stereometric) aspects described too.

Regards,

Oleksandr.

May be the question is too broad and all previous answers are correct. However if the point is in numerical weather prediction per se, then the following book could be of interest:

Mesinger, Arakawa. Numerical methods used in atmospheric models. It is available  online: http://twister.ou.edu/CFD2003/Mesinger_ArakawaGARP.pdf

Generally if you decide to study numerical methods (in particular those used in weather prediction) then my advise is to read at least one book completely to have more or less consistent picture of the field. The above book is just one relevant example. Then you can start getting much more additional information from different parts of different books (in particular many CFD books are relevant, especially the book of Patankar, Numerical Heat Transfer and Fluid Flow, http://www.ewp.rpi.edu/hartford/~ernesto/F2012/CFD/Readings/Patankar-NHTFF-1980.pdf).

As many of the previous answers have alluded to, there are many areas of mathematics  which are required in order to do weather prediction. To keep this post short, let us limit the mathematical areas of focus to the following:

1. partial differential equations: The governing equations of motions are the compressible Navier-Stokes equations. To be more specific, they are the inviscid set which we call the Euler equations. These equations are hyperbolic and so a good understanding of hyperbolic PDEs is a must.

2. ordinary differential equations: Upon discretizing the spatial operators using numerical methods (see below), we end up with ODEs and so a good working knowledge of ODEs is essential.

3. Numerical Analysis: one needs a strong background in numerical analysis in order to understand Taylor series expansions that are the base of most methods used for the discretization of both the spatial operators (e.g. finite difference methods) and the time-derivatives (time-integration).

4. Linear Algebra: once you discretize the equations and solve the resulting ODEs in an implicit fashion, it will require some knowledge of linear algebra in order to solve  the resulting system matrix.  By the way, you will also need linear algebra for Data Assimilation, which is a huge part of weather prediction.

5. Advanced Numerical Methods: Numerical analysis will get you the basic understanding of many of the techniques that are required in weather prediction (e.g., Taylor series, interpolation, solution of ODEs, root finding, and the solution of nonlinear equations) but you will also need a more advanced course in numerical methods to learn how to discretize the spatial and temporal operators more accurately. There are usually advanced courses in the Numerical Methods for Partial Differential equations. In addition, in my department we also teach courses on Galerkin Methods (e.g., continuous and discontinuous finite elements). In fact, you can find a nice review paper on Galerkin methods in weather prediction here: https://www.researchgate.net/publication/274313551_A_Review_of_Element-Based_Galerkin_Methods_for_Numerical_Weather_Prediction._Finite_Elements_Spectral_Elements_and_Discontinuous_Galerkin

6. Finally, I would urge you to take a course on Scientific Computing. In my math department we teach such a course where we teach students the basic concepts in parallel computing (using the Message-Passing Interface API) and OpenCL/CUDA to learn how to exploit graphics processors (GPUs). One needs to develop the computer algorithms with a target computer in mind.

Hope this helps.

Article A Review of Element-Based Galerkin Methods for Numerical Wea...

the probability theory+ statistics+numerical modeling

The computations related to weather forecast is connected to  fluid dynamics. To be more precise, we solve equations related to fluid flow under certain conditions like conservation of mass, energy and continuity equation. The initial state of the atmosphere is defined with various parameters such as temperature, pressure, humidity etc. Then a future predicted state is reached through computations. In view of the non linearity of the relations between various state parameters, the exact solution of equations becomes very tedious. Hence numerical approach is adopted  where in the atmosphere is divided into grids and levels and equations are solved. 

Probability theory+math. statistics+numerical computations

 Actually to predict weather state it is necessary to solve systems of integral equations.

Probability and statistics are essential for model post-processing, statistical analysis and forecasting, ensemble prediction, and verification.

Clearly - Black Magic - is that part of mathematics?.  Or maybe White Magic

Ok you need all sorts of quantitative techniques, and straight out global scale numerical modelling wins.  But you also need satellites, observations, smaller scale numerical modelling, a degree in meteorology, some experience, some advice from those who really know (farmers and sailors) and I'm afraid to say - a good deal of luck to pick those out-of-the box serious events.

If you want to have people listen to it - on TV for instance - helps to be pretty, cute or at least authoritative.

And oh yes - persistence and thick skin - and that's surely part of maths,  You are going to get it wrong more often than right!  But if you manage to tick at least some of these boxes - stick with it,  Its a fulfilling, rewarding and potentially very productive career. Just about everyone is interested and you will meet some interesting people.  And see some interesting weather - and that makes it all worthwhile!  (Ok not all this is maths - but maybe it should be)

Weather prediction is intricately interdisciplinary. The necessary math includes: PDEs,, probability theory, statistics, and machine learning.(well, the last is not really math, but a science).

Stochastic modelling mathematics which is a brack or statistics and probability.

First of all, the partial differential equations and some methods of numerical analysis such as finite difference