This reference

Durran, Dale R. Numerical methods for wave equations in geophysical fluid dynamics. Vol. 32. Springer Science & Business Media, 2013.

could be a good starting point. The point mentioned in the previous post that geometries in atmospheric flows are simple is debatable, as soon as you want to consider orography and avoid coordinate transformation in the vertical direction.

This does not mean that spectral-like methods cannot be applied, but the recent trend is more towards spectral elements or high order finite elements (often DG) rather than the traditional spectral approaches, which however have had historically a key role in improving atmospheric models in the 70's and 80's.

[I do not know your answer, but the following is something worth to mention for your goal!]

In the realm of atmospheric numerical modeling, you usually deal with very large domains, and even for a very coarse resolution you need at least several million nodes/cells. Atmospheric phenomena are 3D and unsteady and turbulence is very important and you have to model state equation for temperature as well! (So you may ask okay why you tell me those facts?)

So they basically use two tricks for their modeling, one is "Down Scaling" and the other is "Higher Order Methods":

Since domains are very large, but FORTUNATELY they have simple geometry (sphere which can transformed to rectangle/cube easily!) In numerical modeling of atmospheric phenomena there is very common to use Spectral Methods (I do not mean "Spectral Element Methods" of FEM families) instead of FDM, FEM, and FVM. Those methods have very high order and you can get "good" result with coarse mesh compared to traditional discretization by FDM, FVM and FEM. If you want to learn more about those methods. Take a look at the book by Mohammed Yousuff Hussaini :

• Spectral Methods for Partial Differential Equations, with D. Gottlieb and R. G. Voigt, SIAM, Philadelphia, 1984.
• Spectral Methods in Fluid Dynamics, with C. Canuto, A. Quarteroni and T. A. Zang, Springer-Verlag, 1987.
• Spectral Methods: Fundamentals in Single Domains, with C. Canuto, A. Quarteroni and T. A. Zang, Springer-Verlag, 2006.
• Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, with C. Canuto, A. Quarteroni and T. A. Zang, Springer-Verlag, 2007.

This reference

Durran, Dale R. Numerical methods for wave equations in geophysical fluid dynamics. Vol. 32. Springer Science & Business Media, 2013.

could be a good starting point. The point mentioned in the previous post that geometries in atmospheric flows are simple is debatable, as soon as you want to consider orography and avoid coordinate transformation in the vertical direction.

This does not mean that spectral-like methods cannot be applied, but the recent trend is more towards spectral elements or high order finite elements (often DG) rather than the traditional spectral approaches, which however have had historically a key role in improving atmospheric models in the 70's and 80's.

Luca Bonaventura Kaveh Zamani Thank you so much for answering my question.

Hi Ananyo,

In addition to the reference suggested by Luca (Durran, 2013), I suggest two more references for numerical implementations:

Hirsch, C. (2007). Numerical computation of internal and external flows: The fundamentals of computational fluid dynamics. Elsevier.

Versteeg, H. K., & Malalasekera, W. (2007). An introduction to computational fluid dynamics: the finite volume method. Pearson education.

and the other two references for atmospheric concepts:

Kaimal, J. C., & Finnigan, J. J. (1994). Atmospheric boundary layer flows: their structure and measurement. Oxford university press.

Shao, Y. (2008). Physics and modelling of wind erosion (Vol. 37). Springer Science & Business Media.

The field of computational methods in atmospheric sciences is vast, as it includes calculations performed for multi-scale systems ranging from cloud nucleation processes to large-scale atmospheric fluid dynamics. The field also includes turbulence studies from simple laboratory systems to complex flows in convective clouds. Since atmospheric processes are affected by electric and magnetic fields, you must also consider the numerical methods for Maxwell's equations in the rotating frame of reference.

To understand the area of computational methods in atmospheric sciences, you need to study many sources. I would suggest starting with the general review of numerous articles published in the Monthly Weather Review from issues published in the 1950s to the present day. You can also consult the Journal of Computational Physics, especially issues published after 1989.

Please also consider the following article presenting the topic of meteorological calculations from the historical point of view:

https://maths.ucd.ie/~plynch/Publications/JCP.pdf

Paran Pourteimouri Janusz Pudykiewicz Thank you for the answers and helping me out.