Abstract
The purpose of this chapter is to use the knowledge acquired in the previous chapters to learn about some slightly more advanced topics. For instance, we sketch ways to construct schemes of higher order accuracy, and ways to solve problems when advection and diffusion are equally important. Furthermore, we consider ways to treat nonlinearities numerically, and ask whether they harbor implications for instability. Since two-way nesting is becoming more and more popular, we also say a few words about smoothing and filtering, and give a detailed presentation of two-way nesting itself. Since the spectral method mentioned in the preface is rather common in global atmospheric models, the chapter ends with a brief description of a one-dimensional application of this method.
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- 1.
Note that \(\sin z/z = 1- z^2/6 + \cdots \), while \(\arcsin z = 1 + z^2/6 + \cdots \), for \(|z|<1\).
- 2.
The algebra is left to the reader, in Exercise 10.8 at the end of this chapter.
- 3.
Note that (6.128) is the acceleration term in the momentum equation for a one-dimensional problem. Hence, nonlinearity is ubiquitous in all realistic atmospheric and oceanographic models.
- 4.
Recall that \(2L=(J_\mathrm{p}+1)\Delta x_\mathrm{p}\) in accordance with (2.49), and hence \(J_\mathrm{pr}=J_\mathrm{p}+1\) when \(b=1\).
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Røed, L.P. (2019). Advanced Topics. In: Atmospheres and Oceans on Computers. Springer Textbooks in Earth Sciences, Geography and Environment. Springer, Cham. https://doi.org/10.1007/978-3-319-93864-6_10
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