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Governing Equations and Approximations Read chapter Read first chapter

2019 | OriginalPaper | Chapter

3. Time Marching Problems

Author : Lars Petter Røed

Published in: Atmospheres and Oceans on Computers

Publisher: Springer International Publishing

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Abstract

The purpose of this chapter is to present some properties inherent in the governing equations listed in Sect. 1.​1. Most problems in the geophysical sciences, including atmospheres, oceans, seas, and lakes, involve solving so-called time marching problems. Typically, the state of the fluid in question is known at one specific time. As postulated by Bjerknes (Meteor Z 21:1–7, 1904) in his comments on weather forecasting (see quote in the preface), the aim is then to predict the state of the fluid at a later time. This is done by solving the governing equations listed in Sect. 1.​1. Such problems are known in mathematics as initial value problems.

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previous chapter Preliminaries
next chapter Diffusion Problem
Footnotes
1
If the dependent variable is a vector, such as the velocity \(\mathbf {v}\), the flux becomes a tensor and (3.1) takes the form \(\partial _t\mathbf {v}+ \nabla \cdot \varvec{\mathscr {F}}= 0\), where \(\varvec{\mathscr {F}}\) is the flux tensor.
 
2
The total energy of a system consists of the internal energy and the mechanical energy. The internal energy is concerned with the heat content of a fluid and is an important part of its thermodynamics. In contrast, the mechanical energy concerns the motion of the fluid and is part of its dynamics.
 
3
It takes its name from Dr. Adolf Eugen Fick who formulated the parameterization given in (3.6) in 1855.
 
4
Geoffrey Ingram Taylor (1886–1975) made fundamental contributions to the study of turbulence, championing the need for a statistical theory and performing the first measurements of the effective diffusivity and viscosity of the atmosphere. He is remembered in the names attributed to several basic fluid flow instabilities (Taylor–Couette, Rayleigh–Taylor, and Saffman–Taylor).
 
5
The velocity variance is twice what is commonly referred to as the eddy kinetic energy.
 
Literature
Arakawa A, Lamb VR (1977) Computational design of the basic dynamical process of the UCLA general circulation model. Methods Comput Phys 17:173–265
Bjerknes V (1904) Das Problem der Vettervorhersage, betrachtet vom Standpunkte der Mechanik und der Physik. Meteor Z 21:1–7
GESAMP (1991) (IMO/FAO/UNESCO/WMO/WHO/IAEA/UN/UNEP Joint Group of Experts on the Scientific Aspects of Marine Pollution), Coastal modelling, Technical report. International Atomic Energy Agency (IAEA), GESAMP Reports and Studies 43, 192 pp
Gill AE (1982) Atmosphere–ocean dynamics. International geophysical series, vol 30. Academic, New York
Hackett B, Røed LP, Gjevik B, Martinsen EA, Eide LI (1995) A review of the metocean modeling project (MOMOP). Part 2: model validation study. In: Lynch DR, Davies AM (eds) Quantitative skill assessment for coastal ocean models. Coastal and estuarine studies, vol 47. American Geophysical Union, Washington, pp 307–327
Lorenz EN (1955) Available potential energy and the maintenance of the general circulation. Tellus 2:157–167
Lynch DR, Davies AM (1995) Quantitative skill assessment for coastal ocean models. Coastal and estuarine studies, vol 47. American Geophysical Union, Washington
Røed LP (1997) Energy diagnostics in a 1\(\frac{1}{2}\)-layer, nonisopycnic model. J Phys Oceanogr 27:1472–1476
Røed LP (1999) A pointwise energy diagnostic scheme for multilayer, nonisopycnic, primitive equation ocean models. Mon Weather Rev 127:1897–1911CrossRef
Metadata
Title
Time Marching Problems
Author
Lars Petter Røed
Copyright Year
2019
Book
Atmospheres and Oceans on Computers

Print ISBN: 978-3-319-93863-9

Electronic ISBN: 978-3-319-93864-6

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-319-93864-6_3