By T.N. Krishnamurti, H.S. Bedi, V. Hardiker, L. Ramaswamy
This introductory ebook on numerical climate prediction focuses on the spectral rework approach, that is an incredible part for international climate forecasts at various operational facilities. for this reason, it's an crucial advisor to the equipment getting used through approximately all significant climate forecast facilities within the usa, England, Japan, India, France, and Australia. The goals of this booklet are to supply a scientific and sequential history for college kids, researchers, and operational climate forecasters as a way to advance accomplished climate forecast versions. The chapter routines permit it for use as a graduate textbook for classes in meteorology in addition.
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Additional info for An Introduction to Global Spectral Modeling, Second Edition (Atmospheric and Oceanographic Sciences Library, 35)
The simultaneous relaxation scheme uses the original values of Ii , j from the previous iteration to calculate values at the next iteration. The sequential relaxation scheme uses new values of Ii , j for calculating values at the next iteration. As will be shown below, the sequential relaxation scheme turns out to be faster than the simultaneous relaxation. The analysis presented here follows a procedure described by Thompson (1961). We now address the simultaneous relaxation scheme, assuming that we are at level m of iteration at grid point (i, j ) .
This scheme is called the centered or the leap-frog scheme. 1). Substituting u ( x, t ) Re[U (t )eikx ] as the solution, we obtain dU iZU dt where Z 0, kc . 36) iZU n . 36). 36) is a second-order differential equation it has a quadratic equation for O . 38) where p Z't . 39) and 12 ip . 40) If U n 1 OU n is to represent an approximation to the true solution, then we must have O o 1 as 't o 0 . For the above roots, since p Z't o 0 as 't o 0 , we have O1 o 1 . However, at the same time, O2 o 1 .
After applying this scheme to the linear wave equation, we obtain 1 §3 · U n 1 U n ikc't ¨ U n U n 1 ¸ . 48) kc . The roots of the above equation are 1/ 2 O1 O2 1ª 3 § 9 2 · 2 «1 i Z't ¨1 Z ('t ) iZ't ¸ 2 «¬ 2 © 4 ¹ º », »¼ 1/ 2 1ª 3 § 9 2 · º 2 «1 i Z't ¨1 Z ('t ) iZ't ¸ » . 50) We note that if 't o 0 , then O1 1 and O2 o 0 . Thus O1 denotes the physical mode and O2 denotes the computational mode. As the time step 't is reduced, the Adams-Bashforth method approaches the true solution.
An Introduction to Global Spectral Modeling, Second Edition (Atmospheric and Oceanographic Sciences Library, 35) by T.N. Krishnamurti, H.S. Bedi, V. Hardiker, L. Ramaswamy