By T.N. Krishnamurti, H.S. Bedi, V. Hardiker, L. Ramaswamy
This introductory ebook on numerical climate prediction focuses on the spectral rework technique, that's an immense part for international climate forecasts at quite a few operational facilities. for that reason, it really is an crucial advisor to the tools getting used by means of approximately all significant climate forecast facilities within the usa, England, Japan, India, France, and Australia. The goals of this e-book are to supply a scientific and sequential history for college students, researchers, and operational climate forecasters so that it will increase entire climate forecast types. The chapter workouts enable it for use as a graduate textbook for classes in meteorology in addition.
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Extra info for An Introduction to Global Spectral Modeling, Second Edition (Atmospheric and Oceanographic Sciences Library, 35)
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 . The solution associated with O1 is an approximation of the true solution and is called the physical mode.
Therefore each wave component is advected at a constant velocity c along the xaxis with no change in amplitude. We now look for an analogous solution for the finitedifference equation, where we can substitute the form u nj Re(U n eikj'x ) , where U n is the amplitude at the nth time level. 5) O U n 1 . 6) or Un Substituting for | U n 1 | | O || U n 2 | and | U n 2 | | O || U n 3 | and following this procedure, we obtain | U n | | O |n | U 0 | . For the scheme to be stable, it is required that | U n | A , where A is some finite number.
49) where we assume I0 I N 1 0 . This is equivalent to saying that the geopotential anomaly vanishes at the top and bottom. 50) * where A and B are N u N coefficient matrices while ) and G are column vectors, that is, An Introduction to Finite Differencing 21 ª I1 º «I » « 2 » « I3 » « » « # » «¬ I N »¼ ) and ª G1 º «G » « 2» « G3 » . « » « # » «¬GN »¼ G* Here Ik and Gk are the values of I and G at a particular level k and a fixed grid point in the horizontal. 50) by the matrix B 1 , we obtain B 1 22 A) B 1 B) B 1G * .