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**Extra info for An Introduction to Global Spectral Modeling, Second Edition (Atmospheric and Oceanographic Sciences Library, 35)**

**Example text**

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 * .