By D R Durran

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**Extra info for Numerical Methods for Wave Equations in Geophysical Fluid Dynamics**

**Example text**

Suppose that the boundary conditions on u and v are 1 . 70) where N is a positive integer. 66) is simply u(x, y) = v(x, y) = O. 70) is 1 . ;-:::y is areal constant. 71) disappears, but the difference between the solutions generated by each boundary condition increases without bound along any line y = Yo > O. Arbitrarily small changes in the amplitude of the imposed boundary values can produce arbitrarily large changes in the amplitude of the interior solution. 70) is evaluated to obtain numerical values for the grid points along y = 0 may generate arbitrarily large perturbations in the interior solution.

44) oz de N ==() dz is the square of the Brunt-Väisälä frequency. Suppose that the reference state is isothermal. Then N 2 and the speed of sound Cs = (cpRT /c v ) I /2 are constant, and the preceding system can be simplified by removing the influence of the decrease in the mean density with height via the transformation _ (/3)1/2, u, Po -)1/2w , w= (- ii = u= - P (/3 )1/2 cpe n', Po _)1/2 (~ :e()'. 46) Note that 9 represents a scaled buoyancy and ir a scaled pressure. 47) 0 0 0 0 Cs ' S = Cs [ 0 0 0 0 ~) o ' 0 ~ 0/3 + l oe).

When the expansion functions form an orthogonal set, the series expansion approach is a spectral method. If the preceding periodic function were to be approx imated by a spectral method using five pieces of data, a natural choice would be the truncated Fourier series at + a2 cosx + a3 sinx + a4 cos 2x + as sin2x. 65) The five Fourier coefficients (at, a2, . ,as) need not be chosen such that the value ofthe Fourier series exactly matches the value of fex) at any specific point in the interval x ::: 2n .