4.4 SphericalHannonics
209
(4.68)-(4.72) become
(4.77)
(4.78)
(4.79)
The preceding system of equat ions for '1/1, X, and <1>' can be closed using the
diagnostic relations
(4.80)
Implementation ofthe Transform Method
The basie strategy used to implement the transform method for sphericalharmonie expansion functions is the same as that used with simpler Fourier series , whieh is to compute binary products in physical space and then transform
the result back to spectral space. During this procedure, those terms involving
derivatives are evaluated to within the truncation error of the spectral approximation using the known properties of the expansion functions. The zonal derivative
of each spherieal harmonie is simply imYm,n' The horizontal Laplacian is easily
evaluated using (4.48), and the meridional derivative can be determined using the
recurrence relation (4.47)
Prognostie equations for the spectral coefficients associated with the vorticity,
the divergence, and the free-surface displacement can be derived as folIows. Let
(4.81)
(4.82)
M N(m)
'I/I(J.., J-L) = a
2
L L 'I/Im .nYm,n(J.., J-L),
m=-Mn=lml
M
N(m)
X(J.., J-L) = a
2
L L Xm,nYm,n(J.., J-L),
m=-Mn=lml
M N(m)
'(J.. , J-L) = L L m,nYm,n(J.., J-L) .
m=-Mn=!ml
Since the nonlinear products in the governing equations (4.77)-(4.79) are U \12'1/1,
V\12'1/1, U4>', V <1>' , and U 2 + V 2, it is also convenient to define expansion coefficients for U and V. These coefficients can be diagnostieally computed from the
expansion coefficients for ' 1/1 and X as folIows. Let Um ,n and Vm,n be the spectral