4.4 Spherical Hannonics
197
~~tEmITIIIJJ
bdEEJITIIJ
EJDJ
[]
FIGURE 4.4. Schematic indication of the distribution of the nodal lines for the spherical
hannonics Ym .n in the set 0 ::: m ::: n :::
3. The horizontal axis in each map is linear in).. and includes the domain -7f ::: ).. :::
tt . The vertical axis is linear in sin B and includesthe domain -ot/2 ::: B :::
tr /2. The m, n index of each mode is indicated in the upper leftcorner of each map.
Thc eigenvalue associated with each Ym.n can be used to define a total wave number by analogy to the situation on a flat plane, where
and the total wave number is (m 2 + n 2 ) 1/ 2. The total wave number associated
with Ym.n is (n
2
+ n) 1/2 ja, whieh is independent of the zonal wave number m.
The nonintuitive absence of m in the forrnula for the total wave number arises,
in part, because the effective meridional wave number of a spherical harmonie
depends on both m and n.
4.4.1 Truncating the Expansion
The zero-order structure of Ym,n (A. J-L) is described by the distribution of its nodal
Iines on the surface of the sphere. The nodal Iines for the those modes for whieh
o m n 3 are schematically diagrammed in Fig. 4.4. The zonal structure of
Ym .n is that of a simple Fourier mode eimJ.., so there are 2m nodallines intersecting
a circle of constant latitude. The distribution of the nodal Iines along a line of
constant longitude is more complex . The forrnula for the meridional structure of
197
~~tEmITIIIJJ
bdEEJITIIJ
EJDJ
[]
FIGURE 4.4. Schematic indication of the distribution of the nodal lines for the spherical
hannonics Ym .n in the set 0 ::: m ::: n :::
3. The horizontal axis in each map is linear in).. and includes the domain -7f ::: ).. :::
tt . The vertical axis is linear in sin B and includesthe domain -ot/2 ::: B :::
tr /2. The m, n index of each mode is indicated in the upper leftcorner of each map.
Thc eigenvalue associated with each Ym.n can be used to define a total wave number by analogy to the situation on a flat plane, where
and the total wave number is (m 2 + n 2 ) 1/ 2. The total wave number associated
with Ym.n is (n
2
+ n) 1/2 ja, whieh is independent of the zonal wave number m.
The nonintuitive absence of m in the forrnula for the total wave number arises,
in part, because the effective meridional wave number of a spherical harmonie
depends on both m and n.
4.4.1 Truncating the Expansion
The zero-order structure of Ym,n (A. J-L) is described by the distribution of its nodal
Iines on the surface of the sphere. The nodal Iines for the those modes for whieh
o m n 3 are schematically diagrammed in Fig. 4.4. The zonal structure of
Ym .n is that of a simple Fourier mode eimJ.., so there are 2m nodallines intersecting
a circle of constant latitude. The distribution of the nodal Iines along a line of
constant longitude is more complex . The forrnula for the meridional structure of