4.4 Spherical Hannonics
207
Each tenn in C m (/-L)Pm ,n(/-L) is therefore a function of the form
(1 - /-L2)(p+r+m)/2Qp ,q(/-L)Qr,s(/-L) Qm,n(/-L) ,
Sinee the indices in the preeeding satisfy p + r + m = 2m , eaeh tenn is a polynomial in /-L of degree
2m + (q - p) + (s - r) + (n - m) = q + s + n.
The degree of the highest-order polynomial in the integrand of (4.65) is the maximum value of q + s + n, which is dependent on the type of truneation used in the
expansions (4.63) and (4.64). In the ease of a triangular truneation, this maximum
is simply 3M, and the exaet evaluation of (4.65) by Gaussian quadrature requires
a minimum of (3M + 1)/2 meridional grid points. In the ease of a rhomboidal
truneation, the maximum value of q + s + n is 3M + P + r + m = SM, and
(SM + 1)/2 meridional grid points are required for an exaet quadrature.
4.4.4 Nonlinear Shallow-Water Equations
Two additional eonsiderations that arise in using spherieal-harmonie expansions
funetions in praetieal applieations are the evaluation of derivatives with respeet
to the meridional eoordinate and the representation of the veetor velocity field.
The treatment of these matters ean be illustrated by eonsidering the algorithm
proposed by Bourke (1972) for integrating the nonlinear shallow -water equations
on a rotating sphere,
au
-
at
+ u . Vu + fk x u + V = 0,
a
-
at
+ V .u = O.
(4.67)
(4.68)
Here u = ui + vj, where u and v are the eastward and northward velocity components, f = 2Q sin (1 is the Corioli s parameter, k is the vertieal unit veetor, is the
gravitational eonstant times the free-surface displaeement, and V is the horizontal
gradient operator.
Prognostic Equations for Vorticity and Divergence
The velocity eomponents u and v are not eonveniently approximated by a series of spherieal harmonie funetions beeause artificial diseontinuities in u and
v are present at the poles unless the wind speed at the pole is zero. This problem arises beeause the direetion defined as "east" switehes by 180 degrees as an
observer traveling northward along a meridian steps aeross the pole. The same
veetor veloeity that is reeorded as " westerly" at a point on the Greenwich rneridian is reeorded as "easterly" at a point on the international dateline. In a similar
way, a southerly velocity beeomes a northerly velocity as the observer erosses the
pole. This diffieulty ean be eommonly avoided by replacing the prognostie equations for u and v by equations for the vortieity and divergenee and by rewriting
207
Each tenn in C m (/-L)Pm ,n(/-L) is therefore a function of the form
(1 - /-L2)(p+r+m)/2Qp ,q(/-L)Qr,s(/-L) Qm,n(/-L) ,
Sinee the indices in the preeeding satisfy p + r + m = 2m , eaeh tenn is a polynomial in /-L of degree
2m + (q - p) + (s - r) + (n - m) = q + s + n.
The degree of the highest-order polynomial in the integrand of (4.65) is the maximum value of q + s + n, which is dependent on the type of truneation used in the
expansions (4.63) and (4.64). In the ease of a triangular truneation, this maximum
is simply 3M, and the exaet evaluation of (4.65) by Gaussian quadrature requires
a minimum of (3M + 1)/2 meridional grid points. In the ease of a rhomboidal
truneation, the maximum value of q + s + n is 3M + P + r + m = SM, and
(SM + 1)/2 meridional grid points are required for an exaet quadrature.
4.4.4 Nonlinear Shallow-Water Equations
Two additional eonsiderations that arise in using spherieal-harmonie expansions
funetions in praetieal applieations are the evaluation of derivatives with respeet
to the meridional eoordinate and the representation of the veetor velocity field.
The treatment of these matters ean be illustrated by eonsidering the algorithm
proposed by Bourke (1972) for integrating the nonlinear shallow -water equations
on a rotating sphere,
au
-
at
+ u . Vu + fk x u + V = 0,
a
-
at
+ V .
(4.67)
(4.68)
Here u = ui + vj, where u and v are the eastward and northward velocity components, f = 2Q sin (1 is the Corioli s parameter, k is the vertieal unit veetor, is the
gravitational eonstant times the free-surface displaeement, and V is the horizontal
gradient operator.
Prognostic Equations for Vorticity and Divergence
The velocity eomponents u and v are not eonveniently approximated by a series of spherieal harmonie funetions beeause artificial diseontinuities in u and
v are present at the poles unless the wind speed at the pole is zero. This problem arises beeause the direetion defined as "east" switehes by 180 degrees as an
observer traveling northward along a meridian steps aeross the pole. The same
veetor veloeity that is reeorded as " westerly" at a point on the Greenwich rneridian is reeorded as "easterly" at a point on the international dateline. In a similar
way, a southerly velocity beeomes a northerly velocity as the observer erosses the
pole. This diffieulty ean be eommonly avoided by replacing the prognostie equations for u and v by equations for the vortieity and divergenee and by rewriting