7.6 PrimitiveEquation Models
381
now satisfy
(7.116)
and
VV 1/1 - a - - --(1nps) = LJ Bme
2
• 8U
RTf 8
A
im>.. .
(7.117)
8a
a 81
m=-M
The spectral form of the tendency equations for the velocity potential, the perturbation temperature, and the surface pressure may be found in Bourke (1974)
and will not be given here. Note that the vertical advection terms in (7.116) and
(7.117) involve the product of three spatially varying functions (since Ö' itself depends on the product of two spatially varying functions). Thestandard transform
method cannot be used to transform these tripie products between wave-number
and physical space without incurring some numerical error. This "aliasing" error
is nevertheless very small (Hoskins and Simmons 1975).
7.6.3 Vertical Differencing
The most significant modifications required to extend the shallow-water algorithm
to a a -coordinate model are those associated with the computation of the vertical
derivatives. The vertical derivatives are computed using finite differences at that
stage of the integration cycle when all the unknown variables are available on the
physical mesh. As in (7.116) and (7.117), the resuits from these finite-difference
computations are then combined with the other binary products computed on the
physical mesh, and the net forcing is transfonned back to wave-number space.
A convenient and widely used vertical discretization for the a-coordinate equations is illustrated in Fig. 7.6 for a model with N vertical levels. The upper and
lower boundaries are located at a = 0 and
N
o = 1 =
k=1
where
is the width of the kth o layer. The stream function, velocity potential,
temperature, and geopotential are defined at the center of each a layer, and the
velocity Ö' is defined at the interface between each layer. The vertical derivatives
appearing in (7.112)-(7.114) involve variables, such as the temperature, that are
defined at the center of each a layer. These derivatives are approximated such that
381
now satisfy
(7.116)
and
VV 1/1 - a - - --(1nps) = LJ Bme
2
• 8U
RTf 8
A
im>.. .
(7.117)
8a
a 81
m=-M
The spectral form of the tendency equations for the velocity potential, the perturbation temperature, and the surface pressure may be found in Bourke (1974)
and will not be given here. Note that the vertical advection terms in (7.116) and
(7.117) involve the product of three spatially varying functions (since Ö' itself depends on the product of two spatially varying functions). Thestandard transform
method cannot be used to transform these tripie products between wave-number
and physical space without incurring some numerical error. This "aliasing" error
is nevertheless very small (Hoskins and Simmons 1975).
7.6.3 Vertical Differencing
The most significant modifications required to extend the shallow-water algorithm
to a a -coordinate model are those associated with the computation of the vertical
derivatives. The vertical derivatives are computed using finite differences at that
stage of the integration cycle when all the unknown variables are available on the
physical mesh. As in (7.116) and (7.117), the resuits from these finite-difference
computations are then combined with the other binary products computed on the
physical mesh, and the net forcing is transfonned back to wave-number space.
A convenient and widely used vertical discretization for the a-coordinate equations is illustrated in Fig. 7.6 for a model with N vertical levels. The upper and
lower boundaries are located at a = 0 and
N
o = 1 =
k=1
where
is the width of the kth o layer. The stream function, velocity potential,
temperature, and geopotential are defined at the center of each a layer, and the
velocity Ö' is defined at the interface between each layer. The vertical derivatives
appearing in (7.112)-(7.114) involve variables, such as the temperature, that are
defined at the center of each a layer. These derivatives are approximated such that