Geography Reference
In-Depth Information
the divergence of the horizontal momentum equation, and assuming that at the
initial time
(∂
∇·
V
/∂t)
t
=
0
=
0
and
(
∇·
V
)
t
=
0
=
0
(13.57)
then the initial wind field is nondivergent and is related to the geopotential field
by the balance equation (11.15).
The balance equation, however, is quadratic in the streamfunction ψ and hence
does not provide a unique horizontal velocity field for a given field. Further-
more, realistic baroclinic motions in the atmosphere are not exactly nondivergent,
but rather have slowly evolving divergent wind components associated with the
secondary circulation discussed in Chapter 6, which are required to maintain the
delicate balance between the mass and velocity fields that is maintained as the flow
evolves.
13.7.2
Nonlinear Normal Mode Initialization
It should be clear from the previous discussion that the process of model initial-
ization should eliminate high-frequency gravity modes while retaining the rela-
tionships among the dynamical and thermodynamic fields that describe the slowly
evolving large-scale flow. A method that has been used commonly to obtain these
objectives is called
normal mode initialization
. Although some modern data assim-
ilation systems no longer require this initialization step, it is worth reviewing to
gain a better understanding of the initialization problem.
The normal modes of a dynamical system are the free modes of oscillation of
the system. If the discretized form of the primitive equations used in a prediction
model are linearized about a state of rest, the normal modes can be calculated. For
each grid point and pressure level there are three normal modes: an eastward and
westward propagating gravity mode and a westward propagating Rossby mode.
The analyzed fields can be represented by a combination of these normal mode
solutions in a manner analogous to the expansion of a field in terms of a finite series
of Fourier components or spherical harmonics. Such a representation is referred
to as a
projection
onto the normal modes.
This method can be illustrated in a simple fashion using the shallow water equa-
tions. We first must rewrite (7.69)-(7.71) to include the nonlinear terms neglected
in Section 7.6:
ε
u
∂u
∂u
∂t
−
∂
∂x
=−
v
∂u
∂x
fv
+
∂x
+
(13.58)
ε
u
∂v
∂v
∂t
+
∂
∂y
=−
v
∂v
∂x
+
∂x
+
fu
(13.59)
c
2
∂u
ε
∂ (u)
∂x
∂
∂t
+
∂v
∂y
∂ (v)
∂x
∂x
+
=−
+
(13.60)
