The basic equations utilized are same as TI2006: the horizontal structure equations of a primitive system using an equatorial β-plane approximation. Dissipation is excluded. We consider a channel extending along longitudinal direction infinitely. At both the north and south boundaries, meridional velocities are set to zero. The x- and y-axes are along the longitudinal and meridional directions, respectively. The equator is located at y=0. By expanding variables in zonal harmonics in the same manner as Dunkerton (1993), non-dimensional perturbation equations linearized about a linear shear flow are
where
,
,
,
and
are the basic zonal velocity, perturbation zonal velocity, meridional
velocity, and geopotential, respectively.
Meridonal velocity
is defined to be out of phase with
and
.
and
are zonal wavenumber and complex frequency, respectively.
The above equations are non-dimensionalized by
the length scale
which is the latitudinal width of
the inertially unstable region,
time scale
,
velocity scale
,
and geopotential scale
;
(
is the Coriolis parameter)
and
.
The only non-dimensional parameter is
,
where
and
are gravitational constant and equivalent depth, respectively.
In this paper, same as TI2006, we adopt
|
as the basic zonal flow (figure 1). In this basic flow, an inertially unstable region exists in the region of 0 ≤ y ≤ 1 (a red shaded area in figure 1). The calculational domain used in most cases is -2 ≤ y ≤ 3 as well as TI2006. In sections 5 and 6, cases in which calculational domains are gradually expanded northward and/or southward are considered. The eigenvalue equations (1)-(3) are solved by discretizing in the y-direction with the grid interval 0.078125. The zonal wavenumber in the range of 0.00 ≤ k ≤ 1.00 is considered. Values of E within the range of -2.50 ≤ log E ≤ 7.50 are considered. |
Figure 1: Meridional profile of the basic zonal wind.
Horizontal and vertical axes are velocity
|
