East Asia and Western Pacific METEOROLOGY AND CLIMATE

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3. Steady-State Solutions
The purpose of obtaining steady-state solutions of (2.4) (or (2.6)) is to understand the balanced
state from this model. First, let us review the basic dynamics for the wavenumber one structure
from Chan and Williams (1987) and Fiorino and Elsberry (1989).
The origin of the wavenumber one gyres is the linear / term that forces the initial vortex
to become asymmetric with positive vorticity tendency west of the vortex center and negative to
the east. Without a mean flow, the vortex can move through nonlinear advection only if there
is a wavenumber one asmmetry. More specifically, the circulation in this asymmetry crosses the
center of the vortex so that it can advect the total vortex. Therefore, the asymmetry is oriented
in the direction of the vortex movement as observed by Fiorino and Elsberry (1989). This effect
is represented by term 2 on the R.H.S. of (2.4), which is the advection of the basic vorticity by
the asymmetric flow. The direction of the maximum amplitude of this term is therefore in the
direction of the asymmetric circulation when the basic vorticity gradient is negative. The advection
of the asymmetric vorticity by the basic symmetric flow (term 1 on R.H.S. of (2.4)) will not move
the vortex center, but it helps to rotate the gyres so that they are aligned with the direction of
movement.
To show the balanced state, vectors representing the maximum magnitude of the terms in the
azimuthal direction at different radii are given in Fig. 2. The prescribed direction of movement
is 120° and the speed is 0.6 (2.8 m/s). The /-term (term 4) always points toward west. The
direction of term 3, which is the forcing term due to advection of the coordinate system, depends
on the direction specified and the basic vorticity gradient. In the inner part of the vortex (Fig. 2
a) where the basic vorticity gradient is negative (vorticity decreases outward from center), term
3 points toward the southeast when the specified direction is to the northwest (opposite to the
specified direction). Term 2, which represents the advection of the basic symmetric vorticity by
the asymmetric flow, is a maximum in the direction of the orientation of the gyres. Term 1, which
represent the advection of the asymmetric vorticity gradient by the symmetric flow is almost in
the opposite direction of term 2. The residual of these two terms balances with the sum of term 3
and term 4.
In the outer part of the vortex (Fig. 2 b), the basic vorticity gradient is positive (vorticity
increases outward). In this situation, term 3 points exactly to the direction of specified movement.
The / term (term 4) continues to point westward. Term 2 is now in the opposite direction of
the gyre's orientation (d^/dr > 0) and term 1 is approximately in the same direction of term 2.
Therefore, from Fig. 2 b, the gyres are oriented in the same direction as the specified movement
of 120° in the outer part. In between, the balance changes gradually from the inner part to the
outer part (not shown).
The steady-state streamfunction corresponding to Fig. 2 is shown in Fig. 3 where the resolution
is Ar == 5 krn. Note the gyres structure in Fig. 3 correspond exactly to this inner balance
(Fig, 2 a)i From previous discussion, one can see that to obtain the gyres oriented in the direction
of the specified movement, the gyres must be determined by the balance in the outer part of the
vortex; If the inner boundary is placed some distance from r =r 0, then the overall asymmetric
streamfunction can be determined by the balance in the outer part instead of the inner part of the