East Asia and Western Pacific METEOROLOGY AND CLIMATE

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equations. The 18 ordinary differential equations are reduced to 18
algebra equations as we set the tendency to zero searching for
stationary solutions.
By using the iteration method, we can identify seven equilibrium
branches in the forcing range 20°K < AT < 80°K. For a given AT ,
however, there can be a maximum of five equilibria. Fig. 3 shows
these branches on the AT vs. AT* diagram. Branches 4 and 5 appear
when AT > 57.5°K, and Branches 1 and 2 appear when AT* > 79.5°K.
Branches 1, 2 and 3 correspond very closely to the equilibria of the
highest truncation case, m=l. On the other hand, Branches 4 and 5
are two new^branches. They appear even for a weaker AT than the
critical AT required for Branches 1 and 2. For these two new
branches, the first mode components still dominate in the mean zonal
flow but all three meridional modes are equally important in the wave
components. Branches 6 and 7 are also two new branches that appear
in very weak thermal forcing (AT < 28.5°K) . For these branches
second mode components are zeros and the third mode wave amplitudes
are larger than amplitudes of the first mode wave.
Linear stability analysis shows that all equilibrium states
are unstable except for Branch 3 under a very weak thermal forcing
of AT < 29°K, and for Branch 7 which appears only when AT < 28.5°K.
Branch 2, 4 and 6 are orographically unstable.
4. MULTIPLE STATISTICAL EQUILIBRIA (TIME MEAN SOLUTIONS)
As we have found all equilibria are unstable unless in a range
of very weak thermal forcing, the results of stationary solutions
cannot be directly applied to interpret the atmospheric phenomena.
We need to find how these (stationary) equilibria are modified by the
instabilities. To answer this question, long-term numerical time
integrations of the spectral equations with three meridional modes
are conducted for fixed values of the thermal forcing a variety of
initial conditions. The Runge-Kutta fourth-order scheme with a onehour
time step is used in the integrations. For fixed values of AT ,
four types of initial conditions are used: a baroclinic zonal flow
(that corresponds to the Hadley equilibrium) , a barotropic zonal
flow, a baroclinic wavy flow, and a barotropic wavy flow.
A statistical equilibrium state is defined as the time-averaged
state over the last 3 1/2 years of a 4-year integration. Several
additional experiments with slightly perturbed equilibrium states as
the initial conditions have also been conducted to examine the
relationship between the statistical equilibria and stationary
equilibria 1 . ' ' '.. ' - ' - , . ' • . • . • ' • • • '•/ ' ' • ,, '. . . ..•• •" . . • • • .•': .. .•
From the numerical integrations, four statistical equilibrium
branches, Branches A, B, B 2 and C, can be identified in the range
20°K < AT < 80°K. However, for a given thermal forcing the model
can produce a maximum of three statistical equilibria. Branch 1S>2 *-&