An important character of the cloud convection in the earth's atmosphere,
which has not been examined in the previous subsections,
is an asymmetry of the relationship between
condensation heating and vertical motion.
Positive heating occurs as condensation heating in the upward motion,
i.e., positive vertical wind,
while negative heating does not usually occur in the downward motion,
i.e., negative vertical wind, except for the cloud areas.
Negative heating shall result from evaporation of
condensate, i.e., cloud particles or raindrops,
which are rapidly removed by the precipitation processes
and are usually absent in the area of downward motion
in the free atmosphere, except for cloud inside.
As a simple model of this asymmetric characteristic around the upward motion
of the moist atmosphere,
there is a formulation that heating occurs only when upward motion w is
positive in (A.6), the evaluation of heating
in the condensation term of heat equation
(e.g., Miyahara, 1987,
Lau and Peng, 1987,
Yoshizaki, 1991).
The nonlinearity which results from the switching of heating
depending on the signature of vertical motion remains
even in the limit of infinitesimal amplitude,
the behavior of solutions
in the physical space can not be described
as a superposition of the behavior of individual modes in
wavenumber space;
the solution of initial value problem is the only way
to explore the behavior of disturbances.
In the following,
we are going to present some solutions of initial value problems,
as an example,
that demonstrate effects of this nonlinearity.
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Fig.A.6:
Examples of time development of geopotential height anomaly
for wave-CISK solutions from an initial value.
(left) development around the equator for
wave-CISK symmetric heating case (A.6)
(middle) development around the equator for
wave-CISK asymmetric heating case, i.e.,
heating occurs only when upward motion in (A.6) is positive.
(right) the same as (middle) but for non-rotating case.
Upper panels are longitude-latitude cross sections,
and lower panels are longitude-height sections, respectively.
Clicking panels launches movies
(animation gif).
Amplitudes are arbitrary.
Note that contour interval and coloring are normalized
by the maximum amplitude in the computational area at each time.
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Fig.A.6 and the movies linked to each panel show
example results of wave-CISK solutions obtained
from (A.1)-(A.5) with a heating profile
suitable for propagating growing disturbances
(¦Ç1=1.5, n2=-1.5)
from the same initial value that is a
circular positive temperature anomaly
placed at the equator of a motionless atmosphere.
Fig.A.6(left) shows the case for symmetric
heat generation for upward motion,
i.e., (A.6) is adopted as condensation heating representation,
resulting heating for upward motion and
cooling for downward motion.
Growing disturbances develop as wave packets propagating
both in the east- and westward directions.
This can be interpreted by decomposing the initial disturbance into
equatorial wave modes;
for the present initial disturbance,
the major wave packets are
an eastward wave packet with Kelvin wave like structure
and
a westward wave packet with gravity wave like structure.
They are growing while dispersively propagating.
Nnote that, althogh a free Kelvin wave is non-dispersive,
a Kelvin wave packet with wave-CISK feedback is
more or less dispersive.
The vertical structures of those wave packets
show that there are phase difference between the upper high pressure
and the lower low pressure,
which is a characteristics of unstable disturbance discussed
in subsection A.2;
we can observe westward phase shift for the Kelvin wave packet and
eastward phase shift for the westward gravity wave packet
as the increase of altitude.
Fig.A.6(middle) shows the case for asymmetric
heat generation for upward motion,
i.e., (A.6) is modified to have
(only positive) condensation heating only for upward motion.
An intereting feature is that
the westward propagating wave packet becomes gradually less visible,
while the eastward propagating wave packet grows as a one-signed solitary
disturbance with upward motion and condensation heating.
These features are caused by the fact that,
among the equatorial free waves, only a Kelvin wave
is non-dispersive and can keep a one-signed solitary shape.
And the numerical result indicates that
there actually is an one-signed solitary unstable solution
driven by heating which is generated by upward motion
associated with an original wave whose shape is homothetic
to the solution itself.
Contrary to the case for symmetric wave-CISK feedback,
the Kelvin wave packet for asymmetric wave-CISK feedback is non-dispersive
in this sense.
For dispersive free waves,
there inevitably appear both of the regions of upward and downward motion,
and because of the appearance of asymmetric heating,
wave packet cannot keep its shape.
The result of asymmetric appearance of growing wave packet
between eastward and westward directions is caused by
rotation of the planet.
Actually, when we remove the effect of rotation
by setting rotation rate to be zero,
the case for asymmetric heating generation also
presents isotropic propagation of growing waves, in the same way for
both eastward and westward, and even northward and southward,
from an initial disturbance as indicated Fig.A.6(right).
This is because, under the assumption of non-rotating and hydrostatic
atmosphere, gravity waves are non-dispersive horizontally.
As is described above,
an unstable disturbance caused by the wave-CISK feedback which is
asymmetric between upward and downward motion appears
as a solitary region of upward motion moving coherently eastward.
The interesting point is that
these characteristics are quite similar to the
features of equatorial precipitation activity obtained
in the results of present GCM experiment,
especially, the results with Kuo scheme as cumulus parameterization.
However, we have to note that,
in our GCM,
there are additional effects, which we have not considered in the simple
system here, for instance,
moisture budget, sea surface process, effect of radiation,
break down of proportionality between heating and upward motion,
and change of basic field due to growth of a disturbance.
