April 1987
A. Abe
The
Looping
Motion
and
the
Asymmetry
247
of Tropical
Cyclone
By Shigeo Abe
Institute of Meteorology, the Defense Academy, Hashirimizu 1-10-20, Yokosuka 239, Japan
(Manuscript received 10 April 1986, in revised form 11 January 1987)
Abstract
We study the looping motion of tropical cyclone by a numerical simulation for the case of no
steering current in order to investigate the relation between the asymmetry and the motion of model
vortex. The dynamically balanced vortex is used initially to avoid an initial unrealistic motion of the
vortex.
The mean winds in each pressure level rotate counterclockwise in the developing stage in both *and *-plane cases. As the mean wind is produced by the asymmetric distribution of winds, the rotation
of the mean wind is considered as an inertial wave propagating on the eyewall. We apply the linear
wave theory to a modified Rankine vortex which has an internal boundary in no gravity field. The
angular velocity of this inertial wave is estimated approximately at a half of the vorticity in the outer
region of the eyewall.
If the distribution of azimuthal wind is in proportion to 1/r approximately, the looping motion
will be absent as the vorticity becomes zero in the outer area of the eyewall. The looping motion is
commonly cyclonic, because the vorticity near the eyewall is positive due to the surface friction and
the lateral mixing of momentum.
1.
Introduction
The looping motion of tropical cyclones has
been observed by Jordan (1966), Lawrence et al.
(1977) and others. Recently, Muramatsu (1986)
has shown the detailed data for the typhoon
8019 whose looping motion was characterized
by the period of 5*8 hr, and the amplitude of
23 km.
Kuo(1969), Jones (1977) and others treated
a tropical cyclone as a solid rotating cylinder
whose motion consisted of a mean motion equal
to the basic current and a trochoidal oscillation.
And they explained this trochoidal motion by
means of some external forces such as magnus
effect and frictional and vortex drag forces in
addition to deflecting force. They concluded
that the direction of the vortex motion always
shifts rightward so far as the deflecting force
acts.
Madala et al. (1975), Kitade (1980) and
others had shown using numerical simulation
c1987
Meteorological
Society
of Japan
that the model cyclone moved commonly northto-westwasrd accompanying a looping motion in
-plane in spite of no steering current . The mo*
tion of a cyclone may be possible by asymmetrical effects of the Coriolis force around the cyclone in *-plane (Holland, 1983). According to
Jones (1977), the looping motion was also observed in *-planein the period of developing stage
whose wind distributions became asymmetry. He
explained the cyclonic trochoidal motion based
on the discussion of Kuo (1969), and pointed
out appearances of the barotropic and inertial
instabilities together with the anti-cyclonic outflow in the upper layer. It is difficult to investigate the relation between the inertial instability
and the looping motion, because there are large
time fluctuations of physical parameters, such as
the central pressure and the mean wind velocity
and these fluctuations strongly relate to the
structure of the model cyclone.
We study some examples for comparatively
regular looping motions in the developing stage
248
Journal
of the
Meteorological
Society
of Japan
Vol . 65, No.
2
in which the inertial instability still does not
appear. In this stage, there are no irregular time
changes for physical parameters as compared
with the stage when the inertial instability is
observed. The looping motion of real typhoons
is observed only near the turning point of their
courses. We will study the relation between the
looping motion and the asymmetric wind using
Kitade's model (1980).
2.
The numerical model
We used Kitade's model in this paper, which
contains four layers in the vertical direction. The
variables of u, v (x-and y-components of velocity) and * (geopotential) appear at 1000, 900,
650, 400 and 150mb, while *(potential temperature), q (mixing ratio of water vapor) and *
(p-velocity) appear at intermediate levels of 950,
775, 525 and 275mb. This staggering simplifies
vertical integration of hydrostatic and continuity
equations. The horizontal gridmesh consists of
the minimum grid interval 0.5* in x and y direction (50km) and the maximum one is given by
3*. The fine meshes (interval 0.5*l.l*)
are
taken in the width of 1100km in x and y direction from the center position at (140*, 30*), and
the coarse grids (interval 3*) are placed in the
both side of these fine meshes. Thus, the domain
of integration extended east-westward 6000km
and north-southward 5000km is sufficiently
wide to avoid the influences of the boundary
conditions which are set to be cyclic at eastern
and western boundaries and *=0 at the southern
and northern boundaries.
The resolution of this grid system is insufficient to represent the detail of tropical cyclone,
in particular, such as the eye and spiral rain
bands. However, it may give some useful insights
concerning the asymmetry and movement of
tropical cyclones. Next, we will mention to some
aspects used in this simulation (refer to Kitade
(1980) in detail).
1) The initial values
The potential temperature as an initial value
in each layer is given horizontally uniform, and
the nondivergent part of the initial vortex is
given by stream function. The geopotential
height c at each level is obtained by the balance
equation. A simple w-equation is used to obtain
Fig.
1.
The
distributions
of
the
azimuthal
mean
of
the initial field of the divergent wind which is
calculated by solving Laplace equation for the
velocity potential. Fig. 1 shows the initial azimuthal mean values of * (azimuthal wind),
(vorticity) and * f in *-plane. It is noteworthy
that the point of * =0 which is related to the
barotropic instability exists near radius 3.5*.
2) Parameterization of cumulus convection
Latent heat release is primary sources of
energy for development of tropical cyclones, and
has an important role to the movement of them.
We used Kitade's scheme (1980) in which the
heating rate by cumulus convection is assumed
to be nearly proportional to the vertical p-velocity at the top of mixed layer. As in Kuo's scheme
(1965) the converged moisture seems to be excessively consumed for the moistening of the
atmosphere, the additional conversion process
from moisture to heating is introduced. In this
formulation it is assumed that a part of cumulus
convection is represented by cumulus mass flux
formulation by Ooyama (1971) and Arakawa
and Schubert (1970). Furthermore it is assumed
that the deep cumulus cloud has a root in the
boundary layer and take the moisture into the
cloud by strong cumulus updraft, and the excess
moisture is consumed to raise the temperature
and moisture in cloud.
3) Other physical process
The cumulus convection transports the horizontal momentum in the vertical direction. This
April
1987
S. Abe
transport is assumed to be nearly equal to the
mementum flux by large-scale upward motion
in case of the linear vertical shear.
We will study the looping motions for the two
cases of constant *-(Case I) and *-plane (Case II).
3.
The trajectories and the time changes of
physical parameters
First, the trajectories of the model cyclones in
Case I (* plane) and II (*-plane) are shown in Fig.
2. In Case I, the model cyclone which moved
initially counterclockwise (its diameter is about
40*km) shifted south-eastward at 54 hr, and deflected southward after 66 hr. In Case II, the cyclone which moved toward north-northeast
initially, traveled northwestward on an average
accompanying counterclockwise rotation as seen
in Case I. It shifted north-northeastward with
irregular looping motion after 54hr.
Fig. 3 shows the distributions of moving speed
of the model cyclone and of vertically and horizontally averaged mean wind within 1.8* radius
of the cyclone. As the determination of the cyclone center is difficult for coarse grid (Here, it
is calculated by the point of the minimum pressure), moving speeds are estimated by 6 hour
mean of hourly data in the trajectory. In either
Cases, the mean winds which are averaged excepting the top and bottom layers are specially
coincident with the moving speed in the early
period. Though it is possible to move northwestward in *-plane for the asymmetry of
Coriolis parameter (Holland, 1983), a symmetric
cyclone has no moving factors in constant *plane. However, looping motions are observed in
both Cases. Since their causes seem to exist in
the model cyclone itself, we will investigate, at
first, the characteristics of the model cyclone.
The time changes of *.5 1000
, *.5 275
, *.5, * 1.1 and
Q.5 are shown in Fig. 4, where (*)
represents
vertical mean and sufficesindicate constant pressure level and measured radius in degrees. In
initial stage, diabatic heatings are small in all
layers, and the surface pressure of the cyclone is
temporarily filled. Other physical factors change
more slowly as the sameas the surface pressure.
All physical factors increase grandually after the
surface pressure reached maximum and the cyclone begins to move north-westward accompanying counterclockwise looping motion. We
denot this period to the developing stage. After
the surface pressure reached minimum, the time
fluctuations of the surface pressure distinctly
appear and other physical factors also reach their
maximum values and fluctuate largely. We investigate mainly the looping motions in the developing stage, because the time changes of physical
factors are smooth and their flucuations are almost not observed.
4.
Fig. 2. The trajectories of the model cyclones for
Case I (constant *-plane at 30*N) and Case II (*plane).
249
The looping motion
It seems to be important for the looping or
trochoidal motion that there exist irregular rotations-asymmetric wind distribution-in a vortex
as seen in the last Section. We will investigate
the existence of these asymmetric flows by the
results or numerical simulation.
Synchronizing with the looping motion of the
trajectories in Fig. 2, the time changes of the
mean winds in each layer rotate clearly counterclockwise as seen in Fig. 5. It is very interesting
that these looping motions appear in the developing period of cyclone when much more heat are
added rather than the initial period as seen in
Fig. 4. The time changes of *' and *' which are
the difference from the azimuthal mean value
are shown on the ring of 1.8* radius in Fig. 6
(Case I is omitted here on account of the same
250
Journal
Fig. 3.
The distribtuions
of the Meteorological
Society
of the moving speed (broken
of Japan
Vol.
65,
No.
2
line) and the mean wind
(solid line) averaged horizontally
and vertically within the area of 1.8*
radius in Cases I and II. The dotted lines are the mean wind averaged midlayers only.
tendency with Case II.). In the initial stage the
steady wave (wavenumber 2) is almost stationary
and the amplitude of disturbances is very small.
The phase shifting rightward in the developing
stage shows that the maximum wind core rotates
counterclockwise. The angular velocity of this
rotation is estimated in the range 0.03*0.1*
10-3 rad.s-1 whose values are much larger in the
later time. The phases of *' and *' are consistent
each other except their signs and their amplitudes are almost constant in spite of the developing stage. The disturbed wave (wavenumber, k=2)
does not propagate again after the minimum surface pressure, and short period disturbances increase rapidly. At the same time, the area of
in Fig. 7 are almost coincident with that of
angular momentum conservation (broken lines in
Fig. 7), and their shears are larger rather than
that of the conservation law. Disturbances in this
stage seem to become unstable and fluctuate
irregularly.
5.
Discussions
Weobserved the time changes of asymmetrical
wind distributions in a model vortex in the last
section. An example of horizontal distribution of
disturbances in the developing stage is shown in
Fig. 8. Disturbances concentrate to the maximum wind area and decrease rapidly to the outer
region. It suggests that these disturbances generate on the eyewall as an azimuthally propagatinertial
instability:*a*+1/r*r(r*)<0,
areform- ing wave on an internal boundary. After the sured in the upper
layer as seen in Table 1 and Fig.
face pressure reaches minimum, disturbances
7. The azimuthal
mean wind distributions
shown
predominate in the outer region rather than in
April
1987
S. Abe
Fig. 4.
The time changes of *,*,*, *
251
and Q in Cases! and II.
252
Journal
Fig. 5.
The
time
changes
of the
of the
mean
Meteorological
Society
winds
level
at each
of Japan
in Cases
Vol.
I and
II. *
is 1 m/s
,5m/s
the
center
taneously
we
stage
consider
which
area
and
appears
65,
No.
and
*
the
near
inertial
the
the
disturbances
do
not
grow
instability
eyewall.
in the
in
spite
simul-
symmetrically
developing
vortex.
What
of causes
are there
for the appear-
of
disturbances
Therefore,
developing
of
the
axi-
a kind
ance
in the
model
vortex
that
develop
from
initially
these
axisymmetric
condi-
2
April 1987
S. Abe
253
Fig. 8. The horizontal distribution of *' at 48 hr in
Case II. The broken circle shows the 1.8* radius on
which ¢' is measured.
Fig. 6. The time change of the disturbances for *' and
' in Case II . (a):*' (m2/s2) at 1000*mb, (b): *'
*
(deg) at 950mb and (c):*' (deg) at 275mb on l.8*
radius circle from the center of minimum pressure.
Fig.
7.
Azimuthal
Case
II. The
momentum
mean
broken
wind
lines
conservation.
distribution
show
the
at 54hr
absolute
angular
in
tions? Anthes (1972) and Jones (1977) have
discussed the development of asymmetric flows
by dynamic instabilities.
First, an axially symmetric barotropic flow
may be unstable with respect to symmetric
where * is the tangential wind in cylindrical coordinates and *a absolute vorticity respectively.
There is little or no evidence that inertial instability plays an important role in the intensification of symmetric model vortex (Anthes,1972).
Although, according to Yanai (1964), when baroclinity is included (inertial intability along isentropes), this type of instability may be important. In our cases, regions of negative absolute
vorticity in azimuthally averaged wind distribution are observed in the upper level as shown in
Fig. 7 and Table 1, and the time of their appearances are consistent with those of observed
strong mean wind (Fig. 5) and time fluctuation
of physical factors (Fig. 4).
Second, the condition for the instability of a
zonally symmetric barotropic flow with respect
to azmuthal perturbations is that the gradient
of the absolute vorticity vanishes at some latitude (Kuo, 1949). The instability criterion expressed in cylindrical coordinates,
254
Journal
of the
Meteorological
Society
of Japan
Table 1. Radial distributions of azimuthally averaged absolute vorticity (x104
150mb level at 12, 36 adn 54hr, f=.729*10-4
*r*a=0
at some
r.
The minimum mean *a is observed at 1.82.6*
radii from the center in the upper level from the
initial time as shown in Table 1. But, it is noticeable that inertial instability does not appear in
the developing stage in our experiment.
Jones (1977) discussed in the case of 10km
fine meshes that there are zones of inertial and
barotropic intabilities near the eyewall and corresponding bulges of the zero relative vorticity
contour rotates cyclonically and migrate outward. It is generally thought that the wavelengths
of greatest barotropic instability are of the order
of a few thousand kilometers in cases where the
geometry is essentially Cartesian (Kuo, 1949).
In the present case, where the instability zone is
about 200km from the vortex center, azimuthal
wavenumber 1 would seem to be the preferred
mode.
In the developing stage, the disturbances concentrate on the eyewall, and their amplitudes
are almost constant from 20hr to 40hr as seen
in Fig. 6 notwithstanding the model vortex develops axisymmetrically. This suggests that the axisymmetrical baroclinity plays an important role
for the development of vortex and the above
barotropic intability is not essential. Disturbances due to barotropic instability concentrate
at the eyewall as an internal boundary and may
be treated as an azimuthally propagating inertial
wave.
The eyewall is almost vertical and the effect
of gravity mostly acts to vertical circulation in
connection with baroclinity. Now, we consider
the modified Rankine vortex which has an internal boundary at radius r0 from the center in the
cylindrical area of radius r1 in no gravity field.
The angular wave velocity of the inertial wave
propagating in azimuthal direction on the internal boundary is given approximately (see Ap-
Vol.
65, No.
2
s-1) in
pendix) by
where suffix 0 represents the value at the interface. The wave velocity consists of the angular
velocity at the interface and the difference of a
half of the vorticities in the inner and the outer
regions. In the second term of (5-1), if *2*1
the wave velocity increases due to larger outer
vorticity. From these results, changes off the
vorticity in the outer region influence strongly
to the propagating velocity of the inertial wave.
If the velocity in the central area is given by
V0=r0* and *=1, * is equal to *2/2, which coincides formally with the frequency givenby Jones
(1977). If azimuthal velocities are proportionate
to 1/r (which seems to be realized on an average
in many tropical cyclones (Syono (1951)), *2
becomes to zero (*0) and the looping motion
is absent. *2>0 is commonly observed in the
outer region of the maximum azimuthal wind
where horizontal mixing of momentum predominates. The values of *2/2 in the outer region are
estimated at 1000mb level in the developing
stage as shown in Table 2, whose values represent
the same order with those observed in Fig. 6.
According to Muramatsu (1986), in the observed example of Typhoon 8019 which consists
of double wind maxima, *2 is estimated 3*4*
10-4 s-1 (9hr) in the inner maximum wind
region and -0.1*10-4
(340hr) in the outer
maximum wind region. The period of 9 hr is
almost equal to the observed period of 5*8 hr.
In examples of simulations, Jones (1977) obtained two examples for moving grid cases of 10
km and 30km. As P/Ps=0.622 level (Ps is surface pressure) in either cases, *2 is 2-4*10-4
s-1 (4*5hr)
near the maximum wind, and
s-1 (30*80hr)
near the
-0.2*-0.5*
radius of 100km from the center. The observed
values are 7*l2 hr in cyclonic and 48*91 hr in
April
1987
Table 2. *2/2
S. Abe
255
values and their equivalent time periods; ( ), in the developing stage at sometime steps.
anticyclonic looping motions respectively. *2=2
10-4 s-1 (15hr) is observed*in the example of
Kurihara et al. (1974), while the observed values
are 10*15hr.
Though the gravity is neglected in the above
discussions, the effect of inertial force near the
center seems to be larger than that of the gravity.
Time changes of physical factors after the minimum surface pressure are variable in addition to
the inertial instability, therefore the above discussions depending only on inertial force may be
insufficient. The zonal wind corresponding to
the inertial instability increases owing to the
convergence of winds in the lowr layer, and this
convergence is proportional to the mean pvelocity in the center region. The limiting values
of * and * in Fig. 4 have a good correspondence
each other. While, the vertical wind shear which
is observed in the fully developed period restrains
ascending motions near the center due to the
difference of inertial forces between the upper
and the lower layers. Therefore, the irregular
cyclic fluctuations of physical parameters after
the minimum surface pressure would occur by
the interaction between the inertial instability
and the reverse effect of the vertical wind shear.
If this interaction acts asymmetrically around the
eyewall, irregular looping motion will appear as
the same as cases in the developing stage. The
looping motion of comparatively long period
(20*50 hr) is commonly observed in the stage
before developing. The short period of that
seems to be related to the double wind maxima,
though observed examples are very few.
6. Conclusions
We studied the characteristics of the looping
motion for vorticiey on *- and *-planes using
numerical
simulation.
The moving velocity of a
vortex is approximately
equal to the mean wind
averaged in the center area containing
the maximum wind as seen in Fig. 3. When the time
change of the mean wind rotates around the
vortex center, the looping motion is observed
specially in the initial developing stage in which
the inertial instability still does not appear.
According to the linear theory, we can discuss
this wavy motion which is considered
to be an
inertial wave caused by the barotropic
instability.
The angular velocity of this wave can be estimated approximately
by a half of the vorticity
in the outer region of the eyewall. The vorticity
is commonly
negative, however it is positive in
almost cases by large lateral mixing of momentum and surface friciton.
Thus both cyclonic
and anticyclonic
looping motions
are possible
to be observed.
The author is indebted to Dr. Kitade who has
given many valuable suggestions for programming
of the model. He also grateful to Mr. Tajima for
employing and programming of the computation
and compilation of the data.
Appendix
There are large temperature difference and
strong wind shear at the eyewall which is almost
vertical near the surface. The looping motion is
considered to occur due to the disturbances on
this discontinuous surface. But it is difficult to
treat strictly the problem of the perturbation
around the eyewall.
Now, we investigate the distrubance propagating azimuthally on a symmetrical rotating vortex,
256
Journal
neglecting
the
gravity
as
the
of the
eyewall
Society
of Japan
Vol.
65,
No.
2
is nearly
vertical.
Using
the
conception
of
Rankine
vortex,
the
basic
system
motion
Meteorological
a modified
is given
as follows:
Fig.
A-1
The
coordinate
system
and
the
basic
wind
fields.
where r0 and rl are the radii of the eyewall and
rigid outer boundary, * an angular velocity of
rigid rotation in I, * Coriolis parameter respectively, and a is an arbitrary constant which is
determined by the boundary condition at the
eyewall (see Fig. A-1).
Thus, the perturbation equations are
where
Z is an absolute
vorticity
which
is assumed
to be constant;
where
Assuming
the
solutions
to
where * is an angular wave velocity and * wavenumber, respectively, we can get the differential
equation for *r(r);
AsV/r-*=0 commonly, (A-4) is solved by the
boundary
conditions;
Thus, we obtain
II, respectively:
the solutions
in the region I and
Frequency
equations
are obtained
from the
dynamic boundary conditions at r=r0 as the same
with the gravity wave,
April
1987
S. Abe
Using V1=V2=Vo at r=r0 and substituting (A-6)
into (A-7), we get the frequency eqs.;
where
Thus, *
becomes
When *1 >*2, C which is called the dynamic
buoyancy affects to instability of the wave
motion. Since the second term in the root is
smaller than the first, *1*2
and F+/F-*l
(if r1*r0 ), a is rewritten approximately to
(A-10) is obtained by the condition of constant
negative vorticity for the basic state, but we can
estimate for the case of positive vorticity in the
same way as *2<O case by different boundary
conditions.
References
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257
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258
Journal
of the Meteorological
Society
of Japan
Vol.
65, No.
台風 の 蛇 行 運 動 と非対 称 性
阿
部
成
雄
(防衛大学校地学教室)
台 風 の 蛇 行 は 数 値 シ ミュ レー シ ョ ン か ら 得 られ た 平 均 風 の 時 間変 化 と密 接 に 関 係 し て い る 。 発 達 期 の 渦
に お い て 各 高 度 の 平 均 風 はf-面,β-面 の いず れ で も 反 時 計 ま わ りに 回 転 し て い る 。 平 均 風 は 渦 系 内 の 非
対 称 風 に よ っ て 生 ず る の で,平
均 風 の 回 転 は 回 転 風 の 擾 乱 成 分 が θ方 向 に 伝 ぱ す る と し て 考 え る こ とが
出 来 る 。 も し こ の 擾 乱 が 眼 の壁 の 境 界 に 生 ず る とす る と,慣 性 波 動 と して の 角 速 度 は 境 界 に お け る 流 れ
の 角 速 度 と壁 の 内 外 域 の 流 れ の 角 速 度 の 差 の 和 と して 与 え ら れ る。 内 域 は ほ と ん ど 剛 体 回 転 な の で,伝
ぱ 角 速 度 は 外 域 の 角 速 度 の み で 与 え られ る。
外 域 で は 一 般 に 風 速 は1/rに 比 例 す る の で,蛇 行 運 動 は 起 ら な い 。 角 運 動 量 を 保 存 す る よ うな 風 速 分
布 の 時 は f/2と な り高 気 圧 性 の 蛇 行 が 起 る が,一 般 に は 地 表 マ サ ツ や,側 面 混 合 で 外 域 で も渦 度 へ は
正 とな り低 気 圧 性 の 蛇 行 を生 ず る こ と に な る 。
2