数理解析研究所講究録 1226 巻 2001 年 171-175

171
Behaviors of synoptic eddies in the atmosphere
釜慶大学校 丁*亨斌・東大海洋研 木村龍治 * (H.B. Cheong and R. Kimura )*Pukyong National Univ. and ORI Univ. of Tokyo

1. Observational analysis
toSynoptic eddies and the background pressure fifields are separated in weather cha1ts at
instabih.ty.
investigate the characteristics of propagation of dismlbances produced by the
The geopotential height data used in this analysis are objectively analysed data set fffom 1985 to 1991
pressure fifiel&
provided by ECMWF. The weather charts for synoptic eddies and thewere made with filtered data of periods from 2.5 to 6.5 days and of periods longer than 7 days,
respectively.
. The correlation
Spatial correlation of synoptic eddies were calculated at abase grid point at$36$ C0lTelation pattems by
pattern depends upon the choice of the base grid point. We). Fig.l is the result made bychanging the longitude of the base grid points by 10 degrees (atto eliminate
composite of all correlation pattems so that all base points are the center of thelongitudinal effects as done by Randel (1988). Since the synoptic eddies are produced by thepressure
to get a wave train of a low andbaroclinic instability of the westerly, it is nottendency
system propagating to ffie east direction along the latitude circle. However, there is afor the wave to propagate equatorward.
$500\mathrm{h}\mathrm{P}\mathrm{a}$$\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{m}\dot{\mathrm{c}}$$\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{k}\Psi \mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}$$45\mathrm{N}$$\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\alpha 1$$45\mathrm{N}$$\mathrm{f}\mathrm{i}\mathrm{f}\mathrm{i}\mathfrak{R}\mathrm{e}$$\mathrm{h}\mathrm{i}\mathrm{g}_{1}$$\mathrm{s}\mathrm{u}\varphi \mathrm{r}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}$$\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{g}\iota$$\subset \mathrm{t}_{\sim}\mathrm{t}\lambda\omega_{l}$$\mathrm{e}^{\mathrm{t}}\mathrm{k}w\mathrm{q}$Fig 1One-point correlation map at.
winter season.
Contour interval is$500\mathrm{h}\mathrm{P}\mathrm{a}$$0.1\mathrm{m}$$45\mathrm{N}$in

Fig 2Fr 伽 uency distributions of angle w.lh zonal
direction, no \eta all.z 伽何 by total fr 伽 uency
divided$\mathrm{b}\mathrm{y}50$.

This tendency was confifimed by calculating the propagation-velocity vectors of ffffie center ofindividual pressure anomalies (both positive and negative) by compaing two weather chalts wiffffi
12-hour interval. Fig.2 shows frequency distributions of angle between each propagation-velocity
vectors and the latitude line (negative values mean equatorward propagation). In ffiis fifigure we

172
sepamte eddies indeveloping stage ffom the decaying stage. Note that the distribution is not: The ffequency of the equat0lward propagation is greater than that of the poleward
propagation. Besides, the deviation fiom the
distribution is greater for the eddies in the
decaying stage than for the eddioe in the developing stage. ffi.s result implies that the equatorward
propagation is caused by the bmtropic process, because the baroclinic
tum out to be barotropic
in the decaying stage.
by thiswe ny to explain his tendency by means of theequation.
$\mathrm{f}1_{1}\mathrm{e}$$\mathrm{s}_{\mathbb{W}}\mathrm{m}\mathrm{e}\mathrm{t}\dot{\mathrm{n}}\mathrm{c}$$\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{n}\cdot \mathrm{c}$$\mathrm{f}\infty$$\mathrm{E}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{u}\iota \mathrm{a}\mathrm{g}\mathrm{e}\mathrm{d}$$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{d}\mathrm{i}_{\mathrm{V}\mathrm{e}1}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{i}_{\mathrm{C}\mathrm{V}\mathrm{O}1}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}$2. propagation wave-packet and phase tilt
The linearized
V0lticity equation with a zonal flflow
written as$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{d}\mathrm{i}_{\mathrm{V}\mathrm{e}1}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}$$\mathrm{b}\mathrm{a}\mathrm{l}0\mathrm{t}\mathrm{m}\mathrm{p}\mathrm{i}\mathrm{c}$$-u$on a sphere is(1)$\frac{\partial\zeta’}{\partial t}=-\frac{-u}{\cos\theta}\frac{\partial\zeta’}{\partial\lambda}-v’\frac{d}{d\theta}(f+\overline{\zeta})$where all variables
king scaled byand lengh by the radius of theealth.
is longitude,is latitude and $f$ is the Coriolis parameter,. Let usintroduce the stream ffinction to exproesand. If Eq.(l) isby
and averaged in the zonal direction,$\mathrm{a}\mathrm{o}\mathrm{e}$$\lambda$$\Omega^{-1}$$\mathrm{n}\mathrm{o}\mathrm{n}\prec \mathrm{l}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\downarrow \mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}$$\theta$$2\Omega\sin\theta$$u= \frac{\partial\varphi}{\partial\theta},v=\frac{1\partial\varphi}{\cos\theta\partial\lambda}$$\zeta=\nabla^{2}\varphi$$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{i}\alpha 1$$\zeta’\cos\theta$(2)$\frac{\partial V}{\partial t}=-2\gamma\cdot\overline{v’\zeta’}\cos\theta$$=+2 \gamma\cdot\frac{d}{d\mu}(\overline{u’v’}\cos^{2}\theta)$

where

$\mu$

is sine of latitude,

and$V=\overline{\zeta^{\prime 2}}$,

(3)$\gamma=\frac{d}{d\mu}(f+\overline{\zeta})$.

${\rm Re}$$N^{\mathrm{i}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e}}$value of$\gamma$makes themean flflow free ffim baroropic instability (Baines, 1976). The zonaly
enstrophy, ,represents the meridionalof amph.tude of eddiooe when they areof only one
zonal wavenumber or an isolated one. We are interested in the propagation of eddies rather than the
wave activity discussed in Held and Hoskins (1985).
$V$$\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{g}\alpha 1$$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{n}\cdot \mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\alpha 1$

3. Meridional phase tilt
the stream function with singe zonal wavenumber$\mathrm{k}\mathrm{t}$$m$

be written as$\varphi(\lambda,\theta)=C_{n;}(\theta)\cos(m\lambda)+S,,,(\theta)\sin(m\lambda)$$=\sqrt{C_{m}^{2}+S_{m}^{2}}\cos\{m\lambda-m_{-}^{-}-(\theta)\}$where$m_{-}^{-}-$

is the longitudinal phase angle at latitude

meridional pdient$\theta$, i.e.,,(4)$m_{-}^{-}-= \tan^{-\mathrm{I}}..\frac{\backslash m}{(m},$. Then, thephase or the phase tilt is expressed as$\frac{\partial_{-}^{-}-}{\partial\theta}=\frac{1}{m^{2}}\frac{\overline{u’v’}\cos\theta}{\overline{\varphi’\underline{)}}}$.
(5)

When a V0lticity anomaly is
letbe a point where, i.e., the location of
the maximum amplitude. If the latitude is
by infinitesimal amount, the longitudinallocation ofthe maximum amplitude will be shifted by.
$\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}4$$(\lambda_{0},\theta_{0})$$\frac{\partial\varphi’}{\partial\lambda}=0$$\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\alpha 1$$\delta\theta$$\delta\lambda$

173$\frac{\partial\varphi’}{\partial\lambda}(\lambda_{0}+\delta\lambda,\theta_{0}+\delta\theta)=\frac{\partial\varphi’}{\partial\lambda}+\frac{\partial^{2}\varphi’}{\partial\lambda^{2}}$where the first term inEq.(6) is multiplied by$\mathrm{r}\mathrm{h}\mathrm{s}$$\varphi’$.

vanishes by definition, and the diffentiation in
and averaged, we get.

$\frac{\delta\lambda}{\delta\theta}\equiv\frac{\partial_{-}^{-}}{\partial\theta}$(6),$\delta\lambda+\frac{\partial^{2}\varphi’}{\partial\lambda\partial\theta}\cdot\delta\theta\equiv 0$$= \frac{\overline{u’v’}\cos\theta}{-_{2’},v’\cos^{2}\theta}$$\mathrm{r}\mathrm{h}\mathrm{s}$is taken at$(\lambda_{0},\theta_{0})$

. If

(7)

.

, Eq.(7) is identical with Eq.(5).
When the stream function is represented by a single wavenumber-SE is as negative.
In the Eq.(5) alld (7), the NE-SW phase tilt is defined as positive andThen, the Eq.(3) can be used as a prognostic equation on the wave-packet propagation in thewhose phase tilt is
meridional direction. Suppose that a wave-packet of zonal wavenumber
exactly NE-SW, , contant with latitude, is located in mid-latitude. In this case the meridional
is positive to the south and negative to the north of ffffie center of it.
gradient of
Eq.(3) this means that the wave-packet will propagate into low latitudes. Therefore, the wave-packet
with NE-SW(NW-SE) phase tilt is expected to propagate into low latitude (high latitude).
$m$$\mathrm{N}\mathrm{W}$$m$$\mathrm{i}.\mathrm{e}.$$\overline{u’v’}\cos\theta$$\mathrm{F}\mathrm{o}\mathrm{m}$Fig.4 Two phase configurations, NW-SE and
NE-SW. Cl is 1/5 of the maximum valueof the initial condition.

Fig.3 Time evolution of . Contour interval is 1/10 of the
maximum of day 0.
$V$

4. Phase b.lt and wave-packet propagation with u $=0,m=6$
and time integrated with anEq.(l) is represented by the spectral method with a truncation
given byappropriate initial condition. We consider an initial vorticity$N$$\mathrm{f}_{1}\mathrm{e}1\mathrm{d}$$\zeta=C\frac{\cos\theta}{\cos\theta_{0}}\cos(m\lambda)\exp\{-(\frac{\theta-\theta_{0}}{10^{\mathrm{o}}})^{2}\}$(8),

and $C=0.1$ . This vorticity field is symmetric with respect to the equator.
shows time evolution of . The
Notice that this initial eddy fifield has no phase tilt.
wave-packet initially located in mid-latitude propagates into low or high latitudes,changing the phase tilt. The initial vorticity field is dispersed by the dispersion relation ofeach modes. The local maximum of the eddy amplitude propagatesof Rossby-Haurwitsz waves;with $m=6$ ,$\theta_{0}=45^{\mathrm{o}}$$\mathrm{F}\mathrm{i}\mathrm{g}.3$$V$$\mathrm{c}\mathrm{o}\mathrm{n}\dot{\mathrm{b}}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{l}\mathrm{y}$$\sigma_{t}’,=-,,\frac{2_{l\mathfrak{l}1}}{|(_{1}+|)}$

174
into high or low latitudes, by the interference of Rossby-Haurwitsz waves. Around the day 73 thewave-packet shows an apparent tendency to propagate toward low latitude, while around the day 48
the wave-packet propagates toward high latitude. Vorticity fields 0fday48 and 73 are shown in Fig.4.
The phase tilt

them is NW-SE and NE-SW,$\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}1\mathrm{y}_{\ovalbox{\tt\small REJECT}}$each panel.

5. Effect of zonal flowby the meridional
If the initial eddy given by
is located in a zonal flflow, ffffie phase is.
shear of it.
shows the time evolutions of the initial eddy in the 4 zonal flflows shown inmost striking feature in the presence ofthe zonal flflow is that once the wave-packet propagates into
higll or low latitude, they are trapped there or reflected and eventually reach a certain latitude. Theseis splitted into two parts by the zonal
features can be explained qualitatively. The wave-packet
flow. The northern part whose phase tilt is-SE propagates )and the southem part whose. During ffffie
ffffie phase is steepend mostand more; the phase tilt of the noffiem part tends toclose W-E and the southem E-W. Before
becomooe NE-SW, ffffie inverse phaseofreaching the pole region the phase tilt of ffffie n0lthem
initial stages, which means the turning of theoetion. The wave-packet is.
phase tilt compared with the wave-packet
into two parts also, but in this case with thewave-packet tends to back to the mid-latitude forming a waveguide. In case of $m=6$ theshear effect dominates over that of latitudinally varing Coriolis effect. The simple prognostic Eq. (3)is use in interpreting propagation Rossby wave-packet in horizontal shear flflow as in Yamagata(1976), where the trajectory ofthe wave-packet was calculated by thepath theory.
$\dot{\mathfrak{n}}\mathrm{l}\mathrm{t}\mathrm{d}$$\mathrm{F}\eta.(8)$$\mathrm{F}\mathrm{i}\mathrm{g}.6$$\mathrm{F}\mathrm{i}\mathrm{g}.5$$\mathrm{T}[\rceil \mathrm{e}$’$‘ \mathrm{A}$$\mathrm{N}\mathrm{W}$$\mathrm{l}\mathrm{e}\mathrm{w}\mathrm{a}\mathrm{f}\mathrm{f}\mathrm{i}$$\mu$$\dot{\mathrm{u}}\mathrm{l}\mathrm{t}$$\mathrm{P}^{\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{i}1\mathrm{t}\mathrm{i}\mathrm{s}\mathrm{N}\mathrm{E}- \mathrm{S}\mathrm{W}}\mathrm{P}^{\mathrm{r}\mathrm{o}}\mathrm{P}^{\mathrm{a}}\Psi^{\mathrm{t}\mathrm{e}\mathrm{s}\Re \mathrm{u}\mathrm{a}\mathrm{t}\mathrm{O}1\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{d}}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{a}\mathrm{g}\mathrm{a}\dot{\mathrm{b}}\mathrm{o}\mathrm{n}$$\mathrm{k}$$\dot{\mathrm{u}}\mathrm{l}\mathrm{t}$$\mathrm{p}\mathfrak{N}$$‘ \mathrm{a}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{a}\mathrm{g}\mathrm{a}\dot{\mathrm{h}}\mathrm{o}\mathrm{n}\mathrm{d}\dot{\mathrm{u}}$’$\mathrm{s}\mathrm{p}\mathrm{h}.\mathrm{t}\mathrm{t}\alpha 1$$‘ \mathrm{A}’$$\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{t}\mathrm{s}\alpha 1$’$\eta 1\mathrm{e}$$‘ \mathrm{a}$$\mathrm{f}\mathrm{l}\mathrm{l}\mathrm{e}$$\mathrm{f}\mathrm{u}\mathrm{l}$$\mathrm{n}\mathrm{y}$6. Packet velocity
cannot be used for the meridional energy
in our
The usual concept ofvelocity’ on the basis ofthe basic concept. The
problem. Instead we can defineinlocation largest amplitude is solely due to the zonal phase propagation
the absence of the zonal flow. In the presence of the shear flow, however, the location of largest
amplitude is determined by both the shear flow and the dispersion of Rossby-Haurwitz waves due to$\Psi^{\mathrm{o}\mathrm{u}}\mathrm{P}^{\mathrm{v}\mathrm{e}1\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y}}$$\mathrm{P}^{1\mathrm{O}}\mathrm{p}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{l}\mathrm{a}\dot{0}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}$$\mathrm{a}‘ \mathrm{p}\mathrm{a}\mathrm{c}\mathrm{k}\mathrm{e}\mathrm{t}$$\mathrm{o}\mathrm{f}\mathrm{R}\mathrm{o}\mathrm{s}\mathrm{s}\mathrm{b}\mathrm{y}- \mathrm{H}\mathrm{a}\mathrm{u}\mathrm{l}\mathrm{W}\mathrm{i}\propto \mathrm{w}\mathrm{a}\mathrm{v}\mathrm{o}\mathrm{o}\mathrm{e}$175depends on the detailed profifile of the zonal flflow asfactor will
the Coriolis factor.
well as the zonal wavenumber of the vorticity field. We define the ‘packet velocity’ as the
propagation speed ofthe location local maximum. Let the vorticity fifield be$\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{i}\cdot \mathrm{a}\mathrm{t}\mathrm{e}$$\mathrm{W}\mathrm{l}\dot{\mathrm{u}}\mathrm{c}\mathrm{h}$$\sigma’=\sum_{\prime r=1}^{l77+N}"\hat{\zeta}’,P’,,(\mu)\cos(m\lambda+\alpha_{l}’,’)$

where$\alpha’,,$,$=\sigma’,$,$t+\alpha_{?}^{l\mathfrak{l}\mathfrak{l}0},,\alpha_{\mathfrak{l}}^{l10}$is the$\ddot{\mathrm{m}}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$phase in the longitude.

(9),$\Pi \mathrm{e}\mathrm{n}$, by$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\cdot \mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$and fffom Eq.

(2)

(10)

$V= \frac{1}{2},\sum_{\prime|,|},\hat{\zeta}_{/}^{l||}\hat{\zeta}_{n}^{n\iota}P^{\prime\uparrow\prime},(\mu)P_{n}^{nt}(\mu)\cos(\alpha^{n\iota},-\alpha_{n}^{n1}\lambda$(11)$\frac{\partial V}{\partial t}=\frac{1}{2}\gamma(\mu),,\sum_{\prime\prime}\cdot\hat{\zeta}_{/}^{n\uparrow}\hat{\zeta}_{l?}^{l1\mathfrak{l}}P’,’(\mu)P_{n}’’’(\mu \mathrm{b}^{n\uparrow\sin(\alpha^{n1}-\alpha_{n}^{n1})},,$

Let$\mu_{()}$

be where$\frac{\gamma_{1’}}{r\gamma u}=0$at$t_{0}$. Then the packet velocity can be written as(12)$V_{\mathrm{k}^{\prime\nu}} \cdot\cos\theta=\frac{\delta\mu}{\delta t}=-\frac{\partial^{2}V}{\partial\mu\partial t}(\frac{\partial^{\underline{7}}V}{\partial\mu^{7}\sim})^{-1}$can be evaluated directly by using the reculSion relation ofdifferentiation with respect tois known, the packet velocity can be calculated for any zonal flflow.
Legendre polynomials. One. The presence offfffie
We show the calculated ‘packet velocity’ ofthe vorticity fifield ofday 73 inenhances ffffiezonal flflow alters the packet velocity to a la1ge extent. The zonal flow ofequatorward propagation ofeddies into low latitude, while does not.

$\mathrm{T}11\mathrm{e}$$\mu$$\mu_{0}$$\mathrm{F}\mathrm{i}\mathrm{g}.7$’$‘ \mathrm{A}$$‘ \mathrm{a}$$\mathrm{c}?1\Pi t$.
$\vee$$t$$\aleph$’, ,Fig 7T 加伽 packd velocity. The le 廿
and represent zonal flow Sp 伽 S.
denotes the case . ho 火 zonal$\mathrm{A}$$\mathrm{e}1\mathrm{S}$$\mathrm{B}$$\mathrm{a}$$\mathrm{b}$$\mathrm{O}$flow$\tilde{\grave{\wedge}}\lambda \mathfrak{B}$References
Baines, P.G.(1976): J. Fluid Mech., 73, 193-213.
Held, I.M. and BJ. Hoskins (1985): Issues in Atmospheric and Oceanic Modelling. Pan A.. Climate
Dynamics, Academic Press, 3-31.
Randel, W.J.(1988): Tellus, 40 A, 257-271.
Yamagata, T. (1976): J. Oceanogr. Soc. JaPan, 32, 162-168.