Mitchell and Vallis (2010)“The transition to superrotation interrestrial atmospheres” のレビュー
樫村 博基（宇宙研）

Mitchell, J. L. & G. K. Vallis (2010) The transition to superrotation in terrestrial atmospheres.
Journal of Geophysical Research-Planets, 115, doi:10.1029/2010JE003587

0 Mitchell & Vallis (2010) の要旨• 簡単乾燥プリミティブモデルの数値実験- 加熱は自転軸対称- 熱ロスビー数を大きくすることで- 循環 が 地球的 → スーパーローテーション へ• 熱ロスビー数が大きいとき- 超回転生成期には、混合順圧傾圧不安定からの波数１の傾圧波が赤道向きの角運動量輸送を担う- 超回転生成後は、波数１の順圧不安定波で維持
2

1 イントロダクション• 自転軸対称な大気は 角運動量 down-gradient な拡散がある場合には超回転できない（Hide 1969）• つまり、up-gradient に輸送する非軸対称擾乱があるときのみ超回転可能...（中略）...

• 金星の超回転を説明するのためにも研究されてきた（Gierasch 1975 他）• 本研究は、超回転を生み出す力学機構とその地球的循環との関係性（パラメタ依存性）に着目する
3

2 数値モデル

MITCHELL AND VA• 乾燥プリミティブ方程式、理想気体、球面（T42）• 東西一様なニュートン加熱・冷却
点線：基準温度場- 安定成層- 緩和時間 40日、4日 (p/ps > 0.7)
• 地表面摩擦：レイリー摩擦- 緩和時間は 1日
2/s （非常に小さい）
Figure 1. Thermal forcing profile used
鉛直拡散係数0.01m•temperatures over the last 360 days of 1
cases. Contours are spaced evenly at intetemperature.
• ４次の超粘性：大きさ不記載
E12008（grid-scaleのエネルギーを散逸させるためとの記述あり）

others in the context of superrotation on Venus.
Held & Suarez (1987) と同じような設定
eddy viscosity would tend to bring the fluid into so4

factorgivingitalmodel.
to Rayleighradius,
friction H
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is
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3nondimensionalize
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2006, section 2.2.5] as above equations
alsobycontributesibbD/(2Wa).
on;運動方程式
thus,
U
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u= wa/(UH);bo Hb(u, v)/U; w
F = F/(2WUa);
t
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the@t speed, a the planetary radius, H the scalen horizontal windhvaryadimensionalpaheight(u,andWtherotation
rate.
For
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f
wa ˆ
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where
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o
H
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ð2Þ
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0
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r
meridional
and
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;
An
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thermodynamic and mass continuity equations
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þ RoT ðbv " rbuÞ þ f # b
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[9] Denoting
in providing
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r
b
nondimensionalize
the above equations
bysetting
(b
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)
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ð
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r
the
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where
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Rossby number
RoT ¼bt = 2Wt;:
¼ f = f/(2W)
ð3aÞ
mber
Ethermal¼w
b = ;wa/(UH);(u,Av)/U;F = F/(2WUa);
=
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three
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c
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E 熱ロスビー数¼ radius,
horizontal wind speed,a theplanetary
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and
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the
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we5the⌧ˆ=2⌦⌧
A height
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nondimensionalparameters
appearsinthe他にもなどが無次元パラメータUð RT0 DH Þ

4 数値実験• 地球に比べて- 金星は Ω が非常に小さい- タイタンは Ω が小さい & a が小さい
• モデルコードは有次元なので、Ω を変えるとE,E12008^τ も変わるので面倒MITCHELL AND VALLIS: TRANSITION T• 本実験では a を変えることで ROT を変える
Table 1. Parameters for Experiment Designa
RoT
a
W
To
DH
a

0.02

1.3

10.5

6.4 × 106 m
7 × 10−5 s−1
285 K
0.2

8 × 105 m–––

2.8 × 105 m–––

A dash indicates the same value as that of the entry on its left.

that altho
it may nitself. Th
control p
Hide, 19
studies o[10] In
we repor
The Ekm6

08

4 1 準定常後の平均温度場／流速場
MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008

実践：平均温度場
点線：基準温度場

E12008E12008E12008MITCHELLMITCHELLMITCHELLAND VALLIS:
AND AND
VALLIS:
TRANSITIONVALLIS:
TRANSITION
TRANSITION
TO SUPERROTATIONTO SUPERROTATIONTO SUPERROTATION

E12008E12008E12008

平均流速場
Figure 1. Thermal forcing profile used for the experiments (dotted) and the zonal and time mean
temperatures over the last 360 days of 1080 day integrations (gray line) for each of our three modelcases. Contours are spaced evenly at intervals of Dh/4 from 0.7 To to To, where To is the global mean
temperature.

s in the context of superrotation on Venus. Such an
viscosity would tend to bring the fluid into solid body
T to superrotation when coupledon, potentially leading
steady overturning circulation. This effect has beenked, at least in part, by a number of subsequent studies
eing the cause of superrotation on Venus and Titan
sow and Williams, 1979; Del Genio and Zhou, 1996;
din et al., 1995; Luz et al., 2003; Lee et al., 2007], with

with few essential differences found). The main differencesfrom a comprehensive GCM are that our model has nocondensate (for example water vapor for Earth, methane for
Titan) and that the radiative effects are represented by a
Newtonian cooling. These simplifications are severe, butallow us to focus on the dynamical aspects of the circulations of the systems. The simplest formulation of diabatic
atmospheric processes is to use linear relaxation, or New-Ro ：大ほど、南北に温度一定の範囲が広がる→ハドレー循環の拡大に対応
7

4 1 準定常後の平均温度場／流速場

E12008E1200808 E12008

MITCHELLMITCHELLMITCHELLAND VALLIS:AND
AND
VALLIS:
TRANSITIONVALLIS:
TRANSITION
TRANSITION
TO SUPERROTATIONTOTO SUPERROTATIONMITCHELLAND
VALLIS:
TRANSITIONTOSUPERROTATIONSUPERROTATION

E12008E12008E12008E12008

平均流速場

平均子午面循環• RoT: 0.02→1.3- ハドレーセルの拡大、フェレルセルの極側シフト• RoT: 10.5
- 境界層 (p/ps > 0.7)の中に、弱い- 上空では傾いている ← 対称（傾斜）不安定
Figure 1. Thermal forcing profile used for the experiments (dotted) and the zonal and time mean
temperatures over the last 360 days of 1080 day integrations (gray line) for each of our three modelcases. Contours are spaced evenly at intervals of Dh/4 from 0.7 To to To, where To is the global mean
temperature.

s in the context
superrotation
on mean
Venus.
Such
an
with
few
essential
found).
FigureFigure2.ofFigure
Zonal
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and2. Zonal
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eing the cause of superrotation on Venus and Titan allow us to focus on the dynamical aspects of the circulasow and Williams, 1979; Del Genio and Zhou, 1996; tions of the systems. The simplest formulation of diabatic
dintureet al.,
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平均東西風• Ro : 1.3 は超回転への過渡的状態

8

4 1 準定常後の局所ロスビー数

Figure 2. Zonal and time mean mass stream function (shaded) and zonal winds (contours, zero linebold) for RoT = 0.02, 1.3, and 10.5. (top) The full three‐dimensional simulations and (bottom) axisymmetric simulations of the same suite of parameters. Zonal wind contours begin at 0 m s−1 (bold line)
and are spaced by 10 m s−1 increments. Mass stream function contour ranges are reduced by a factorof 10 for each increase in Ro, and the maximum mass flux for each case is marked in white text (Tg s−1).

• ロスビー変形半径は Ld ~ 1500 km ~ aEarth/4• a < Ld のときに傾圧不安定は抑制される- すなわち Ro > 16RoEarth =ū 0.32
- 局所ロスビー数 Ro ⌘ | f |a cos
ture gradient, now isolated to regions poleward of 60°N/S
latitude. Thus the state of the atmosphere becomes morebarotropic as RoT increases.
[14] Figure 2 shows the time and zonal mean overturning
circulation (shaded) and zonal winds (contours) for our
numerical experiments. Figure 2 (top) displays the full
three‐dimensional simulations, while Figure 2 (bottom)

shows axisymmetric versions of the same three cases. The
latter were obtained by initializing the three‐dimensionalmodel without initial seed perturbations, and were verified
to be very nearly axisymmetric throughout the simulations.
The color scale of the overturning circulation has been
reduced by a factor of 10 between successive increases in
RoT while the scale of zonal winds is fixed (10 m s−1 con-Figure 3. An estimate of the local Rossby number, Ro ≈ u/(∣f ∣a cos’) with zonal mean zonal wind u,(absolute value of the) Coriolis parameter ∣ f ∣, planetary radius a, and latitude ’ for our three standard
cases averaged over the last 360 days of 1080 day integrations. Shaded contours are on the same color
scale in each panel and are spaced from 0 to 3 by 0.5. The line spacing changes for each panel: lines
are spaced from 0 to 0.1 by 0.02 for RoT = 0.02, from 0 to 2 by 0.5 for RoT = 1.3, and from 0 to 5
by 0.5 for RoT = 10.5. The bold line marks Ro = 0.

Roはまだ小さい→傾圧不安定が起こりうる4 of 179

4 1 PVとジオポテンシャルハイト
E12008MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008Figure 4. Zonal and time mean zonal winds (solid line, m s−1) and potential vorticity (dotted line) for thesteady state of RoT = 0.02, 1.3, and 10.5 and at the (top) 200 and (bottom) 500 hPa levels.
tour spacing; zero wind line bold). Between RoT = 0.02 and
RoT = 1.3, the Hadley cell for the three‐dimensional simulations widens and substantially weakens while the indirect
cell due to eddy motions in the mid latitudes is pushedpoleward. The Hadley cell in the RoT = 10.5 case is substantially weaker aside from a viscous cell contained in theboundary layer (pressures greater than p/ps = 0.7). The mean
circulation above the boundary layer in this case slants
upward and poleward away from the equator. This slant
structure is characteristic of symmetric instability arisingfrom the misalignment of contours of constant angularmomentum and isentropes.
[15] The zonal mean zonal winds in Figure 2 (top) transition from an Earth‐like, nonsuperrotating regime at RoT =滑らか

実線：東西風
点線：PV0.02 to a fully superrotating regime at RoT = 10.5. (Surface
zonal winds in the RoT = 10.5 case are all easterly becauseof the use of a Rayleigh drag in the lowest few model layers
and not just at the ground; however, the integrated torqueover the boundary layer is zero, as is required for a steadystate.) The transition between regimes occurs near the RoT =1.3 case, in which the zonal wind has Earth‐like isolated
midlatitude jets and also equatorial superrotation. This is to
be expected because, as Ro increases past unity, the scale ofbaroclinic turbulence (which has a significant impact on thezonal wind structure) can no longer fit within the confines ofthe spherical geometry of the planet as also noted byWilliams [2003].

階段状

滑らか

Figure 4. Zonal and time mean zonal winds (solid line, m s−1) and potential vorticity (dotted line) for thesteady state of RoT = 0.02, 1.3, and 10.5 and at the (top) 200 and (bottom) 500 hPa levels.
tour spacing; zero wind line bold). Between RoT = 0.02 and 0.02 to a fully superrotating regime at RoT = 10.5. (Surfaceジオポテンシャルハイト
RoT = 1.3, the Hadley cell for the three‐dimensional simu- zonal winds in the RoT = 10.5 case are all easterly because
lations widens and substantially weakens while the indirect of the use of a Rayleigh drag in the lowest few model layers
cell due to eddy motions in the mid latitudes is pushed and not just at the ground; however, the integrated torquepoleward. The Hadley cell in the RoT = 10.5 case is sub- over the boundary layer is zero, as is required for a steadystantially weaker aside from a viscous cell contained in the state.) The transition between regimes occurs near the RoT =boundary layer (pressures greater than p/ps = 0.7). The mean 1.3 case, in which the zonal wind has Earth‐like isolated
circulation above the boundary layer in this case slants midlatitude jets and also equatorial superrotation. This is to
upward and poleward away from the equator. This slant be expected because, as Ro increases past unity, the scale ofstructure is characteristic of symmetric instability arising baroclinic turbulence (which has a significant impact on thefrom the misalignment of contours of constant angular zonal wind structure) can no longer fit within the confines ofmomentum and isentropes.
the spherical geometry of the planet as also noted by
Figure5.
Snapshots
at
day
1080
of
the
400
hPa
level geopotential
10[15] The zonal mean zonal winds in Figure 2 (top) tran- Williams[2003]. height in our three standard cases.
Contours are spaced evenly from 900 to 980 m in increments of 10 m.
sition from an Earth‐like, nonsuperrotating regime at RoT =

波数４

波数１

波数１

4 2 軸対称テスト
E12008MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008• 超回転しない ← 非軸対称擾乱が不可欠• Held & Hou (1980) の理論にしたがう• 子午面循環の強さは、RoTが大きいほうがよく一致
Figure 2. Zonal and time mean mass stream function (shaded) and zonal winds (contours, zero linebold) for RoT = 0.02, 1.3, and 10.5. (top) The full three‐dimensional simulations and (bottom) axisymmetric simulations of the same suite of parameters. Zonal wind contours begin at 0 m s−1 (bold line)
and are spaced by 10 m s−1 increments. Mass stream function contour ranges are reduced by a factorof 10 for each increase in Ro, and the maximum mass flux for each case is marked in white text (Tg s−1).

ture gradient, now isolated to regions poleward of 60°N/S
latitude. Thus the state of the atmosphere becomes morebarotropic as RoT increases.
[14] Figure 2 shows the time and zonal mean overturning
circulation (shaded) and zonal winds (contours) for our
numerical experiments. Figure 2 (top) displays the fullshows axisymmetric versions of the same three cases. The
latter were obtained by initializing the three‐dimensionalmodel without initial seed perturbations, and were verified
to be very nearly axisymmetric throughout the simulations.
The color scale of the overturning circulation has been
reduced by a factor of 10 between successive increases in11

4 3 スポンジテスト• 非軸対称擾乱を強く減衰させる(τ ~ 10min)スポンジを高緯度あるいは低緯度に入れた実験 (RoT = 10.5)
MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008［45°以極］E12008［赤道南北3°］

MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008E12008Figure 6. Hadley cell and zonal mean zonal winds for test simulations of the RoT = 10.5 case withsponge layers (contouring is as described in Figure 2 for the RoT = 10.5 case). (left) The control casewith no sponge applied. (middle) A sponge applied to latitudes poleward of 45°N/S latitude. (right) A
sponge applied at the closest two grid cells to the equator, corresponding to the area between 3°N and 3°S
latitude.

Hi-lat: 超回転が弱い→高緯度が重要
Low-lat: 超回転しない→低緯度はより重要[16] To see what causes baroclinic instabilities to become
inefficient, compare the deformation radius (which largely
determines the scale at which baroclinic cyclogenesis occurs)

nonuniform PV, while at RoT = 1.3 the PV becomes nearlypiecewise homogeneous, forming staircase‐like structures.
The RoT = 10.5 case has very nonuniform PV at all levels.

12

5 1 ＥＰフラックス（波活動）E12008MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008

濃色：加速
淡色：減速

• スピンアップ時- EPフラックスは極向き、赤道域を加速

Figure 7. EP fluxes and their divergences. (left) The RoT = 0.02 (Earth‐like) case after it has reached astatistically steady state (during days 1000 to 1080). (middle) The RoT = 10.5 case (Titan‐like), during
active spin‐up, of the layer near p/ps = 0.4 (days 215 to 305), and (right) the RoT = 10.5 case after reaching statistically steady state (days 1000 to 1080).

• 準定常状態- 波活動は小さい、赤道域の加速も小さい

4.3. Sponge Cases[22] Previous studies of superrotation have often attributed
equatorial eddy momentum convergence to high‐latitude
barotropic instabilities feeding off of horizontal shear of thejets (see, e.g., Williams [2003] or Hourdin et al. [1995]).
However, the eddies that produce the superrotation cannot be
Rossby waves emanating from high latitudes; if such waves
were to radiate from a high‐latitude source region and breakthus decelerating equatorial winds. We now demonstrate in aset of test simulations that the low latitudes play an equallyimportant role in genesis of the disturbances that give rise tosuperrotation.
[23] In these test simulations, we impose a damping, or“sponge” on the nonaxisymmetric component of all fields,
but only in a select band of latitudes. The sponge takes theform of a linear damping of the nonaxisymmetric component at a timescale of 10 minutes, which results in an13to radiate from a high‐latitude source region and breakw latitudes, they would deposit westward momentum

form of a linear damping of the nonaxisymmetric component at a timescale of 10 minutes, which results in an
essentially axisymmetric state in the latitude band where the

5 1 波・平均流相互作用？

中高緯度で傾圧不安定
中高緯度からのロスビー波はFigure 8. Schematic representation of eddy mean flow interaction in the (left) Earth‐like and (right)
Titan‐like regimes. “Wave source” regions are intended←そもそも存在しない、またはto represent the location in latitude and char→上空、赤道に伝播し砕ける
acteristic phase speed of disturbances relative to the zonal wind. In so far as dissipative effects areotherwise small, meridional propagation of these disturbances
continues until either a critical layer is→低緯度を減速させる（減速機構）←平均流と作用しない
reached, denoted by arrows intersecting the zonal wind, or they are reflected by the vorticity gradient,denoted by arrows bounded by horizontal lines. (The generation of Rossby waves at the equator in
the Earth’s current atmosphere is, in fact, relatively weak.)14

spin‐up. Dark (light) regions correspond to positive (negative)
eddy
momentumflux.
5 渦運動量フラックス Cospectra
2given by
D ! 0
"
E
0
*
Kn;! ¼ 2 Re Un;! Vn;! ;ð4Þ

は空間・時間方向でフーリエ変換した
n,ω V’n,ω are• U’(U′,
where
V′)
the fourier transforms of the nonaxisymmetriccomponents of (u, v), starred quantities are complex
係数（＊は共役複素数）
conjugates, “Re” refers to the real component, and angle
RoT = 10.5 を解析•
brackets denote time averaging over a given window. Fouriertransforms are performed by direct FFT in space and time
resulting in fields gridded in zonal wave number, n, andfrequency, w [Hayashi, 1971]. Positive (negative) frequencies represent westward (eastward) propagating waves. Time15

5 400 hPa, 赤道上での @Kn,! /@y2.1

MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

成長期∂Kn,w/∂y at the 400 hPa level
and at the 準定常状態
r ranges are the same in each panel (their-0.1 d-1-0.6 d-1-0.25 d-1
-0.6 d-1

E12008青色：加速-0.25 d-1

he turbulence accelerates surface winds in middecelerates the upper level winds, and radiates
aves
where
are
and
Figure 9.equatorward
Divergence of the eddy momentum
fluxtheycospectrum,∂K absorbed/∂y at the 400 hPa level
and at the
どの段階も波数１が卓越•
equatorfor
Ro = 10.5 averaged over the days indicated. Contour ranges are the same in each panel (theirsubtropicalwinds. Figure 7 (middle) displays themagnitudes are arbitrary).
on phaseof the equatorial winds of the RoT = 10.5
• 位相速度に変換して、平均流と比べる必要がある
is which
applied. These
test simulations
are a
aimed
at bulence; the turbulence
accelerates
surfacewinds in midng
time
there
is
divergenceofEP
fluxesatanding the influence of equatorial versus high‐ latitudes, decelerates the upper level winds, and radiates
eddieson thezonal levels
mean zonal (p/pflow. The =first0.4
of Rossby
waves
equatorward
where
they
are absorbed and
and
0.7).
There
is
no
d
700
hPa
cases labeled “Hi‐lat sponge” in Figure s
6 has the decelerate subtropical winds. Figure 7 (middle) displays theappliedtoregions poleward ofof
45°N/S
latitude. acceleration
phase of theInstead,equatorial windsaof the Ro = 10.5oradivergenceverticalEP
fluxes.
mean zonal winds in this high‐latitude sponge test case, during which time there is a divergence of EP fluxes at= 0.4 and 0.7). There is no
andequatorial
700 hPa levels (p/p zonal
weaklysuperrotating at theacceleration
equator, lending the 400
16ecee onlyiscontributingofof a high‐latitude influence on the development of evidence for a divergence of vertical EP fluxes. Instead, a
n,w

T

T

s

MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E120085 2.2 400 hPa, RoT = 10.5

濃色：
正の運動量フラックス
減速

加速

加速

Critical layer

淡色：
負の運動量フラックス

平均東西風

加速

Figure 10. Phase speed‐latitude plot of eddy momentum
flux cospectra at the 400 hPa level (shaded) and zonal

Figure 11. Same as17Figu

TRANSITION TO SUPERROTATIONum
nal

E120085 2.2 700 hPa, RoT = 10.5

濃色：
正の運動量フラックス
加速

加速

淡色：
負の運動量フラックス

平均東西風
境界層（レイリー摩擦）が東西風の成長を妨げている

加速

Figure 11. Same as Figure 10 at the 700 hPa level.

18

agating waves. Timespace by smoothingee Randel and Held,
2.2
umber 1
umbers interact with(left) displays thecospectrum ∂Kn,w /∂yr for the entire first
tive/blue divergenceean flow). It is clear
ng the divergence ofquencies of −0.1 d−1

disturbances, critical layers are oriented along latitude circles where the zonal mean zonal wind u ≈ c. It is therefore

5 準定常状態

加速なし

vergence of Kn,w at
260 of the RoT =se of spin‐up. Wave
arly dominate the, predominantly at−1. Figure 9 (right)

加速

Figure 12. Eddy momentum flux cospectra at (left) 400 and• 上空では、加速はほとんどない(right) 700 hPa in the statistically steady state. Note that the
global wave now has an equatorial phase speed of ∼5 m s .
- →それでも超回転を維持できる−1

9 of 17• 境界層上端では加速がある19

5 3 超回転を生成する波の構造
E12008MITCH• これまでの診断で分かったこと-1
波数１、振動数-0.6d- 波は平均流に対して、‣ 低緯度で東進‣ 高緯度で西進- コヒーレントな緯度構造は変わらない

20

波数１、振動数 0.6 でフィルタリングした
陰影：
400 hPaジオポテンシャル
等値線：PV
青線：平均流×５ m/s•130-210日で低緯度の成分(陰影)と高緯度成分が同調(phase)しはじめる•230-290日で両成分とも強くなり、赤道の平均流が加速される•その後、両成分とも弱くなり、同調が外れ、平均流加速が止む21

•赤道のV字構造はケルビ

ン波(重力波)の特長だが、その位相速度(12 m/s)は
重力波のそれと一致せず•40°以極では、波は平均流に対して西進しているが、赤道の成分と逆位相を保っている22

ジオポテンシャルハイトの偏差（フィルタ済み）270日
MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008

経度-高度断面

複雑：鉛直シアーが原因

傾圧第１モード

経度-緯度断面e 14. Vertical and horizontal slices of zonally asymmetric geopotential filtered for zonal wave
er 1 and frequency −0.6 d−1 (shaded) during spin‐up (day 270) at the latitudes or levels indicated.
urs are spaced evenly from −0.5 to 0.5 m in increments of 0.1 m.

傾きが大きくなる ←コヒーレントe 14. Vertical and horizontal slices of zonally asymmetric geopotential filtered for zonal wave
−1
(shaded)during(day270) influenceat the latitudes
or levels winds,
indicated.
er1
andfrequency−0.6dbaroclinic turbulence originatingin themid- spin‐up
Rossbywave’son equatorial
and in par-23

ジオポテンシャルハイトの偏差（フィルタ済み）準定常後
E12008MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008E12008MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION

E12008

経度-高度断面

順圧構造
境界層

経度-緯度断面境界層

Figure 15. Same as in Figure 14 for the global wave, now with frequency −0.25 d−1, in the statisticallysteady state (day 1430).

Uniform structure

Figure 15. Same as in Figure 14 for the global wave, now with frequency −0.25 d−1, in the statisticallysteady state (day 1430).

24

ame as in Figure 14 for the global wave, now with frequency −0.25 d , in the statisticallyay 1430).

5 4 帯状平均PVの構造
実線：東西風 点線：PV

E12008PVの南北勾配MITCHELL AND VALLIS: TR
ピンク色はゼロ線

順圧不安定

傾圧不安定

400 hPaame as in Figure 4 for snapshots during spin‐up (on days indicated) of the RoT = 10.5 case
12 of 1725

5 4 準定常後
準定常後
順圧不安定

flux convergence is able to maintain the superrotating state.
Section 5.5 contains further discussion of the mechanism
maintaining superrotation.

5.3. Structure of the Global Wave Producing
Superrotation
E12008 [37] The spectral diagnostics
MITCHELL in
AND
VALLIS:
TRANSITION
Figures
9–12
together TO SUPERROTA
reveal the disturbance responsible for equatorial momentum
700
convergence during spin‐up has zonal wave number one
andhPa level deposits p
The equatorial eddy mom
frequency −0.6 d−1. Figure 10 also shows the disturbance
400 hPa level duri
propagates both eastward and westward relative to the thezonal
at this levemean zonal wind, depending on the latitude, but,atmosphere
imtional torques from bo
portantly, the disturbance retains a coherent latitudinal
confined
to p > 700 hP
structure. Figure 13 displays a sequence of snapshots ofthecirculation (400 hPa geopotential (shaded) and potential vorticityoverturning(PV,andcontours) during spin‐up filtered for zonal wave number
onedoes not experien
activity (as can be seen
and frequency −0.6 d−1 (the zonal mean zonal wind multirelatively weak and disor
flux convergence is able
Section 5.5 contains fu
maintaining superrotatio

re 17. ZonalPVの反転は少ない
and time mean PV gradient, ∂q/∂y, during
an active phase of spin‐up and (bottom) in the steadyfor the RoT = 10.5 case. The zero PV gradient line is
ked with a magenta line, with warm (cool) colors indicatpositive (negative) PV gradient. The 295 K, 300 K, andK isentropes are overplotted in black.

ances dominate the eddy momentum fluxes at this, with some important differences. The faster wave
erges momentum to the equator as it does at the 400
level, but it does so globally without a reversal in the

スピンアップ5.3. Structure of theSuperrotation[37] The spectral dia
reveal the disturbance reconvergence during spin
frequency −0.6 d−1. Fig
propagates both eastwar
mean zonal wind, dep
portantly, the disturban
structure. Figure 13 disp
400 hPa geopotential (scontours) during spin‐up
and frequency −0.6 d−1
26

400 hPa geopotential (shaded) and potential vorticity (PV,contours) during spin‐up filtered for zonal wave number one
and frequency −0.6 d−1 (the zonal mean zonal wind multi-

5 5 順圧不安定 [47]1) 渦度勾配があるところでロスビー波が生じる
2) 渦度勾配が強ければロスビー波は edge wave になる
3)Figure
もしwave
が赤道と高緯度に生じ、十分に近け
17. edge
Zonal and timemean PV
gradient, ∂q/∂y, during(top) an active phase of spin‐up and (bottom) in the steady
れば、それらはlockzero PV gradientlineし、相互作用して不安定
isstatefor the Ro = 10.5 case. Thephasemarked with a magenta line, with warm (cool) colors indicatingを作りうる
positive (negative) PV gradient. The 295 K, 300 K, andT

305 K isentropes are overplotted in black.

4) 東西風は高緯度で強いので、phase lock するためには
turbancesdominate the eddy momentum fluxes at this
波は平均流に対して逆に伝播する必要がある
level, with some important differences. The faster wave
converges momentum to the equator as it does at the 400hPa level, but it does so globally without a reversal in the
direction of momentum flux (as occurs at the 400 hPa
level poleward of 40°N/S latitudes). Momentum fluxes atthe 700 hPa level due to the slower wave are also oppositely oriented relative to those at the 400 hPa level. The
700 hPa level resides at the top of our boundary layer, anda frictional torque prevents equatorial superrotation fromdeveloping despite strong eddy momentum flux convergence there.
[36] Figure 12 displays the 400 and 700 hPa (Figures 12,
left, and 12, right, respectively) eddy momentum flux
cospectra in the statistically steady state, i.e., after super-5) ロスビー波がそうなるには、PV勾配が逆符号であることが必要
6) さらに、相互作用する edge wave は南北に広がっていなければならないので、波数１の構造があらわれる
Figure 18. Schematic of barotropic instability during spin‐
up of the RoT = 10.5 case. The meridional structure of the27
zonal mean zonal wind is shown as the line. Interacting edge

こういうこと？
Vallis (2006) §6.2.4 Interacting edge waves producing instability
Instability of Parallel Shear Flow267

6.2 Instability of Parallel Shear Flow

渦度：非線形計算

基本場（u）

269

to

8

y=b

................ y=a .................................

270..........................

y=-a .......................
Fig. 6.6 A sequence of plots of the vorticity, at equal time intervals, from a numericalsolution of the nonlinear vorticity equation (6.12), with initial conditions as in Fig. 6.4,
plus a very small random perturbation. Time increases first down the left column, thendown the right column. The solution is obtained in a rectangular (4 ⇥ 1) domain, withperiodic conditions in the x -direction and slippery walls at y D .0; 1/. The initial shear
is confined to a region 0:4 < y < 0:6 so that a D 0:1
y, =and-bthe maximum instability
thus occurs for a wavelength of 1:57. For a domain of length 4, this corresponds to anondimensional wavenumber of 2:55. Since the periodic domain quantizes the allowable
wavenumbers, the maximum instability is at wavenumber 3, and this is what emerges.
Only in the first two or three frames is the linear approximation valid.

全体

8

to -

Chapter 6. Barotropic and Baroclinic Instability

u

流線関数

偏差
Fig. 6.4 Barotropically unstable velocity profiles. In the simplest case,on the left, a region of shear is sandwiched between two infinite regions
three in this problem. Eventually, the perturbation grows sufficiently that the linearof constant velocity. The edge waves
at y areDno˙aanpanels of Fig. 6.6, vortices
longerinteract
validand, andas istoseenproduce
in thestreamfunction
lower threeFig.equations
6.7 Total streamfunction(top panel)
perturbationand numericalpinch off.
Eventually,
the 6.6,
vorticesinteract
and the flow develops into twofromformthe same
calculation
as in Fig.
at a timecorresponding
instability. If a D 0, then the situation
corresponds
to
that
of
Fig.
6.1,
givingto thesecond frame.
Positive values(a clockwise
circulation)
dimensional
turbulence,
consideredin chapter
8. are solid lines,
28
and negative values are dashed. The perturbation pattern grows exponenKelvin-Helmholtz instability. In the caseontheright,
theflow
isboundedtially, but is locked in place.

6 太陽系のなかの超回転• 金星は RoT ~ 1200
• タイタンはRoT ~ 30
• 火星はRoT ~ 0.1• タイタンの超回転メカニズムは本実験に近いかも• 火星は、transition (RoT = 1 )に近いけど、将来的に自転角速度が変わって超回転になることはな
さそう29E12008

7 Conclusions

LL AND VALLIS: TRANSITION TO SUPERROTATIONMITCHELL AND VALLI

E12008• 地球型惑星の大気循環はRoT>1で超回転になる- RoT<1なら、惑星規模より小さいロスビー波が中緯度でうまれ、Figure 4. Zonal and time mean zonal winds (solid line, m s−1) and potential vorticity (dotted line) for thesteady state of RoT = 0.02, 1.3, and 10.5 and at the (top) 200 and (bottom) 500 hPa levels.

赤道側に伝播して壊れる→赤道の西風を減速→超回転にならない- RoT>1なら、傾圧不安定のスケールが惑星規模よりも大きくなり
ロスビー波の生成源が弱くなる。波数１が卓越する- 成長期には傾圧・順圧の両不安定が効いてくるが、準定常後は、
E12008MITCHELL AND VALLIS: TRANSITION TO SUPERROTATION
順圧不安定のみが関係する

spacing; zero wind line bold). Between RoT = 0.02 and 0.02 to a fully superrotating regime at RoT = 10.5. (Surface= 1.3, the Hadley cell for the three‐dimensional simu- zonal winds in the RoT = 10.5 case are all easterly because
ns widens and substantially weakens while the indirect of the use of a Rayleigh drag in the lowest few model layers
due to eddy motions in the mid latitudes is pushed and not just at the ground; however, the integrated torqueward. The Hadley cell in the RoT = 10.5 case is sub- over the boundary layer is zero, as is required for a steadyially weaker aside from a viscous cell contained in the state.) The transition between regimes occurs near the RoT =ndary layer (pressures greater than p/ps = 0.7). The mean 1.3 case, in which the zonal wind has Earth‐like isolated
lation above the boundary layer in this case slants midlatitude jets and also equatorial superrotation. This is toard and poleward away from the equator. This slant be expected because, as Ro increases past unity, the scale ofture is characteristic of symmetric instability arising baroclinic turbulence (which has a significant impact on thethe misalignment of contours of constant angular zonal wind structure) can no longer fit within the confines ofmentum and isentropes.
the spherical geometry of the planet as also noted byan zonal winds (solid line, m s−1) and potential vorticity (dotted line) for the] The zonal mean zonal winds in Figure 2 (top) tran- Williams [2003].
3, and 10.5 and at the (top) 200 and (bottom) 500 hPa levels.
n from an Earth‐like, nonsuperrotating regime at RoT =ween RoT = 0.02 ande‐dimensional simuns while the indirect
latitudes is pushed= 10.5 case is subcell contained in thep/ps = 0.7). The meanin this case slants
equator. This slant
ic instability arisingof constant angular

Figure 2 (top) tranting regime at RoT =0.02 to a fully superrotating regime at RoT = 10.5. (Surface
zonal winds in the RoT = 10.5 case are all easterly becauseof the use of a Rayleigh drag in the lowest few model layers
and not just at the ground; however, the integrated torqueover the boundary layer is zero, as is required for a steadystate.) The transition between regimes occurs near the RoT =1.3 case, in which the zonal wind has Earth‐like isolated
midlatitude jets and also equatorial superrotation. This is to
be expected because, as Ro increases past unity, the scale ofbaroclinic turbulence (which has a significant impact on thezonal wind structure) can no longer fit within the confines ofthe spherical geometry of the planet as also noted byWilliams [2003].

Figure 5. Snapshots at day 1080 of the 400 hPa level geopotential height in our three standard cases.
Contours are spaced evenly from 900 to 980 m in increments of 10 m.
5 of 17

成長期

準定常後

順圧不安定
傾
圧
不
安
定

700 hPa level deposits prograde mo
The equatorial eddy momentum flux
the 400 hPa level during this phatmosphere at this level does not dtional torques
from boundary laye
順圧不安定
confined to p > 700 hPa), has nooverturning circulation (as can be se
and does not experience a vertic
activity (as can be seen in Figure 7,
relatively weak and disorganized hor
flux convergence is able to maintain
Section 5.5 contains further discus
maintaining superrotation.

5.3. Structure of the Global
30 Wa
Superrotation

S レビューのまとめ• 分かったこと- 単純な軸対称な加熱でもRoTが大きければ超回転になる- 超回転生成期には順圧不安定・傾圧不安定の両者が見られる- 準定常後には順圧不安定のみ- いずれの段階でも、波数１が卓越する- 循環形態の transition は RoT ~ 1• よく分からないこと- 生成期の順圧不安定と傾圧不安定との関わり- ２つのedge waveの相互作用→不安定は予見可能か
31