惑星⼤気の熱的散逸 - 再考
寺⽥ 直樹（東北⼤）2017年7⽉18⽇
CPSセミナー1

Outline• イントロダクション• 惑星⼤気の熱的散逸• Jeans escape• Hydrodynamic escape• Slow hydrodynamic escape• 過去の地球⼤気• 過去の⽕星⼤気

⼤膨張？？
保持できず？

• ⼤気波動（⼤気重⼒波）の影響• MAVEN/NGIMS観測• DSMC計算
2

The Astrophysical Journal Letters, 729:L24 (5pp), 2011 March 10

Volkov et al.

熱的散逸の古典的説明
Escape parameter : λ=GMm/kTexorexo
太陽EUV放射などによる上層⼤気の加熱• Jeans escape• λ >~10 ??
• 外圏底付近で⼤気の平均熱速
度が脱出速度より⼩さい• マクスウェル分布の⾼速成分
のみが流出• fJeans ∝ ~Texo-1/2exp(-a/Texo)• Hydrodynamic escape• λ <~2 ??
• ⼤気のバルク速度が外圏底
以下で超⾳速に加速• ⼤気のblowoff（energylimited or diffusion-limited）!
Figure 1. (a) Dimensionless properties: number density n/n0 , (b) local Mach number Ma = u/ (5/3)kT /m, (c) T|| /T0 , (d) T⊥ /T|| vs. r/R0 for HS gas at
Kn0 = 0.001, λ0 = 0, 1, 2, 3, 10. For λ0 ! 2 results are close to an isentropic expansion. Results fully scale with λ0 and Kn0 : simulations performed for fixed
m = 4.65 × 10−26 kg, σ = 7.1 × 10−15 cm2 (cross section for N2 at 90 K; Bird 1994), T0 = 100 K, R0 = 1000 km, while M and n0 were varied.

3

λ0 = 1Kn0 = 0.001

Normalized distribution function

for an atomic gas, in which there is a critical point, beyond
which supersonic expansion can occur, for every λ0 .
Figure 2 shows the change in the parallel velocity distribution
at r/R0 = 10 for planets having different λ0 . For λ0 ! 2, thedistribution is shifted toward large v ∥ and u is close to the mostprobable molecular velocity. For λ0 " 3, the most probable v ∥is seen to be close to zero and escape is provided by the upward
moving, high-speed molecules. This fact together with T⊥ ̸= T||highlights the non-equilibrium nature of the flow. In addition, for
λ0 " 3 the flow velocity and the molecular escape rate, Φ, cannot
be associated with the maximum in the velocity distribution.
The effect of the change in the nature of the flow on Φ isshown in Figure 3. For the HS model, the√ ratio of Φ to the“evaporation” rate at R0 , Φ0,0 = 4πR02 n0 kT0 /(2πm) drops
dramatically between λ0 = 2 and 3 for both Kn0 = 0.001 and
0.0003. Up to λ0 = 2, the ratio is very close to ∼0.82, the rate
found in the absence of gravity as in a comet-like expansion(e.g., Cong & Bird 1978; Crifo et al. 2002; Tenishev et al.
2008). The ∼0.82 is due to collisions in the Knudsen layer
causing the return flow. As the atmospheric outflow is choked

λ0 = 0λ0 = 2r / R0 = 102
λ0 = 3λ0 = 101-2

Maxwellian
distribution on
the source surface
at temperature T0-1

01
Parallel velocity v|| / C0

速度分布関数（DSMC計算例）
3[Volkov et al., 2011]
2

Figure 2. Normalized distributions of parallel velocity v|| /C0 in HS gas for熱的散逸・⾮熱的散逸

熱的散逸：

⾮熱的散逸：• ⽐較的よく分かっている• 2つのend-member近似• 過程が複雑、かつ、複数の過程が同程度に重要となることが多い• MAVEN等の探査機観測は、
4
主にこちらにフォーカス（Jeans escape, Hydrodynamic
escape）は、理論式も⽐較

的シンプル

探査機観測• MAVENの成果例• Sputtering（⾮熱的散逸）によるAr流出量の⾒積り[Jakosky et al., 2017]• 現在から過去40億年前までを集中的に調査中• 初期数億年間は？

• より激しい散逸（太陽EUV放射は10-100倍）• 熱的散逸＋⾮熱的散逸

5

熱的散逸の理論の進展
Escape parameter : λ=GMm/kTexorexo• Jeans escape• λ >~10 ??
• 外圏底付近で⼤気の平均熱速
度が脱出速度より⼩さい• マクスウェル分布の⾼速成分
のみが流出• fJeans ∝ ~Texo-1/2exp(-a/Texo)• Hydrodynamic escape？

• λ <~2 ??
• ⼤気のバルク速度が外圏底
以下で超⾳速に加速• ⼤気のblowoff（energylimited or diffusion-limited）• Slow hydrodynamic escape (SHE) ??
• λ = 2~10 ??
• ⼤気はnon-hydrostaticで、上向き流速(hydrodynamic flow)を持つ• 外圏底以下では亜⾳速のまま• fJeansよりも⼤規模な流出??
[e.g., Strobel 2008]

初期地球で
Texo=5000~10000 K
m=16mpのとき
λ=5~10.
初期⽕星や現在のTitanなどもSHE ??
6

初期地球⼤気は⼤膨張？
E05008TIAN ET AL.: HYDRODYNAMIC PLANETARY THERMOSPHERE

E05008外圏底⾼度

λ=3~10 → SHE?

Figure 6. Temperature profiles for different solar EUV flux cases (normalized to present-day solarmean energy
flux !1 times
which represents solar EUV energy flux !5.1 mW/m ). It is• SHEモデル[Tian
et present
al.,EUV,
2008]
shown that when solar EUV energy flux exceeds certain critical value, the upper part of the thermosphere2begins to cool as a result of the increasingly significant adiabatic cooling effect. Beyond the critical flux
• 太陽EUV放射が6倍（約35億年前）で外圏底が10,000kmに？
(!5 times present EUV in this plot), the higher the energy input into the thermosphere, the lower theexobase temperature. This behavior is typical in the studies of hydrodynamic flow in planetary• 太陽EUV放射が20倍（約41億年前）で外圏底が100,000kmに？？
atmospheres.
7• 膨張、断熱冷却が重要となる
cooling mechanisms in this model at both solar minimum andsolar maximum are close to those in previous works.

GAIT, the peak density of O+ reaches a local maximum at asolar EUV energy flux of !11 mW/m2 and decreases

early Mars: (left) neutral temperature profiles and (right)[12] Because of Mars’ weak gravity, the planet could also
number density profiles. Blue, green, and red colors have lost significant atmosphere through impact erosion
represent different solar EUV energy flux levels. Curve [Melosh and Vickery, 1989], which could have further
styles represent different atmospheric pressures. The black reduced the total CO2 inventory. This only strengthens our
curves in Figure 3 (right) are the number density profile of conclusion that Mars could not have sustained a dense CO2
CO2 (dashed), atomic carbon (solid), and atomic oxygen atmosphere for time scale of #10 Myrs, and thus was(dotted) in Case 3.
initially cold. (With few greenhouse gases present, theL02205TIAN ET AL.: CARBON ESCAPE FROM EARLY MARS
L02205

⽕星は初期数億年間、⼤気を保持できなかった？

4.1 Ga (dashed curve in Figure 4) and could have reached
3 bars as the Tharsis CO2 inventory, the upper solid line in
1012 cm!2 s!1 at 4.5 Ga. The corresponding timescales
Figure 4 is obtained. The lower solid line in Figure 4 is for
EUV
1043
carbonrequired to lose 1 bar of CO2 from Mars (3.7 "20
20 bars as the total CO2 inventory and 1.5 bars as theatoms) are 10 and 1 Myrs respectively. Hence, these dense
Tharsis CO2 inventory.
atmospheres could not have been maintained during the
散逸率For purposes of comparison, thedot-dashed line marks the CO2 outgassing rate for present
early Noachian unless the loss of CO2 was balanced byO
Earth [Sleep and Zahnle, 2001]. By these estimates, this is
rapid rates of volcanic outgassing.
roughly equal to the Martian outgassing rate at #4 Ga.
[10] We estimate CO2 outgassing rates on MarsCduring
the early Noachian in the following manner: At present,
Using these assumptions of the outgassing
history, carbon
脱ガスboth Earth and Venus have about 100 bar of CO2 at theirloss rate from Mars was greater than the CO2 outgassing
による
surfaces. On Venus, it is in the atmosphere; on Earth, it israte throughout the early Noachian - a dense
early Noachian
供給率
mostly in carbonate rocks. 100 bar of CO2 on the Earth isMartian atmosphere could not have been formed. Outgas#7 " 1045 molecules. Mars has about 1/10th of Earth’s
sing of CO2 might have been episodic instead of continumass, 1/5th of Earth’s surface area, and 40% of Earth’sous. Nevertheless, considering the short time scale (1#10
gravity. Hence, if Mars was formed from the same material
Myrs for 1 bar CO2) for carbon to escape from early Mars,
as Earth, its total CO2 reservoir could have been as much as
the total time during which a dense CO2 atmosphere could
7.5 " 1044 molecules, or 20 bars. This is consistent with the Figure
betweenatmosphericescape haveand beenTian
et al.
[2009]
have 4.
beenComparison
maintainedprior
to
4.1 Ga should
estimate based on geomorphology [Carr, 1986]. Consider- outgassing
on early Mars. The dashed curve represents the
Figure3. Mars
The formed
calculated
upper
brief.
ing that
farther
fromatmospheric
the Sun thanstructure
the Earthof thermal
escape flux of atomic carbon from a CO2earlyMars:(left)
neutraltemperature
profilesand(right)12] Because of Mars’ weak gravity, the planet could also
[
and
thus
could
containmaterials
withvolatile
content
2"• 約40億年以上前ではCO2が効率的に解離。15μm放射冷却が効きにくくなる
dominated atmosphere. The triangles represent the thermal
45
numberdensity
profiles.
Blue,40-bar
green,
redcolors escapehave fluxes
lost significant
atmospherethroughwith
impactmolericher than
Earth,
a maximum(1.5and" 10
of carbon froman atmospherean H2erosion
is possible.
Mars’ Curve
CO2 mixing
cules)
total
CO2 inventory• 酸素原⼦（O）や炭素原⼦（C）はSHE
or HEで⼤規模流出？
representdifferent
solar EUV
energyIf mostflux oflevels.
[Melosh1989],
which
couldH haveratio andof (3 –Vickery,5) " 10!4and with
associatedescapefurther
10
!2
!1
were
emplacedinits
atmosphere
at
4.5
Ga,
immediately
styles represent
different
atmospheric pressures. The black fluxes
strengthens
reducedtheordertotalofCO
cm s .This
The only
two solidlines ourof the102 inventory.
• λO=23@ 10EUV
followingtheplanet’s
formation,
theentireinventory
could
curves in •Figure
3 (right)
are @the20EUV
number density profile of represent
conclusionthat COMars
could nothave sustained
dense CO2
outgassing
fluxes ona earlypossible
2 volcanic
λlost
λO40
=2.4
C=1.8,
havebeen
withinMyrs.
The dash-dottedhorizontal
marksMyrs,
CO2 outgassing
CO2 (dashed), atomic carbon (solid), and atomic oxygen Mars.
atmosphere
for timescale lineof #10and thus was
12
10
!2 !1
11
]
Alternatively,
Mars
could
have
releasedCO
gradu[
2moles
per year,
2.3 " 10 cmgasess ) present,on the theflux(6 " 10cold.
• 初期数億年間は、⼤気を保持できなかった？
その後に⼤気を形成？？
(dotted)in Case 3.
initially(With
fewor greenhouseally and thus avoided the initial fast volatile loss episode. It present Earth. The dotted curve represents 10%8 of theis estimated that #1.5 bar of CO2 was released volcanically energy-limited escape flux of carbon, Flim = eFEUV , where
4Egrav

SHE ?

SHEの流体モデルの問題点:
上側境界条件の影響

E05008Tian et al. [2008]
Tian [2013]
⾮熱的散逸などの効果を考慮して、上側境界（外圏底）でジーンズ流出速度を仮定。
上側境界からの散逸速度を3, 10, 20倍にF. Tian / Earth and Planetary Science Letters 379 (2013) 104–107
105
TIAN ET AL.: HYDRODYNAMIC PLANETARY THERMOSPHERE
E05008> 5 EUVでhydrodynamic
flow？

When a planetary atmosphere is exposed to intense stellar XUVphoton flux, which occurs on terrestrial planets during their early
evolution histories, close-in exoplanets, and small dwarf planets
such as present Pluto, the upper atmosphere is heated and temperature rises and the atmosphere expands. For this scenario tooccur, thermal conduction through the lower boundary must be
ess than the net heating. When the atmosphere expands to large
distance, the gravity of the planet at the exobase, the top of the atmosphere, becomes weak enough and major atmospheric species
escape more eﬃciently through either thermal or nonthermal processes. When the escape of major atmospheric species is eﬃcient,
he upper atmosphere flows outward and the adiabatic cooling associated with the expansion of the rapidly escaping atmospherebecomes a dominant part of the energy budget of planetary atmospheres – the hydrodynamic regime or a hydrodynamic planeFigure 6. Temperature
different solar
EUV fluxBecause
cases (normalized
present-day solar
ary atmosphere(Tian profiles
et al.,for 2008a,
2008b).
the todiffusionmean energy flux !1 times present EUV, which represents solar EUV energy flux !5.1 mW/m2). It is
imitedshown
theorydoes
considerenergyrequired
support
rapid
that whensolar notEUV energy
flux exceeds
certain criticalvalue, theto
upper
part of the thermospherebeginstocool
asaresultoftheincreasingly
significant
adiabaticcooling
effect.
Beyondtheescape,(!5it timescannotprovide us a good estimate on escape rate of criticalma- fluxpresent EUV in this plot), the higher the energy input into the thermosphere, the lower theexobase temperature.
This behavior is typical in the studies of hydrodynamic flow in planetaryor atmosphericspecies.
atmospheres.
Note that there is a difference between the above-mentioned

境界条件を変えると、⼤気はあまり膨らまない？

Fig. 1. Upper atmosphere structures of the Earth under 5 times present XUV radiation level with different escape effusion velocities at the exobase level,9which are
where the curves end.

熱的散逸の全粒⼦モデル
The Astrophysical Journal Letters, 729:L24 (5pp), 2011 March 10

HS, Kn0 = 0.001
VHS-LB, Kn0 = 0.001
HS, Kn0 = 0.0003
HS, Kn0 = 0.0003

10-1
10

101

Transition region in VHS-LB gas-2Φ / Φ0,010ΦJeans / Φ0,0Transition region in HS gas

-5

0

2

4

6
8
10
Jeans parameter λ0

12

100

HS : Hard Sphere原⼦モデル
10-1
VHS : Variable Hard Sphere分
10-2
⼦モデル

10-3
10-4

Dimensionless escape rate Φ / Φ0,0

Dimensionless escape rate

Volkov et al. [2011]の
100
DSMCモデル（全粒⼦モデル）

14

10

-3

10-4
10

-5

10-6 -4
10

10-3

1
Knu

ただし、VolkovモデルはBlowoffからJeansescapeへの切
Figure4. Dimensionless escape rate
Figure 3. Dimensionless escape rate Φ/Φ0,0 (solid curves) and Jeans escape• 加熱は下端のみ
HS molecules.
り替えは急激。
rate ΦJeans /Φ0,0 evaluated at the nominal exobase (dashed curve)vs. λ0 at
Kn0 = 0.001 (circles and triangles) and Kn0 = 0.0003 (squares). Square• 単⼀組成⼤気
SHEは短期間しか起こらない？
and triangles are for a gas composed of HS molecules; circle symbols: for• 放射冷却や光化学反応は考慮せず
a diatomic gas composed of VHS molecules with viscosity index equal to 1
10-2
10
and two internal degrees √of freedom, describedby the Larsen–Borgnakke
Kn(r) = 0.1→ これらを改善したDSMCモデルを構築[K. Terada
et al., 2016] (LB)
2model. Φ0,0 = 4πR0 n0 kT0 /(2πm) is the evaporation rate on R0 . Verticallines indicate transition regions.

-2

熱的散逸の全粒⼦モデル• 今⽇は改善した全粒⼦モデル [K. Terada et al., 2016]
を⽤いて、熱的散逸のレジーム間遷移を議論• ではなくて、• MAVENの観測データ解析 [Terada et al., 2017] と全粒⼦モデル [K. Terada et al., 2016] を⽤いて、熱的散逸に⼤気重⼒波が及ぼす影響を議論• 太陽コロナの加熱に波動（磁気流体波）は重要。
惑星上層⼤気では？
11

エネルギー流⼊/流出の⾼度分布
エネルギー流⼊（太陽放Magnetopause / ionopause
Exobase

上に⾏くほど分⼦数は減少。
1分⼦あたりが受け取るエネルギーは⼤きくなる

無衝突⼤気

射・太陽⾵）エネル
ギー流出

太陽EUVエネルギー
Kn=0.01

Homopause
エネルギー
流⼊（⼤気波動など）

外圏底温度の決定、
上向き流速は形成？

衝突⼤気

Kn=1

クヌーセン数：Kn = lmfp / HS

分⼦拡散
上向き流速・断熱冷却
⼤気波動
放射
熱伝導

エネル
ギー流出
12

エネルギー流⼊/流出の⾼度分布
エネルギー流⼊（太陽放Magnetopause / ionopause
Exobase

上に⾏くほど分⼦数は減少。
1分⼦あたりが受け取るエネルギーは⼤きくなる

無衝突⼤気

射・太陽⾵）エネル
ギー流出

太陽EUVエネルギー
Kn=0.01

Homopause
エネルギー
流⼊（⼤気波動など）

外圏底温度の決定、
上向き流速は形成？

衝突⼤気

Kn=1

クヌーセン数：Kn = lmfp / HS

分⼦拡散
上向き流速・断熱冷却
⼤気波動
放射
熱伝導

エネル
ギー流出
13

エネルギー流⼊/流出の⾼度分布
熱的散逸/⾮熱的散逸
エネルギー
流⼊（太陽放太陽⾵によるピックアップ
Magnetopause / ionopause
プラズマ波動
ExobaseKn=1
プラズマ粒⼦降込
太陽EUVエネルギー（⼀部は⾮熱的）ジュール加熱（磁気圏電流）
Kn=0.01
Homopause
FUV, IRエネルギー
エネルギー
流⼊（⼤気波動など）

外圏底温度の決定、
上向き流速は形成？

無衝突⼤気

上に⾏くほど分⼦数は減少。
1分⼦あたりが受け取るエネルギーは⼤きくなる

衝突⼤気

射・太陽⾵）

エネル
ギー流出

クヌーセン数：Kn = lmfp / HS

分⼦拡散
上向き流速・断熱冷却
⼤気波動・プラズマ波動
放射
熱伝導

エネル
ギー流出
14

熱的散逸に⼤気重⼒波が及ぼす影響• MAVEN/NGIMS観測• DSMC (Direct Simulation Monte-Carlo) 計算

15

Possible effects of GWs on
atmospheric escape• Thermal effect: Thermal balance
in upper thermosphere èEscapeof light components[e.g., Walterscheid et al., 2013]• Dynamical effect: Large-scale
flow in MTI regions[e.g., Medvedev et al., 2011]

Dynamical effect of GWs. GCM runs without(left) and with (right) GWs. Red curves are
meridional stream functions. Color shadeindicates GW drag [Medvedev et al., 2011]• Compositional (mixing) effect:
Homopause altitude èEscaperatio of minor components[e.g., Imamura et al., 2016]

Less observational constraints 16

MAVEN/NGIMS observation

The Mars Atmosphere and Volatile Evolution (MAVEN) Mission• In-situ neutral gas observation in theMartian upper thermosphere [cf. Mahaffy etal., 2015]• Below 500 km alt.
• Periapsis is usually ~150 km alt.

(sometimes goes down to ~130 km during deep dips)• Ar density profiles are used to analyze
wavelike perturbations• Density range between 105 - 107 cm-3
(around exobase) is analyzed• Ar and CO2 behave similarly: Mass is similar,
density is about 60 times smaller

Fig. 16 Nominal MAVEN orbit shown to scale with Mars

The periapsis is targeted toward an atmospheric density rather than an altitude, withthe density range within this corridor being between 0.05–0.15 kg/km3 . Actual altitudevaries depending on season, location, and surface elevation. Periodic Orbital Trim Maneuvers (OTMs) are carried out in order to keep the periapsis within this density corridor.
The apoapsis altitude, orbital period, and orbital inclination (75°) were chosen to provide
appropriate rates of precession of the orbit in both local time and latitude of periapsis as
driven by the non-symmetric gravitational field. With the orbit precessing, observations canbe made at all local solar times and at all latitudes between +/ − 75◦ . Figure 17 shows thetrace of periapsis latitude and local solar time throughout the mission.
Each orbit is divided up into four orbital segments, with different observations or modes
planned for~200
each segment. Figure 18 shows this graphically. The four segments consist of:• Apparent wavelengths between ~100 and– Periapsis segment. Observations emphasize composition and structure from the nominal~500 km are analyzed
periapsis altitude of 150 km up to an altitude of 500 km.

• Period analyzed• Feb. 11, 2015 - Mar. 31, 2016

– Inbound/outbound segments. Observations emphasize measurements of the hot corona
and of escaping species at intermediate altitudes.
– Apoapsis segments. Observations emphasize quasi-global IUVS observations of theplanet’s disk.

Nominal trajectory of MAVEN

During science operations, there are a limited number of modes in which the spacecraft
can operate. Each mode consists of a particular orientation of
17the spacecraft and of the APP
during each of the legs in the orbit. While one can imagine wanting infinite flexibility in
combining different observations on the different legs, there actually are a limited numberof useful combinations; we simplified the planning and implementation of the sequences by

Examples of NGIMS observation in theMartian upper thermosphere
SZA~25 deg.

SZA~40-65 deg.

SZA~75-85 deg.

SZA~100-120 deg.

~exobase
18

Amplitude of small-scale waves• Wave amplitudes in the Martian
upper thermosphere (> ~150 kmalt.) are ~10-20 % on average,
and sometimes > 40 %• Higher amplitudes on nightside
R = 0.7, if daily-averaged GW amplitudes are used.

19

Saturation of GWs?

No.
1]
Atmospheric OBSERVATIONS
gravity waves observed by theradar and GPS WAVESROENGLANDET AL: MAVEN
OFMUGRAVITY
AT MARS

altitude [km]

300

21

Typical MAVEN orbit
near periapsis

250

~1:10

200
150
0

500

1000

1500

2000

2500

3000

[km]

[England et al., 2016]

Fig. 10. A model of a vertical wave number spectrum forsaturated gravity waves in the troposphere (T), stratosphere(S) and mesosphere (M). The spectral density in low wave
numbers (unsaturated) increase with altitude ranges, while thatin high wave number portion does not change, as the waves are0
saturated.

[Tsuda, 2014]• Slope in power spectral density (PSD) of ρ’/ρ changes around“apparent wavelength” of ~200 km.
Fig. 9. A conceptual proﬁle for the amplitude growth ofwind velocity for upward propagating gravity waves
instability phenomena
cannot be clearlyobserved•horizontal“Apparent
wavelength” means wavelength
measuredalong
thewith long (top) and short (bottom) vertical wavelengths. The
for individual waves, statistical analysis using thevertical
gradients of thewind velocity are also compared in the
MAVEN’strack
20
vertical wave number spectra enables us to under38)

right panels.

stand the wave attenuation process.

3500

2

(1)

3

where u' is the horizontal perturbation velocity, u0 is the horizontal mean flow

4velocity, and c is the horizontal phase speed of the wave. Breaking/saturation due to5shear instability is neglected here, since it has a higher threshold for a wave having

6

intrinsic frequency much higher than the inertial frequency [Fritts and Rastogi, 1985].

Saturation of GWs?

7 Using equation (1) and the polarization relations for non-dissipative GWs [Hines, 1960],• A theoreticalthreshold for breaking/saturation
we obtain the corresponding threshold of convective instability for the perturbationof 8a GW
due to convective inst.
2
ʹ as
9 numberdensity
N) gk
i
γ
−1
ʹ
N
ω γ −1 mg 1 (1)(x1• 31 u' > c <− u0
⋅ I =

2•
10

N

ω γ k Hgkkγ k T kz

(3)

2
I i z2H +x i γ −1xgkx2 ω I B 0
N ʹ 0ω I ( kz −
)
(
)
>
⋅
,
32 N 0 ω I (γ kz + i (γ − 2) 2H ) Hgkx kx

(1)
(2)
(2)
3
where u' is the horizontal perturbation velocity, u0 is the horizontal mean flow
γ N−1 ismgthe 1mean number density, i is the imaginary unit, k and k are the
11
where
33
~
z
0
λz limit of kphase4
velocity,
and c is(shortthe horizontal
speed of the xwave. Breaking/saturation
due toz>>1/H)γ kBT0 kz
12 34 horizontal and vertical wavenumbers, respectively, ω I = kx ( c − u0 ) is the intrinsic
5
shear instability is neglected here, since it has a higher threshold for a wave having• Mostthe observed
GWγ amplitudes
would
be g is the gravity of the planet,
13 offrequencyof the wave,
is the heat capacity
ratio,
6 35 intrinsic
frequencymuchhigherthanthekinertial[Fritts
Rastogi, 1985].
Hfrequency
k x theCandthesaturation
shortverticalwavelength, kz >>, andexplainedwith
λz=20-40km
14
the ofBoltzmannmean
kB islimit
H =inkby
T0 isz >> 1constant,s >> ω I kz ,BT0 mg is the scale height,
7 36 large
Usingequation(1)andthesoundfor non-dissipative[Hines,whereismolecularthespeed.ofEquation(3)simplicity,
indicatesthat
the threshold
Cand
γ gH
N ʹ 1960],
15
temperature,isrequiretheweighttherelations
gas. For
weassumethat GWsfor
s = m• Someamplitudes
λpolarization
z<100-200 km
From eq.(2)16similarly
to the main
(CO2This),ofhence
small differencesin wave
isobtaininversely
proportional
to composition
T0 inthreshold
this limit.
approximation
is valid
forfor
T0 >the~150
K
8 37 Ar
webehavesthecorresponding
convective
instability
perturbation

λz =40 km
17 38 amplitude
andAr and CO [England et al., 2016] are neglected.
Thekm
λz =phase
20 ~ between
40
when density(3) yields an
λz =209
numberas for which2 kz > 1 H is satisfied and equation
N ʹ km,18 solutionsprocessesof equation (2)calculated
withincluded
Mars’ parameters of g = 3.7 m/s2, m = 44
• Dissipation
need
to
be
39 error less than 10%. Figure 2d shows that a large part of the GWs distribute around the19amu, and 2γ = 1.3 are displayed in Figure
2d with green long dashed, dashed,2• 40
Molecularradiative
damping,(and ionkcurves
−
i
2H
+
i
γ
−1
gk
λ
=
20
ʹ ω I diffusion,
obtainedfor
Nthreshold
ω
(
)
(
)
z
zx
I ~ 40 km, suggesting that their amplitudes are<⋅
10,
20 dash-dot-dot-dot, dash-dot, and dotted curves for a set of horizontal ( λ x ) and verticaldrag)
N 0limited
ω I (byγ kthe(γ − 2) 2Hbreakdown/saturation) Hgkx kx
41
for moderately long vertical wavelengthz + iconvective
21

22
11

42

( λz ) wavelengths of ( λ x , λz ) = (>20 km, 20 km), (>40 km, 40 km), (>100 km, 100

modes of λ = 20 ~ 40 km.

zkm),(200 Nkm, 200km),meanand (400
km, 200density,km), respectively.
Noteimaginarythat equationunit,(2) is k
i is thewhereis thenumber(2)

21

and k are theFull-particle DSMC simulations(molecular diffusion and radiative damping included)[K. Terada et al., 2016]• A full-particle code• 2-D version is used in this study

300

altitude [km]• A thermospheric DSMC (Direct
Simulation Monte-Carlo) model• Lower boundary condition• From a GW dispersion relation

200
180

140
10

• lmfp = 10m ~ 10km
• Δx = 10m ~ 1km• Δz = 10m ~ 10km

300
280
260

altitude [km]• >~100 particles/cell

220

160

• Grid spacing• No. of particles in a cellH
280 CO2
N2
260
O
240

100
1000 10000 100000
number of particles in a cellmean free pathvertical grid spacing
horizontal grid spacing

240
220
200
180
160
140
0.01

0.1

1
[km]

10

22

MD simulation

calculated orbit of O interacted with O(collision energy = 0.03 eV)numerical integration scheme

10

interval of timeΔt = 3 × 10-18 ~ 1 × 10-17 s

cutoff point

€

−19

$
χ = cos &%(v − v )
x

σOO

-5
-10
2

2

(v − v ) + (v − v )xGCxdiffusion cross section
∞

€

y

N

deflection angle
−1

5

0

F ≤ 1.0 × 10€

[Angstrom]

Verlet algorithm

xGC

y

yGC

2

x

target particle
-5

'
)
(

0
5
[Angstrom]

10vx, vxGC, vy, vyGC :value at cutoff point

15×10 −10

σM ≡ 2π ∫ 0 (1 − cos χ)bdb ≈ 2π ∑ (1 − cos χ )bΔb

viscosity cross section∞

b =0

15×10 −10σµ ≡ 2π ∫ 0 (1 − cos χ)bdb ≈ 2π ∑ (1 − cos χ )bΔb
2

2

b =0

Δb = 1.0 × 10-13 m
23

15

Full-particle DSMC simulations(molecular diffusion and radiative damping included)

GWs with λz=30 km

GWs with λz=215 km

外圏底

λz=200 km or more is required to explain high altitude(>200 km alt.) perturbations.

24Full-particle DSMC simulations(molecular diffusion and radiative damping included)

MAVEN observation

DSMC simulation (λz=215 km)

density variation

altitude

altitude [km]

260km

190km

Temporal variations may be important to explain observed
vertical profiles.
25

Full-particle DSMC simulations(molecular diffusion and radiative damping included)

DSMC simulation (λz=215 km)
300

250

250

altitude [km]

300

200
H
CO2
N2
O
DEGREES PER DAY [K/day]

150

300

250

200

0

100

200

300

150
400

0

Heat budget
the Martian upper
density in[%/day]atmosphere [Bougher et al., 1988]

200

400
600
temperature [K/day]

800

1000

300

Wave heating is comparable 250
to or even larger than the
EUV heating rate.
26
altitude [km]

-100

200

200

ness temperatures osupposedto be equa
Forthecalculation
oftheheatloss
rateqco2owing
to for the 14-/x band of
infraredradiationof CO2in the15-/xband,excitationof the bands of NO and
CO2(01ø0)statesin collisions
withheavyparticles(including constants for the N
oxygen atoms) which is effective at z >• 90 km and collisional
O3(00ø1)vibrationa
and radiative de-excitation processesand absorption ofs-l)radiation are taken into account. Absorption of radiation in
the 15-/xband of CO2 may result in a considerablek(Nz reductionof the rate of radiative coolingat the heightsz •< 100 km. For
k(CO2-O2)= 3 x k(Cthe accuratedescriptionof this effect a numericalsolutionof
the radiative transfer boundaryvalue problemis needed.Ank(O3(01ø0)- O) =
approximate estimate of the role of absorptioncan, howevaccording to Gldnz
er, be obtained in a rather simple way by neglecting the[1975], West et al. [1exchangeofradiation
betweenthethermosphericlayers,• 恐らく、CO2 15μm放射冷却の温度依存のため constant for the relai.e., using 'cool-to-space" approximation[Dickinson, 1972]
sionswith atomic ox
experiment. The rate

2.2 Radiative Coolingin the 15-1aBand

熱的散逸率への影響• ⻑い鉛直波⻑をもつ⼤気重⼒波（λ >200 km）
は、⽕星の上部熱圏で主加熱源になりうる• しかしながら、上部熱圏の温度上昇率は~10%程度に留まる

qco
[erg]
Lcm3s
=1.33
x10_13ßg•0.exp(96•0)- • ß[co2] [ 1978a, b].

been assumedx(•kM[M])xF(•,h)
(14)
3.1

M

to be

3.

RE

Radiative Cool

Let us consider aFO', k) =0.5h L(•)
1 - h + 0.5 hL0-)

k =[Gordiets
et al., 1982]
models that corresp
A •o
+
M

(15a)

hydrostatic and hea
heating by UV radi27
average values.
3.1.1. Model 1.

熱的散逸率への影響• 現在の⽕星では、⽔素のジーンズ流出率が多少（~50%）増える程度。⼤きな影響は無い。
• しかし初期⽕星では、CO2が解離して[Tian et al.,2009]、放射冷却が効きにくくなる。λの急減少。
• エスケープパラメタ（ λ=GMm/kTexorexo）が2倍変わると影響は⼤きい。HE, SHE期間の増加• cf. ⽕星が⼤気を保持しにくい主な原因は、⾦星や地球より表⾯重⼒が2-3倍⼩さいため28

A gap between lower and upper
thermospheres?
MGS observation inlower thermosphere[Creasey et al., 2006]

MAVEN observation in
upper thermosphere[Terada et al., 2017]
1659 orbits

Dayside1500 LT~60°S
35 orbits
Nightside
0200 LT~60°S
30 orbits

Day-night variation is opposite

Gap?

T0 (or saturation?)
mainly controls
29Full-particle DSMC simulations
Effect from above(effect of precipitating ions on thermospheric temperature)

May explain high altitude (>200 km alt.) perturbations. But, 10~100
times higher energy precipitation is required than MAVEN obs.30

まとめ• ⻑い鉛直波⻑をもつ⼤気重⼒波（λz >200 km）
は、⽕星の上部熱圏で主加熱源になりうる• 現在の⽕星では、⽔素のジーンズ流出率が多少（~50%）増える程度• しかし初期⽕星では、CO2が解離して[Tian et al.,2009]、放射冷却が効きにくくなる。λの急減少、
HE, SHE期間の増加。
• 熱的散逸率をさらに増加させ、初期数億年間に⽕星が⼤気を保持できなかった可能性がさらに増加する。
• 現在の⽕星の下部熱圏と上部熱圏で、⼤気重⼒波の分布が異なる。⼆次的重⼒波の⽣成など、
⻑波⻑重⼒波（λz >200 km）の⽣成機構を引き続き調査する必要がある。
31