The dynamics of planetary atmospheres
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![Page 1: The dynamics of planetary atmospheres](https://reader035.fdocuments.in/reader035/viewer/2022062314/56812ef2550346895d949233/html5/thumbnails/1.jpg)
The dynamics of planetary atmospheres
Theoretical concept is formulated in the
Geophysical Fluid Dynamics – the dynamics of stratified fluid on a rotating sphere
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Solar flux
Hadley cellcirculation
convective flow westerly jet
easterly (passat) flow
tidal buldge
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“Primitive” equations:fluid on a rotating sphere
.
,0
,
,22
RTpdt
d
fqdt
dp
dt
dTC
gpt
V
v
vωvvvv
momentum
energy
continuityeqn. of state
modifications
vvvv 22
22
1
p
t
vrot,2 absolute vorticity
algeopotenti ,g
Boussinesq approximation: 0dt
deverywhere but gravity
gpdt
d vωv
2
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Spherical coordinate system:
z, w
x, u
y, v
cos2
cos2
)cossin(2
uz
pg
dt
dw
uy
p
dt
dv
vwx
p
dt
du
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sin2cos2 f Coriolis parameter
Useful numbers
yR
f 00 cos2
sin2
0cos2 R
Rossby parameter
-plane approximation
Rossby numberfL
VRo
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Adjustments
p
fz
wt
uku
uuu
.
,ctg
1
,
z
pg
z
pgfu
dt
dw
pf
pf
g
ku
uk
equation of horizontal motion
geostrophic approximaton
hydrostatic approximation
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Ro << 1 thermal wind equation
Tz
p
fT
gT
z
pp
z
T
fp
R
z g
kkku
Opposite case: Ro >> 1
., 21
pRu cc k cyclostrophic approximation
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Wave motion in the atmosphere
•Elasticity sound•Buoyancy gravity waves•Coriolis force inertial waves
Planetary-scale forced oscillations of any type are called tides
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acoustic waves
.
0,,
,
2002
2
000
ppdt
p
constconstp
pdt
dpp
dt
d
v
vv
.
,
,0
000
200
2
0
0
pk
kuc
pppdx
udt
uif
x
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gravity waves
.0
.)0()(
,
2
2
0
0
02
2
zdz
dg
dt
zd
dz
dzz
gdt
zd
dz
dgg
2Brunt-Vajsala frequency
-potential temperature
For non-vertically propagating waves, 2
222
V
Hg k
k
Group velocity .2 HVV
gg kk
kc ki
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Rossby waves
.0
0
0
y
v
x
u
p
yfv
x
vU
t
v
p
xfv
x
uU
t
u
Search for the solution in form
.)(
~)(~)(~
)( kxtiAeyP
P
yv
v
yu
u
.0~
~
0~
~~)(
0~~~)(
y
vuik
y
PufvkUi
PikvfukUi
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.sincos~
issolution general
,0~~
or
,0~~~
21
22
21
21
22
2
22
2
ykcU
CykcU
Cv
vkcUy
v
vkvky
vkU
Eliminating P and use incompressibility,
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Solar flux
Solsticial circulation on Mars
westerly jet
thermal tide
Rossby waves
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0
180
Water vapor column:Ls = 92, North hemisphere [kg/m2]
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0
180
Water vapor column:Ls = 113, North hemisphere [kg/m2]
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Water vapor column:Ls = 143, North hemisphere [kg/m2]
0
180
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OMEGA observations of the residual seasonal frost around South pole. Ls = 236-243
1.5 m band depth
0.02
0.04
0.06
0.08
0.1
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Ls=93-97 Ls=113-115 Ls=127-136
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Cyclonic vortices Anticyclonic vortex
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Venus case (symmetric mode)
convective flow
superrotation thermal tide
multiple Hadley cells?
VIRTIS image of the Southern Polar vortex @ 5.0 µm
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VIRTIS image of the Southern Polar vortex @ 5.0 µm
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Titan: another superrotating atmosphere
Mingalev et al., 2008
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Vertical velocity on Titan: a signature of thermal tide
0 30 60 90 120 150 180 210 240 270 300 330 360-90
-75
-60
-45
-30
-150
15
30
45
60
7590
longitude
lati
tud
e
z=550 km
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
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Giant planetszones
belts
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Hot spot
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Polar vortex
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32 km/pix
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