The dynamics of planetary atmospheres

Theoretical concept is formulated in the

Geophysical Fluid Dynamics – the dynamics of stratified fluid on a rotating sphere

Solar flux

Hadley cellcirculation

convective flow westerly jet

easterly (passat) flow

tidal buldge

“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

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

sin2cos2 f Coriolis parameter

Useful numbers

yR

f 00 cos2

sin2

0cos2 R

Rossby parameter

-plane approximation

Rossby numberfL

VRo

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

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

Wave motion in the atmosphere

•Elasticity sound•Buoyancy gravity waves•Coriolis force inertial waves

Planetary-scale forced oscillations of any type are called tides

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

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

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

.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,

Solar flux

Solsticial circulation on Mars

westerly jet

thermal tide

Rossby waves

0

180

Water vapor column:Ls = 92, North hemisphere [kg/m2]

0

180

Water vapor column:Ls = 113, North hemisphere [kg/m2]

Water vapor column:Ls = 143, North hemisphere [kg/m2]

0

180

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

Ls=93-97 Ls=113-115 Ls=127-136

Cyclonic vortices Anticyclonic vortex

Venus case (symmetric mode)

convective flow

superrotation thermal tide

multiple Hadley cells?

VIRTIS image of the Southern Polar vortex @ 5.0 µm

VIRTIS image of the Southern Polar vortex @ 5.0 µm

Titan: another superrotating atmosphere

Mingalev et al., 2008

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

Giant planetszones

belts

Hot spot

Polar vortex

32 km/pix