eart164: planetary atmospheres

F.Nimmo EART164 Spring 11

EART164: PLANETARY ATMOSPHERES

Francis Nimmo

F.Nimmo EART164 Spring 11

Last Week – Radiative Transfer• Black body radiation, Planck function, Wien’s law• Absorption, emission, opacity, optical depth• Intensity, flux• Radiative diffusion, convection vs. conduction• Greenhouse effect• Radiative time constant

F.Nimmo EART164 Spring 11

Radiative transfer equations

dzIdI

3

3

16)(

T

z

TzF

4 40

3( ) 1

2T T

)1( AF

TC

solar

gP

p

Absorption:

Optical depth:

Radiative Diffusion:

Rad. time constant:

Greenhouse effect: eqTT

4/10 2

1

F.Nimmo EART164 Spring 11

Next 2 Weeks – Dynamics• Mostly focused on large-scale, long-term patterns of

motion in the atmosphere• What drives them? What do they tell us about

conditions within the atmosphere?• Three main topics:

– Steady flows (winds)

– Boundary layers and turbulence

– Waves

• See Taylor chapter 8• Wallace & Hobbs, 2006, chapter 7 also useful• Many of my derivations are going to be simplified!

F.Nimmo EART164 Spring 11

Dynamics at work

13,000 km

24 Jupiter rotations

30,000 km

F.Nimmo EART164 Spring 11

Other examplesSaturn

Venus

Titan

F.Nimmo EART164 Spring 11

Definitions & Reminders• “Easterly” means “flowing from the east” i.e.

an westwards flow.

• Eastwards is always in the direction of spin

x

y

u

v

“zonal/azimuthal”

“meridional”

N

E

TR

P g

dP = - g dzHydrostatic:

Ideal gas:

R is planetary radius, Rg is gas constantH is scale height

R

F.Nimmo EART164 Spring 11

Coriolis Effect• Coriolis effect – objects moving on a rotating

planet get deflected (e.g. cyclones)• Why? Angular momentum – as an object moves

further away from the pole, r increases, so to conserve angular momentum decreases (it moves backwards relative to the rotation rate)

• Coriolis accel. = - 2 x v (cross product)

= 2v sin()• How important is the Coriolis effect? is latitude

sin2 L

v is a measure of its importance (Rossby number)

e.g. Jupiter v~100 m/s, L~10,000km we get ~0.03 so important

Deflection to rightin N hemisphere

F.Nimmo EART164 Spring 11

1. Winds

F.Nimmo EART164 Spring 11

Hadley Cells• Coriolis effect is complicated by fact that parcels of

atmosphere rise and fall due to buoyancy (equator is hotter than the poles)High altitude winds

Surface winds• The result is that the atmosphere is

broken up into several Hadley cells (see diagram)

• How many cells depends on the Rossby number (i.e. rotation rate)

Fast rotator e.g. Jupiter Med. rotator e.g. Earth

Ro~0.1

Slow rotator e.g. Venus

Ro~50Ro~0.03(assumes v=100 m/s)

cold

hot

F.Nimmo EART164 Spring 11

Equatorial easterlies (trade winds)

F.Nimmo EART164 Spring 11

Zonal Winds

Schematic explanationfor alternating wind directions.

Note that this problem is not understood in detail.

F.Nimmo EART164 Spring 11

Really slow rotators• A sufficiently slowly rotating body will

experience Tday-night > Tpole-equator

• In this case, you get thermal tides (day-> night)

coldhot

• Important in the upper atmosphere of Venus

• Likely to be important for some exoplanets (“hot Jupiters”) – why?

F.Nimmo EART164 Spring 11

Thermal tides• These are winds which can blow from the hot (sunlit)

to the cold (shadowed) side of a planet

Extrasolar planet (“hot Jupiter”)

Solar energy added =

Atmospheric heat capacity =Where’s this from?

So the temp. change relative to background temperature

t=rotation period, R=planet radius, r=distance (AU)

Small at Venus’ surface (0.4%), larger for Mars (38%)

tr

FAR E

22 )1(

4R2CpP/g

trPTC

gFA

T

T

p

E24

)1(

F.Nimmo EART164 Spring 11

Governing equation

• Normally neglect planetary curvature and treat the situation as Cartesian:

1ˆ2 sin

dvP z v F

dt

xFfvx

P

dt

du

1

yFfuy

P

dt

dv

1

f =2sin (Units: s-1)

u=zonal velocity (x-direction)v=meridional velocity (y-direction)

• Winds are affected primarily by pressure gradients, Coriolis effect, and friction (with the surface, if present):

F.Nimmo EART164 Spring 11

Geostrophic balance

• In steady state, neglecting friction we can balance pressure gradients and Coriolis:

1

2 sin

Pv

x

• The result is that winds flow along isobars and will form cyclones or anti-cyclones

• What are wind speeds on Earth?• How do they change with latitude?

L L

Hisobars

pressure

Coriolis

wind

xFfvx

P

dt

du

1

Flow is perpendicular to the pressure gradient!

F.Nimmo EART164 Spring 11

Rossby number

• For geostrophy to apply, the first term on the LHS must be small compared to the second

• Assuming u~v and taking the ratio we get

y

Pfu

dt

dv

1

fL

u

fu

tuRo

/~

• This is called the Rossby number

• It tells us the importance of the Coriolis effect

• For small Ro, geostrophy is a good assumption

F.Nimmo EART164 Spring 11

Rossby deformation radius• Short distance flows travel parallel to pressure gradient• Long distance flows are curved because of the Coriolis

effect (geostrophy dominates when Ro<1)• The deformation radius is the changeover distance• It controls the characteristic scale of features such as

weather fronts

• At its simplest, the deformation radius Rd is (why?)prop

d

vR

f

• Here vprop is the propagation velocity of the particular kind of feature we’re interested in

• E.g. gravity waves propagate with vprop=(gH)1/2

Taylor’s analysis on p.171is dimensionally incorrect

F.Nimmo EART164 Spring 11

Ekman Layers• Geostrophic flow is influenced by boundaries (e.g.

the ground)• The ground exerts a drag on the overlying air

xFfvx

P

dt

du

1

• This drag deflects the air in a near-surface layer known as the boundary layer (to the left of the predicted direction in the northern hemisphere)

• The velocity is zero at the surface

Hisobars

pressure

Coriolis

with drag

no drag

F.Nimmo EART164 Spring 11

Ekman Spiral• The effective thickness of this layer is

2/1

where is the rotation angular frequency and is the (effective) viscosity in m2s-1

• The wind direction and magnitude changes with altitude in an Ekman spiral:

Expected geostrophicflow direction

Actual flow directions

Increasing altitude

F.Nimmo EART164 Spring 11

Cyclostrophic balance• The centrifugal force (u2/r) arises when an air packet

follows a curved trajectory. This is different from the Coriolis force, which is due to moving on a rotating body.

• Normally we ignore the centrifugal force, but on slow rotators (e.g. Venus) it can be important

• E.g. zonal winds follow a curved trajectory determined by the latitude and planetary radius

R

u

• If we balance the centrifugal force against the poleward pressure gradient, we get zonal winds with speeds decreasing towards the pole:

TRu g

tan2

F.Nimmo EART164 Spring 11

“Gradient winds”• In some cases both the centrifugal (u2/r) and the Coriolis

(2x u) accelerations may be important• The combined accelerations are then balanced by the

pressure gradient• Depending on the flow direction, these gradient winds can

be either stronger or weaker than pure geostrophic winds

Insert diagram hereWallace & HobbsCh. 7

F.Nimmo EART164 Spring 11

Thermal winds• Source of pressure gradients is temperature gradients• If we combine hydrostatic equilibrium (vertical) with

geostrophic equilibrium (horizontal) we get:

u g T

z fT y

N

x

y

z

u(z)

hot

cold

This is not obvious. The key physical result is that the slopes of constant pressure surfaces get steeper at higher altitudes (see below)

Example: On Earth, mid-latitude easterly winds get stronger with altitude. Why?

P2

P1

P2

P1

hotcold

Large HSmall

H

F.Nimmo EART164 Spring 11

Mars dynamics example• Combining thermal winds and angular momentum

conservation (slightly different approach to Taylor)• Angular momentum: zonal velocity increases polewards• Thermal wind: zonal velocity increases with altitude

2

~y

uR

2

~u y

z RH

~2

u g T gR T

z fT y yT y

R

u y

so

4

0 expy

T T

1/ 42

2

R Hg

Does this make sense?

Latitudinal extent?Venus vs. Earth vs. Mars

F.Nimmo EART164 Spring 11

Key Concepts

• Hadley cell, zonal & meridional circulation• Coriolis effect, Rossby number, deformation radius• Thermal tides• Geostrophic and cyclostrophic balance, gradient winds• Thermal winds

xFvx

P

dt

du

sin21

sin2

L

uRo

u g T

z fT y