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

Absorption: Optical depth: Greenhouse effect:

T

4

dI

 

d

t

dz

   =  ar

I



dz



T

0 4    1  3 2     Radiative Diffusion:

F

(

z

)  Rad. time constant:  16 

T C

3  

z

 

g T p F solar

( 1 

A

) 

T

 3

T

0  1 2 1 / 4

T eq

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

Saturn

Other examples

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 Ideal gas: Hydrostatic:

P

 

R g T



dP = -



g dz R H

is planetary radius,

R g

is scale height is gas constant

y

N

“meridional”

v R

f

x u

“zonal/ azimuthal”

E

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 w decreases (it moves backwards relative to the rotation rate) Deflection to

right

• Coriolis accel. = - 2

= 2

W x

v

(cross product) W

v sin(

f

)

f is latitude in N hemisphere • How important is the Coriolis effect?

v

is a measure of its importance (Rossby 2

L

W sin f number) e.g. Jupiter

v

~100 m/s,

L

~10,000km we get ~0.03 so important F.Nimmo EART164 Spring 11

1. Winds

F.Nimmo EART164 Spring 11

Hadley Cells

• Coriolis effect is complicated by fact that parcels of hot atmosphere rise and fall due to buoyancy (equator is High altitude winds Surface winds hotter than the poles) cold • 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) Slow rotator e.g. Venus Fast rotator e.g. Jupiter Med. rotator e.g. Earth

Ro

~0.03

(assumes

v

=100 m/s)

Ro

~0.1

Ro

~50 F.Nimmo EART164 Spring 11

Equatorial easterlies (trade winds) F.Nimmo EART164 Spring 11

Zonal Winds

Schematic explanation for 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 D

T

day-night > D

T

pole-equator • In this case, you get

thermal tides

(day-> night) hot cold • 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 Solar energy added = 

R

2 ( 1 

A

)

F E

2

t r t

=rotation period,

R

=planet radius,

r

=distance (AU) Atmospheric heat capacity = Where’s this from?

4



R 2 C p P/g

Extrasolar planet (“hot Jupiter”) So the temp. change relative to background temperature D

T T

 ( 1 

A

)

gF

4

PTC E p r

2

t

Small at Venus’ surface (0.4%), larger for Mars (38%) F.Nimmo EART164 Spring 11

Governing equation

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

d v dt



P



z

ˆ 

F

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

du dt

  1  

P



x



dv dt

  1  

P



y



fv



F x fu



F y f

=2 W sin f (Units: s -1 )

u

=zonal velocity (

x

direction)

v

=meridional velocity (

y

-direction) F.Nimmo EART164 Spring 11

Geostrophic balance

du dt

  1  

P



x



fv



F x

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

v

 1 2  W sin f 

P



x

Flow is

perpendicular

to the pressure gradient!

L L wind Coriolis H pressure isobars • 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?

F.Nimmo EART164 Spring 11

Rossby number

dv dt



fu

  1  

P



y

• 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

Ro

~

u

/

t



u fu fL

• 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

R d

is (why?)

R d



v prop f

Taylor’s analysis on p.171

is dimensionally incorrect • Here

v prop

is the propagation velocity of the particular kind of feature we’re interested in • E.g. gravity waves propagate with

v prop =(gH) 1/2

F.Nimmo EART164 Spring 11

Ekman Layers

• Geostrophic flow is influenced by boundaries (e.g. the ground) • The ground exerts a

drag du dt

  1   on the overlying air 

P x



fv



F x

with drag no drag Coriolis H pressure isobars • 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 F.Nimmo EART164 Spring 11

Ekman Spiral

• The effective thickness d of this layer is d   W  1 / 2 where W is the rotation angular frequency and (effective) viscosity in m 2 s -1  is the • The wind direction and magnitude changes with altitude in an Ekman spiral : Actual flow directions Increasing altitude Expected geostrophic flow direction F.Nimmo EART164 Spring 11

Cyclostrophic balance

• The centrifugal force (

u 2 /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

u R

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

u

2  

R g

tan f 

T

 f F.Nimmo EART164 Spring 11

“Gradient winds”

• In some cases both the centrifugal (

u 2 /r

) and the Coriolis (

2

W x

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 here Wallace & Hobbs Ch. 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:

N

y

 

u z

  cold

z g fT



T



y u(z)

Small

H

P This is

not

obvious. The key physical result is that the slopes of constant pressure surfaces get steeper at higher altitudes (see below) P 2 2 P 1 Large

H

P 1 hot

x

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

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

R

f

u



y u

~

y

2 W

R



u



z

~ 

g fT

  so

T y

  

u



z

~

y

2

RH

W 2

gR

W

yT



T



y T



T

0 exp    d 4   d W 2  1/ 4  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

Ro



u

2

L

W sin f

du dt

  1  

P



x

 2 W sin f

v



F x



u



z

 

g fT



T



y

F.Nimmo EART164 Spring 11