eart164: planetary atmospheresfnimmo/eart164/week6_dynamics_part1.pdf · “meridional” f n e ......

F.Nimmo EART164 Spring 11

EART164: PLANETARY

ATMOSPHERES

Francis Nimmo


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


Radiative transfer equations

dt

dz=ar

dzIdI

3

3

16)(

T

z

TzF

4 4

0

3( ) 1

2T T

)1( AF

TC

solar

gP

p

Absorption:

Optical depth:

Radiative

Diffusion:

Rad. time constant:

Greenhouse

effect: eqTT

4/102

1


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!


Dynamics at work

13,000 km

24 Jupiter rotations

30,000 km

PIA02863.mov


Other examples Saturn

Venus

Titan


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”

f

N

E

TRP

g

dP = - g dz Hydrostatic:

Ideal gas:

R is planetary radius, Rg is gas constant

H is scale height

R


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)

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

= 2 W v sin(f)

• How important is the Coriolis effect?

f is latitude

fsin2 WL

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 right

in N hemisphere


1. Winds


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~50 Ro~0.03

(assumes v=100 m/s)

cold

hot


Equatorial easterlies (trade winds)


Zonal Winds

Schematic explanation

for alternating wind directions.

Note that this problem is not

understood in detail.


Really slow rotators • A sufficiently slowly rotating body will

experience DTday-night > DTpole-equator

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

cold hot

• Important in the upper atmosphere of Venus

• Likely to be important for some exoplanets

(“hot Jupiters”) – why?


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

2

2 )1(

4R2CpP/g

trPTC

gFA

T

T

p

E

24)1(

D


Governing equation

• Normally neglect planetary curvature and treat the

situation as Cartesian:

1

ˆ2 sin dv

P z v Fdt

f

W

xFfvx

P

dt

du

1

yFfuy

P

dt

dv

1

f =2Wsin f (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):


Geostrophic balance

• In steady state, neglecting friction we can balance

pressure gradients and Coriolis:

1

2 sin

Pv

x f

W

• 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

H isobars

pressure

Coriolis

wind

xFfvx

P

dt

du

1

Flow is perpendicular to

the pressure gradient!


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


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.171

is dimensionally incorrect


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

the ground)

• The ground exerts a drag on the overlying air

xFfv

x

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

H isobars

pressure

Coriolis

with drag

no drag


Ekman Spiral • The effective thickness d of this layer is

2/1

W

d

where W 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 geostrophic

flow direction

Actual flow directions

Increasing

altitude


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

f

R

u

• If we balance the centrifugal force

against the poleward pressure

gradient, we get zonal winds with

speeds decreasing towards the pole: ff

TRu

g

tan

2


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

(2W 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


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

hot cold

Large

H Small

H


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

W2

~u y

z RH

W

~2

u g T gR T

z fT y yT y

W f R

u

y

so

4

0 expy

T Td

1/ 42

2

R Hgd

W

Does this

make sense?

Latitudinal extent?Venus vs. Earth vs. Mars


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

duW

f

sin2

1

fsin2 W

L

uRo

u g T

z fT y

eart164: planetary atmospheresfnimmo/eart164/week6_dynamics_part1.pdf · “meridional” f n e ......

Documents