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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
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
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
Other examples Saturn
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”
f
N
E
TRP
g
dP = - g dz Hydrostatic:
Ideal gas:
R is planetary radius, Rg is gas constant
H 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
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
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~50 Ro~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 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 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?
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
2
2 )1(
4R2CpP/g
trPTC
gFA
T
T
p
E
24)1(
D
F.Nimmo EART164 Spring 11
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):
F.Nimmo EART164 Spring 11
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!
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.171
is 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
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
F.Nimmo EART164 Spring 11
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
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
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
F.Nimmo EART164 Spring 11
“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
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
hot cold
Large
H Small
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
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
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
duW
f
sin2
1
fsin2 W
L
uRo
u g T
z fT y