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
dI    I  dz
dt
= ar
Optical depth:
dz
Absorption:
Greenhouse
effect:
 3 
T ( )  T 1   
 2 
Radiative
Diffusion:
16 T T
F ( z)  
3 z 
4
4
0
Cp
Rad. time constant:
1
T0  1/ 4 Teq
2
3
 T
P
g
Fsolar (1  A)
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
30,000 km
24 Jupiter rotations
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
RgT
Ideal gas: P 

Hydrostatic: dP = -  g dz
N “meridional”
y
v
f
R is planetary radius, Rg is gas constant
H is scale height
R
x E
u
“zonal/
azimuthal”
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
Deflection to right
the rotation rate)
in N hemisphere
• Coriolis accel. = - 2 W x v (cross product)
= 2 W v sin(f)
f is latitude
• How important is the Coriolis effect?
v
2 LW sin f
is a measure of its importance (Rossby
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
atmosphere rise and fall due to buoyancy (equator is
High altitude winds
hotter than the poles)
Surface winds
• The result is that the atmosphere is
cold
broken up into several Hadley
hot
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~0.03
(assumes v=100 m/s)
Ro~0.1
Slow rotator e.g. Venus
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 DTday-night > DTpole-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 =
FE
R (1  A) 2 t
r
2
t=rotation period, R=planet radius, r=distance (AU)
Atmospheric heat capacity = 4R2CpP/g
Where’s this from?
Extrasolar planet (“hot Jupiter”)
So the temp. change relative to background temperature
DT
gFE
 (1  A)
t
2
T
4PTCp r
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):
dv
1
  P  2W sin f  zˆ  v   F
dt

• Normally neglect planetary curvature and treat the
situation as Cartesian:
f =2Wsin f
du
1 P

 fv  Fx
dt
 x
dv
1 P

 fu  Fy
dt
 y
(Units: s-1)
u=zonal velocity (xdirection)
v=meridional velocity
(y-direction)
F.Nimmo EART164 Spring 11
Geostrophic balance
du
1 P

 fv  Fx
dt
 x
• In steady state, neglecting friction we can balance
pressure gradients and Coriolis:
1
P
Flow is perpendicular to
v
the pressure gradient!
2 W sin f x
L
L
wind
pressure
Coriolis
H
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
1 P
 fu  
dt
 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
u/t u
Ro ~

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 Rd is (why?)
Rd 
v prop
f
Taylor’s analysis on p.171
is dimensionally incorrect
• 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
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
du
1 P

 fv  Fx
dt
 x
with drag
no drag
pressure
Coriolis
H
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  is the
(effective) viscosity in m2s-1
• 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 (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
u
the latitude and planetary radius
R
• If we balance the centrifugal force
against the poleward pressure
gradient, we get zonal winds with
speeds decreasing towards the pole:
f
T
u 
 tanf f
2
Rg
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
This is not obvious. The key
physical result is that the
slopes of constant pressure
surfaces get steeper at higher
altitudes (see below)
P2
z
N
y cold
Small
H
u(z)
hot
x
P2
P1
P1
cold
Large
H
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
y2
u~
W
R
u
R
f

y
u
y2
~
W
z RH
so
u
g T
gR T
~

z
fT y
2WyT y
  y 4 
T  T0 exp     
 d  


1/ 4
 R Hg 
d 

2
W


2
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
u
Ro 
2 LW sin f
du
1 P

 2W sin fv  Fx
dt
 x
u
g T

z
fT y
F.Nimmo EART164 Spring 11