eart164: planetary atmospheres
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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