eart164: planetary atmospheresfnimmo/eart164/week6_dynamics_part1.pdf · “meridional” f n e ......
TRANSCRIPT
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F.Nimmo EART164 Spring 11
EART164: PLANETARY
ATMOSPHERES
Francis Nimmo
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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
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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
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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!
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F.Nimmo EART164 Spring 11
Other examples Saturn
Venus
Titan
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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
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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
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F.Nimmo EART164 Spring 11
1. Winds
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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
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F.Nimmo EART164 Spring 11
Equatorial easterlies (trade winds)
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F.Nimmo EART164 Spring 11
Zonal Winds
Schematic explanation
for alternating wind directions.
Note that this problem is not
understood in detail.
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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?
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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
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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):
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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!
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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
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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
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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
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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
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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
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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
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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
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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
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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