Atmospheric Physics I
8. Global Circulation & Waves
1
18.10.05
introduction
2
25.10.05
vertical structure, stability
-
01.11.05
holiday
3
08.11.05
radiation transfer, basic laws
4
15.11.05
radiation balance at surface, greenhouse effect
5
22.11.05
atmospheric dynamics, Navier Stokes equation
6
29.11.05
vorticity, potential and absolute vorticity
7
06.12.05
thermal wind & waves
8
13.12.05
global circulation & waves
9
20.12.05
molecular & turbulent diffusion
10
10.01.06
planetary boundary layer
11
17.01.06
variations of global circulation (ENSO, ...)
12
24.01.06
stratosphere troposphere exchange
13
31.01.06
upper atmosphere: air glow and aurora
14
07.02.06
to be seen
15
14.02.06
to be seen
Review
Barotropic & baroclinic atmospheres
Thermal wind
Direct convection
Connection to generation of vorticity
Atmospheric waves: Rossby waves
Gravitational Waves in Spain
Revisited
gravitational waves over Spain as reflected in the formation of
cloud bands, picture from D. Etling (2002)
Pressure Tendency
local p-changes are influencing weather
lead to changes in dynamic equilibrium
and may lead to the development of cyclones and
anticyclones
Pressure Tendency
From the hydrostatic pressure
p(z) =
!∞
gρ dz"
z
we can immediately derive an
expression for the p-tendency
!
∂p
∂t
"
z
∂
=
∂t
#∞
gρ dz" =
z
#∞
z
g
∂ρ "
dz
∂t
using the continuity equation
!
∂p
∂t
=−
"
z
!∞
z
=−
#∞
g∇ · (ρ#u) dz$
z
g∇h · (ρ"uh )dz$ −
!∞
z
g
∂
· (ρw) dz$
∂z
We can integrate the second
term
!
∂p
∂t
"
z
=−
#∞
g∇h · (ρ#uh )dz$ + (gρw)z
z
and conclude that two factors
contribute
the divergence of the
horizontal mass flow above z
and
the direct mass flow through
the z-niveau.
Pressure Tendency
Consider horizontal divergence
∇h · (ρ"uh ) = "uh · ∇h ρ + ρ∇h · "
uh
This leads to the barometric
tendency equation
!
∂p
∂t
"
=−
z
−g
!∞
#∞
gρ(∇h · #
uh ) dz$ −
z
(!uh · ∇h ρ)dz$ + g(ρw)z
z
Note, only ageostrophic wind can
cause any divergence
g∇h · (ρ"uh ) = ∇h ·
!
"
1"
k × ∇h p = 0
f
from H. Pichler (1997):
Dynamik d. Atmosphäre
Global Circulation
1. Differential heating between equatorial regions
and poles drives meridional heat flow in
atmosphere & oceans
2. Conservation of total angular momentum in the
Earth-Atmosphere system requires regions of
both easterly and westerly winds (Hadley 1735)
Meridional Radiation
Imbalance
solar irradiance,
planetary albedo &
longwave emission
determine radiation budget,
that shows dependence on
latitude
Pronounced imbalance in the
radiation budget between
low and high latitudes
picture taken from Houghton (2000)
Meridional Energy
Transfer
global oceanic circulation
from H. Kraus (2000) Die
Atmosphäre der Erde
A Single Convection Cell ?
Can there be a single convection cell ?
from Roedel (2000)
Global Circulation
Hadley Circulation: G. Hadley 1735: “Concerning the Cause of the
General Trade Winds”, Phil. Trans. 39 Roy. Soc. London, 58.
Angular Momentum
Conservation
Single cell implies easterly winds everywhere
Friction would cause slow down Earths rotation
Since rotation keeps on going, there must be zones
of opposite wind direction
Single cell model fails due to rotation of Earth
Dishpan Experiment
Streak photographs of metal
particles in rotating fluid, which is
heated at the outside (ΔT = 9K):
Ω = 0.41, 1.07, 1.21, 3.22, 3.91 and
6.4 s-1 for the sequence a) - f),
after Hide & Mason (1975)
Hadleys Arguments (1735)
1. Sun as driving force of direct circulation
" this has been recognized earlier on
2. Coriolis force
Hadleys Arguments (1735)
3. angular momentum conservation
4. existence of circulation cell between subtropics
and equator
Hadley Cell
convective uprise of air,
amplified by high latent heat
content
surface winds towards ITC are
affected by Coriolis force and
are steady easterly winds
anti trade wind velocities are
small and superposed by a fast
zonal component, that
culminates in the STJ
from Roedel (2000)
Sub Tropic Jet (STJ)
We can understand velocities in the jet roughly being due
to angular momentum conservation
Consider angular momentum
(z-component) per mass
L = Θ(Ω + ω)2
Because
Θ = (R cos ϕ)2 ,
ω = v/(R cos ϕ)
angular momentum
conservation requires const.
v
L = (R cos ϕ)2 (Ω +
)
R cos ϕ
At equator we start with v = 0
!"
L = ΩR2
thus
v
R Ω = (R cos ϕ) (Ω +
)
R cos ϕ
2
2
and
RΩ(1 − cos2 ϕ)
v=
cos ϕ
= RΩ sin ϕ tan ϕ
gives roughly 130 m/s, in reality
its about 3 - 4 times less
Three Band General
Circulation
from Roedel (2000)
Zonally averaged
pressure
Roedel (2000)
Global Circulation from
Satellite
IR photograph from satellite
Meteosat-7 (25.April 1999,
12:00 UTC)
taken from H. Kraus (2000)
“Die Atmosphäre der Erde”
Meridional profile of wind
velocities
from Roedel (2000)
Atmospheric Mixing Times
Zonal (O-W),
meridional (N-S) and
interhemispheric mixing
times (D.J. Jacob 1999)
Zone of the westerlies
Interpretation of the westerlies as
a thermal wind leads to a gradient
of 1 - 3 ms-1/km which is in
agreement with observation
BUT westerly winds have highly
complex and variable structure
(formation of cyclones) which is
linked to atmospheric waves that
lead to disturbances.
planetary (barotropic / Rossby)
waves at the 500 mb level
(schematic)
Baroclinic Waves
quasi barotropic niveau only at z = 5 - 6 km
barotropic waves possess instabilities that cannot
lead to the formation of zyclones and
anticyclones
other amplification mechanism is required
generally, however, atmosphere is baroclinic and
not divergence free.
Baroclinic Waves
Understanding the dynamic character of cyclogenesis
Vorticity equation
∂η
= η∇h#u + #u ∇h η = 0
∂t
Consider stationary wave
∇h (η · "
u) = η∇h"
u +"
u ∇h η = 0
which implies
!u · ∇h η
∇h!u = −
η
2u∆η
As a rough estimate, we obtain ∇h!u ≈ −
λη
from Roedel (2000)
Front Systems
form when air masses of
different T meet and
then form a confined
region of transition
characterized by large Tgradients and small
extensions (10 km)
high wind speeds (du/dz)
follow from thermal wind
considerations
Angle of inclination
Formula of Margules (1906):
f ∆u
tan α = T
g ∆T
Consider pressure at interface:
α
dp = ∇pc · d!s = ∇pw · d!s
from Roedel (2000)
!
∂p
∂y
"
· dy +
c
!
∂p
∂y
"
!
∂p
∂z
"
· dy +
w
· dz =
c
!
∂p
∂z
"
w
· dz
thus
! "
∂p
∂y
−
∂p
∂z c
−
tan α = ! "w
! "
∂p
∂y
! "c
∂p
∂z w
Angle of inclination
Use barometric formula and
geostrophic wind equation
∂p
= −gρ
∂z
∂p
= −fρu
∂y
α
!"
! "
! "
f ρu w − ρu c
tan α =
g
ρc − ρw
ρ∝
1
T
"#
from Roedel (2000)
thus
f Tc uw − Tw uc
tan α =
g Tw − Tc
f ∆u
≈ T
g ∆T
typically, ∆u = 30 ms−1 , ∆T = 10 K
"#
α = 0.5◦
Warm and Coldfront
from Roedel (2000)
Cyclones & Anticyclones
Positive feedback
between wave
amplitudes and
formation of
divergences leads to
cyclogenesis
Cyclogenesis
Baroclinic Instability
A schematic picture of a baroclinic atmosphere that is
vertically stable θ1 < θ2 < θ3
Two effects lead to enhanced
instability wrt to a barotropic
atmosphere
static effect: Though vertically
stable, elongation within the
angular sector α leads to
instability
from Etling (2001)
dynamic (inertial) effect: Due to
thermal wind shear, elongation in
different z-niveaus leads to
imbalance of local geostrophic
force balance and an additional
horizontal elongation occurs,
effectively enlarging the original
sector of instability