v - Institut für Umweltphysik

Atmospheric Composition, Structure, Waves &
Circulation
Prof. Dr. K. Pfeilsticker
Institut für Umweltphysik
Universität Heidelberg
INF 229
69120 Heidelberg
(May 30th, 2012)
Outline:
• Introduction
•Atmospheric Structure, composition& circulation revisited
• Waves as building blocks of dynamical processes
• Sound waves
• Capillary waves
• Gravity waves
• Rossby waves
• Kelvin waves
•Summary
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Atmospheric Structure
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Atmospheric Structure
The atmospheric vertical structure can be defined according to
1.) electro-magnetic behavior, c.f. charge density which historically came first into
play
•
Lower and middle atmosphere (103 ions/cm3)
•
Ionosphere ( > ~ 65 km) with the D, E, F, layers (105 ions and e-/cm3)
2.) its composition, c.f.,
•
Homosphere: vertically ‚well‘ mixed !
•
Heterosphere: gravitational separation becomes important
3.) its temperature
•
Troposphere (0 – 10/18 km, dominated by moist convection)
•
Stratosphere (10/18 – 50 km, RT-dominated, and O3 , ... absorption)
•
Mesosphere (50 – 90 km, RT-dominated but occasionally gravity wave
(energy) deposition)
•
Thermosphere (90 – 500 km, RT-dominated)
•
Exosphere ( > 500 km, EM-dominated)
These layers are further separated by so-called ‚pauses‘.
4.) In the recent past, other criteria for the devision of the atmosphere into layers came
also into play. This includes criteria such as certain values for the potential
vorticity, or ozone concentration, ....
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Atmospheric Composition/Major Gases
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Some Useful Notation and Numbers
1.
2.
3.
4.
5.
6.
Mean molecular weight: Mmol = {[N2]·MN2 + [O2]·MO2 +….}/{[N2] + [O2] + ...) Mmol =
~ 28.84 g/mol
Scale height: In lecture dynamics1 we introduced the hydrostatic equation in
1D; dp = - ρ·g·dz = dp/p = - (Mmol·g)/(R·T)·dz p (z) = po· exp(- Mmol·g/(R·T)·z) = po·
exp(- z/zo) with the scale height (e-fold distance) zo = R·T/(Mmol·g) ~ 7770 m for T
= 220 K. This implies that the atmospheric pressure ½’s for every 5 km increase
in atmospheric height.
Molecular concentration, [no]: Taking the ideal gas law no = (p·Nmol)/(R·T) for
standard condition (p = 1013 mbar, T = 273.3 K) [no] = 2.68·1019/cm3
Molecular column density, [N]: Integration of the hydrostatic equation from the
surface to infinity leads to [N]T = no·zo = 2.08·1025/cm2.
Total atmospheric mass, [Mtotal]: Taking [N]T and integrating over the whole the
earth surface (A = 5.09·1018 cm2) leads to [Mtotal]= 1.76·1020 mol = 5.09·1018 kg.
Potential temperature, [Θ
Θ]: Since total thermal energy is frequently conserved in
atmospheric dynamics, the potential temperature Tpot – defined as an air mass’
T at standard condition - is a useful concept. Considering an adiabatic process
we obtain Θ = To·(po/p)κ-1 (with Cp/Cv ≈ 7/5).
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
General Circulation: Major Tropospheric Cells
Surface Winds
General Circulation
see http://www.youtube.com/watch?v=qh011eAYjAA&feature=related
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Typical Scales (Time, Space) in Atmospheric Dynamics
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Stratospheric Circulation (Modern View)
A more modern view of the stratospheric circulation was suggested by Holton in
1995. Thereby, beside the action of diabatic processes, the action of breaking
(mostly planetary) waves has been more recognized.
It is now thought, that large
scale waves in midlatitudes drive a pump that
sucks air out of the tropical
pipe and pushes it to more
polar latitudes. The action
of diabatic processes is
then more to adjust the air
masses to the ambient
radiation conditions.
Also, a regime (called
stratospheric underworld,
or LS with Θ < 380 K),
relative isolated from the
‚overworld‘ is now being
recognized. It is separated
from the troposphere and
stratosphere by different Θ
and PV values.
Klaus Pfeilsticker
Tropical pipe
Institut für Umweltphysik
University of Heidelberg
Waves as building blocks of dynamical processes
We recall some simple physical facts about unforced and undamped waves:
1.) a wave equations of unforced a freely moving (undamped) oscillations
combine spatial (x) and temporal (t) second derivatives of amplitudes Ψ, i.e.
2
∂ 2ψ
2 ∂ ψ
−c ⋅ 2 = 0
∂t 2
∂x
or
equivalently
∂ 2ψ
− c ⋅ k 2 ⋅ψ = 0
∂t∂x
(1)
where c = ω/k is the phase velocity, k (= 2π/λ) the wavenumber, and vg = ∂ω/∂
∂k
the group velocity.
2.) The solution is then given by
ψ = ψ o ⋅ sin( w ⋅ t − k ⋅ x)
(2)
and any combination of the equation which fullfills c = ω/k. We also note that in 3dim
ψ = ψ o ⋅ sin(w ⋅ t − k ⋅ x)
with
c=−
ω
k
Klaus Pfeilsticker
2
⋅k
(3)
and
vg = −∇ k ω
Institut für Umweltphysik
University of Heidelberg
(1) Sound Waves
Sound waves are due to isentropic compression and relaxation of air, which
propagates longitudinal through shearless media.
1.) In order to formulate the wave equation, we start with the continuity equation
(1) and the total differential of the isentrop (Poisson) equation (2) (ρ
ρ being the
air density)
•
ρ
+ ∇ ⋅ v = 0 (Kont.gl. ) (1)
ρ
p Rc
•
•
d[Θ ⋅ ( ) p ]
dT
po
p ρ
=
→ (1− R ) ⋅ − = 0 (Poisson ) (2)
cp p ρ
dt
dt
2.) We further consider departures of the pressure from the static mean po, i.e., p = po+
p‘ with po >> p‘ and assume a media at rest, i.e. dp/dt ≈∂p’/∂
≈∂ ∂t. By eliminating all terms
containing ρ in equation (1) and (2) (in which we descard the prime), we obtain
•
(1− R ) ⋅ p+ po ⋅ ∇ ⋅ v = 0
cp
(3)
3.) Next, we take into account the (linearized) equation of motion (Navier-Stokes)
ρ
∂v
+ ∇p = 0
∂t
Klaus Pfeilsticker
(4)
Institut für Umweltphysik
University of Heidelberg
(1) Sound Waves
4.) and we apply the operators ∂.. /∂
∂t to equation (3) and ∇ to equation (4)
∂2 p
∂(∇ ⋅ v)
R
(1− ) ⋅ 2 + po ⋅
=0
c p ∂t
∂t
ρ∇
and
∂v
+ ∆p = 0
∂t
(5)
from which we elimate the term ∂ ../∂
∂t ∇v. Finally, we arrive at the wave equation
for sound waves
∂2p
po
R
(1 −
)⋅
−
∆p = 0
cp
∂t 2
ρo
or
po
∂2p
−
∆p = 0
2
R
∂t
ρ o ⋅ (1 −
)
cp
(6 )
Hence we get for a ideal gas at To = 300 K (cp= cv+ R = 7/2·R), the sound velocity c
c =
2
po
ρo ⋅ (1− R )
cp
=
cp
⋅
R
cv M mol
⋅ To (= v2g ) = (348
m2
)
s
(6)
Discussion: We note in particular that since c = vg´sound waves are not (ideally)
dispersive ! The dispersion relation is obtained by inserting the solution p =
B⋅⋅exp[i(kx-ω⋅
ω⋅t)]
ω⋅t)]
ω⋅ and v= A⋅⋅exp[i(kx-ω⋅
ω⋅ into equation (3) and (4), respectively.
Sound wave dispersion is only due to an-isentropy by internal friction, which in
general increases with smaller wavelengths hence larger frequencies.
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(2) Capillary Waves
Capillary waves propagate in (deep) media with surface
tension σ, bended by a surface of radius r, which excerts a
pressure ps = σ/r perpendicular to the surface. In a linear
approximation, the bending 1/r can be approximated by the
second derivative y“ of the surface y(x,t), = yo·cos (k·z – ω·t) ,
i.e. y“ = - k2y and p = -k2·σ·y. The volume force is then F/V = p·k
= -k3·σ·y. This volume force needs to be balanced by the
inertial force assuming the density remains constant, i.e.
- ρ·∂2y/∂t2 = - ρ·ω
ω2·y or k3·σ·y = ρ·ω
ω2·y, or
ω =
2
σ 3
⋅k
ρ
(1)
Therefore the phase velocity c = ω/k gets
c=
σ
⋅k
ρ
( 2)
and the group velocity vg = ∂ω
ω/∂k
vg =
3 σ
⋅
⋅k
2 ρ
( 2)
Discussion: We learn that waves of large λ‘s (small k) move slower than waves with small
λ‘s (anormalous dispersion) ! For pure water the minimum speed occurs at λ = 1.73 cm,
and for larger λ gravity waves occur, see http://www.youtube.com/watch?v=ZGsufNHG3x8
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(3) Gravity-Waves
Gravity waves (at Brunt-Vaisälä frequency the cloud ripples) embedded in the
subtropical jet stream
see http://www.youtube.com/watch?v=yXnkzeCU3bE
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(3) Gravity-Waves – Static Stability
stable
Γs < Γd=i
Klaus Pfeilsticker
indifferent
Γs = Γd=i
instable
Γs > Γd=i
Institut für Umweltphysik
University of Heidelberg
(3) Gravity Waves
Now we consider gravity waves, which are due to buoyancy forces. In a static
atmosphere, we have dpo/dz = - g·ρ
ρo with p = po + p’, and ρ= ρo + ρ’ as before.
1.) Buoyancy leads to an acceleration b between the surrounding (s) and the
isentrope (i) (for which - dT/dz|i = g⋅⋅mMol/cp = 10 K/km)
db = −g ⋅
ρ'
∂ρ
∂ρ
= −g ⋅{( )i − ( )s } (1)
ρo
ρo
ρo
2.) Further considering the total differential
ideal gas law (2) and adiabatic equation (3)
p= ρ⋅
RT
dp dρ dT
→
=
+
ρ
M mol
p
T
p c
Θ = T ⋅( o ) p
p
R
( 2)
dΘ dT c p dp
→
=
− ⋅
Θ
T
R p
z
surrounding
isentrope
(3)
from both (2 and 3) we eliminate T:
dρ
dp dΘ
= (1 − ) ⋅ −
ρ
R p
Θ
cp
Klaus Pfeilsticker
( 4)
ρ‘
∂ρi
ρ
∂ρs (= ρ‘ + ∂ρ i)
Institut für Umweltphysik
University of Heidelberg
(3) Gravity Waves - Brunt Väisälä Waves
3.) Further we consider equation (4) for an isentrope (i) (dΘ
Θ/Θ
Θ = 0) and the
surrounding air (s), and we arrive at
dρ
i
|
ρ
= (1 −
cp
R
)⋅
dp
p
and
dρ
s
|
ρ
= (1 −
cp
R
)⋅
dp dΘ s
−
|
p
Θ
(5)
Noting that the pressure change dp/p is the same for (i) and (s). We insert
(5) into (1), and arrive at
db = −g ⋅{(
∂ρ
ρo
)i − (
∂ρ
ρo
)s } = −g ⋅
δΘ s
| ≈ −N
Θ
2
⋅ dz
(6)
The ‚N‘ on left hand side can be readily indentified with the ‚Brunt Väisälä‘
frequency (which is a measure of stability, see above), if we interpret δΘ/
δΘ ∂z|s
with the change in potential temperature Θ with height (- dT/dz|i), To with the
ambient temperature and Γs = - dT/dz|s with the lapse rate in the ambient air
N2 =
g δΘ s
Γ −Γ
⋅ | = g⋅ i s
Θ δz
To
(7)
Typical values of N are of the order of 10-2 – 10-3/s, assuming To = 270 K,
and Γs = - 9 K/km. This translate into wavelengths of some kilometers
assuming a horizontal wind velocity of 30 m/s !
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(3) Gravity Waves Dispersion
4.) Wave equation: We recall equation 6 and a linearized form of the equation
of motion in horizontal (x) and vertical (z) with a velocities (u,v,w) yields
db = − N 2 ⋅ dz
∂u 1 ∂p '
+
=0
∂t ρ o ∂x
→
in
∂w
1 ∂p '
−b+
=0
∂t
ρ o ∂z
∂z
∂b
= −N 2 ⋅ = −N 2 ⋅ w
∂t
∂t
the
in
horizontal
the
with
N2 = g⋅
Γi − Γ
(6)
T
(7 )
vertical
(8)
and assume divergence-free conditions (div(v)= ∂u/∂
∂x+ ∂w/∂
∂z=0).
5.) Eliminating p‘ by applying the operators ∂../∂
∂z and ∂../∂
∂x to equation (7) and
(8), respectively, yields
∂ ∂w ∂b ∂ ∂u
− −
=0
∂x ∂t ∂x ∂z ∂t
(9)
and after differentiating ∂../∂
∂t and substituting equation (6) into (9)
∂ ∂2w
∂ ∂ 2u
2 ∂w
+N
−
=0
2
2
∂x ∂t
∂x ∂z ∂t
Klaus Pfeilsticker
(10)
Institut für Umweltphysik
University of Heidelberg
(3) Gravity Waves Dispersion
5.) Finally, we differentiate equation (10) again with respect to ∂.../∂
∂x and recall
that ∂u/∂
∂x = - ∂w/∂
∂z = 0 (continuity equation), we obtain the wave equation for
vertically propagating gravity waves
2
∂ ∂2
∂2
2 ∂
[ 2 ( 2 + 2)+ N
]w = 0
2
∂t ∂x ∂z
∂x
(11)
6.) Inserting the solution w = wo⋅cos(k⋅⋅x+µ⋅
µ⋅z+ω⋅
µ⋅ ω⋅t),
ω⋅ we obtain the following
dispersion equation
ω=±
N
1+ µ k
2
2
(12)
Discussion:
We note that (12) has 2 solutions for the
phase velocity c = (ω
ω/k, ω/µ
µ)
and
the group velocity vg = -(∂ω
∂ω/∂
∂ω ∂k, ∂ω/∂µ
∂ω ∂µ).
∂µ
The largest ω are obtained for k>>µ
µ, i.e.
horizontal waves. Gravity waves with
large λ‘s move more rapid than with
small λ‘s (normal dispersion) !
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Gravity Waves in the Mesosphere
Upper Figure: T- deviations from
the nightly mean for two
consecutive nights over
Kühlungsborn (54o N) on Nov. 11–
12 and 12–13, 2003. The dashed
lines indicate the phase
propagation. The dotted lines
indicate the transition heights
between different measurement
methods. The gap in the RayleighMie-Raman lidar data at 0300 UT
(left plot) is due to technical
reasons.
Lower Figure: Mean wavelet
spectrum of vertical wavelengths
for the temperature profiles
shown in the upper Figure .
Pluses indicate local amplitude
maxima. The hatched area
indicates the ‘‘cone of influence.’’
(Rauthe et al., JGR, 111, D24108,
2006.
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Gravity Waves
Discussion: Brunt Väisälä waves can be regarded
as building blocks for the generation of gravity
waves. Gravity waves can emanate from air flowing
over mountain ridges, or by the air masses
displaced at a front. They may propagate in the
horizontal and vertical (with the group/phase
velocity‘s major components being directed
upwards/horizontally). As matter of fact, the
amplitude of upwardly propagating gravity waves
increase due to the decreasing air pressure.
They may then give rise to
• straits of clouds (see the figure before) in the
troposphere
• mountain lee polar stratospheric clouds (see
the Figure on the left) in the stratosphere
• or they may even break
• in upper stratosphere (in summer)
• or in the mesosphere (in winter)
depositing wave energy there.
Note that gravity waves also exist in water!
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Barotropic and baroclinic Rossby &Kelvin Waves
• Rossby Waves: Due to conservation of angular momentum (internal and Earth
angular vorticity)
• Kelvin Waves: Geostrophic (Coriolis force eq. pressure force) waves, which travel
along boundaries (coasts, equator, tropopause..)
Barotropic
Baroclinic
Barotropic waves (grad (p) ║ grad (T) or grad(ρ)) propagate fast (many m/s)
Baroclinic waves (grad (p) ≠ grad (T) or grad(ρ)) are slower (several cm/s)
Institut für Umweltphysik
University of Heidelberg
(4) Rossby Waves in the mid Latitude) Westwind Zone
We will see that the meridional thermal gradient drives the jet streams
in subtropics and polar region. They separate warm regions (highs) in the South
from cooler region (lows) in North.
The jet stream often shows a wavelike
pattern-called Rossby (1930) waves
(along the west/east axis). Rossby waves
are due to conservation of total angular
momentum - called (potential) vorticity
(or PV) - of a fluid on the rotating Earth. It
is the sum of PV = Earth + internal
rotation Rossby waves are erected by
the flow of air masses over vertical
extended obstacles such as mountain
ridges, (c.f. Rockies, Scandinavian
mountains, Ural) or strong high
pressure systems. Forced excursions of the jet to the South/North leads to
weakening/strenghening of the Coriolis component to total PV, which – assuming PV
conservation has to be compensated for by a corresponding
strengthening/weakening of the internal rotation, i.e. a strengthening/weakening of
winds around highs/lows.
see http://www.youtube.com/watch?v=DcePOGB6L8k&feature=related
and http://www.youtube.com/watch?v=fmNyygMtZ9g&feature=related
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Total Angular Moment&Potential Vorticity (PV) of a Geofluid
1.) In the lecture MKEP, we introduced the NS-equation:
r
r
r
r
r
r
dv ∂v r r r ∂v 1
=
+ v ⋅∇ ⋅v =
− grad (v 2 ) − v × rot (v )
dt ∂t
∂t 2
r
1
v
= g − grad ( p ) − 2 ⋅ (Ω × v ) +ν ⋅ ∆ ⋅ v (1)
( )
ρ
2.) Now we examine its rotational components, i.e., we apply the operator ‚rot‘ to
the NS and further note that rot[grad(...)] = 0 and div[rot(...)] = 0. Hence the terms
grad(p), grad(v2), and g = grad(Φ
Φ) vanish, and by noting that ∇×(A×
∇× ×B) = - ∇×(B×
∇× ×A),
the equation
reads
r
∇×
r
r
r
∂v
v
− ∇ × ( v × ( ∇ × v )) + 2 ⋅ ∇ × ( Ω × v ) = ν ⋅ ∇ × ( ∆ ⋅ v )
∂t
3.) We call ∇×v
∇× = ζ the field‘s vorticity
r
r
r
∂ς
− ∇ × (v × ς ) + 2 ⋅ ∇ × (Ω × v ) = ν ⋅ ∇ × ( ∆ ⋅ v )
∂t
(2)
4.) and recall the similarity of ζ and Ω by adding the term 2·∂
∂ Ω/∂
∂t = 0 and by
noting that ∇×(A×
∇× ×B) = - ∇×(B×
∇× ×A), i.e. we obtain
r
r
∂ς
∂Ω
v
+ ∇ × (ς × v ) + 2 ⋅ (
= 0) + 2 ⋅ ∇ × (Ω × v ) = ν ⋅ ∇ × (∆ ⋅ v )
∂t
∂t
Klaus Pfeilsticker
(3)
Institut für Umweltphysik
University of Heidelberg
Total Angular Moment&Potential Vorticity (PV) of a Geofluid
5.) Finally, we call the total angular moment total vorticity η(=ξ
ξ+2⋅Ω
⋅Ω),
⋅Ω and note
that the right hand side is due to vorticity dissipation by internal friction, or
more general due to internal or external forces, FR/ρ
ρ (see below). We thus
arrive with Ertel‘s PV equation in its most general form
r
∂η
1
+ ∇ × (η × v ) = ⋅ ∇ × F (4)
∂t
ρ
Discussion: We further note the following identity
r
r
r
r r
∇ × (η × v ) = (v ⋅ ∇)η − (η ⋅ ∇)v + η (∇v ) − v ⋅ (∇η )
and
r
r
r
(η ⋅ ∇)v = gradη v = + η ⋅ grad (v ) ⋅ eη = 0
r
v ⋅ (∇η ) = 0
sin ce
sin ce
r
grad (v ) ⊥ eη
div(rot..) = 0
i.e., we arrive with an equivalent formulation of the PV equation than given by (4)
r
∂η r
1
+ (v ⋅ ∇)η + η (∇v ) = ⋅ ∇ × F
∂t
ρ
Klaus Pfeilsticker
(5)
Institut für Umweltphysik
University of Heidelberg
Angular Moment (Potential Vorticity - PV) Conservation
2.) divergence or
convergence ↓
1.) friction ↓
r
∂η r
1
∂η r
dη
+ (v ⋅ ∇)η + [η (∇v ) = 0] = [ ⋅ ∇ × F = 0] =
+ (v ⋅ ∇)η =
ρ
dt
∂t
∂t
Discussion: Finally, it becomes evident that the PV, η is conserved if - and only if (1) the internal/external forces (the term on the right hand side of equation 5) are
not acting (or are very small) and
(2) the field is divergence/convergence free, i.e., div(v) = 0
⇒
dη
=0
dt
or
r
∇ × v + 2 ⋅ Ω sin(θ ) = const
(6)
Equation (6) is called Ertel‘s PV (conservation) equation.
Discussion: Evidently, both criteria are only fullfilled if dissipation processes, for
example driven by atmospheric waves are negligible or where boundaries do not
disturb a freely flowing flow. Such condition are sometimes met in the free
troposphere, and more frequently in the stratosphere (which is far from a friction
causing surface) !
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Air flow across a cyclone/anticlone (div (v) ≠ 0) :
top of the low
pressure system
(at 300 mbar)
PBL
(at 900 mbar)
Low pressure
at ground
Fig. Convergence/divergence of air masses around low/high pressure systems
Question: How is net PV produced in a baroclinic system (for the answer see next
slide)?
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Net production of PV in direct thermal (baroclinic) circulation
1.) Circulation ζ is connected to the rotation of a vector field via
v v
1 r r
ζ = ∇ × v = lim ∫ v ⋅ ds
A→ 0 A
A
2.) Acceleration by pressure gradients
r r
dζ
d r r
1 r r
1
= ∫ v ds = ∫ a ds = − ∫ ∇p ds = − ∫ dp
dt dt
ρ
ρ
3.) Separation of the path integral into the
2
T2
sections shown in the figure
3
4
1
2 1
dζ
1
1
1 

= − ∫ dp + ∫ dp + ∫ dp + ∫ dp 
dt
ρ
ρ
ρ 
2
3
4
1 ρ
3
1
T1 > T2
p1 >p2
p2
4
T1
p1
4.) The second and forth integrals are zero since dp = 0 on isobars. With the ideal gas
law, ρ = (Mp)/(RT), this yields:
dζ
R
=−
dt
M
p1
p2
 p2 dp

dp
R
dp
 T2

+
T
=
−
⋅
∆
T
⋅
1∫
∫p
 p∫ p

p
M
p
p
2
1
 1

5.) Baroclinic stratification leads to a circulation of the wind vector field (in the example
shown in the figure, a counterclockwise rotation of the velocity field occurs subtropical and polar jet
Institut für Umweltphysik
University of Heidelberg
Hadley Circulation
Θ = 380 K
18
tropopause
Altitude (km)
14
Θ = 330 K
10
release of latent heat
Tropical low pressure
(convergence)
Subtropical high
Pressure (divergence)
run601-710.MOV
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(4) Barotropic Rossby Waves
We now consider the meandering polar jet stream
around the globe, and recall that the jet is due
to the large meridional thermal gradient
around mid-latitudes. The jet has a largely
dominating velocity component in west-east
direction (x) and a much smaller component in
south-north (y) direction, i.e. vy < vx. The vycomponent is frequently induced during
overflow of the jet over larger obstacles
(Rocky mountains, stable high pressure
systems, et cetera...). We assume further that
the flow is divergence-free, i.e. div(v) = 0.
Polar jet
Wave equation:
1.) PV conservation yields for the z-component:
r
r
r
dη
dζ d [2 ⋅ Ω ⋅ sin(Θ)] dζ d [2 ⋅ Ω ⋅ sin(Θ)] dy dζ
d [sin(Θ)]
⋅
=
+
⋅
2
⋅
Ω
⋅
=
0
=
+
=
+
vy
|
dt z
dt
dt
dt
dy
dt dt
dy
r
r
dζ
dΘ dζ
=
+ v y ⋅ 2 ⋅ Ω ⋅ cos(Θ)
=
+ v y ⋅ 2 ⋅ Ω ⋅ cos(Θ) / R = 0 sin ce dy = R ⋅ dΘ (1)
dt
dy
dt
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(4) Rossby Waves
Discussion: We consider for the PV conservation (dη
η/dt = dζ
ζ/dt +vy·2·|Ω
Ω|·cos(Θ
Θ)/R) = 0
the following cases and recall that, ζ|z ~ ∂vy/∂
∂x, and hence
in the horizontal:
1. vy < 0 → dζ
ζ/dt > 0, which is a gain in
internal PV (an angular acceleration
opposite to the Earth rotation)
2. vy > 0 → dζ
ζ/dt < 0, which is a loss in
internal PV (an angular acceleration
in the same direction than the Earth
rotation)
in the vertical:
1. PV conservation (including mass
continuity) leads subsequently to a
disturbance (c.f. a mountain ridge)
to a street of strong cyclones and
weaker anticyclones !
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(4) Rossby Waves - Wavelength
2.) We further assume, that the velocity changes in north/south (y) direction
are negligible (2), i.e.,
∂vx ∂v y
=
=0
∂y
∂y
(2)
and consider the total differential of ζ with respect to time
∂ v y ∂ vx
∂ vy
ζz =
−(
= 0) =
∂x
∂y
∂x
∂ζ ( x, y )
∂ζ z
∂ζ z
∂ζ z
dζ
=
+
(
v
⋅
∇
)
=
+
+
(
=0
ζ
vx
vy
|
|
z
z
dt
∂t
∂t
∂x
∂y
∂ vy
∂2 vy
=
+ vx 2
∂t∂x
∂x
due
to
(2))
(3)
3.) Inserting (3) in (1) yields, the Rossby wave equation
∂2 vy ∂2 vy
dζ
= − v y ⋅ 2 ⋅ Ω ⋅ cos(Θ) / R = v x ⋅ 2 +
dt
∂x
∂x ⋅ ∂t
Klaus Pfeilsticker
(4)
Institut für Umweltphysik
University of Heidelberg
(4) Rossby Waves - Dispersion
4.) We insert the solution vy= a·sin(ω
ω·t-k·x) into equation (4) while keeping vx =
constant and we obtain the following dispersion relation
− 2 ⋅ Ω ⋅ cos(Θ) / R = k ⋅ ω − v x ⋅ k 2
(5)
which leads to the phase velocity c (= ω/k), or to the group velocity vg (= ∂ω/∂
∂ω ∂k)
1
)
k2
1
v g = v x + (2 ⋅ Ω ⋅ cos(Θ) / R) ⋅ 2 )
k
c = v x − (2 ⋅ Ω ⋅ cos(Θ) / R) ⋅
phase velocity
group velocity
Considering the stationary part (c= 0) only, leads to a typical wavelength, λR for
atmospheric Rossby waves (R = 6370 km, vx= 15 m/s, Θ = 60o) of
λR = 2 ⋅ π ⋅
Klaus Pfeilsticker
vx ⋅ R
≈ 7200 km
2 ⋅ Ω ⋅ cos(Θ)
(6)
Institut für Umweltphysik
University of Heidelberg
(4) Rossby Waves-Dispersion
We note, that since for all k‘s vg> 0, Rossby waves move from west to east
(‚changing weather – almost - always comes from the west‘), but its phase
moves the same way only for small wavelengths (λ
λ < λR), but opposite for
large wavelengths (λ
λ > λR).
Dispersion relation:
vg, c
eastwards
vg
vx
0
c
1/λ
λR
k
Rossby.mov
westwards
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(5) Kelvin waves: The geostrophic approximation
•
Kelvin waves stem from a geostrophic balance:
−
•
1
v
grad ( p ) + 2 ⋅ (v × Ω) = 0
ρ
The pressure force perpendicular to the coast (x direction) is
compensated by the Coriolis component coming from the flow
parallel to the coast (y direction). Thus Klevin waves tend to
propagate along a boundary.
see for equatorial eqkel.mpg and for coastal kel.mpg Kelvin waves
Institut für Umweltphysik
University of Heidelberg
(5) Kelvin waves: The wave equation
Geostrophic waves (Coriolis force equals pressure force), which travel at boundaries
(coasts, equator, tropopause..). We start with the NS equation and a notation as before
r
r
r r r
∂v
dv
1
=
+ v ⋅ ∇ ⋅ v = g − grad
dt
∂t
ρ
(
)
( p ) + 2 ⋅ ( vv × Ω ) + ν
r
⋅∆ ⋅v
(1)
From the scale analysis, we see that (for Kelvin waves) the horizontal scale is much larger
than the vertical scale. With Ψ(z) being the surface elevation, f=2·Ω·sin(Θ
Θ) the NS can be
approximated by
∂u
∂ψ ( z )
∂v
∂ψ ( z )
− f ⋅v = g ⋅
( 2)
+ f ⋅u = g ⋅
( 3)
∂y
∂t
∂x
∂t
and the continuity equation for div(v) = 0 yields
(
∂u ∂v
1 ∂ψ ( z )
+ )+
=0
∂ x ∂y
H ∂t
Assuming u, v, and Ψ(z) have solutions ~ exp(i[k·x + l·y - ω·t]) then eq. (2), (3) and (4)
yields
− i ⋅ ω ⋅ u − f ⋅ v = − i ⋅ g ⋅ k ⋅ψ ( z )
− i ⋅ ω ⋅ v + f ⋅ u = − i ⋅ g ⋅ l ⋅ψ ( z )
− iω ⋅ψ ( z ) + i ⋅ H ⋅ ( ku + lv ) = 0
which only solve with non zero solution for the following dispersion equation
2
ω 2 = f + (k 2 + l 2) ⋅ g ⋅ H
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(5) Kelvin waves: The wave equation
Again we see for the phase velocity vphase = +/- (ω/k, ω/l) (in one component)
v phase
ω
2
(k 2 + l 2)
f
= = ± g⋅H ⋅ ( 2
+
)
2
k
k
k ⋅g⋅H
and the group velocity vgroup (in one component)
k ⋅ g ⋅h
∂ω
=±
vg =
2
∂k
f
+ ( k 2 + l 2 ))
(
g ⋅h
Evidently there are two regimes f/√gH > k, or f/√gH < k. Since we can identify H with the
near costal water depth (~2.5m) √gH ~ 5 m/s and f/√gH ~2·10-5/m, we arrive at limiting
wavelengths λg for the both regimes, i.e. λg ∞ for φ=00 or roughly 100 km for φ = 600.
Evidently, for physical meaningful solutions λ should be smaller i.e. k > f/√gH. Therefore
oceanic Kelvin wave may have the following phase and group velocities:
v phase =
ω
k
= ± g⋅H ⋅
vg =
( k 2 + l 2)
k
2
) → v phase ~ ± 2 ⋅ g ⋅ H
k ⋅ g ⋅h
∂ω
~±
=±
2
2
∂k
(k + l )
g ⋅h
2
Assuming k ~ l and inserting some numbers, we get vphase ~ 10 m/s, and vgroup ~ 3.5 m/s, H
= 2.5 m, ω = 10-4/s and λ = ~ 50 km.
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
(5) Kelvin waves: Propagation at the coast
i1087-3562-3-1-1-m02.mpg
The sea level anomaly from the NRL global ocean model (courtesy H. Hurlburt). The
climatology was removed. The yellow band along the coast of N. America is the
product of the coastal Kelvin wave. In this model, the speed of propagation is 2-3 m/s.
Institut für Umweltphysik
University of Heidelberg
(5) Kelvin waves: Propagation along the equator
•
NH counterclockwise (CCW)
propagation
•
SH clockwise (CW) propagation
•
Kelvin wave offer a way of building time
cycles into ocean (climate oscillators)
(see later in the lecture)
Institut für Umweltphysik
University of Heidelberg
(5) Equatorial Kelvin waves
•
Satellite altimetry from TOPEX/Poseidon
•
Scenes are 10 days apart
see http://www.youtube.com/watch?v=ELDkYJWHNiU&feature=related
Institut für Umweltphysik
University of Heidelberg
(5) Equatorial Kelvin waves
In the Pacific Kelvin waves propagate in 4 months 13,000 km eastwards vphase = 1.3 m/s
Institut für Umweltphysik
University of Heidelberg
Non-linear coupling of Rossby& Kelvin waves: El Niño conditions
Recognize Rossby wave propagate from East to West and Kelvin waves from West to East!
Institut für Umweltphysik
University of Heidelberg
Summary: Waves
Frequency distribution of ocean
(surface) waves
Klaus Pfeilsticker
Dispersion relation of ocean (surface) waves
Institut für Umweltphysik
University of Heidelberg
Typical Scales (Time, Space) in Atmospheric Dynamics
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg
Conclusion: Take-home messages
We have seen, that
1. the atmosphere can be vertically devided according to different
criteria. Its thermal division is mostly used to structure the
atmosphere.
2. the atmospheric and oceanic circulation is driven by
•
thermal gradients caused by radiation,humidity (the Hadley
circulation) salinity (thermohaline circulation) and the west wind
– Ferrel – circulation in the troposphere, or the Brewer-Dobson
circulation in the stratosphere
but mass conservation, pressure gradients induced by solar radiation
and gravity together with the
•
Coriolis force (for trade winds and Kelvin waves)
•
PV (potential vorticity) conservation (Rossby waves)
and finally
•
non-linear wave interaction (gravity&planetary&Kelvin waves)
are important too.
⇒ for more info see the lecturer notes under:
http://www.iup.uni-heidelberg.de/institut/studium/
Klaus Pfeilsticker
Institut für Umweltphysik
University of Heidelberg