1. No category

Physics of the Atmosphere
Physik der Atmosphäre
WS 2014/15
Ulrich Platt
Institut f. Umweltphysik
SR 108/110 , INF 229, Mi, 09:15 - 10:45
Übungen zur Atmosphärenphysik (MVEnv1.2)
SR 108/110, INF 229, Mi, 11:00 - 12:00
[email protected]
Last Week
• The Navier-Stokes Equation describes the
conservation of momentum in fluids
• Frontal zones and fronts are an important
phenomenon in the Earth‘s atmosphere
• Fronts are strongly tilted  Formula of Margules
• High- and low pressure systems form in the
descending and ascending branch of baroclinc waves
Contents
15.10.2014
Introduction – Literature, Structure of the atmosphere
22.10.2014
Atmospheric Radiation: Basics, Scattering
29.10.2014
Atmospheric radiation: Energy budget of the atmosphere - Climate
5.11.2014
Global circulation (Fronts, Rossby Waves)
12.11.2014
The Atmospheric Boundary Layer
19.11.2014
The Global Hydrological Cycle
26.11.2014
The Carbon Cycle
3.12.2014
Atmospheric Aerosol
10.12.2014
Gas-Phase Chemistry: Reaction Kinetics
17.12.2014
Ozone and Free Radicals
14.01.2015
Nitrogen, Sulfur, and Halogen Cycles
21.01.2015
The Stratosphere: Physics (Radiation and Circulation)
28.01.2015
The Stratosphere: Chemistry – 1 (Chapman Cycle + Extensions)
4.02.2015
11.02.2015
The Stratosphere: Chemistry – 2 (Ozone Hole)
Measurement Techniques for Atmospheric Composition
Characterisation of the
Planetary Boundary Layer (PBL)
•
General definition of the PBL:
The layer which is influenced by surface friction
In this layer the shear stress τ is nearly constant with altitude
• The PBL is the part of the atmosphere which is in direct contact to the
•
•
•
•
•
Earth‘s (or ocean) surface
Here the exchange of scalar and vectorial tracers (heat, momentum,
gases) between surface and atmosphere occurs
The lowermost layer (thickness in the order of mm) is governed by
molecular diffusion  Molecular-viscous layer
Above the molecular-viscous layer, turbulence is the dominant transport
process
Size of turbulent „eddies“ increases with altitude
Questions: What is the flux from the surface to the free atmosphere above
the PBL? How does it depend on:
– Shear stress
– Temperature profile and vertical stability (vertical temperature gradient)
– Surface roughness
Boundary Layer in a Wind Tunnel
Seeds Dispersed in Olive Oil
http://efd.safl.umn.edu/research/wind_tunnel/
Boundary Layer
on the Wing of an Aircraft
Evolution of the Atmospheric Boundary Layer
The Structure of the Planetary Boundary Layer
Classification by
shear stress:
104
Free atmosphere
0
103
z [m]
• Molecular-viscous layer
(z ~ mm):
– v(z=0) = 0
– τ = -ρ  ∂vx/ ∂z
≡ const
• Prandtl layer (z~10-100 m):
– τ = -ρ K ∂vx/ ∂z
= -ρ u*2 ≡ const;
K = K(z)
• Ekman layer (z ~ 1 km):
– τ linearly decreasing
with altitude, change
in wind direction
• Free atmosphere
(z > 1 km):
– τ≈0
Ekman - layer
102
101
100
Prandtl - layer
  const
0
10-1
10-2
10-3
Molecular – viscous layer
Planetary
Boundary
Layer
Wind Profile in the PBL (1)
low p
•
Fp
In the free atmosphere (free of friction), the wind
is geostrophic (i.e., parallel to isobars due to the
vg
balance between pressure gradient and Coriolis
force)
•
high p
low p
Close to the surface, friction will cause a
FC
Fp
vr
deviation of the wind direction from geostrophic
solution (flow from high to low pressure)
FR
vg
high p FC
•
Consequences:
–
wind speed increases with altitude
–
wind direction canges with altitude
165 m
330 m
500 m
0 m
in form of a spiral, the so-called
Ekman Spiral
750 m
950 m
V g e o s tro p h is ch
Wind Profile in the PBL (2)
Close to the surface friction reduces the wind speed to levels well below the
geostrophic speed vg. Since (Fc  v) the influence of the Coriolis force is reduced.
The direction of the friction force is opposite to the direction of the wind the, therefore
close to the ground the wind will turn into the direction of the pressure gradient.

FC
A

FP

FR

v

FF
B

FC
C

FC

FP

v

FF

FC

FP

v

FC
A) Close to the ground the friction force is relatively large, v points approximately in the direction of
pressure gradient force.
B) In intermediate altitudes there is already a considerable angle between FP and v.
C) In the geostrophic case (at several 100 m altitude) the friction force can be neglected and FC is
anti parallel to FP. The air parcel moves at right angle to the pressure gradient force.
Vertical Wind Profile in the Boundary Layer
- Neutral Conditions Very close to the surface the wind velocity is determined by molecular friction
(kinematic viscosity υ), the velocity profile is linear. Shear Stress:
 xz
dv x
dv x
 
  
 const
dz
dz
Inspecting the dimension of the expression we find that   (Velocity)2


2
2
 N
dv x  kg m
kgm
kg m 
m
kg m 

 3  2 
 2  2 2  3  2   
 3 
dz  m
s s m m 
s m
m
s 
s 
m
v2 

2
 xz
Calling this velocity „Friction Velocity“ u* we may write:
 xz
dv x
  
   u *2  const  u * 
dz
2
dv x
dv
u

  u2*  x  *  v x (z) 
dz
dz 


 const

z dv
0
x
dz
dz 
z
u2*
0


dz 
u2*

z
How to Measure the Shear Stress
No Wind
Wind
force
Turbulence near the Surface
From z  few mm turbulence sets in:
In the turbulent regime we set:
 xz
dv x
  K z
   u *2
dz
2
vx z
5 m
Re  z 
 1000,   1.5 10

s
5  3 103
Re  0.003m 
 1000
5
1.5 10
Laminar regime
for comparison:
 xz   
dv x
dz
The “Turbulent Diffusion Constant” Kz=Kz(z) will certainly increase with
height, since close to the surface only small eddies can exist
(c.f. Kolmogorow – theory).
We thus assume:
K z  z   u* z
with   0.4 von Kármán constant
Laminar regime
for comparison:
  const.
Vertical Wind Profile in the Boundary Layer - 2
Vertical wind velocity – profile:
dv x u*2
u*2
u*



dz K z  u* z  z
After integration we obtain vx(z)
under the assumption that
τ = ρ u*2 = const:
log.
profile
transition
region
z
u*
u*  z 
v x ( z)  
dz  ln  
z
  z0 
z0
with the Roughness Parameter z0
depending on the surface properties.
For aerodynamically smooth surfaces,
z0 is given by z0 ≈ υ/9u*
linear
profile
Vertical Wind Profile in the Boundary Layer - 3
• Usually the surface wind is driven by the wind in the free
atmosphere
• Assume that the velocity vr(zr) at a refererence altitude zr
is known (e.g., geostrophic wind)
• Thus we have:
u *  zr 
u*  z 
v r ( zr )  ln   and v x ( z)  ln  
  z0 
  z0 
• With u* = const this yields:
ln  z / z 0 
v ( z)  v r
ln  zr / z 0 
Wind Profiles for Different Surface Roughness
Rough surfaces:
1)
Earth surface no longer reference height  Zero Point displacement d
2)
Interpret integration constant as „Roughness Parameter“ z0
u*  z  d 
v z 
ln 


z
 0 
Surface
z0 /mm
snow
0.1 - 1
grass
1 - 10
cereals
50 - 100
forest
500-1000
city
1000-5000
z0=10-2 mm
z0=1 mm
z0=100 mm
Transition of Wind Profiles
Change of vertical wind profile at the boundary
rough  smooth surface
rough
smooth
• An ‚inner boundary
layer‘ forms as a
transition between both
wind profiles
• The upward
propagation of this
inner boundary can be
described by turbulent
diffusion
• Height of boundary
given by implicit
equation (see Roedel):
x
 ( x)  const
ln  ( x) z0 
The non-neutral PBL
• For the neutral PBL, measurement of wind
profile is sufficient for a complete description of
dynamics
• This is not valid anymore if the PBL is
– Unstable: increased vertical exchange, larger
diffusion coefficients, smaller gradients
– Stable: reduced vertical exchange, stronger
gradients, eventually (during strong inversions)
complete surpression of turbulent mixing
• In these cases, the buoancy of air parcels in
relation to shear forces needs to be considered
Influence of Water Vapour
on Vertical Stability
• So far, we have only considered the release of latent heat on vertical
stability
• Even without condensation water vapour also influences vertical
stability because moist air is less dense than dry air
• Ratio of molar masses of air and water vapour:
M air
 1.61
Mw
• Density fluctuations under consideration of water vapour with
density ρw:
 M air
d
d

' 
 '
 ' w    '  1 
d
d w

MW



  ' w    ' 0.61 ' w


Influence of Water Vapour on Vertical Stability
•
The density fluctuations of moist air lead to energy production due to buoyancy forces:


 A  g  ' v ' z  g   ' v ' z  0.61 ' w v ' z 


•
The first term in brackets describes the flux of sensible heat, for which we had already
inferred

H
 ' v 'z 

cp 
•
The last term in brackets describes the turbulent flux of latent heat with the evaporation
heat L:
Hl   ' w v ' z L
•
Thus the turbulent energy production rate becomes
 H
Hl 
A  g 
 ' v ' z  0.61 
c

L 
 p
•
•
Over land: Hl ≈ H, contribution of water vapour to production of turbulent energy only
several percent
Over ocean: Hl ≈ 9H, contribution of water vapour to production of turbulent energy similar
to contribution of thermal convection
Transport of Trace Species in the Atmospheric
Boundary Layer
Trace Gas Flux JC: J C  K  z   D  
JC
dc


K  z   D 
dz
dc
dz
Integration and Division by JC yields:
z2
c z 2   c z1 
 R12 
 JC
dz
z K z   D 
1
where R12 denotes the transfer resistance for trace gas transport
between the altitude levels z2, z1.
Its reciprocal is the transfer velocity: 1/R12 = v12 (or „piston velocity“)
Transfer resistances are additive:
z2
z
3
dz
dz
R12  R23  


K  z   D  z2 K  z   D 
z1
z3
dz
z K z   D   R13
1
The Trace Gas Profile
The trace species –Vertical profile at a given (height independent)
vertical flux of the trace species JC :
At sufficient distance from the ground (at neutral layering) we have:
K = u*z + D  u*z
z2
c  z 2   c  z1 
z
2
dz
dz
JC  z 2 
 JC  
 JC  

ln  
u  z u*   z 1 
K z  D
z1   
z1 *
thus c(z)  ln(z)
The transfer resistance R12 between two altitude layers (z1, z2):
R12 
c  z 2   c  z1 
JC
 z2 
ln  
z1 


u* 
The Transfer Resistance
R10, 100
R1, 10
R0.1, 1
Each decade in z (0.1 – 1m, 1m – 10m, ...)
represents the same resistance for the trace
species.
Thus Rges =  ?
R0.01, 0.1
R0.001, 0.01
Rlaminar
RG
Flux Measurements
1) Determine Transfer resistance from:
R10, 100
R1, 10
c(z2)
v(z2)
c(z1)
v(z1)
R0.1, 1
R0.01, 0.1
z 
ln  2 
c  z 2   c  z1 
z
R12 
  1
JC
u* 
v ( z1 )  v ( z 2 ) 
2) Calculate flux:
c  z 2   c  z1  c
JC 

R12
R12
R0.001, 0.01
Rlaminar
RG
u *  z2 
v ( z1 )  v ( z 2 )
ln    u* 

  z1 
 z2 
ln  
 z1 
u* 

 c 
 z2 
ln  
 z1 
v ( z1 )  v ( z 2 ) 2

 z2 
ln  
 z1 
z 
ln  2 
 z1 
 c 
v ( z1 )  v ( z 2 )   2
  z2 
ln   
  z1  
2
 c
Vertical Flux – Example: NO2
C. Volpe Horii, J.W. Munger, and S.C. Wofsy,
M. Zahniser, D. Nelson, and J. B. McManus
(2004), Fluxes of nitrogen oxides over a
temperate deciduous forest, J. Geophys. Res.
109, D08305, doi:10.1029/2003JD004326.
Gas Exchange Atmosphere - Ocean
Basic Gas Flux Equation
Gas Flux (outward is positive):
F = kL (Cl -  Cg)
kL:
Cl:
Cg:
:
Gas transfer velocity, also called piston velocity, gas exchange coefficient
or deposition velocity. kL = 1/RL
Concentration in water near the surface
Concentration in air near the surface
Solubility of the gas in water (Cl/Cg)equilibrium
Time Scale considerations:
- Characteristic time scale of gas transfer ( = h/k) is on order of weeks
- Forcing function change on order of hours.
In order to quantify gas fluxes on a regional or global scale
we must have synoptic and co-located estimates of gas concentrations
and forcing function.
Conceptual view of air-sea gas exchange of inert gases
Rick Wanninkhof
Basic Conceptual Model
Gas Phase:
F= kg(Csg-Cg)
Cg
Csg
Csl
Cl
Csg=Csl
Water Phase:
F=kl(Cl-Csl)
Conceptual view of air-sea gas exchange of inert gases
Rick Wanninkhof
Air/water Resistance
Magnitude of typical Ostwald solubility coefficients:
He ≈ 0.01
O2 ≈ 0.03
CO2 ≈ 0.7
DMS ≈ 10
CH3Br ≈ 10
PCB's ≈ 100-1000
H2O ≈ ∞
Water side resistance
Air and water side resistance of importance
Air side resistance
Conceptual view of air-sea gas exchange of inert gases
Rick Wanninkhof
Global CO2 Budget 1990-2000 (PgC a-1)
Fossil Emissions
+6.3 ± 0.4
Atmospheric Increase
+3.1 ± 0.1
Ocean
-1.9 ± 0.7
Net Land
-1.2 ± 0.8
Emissions from Changes
in Land Use
+0.5 - +3.0
Residue ("Missing Sink")
-0.9 - -5.0
Positive values: flux into atmosphere
IPCC-TAR, Prentice et al., 2001
Oceanic Carbon Cycle
Transport Mechanisms:
• Advection and mixing through
ocean currents (“Solubility Pump”)
Marine biological “pumps”:
• Organic carbon
• Carbonates
‘AEOLOTRON’ – The Heidelberg Wind-Wave Facility
circular wind-wave flume
Institute of Environmental Physics,
University of Heidelberg:
• Diameter: 10 m (Perimeter: 29.2 m)
• Width: 0.6 m
B. Jähne,
• Height: 2.4 m
M. Schmidt.
R. Rocholz
• Water depth: 1.2 m
(2005),
2
• Surface area: 18.4 m
• Water volume: 21000 l
• Wind speed up to 14 m/s
•
•
•
•
•
•
Thermal imaging: passive und active, spectroscopy
Fourier-Transform-Spectroscopy ( FTIR )
Gas Chromatography ( He, H2 )
Mass balance methods ( CO2, CH4, F12, N2O )
Wind waves (slope)
Water- and wind current, temperature, humidity
Setup for the Wave State Measurement at the AELOTRON
Imaging Slope / Height Gauge
digital image processing
wave state:
slope  refraction at the surface
height  absorption in the water body
area extended light source
B. Jähne, M. Schmidt and R.
Rocholz (2005), Combined
optical slope/height measurements of short wind waves:
principle and calibration, Meas.
Sci. Technol. 16,1937-1944.
B. Jähne,
M. Schmidt.
R. Rocholz
(2005),
Measures of the Wave State
y
x
water surface reconstruction
slope in x
S= y/x
• slope saturation spectra:
slope in y
plus height information
• surface roughness
• <s2> as a better parameter
for gas transfer
velocities
Parameters influencing air sea gas exchange
B. Jähne,
M. Schmidt.
R. Rocholz
(2005),
wind
turbulence
wind waves
bubbles
surfactants
in order to improve the parameterizations and the models of gas exchange, the different
transport mechanisms have to be understood in detail and quantitatively measured.
Scaling Parameters of Transport Processes
Across the Sea-Surface Microlayer
flux scale
F m
k
C s
vertical space z  D

k
scale
Typical microlayer thickness:
~ 20 – 200 m for diffusive sublayer (gas)
~ 400 m – 2 mm thermal sublayer (heat)
~ 0.5 – 5 mm viscous sublayer (momentum)
time
scale
z D
t   2
k k
All concepts (transport models, scaling,
parameters) apply for transport of
momentum, heat, and mass due to
similarity of transport equations
B. Jähne,
M. Schmidt.
R. Rocholz
(2005),
experimentally extreme difficult (e.g. wavy surface)
typical time scale: 0.1 – 10 sec
Parameterizations of the gas transfer velocity
Wanninkhof, [1992]:
Wanninkhof & McGillis, [1999]:
Nightingale, [2000]:
Quadratic fit of natural 14C
disequilibrium and bomb 14C
inventory methods.
Cubic fit for transfer rates
GASEX 1998 CO2 covariance
methods.
Best fit (quadratic) to North Sea 92,
93, Georges Banks 97, 98 data,
3He/SF deliberate tracer studies.
6
Parameterizations versus measurements
Wind speed is not the only parameter influencing air-sea gas transfer:
• Transfer rate is correlated with mean square slope of short wind waves, e.g. in
[Jähne, 1980]
• Surface films lead to strong decrease in transfer rate, e.g. in [Frew et al., 1990]
• Fetch conditions have to be taken into account to infer from the wind speed to the
sea state e.g. in [Woolf, 2005]
• Bubble mediated transport: air entrainment due to wave breaking, e.g. in [Woolf, 1987]
Global Air – Sea Flux of CO2
Estimation of the global exchange rate between
ocean and atmosphere utilizing radar backscatter
Cooperation with D. M. Glover, N. M. Frew,
and S. J. McCue, Woods Hole Oceanographic
Institution, Woods Hole, MA, USA:
“Estimating regional and global air-sea gas exchange
rates using the dual-frequency TOPEX and JASON-1
altimeters”
Estimation of the Global gas transfer velocity based on
parameterization with mean square slope of short wind
waves.
Summary
•
•
•
•
The planetary boundary layer is the layer where surface friction has an impact (τ ≠ 0). It
can be subdivided into different regimes:
–
Molecular-viscous layer governed by molecular diffusion
–
Prandl- layer, where shear stress is constant with altitude
–
Ekman- layer, where shear stress decreases with altitude
(until it is zero in the free atmosphere)
Basic assumption: Turbulent diffusion coefficient is proportional to altitude  Logarithmic
wind profile
Water vapour has an impact on vertical stability not only due to the release of latent heat,
but also due to its lower density
The transport of scalar tracers in the boundary layer can be parameterised with the
transfer resistance R or the piston velocity v12:
z2
R 12 
z1
•
•
•
dz
z)
 K(
or
v 12 
1
R 12
In the turbulent regime, the transfer resistance is proportional to the logarithmic ratio of
the altitude difference
Air/sea gas exchange is a very important issue in the chemistry and climate of the
atmosphere (how much anthropogenic CO2 is taken up by the oceans?)
It can be investigated using wind-wave facilities, such as the Aelotron at the IUP
The non-neutral PBL
Bouyant forces
• Buoyant forces Fb due to turbulence are caused by
density fluctuations:
     ' and thus Fb   ' g and dW  g  ' dz '
• Thus the turbulent power density (per volume) due to
buoancy is:
dW
dz '
A 
 g  '
 g  ' v ' z
dt
dt
• Density fluctuations are caused by:
– Temperature fluctuations
– Fluctuations in water vapour content
(due to the smaller density of moist air, not due to the release of
latent heat!)
The non-neutral PBL
Bouyant forces
• Express density fluctuations as fluctuations of the
potential temperature
'
• Since ρ = const/θ , we have  '  

• Thus the turbulent power density becomes:
g   ' v ' z gH
 A  g  ' v ' z 


cp 
with the turbulent heat flux:
H   cp  ' v 'z
The non-neutral PBL
Shear stress
• Work done due to shear stress per unit area is dW = τ dx
• Thus the power per unit area (= Energy flux) is
dW
dx

  v x   u*2 v x
dt
dt
• The negative divergence of the energy flux yields the
power (energy production) per volume due to shear
stress:
d2 W
2 dv x
s  
  u*
dt dz
dz
The non-neutral PBL
Shear stress
• For neutral conditions, we had (logarithmic wind profile)
dv x u*

dz  z
• For the general case of non-neutral conditions, a
correction function Φ(H, u*, z) is introduced:
dv x u*

 (H, u* , z)
dz  z
with
current gradient of wind velocity
 (H, u* , z) 
gradient of wind velocity for neutral conditions
• Thus the energy production due to shear stress becomes:
3
dv

u
 s   u*2 x  *  (H, u* , z)
dz
z
The non-neutral PBL
Richardson Number and Monin-Obuchow Length
•
The Flux-Richardson number is defined as the negative ratio of energy
production rates due to thermal forces and due to shear stress:
A
Rf  
s
•
Sign of Rf:
– Rf > 0 for stable conditions
– Rf = 0 for neutral conditions
– Rf < 0 for labile conditions
•
It has been shown empirically that Rf
and Φ only depend on altitiude z and a
scale length called
Monin-Obuchow Length L*:
z
Rf 
L *  (z / L * )
•
L* is (in first approximation) independent
from altitude
Monin-Obuchow length as a
function of 10 m wind speed and
turbulent heat flux
The non-neutral PBL
Richardson Number
• The Flux-Richardson number is (or rather was) difficult to measure
(simultaneous measurement of heat flux and shear stress).
• A quantity more easy to measure (only temperature and wind speed
profile necessary) is the Richardson-Number, given by
Ri 
g  d / dz 
  dv x / dz 
2
• The Richardson-Number is related
to the Flux-Richardson number vial
the ratio of turbulent diffusion
coefficients for heat and
momentum, KH and K, respectively:
RF  Ri
KH
K
Labile
Stable
The non-neutral PBL
Richardson Number as a Funciton of z/L*
The non-neutral PBL
Turbulent Diffusion Coefficient
• The diffusion coefficient for momentum can be obtained from the
definition of shear stress

 u 2*
dv x
  K
dz
and the vertical wind profile:
dv x u *

(z / L * )
dz
z
yielding:
 u*z
K
(z / L * )
Labile
Stable
Characterisation of Vertical Exchange
Convective Energy Produktion Rate A
Turbulent Heat Flux H
Monin-Obuchow-Length L*
Richardson-Number Rf, Ri
Stratification
stable neutral labile
0
+
0
+
+

+
0
-