Vertical stability - Institut für Umweltphysik

Physik der Atmosphäre Atmosph re I
WS 2008/09
Ulrich Platt
Institut f. Umweltphysik
R. 424
Ulrich.Platt@iup.uni-heidelberg.de
Ulrich.Platt@iup.uni heidelberg.de
Physik der Atmosphäre I

Last week
• Atmosphere crucial for live
• Atmosphere is a complex and non-linear
system, interacting with the other geophysical
compartments (ocean, land, ice sheet…)
• The primary source of energy is solar radiation
• Strongest variability (T, p, …) is in the vertical
• Large range of spatial and temporal scales
• Hydrostatic equation:
p
= p0e
Physik der Atmosphäre I
−
z
H

Contents
7.10.08 Introduction – Literature - Vertical structure of the atmosphere
14.10.08 Adiabatic processes - Vertical stability
21.10.08 Atmospheric radiation: Absorption, scattering, emission
28.10.08 Atmospheric radiation: The energy budget of the atmosphere
4.11.08 Atmospheric dynamics: Navier-Stokes equation, continuity equation
Atmospheric dynamics: Pressure gradients and wind fields, thermal
11.11.08
wind
18.11.08 Atmospheric dynamics: Vorticity
25.11.08 Atmospheric dynamics: The planetary boundary layer
2.12.08 Atmospheric circulation: Global circulation patterns, planetary waves
9.12.08 Atmospheric circulation: Pressure systems, Hadley cell
13.12.08 Diffusion and turbulence: Molecular diffusion, basics of turbulence
13.01.09 Diffusion and turbulence: Theorem of Taylor, correlated fluctuations
20.01.09 Diffusion and turbulence: Diffusion of scalar tracers
27.01.09 Near-surface dynamics: Wind profile, influence of surface friction
Physik der Atmosphäre I

Outline for Today
1. Basic thermodynamic definitions
2. Dry-adiabatic lapse rate
3. Moist-adiabatic lapse rate
4. The potential temperature
5. The equivalent temperature
6. Vertical stability
7. Buoyancy oscillations
Physik der Atmosphäre I

The vertical structure of the atmosphere
Physik der Atmosphäre I

How to determine the
tropospheric temperature profile
1. Atmosphere is heated by the Earth‘s surface
2. Cooling/heating rates of the air by emission/absorption of
radiation are much smaller than typical transport times
3. Transport processes are (nearly) adiabatic
4. The condensation of water vapour is an additional source
of heat
5. To determine the tropospheric temperature profile, we
need
a. the ideal gas law
b. the first law of thermodynamics
c. the hydrostatic equation
Physik der Atmosphäre I

Ideal gas equation
• Ideal gas:
– molecules have no volume
– no collisions between gas molecules
– error for assuming that atmospheric gases are ideal gases is small
• Ideal gas equation
– p: pressure
– V: volume
– n: amount of gas in moles
– T: absolute temperature
– R: gas constant (8.314 J/mol K)
pV= nRT
n
p= RT= ρ RT
V
• Extensive properties:
– proportional to size of system, e.g. n, V
– „2x size 2x property“
• Intensive property:
– independent of mass: p,T, ρ
– „intensify“ extensive property by dividing by mass or volume “specific property”, e.g. ρ = m/V
Physik der Atmosphäre I

First law of thermodynamics
• First law (1. Hauptsatz):
dU = dQ +
dW
• dQ – heat added to the system
• dW – work done ON the system
• dU – change in internal energy
• Work done on system: compression of air
dW =−p⋅dV • Internal energy of an ideal gas is independent of
volume and proportional to temperature
dU = Cv
dT
• Cv – heat capacity (at constant volume)
⇒ dQ = dU − dW = pdV + CvdT Physik der Atmosphäre I

First law of thermodynamics (contd contd.) .)
1. First law (1. Hauptsatz):
dQ= pdV+ C dT⇒ pdV= dQ−CdT 2. Ideal gas law:
p V = n RT
⇒ pdV
+ V dp =
⇒
dQ V dp nR
Cv
dT ) ( + + − =
• Convert to intensive
quantities by dividing by n:
V
= − dp + ( R + cv
) dT
n
– q = Q/n: heat per mole
– c v = C v /n: specific heat capacity
– c p = c v + R: specific heat capacity at
constant pressure
v v
RT
p
nRdT
dp
dq p
=
−
Physik der Atmosphäre I
+
c
dT

Dry-adiabatic
Dry adiabatic lapse rate
1. First law (1. Hauptsatz):
2. Hydrostatic equation:
⇒
=
−
RT
p
dp
+
c
dq p
p
=
p
0
e
M g
− z
RT
⇔
dp
p
dT
=
−
dq = M gdz
+ c pdT
M g
RT
• Movement of air parcel is assumed to be adiabatic, i.e. no
heat is exchanged with the environment: dq = 0. This is
justified since cooling rates in the troposphere are much
smaller than typical transport times.
⇒
dT
c p
=
−M
gdz
dz
Physik der Atmosphäre I

Dry-adiabatic
Dry adiabatic lapse rate (contd ( contd.) .)
• Temperature gradient in the atmosphere:
dT
c p
= −M
gdz
⇒
Values for air:
– c p = 28.97 J/(K mole)
– M = 28.97 g/mole
– g = 9.81 m/s 2
• Dry-adiabatic lapse rate:
⎛ dT ⎞
Γ ≡ −⎜
⎟
⎝ dz ⎠
adiabatic
K
= 0. 00981 ≈ 1K
m
– The temperature gradient is given by Γ only if no additional heat
sources (condensation, absorption of radiation) are present (dq=0)
• Actual temperature gradient present in the atmosphere:
γ ≡
⎛
−⎜
⎝
dT ⎞
⎟⎠
dz
actual
dT
dz
=
−
M g
c p
Physik der Atmosphäre I
100m

Atmospheric Energy
• Kinetic energy (per volume):
– E=½ ρ air v 2 , ρ air = 1.293 kg m -3 , v = 10 ms -1
E = 65 J m -3
• Sensible heat (per volume):
– Q s = ρ air c p ∆T, c p = 10 3 J kg -1 K -1 , ∆T = 10K
Q s = 1.3x10 4 J m -3
• Latent heat (per volume):
– Q L = ρ air L q, L = 2.256x10 6 J kg -1 , q = 10 g kg -1
Q L = 2.9x10 4 J m -3
Phase transfer processes (e.g. convection, cloud
formation) play major roles in energy balance
Physik der Atmosphäre I

• Vapour pressure [hPa]:
Moisture parameters
– partial pressure of H 2 O vapour: e
– p=Σ i p i =p dry +e
– assume H 2 O vapour to be an ideal gas: e = ρ v R v T
• Saturation vapour pressure [hPa]: e*
• Absolute humidity [g m -3 ]:
a = ρ v = m v / V
• Specific humidity [kg/kg=1]:
q = ρ v / ρ air , with ρ air density of moist air
• Relative humidity [1]:
f = e / e*
• Dew point: T @ e = e* (p=const)
Physik der Atmosphäre I

The Moist-Adiabatic
Moist Adiabatic Lapse Rate
• Condensation of water vapour releases heat, i.e. dQ ≠ 0:
dQ = −Ldmw
– L ≈ 2250 J/g: specific evaporation heat
– dm w : change in mass of gaseous water vapour
• Convert to intensive quantities by dividing by n:
dq
= −
L
n
dm
w
V RT
= −L
dρ
w = −L
dρw
= −L
n p
– ρ w = m w /V is the saturation water vapour concentration
• Re-ordering yields the moist-adiabatic lapse rate:
w
p
Physik der pAtmosphäre dT I
RT
p
dρw
dT
RT
dρw
dq = M gdz
+ c p dT ⇒ −L
dT = c p dT + M gdz
p dT
⎛ dρ
⎞
R T
⇒ ⎜− − ⎟ =
⎝ p dT ⎠
w L cpdT Mgdz dT Mg
=−
dz RLT dρ
c + ⋅
dT

Saturation vapour pressure
Magnus formula for e* [hPa]:
e water
e ice
* =
* =
6.
11hPa
6.
11hPa
⎛ 17.
1(
T − 273K)
⋅exp⎜
⎝ 235 + ( T − 273K)
⎛ 22.
4(
T − 273K)
⋅exp⎜
⎝ 272 + ( T − 273K)
rule of thumb:
∆T=10K ∆e ≈ 2 e
Bergeron-Findeisen process
⎞
⎟
⎠
⎞
⎟
⎠
Physik der Atmosphäre I
Wallace and Hobbs, 2006

Water vapour saturation curve
Water vapour partial pressure E (right axis) and water vapour density ρ w (left axis) over
liquid water as a function of temperature. Adapted from Roedel (1992).
Physik der Atmosphäre I

Moist-adiabatic
Moist adiabatic lapse rate
Moist-adiabatic lapse rate as a function of air temperature, with pressure
as parameter (Roedel, 1994).
Physik der Atmosphäre I

The (Alpine) Föhn F hn
Inflow of moist air cooling, precipitation heating outflow of warm, dry air
Source: Roedel
Physik der Atmosphäre I

Potential Temperature
Potential temperature Θ = temperature of an air parcel
if it would be adiabatically compressed to a pressure of 1013 mBar
• 1st law of thermodynamics:
RT
( c p − cv
) T
= − dp + c p dT = − dp + c
p
p
• dq = 0 (adiabatic process) yields:
dp dT
( c p − cv
) = c p
p T
• Integration yields:
dq p
T
p
T
0
− 1 −1
0
= κ k
κ κ
p
with κ = c p /c v and the surface pressure p 0 = 1013 mBar.
• Θ ≡ T 0 is the temperature of the air parcel after compression from p to p 0 :
Θ
=
⎛
T
⎜
⎝
p
p
0
⎞
⎟
⎠
κ −1
κ
dT
with (κ-1)/κ ≈ 0.286 in air
Physik der Atmosphäre I

Höhe (Km)
Strahlungsheizung durch O 2 (UV)
Strahlungskühlung durch H 2 O & CO 2
Strahlungsheizung durch O 3 (UV)
Θ=800K
Strahlungskühlung (IR) durch H2O & CO2 Konvektion
Θ=1900K
Temperatur (K)
Θ=340K
Physik der Atmosphäre I
Druck (mBar)
Homosphäre

Equivalent Temperature
Equivalent temperature T eq = temperature of a moist air parcel
if all water vapour would condense
• Release of latent heat per volume...
∆
q v
= ρ L
• .. leads to an increase in internal energy:
u air c p T ∆ = ∆ ρ
• ∆u = ∆q Temperature increase:
∆
ρ L
ρ c
v T = =
air
p
q
L
c
p
with water vapour mixing ratio q = ρ v /ρ air
• Equivalent and equivalent-potential temperature:
T = T + q
eq
L
c
p
Θ
eq
= Θ + q
L
c
p
Physik der Atmosphäre I
Example:
• L ≈ 2250 J/g
• c p ≈ 1 J/(g K)
• T = 15°C
• Specific humidity: 70%,
corresponding to q ≈ 12.4⋅10 -3
∆T ≈ 31 K
Equivalent temperature T eq ≈ 46°C

Potential temperature and vertical stability
• From the hydrostatic equation, we have
p
=
p
• Total differential of the potential temperature:
dΘ
=
Θ
0
e
dT
T
M g
− z
RT
• Thus, with
−
R
c
⇔
p
dp
p
dp
p
=
= −
dT
T
dΘ
Θ ⎛ dT ⎞ Θ
= ⎜ + Γ⎟
=
dz T ⎝ dz ⎠ T
M g
RT
M g
+ dz
c T
Γ = ( g)
( c T )
M p
p
dz
( Γ −γ
)
where γ = − dT dz is the actual temperature gradient.
Physik der Atmosphäre I

Vertical stability
• γ = Γ, dΘ/dz = 0: neutral
• γ < Γ, dΘ/dz > 0: stable
• γ > Γ, dΘ/dz < 0: unstable
Physik der Atmosphäre I

z
z 2
z 1
z
z 2
z 1
dΘ
>
dz
T(z)
Potential Temperature and Vertical Stability
0 Stabile Schichtung
F
F
Θ P
Θ P
F
F
Θ(z)
∆z
Θ
∆z
Physik der Atmosphäre I
Θ
Auslenkung eines Luftpakets nach
oben (von z 2 nach z 3 ):
Θ P =const.
Θ P < Θ(z 3 )
Dichte des Luftpaketes größer
als Dichte der umgebenden Luft
Rücktreibende Kraft
dΘ
<
dz
0 Labile Schichtung
Auslenkung eines Luftpakets nach oben
(von z2 nach z3 ):
ΘP =const.
ΘP > Θ(z3 )
Dichte des Luftpaketes geringer als
Dichte der umgebenden Luft
Resultierende Kraft nach oben, treibt
Luftpaket weiter von der
ursprünglichen Lage weg

Dry and moist stability
Physik der Atmosphäre I

Stüve St ve diagram (1)
http://www.csun.edu/~hmc60533/CSUN_103/weather_exercises/soundings/smog_and_inversions/Understanding%20Stuve_v3.htm
Physik der Atmosphäre I

Isobars
Stüve St ve diagram (2)
Air temperature
Inversion
Wind speed
and direction
Temperature [K]
Pressure Altitude Temperature [°C]
Isotherms
Physik der Atmosphäre I

Stüve St ve diagram (3)
Dew point temperature
Saturation water vapour mixing ratio [g water/kg air]
Physik der Atmosphäre I

Dry
lapse rate
Stüve St ve diagram (4)
Condensation
starts
Physik der Atmosphäre I
Yellow line:
Temperature of an
air parcel lifted up
adiabatically from
950 mBar.
Initial water vapour
content: 8g/kg
(inferred from dew
point temperature).
Air parcel is
undersaturated and
thus follows the dry
adiabatic
temperature gradient
(~1K/100m)
Condensation starts
at 800mBar, where
saturation water
vapour mixing ratio
equals water vapour
content.

Dry
lapse rate
Stüve St ve diagram (5)
Moist
lapse rate
Temperature of rising air parcel is always below actual temperature
Stable conditions
Physik der Atmosphäre I
Yellow line:
Temperature of an
air parcel lifted up
adiabatically from
950 mBar.
Initial water vapour
content: 8g/kg
(inferred from dew
point temperature).
Air parcel is
undersaturated and
thus follows the dry
adiabatic
temperature gradient
(~1K/100m)
Condensation starts
at 800mBar, where
saturation water
vapour mixing ratio
equals water vapour
content.
Now the air parcel
follows the moist
adiabat due to
release of latent
heat.

Stüve St ve diagram (6)
Temperature of rising air parcel is above actual temperature
Unstable conditions, convection
Physik der Atmosphäre I
Yellow line:
Temperature of an
air parcel lifted up
adiabatically from
950 mBar.
Dew point
temperature is equal
to actual temperature
up to 750 mBar
condensation,
formation of clouds.

Brunt-Väis Brunt isälä or buoyancy oscillations
How can atmospheric stability be defined quantitatively?
• Equation of motion for the buoyancy of an air parcel:
dv z
( )
ρ −
dt
= g ρ * ρ
ρ: density of air parcel; ρ*: density of surrounding air
• ρ proportional to 1/T:
dvz dt
*
1 T −1
T T −T
= g = g *
1 T T
• Adiabatic and actual lapse rate:
*
T ( z)
= T0
− Γ∆z
T ( z)
= T0
− γ∆z
• It follows:
dvz γ − Γ
= g ∆z
*
dt T
• Potential temperature:
dΘ Θ dvz
g dΘ
= ( Γ − γ ) ⇒ = − ∆z
*
dz T
dt Θ dz
*
Physik der Atmosphäre I

Brunt-Väis Brunt isälä or buoyancy oscillations
How can atmospheric stability be defined quantitatively?
• Equation of motion for the buoyancy of an air parcel:
dvz 2
ρ = −ρ
B ∆z
dt
with
2 g dΘ
B =
Θ dz
• This is the equation of an harmonic oscillator with the
Brunt-Väisälä frequency
f
B 1
= =
2π
2π
g dΘ
Θ dz
• For dΘ/dz > 0 (stable conditions), this leads to buoyancy oscillations.
• Typical periodic times:
– 15 minutes for γ = 0.8 (stable conditions)
– 5 minutes for γ = 0 (isotherm stratification)
Physik der Atmosphäre I

Lee waves
Buoyancy oscillations downwind of mountains
„Lenticular clouds“
Physik der Atmosphäre I
Formation of lee waves

Lee waves
… can propagate up to the stratosphere
Polar stratospheric clouds
observed in Kiruna, northern
Sweden.
These clouds are formed in the
ascending branch of the lee
waves occuring downwind of
the Scandinavian mountains.
(Photo courtesy of C.F. Enell)
Physik der Atmosphäre I

Summary
• Rising air cools down due to adiabatic expansion
– Dry-adiabatic lapse rate: Γ ≈ 1K/100m
• Condensation of water vapour leads to the release of latent heat
– Moist-adiabatic lapse rate: (0.5 – 1)K/100m
• Potential temperature Θ = temperature of air parcel, if compressed
adiabatically to 1013 mBar
• Vertical stability of the atmosphere is determined by the actual
vertical temperature gradient compared to the dry/moist lapse rate:
– dΘ/dz = 0: neutral
– dΘ/dz > 0: stable
– dΘ/dz < 0: unstable
• Buoyancy oscillations (Brunt-Väisälä oscillations) can occur for stable
conditions (dΘ/dz > 0)
Formation of lee waves downwind of mountains
Physik der Atmosphäre I