thermodynamics-moist air-l2
TRANSCRIPT
In-‐class exercise:
When westerly wind from Pacific arrive California coast, the air would be li>ed by Sierra Nevada. Assume that air temperature is T=25°C and Td=20°C, the mountain top is 1500 m, use the Skew T-‐lnp chart to determine:
a) Determine the cloud base (LCL) as air rise on windward side. b) Determine air temperature at the top of the mountain. c) Determine temperature of the air at the based of leeside for
pseudoadiabaPc process and saturated moist adiabaPc process, respecPvely. Assume that the base of the windside is 0 m above sea level, the base of the leeside is 500 m or 950 hPa above sea level. Latent heat of vaporizaPon, Lv=2.5X106 J kg-‐1.
a. Find LCL: follw θ line unPl it intersects with 20C isothermal at 950 hPa b. Above LCL, T change follows θe line. T at mountain top is determined by intersects
between θe at 1500 m or 850 hPa and isothermal line, which is 15C, c. If the air followed pseudoadiabaPc process when it rises, T would follows θ line that
intersects with θe at 850 hPa as it flows downhill to the boZom of the leeside because there is no condensaPon. T at 1000 hPa is 30C, 5C warmer than it’s iniPal T at boZom of windward side of the mountain.
d. If the air followed saturated moist adiabaPc process, T would follow θe line back to LCL and then follow the θ line back to 1000 hPa. T=25C, the same as T at the boZom of windward side. The process is reversible.
25C
LCL=950hPa
Top=1500m=850hPa Ttop=12C
Tlee,pseudo=30C
Tlee,sat=25C
θ θε
T ws
4.6 StaPc stability:
Sta$c stability of unsaturated air – Determines whether a verPcally displayed air parcel would oscillated around its original
locaPon or would convect. For example, – When the ambient lapse rate, Γ< Γd, an air parcel would be colder and less buoyant
than the ambient air when it is displayed upward, thus it would conPnue to move upward without any forcing. Likewise, it would be cooler and heavier than the ambient air when it is displayed downward. Thus, it would conPnue to sink downward without any external forcing. The ambient atmosphere is stable for convecPon.
– When , Γ> Γd, an air parcel would be warmer and lighter than the ambient air when it is displayed upward, thus it would sink with external forcing,. Likewise, it would be warmer and lighter than the ambient air when it is displayed downward, consequently, rise. Thus, this verPcally displaced air parcel would oscillate around its original locaPon. The ambient atmosphere is un stable for convecPon.
How do we quantify the influence of Γ on static instability? • Stability of unsaturated air can be quanPfied by
– T’-‐T: temperature difference between the convecPng parcel and ambient atmosphere T’-‐T>0, unstable, T’-‐T<0, stable, T’-‐T=0, neutral
– Γd-‐Γ: lapse rate difference between dry adiabaPc process and ambient
atmosphere Γd-‐Γ<0 or Γ>Γd: unstable, only occurs near surface in summer desert Γ<Γd: stable, occurs most of Pme in reality, Γ=Γd: neutral
– dθ/dz: verPcal slope of potenPal temperature
dθ/dz>0: stable, occurs most of Pme in reality, dθ/dz<0: unstable, only occurs near surface in summer desert dθ/dz=0: neutral
Why?
ρ 'w = FB where w ≡dzdt
is the veritcal velocity, FB is buoyancy force as
determined by density difference between the air parcel and the ambient air, i.e., ρ'-ρ, where ρ' is density of the rising/sinking air parcel, ρ is density of the ambient air
dwdt
=FBρ '= −g ρ'-ρ
ρ '= −g
P/R 1T'−
1T
#
$%
&
'(
P/R 1T'#
$%
&
'(
= gT '−TT
For a rising air parcel: If T'-T>0, dz/dt>0, unstable, If T'-T<0, dz/dt<0, stable
Or dwdt
= gT '−TT
= gTo −Γdz '− (To −Γ z ')T
=gz 'T
Γ−Γd( )
where z'=z-zo zo : original height of the air parcel
In-‐class exercise: show
€
1θ∂θ∂z
=1T(Γd −Γ)
The following derivaPon shows that air rises, i.e., convects, when Γ>Γd.
€
Thus, ∂θ∂z
> 0 means Γd − Γ > 0 or Γ < Γd stable
€
because θ = T po
p#
$ %
&
' (
R / c p
, lnθ = lnT - RCp
ln p +RCp
ln poconstant
Cpdθθ
= CpdTT− R
dpp
= CpdTT
+ Rρgdzp
= CpdTT
+gdzT
ideal gas law
1θdθdz
=1T
dTdz
+gCP
#
$ %
&
' ( =
1TΓd −Γ( )
How does dry stability affect weather and climate?
• StaPc stability affect gravity wave generated by mountains or weather disturbances. The wave length (or frequency) depends on stability, or buoyancy oscillaPon.
• Waves would develop if the dynamic wave length is comparable to that corresponding to buoyancy frequency.
• The buoyancy wave frequency is determined by
Stable atmosphere
dwdt
=dwdt
= −gTΓd −Γ( ) z ' where z'=z-zo
d2z 'dt2 + N 2z '=0 wave equation
where N2 ≡gTΓd −Γ( )
referred to as the Brunt-Vaisala frequencyor buoyancy frequency
Gravity wave induced clouds under stable condiPon:
What determines the wavelength or the space between cloud rolls?
Example:
• Meteorological measurements show that the lapse rate of the boundary layer is 5 C/km, the temperature is 27C and wind speed is 10 m/s over coast of California. What would be distance between rolls of boundary layer clouds?
• The buoyancy frequency would be N2=10 m/s/(273+27)K[10K/km-‐5K/km]/1000m, N=1.29X10-‐2 s-‐1.
• The gravity waves that form rolls of clouds would have frequency similar to N, thus
Τ=L/U=2π/Ν L=2πU/Ν=2X3.14X10m/s/1.29X10-‐2 s=4.87X103m=4.87 km The distance between rolls of clouds would be 4.87 km. What would happen to the distance between roll of clouds a. if wind speed increases to 20 m/s? 9.74 km b. Lapse rate increases to 7C/km and wind remains 10m/s? 6.28 km
Stability of saturated air:
• Γ<Γd: the atmosphere would be stable for dry convection, however, if – Γ>Γm, or dθe/dz>0, the atmosphere would be unstable for saturated air,
referred to as conditional unstable; – Γ<Γm, or dθe/dz<0, the atmosphere would be stable for saturated air, in this
case, referred to absolute stable
Γd
Γm
LFC: the level of free convection, referred to the height above which the convecting air would be warmer than the ambient atmosphere, therefore, would rise spontaneously.
Γ
LCL
LFC
Convective Available Potential Energy • Vertical integrated buoyancy from the level of free convection to the
limit of convection
Γd: dry adiabatic
Γm: moisture adiabatic
Γ: measured by radiosondes
zi
LCL
LFC
LOC
CAPE
T
B
unstable for dryconvection
unstable for moistconvection
CINE
positvie buoyant
negative buoyant
Updraft velocity: If CAPE were 100% converted to kinetic energy and without frictional drag, then the updraft velocity at the LOC would be:
w = 2 • CAPE Trigger mechanism: A lift of surface air to the LFC, such as fronts, dry lines, sea-breeze
fronts, gust fronts from other thunderstorms, atmospheric buoyancy waves, mountains and localized regions of excess surface heating.
How strong the trigger, i.e., the initial lifting, is required depending on the magnitude of negative buoyancy a convective parcel has to overcome between the surface and the LFC. This negative buoyancy is referred to as the Convective Inhibition Energy (CINE).
Γd: dry adiabatic
Γm: moisture adiabatic
Γ: measured by radiosondes
zi
LCL
LFC
LOC
CAPE
T
B
unstable for dryconvection
unstable for moistconvection
CINE
positvie buoyant
negative buoyant€
CINE = −g (Δz •i= sfc
LFC
∑ Tc −TeTe
)i
Discussion: • Why does occurrence of convecPon strongly depends on surface air humidity in the tropics?