Moist adiabatic processes
An adiabatic process in moist, saturated
air is called
MOIST ADIABATIC PROCESS
This process is significantly different
form that in the dry or non-saturated air
1
P
Dry adiabat
Moist adiabat
Lifting
Pk
Condensation level
T
S0=Sm, Tk,
So, Ti0,
RH=100%
RH<100%
RH = 100%
Continuous ascent results in further
temperature fall and water vapor
condensation that makes the rate of the
temperature fall less than 1°/100 m.
S m < S 0 ; Ti < Tk
RH = 100%
S 0 = S m ; Ti = Tk
So, Ti0, RH<100%
dTi
= −10 / 100m
dz
dSi
=0
dz
d (RH )
>0
dz
Condensation level
Initial level
The rate of temperature
variation of the ascending
saturated air without heat
influx or outflow is called
MOIST ADIABATIC LAPSE
2
RATE
From the above reasoning it follows:
•Temperature of an ascending parcel of air decreases with height,
but slower that at dry adiabatic process ( γ m.a. < γ a ).
• Due to condensation, the particle specific humidity Sm
decreases with height
• Relative humidity remains equal to 100%.
Adiabatic ascent of the moist air till attaining saturated state is
called DRY STAGE.
Further ascending of the saturated air above the condensation
level is called MOIST STAGE
P
Dry adiabat
Moist stage
Condensation level
Dry stage
T
3
First law of thermodynamics for the
moist, saturated air
Suppose a parcel of the saturated air has got some amount of
heat dq. This heat will be laid out for:
• Inner energy increase
• Expansion work
• Evaporation of some amount of water
Reason for evaporation
dq > 0
dTi > 0
RH = 100% ⇒ RH < 100%
RH = 100%
The parcel becomes non-saturated + dS m evaporation
dP
dq = cv dTi + pdvi + LdS m As we know, pdvi = − RTi
P
dP
dq = c p dTi − RTi
+ LdS m
P
4
For adiabatic process
cv dTi + pdvi + LdS m = 0
Accounting for static equation,
we get:
gTi
c p dTi +
dz + LdS m = 0 ÷ c p dz
Te
dTi gTi
L dS m
+
+
=0
dz c pTe c p dz
γ m.a
L dS m
= γa +
c p dz
dP
+ LdS m = 0
c p dTi − RTi
P
Since
dTi
T
g
= γ m.a . ; = γ a ; i ≈ 1
dz
cp
Te
dS m
<0
dz
γ a = const
γ m.a ≠ const
γ m.a < γ a
Value of the moist adiabatic
lapse rate depends on
pressure and temperature only
and does not depend on
5
humidity
E
S m = 0,622
P
ln S m = ln 0,622 + ln E − ln P
1 dS m 1 dE 1 dP
=
−
S m dz
E dz P dz
1 dS m 1 dE dTi 1 dP
=
−
S m dz
E dTi dz P dz
dS m
S m dE
Sm g
=−
γ m.a +
dz
E dTi
RTe
γ m. a
L dS m
=γa +
c p dz
γ m. a
E = E (T )
1 dP
g dTi
−
=
;
= −γ m.a
P dz RTe dz
E ⎡ g γ m.a dE ⎤
= 0,622 ⎢
−
P ⎣ RTi
E dTi ⎥⎦
L E⎡ g
1 dE ⎤
= γ a + 0,622
− γ m.a
⎢
c p P ⎣ RTi
E dTi ⎥⎦
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γ m.a
L E⎡ g
1 dE ⎤
= γ a + 0,622
− γ m.a
⎢
c p P ⎣ RTi
E dTi ⎥⎦
Opening brackets and solving the equation with respect to
we obtain
L E g
c p P RTe
=
L 1 dE
1 + 0,622
c p P dTi
γ a + 0,622
γ m. a
L = 2,5 × 106 J
kg
γ m. a
γ m.a ,
dE L E
=
dT Rw T 2
LE
P + 0,622
RTe
= γa
L2 E
P + 0,622
c p RwTi 2
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Values of the moist adiabatic lapse rate
at different temperature and pressure
Pressure
hPa
T°C
-50
-20
0
10
20
30
1000
0,966
0,856
0,658
0,532
0,435
0,363
800
0,964
0,831
0,614
0,489
0,398
0,335
600
0,960
0,793
0,557
0,436
0,356
0,303
400
0,952
0,730
0,478
0,371
0,307
0,267
200
0,928
0,597
0,361
0,286
0,247
0,223
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Criterion of instability for the moist air
γ > γa
γ <γa
γ = γa
γ > γ m.a
γ < γ m.a
γ = γ m.a
Unstable atmosphere
Stable atmosphere
Dry, non-saturated air
Neutral atmosphere
γ a > γ m. a
Unstable atmosphere
Stable atmosphere
Moist, saturated air
Neutral atmosphere
γ > γ . a > γ m. a
Absolute instability
γ < γ m.a < γ a
Absolute stability
γ a > γ > γ m.a
Combine criterion of instability
Conditional instability
The air is unstable, it is saturated
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Some additional information
Equivalent-potential temperature is the potential
temperature of an air parcel, the water vapor containing in it
had been condensed due to adiabatic ascent and the heat
obtained has been laid out to rise up the air parcel temperature,
Θ+dθ, e=0
Pseudo-potential
temperature
Initial level
Θ, e
Θe, e=0
Θp.p, e=0
1000 hPa
Equivalent-potential temperature
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