Thermodynamics Notes Sections 5.5-5.7

(mostly from Thermo_for_slides.tex)

5.5 Changes of phase and latent heats

In the atmosphere, liquid water and water vapor can coexist inthermodynamic equilibrium, as can ice and water vapor.

• Vapor, liquid, and solid are called phases.

• The following phase changes of water substance occur in theatmosphere: evaporation, condensation, sublimation,deposition, melting, and freezing.

– Evaporation occurs when liquid water changes to watervapor, while condensation is the opposite.

– Sublimation occurs when ice changes to water vapor, whiledeposition is the opposite.

– Melting occurs when ice changes to liquid water, whilefreezing is the opposite.

• Both condensation and freezing require nuclei to initiate thephase change.

• If su�cient nuclei are not present, supersaturation of watervapor, or supercooling of liquid water, may occur.

• Cloud condensation nuclei are usually abundant, sosupersaturation is negligible in clouds without ice.

• Ice nuclei are not abundant, so supercooling of liquid water istypical.

34

5.5 Changes of phase and latent heats

In the atmosphere, liquid water and water vapor can coexist inthermodynamic equilibrium, as can ice and water vapor.

• Vapor, liquid, and solid are called phases.

• The following phase changes of water substance occur in theatmosphere: evaporation, condensation, sublimation,deposition, melting, and freezing.

– Evaporation occurs when liquid water changes to watervapor, while condensation is the opposite.

– Sublimation occurs when ice changes to water vapor, whiledeposition is the opposite.

– Melting occurs when ice changes to liquid water, whilefreezing is the opposite.

• Both condensation and freezing require nuclei to initiate thephase change.

• If su�cient nuclei are not present, supersaturation of watervapor, or supercooling of liquid water, may occur.

• Cloud condensation nuclei are usually abundant, sosupersaturation is negligible in clouds without ice.

• Ice nuclei are not abundant, so supercooling of liquid water istypical.

34

Heating or cooling of the environment occurs during phase changes,even though the phase change is isothermal.

• The energy transferred is called the latent heat.

• Cooling occurs during evaporation, sublimation, and melting.

• Heating occurs during condensation, deposition, and freezing.

At 0�C

• the latent heat of evaporation is Le = 2.5⇥ 106 J kg�1,

• the latent heat of melting is Lm = 0.334⇥ 106 J kg�1, and

• the latent heat of sublimation is Ls = 2.834⇥ 106 J kg�1.

35

5.6 Adiabatic processes of saturated air

When condensation occurs during ascent, the latent heat that isreleased significantly reduces the rate of temperature decrease due toadiabatic expansion. Consider two cases:

All condensed water remains suspended. This is called amoist adiabatic or saturation adiabatic process, and isreversible.

All condensed water falls out of the parcel immediately.This is called a pseudo-adiabatic process, and isirreversible.

The real situation lies between these two extremes.

Note that the rate of cooling in a pseudo-adiabatic process isessentially equal to that in a truly moist adiabatic one.

36

We will now consider a pseudo-adiabatic process in which there issaturation but not supersaturation.

The amount of water vapor condensed is then �dws and the latentheating is �Ldws.

(Here we use L = Le for simplicity.)

The first law for the mixture of dry air and water vapor is

�Ldws = cp dT �RTdp

p. (29)

Since ws and es are known functions of T and p, this is a di↵erentialrelationship between T and p during a pseudo-adiabatic process.

We will use the definition of potential temperature given by

T

✓=

✓p

p0

◆

.

to write another form of the first law. Take the logarithm to get

lnT

✓=

R

cpln

p

p0.

Di↵erentiate this to obtain

d lnT � d ln ✓ =R

cp(d ln p� d ln p0),

which becomesdT

T� d✓

✓=

R

cp

dp

p.

Rearrange this to get

cpd✓

✓= cp

dT

T�R

dp

p. (30)

37

pn

pn+1

dry adiabat

satu

ratio

n a

dia

bat

satu

ratio

n m

ixin

g ra

tio

✓n+1

✓n

✓⇤

�✓ �wwn+1

wn

w⇤

Saturation Adjustment Algorithm

1. Adiabatic. No phase changes involving cloud droplets (C=0):

2. Isobaric. Only phase changes involving cloud droplets operate (|C| > 0):

��, w� � �n+1, wn+1

�n, wn � ��, w�

Saturated Adiabatic Ascent

pn

pn+1

dry adiabat

satu

ratio

n a

dia

bat

satu

ratio

n m

ixin

g ra

tio

✓n+1

✓n

✓⇤

�✓ �wwn+1

wn

w⇤

Saturation Adjustment Algorithm

2. Isobaric. Only phase changes involving cloud droplets operate (|C| > 0):

�n+1, wn+1 =(�� + ��, w� + �w)

Conservation of energy(First Law of Thermodynamics):0 = cp⇥�� + L�w or0 = cp�T + L�w

Adjusted state is exactly saturated:wn+1 = ws(⇥�n+1, pn+1) orwn+1 = ws(Tn+1, pn+1)

Saturated Adiabatic Ascent

We will now consider a pseudo-adiabatic process in which there issaturation but not supersaturation.

The amount of water vapor condensed is then �dws and the latentheating is �Ldws.

(Here we use L = Le for simplicity.)

The first law for the mixture of dry air and water vapor is

�Ldws = cp dT �RTdp

p. (29)

Since ws and es are known functions of T and p, this is a di↵erentialrelationship between T and p during a pseudo-adiabatic process.

We will use the definition of potential temperature given by

T

✓=

✓p

p0

◆

.

to write another form of the first law. Take the logarithm to get

lnT

✓=

R

cpln

p

p0.

Di↵erentiate this to obtain

d lnT � d ln ✓ =R

cp(d ln p� d ln p0),

which becomesdT

T� d✓

✓=

R

cp

dp

p.

Rearrange this to get

cpd✓

✓= cp

dT

T�R

dp

p. (30)

37

We will now consider a pseudo-adiabatic process in which there issaturation but not supersaturation.

The amount of water vapor condensed is then �dws and the latentheating is �Ldws.

(Here we use L = Le for simplicity.)

The first law for the mixture of dry air and water vapor is

�Ldws = cp dT �RTdp

p. (29)

Since ws and es are known functions of T and p, this is a di↵erentialrelationship between T and p during a pseudo-adiabatic process.

We will use the definition of potential temperature given by

T

✓=

✓p

p0

◆

.

to write another form of the first law. Take the logarithm to get

lnT

✓=

R

cpln

p

p0.

Di↵erentiate this to obtain

d lnT � d ln ✓ =R

cp(d ln p� d ln p0),

which becomesdT

T� d✓

✓=

R

cp

dp

p.

Rearrange this to get

cpd✓

✓= cp

dT

T�R

dp

p. (30)

37

We will now consider a pseudo-adiabatic process in which there issaturation but not supersaturation.

The amount of water vapor condensed is then �dws and the latentheating is �Ldws.

(Here we use L = Le for simplicity.)

The first law for the mixture of dry air and water vapor is

�Ldws = cp dT �RTdp

p. (29)

Since ws and es are known functions of T and p, this is a di↵erentialrelationship between T and p during a pseudo-adiabatic process.

We will use the definition of potential temperature given by

T

✓=

✓p

p0

◆

.

to write another form of the first law. Take the logarithm to get

lnT

✓=

R

cpln

p

p0.

Di↵erentiate this to obtain

d lnT � d ln ✓ =R

cp(d ln p� d ln p0),

which becomesdT

T� d✓

✓=

R

cp

dp

p.

Rearrange this to get

cpd✓

✓= cp

dT

T�R

dp

p. (30)

37By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

5.8 Skew T -log p diagram

Figure 7: Depiction of thermodynamic variables and processes on askew T -log p diagram (Bohren and Albrecht 1998).

41

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

5.8 Skew T -log p diagram

Figure 7: Depiction of thermodynamic variables and processes on askew T -log p diagram (Bohren and Albrecht 1998).

41

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

5.8 Skew T -log p diagram

Figure 7: Depiction of thermodynamic variables and processes on askew T -log p diagram (Bohren and Albrecht 1998).

41

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

By comparing (29) divided by T and (30), we see that the first law of

thermodynamics for a pseudo-adiabatic process is

�Ldws

T= cp

d✓

✓.

It can be shown that (see Wallace and Hobbs, Second Edition,Problem 3.52)

d

⇣ws

T

⌘⇡ dws

T,

so

�Ld

⇣ws

T

⌘⇡ cp

d✓

✓= cp d ln ✓.

Integrate this from the original, saturated state (T,ws(T, p), ✓(T, p))to a state where ws = 0 and ✓ = ✓e:

�Z 0

ws/TLd

⇣ws

T

⌘⇡

Z ✓e

✓cp d ln ✓

to getLws

cpT= ln(✓e/✓),

then exponentiate and rearrange to obtain

✓ = ✓e exp(�Lws/cpT ).

This describes a pseudo-adiabat which is characterized by ✓e, theequivalent potential temperature:

✓e = ✓ exp(Lws/cpT ). (31)

38

5.7 More moisture variables

The thermodynamic processes that define the following four variablesare easily visualized on a a skew T -log p diagram, as shown in Fig. 7.

Equivalent potential temperature, ✓e The potential temperatureof a parcel that has ascended pseudo-adiabatically until allwater vapor has been condensed. Eq. (31):

✓e = ✓ exp(Lws/cpT ) ⇡ ✓ + Lws/cp.

Equivalent temperature, Te The temperature of a parcel that hasfirst ascended pseudo-adiabatically until all water vapor hasbeen condensed, then descended (dry adiabatically) to itsoriginal pressure:

Te = ✓e

✓p

p0

◆R/cp

= T exp(Lws/cpT ) ⇡ T + Lws/cp.

Wet-bulb temperature, Tw (i) The temperature of a parcel thathas been isobarically cooled by evaporation until saturated.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to its original pressure.

Wet-bulb potential temperature, ✓w (i) The temperature of aparcel that has first been isobarically cooled by evaporationuntil saturated, then descended moist (saturated) adiabaticallyto 1000 hPa.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to 1000 hPa.

39

5.7 More moisture variables

The thermodynamic processes that define the following four variablesare easily visualized on a a skew T -log p diagram, as shown in Fig. 7.

Equivalent potential temperature, ✓e The potential temperatureof a parcel that has ascended pseudo-adiabatically until allwater vapor has been condensed. Eq. (31):

✓e = ✓ exp(Lws/cpT ) ⇡ ✓ + Lws/cp.

Equivalent temperature, Te The temperature of a parcel that hasfirst ascended pseudo-adiabatically until all water vapor hasbeen condensed, then descended (dry adiabatically) to itsoriginal pressure:

Te = ✓e

✓p

p0

◆R/cp

= T exp(Lws/cpT ) ⇡ T + Lws/cp.

Wet-bulb temperature, Tw (i) The temperature of a parcel thathas been isobarically cooled by evaporation until saturated.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to its original pressure.

Wet-bulb potential temperature, ✓w (i) The temperature of aparcel that has first been isobarically cooled by evaporationuntil saturated, then descended moist (saturated) adiabaticallyto 1000 hPa.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to 1000 hPa.

39

5.7 More moisture variables

The thermodynamic processes that define the following four variablesare easily visualized on a a skew T -log p diagram, as shown in Fig. 7.

Equivalent potential temperature, ✓e The potential temperatureof a parcel that has ascended pseudo-adiabatically until allwater vapor has been condensed. Eq. (31):

✓e = ✓ exp(Lws/cpT ) ⇡ ✓ + Lws/cp.

Equivalent temperature, Te The temperature of a parcel that hasfirst ascended pseudo-adiabatically until all water vapor hasbeen condensed, then descended (dry adiabatically) to itsoriginal pressure:

Te = ✓e

✓p

p0

◆R/cp

= T exp(Lws/cpT ) ⇡ T + Lws/cp.

Wet-bulb temperature, Tw (i) The temperature of a parcel thathas been isobarically cooled by evaporation until saturated.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to its original pressure.

Wet-bulb potential temperature, ✓w (i) The temperature of aparcel that has first been isobarically cooled by evaporationuntil saturated, then descended moist (saturated) adiabaticallyto 1000 hPa.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to 1000 hPa.

39

5.8 Skew T -log p diagram

Figure 7: Depiction of thermodynamic variables and processes on askew T -log p diagram (Bohren and Albrecht 1998).

41

5.8 Skew T -log p diagram

Figure 7: Depiction of thermodynamic variables and processes on askew T -log p diagram (Bohren and Albrecht 1998).

41

5.7 More moisture variables

The thermodynamic processes that define the following four variablesare easily visualized on a a skew T -log p diagram, as shown in Fig. 7.

Equivalent potential temperature, ✓e The potential temperatureof a parcel that has ascended pseudo-adiabatically until allwater vapor has been condensed. Eq. (31):

✓e = ✓ exp(Lws/cpT ) ⇡ ✓ + Lws/cp.

Equivalent temperature, Te The temperature of a parcel that hasfirst ascended pseudo-adiabatically until all water vapor hasbeen condensed, then descended (dry adiabatically) to itsoriginal pressure:

Te = ✓e

✓p

p0

◆R/cp

= T exp(Lws/cpT ) ⇡ T + Lws/cp.

Wet-bulb temperature, Tw (i) The temperature of a parcel thathas been isobarically cooled by evaporation until saturated.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to its original pressure.

Wet-bulb potential temperature, ✓w (i) The temperature of aparcel that has first been isobarically cooled by evaporationuntil saturated, then descended moist (saturated) adiabaticallyto 1000 hPa.(ii) The temperature of a parcel that has first ascended dryadiabatically to its LCL, then descended moist (saturated)adiabatically to 1000 hPa.

39

5.8 Skew T -log p diagram

Figure 7: Depiction of thermodynamic variables and processes on askew T -log p diagram (Bohren and Albrecht 1998).

41

Liquid water mixing ratio, wl The mass of liquid water(droplets) per unit mass of dry air.

Total water mixing ratio, wt The mass of water vapor plus liquidwater (droplets) per unit mass of dry air: wt = w + wl.

• During a reversible process, the total water mixing ratio

(wt = w + wl) in a parcel remains constant.

• During a pseudo-adiabatic process, any condensed waterimmediately falls out of the parcel (as precipitation) so that theliquid water mixing ratio (wl) is always zero.

• Naturally occurring processes are usually neither exactlyreversible nor pseudo-adiabatic, but somewhere in between:

Some, but not all, of the condensed water falls out ofthe parcel as precipitation so that the liquid water

mixing ratio may be greater than zero, but the total

water mixing ratio is reduced by the loss due toprecipitation.

• Given the total water mixing ratio, one can then determine theremaining unknown mixing ratios:

– We will assume that the parcel is either exactly saturated,or unsaturated with no liquid water.

– If a parcel is exactly saturated: w = ws(T, p), sowl = wt � ws(T, p).

– If a parcel is unsaturated with no liquid water: wl = 0, sow = wt.

40

Liquid water mixing ratio, wl The mass of liquid water(droplets) per unit mass of dry air.

Total water mixing ratio, wt The mass of water vapor plus liquidwater (droplets) per unit mass of dry air: wt = w + wl.

• During a reversible process, the total water mixing ratio

(wt = w + wl) in a parcel remains constant.

• During a pseudo-adiabatic process, any condensed waterimmediately falls out of the parcel (as precipitation) so that theliquid water mixing ratio (wl) is always zero.

• Naturally occurring processes are usually neither exactlyreversible nor pseudo-adiabatic, but somewhere in between:

Some, but not all, of the condensed water falls out ofthe parcel as precipitation so that the liquid water

mixing ratio may be greater than zero, but the total

water mixing ratio is reduced by the loss due toprecipitation.

• Given the total water mixing ratio, one can then determine theremaining unknown mixing ratios:

– We will assume that the parcel is either exactly saturated,or unsaturated with no liquid water.

– If a parcel is exactly saturated: w = ws(T, p), sowl = wt � ws(T, p).

– If a parcel is unsaturated with no liquid water: wl = 0, sow = wt.

40

Liquid water mixing ratio, wl The mass of liquid water(droplets) per unit mass of dry air.

Total water mixing ratio, wt The mass of water vapor plus liquidwater (droplets) per unit mass of dry air: wt = w + wl.

• During a reversible process, the total water mixing ratio

(wt = w + wl) in a parcel remains constant.

• During a pseudo-adiabatic process, any condensed waterimmediately falls out of the parcel (as precipitation) so that theliquid water mixing ratio (wl) is always zero.

• Naturally occurring processes are usually neither exactlyreversible nor pseudo-adiabatic, but somewhere in between:

Some, but not all, of the condensed water falls out ofthe parcel as precipitation so that the liquid water

mixing ratio may be greater than zero, but the total

water mixing ratio is reduced by the loss due toprecipitation.

• Given the total water mixing ratio, one can then determine theremaining unknown mixing ratios:

– We will assume that the parcel is either exactly saturated,or unsaturated with no liquid water.

– If a parcel is exactly saturated: w = ws(T, p), sowl = wt � ws(T, p).

– If a parcel is unsaturated with no liquid water: wl = 0, sow = wt.

40

pn

pn+1

dry adiabat

satu

ratio

n a

dia

bat

satu

ratio

n m

ixin

g ra

tio

✓n+1

✓n

✓⇤

�✓ �wwn+1

wn

w⇤

Saturation Adjustment AlgorithmSaturated Adiabatic Ascent

total water (if reversible)water vapor

condensed water

Reversible: liquid water = condensed water

total water is constant

Irreversible: liquid water < condensed water

total water decreases

Liquid water mixing ratio, wl The mass of liquid water(droplets) per unit mass of dry air.

Total water mixing ratio, wt The mass of water vapor plus liquidwater (droplets) per unit mass of dry air: wt = w + wl.

• During a reversible process, the total water mixing ratio

(wt = w + wl) in a parcel remains constant.

• During a pseudo-adiabatic process, any condensed waterimmediately falls out of the parcel (as precipitation) so that theliquid water mixing ratio (wl) is always zero.

• Naturally occurring processes are usually neither exactlyreversible nor pseudo-adiabatic, but somewhere in between:

Some, but not all, of the condensed water falls out ofthe parcel as precipitation so that the liquid water

mixing ratio may be greater than zero, but the total

water mixing ratio is reduced by the loss due toprecipitation.

• Given the total water mixing ratio, one can then determine theremaining unknown mixing ratios:

– We will assume that the parcel is either exactly saturated,or unsaturated with no liquid water.

– If a parcel is exactly saturated: w = ws(T, p), sowl = wt � ws(T, p).

– If a parcel is unsaturated with no liquid water: wl = 0, sow = wt.

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Liquid water mixing ratio, wl The mass of liquid water(droplets) per unit mass of dry air.

Total water mixing ratio, wt The mass of water vapor plus liquidwater (droplets) per unit mass of dry air: wt = w + wl.

• During a reversible process, the total water mixing ratio

(wt = w + wl) in a parcel remains constant.

• During a pseudo-adiabatic process, any condensed waterimmediately falls out of the parcel (as precipitation) so that theliquid water mixing ratio (wl) is always zero.

• Naturally occurring processes are usually neither exactlyreversible nor pseudo-adiabatic, but somewhere in between:

Some, but not all, of the condensed water falls out ofthe parcel as precipitation so that the liquid water

mixing ratio may be greater than zero, but the total

water mixing ratio is reduced by the loss due toprecipitation.

• Given the total water mixing ratio, one can then determine theremaining unknown mixing ratios:

– We will assume that the parcel is either exactly saturated,or unsaturated with no liquid water.

– If a parcel is exactly saturated: w = ws(T, p), sowl = wt � ws(T, p).

– If a parcel is unsaturated with no liquid water: wl = 0, sow = wt.

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