LECTURE OUTLINE 1. Ways of reaching saturation

LECTURE OUTLINE

1.

Ways of reaching saturation

• vertical motion

Fundamentals of Atmospheric Physics, M.L. Salby; Salby

A Short Course in Cloud Physics, R.R. Rogers and M.K. Yau; R&Y

Thermodynamics of Atmospheres and Oceanes,

J.A. Curry and P.J. Webster; C&W

Salby, Chapter 5 C&W, Chapter 6

R&Y, Chapter 4

/34 3

Chapter 2: Clouds as Fluids

Siebesma, A., Bony, S., Jakob, C., & Stevens, B. (Eds.). (2020). Clouds and Climate: Climate Science's Greatest Challenge. Cambridge: Cambridge University Press. doi:10.1017/9781107447738

THERMODYNAMIC IN VERTICAL MOTION

1.

Lifting condensation level (LCL)

2.

Dew-point temperature variation in vertical motion

Pseudo-adiabatic process

§ saturated adiabatic lapse rate

§ water condensed in pesudo-adiabatic process

/34 5

A parcel that moves vertically expands or contracts to preserve its mechanical equilibrium (adjust its pressure to the environmental pressure). It results in work being performed.

Compensating this work is a change of internal energy, which alters the temperature and hence the saturation vapor pressure.

The change of saturation vapor pressure (assuming L_lv=const):

The saturation specific humidity varies with pressure and temperature:

The saturation specific humidityincreases with decreasing pressure (increase of altitude).

However, q_s decreases sharply with decreasing temperature, which likewise accompanies upward motion.

Even though an ascending parcel’s pressure decreases exponentially with altitude, the temperature dependence prevails, so itssaturation specific humidity decreases

monotonically with altitude.

ln 𝑒_!

𝑒_!" = −𝐿_#$

𝑅_$ 1

𝑇 − 1 𝑇_"

𝑞_! = 𝜀 𝑒_! 𝑝 𝑞_!" = 𝜀 𝑒_!"

𝑝_"

𝑞_!

𝑞_!" = 𝑒_!- 𝑒_!"

𝑝-

𝑝_" = exp −𝐿_#$

𝑅_$ 1

𝑇 − 1 𝑇_"

𝑝𝑝_"

Lifting condensation

level

/34 7

As it rises, the parcel performs work at the expense of its internal energy.

The parcel’s temperature decreases at the dry adiabatic lapse rate G_d.

The decrease of temperature is attended by a reduction of saturation specific humidity 𝑞_!. The parcel’s actual specific humidity 𝑞_$ and potential temperature q remain constant.

Sufficient upward displacement will reduce the saturation specific humidity to the actual specific humidity.

The elevation where 𝑞_$ = 𝑞_! for the first time is referred to as the lifting condensation level (LCL).

q_v z

q_v0

q_v q_s

q_s(z) q_v(z)

LCL

Dq(z) Consider a moist air parcel ascending in thermal convection.

Under unsaturated conditions, the parcel’s specific humidity and saturation specific humidity satisfy 𝑞_$ < 𝑞_!

The lifting condensation level defines the base of cumulus clouds that are fueled by air originating at the surface.

Below the LCL, the parcel’s thermodynamic behavior can be regarded as adiabatic because the timescale for vertical motion (from minutes in cumulus convection to 1 day in sloping convection) is small compared to the characteristic timescale for heat transfer.

q_v z

q_v0

q_v q_s

q_s(z) q_v(z)

LCL

Dq(z)

Position of the LCL

/34 9

As air expands adiabatically and cools, the relative humidity (f) increases as the temperature and saturation mixing ratio decrease.

We will find coordinates of the LCL 𝑇_%&%, 𝑝_%&% – the point where the air becomes saturated.

Using Dalton’s law of partial pressure, that states that the total pressure exerted by a

mixture of gases is equal to the sum of the partial pressures that would be exerted by each constituents alone if it filled the entire volume at the temperature of the mixture.

Therefore: 𝑑 ln 𝑝 = 𝑑 ln 𝑒 . 𝑓 = 𝑒

𝑒_!

𝑑 ln 𝑓 = 𝑑 ln 𝑒 − 𝑑 ln 𝑒_!

𝑝_' = 𝑚_'𝑅_'𝑇 𝑉 𝑑𝑝_'

𝑝_' = 𝑑𝑇 𝑇

The First Law of thermodynamics for an adiabatic process in enthalpy form:

Using the Clausius-Clapeyron equation:

Because 𝑑 ln 𝑝 = 𝑑 ln 𝑒 :

The change of relative humidity fulfils the following equation:

𝑐₍𝑑𝑇 = 𝑣𝑑𝑝 𝑐₍𝑑𝑇 = 𝑅𝑇

𝑝 𝑑𝑝 𝑑 ln 𝑝 = 𝑐₍

𝑅 𝑑 ln 𝑇 𝑑 ln 𝑒 = 𝑐₍

𝑅 𝑑 ln 𝑇 𝑑 ln 𝑒_! = 𝐿_#$

𝑅_$𝑇𝑑 ln 𝑇

𝑑 ln 𝑓 = 𝑑 ln 𝑒 − 𝑑 ln 𝑒_!

𝑑 ln 𝑓 = 𝑐₍

𝑅 𝑑 ln 𝑇 − 𝐿_#$

𝑅_$𝑇 𝑑 ln 𝑇

/34 11

We will integrate the equation from initial conditions to conditions where saturation is attained, indicated by f=1 and 𝑇 = 𝑇_%&%, where 𝑇_%&% is the saturation temperature. We will assume that L_lv=const.

Equation can be solved numerically to obtain 𝑇_%&%.

The saturation pressure can be obtained from the dry adiabat equation:

9

)

*

𝑑 ln 𝑓⁺ = 9

, ,_!"!

𝑐₍

𝑅 − 𝜀𝐿_#$

𝑅𝑇 𝑑 ln 𝑇⁺

− ln 𝑓 = 𝑐₍

𝑅 ln𝑇_%&%

𝑇 + 𝜀𝐿_#$

𝑅

1

𝑇_%&% − 1 𝑇

𝑝_%&% = 𝑝 𝑇_%&%

𝑇

._#- /

An approximate equation for 𝑇_%&% given initial values of T and f, is given by Bolton (1980).

f T Ts ps

0,1 0 -16 815

0,1 10 -7 806

0,1 20 2 797

0,1 30 10 788

0,3 0 -8 896

0,3 10 1 891

0,3 20 10 886

0,3 30 19 880

0,5 0 -5 938

0,5 10 5 935

0,5 20 14 932

0,5 30 24 929

0,7 0 -3 968

0,7 10 7 966

0,7 20 17 964

0,7 30 27 962

𝑇_%&% = 1

𝑇 − 55 −1 ln 𝑓 2840

+ 55, 𝑇 = 𝐾

Variation of dew-point

temperature with altitude

During adiabatic ascent, the water vapor specific humidity, qv, remains constant until saturation occurs.

The dew-point temperature decreases slightly during the ascent as pressure decreases.

We will calculate how the dew-point temperature changes during ascent of non-saturated adiabatic parcel.

The dew-point temperature fullfils the equation:

The hydrostatic equation.

From the Dalton’s law: 𝑑 ln 𝑝 = 𝑑 ln 𝑒

For typical atmospheric values dT_d/dz is

approximately 1/6 of the dry adiabatic lapse rate.

𝑑 ln 𝑒 = 𝜀𝐿_#$

𝑅𝑇₀¹ 𝑑𝑇₀ 𝑑𝑝 = −𝜌𝑔𝑑𝑧 ⟶ 𝑑 ln 𝑝 = − 𝑔

𝑅𝑇𝑑𝑧 𝑝 = 𝑅𝑇𝜌

𝑑 ln 𝑒 = − 𝑔 𝑅𝑇 𝑑𝑧 𝜀𝐿_#$

𝑅𝑇₀¹ 𝑑𝑇₀ = − 𝑔 𝑅𝑇 𝑑𝑧 𝑑𝑇₀

𝑑𝑧 = − 𝑇₀¹𝑔

𝜀𝐿_#$𝑇 E 𝑐₍

𝑐₍ ≅ − 𝑇₀¹𝑐₍

𝜀𝐿_#$𝑇 E Γ₀ Γ₀ = 𝑔

𝑐₍₀ ≈ 𝑔 𝑐₍

𝑇_!^"𝑐_#

𝜀𝐿_$%𝑇 ≅ 280^" ) 1004

0.622 ) 2.5 ) 10^& ) 300 = 0.166 ≈ ⁄1 6

/34 15

Equation of change of the dew-point temperature can be written in form of a change of dew-point deficit: T-T_d

When T=T_d, the saturation level has been reached, and a value of z_LCL can be determined by integrating form initial values (0,T₀-T_d0) to the saturation state (z_LCL,0).

T₀-T_d0 is the dew-point depression at the surface.

At saturation level, T becomes equal to T_d (and to T_LCL).

The lifting condensation level (LCL), z_LCL corresponds to the level of the saturation pressure p_s.

𝑑𝑇₀

𝑑𝑧 = − 𝑇₀¹𝑐₍

𝜀𝐿_#$𝑇 E Γ₀ ⟶ 𝑑𝑇

𝑑𝑧 − 𝑑𝑇₀

𝑑𝑧 = −Γ₀ + 𝑇₀¹𝑐₍ 𝜀𝐿_#$𝑇 E Γ₀ 𝑑 𝑇 − 𝑇₀

𝑑𝑧 = − 1 − 𝑇₀¹𝑐₍

𝜀𝐿_#$𝑇 E Γ₀

9

,_$2,_%$

"

𝑑 𝑇 − 𝑇₀ = − 9

"

3_!"!

1 − 𝑇₀¹𝑐₍

𝜀𝐿_#$𝑇 Γ₀𝑑𝑧

For a parcel lifted from the surface, the value of z_LCL can be estimated (assuming 𝑑𝑇₀⁄𝑑𝑧 = − ⁄1 6 Γ₀):

Calculation of the lifting condensation level provides a good estimate of the cloud height for clouds that form by adiabatic ascent.

𝑧_%&% ≈ 0.12 𝑇_" − 𝑇_0" km

ADIABATIC

AND PSEUDO-ADIABATIC PROCESSES

1.

Wet adiabatic lapse rate / saturated adiabatic lapse rate

Pseudo-adiabatic process

3.

Water condensed in pseudo-adiabatic process

Wet adiabatic lapse rate

𝑑ℎ = 𝑐₍𝑑𝑇 + 𝐿_#$𝑑𝑞_$

𝑑ℎ = 𝛿𝑞 + 𝑣𝑑𝑝

First Law of thermodynamics for adiabatic processes

𝑐₍𝑑𝑇 + 𝐿_#$𝑑𝑞_$ − 𝑣𝑑𝑝 = 0

Water vapor is saturated 𝑞_$ = 𝑞_! 𝑇, 𝑝

𝑑𝑞_$ = 𝑑𝑞_! = 𝜕𝑞_!

𝜕𝑇 𝑑𝑇 + 𝜕𝑞_!

𝜕𝑝 𝑑𝑝

If expansion work occurs fast enough the heat transfer with the environment remains negligible.

If no moisture precipitates out, the parcel is closed and and its behavior above the LCL is described by a reversible saturated adiabatic process.

/34 19

𝑐₍𝑑𝑇 + 𝐿_#$ 𝑞_!

𝑇 𝛽_,𝑑𝑇 − 𝐿_#$ 𝑞_!

𝑝 𝛽₍𝑑𝑝 − 𝑣𝑑𝑝 = 0 𝑑𝑝 = −𝜌𝑔𝑑𝑧

𝑣𝑑𝑝 = −𝑔𝑑𝑧 𝑑𝑝 = − 𝑝

𝑅𝑇𝑔𝑑𝑧 𝑐₍ + 𝐿_#$ 𝑞_!

𝑇 𝛽_, 𝑑𝑇 + 𝐿_#$ 𝑞_!

𝑅𝑇𝛽₍ + 1 𝑔𝑑𝑧 = 0

Γ_! = −𝑑𝑇

𝑑𝑧 = 𝑔 1 + 𝑞_!𝛽₍ 𝐿_#$

𝑅𝑇 𝑐₍ + 𝑞_!𝛽_, 𝐿_#$

𝑇 Saturated moist adiabatic lapse rate:

It is convenient to express the partial derivatives of 𝑞_! as logarithmic partial derivatives:

𝜕𝑞_!

𝜕𝑇 = 𝑞_! 𝑇

𝜕 ln 𝑞_!

𝜕 ln 𝑇

𝜕𝑞_!

𝜕𝑝 = 𝑞_! 𝑝

𝜕 ln 𝑞_!

𝜕 ln 𝑝 𝛽_, = 𝜕 ln 𝑞_!

𝜕 ln 𝑇 𝛽₍ = −𝜕 ln 𝑞_!

𝜕 ln 𝑝 𝑑𝑞_! = 𝑞_!

𝑇 𝛽_,𝑑𝑇 − 𝑞_!

𝑝 𝛽₍𝑑𝑝

We will calculate 𝛽₍ and 𝛽_,.

𝑞_! = 𝜀𝑒_!

𝑝 − 1 − 𝜀 𝑒_! = 𝜀 𝑒_! 𝑝

1 − 1 − 𝜀 𝑒_! 𝑝

𝜕𝑞_!

𝜕𝑝 = −𝑞_!

𝑝 1 + 1 − 𝜀 𝜀 𝑞_! 𝛽₍ = −𝜕 ln 𝑞_!

𝜕 ln 𝑝 = 1 + 1

𝜀 − 1 𝑞_! ≈ 1

𝜕𝑞_!

𝜕𝑇 = 1 𝑒_!

𝑑𝑒_!

𝑑𝑇 𝑞_! + 1 − 𝜀 𝜀

1 𝑒_!

𝑑𝑒_! 𝑑𝑇 𝑞_!¹ 𝛽_, = 𝜕 ln 𝑞_!

𝜕 ln 𝑇 = 𝑇𝑑 ln 𝑒_!

𝑑𝑇 1 + 1 − 𝜀

𝜀 𝑞_! = 𝑇𝑑 ln 𝑒_! 𝑑𝑇 𝛽₍ 𝛽_, = 𝐿_#$

𝑅_$𝑇 𝛽₍ ≈ 5 400 𝐾 𝑇

𝑑 ln 𝑒_!

𝑑𝑇 = 𝐿_#$

𝑅_$𝑇¹ Clausius-Clapeyron equation

𝛽₍

𝛽_,

Saturated moist adiabatic lapse rate

/34 21

Γ_! = 𝑔 1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 𝑐₍ + 𝑞_!𝛽_, 𝐿_#$

𝑇

Γ_! = 𝑐₍₀ 𝑐₍₀

𝑔 𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇

Γ₀ = 𝑔 𝑐₍₀

Γ_! ≡ −𝑑𝑇

𝑑𝑧 = 𝛾Γ₀ 𝛾 = 𝑐₍₀

𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇

𝑐₍ = 𝑞₀𝑐₍₀ + 𝑞_!𝑐_($ + 𝑞_#𝑐_# 𝑅 = 𝑞₀𝑅₀ + 𝑞_!𝑅_$

𝛽_, = 𝐿_#$

𝑅_$𝑇 𝛽₍ ≈ 5 400 𝐾 𝛾 ≤ 1 𝑇

If no moisture precipitates out, the parcel is closed and and its behavior above the LCL is described by a reversible saturated adiabatic process.

The process depends weakly on the abundance of condensate present (e.g., on how much the system’s enthalpy is represented by condensate).

Because the condensate is present is present only in trace abundance, the variation of condensate unnecessarily complicates the parcel’s description under saturated conditions.

A simplification is proposed

Pseudo-adiabatic process: the system is treated as open and condensate is removed (added) immediately after (before) it is produced (destroyed).

/34 23

A pseudo-adiabatic change of state may be constructed in two legs:

1. Reversible saturated adiabatic expansion (compression), which results in the

production (destruction) of condensate of mass dm_c and a commensurate release (absorption) of latent heat to (from) the gas phase

2. Removal (addition) of condensate of mass dm_c

Wet adiabatic vs pseudo-adiabatic lapse rate

Γ_! = 𝛾Γ₀ 𝛾 = 𝑐₍₀ 𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇

𝑐₍ = 𝑞₀𝑐₍₀ + 𝑞_!𝑐_($ + 𝑞_#𝑐_# 𝑅 = 𝑞₀𝑅₀ + 𝑞_!𝑅_$

Wet adiabatic lapse rate:

𝑞₀ + 𝑞_! + 𝑞_# = 1 and 𝑞₀ = 𝑐𝑜𝑛𝑠𝑡 , 𝑞_! + 𝑞_# = 𝑐𝑜𝑛𝑠𝑡

Pseudo-adiabatic lapse rate:

𝑞_# = 0, 𝑞₀+𝑞_! = 1

Simplified version of Γ _!

/34 25

Γ_! = 𝛾Γ₀ 𝛾 = 𝑐₍₀ 𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇

⟶ 𝛾 = 1 + 𝑞_!𝐿_#$

𝑅₀𝑇 1 + 𝑞_!𝐿¹_#$

𝑐₍₀𝑅_$𝑇¹ In many textbooks a simplified version of Γ_! is presented.

𝑅 ⟶ 𝑅₀ 𝑐₍ ⟶ 𝑐₍₀

Non-dimensional lapse rate 𝛾

• pressure: 1 000 hPa

• the air initially saturated at 300 K

Γ_! = 𝛾Γ₀

𝛾 = 𝑐₍₀ 𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇 Γ_! < Γ₀

Non-dimensional lapse rate 𝛾

• pressure: 800 hPa

• the air initially saturated at 300 K

/34 27

Γ_! = 𝛾Γ₀

𝛾 = 𝑐₍₀ 𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇 Γ_! < Γ₀

Non-dimensional lapse rate 𝛾

• pressure: 1000 hPa

• pressure: 800 hPa

• the air initially saturated at 300 K

Γ_! = 𝛾Γ₀

𝛾 = 𝑐₍₀ 𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇 Γ_! < Γ₀

Difference between pseudo-adiabatic and simplified pseudo-

adiabatic normalised lapse rates

• pressure: 1 000 hPa

• the air initially saturated at 300 K

/34 29

Γ_! = 𝛾Γ₀

𝛾 = 𝑐₍₀ 𝑐₍

1 + 𝑞_!𝛽_, 𝑅_$ 𝑅 1 + 𝑞_!𝛽_, 𝐿_#$

𝑐₍𝑇 Γ_! < Γ₀

Water condensed in adiabatic process

/34 31

Adiabatic enthalpy equation of a closed system:

dry air, water vapor, condensed water: 𝑑ℎ = 𝑐₍𝑑𝑇 + 𝐿_#$𝑑𝑞_! ; 𝑑ℎ = 𝛿𝑞 + 𝑣𝑑𝑝 0 = 𝑐₍𝑑𝑇 + 𝐿_#$𝑑𝑞_! − 𝑣𝑑𝑝

𝑑𝑞_# = −𝑑𝑞_! = 𝑐₍

𝐿_#$ 𝑑𝑇 − 𝑣

𝐿_#$ 𝑑𝑝 𝑑𝑝 = −𝑔

𝑣 𝑑𝑧 𝑑𝑞_# = 𝑐₍

𝐿_#$ 𝑑𝑇 + 𝑔 𝐿_#$ 𝑑𝑧

𝑑𝑞_# = 𝑐₍ 𝐿_#$

𝑑𝑇

𝑑𝑧 + 𝑔

𝑐₍ 𝑑𝑧 Γ₀ = 𝑔

𝑐₍₀ ≈ 𝑔

𝑐₍ , Γ_! = −𝑑𝑇 𝑑𝑧

𝑑𝑞_# ≅ 𝑐₍

𝐿_#$ Γ₀ − Γ_! 𝑑𝑧

The amount of water condensed in a rising adiabatic parcel increases with the height above the cloud base and increases with increasing temperature at the cloud base.

For shallow clouds (cloud depth not bigger than ca. 300-500 m, for instance stratocumulus clouds) it can be assumed that the amount of condensed water increases linearly with height above the cloud base (h).

The rate of this increase (𝑐₄) is approximately constant and depends on temperature and pressure at the cloud base.

Liquid Water Content (LWC) is the amount of liquid water per unit volume:

For shallow clouds one can assume that the air density is constant, therefore:

𝐿𝑊𝐶 = 𝑐₅ 𝑇, 𝑝 E ℎ ; 𝑐₅ = 𝜌 𝑐₍

𝐿_#$ Γ₀ − Γ_! 𝑔 𝑚⁶ 𝑞_# ℎ = 𝑐₄ 𝑇, 𝑝 E ℎ ; 𝑐₄ = 𝑐₍

𝐿_#$ Γ₀ − Γ_! 𝑔 𝑘𝑔 E 𝑚

𝐿𝑊𝐶 = 𝑞_# E 𝜌 ; 𝜌 − density of the air

/34 33

20m 60m 100m 140m 180m

17m 51m 85m 119m 153m N=50 cm^-3

H_base=1278 m N=255 cm^-3

H_base=844 m

𝑐₅ = 1.9 E 10²⁷ 𝑔

𝑚⁶ 𝑐₅ = 2 E 10²⁷ 𝑔

𝑚⁶

𝑐_! = 1.7 & 10^"# 𝑔 𝑘𝑔 & 𝑚