• dry air + water vapor will be referred to as a moist air

THERMODYNAMICS

Atmosphere

A multi-Component Multi-Phase System

• The gas phase atmospheric constituents: major gases, fixed proportions by volume (dry air)

– Nitrogen (N₂) 78,08 % – Oxygen (O₂) 21,05 % – Argon (Ar) 0,934 %

• Variable vapors and minor gases:

– Carbon dioxide (CO₂)

– Neon, Helium, Nitrous Oxide, Ozon, Methan, Sulfur compounds, organics – Water vapor (0 – 4 % by volume)

• Water marks motions in the lower atmosphere on all time and spatial scales because:

– of its proclivity to change phase and the manner in which these phase changes affect the local temperature

– water foster diverse interactions with radiant energy.

/49 2

• In cloud physics we consider:

– dry air, i.e. a gas that is a mixture of nitrogen and oxygen; all other components are in negligible amount

• The molecular mass of dry air is 28.96 g/mol.

– water than can exist in three phases: vapor, liquid and ice

• dry air + water vapor will be referred to as a moist air

• dry air + water vapor and/or liquid water and/or ice is called moist

atmosphere.

Notation

• The basic theromdynamic properties of the atmosphere depend on its component parts. These are defined in terms of their mass. For an equilibrium system four constituents of the moist atmosphere can be defined:

– d – dry air

– v – water vapor – l – liquid water – i – ice

• Total mass of the system: m=m_d+m_v+m_l+m_i

• To describe the amount of matter we use:

– Specific mass: q_x=m_x /m

• It is useful in a thermodynamic description for an equilibrium system (the normalizing mass is invariant as long as the basic flow describes the motion of the dry air)

– Mixing ratio: r_x=m_x /m_d

• It is helpful when considering the possibility that different components of a mass element have their own velocity

/49 4

• We distinguish between:

– equilibrium condensed phases associated with clouds, which evolve with the thermodynamic state in a more or less reversible way, and

– larger hydrometeors which evolve in irreversible way

• Larger hydrometeors, like rain-drops and most forms of ice, develop through irreversible microphysical processes such as the collision and coalescence of water droplets. These are more difficult to approximate as an equilibrium phase. Because they are larger, non-equilibrium phases of water in the atmosphere are also more dilute and short-lived

• The present lecture (Thermodynamics) focuses on reversible thermodynamic processes.

/49 6

•

Thermodynamic systems in local equilibrium are identified with mass elements, sometimes referred to as air- or fluid-parcels.

•

Formally the concept gains validity for mass elements small enough that the volume they occupy encompasses a scale much smaller than the scale over which thermodynamic properties vary, but much larger than the mean-free path.

•

Diffusion rapidly homogenizes the atmosphere on scales smaller than the Kolmogorov lenght scale, h

n

e

–

n

e

•

In vigorous cumulus clouds: e

n

×

–

h

of an air molecule, making the concept of an air parcel a useful one.

•

Typically the specific condensate mass within a cloud is less than 1 g kg

; much less than the specific mass of water vapor

•

What is the volume fraction of the water condensate?

•

The dilutness of condensate can come into conflict with the concept of an equilibrium thermodynamic system, as on the Kolmogorov scale, the

condensate is not continuously distributed.

6 3

3

d l

d 3 l

3 l d

d d l

l d

l

m 10 1cm V

V

kg 1 g m

; m cm 1 g m ;

1 kg m m V

V

= -

=

=

=

=

×

=

r r

r r

/49 8

•

In presence of the condensate (droplets, ice) that is not continously distributed the concept of air parcel should be enlarged.

•

One often imagines air parcel as being on the scale of arround 1 m

.

•

Strictly speaking volumes of air this large cannot be thought of in terms of a

single temperature, but the error of this approximation is typically much less

than those associated with other approximations invoked in the description of

such systems.

Subscripts notation

d dry air

v water vapor

l liquid water

i solid water (ice)

c condensate

t total water (irrespective of phase) s saturated state, or process

e equivalent (all condensate) reference state l liquid-free (all vapor) reference state

Description of water vapor content

/49 10

•

•

•

•

𝜚_!= _#^"

!$ = 𝜀 _#$^"

𝜚_%= ^&_#$^" = &'"

𝑞 = 𝑚_! #$

𝑚 = 𝜌_!

𝜌 = 𝜌_!

𝜌_" + 𝜌_! = 𝜀 𝑒

𝑝 − 𝑒 + 𝜀 * 𝑒 = 𝜀 𝑒

𝑝 − 1 − 𝜀 𝑒 ≈ 𝜀 𝑒 𝑝

𝑟 = 𝑚_!

𝑚_" = 𝜌_!

𝜌_" = 𝜌_!

𝜌_" = 𝜀 𝑒

𝑝 − 𝑒 ≈ 𝜀 𝑒

𝑝 = 𝑞 𝑟 = 𝑚_!

𝑚_" = 𝑚_!

𝑚 − 𝑚_! = 𝑞

1 − 𝑞 ≅ 𝑞

𝜀 = 𝑅_"

𝑅_!

Equation of State

Defining the density of the gaseous/vapor mixture as

r=(m_d+m_v)/V allows one to formulate the equation of state as:

d dry air

v water vapor

l liquid water

i solid water (ice)

c condensate

t total water (irrespective of phase) s saturated state, or process

e equivalent (all condensate) reference state l liquid-free (all vapor) reference state

Taking an air-parcel to be comprised of an ideal mixture of ideal gases, perhaps in the presence of condensate, the equation of state is that for an ideal gas of variable composition, such that:

To avoid dealing with a state dependent gas constant it is customary to define a density temperature, T𝜌, such that:

𝑝 = 𝑝_"+𝑝_! = ^#"_%^$" + ^##_%^$# 𝑇

𝑅_# = 287 𝐽𝐾^$%𝑘𝑔^$% 𝑅_&= 461 𝐽𝐾^$%𝑘𝑔^$%

𝑅 = 𝑞_"𝑅_" + 𝑞_!𝑅_!= 1 − 𝑞_! − 𝑞_& 𝑅_" + 𝑞_!𝑅_!

= 𝑅_" 1 + 𝑅_!

𝑅_" − 1 * 𝑞_! − 𝑞_&

𝑝 = 𝜌𝑅𝑇

𝑝 = 𝜌𝑅_"𝑇_' 𝑇_' = 𝑇 1 + 𝜀⁽𝑞_! − 𝑞_& ^{𝜀' =} ₍⁽_"^! − 1 ≈ 0.608 𝜀 = ⁽₍^"

! ≈ 0.622 𝑚 = 𝑚_# + 𝑚_& + 𝑚_# /𝑚

1 = 𝑞_# + 𝑞_& + 𝑞₎ 𝑞_# = 1 − 𝑞_& − 𝑞₎

Density temperature, virtual temperature

/49 12

In the absence of water the density temperature is the air temperature.

It can be interpreted as the temperature of a dry air parcel having the same density and pressure as the given air parcel.

In the literature the density temperature is often called the virtual temperature.

𝑇_' = 𝑇 1 + 𝜀⁽𝑞_! − 𝑞_&

𝑇_! = 𝑇 1 + 𝜀⁽𝑞_! = 𝑇 1 + 0.608𝑞_!

Buoyancy

Given the pressure, the density temperature determines the density and thus is important to the concept of the buoyancy. The buoyancy (b) of a fluid parcel can be measured by the extent to which its density differs from a background or reference density.

Assume that locally the density is given in terms of a deviation from such a reference state density:

The approximation of the buoyancy in terms of the density temperature follows from the assumption that the relative change in pressure is small compared to the relative change in density.

𝜌 = 𝜌₎ + 𝜌⁽

𝑏 ≡ −𝑔 𝜌⁽

𝜌₎ ≈ 𝑔 𝑇⁽

𝑇₎ + 𝑅⁽

𝑅₎ = 𝑔 𝑇_'⁽ 𝑇_')

𝑝 = 𝜌𝑅𝑇

Thermodynamic functions of state

Enthalpy Entropy

Gibbs free energy

Enthalpy

For an atmospheric system it proves useful to usetemperature and pressure to describe the state of the system.

The First Law becomes (h – specific enthalpy):

q = dh −υdp

h = q_dh_d + q_vh_v+ q_lh_l + q_ih_i c_p = ∂h ∂T

( )

h = h₀ + q

(

)

Limiting our considerations to ideal gases implies that enthalpy depends ONLY on temperature.

For water and ice the subscript ‘p’ for specific heat is omitted because water and ice are non-compressible.

Physically only enthalpy differences are relevant.

Enthalpy

/49 16

L_υ = h_υ − h_l

Vaporisation enthalpy (latent heat):

It proves useful to rewrite the expression for the enthalpy in terms of the phase-change enthalpies, so that it only refers to the reference enthalpy of one of the phases :

• using L_v to substitute h_vin expression:

• using L_v to substitute h_l

• The subscripts ‘e’ and ‘l’ serves as a reminder of which reference state has been adopted.

• Both expressions of enthalpy are simillar if:

h = q_dh_d + q_υh_υ + q_lh_l

h_e = c_eT + q_vL_v c_e = c_d + q_t

(

)

h_ℓ = c_ℓT − q_lL_v c_ℓ = c_d + q_t + c

(

)

𝑐_ℓ − 𝑐₊ =𝑞_,𝐿_! = 𝑞_, 𝑐_! − 𝑐_- 𝑇

The Second Law postulates the existence of an entropy state function, S, defined by the property that in equilibrium the state of the system is that which maximizes the entropy function. Such a function has the property that q≤Tds

As an extensive state function the entropy, like the enthalpy, can be decomposed into its constituent parts:

Entropy

Tds = dh −υdp

s = q_ds_d + q_υs_υ + q_ls_l + q_is_i

Dry air and water vapor entropy

/49 18

dh = Tds +υdp

dh = c_ddT, υ = R_dT p_d ds = c_dd lnT − R_dd ln p_d

s_d = s_d,0 + c_dln T T

(

)

(

)

s_d,0 is the reference entropy of dry air at the temperature T₀ and pressure p₀.

s_υ = s_υ,0 + c_υln T T

(

)

(

)

It is assumed that the specific heats are constant between T and T₀. Dry air (ideal gas):

Water vapor (ideal gas):

Entropy for condensed phases

The condensate is assumed to be ideal so that changes in pressure do not contribute to entropy.

s_l = s_l,0 + c_lln T T

(

)

The general expression for the composite entropy with respect to the equivalent

reference state

/49 20

s_e = q_ds_d + q_υs_υ + q_ls_l q_l = q_t − q_υ

= q_ds_d + q_ts_l + q_υ

(

)

= q_ds_d,0 + q_ts_l,0 + q_dc_d ln T

T₀ − q_dR_d ln p_d

p₀ + q_tc_l ln T

T₀ + q_υ

(

)

s_e = s_e,0 + c_eln T

T₀ − R_eln p_d

p₀ + q_υ

(

)

s_e,0 = q_ds_d,0 + q_ts_l,0 = s_d,0 + q_t

(

)

c_e = q_dc_d + q_tc_l = c_d + q_t

(

)

R_e = q_dR_d

s_e,0 is determined by the amount of water in the system and the reference state temperature and pressure denoted by T₀ and p₀ respectively

Entropy for the liquid-free reference state

s_ℓ = q_ds_d + q_υs_υ + q_ls_l q_υ = q_t − q_l

= q_ds_d + q_ts_υ − q_l

(

)

= q_ds_d,0 + q_ts_υ,0 + q_dc_d ln T

T₀ − q_dR_d ln p_d

p₀ + q_tc_υ ln T

T₀ − qR_υt ln p_υ

p₀ − q_l

(

)

s_ℓ = s_ℓ,0 + c_eln T

T₀ − q_dR_d ln p_d

p₀ − q_tR_υ ln p_υ

p₀ − q_l

(

)

s_ℓ,0 = s_d,0 + q_t

(

)

c_ℓ = q_dc_d + q_tc_υ = c_d + q_t

(

)

Gibbs free energy

/49 22

For a closed isobaric and isothermal system it follows from equation that for a reversible system

Which introduces the Gibbs free energy, or Gibbs potential as

i.e. the energy available to do work in anisothermaland isobaricsystem.

From this definition it follows that the difference in the Gibbs energy of two constituents is related to the differencies in their enthalpies and entropies:

From the postulates of thermodynamics, whereby in equilibrium H and T adopt values that maximize S, it follows that the Gibbs free energy of a system in equilibrium is a minimum.

In equilibrium the specific Gibbs energy of each phase must be equal, otherwise a redistribution of the mass between the phases could lower the total Gibbs energy.

TdS = dH −Vdp 0 = d H − TS

( )

G = H − TS

g_υ − g_l = h_υ − h_l − T (s_υ − s_l)

The Clausius-Clapeyron Equation

g = h − Ts

dg = dh − sdT − Tds = υdp − sdT

Tds = dh −υdp

dg_υ =υ_υdp − s_υdT dg_l =υ_ldp − s_ldT dg_υ = dg_l

dp = s_υ − s_l υ_υ −υ_l dT

υ_υ = R_υT

p υ_υ >>υ_l L_υ

T = s_υ − s_l

Clapeyron equation

Clausius-Clapeyron equation Saturated water vapor

pressure is often denoted by e_s d(ln e_s) = L_v

R_υT² dT

Specific Gibbs free energy

In equilibrium the specific Gibbs energy of each phase must be equal

Saturated water vapor pressure

/49 24

•

•

𝑅_! 1

𝑇 − 1 𝑇₎

𝑒_. = 𝑒_.) 𝑇 𝑇₎

∆&/

$_#

exp −𝐿_!) − ∆𝑐𝑇₎ 𝑅_!

1

𝑇 − 1

𝑇_{) 𝑇* = 273.15 𝐾}

𝑒_+* = 611 𝑃𝑎

𝐿_&* = 2.5 = 10^, 𝐽 = 𝑘𝑔^$%

∆𝑐 = 𝑐_-& − 𝑐_. = −2317 J = 𝑘𝑔^$% 𝐾^$%

d(ln e_s) = L_v

R_υT² dT

The Clausius-Clapeyron equation

• The Clausius-Clapeyron equation very effectively delimits the distribution of water throughout the atmosphere because:

– The atmosphere sits atop a reservoir of water, which endeavors to bring the air above it into

saturation, but how much moisture can be maintained in an air parcel is strongly constrained by the saturation value.

• If the amount of moisture exceeds the saturation value it condenses, and condensate is effectively removed by precipitation by the system.

• Hence the saturation specific humidity limits the amount of water in the atmosphere.

• Because the Clausius-Clapeyron equation so strongly controls the distribution of water in the atmosphere, if one had to single out a particular equation as being the most important for the functioning of Earth’s climate, it would be this equation.

/49 26 Saturation vapor pressure over liquid (solid line) and ice (dashed line).

Colored circles and lines show vapor pressure in the atmosphere, binned according to temperature for

different pressure levels (900 hPa, black; 700 hPa, blue, 500 hPa, orange, 300 hPa, red).

At T = 0C the saturation vapor pressure is 610.15 Pa.

At T = 30C the saturation vapor pressure over liquid water is 50.8 Pa as compared to 38.0 Pa over ice at the same temperature.

Saturation with respect to liquid for T < 0C is relevant because super- cooled water is often present in the atmosphere, with homogeneous nucleation of ice particles first occurring at about T = -38C.

Potential temperatures

Potential temperatures

/49 28

The first and second laws dictate how temperature changes between a given state, and a reference state defined by its pressure p_ϑ, and phase distribution, given by the triplet

{p_ϑ,q_v,q_l}.

The temperature in this reference state is called the potential temperature (denoted by q ) as it measures the temperature the system would have to have in the reference state for the entropy of this state to be identical to that of the given state.

Potential temperatures are invariant under an isentropic process, but their properties and absolute values depend on the choice of the reference.

The potential temperature, rather than the sensible temperature, is often preferred as a state variable because it is invariant for reversible transformations of the air parcel.

The potential temperature provides a convenient way to compare air parcels in different parts of the atmosphere, where for instance the pressure or humidity may vary.

potential temperature q

Reference state {pϑ ,qv,ql}= {10⁵ Pa,0,0}.

For this reference state we equate the equation for s_e with the value of T₀ (𝜃) chosen so that s_e,0=s_e .

𝑠₊ = 𝑠_+,) + 𝑐₊𝑙𝑛𝑇

𝜃 − 𝑅₊𝑙𝑛𝑝_"

𝑝₂ + 𝑞_! 𝑠_! − 𝑠_- 𝑠₊ = 𝑠_+,) → 𝑐₊𝑙𝑛𝜃 = 𝑐₊𝑙𝑛𝑇 − 𝑅₊𝑙𝑛𝑝_"

𝑝₂ 𝑐₊ = 𝑞_"𝑐_" + 𝑞_,𝑐_- = 𝑞_"𝑐_"

𝑅₊ = 𝑞_"𝑅_"

𝑞_! = 0

𝑞_, = 0

𝜽 = 𝑻

𝒑

𝜿 𝜅 = ^$_&^" , 𝑝₂ = 1000 ℎ𝑃𝑎

Equivalent potential temperature q _e

/49 30

Reference state {pϑ ,qv,ql}= {10⁵ Pa,0,qt}

All the vapor is condensed into liquid at p_ϑ=1000 hPa.

For historical reasons this particular reference state is called an equivalent state and denoted by subscript ‘e’.

For the equivalent reference state we equate the equation for se with the value of T0

chosen so that s_e,0=s_e .

𝑠₊ = 𝑠_+,) + 𝑐₊𝑙𝑛𝑇

𝜃 − 𝑅₊𝑙𝑛𝑝_"

𝑝₂ + 𝑞_! 𝑠_! − 𝑠_- 𝑠₊ = 𝑠_+,) → 𝑐₊𝑙𝑛𝜃 = 𝑐₊𝑙𝑛𝑇 − 𝑅₊𝑙𝑛𝑝_"

𝑝₂ + 𝑞_! 𝑠_! − 𝑠_-

Equivalent potential temperature…..

• We express the pressure of the dry air in terms of the total pressure and the specific humidity

• We express the entropy difference (s_v-s_l) relative to the vapor entropy in saturation p_d = R_dT ρ_d

p = RT ρ R = q_dR_d + q_υR_υ p_d = p q_dR_d

R

⎛

⎝⎜ ⎞

⎠⎟ = p R_e R

⎛

⎝⎜ ⎞

⎠⎟

s_υ − s_l = s_υ − s_s + s_s − s_l

= s_υ − s_s + L_υ T s_υ − s_s = c_υ ln T

θ_e − R_υ ln p_υ

p_θ − c_υ ln T

θ_e + R_υ ln p_s p_θ

= −R_υ ln p_υ

p_s = −R_υ lnϕ φ – relative humidity

Equivalent potential temperature

…..

/49 32

c_e lnθ_e = c_elnT − R_e ln p p_θ ⋅ R_e

R

⎛

⎝⎜ ⎞

⎠⎟ − qυR_υ lnϕ + q_υL_υ T

θ_e = T p_θ p

⎛

⎝⎜ ⎞

⎠⎟

R_e c_e

Ω_eexp q_υL_υ c_eT

⎛

⎝⎜ ⎞

⎠⎟ Ω_e = R

R_e

⎛

⎝⎜ ⎞

⎠⎟

R_e c_e

ϕ⁻

q_υR_υ c_e

R_e = q_dR_d

c_e = c_d + q_t

(

)

R = R_d + q_t

(

)

Ω_e is a factor that is very near unity, and (because q_t<<1) depends only very weakly on the thermodynamic state.

θ = T p_θ p

⎛

⎝⎜ ⎞

⎠⎟

R_d c_d

For q_t=0 (dry air) the equivalent potential temperature gets a simpler form

Liquid water potential temperature

The counterpart to the equivalent potential temperature is the liquid-water potential temperature, which is the temperature an air parcel would have if were reversibly brought to the {p_ϑ ,q_v,0} reference state.

For the liquid-free reference state we equate the equation for s_l with the value of T₀ chosen so that s_l,0=s_l .

θ_ℓ = T p_θ p

⎛

⎝⎜ ⎞

⎠⎟

R_l c_ℓ

Ω_ℓexp −q_lL_υ c_ℓT

⎛

⎝⎜ ⎞

⎠⎟ Ω_ℓ = R

q_dR_d

⎛

⎝⎜ ⎞

⎠⎟

q_dR_d c_l

R q_υR_υ

⎛

⎝⎜ ⎞

⎠⎟

q_tR_υ c_l

R_ℓ = R_d + q_t

(

)

c_ℓ = c_d + q_t

(

)

R = R_d + q_t

(

)

Many of the nuances that moisture brings to thermodynamic descriptions can be neglected when considering small perturbations about a given state.

Enthalpies h_eand h_l both describe the enthalpy of moist liquid-vapor system, hence in general he=hl.

However if, as is common, one assumes that:

large differences between h_l and h_ebecome apparent. These differences are however differences only in the absolute sense. In saturated case dqv=dqs=-dql

perturbations in general are similar.

It is customary to define many of the moist thermodynamic variables in an approximate form apprropriate to the consideration of small perturbations.

Approximate forms: q _{e ,} q _l

/49

÷ 34

÷ ø ö çç

è æ-

×

=

÷÷ ø ö çç

è

× æ

=

T c

q exp L

T c

q exp L

p l l v

p v e v

q q

q q

θ = T p_θ p

⎛

⎝⎜ ⎞

⎠⎟

R_d c_p

c_e ≈ c_d = c_p c_l ≈ c_d = c_p

where

Saturated equivalent potential temperature

÷÷ ø ö çç

è

× æ

» c T

q exp L

p s s q v

q

q_s is only a function of temperature and pressure, so it measures the thermal structure of the atmosphere.

q_s is constant following a saturated pseudo-adiabat.

The wordpseudo arises because formally the process corresponding to constant q_s is similar to a reversible adiabat, but the removal of condensate upon condensation, as

implied by the use of q_s instead of q_t implies a loss of condensate enthalpy by the system, hence it is not truly adiabatic.

The difference between q_s and q_e measures the subsaturation, as q_e < q_s.

Adiabatic lapse rate

/49 36

The temperature of an air parcel moving vertically changes due to expansion work.

It is useful to write the first law in form of enthalpy.

c_pdT −vdp = 0

Hydrostatic equation: dp = −ρgdz

c_pdT + gdz = 0

dh= c_pdT =δq + vdp

ρ = 1 v

in adiabatic process dq=0

dp = − g

v dz ⇒vdp = −gdz

Γ_d = − dT

dz = g

c_p Γ_d = g

c_p = 9,81^m_s2

1004 ^J_{kg K} ≈ 9,8 K km

Saturated moist adiabatic lapse rate

𝑐₃ = 𝑞_"𝑐_" + 𝑞_.𝑐_! + 𝑞_-𝑐_- 𝑅 = 𝑞_"𝑅_" + 𝑞_.𝑅_!

Γ_" = 𝑔 𝑐_"

Γ_. ≡ − M^"4

"5 2_$=𝛾Γ_" 𝛾 ≡ 1 + 𝑞_.𝛽₄ 𝑅_!

𝑅 1 + 𝑞_.𝛽₄ 𝐿_!

𝑐₃𝑇

𝛽₄ ≡ 𝜕 ln 𝑞_.

𝜕 ln 𝑇 = 𝐿_! 𝑅_!𝑇

𝜕 ln 𝑞_.

𝜕 ln 𝑝 ≈ 5400 𝐾 𝑇

Non-dimensional lapse rate γ

• pressure: 1 000 hPa

• the air initially saturated at 300 K

• solid curve – pseudo-adiabat

• dashed curve – adiabatic lapse rate

/49 38

Γ_s ≡ −dT

dz _θe =γΓ_d

γ ≡ c_d c_p

1+ q_sβ_T R_υ R

⎛

⎝⎜ ⎞

⎠⎟ 1+ q_sβ_T L_υ

c_pT

⎛

⎝⎜⎜ ⎞

⎠⎟⎟

⎡

⎣

⎢

⎢⎢

⎢⎢

⎤

⎦

⎥

⎥⎥

⎥⎥

β_T ≈ 5400K T

Non-dimensional lapse rate γ

• pressure: 800 hPa

• the air initially saturated at 300 K

• solid curve – pseudo-adiabat

• dashed curve – adiabatic lapse rate

Γ_s ≡ −dT

dz _θe =γΓ_d

γ ≡ c_d c_p

1+ q_sβ_T R_υ R

⎛

⎝⎜ ⎞

⎠⎟ 1+ q_sβ_T L_υ

c_pT

⎛

⎝⎜⎜ ⎞

⎠⎟⎟

⎡

⎣

⎢

⎢⎢

⎢⎢

⎤

⎦

⎥

⎥⎥

⎥⎥

β_T ≈ 5400K T

Water condensated in pseudo- adiabatic process

/49 40

Equation for enthalpy of a composite system for

adiabatic process. dh = q +υdp dh_ℓ = c_ℓdT − L_υdq_l − q_ldL_υ q = 0, c_p = q_dc_d + q_υc_υ + q_lc_l

0 = c_pdT − L_υdq_l −υdp

c_pdT − L_υdq_l −υdp = 0 υdp = −gdz L_υdq_l = c_pdT + gdz

dq_l = c_p L_υ

dT dz + g

c_p

⎛

⎝⎜⎜ ⎞

⎠⎟⎟dz Γ_d = g

c_p , Γ_s = −dT dz dq_l = c_p

L_υ

(

)

LWC = q_l ⋅ρ d LWC

( )

v

Γ_d − Γ_s

( )

𝑖𝑓 𝜌 ≅ 𝑐𝑜𝑛𝑠𝑡 Liquid water content 𝐿𝑊𝐶 = ^#_%^%

Rate of condensation

d LWC

( )

(

)

c_w(T, p) = c_p

L_υρ Γ_d

(

)

c_p

L_υρ Γ_d ≈ 10³

2.5⋅10⁶⋅110⁻²kg / m⁴ = 4 ⋅10⁻³g / m⁴ c_w = 1.5 − 2.2 ⋅10⁻³g / m⁴

For shallow clouds the liquid water content varies linearly with height

/49 42

p=1000 hPa

p=800 hPa

/49 44

1 − 𝛾 ≈ 0.45 1 − 𝛾 ≈ 0.5

p=1000 hPa, T=20C p=800 hPa, T=10C

EUCREX

/49 46 Pawlowska et al., Atmos. Res. 2000

ACE-2

/49 48 Brenguier, Pawlowska, Schueller, JGR. 2003

EUCAARI - IMPACT