Thermodynamics of moist air

Chapter 2

Thermodynamics of moist air

Thermodynamics of moist and cloudy air

¾Literature:

•

•

•

•

•

The equation of state for moist unsaturated air

¾ The state of a sample of moist air is characterized by its:

• Pressure, p

• Absolute temperature, T

• Density, ρ(orspecific volume α= 1/ρ), and

• Some measure of its moisture content, e.g.

• Thewater vapour mixing ratio, r, defined as the mass of water vapour in the sample per unit mass of dry air.

¾ These quantities are connected by theequation of state.

Equation of state:

p α = R T

( 1 + r / ) / ( ε 1 + ε ) ≈ R T

( 1 0 61 + . r )

specific gas constant for dry air

specific gas constant water vapour

ε

= R

/R

= 0.662

r is normally expressed in g/kg, but must be in kg/kg in any formula!

Other moisture variables

¾ The vapour pressure, e = rp/(ε+ r)which is the partial pressure of water vapour.

¾ The relative humidity, RH = 100 ×e/e*(T).

• e* = e*(T)is the saturation vapour pressure -the maximum vapour that an air parcel can hold without condensation taking place

¾ The specific humidity, q = r/(1 + r), which is the mass of water vapour per unit mass of moist air.

More moisture variables

¾ The dew-point temperature, T_d, which is the temperature at which an air parcel first becomes saturated as it is cooled isobarically.

¾ The wet-bulb temperature, T_w, which is the temperature at which an air parcel becomes saturated when it is cooled isobarically by evaporating water into it. The latent heat of evaporation is extracted from the air parcel.

The vertical distribution of rand RHobtained from a radiosonde sounding on a humid summers day in central Europe.

Aerological (or thermodynamic) diagrams

.

T = constant p

α

.

ln p

T = constant

.

(p, Τ_d) ln p

T = constant

.

(p, Τ)

Aerological diagram with plotted sounding

The effect of the water vapour on density is often taken into account in the equation of state through the definition of the virtual temperature:

T_v =T(1+r/ ) / (ε 1+ε)≈T(1 0 61+ . r)

Then, the density of a sample of moist airis characterized by its pressureand its virtual temperature, i.e.

Moist air(r > 0) has a larger virtual temperature than dry air (r = 0) => the presence of moisture decreases the density of air --- important when considering the buoyancy of an air parcel!

ρ = p RT_v

The equation of state for cloudy air

specific volume = (total volume)/(total mass)

( ) ( )

α = V_a +V_l+V_i / M_d +M_v +M_l+M_i

( ) ( )

α α= _d 1+r_l(α α_l / _d)+r_i(α α_i / _d) / 1+r_T

α= ε

+ = +

+ = +

+ R T

p r

R T p

p e

p r

R T p

r r

d

d T

d d

d T

d

T

1 1

1 1

1 1

/

total mixing ratio of water substance

Divide byM_d

ε= R_d/R_v= 0.622

specific heat of water vapour

defines the density temperature for cloudy air:

T_ρ= T(1 + r/ε)/(1 + r_T)

The first law of thermodynamics

The first law of thermodynamics for a sample of moist air may be written alternatively as

dq c dT pd c dT dp

v p

= ′ +

= ′ − α α

, ,

The quantities c_vd( ≈1410 J kg⁻¹K⁻¹)and c_pd( ≈1870 J kg⁻¹K⁻¹) are the corresponding values for dry air.

the heat supplied per unit mass to

the air sample specific heats of the air sample

incremental changes in temperature, pressure and specific volume

v vd p pd

c′ =c (1 0.94r)+ c′ =c (1 0.85r)+

Adiabatic processes

An adiabatic processis one in which there is zero heat input (dq = 0); in particular, heat generated by frictional dissipation is ignored.

( )

dlnT = R′ ′/c_p dlnp

If the process is also reversible and unsaturated,r is a constant.

ln T =(R / c ) ln p ln A′ ′p +

′ = + +

R R_d(1 r/ ) / (ε 1 r)

DefineA such that, when pequals some standard pressure,p_o, usually taken to be1000 mb, T = θ.

The equation can be integrated exactly (ignoring the small temperature dependence of c'_p) to give:

a constant

θ

ε _κ

=

F

HG I

KJ

F

HG I

KJ

F

HG I

KJ

′

′ + ′

T p

p T p

p T p

p

o o pv pd o

R cp

Rd cpd

r

rc c

1 1

/

/

The variation inκis< 1 % and is usually ignored

′ = −

κ κ(

1 0 24r . )

κ = R_d /c_pd We define the virtual potential temperature,θ_vby

θ

κ

v v

T p

= F p

HG I

KJ

The quantityθis called thepotential temperatureand is given by:

The potential temperature

Enthalpy

The first law of thermodynamics can be expressed as dq=d(u+ α − αp ) dp = dk dp− α

The enthalpy is a measure of heat content at constant pressure.

k = u + pαis called the specific enthalpy.

For an ideal gas,k = c_pT.

Water substance and phase changes

The latent heat associated with a phase transition of a substance is defined as the difference between the heat contents, or

enthalpies, of the two phases; i.e.

L_i,ii= k_ii- k_i,

The dependence of L_i,iionT andp may be obtained by

differentiation using k = u + pα (for details see e.g. E94, p114).

It turns out that: L_i,ii= L_i,iio+ (c_pii- c_pi)(T −273.16 K)

L_i,iiois the latent heat at the so-called triple point(T = 273.16 K

= 0.01^oC, ande = 6.112 mb) where all three phases of water substance are in equilibrium.

subscripts refer to the two phases

eis the water vapour pressure

The pressure and temperature at which two phases are in equilibrium are governed by the Clausius-Clapeyron equation:

dp dT

L

ii i T(

i ii

ii i

⎛

⎝⎜ ⎞

⎠⎟ =

, −

,

).

α α

For liquid-vapour equilibrium, α_l<< α_v, and using the ideal gas law for vapour, we obtain an equation for the saturation vapour pressure, e*(T):

de dT

L e R T

v v

* *

= ₂

The Clausius-Clapeyron equation

L_v= the latent heat of vaporization.

This equation may be integrated for e*(T).

de dT

L e R T

v v

*

*

2

A more accurate empirical formulais (see E94, p117):

ln *e = 53 67957 6743 769. − . /T−4 8451. lnT

A corresponding expression for ice-vapour equilibriumis:

ln *e =23 33086 6111 72784. − . /T+0 15215. lnT

These formulae are used to calculate the water vapour content of a sample of air. If the airsample is unsaturated, the dew point temperature(or ice point temperature) must be used.

e* in mb and T in K

Moist enthalpy

The moist enthalpy is conserved in an isobaric process as long asdq = 0 anddr_T= 0.

¾ Define the moist enthalpy kof a sample of cloudy air by the formula:

M_dk = M_dk_d + M_vk_v+ M_Lk_L

¾ Then k = k_d + rk_v + r_Lk_L, expressed per unit mass of dry air.

¾ But L_v(T) = k_v- k_L, => k = k_d + L_vr + r_Tk_L

¾ r_T= r + r_L.

¾ Since k_d= c_pdTand k_L= c_LT,

k = (c_pd+ r_Tc_L)T + L_vr

c_Lis the specific heat of liquid water

Entropy

An excellent reference is Chapter 4 of the book:

C. F. Bohren & B. A. Albrecht

Atmospheric Thermodynamics Oxford University Press

The moist entropy and related quantities

¾ In analogy with the definition of k, we define the total specific entropy per unit mass of dry airas:

s = s_d+ rs_v+ r_Ls_L

¾ s_d, s_v, and s_Lare the specific entropies of dry air, water vapour and liquid water, respectively.

¾ The equality of the Gibbs free energyin phase equilibrium between water vapour and liquid water leads to the

expression:

L_v = T s( ^*_v −s_L)

the saturation specific entropy of water vapour

s s r s L r

T r s s

d

v

v v

= + T L+ +

(

)

s_d =c_pdlnT−R_dlnp_d

s

c

ln T

R

ln e

s

c

ln T

R

ln e

s

= c

ln T

L_v = T s( _v^* −s_L)

d v L L

s

s rs

r s

Some algebra

s c r c T R p L r

T rR RH

pd L d d

v

v

=

(

) ln

ln

ln ( )

s c r c T R p L r

T rR RH

pd L d d v

v

=( + T ) ln − ln + − ln ( )

¾ sis conservedunder reversible moist adiabatic transformations

¾ It is not conservedwhen

• dq≠0: radiative heating or cooling, or conduction

• dr≠0: by external evaporation, precipitation loss

• by evaporation of rain into unsaturated air

Conservation of entropy

(

)

s= c +r c lnθ −R ln p Put

This defines theequivalent potential temperature, θ_e

R /(cd pd r c )T L

rR /(c r c ) v pd T L

o v

e

d pd T L

p L r

T (RH) exp

p (c r c )T

+ − + ⎡ ⎤

⎛ ⎞

θ = ⎜⎝ ⎟⎠ ⎢⎢⎣ + ⎥⎥⎦

Equivalent potential temperature

¾ Entropysis defined by a process in which heat and water substance are added to airwhich is kept just saturated by reversible evaporationfrom a planar water surface as its

temperature is raisedfrom a reference temperature and partial pressure p_dto the state (T, p_d).

¾ sis conserved underreversible moist adiabatic transformations.

(T_ref, p_d) (T, p_d)

Entropy

T_ref T

R /(c r c ) d pd T L

rR /(cv pd r c )T L

o v

e

d pd T L

p L r

T (RH) exp

p (c r c )T

+

− + ⎡ ⎤

⎛ ⎞

θ = ⎜⎝ ⎟⎠ ⎢⎢⎣ + ⎥⎥⎦

Equivalent potential temperature

¾ For dry air(r = 0, r_L= 0, r_T= 0), θ_ereduces toθ.

¾ Note that θ_eisnota state variable →isopleths of constant θ_ecannot be plotted on an aerological diagram.

= 1 = 1

= p

*

d v

pd T pv d

d

dT dp L r

ds (c r c ) R d

T p T

⎛ ⎞

= + − + ⎜ ⎟

⎝ ⎠

Differential form for

s

¾ An alternative form is, using 0 = dr^*+ dr_L

ds c r c dT

T r R de

e R dp

p d L r

pd T pv T v d d T

d

=⁽ + ⁾ − ^**− −

FH IK

¾ The differential form of sfor (saturated) cloudy air (RH = 1, r_L≥0) is

¾ Betts showed that the two forms can be written more symmetrically as:

ds

r c d

c dT T

L

T dq R dp

T p

pm e e

pm v

1+ = θ = + − m

θ

*

*

*

ds

r c d

c dT T

L

T dq R dp

T p

pm l

l

pm v

l m

1+ = θ = − −

θ

(1 + r_T)c_pm= (c_pd+ r^*c_pv+ r_Lc_l) (1 + r_T)R_m = (R_d+ R_v)

q^*= r^* /(1 + r_T) q_l= r_L/(1 + r_T)

Alternative forms

where

¾ the saturation equivalent potential temperature,θ_e* ds

r c d

c dT T

L

T dq R dp

T p

pm e e

pm v

1+ = θ = + − m

θ

*

*

*

ds

r c d

c dT T

L

T dq R dp

T p

pm l

l

pm v

l m

1+ = θ = − −

θ The equations

define

¾ the liquid-water potential temperature,θ_l

c d

c d L

T dr

pd e e

pd v

θ θ

θ θ

*

*

= + *

and

v L

pd pd L

L

d d L

c c dr

T θ = θ−

θ θ

ds

r c d

c dT T

L

T dq R dp

T p

pm e e

pm

v

1+ = θ = + − m

θ

*

*

*

ds

r c d

c dT T

L

T dq R dp

T p

pm l

l

pm v

l m

1+ = θ = − −

θ

Let c_pm≈c_pd, R_m≈R_d then

Approximate forms of θ

* and

Assume that

(L_v/T)dr^*≈d(L_vr^*/T) and (L_v/T)dr_L≈d(L_vr_L/T) Then

* v

e

pd

exp L r*

c T

⎛ ⎞

θ ≈ θ ⎜⎜⎝ ⎟⎟⎠ c d

c d L

T dr

pd e e

pd v

θ θ

θ θ

*

*

= + *

R /(c r c ) d pd T L

* o v

e

d pd T L

p L r

T exp

p (c r c )T

+ ⎡ ⎤

⎛ ⎞

θ = ⎜⎝ ⎟⎠ ⎢⎢⎣ + ⎥⎥⎦ c/f

L

exp (-L r /c T)

θ ≈ θ

Note that when r_L= 0, θ_L= θ

v L

pd pd L

L

L

d d

c c dr

T θ = θ−

θ θ

→theliquid water potential temperature is the potential temperature attained by the evaporation of all liquid water in an air parcel through reversible moist descent.

L

exp (-L r /c T)

θ ≈ θ

Anunapproximatedexpression forθ_l is:

o L L v L

L

T T pd T pv

p r r L r

T 1 1 exp

p r r (c r c )T

χ −γ

χ⎛ ⎞ ⎛ ⎞ ⎡ ⎤

⎛ ⎞ −

θ ≡ ⎜⎝ ⎟⎠ ⎝⎜ −ε + ⎟ ⎜⎠ ⎝ − ⎟⎠ ⎢⎢⎣ + ⎥⎥⎦

where χ ≡

a

f c

h

γ ≡r R_{T v}/

c

h

Theliquid-water virtual potential temperatureθ_lvis defined by:

1 1

o L L L v L

Lv v

T T T pd T pv

p r r r L r

T 1 1 1 exp

p 1 r r r (c r c )T

χ− χ−

χ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎡ ⎤

⎛ ⎞ −

θ ≡ ⎜⎝ ⎟⎠ ⎝⎜ − + ⎟⎜⎠⎝ −ε + ⎟ ⎜⎠ ⎝ − ⎟⎠ ⎢⎢⎣ + ⎥⎥⎦

¾ Note that θ_Lv= θ_vwhen r_L= 0

The liquid-water virtual potential temperature

¾ θ_e, θ_L,andθ_Lvareconserved in reversible adiabatic processes involving changes in state of unsaturated or cloudy air.

¾ θ_e, θ_L,andθ_Lvare notfunctions of state - they depend on p, T, r andr_L

¾ →Curves representing reversible, adiabatic processes cannotbe plotted in an aerological diagram

¾ In a saturated process,r = r^*(p, T)

Some notes

θ_ep ^o

r

LCL

T p

p r r

=

F

HG I

KJ

F

HG I

L KJ

NM O

QP

− 0 2854 1 0 28

1 0 81 3376 2 54

. / ( . )

exp ( . ) .

¾ Approximation:neglect the liquid water content (setr_L= 0).

¾ This makes the approximate forms ofθ_e, θ_L,andθ_Lv functions of state which can be plotted in an aerological diagram.

¾ Adiabatic processes where the liquid water is ignored are calledpseudo-adiabatic processes. They are not reversible!

¾ The formula θ_e*≈ θexp (L_vr^*/c_pdT) is an approximation for the pseudo-equivalent potential temperatureθ_ep. A more accurate formula is:

The pseudo-equivalent potential temperature

Temperature at the LCL

T T

T T

LCL d

K d

=

L

NM O

QP

1 −

56 800 56

ln( / ) 1

T_KandT_din degreesK

¾ T_LCLis given accurately (within 0.1°C) by the empirical formula:

The Lifting Condensation Level Temperature

[Formula due to Bolton, MWR, 1980]

¾ If an air parcel is lifted pseudo-adiabatically to the high atmosphere until all the water vapour has condensed out, θ_ep= θ.

¾ Thepseudo-equivalent potential temperature is the potential temperature that an air parcel would attain if raised pseudo-adiabatically to a level at which all the water vapour were condensed out.

¾ The isopleths ofθ_epcan be plotted on an aerological diagram. These are sometimes labelled by their temperature at1000 mbwhich is called thewet-bulb potential temperature,θ_w.

¾ Lift a parcel of unsaturated air adiabatically

¾ it expands and cools, conserving itsθ_v

¾ its Tdecreaseswith height at the dry adiabatic lapse rate,Γ_d:

d

dq 0 pd pv pd

dT g 1 r

dz ₌ c 1 r(c / c )

⎛ ⎞ +

Γ = −⎜⎝ ⎟⎠ = +

¾ Note thatr is conserved, butr* decreases becausee^*(T) decreases more rapidly thanp.

¾ Saturation occurs at thelifting condensation level(LCL) whenT = T_LCLandr = r^*(T_LCL, p).

The adiabatic lapse rate

Above the LCL, the rate of which its temperature falls,Γ_m, is less thanΓ_dbecause condensation releases latent heat.

T m

s pv

pd

v d

2 v L

L 2

pd pv v pd pv

pd

1 r

dT g

dz c c

1 rc 1 L r

R T

L (1 r / )r 1 r c

c rc R T (c rc )

⎛ ⎞ +

Γ ≡ −⎜⎝ ⎟⎠ = + ×

⎡ ⎤

⎢ + ⎥

⎢ ⎥

⎢ + + + ε ⎥

⎢ + + ⎥

⎢ ⎥

⎣ ⎦

For reversible ascent:

Whenr_Tis small, the ratioΓ_m/Γ_dis only slightly less than unity, but when the atmosphere is very moist, it may be appreciably less than unity.

The moist static energy and related quantities

The first law gives

dq = dk − α_ddp

where dqis expressed per unit mass of dry air.

Adiabatic process(dq = 0) → dk − α_ddp = 0 α_d= α(1 + r_T)

For ahydrostatic pressure change,αdp = −gdz.

Under these conditions:

pd TL v T

dh≡(c +r c )dT+d(L r) (1 r )gdz+ + =0

Ifr_Tis conserved, we can integrate

h=(c_pd +r c T_{T l}) +L r_v + +(1 r gz_T) =constan .t

¾ The quantityhis calledthe moist static energy.

¾ hisconserved for adiabatic, saturated or unsaturated transformations in whichmass is conserved and in which the pressure change is strictly hydrostatic.

¾ h is a measure of the total energy:

(internal + latent + potential)

Some notes

¾ Define thedry static energy,h_d.

¾ Putr_T= r =>

d pd L

h = (c + rc )T + + (1 r)gz.

¾ Thisis conserved in hydrostatic unsaturated transformations.

¾ hand h_d are very closely related to θ_eand θ.

The dry static energy

¾ Define two forms of static energy related toθ_Landθ_Lv.

¾ These are theliquid water static energy:

h_w =(c_pd+r c_{T pv})T−L r_{v L}+ +(1 r gz_T) and thevirtual liquid water static energy

The (virtual) liquid water static energy

v L T

Lv pd

T L T

L r

h c r T gz

r r

1 r

⎛ ε + ⎞

= ⎜⎝ε + − ⎟⎠ − + +

¾ h_Lvis almostprecisely conserved following slow adiabatic displacements.

¾ If r_L= 0, h_Lv= c_pdT_v+ gz (just asθ_Lreduces toθ).

Vertical profiles of dry conserved variables

z (km)

θ, θv Dry static energy

deg K 10⁵J/kg

z (km)

Vertical profiles of moist conserved variables

z (km)

θ, pseudo θe Moist static energy

deg K 10⁵J/kg

z (km)

The stability of the atmosphere

¾ Consider the vertical displacement of an air parcel from its equilibrium position

¾ Calculate the buoyancy force at its new position

¾ Consider first an infinitesimal displacementξ; later we consider finite-amplitudedisplacements

¾ Parcel motion is governed by the vertical momentum equation

2 2

d b

dt ξ=

is thebuoyancy force per unit mass

p a p a

p a

b( ) g⎛ρ − ρ ⎞ g⎛α − α ⎞ ξ = − ⎜⎜⎝ ρ ⎟⎟⎠= ⎜⎝ α ⎟⎠

The motion equation for small displacementsis:

d

dt N

2 2

2 0

ξ+ ξ= where ^{N2 b}

z

= −∂

∂ Newton's law for an air parcel:

2 2

d buoyancy force dt unit mass

ξ =

z 0

buoyancy force b : b( )

unit mass z

ξ = ∂ ξ +

∂

2 2

z 0

d b

dt z

0

∂

The motion equation for small displacementsis:

d

dt N

2 2

2 0

ξ+ ξ=

where N^{2 b}

z

= −∂

∂

For an unsaturated displacement, θ_vpis conserved and we can write

vp va vp va

va va

T T

b( ) g g

T

− θ − θ

⎛ ⎞ ⎛ ⎞

ξ = ⎜⎝ ⎟⎠= ⎜⎝ θ ⎟⎠

Sinceθ_vp=constant≅ θ_va,

vp

2 va va

2

va va

b g

N g

z z z

∂ θ ∂θ ∂θ

= − = ≅

∂ θ ∂ θ ∂

¾ Parcel displacement is:

• stable if ∂θ_va/∂z > 0

• unstable if ∂θ_va/∂z < 0

• neutrally-stable if ∂θ_va/∂z = 0

¾ A layer of airis stable,unstable, or neutrally-stableif these criteria are satisfied in the layer.

Stability criteria

¾ In a saturated(cloudy) layer of air, the appropriate conserved quantity is the moist entropys(or the equivalent potential temperature,θ_e)

¾ Must use thedensity temperatureto calculateb.

¾ Replace α_p in b by the moist entropy,s.

¾ In this case

( )

2 T

m L m

T

r

1 s

N c ln T g

1 r z z

∂

⎡ ∂ ⎤

= + ⎢⎣Γ ∂ − Γ + ∂ ⎥⎦

¾ A layer of cloudy airisstableto infinitesimalparcel displacements ifs(orθ_e) increasesupwards and the total water (r_T) decreasesupwards. It isunstableif θ_edecreases upwards andr_Tincreasesupwards.

¾ The stability criterion does not tell us anything about the finite-amplitude instability of an unsaturated layerof air that leads to clouds.

¾ Parcel method okay, but must consider finite displacements of parcels originating from the unsaturated layer.

Some notes

Potential Instability

¾ A layer of air may bestable if it remains dry, butunstable if lifted sufficientlyto become saturated.

¾ Such a layer is referred to aspotentially unstable.

¾ The criterion for instability is thatdθ_e/dz < 0.

stable

unstable

unsaturated

saturated/cloudy lift

Conditional Instability

¾ The typical situation is that in which a displacement is stable provided the parcel remains unsaturated, but which ultimately becomes unstableif saturation occurs.

¾ This situation is referred to asconditional instability.

¾ To check for conditional instability, we examine the

buoyancy of an initially-unsaturated parcel as a function of height as the parcel is lifted through the troposphere,

assuming some thermodynamic process(e.g. reversible moist adiabatic ascent, or pseudo-adiabatic ascent).

¾ If there is some height at which the buoyancy is positive, we say that the displacement isconditionally-unstable.

¾ If some parcels in an unsaturated atmosphere are conditionally-unstable, we say thatthe atmosphereis conditionally-unstable.

¾ Conditional instability is the mechanism responsible for the formation ofdeep cumulus clouds.

¾ Whether or not the instability is released depends on whether or not the parcel is lifted high enough.

¾ Put another way,the release of conditional instability requires a finite-amplitude trigger.

¾ The conventional way to investigate the presence of conditional instability is through the use of an aerological diagram.

pressure (mb)

200

300

500

700 850 1000

10 g/kg LFC LCL 20 ^oC 30 ^oC LNB

dry adiabat

pseudo- adiabat

The positive area(PA)

PA u_LNB u_LFC T_vp T_va R d_d p

pLNB pLFC

= ^{12 2} −^{12 2} =

z c

h

The negative area(NA) orconvective inhibition(CIN)

( )

parcel pLFC p

vp va d

NA=CIN=

∫

Positive and Negative Area Convective Inhibition (CIN)

We can define also thedowndraught convective available potential energy(DCAPE)

DCAPE_{i p}^p R T_{d a} T_p d p

i

=

z

The integrated CAPE(ICAPE) is the vertical mass-weighted integral of CAPE for all parcels with CAPE in a column.

Theconvective available potential energyorCAPEis the net amount of energy that can be released by lifting the parcel from its original level to itsLNB.

CAPE = PA − NA

Convective Available Potential Energy - CAPE

z (km) z (km)

ms⁻¹

Pseudo θ_e Reversible θ_e

ms⁻¹

z (km) z (km)

z_Lkm z_Lkm

Buoyancy Liquid water

reversible

pseudo-adiabatic reversible

with ice

Buoyancy (^oC)

Height (km)

Downdraught convective available potential energy (DCAPE)

DCAPE

R T

T

d p

i

=

z

⁽

^{) ln}

700

850

1000

r* = 6 g/kg

T_d T q_w= 20^oC

LCL

800 mb, T = 12.3^oC

DCAPE T_w

T_d

T_ρp T_ρa

Buoyancy and θ

z Lifted parcel Environment

T_o(z), r_o(z), p(z), ρ_o(z) T_vp ρ_p

p o

o

( )

b g ρ − ρ

= − ρ

vp vo

o

(T T ) b g

T

= − or

Lifted parcel Environment

T_o(z), r_o(z), p(z), ρ_o(z) T_ρp ρ_p

p vo

o

(T T ) b g

T

ρ −

=

sgn (b) = sgn { T_p(1 + εr_p) −T_o(1 + εr_o) = T_p−T_o+ ε[T_pr_p−T_or_o]}

ε= 0.61 Below the LCL (T_ρp= T_vp)

sgn (b) = sgn [T_p(1 + εr^*(p,T_p)) −T_o(1 + εr_o)]

= sgn [T_p(1 + εr^*(p,T_p) −T_o(1 + εr^*(p,T_o)) + εT_o(r^*(p,T_o) −r_o)]

At the LCL (T_ρp= T_vp)

sgn (b) = sgn [T_p(1 + εr^*(p,T_p) −T_o(1 + εr^*(p,T_o)) + εT_o(r^*(p,T_o) −r_o)]

SinceT_vis a monotonic function ofθ_e, ^{* *}

ep eo

b∝ θ − θ + Δ( ) small z

*

eo eo

θ θ

LFC LCL parcel saturated

dry adiabat moist adiabat

A (T θ r) C

(T_w θ_ε r_w) B

(T_d r)

D

(T_LCL, p_LCL) LCL or saturation level

(T_L θ_L r_L)

E (T_L θ_e* r*) F

(T_dr_T) (r_T= r* + r_L) G

r* - isopleth

P= p_LCL– p₂> 0

P= p_LCL– p₂< 0 p₁

p₂

After Betts

The saturation point

¾ The point (T_LCL, p_LCL)is referred to as thesaturation point.

¾ The saturation point of an air parcel is aconserved quantity in the absence of diabaticormixing processes.

¾ One can plot the saturation points of parcels from a radiosonde sounding on an aerological diagram.

¾ The state of a parcel of air is characterized by its saturation point and the departure of the actual pressure from the saturation pressure, i.e. P= p_LCL– p.

¾ IfP > 0, the parcel is cloudy, ifP < 0it is unsaturated.

The End