Meteorology ENV 2A23 Stability and Moisture Lectures.

Meteorology ENV 2A23

Stability and Moisture

Lectures

• If ∂θ/∂z = 0, the atmosphere is said to be neutral,or neutrally stratified

• the lapse rate is equal to the dry adiabatic lapse rate (DALR)• Γd = g/cp ≈10 K km-1 i.e. the temperature decreases by 10 K

every km.

.

• If ∂θ/∂z > 0, i.e. θ increases with height, the atmosphere is said to be stable, statically stable, or stably stratified.

• Γ < Γd

• If an air parcel is moved upwards adiabatically, it will follow the DALR, so be colder (therefore denser) than its environment and so sink back down.

• Conversely …• In other word, the atmosphere is stable to small perturbations.

• If ∂θ/∂z < 0, i.e. θ decreases with height, the atmosphere is said to be unstable, statically unstable, or unstably stratified.

• Γ > Γd

• If an air parcel is moved upwards adiabatically, it will follow the DALR, be warmer (less dense) than its environment and so keep on rising.

• In other word, the atmosphere is unstable to small perturbations.

If ∂θ/∂z = 0, the atmosphere is said to be neutral,or neutrally stratified, and the lapse rate is equal to the dry adiabatic lapse rate (DALR) Γd ~= 10 K km-1 i.e. the

temperature decreases by 10 K every km.

If ∂θ/∂z > 0, i.e. θ increases with height, the atmosphere is said to be stable, statically stable, or stably stratified.

If ∂θ/∂z < 0, i.e. θ decreases with height, the atmosphere is said to be unstable, statically unstable, or unstably stratified.

• The observed atmospheric lapse rate is closer to the saturated adiabatic lapse rate than the dry adiabatic lapse rate

• Typically Γ ≈ 7 K km-1

• The difference is due to moisture…

Brunt-Väisälä frequency

• Another measure of stability

• Also known as static stability frequency

• N is the frequency with which a vertically displaced air parcel would oscillate

• Typically N = 10-2 s-1

z

gN

2

Moisture in the Atmosphere

• See Ahrens Chapter 4

Water can exist in 3 phases in the atmosphere:

• Gas – water vapour

• Liquid • Solid


• Gas • Liquid

– Cloud droplets– Rain drops

• Solid


• Gas • Liquid • Solid

– Ice crystals– Snow– Hail

Phase-transition equilibria

Fig – Tsonis, Ch 6.

• When water changes phase a large amount of energy, latent heat, is either released or must be supplied.

• In liquid or solid state the H2O molecules tend to align themselves by charge (to minimise their potential energy)

• In gas phase water behaves as an ideal gas.

http://www.its.caltech.edu/~atomic/snowcrystals/ice/ice.htm

Latent heat

Solid (ice) Liquid (water)

Gas(watervapour)

VAPOURISATIO

N

evap

orat

ioncond

ensa

tion

SUBLIMATION

sublimation

FUSION

freezing

melting

Latent heat of fusion Lf = 3.34x105 J kg-1

Latent heat of sublimation Ls = 2.83x106 J kg-1

Latent heat of vapourisation Lv = 2.50x106 J kg-1

Note all latent heats are functions of temperature, values here are for T=0oC

• Lv must be supplied from the environment when water goes from liquid to gas– e.g. during evaporation, heat is taken out of the ocean

and the ocean cools

• Lv is released to the environment when water goes from gas to liquid, – e.g. condensation in cumulus clouds.– In the tropics this is a major driver of atmospheric

motion– Also in tropical cyclones

Hydrological cycle

Measures of Water Vapour I• There are many ways of measuring and expressing water

content of the atmosphere.

• One of the simplest measures is as a mixing ratio:

d

w

m

mr

Mass of water vapour/ mass of dry air

Units: kg/kg, or more usually g/kg

Measures of Water Vapour I• There are many ways of measuring and expressing water

content of the atmosphere.

• One of the simplest measures is as a mixing ratio:

d

w

m

mr

d

w

d

w

d

w

Vm

Vm

m

mr

/

/

Mass of water vapour/ mass of dry air


Measures of Water Vapour I• Almost equal to r is q the specific humidity:

mass of water vapour/ total mass of air


w

wd

w

wd

w

mm

mq


rr

rq

wd

w

1



w

wd

w

wd

w

mm

mq

,as r is small


rr

rq

wd

w

1



w

wd

w

wd

w

mm

mq

,as r is small

• q and r are constant for an “air parcel” if further water is not added or subtracted.

Climatological distributions of q and r:

Fig - Hartmann

Climatological distributions of q and r:

Equation of state for moist air

• For thermodynamic purposes we can treat water vapour as a separate gas (c.f. Dalton’s Law)

• The partial pressure of water vapour will obey ideal gas law:

e = (pw) =ρwRwT,

where ρw is density w.v.

Rw is specific gas constant for w.v.

Rw = R*/Mw = 461 J kg-1 K-1

• The equation of state for dry air is

p - e = ρdRdT


• Moisture can be measured as the ratio of the partial pressure of water vapour (e) to that of dry air (p-e):

r

M

Mr

TR

TR

ep

e

w

d

dd

ww

, where ε = Mw/Md = 18.02/28.99 = 0.6222


• We can combine ideal gas laws for dry air & water vapour:

Tr

rwhereTTRp

Tr

rRp

Tr

Rr

p

Tr

Rp

TRRp

d

d

d

dd

wwdd

1

1,

1

1

)1()1(

)1(

)(

**

T* is the virtual temperature


• Note: T* is just a convenience, it is not a true thermodynamic variable, i.e. it is not a measure of the internal energy of a moist air parcel.

Tr

rwhereTTRp d

1

1, **

Saturated vapour pressure

• Consider a closed system of fixed volume, which contains some liquid water

• The average internal energy of the water molecules in the liquid is proportional to the temperature.

• Imagine the no. molecules in vapour state increases, i.e. vapour pressure (e) increases, then no. of molecules re-entering liquid will increase.

• At a certain critical vapour pressure, system will come into equilibrium,– i.e. condensation = evaporation

• e = esat, the saturated vapour pressure liquid

vapour

Saturated vapour pressure

• As the internal energy of the liquid water molecules depends on temperature, it is intuitive that esat will increase with temperature.

• Determined by the Claussius Clapeyron equation

)( lw

vs

T

L

dT

de

es is a strong function of temperature and be using the first and second laws of

thermodynamics it can be shown that:

where Lv is the latent heat of vapourisation,

αw is the specific volume of water vapour, and

αl is the specific volume of liquid.

This is the Clausius-Clapeyron equation.

)( lw

vs

T

L

dT

de

2TR

eL

dT

de

w

svs







It can be simplified, given αw >> αl to be

)( lw

vs

T

L

dT

de

2TR

eL

dT

de

w

svs

TTR

Lee

wss

11exp

00







where e = es0 at T = T0. If T0 is

taken as 0 oC (the triple point) then es0 = 6.11 mb.


Can be solved to give:

)( lw

vs

T

L

dT

de

2TR

eL

dT

de

w

svs

TTR

Lee

wss

11exp

00

'exp

20

0 TTR

Lee

wss







where e = es0 at T = T0. If T0 is

taken as 0 oC (the triple point) then es0 = 6.11 mb.

where T’ = T-T0.


Can be solved to give:

Or approximated as

Vapour pressure & boiling

• As water boils, bubbles of vapour rise to the surface and escape

esat = atmospheric pressure

and esat proportional to T where lower atmos. pressure lower esat

lower boiling point food takes longer to cook

Ahrens – Ch 5

Measures of Water Vapour II

• Relative humidity

• RH = vapour pressure/ saturated vapour pressure

100sate

eRH


• Dew Point Temperature (TD)

• TD is the temperature at which air becomes saturated if cooled at constant pressure.

• Hence e(T) = esat(TD)


• Wet Bulb Temperature (TW)

• TW is the lowest that can be reached by evaporating water into the air (at constant pressure)

• TD ≤ TW ≤ T, with equality at saturation

• TW is the temperature of an evaporating cloud droplet of rain drop

• It is a good measure of how comfortable for humans, as for TW << 37 oC allows lots of evaporation to take place.


• Wet Bulb Temperature (TW)

• TW is the lowest that can be reached by evaporating water into the air (at constant pressure)

• TD ≤ TW ≤ T, with equality at saturation

• TW is the temperature of an evaporating cloud droplet of rain drop

• It is a good measure of how comfortable for humans, as for TW << 37 oC allows lots of evaporation to take place.

2

)()(

TRTLr

L

crTr

TT

w

sp

sW

Moist Adiabatic Processes

• The relatively small amounts of water vapour in the atmosphere means that, in practice, the moist adiabatic lapse rate for unsaturated air is close to the dry adiabatic lapse rate.

• It can be shown that:

i.e. Γm ≈ Γd , as r is small.

)9.01( rc

gd

pmm

Saturated Adiabatic Processes• For saturated air, the effects of

latent heat release when water changes phase are important.

• Consider an air parcel, rising adiabatically:

• Condensation takes place and latent heat is released:

the (saturated) adiabatic lapse rate is lower than the dry adiabatic lapse rate,

i.e. Γs < Γd.

Saturated Adiabatic Processes

• From the first law of thermodynamics (and using hydrostatic balance) it is possible to show that the saturated adiabatic lapse rate is

• Typically Γs is 6-7 K/km in the troposphere

• Note Γs(T,p)

• Hence saturation adiabats, i.e. lines that follow the saturated adiabatic lapse rate, are marked as curved lines on a tephigram.

• A saturated air parcel will follow a saturated adiabat.

dTde

pCL

RTpLe

s

p

sd

s

1

1

plus saturated adiabatics (lines of constant equivalent potential temperature).

Plus the international standard atmosphere.

Normand’s Rule

• Uses a tephigram to find the condensation level (or lifting condensation level LCL) for an “air parcel”

• Lets follow an air parcel rising through the atmosphere:

• Air parcel at p0 has temp. T & dew point temp. TD

• If unsaturated the air parcel rises along DALR

• At pN it becomes saturated (thereafter it follows saturated adiabat)

• Normand’s point, at pressure pN, is the intersection of r line from TD and the DALR from T.

• This is the LCL, the Lifting Condensation Level.

• From the LCL, if the air parcel rises, it will follow saturated adiabat.

Normand’s Rule

• From LCL following saturated adiabat down tells us temperature air would have if water is continually evaporated into it so it remains saturated; i.e. the wet bulb temperature TW

• Hence saturated adiabats are lines of constant wet bulb potential temperature θW

• Higher up, θW and θ curves become parallel.

• θW asymptotes to the equivalent potential temperature θe

• θe is the potential temp an air parcel would have if all its water vapour were condensed out.

Normand’s Rule

• θW and θe are conserved by an air parcel, if there are no external heat sources or sinks

• Hence θW and θe are often used to indentify “air masses”.

Normand’s Rule

Atmospheric Stability II

• We can now reconsider atmospheric stability for a moist atmosphere

• Latent heat release is important for modifying the stability of the atmosphere

• Recall for a dry atmosphere:

If ∂θ/∂z = 0, the atmosphere is said to be neutral,or neutrally stratified, and the lapse rate is equal to the dry adiabatic lapse rate (DALR) Γd ~= 10 K km-1 i.e. the

temperature decreases by 10 K every km.

If ∂θ/∂z > 0, i.e. θ increases with height, the atmosphere is said to be stable, statically stable, or stably stratified.

If ∂θ/∂z < 0, i.e. θ decreases with height, the atmosphere is said to be unstable, statically unstable, or unstably stratified.

(1) If Γ > Γd or ∂θ/∂z < 0,the atmosphere is said to be unstable, statically unstable, or unstably stratified.

(2) If Γ = Γd or ∂θ/∂z = 0, the atmosphere is said to be neutral,or neutrally stratified

(3) If Γs < Γ < Γd or ∂θ/∂z > 0 and ∂θW/∂z < 0 (or ∂θe/∂z < 0) the atmosphere is said to be conditionally unstable

(4) If Γs = Γ or ∂θW/∂z = 0 (or ∂θe/∂z = 0) the atmosphere is said to be saturated neutral or neutral to moist convection.

(5) If Γ < Γs or ∂θw/∂z > 0 (or ∂θe/∂z > 0) the atmosphere is said to be absolutely stable

Cloud Development

• Clouds form as air rises, expands and cools– Once T cools to TD then condensation can

take place– Water vapour condenses onto particles

(called cloud condensation nuclei) in the atmosphere to form cloud droplets

• What causes the air to rise?

Cloud Development

• Convectively forced cloud development

• CCL (Cloud Condensation Level) is point when T = TD

• Upward motion in thermal

• Downward motion to sides– Continuity– Colder, denser air at

cloud edge due to evaporation

• The erosion due to evaporation and entrainment leads to the clumpy form of convective cumulus

• Downward motion (subsidence) is slower, hence occurs over a larger area

• Subsidence inhibits convection

• Shade surface, so cut off further clouds, until blown away

• Cumulus advected by wind

Trade-wind cumulus clouds over the South Pacific Ocean


Cumulus over northern Portugal


Cumulus over northern Portugal

Isle of Wight Small cumulus clouds over East Africa.

Development of cumulus cloud

• Environmental stability is critical for depth of Cumulus development

• Cumulonimbus can reach as high as the tropopause, where the high stability of the stratosphere blocks further ascent and causes spreading into an ‘anvil’

Cumulus humilis Cumulus congestus Cumulonimbus

Spectacular cumulonimbus anvil

Development of Topographic Cloud

Orographic stratus. View south to Quinag, north-west Scotland, from B869 road, 16:15 GMT.

Orographic cumulus and the Matterhorn

Foehn conditions, Leukerbad, Switzerland. View northwards from the southern outskirts of Leukerbad towards the Gemmi Pass, which is covered in cloud. The mountain almost completely covered in cloud in the centre of the picture is 2,600m high. The white mountain (Rinderhorn) is about 3,450m high.

Development of lenticular clouds

Altocumulus lenticularis.

Cumulus, cirrus, cirrocumulus and altocumulus lenticularis. Picture from the Royal Meteorological Society's historic Clarke and Cave Collection.

Further Reading for reading week:

C. Donald Ahrens (2007) Meteorology Today. Eighth Edition, Brooks/Cole. ISBN 0-534-37201-5

Chapter1. Earth and its atmosphere – composition, vertical structure, instruments,

etc2. Energy: Warming the Earth and Atmosphere – temperature, heat

transfer, radiation, radiative processes, the sun.3. Seasonal and daily temperatures – orbital patterns etc.4. Atmospheric Moisture – water in the atmosphere, measures of water, etc5. Condensation: dew, fog and clouds6. Stability and cloud development – stable air, unstable air, tephigrams,

cloud development.

Meteorology ENV 2A23 Stability and Moisture Lectures.

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