# Stability and Moisture Lectures

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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.

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

Moisture in the Atmosphere
See Ahrens Chapter 4

Water can exist in 3 phases in the atmosphere:
Gas water vapour Liquid Solid

Water can exist in 3 phases in the atmosphere:
Gas Liquid Cloud droplets Rain drops Solid

Water can exist in 3 phases in the atmosphere:
Gas Liquid Solid Ice crystals Snow Hail

Phase-transition equilibria
Fig – Tsonis, Ch 6.

Latent heat 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.

Latent heat of fusion Lf = 3.34x105 J kg-1
melting FUSION Solid (ice) Liquid (water) freezing condensation VAPOURISATION SUBLIMATION Latent heat of sublimation Ls = 2.83x106 J kg-1 sublimation evaporation Latent heat of vapourisation Lv = 2.50x106 J kg-1 Gas (water vapour) 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: 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: Mass of water vapour/ mass of dry air Units: kg/kg, or more usually g/kg

Measures of Water Vapour I
Almost equal to r is q the specific humidity: mass of water vapour/ total mass of air Units: kg/kg, or more usually g/kg

Measures of Water Vapour I
Almost equal to r is q the specific humidity: mass of water vapour/ total mass of air Units: kg/kg, or more usually g/kg ,as r is small

Measures of Water Vapour I
Almost equal to r is q the specific humidity: mass of water vapour/ total mass of air Units: kg/kg, or more usually g/kg ,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:

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

Equation of state for moist air

Equation of state for moist air
Moisture can be measured as the ratio of the partial pressure of water vapour (e) to that of dry air (p-e): , where ε = Mw/Md = 18.02/28.99 =

Equation of state for moist air
We can combine ideal gas laws for dry air & water vapour: T* is the virtual temperature

Equation of state for moist air
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.

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 vapour liquid

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

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.

where Lv is the latent heat of vapourisation,
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. It can be simplified, given αw >> αl to be

where Lv is the latent heat of vapourisation,
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. It can be simplified, given αw >> αl to be 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:

where Lv is the latent heat of vapourisation,
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. It can be simplified, given αw >> αl to be 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: Or approximated as where T’ = T-T0.

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

Measures of Water Vapour II
Dew Point Temperature (TD) TD is the temperature at which air becomes saturated if cooled at constant pressure. Hence e(T) = esat(TD)

Measures of Water Vapour II
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.

Measures of Water Vapour II
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.

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.

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.

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.

plus saturated adiabatics (lines of constant equivalent potential temperature) a quantity which is conserved for a saturated air parcel 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:

Normand’s Rule 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.

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, i.e. θ decreases with height, the atmosphere is said to be unstable, statically unstable, or unstably stratified. 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.

(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

Trade-wind cumulus clouds over the South Pacific Ocean
Cumulus over northern Portugal

Trade-wind cumulus clouds over the South Pacific Ocean
Cumulus over northern Portugal Small cumulus clouds over East Africa. Isle of Wight

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’

Cumulonimbus Cumulus congestus Cumulus humilis

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.