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GEU 0047: Meteorology Lecture 6 Stability and Cloud Development

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- Equilibrium’s 2 States:
- Stable
- Unstable
Perturbed from its initial state, an object can either tend to

return to original point (A. stable) or deviate away (B. unstable)

Atmospheric Stability

- If perturbed vertically, a stable air parcel will tend to go back to its original altitude, whereas an unstable parcel will usually accelerate away vertically.
- If rising/sinking, the air parcel tends to cool down/warm up, due to the change in ambient pressure with altitude.
- Remember e-1 change for every 7.29 km (scale height Hp)
P = Po e-z/Hp

Γd: a constant

Dashed lines: unsaturated parcel’s lapse rate

Solid lines: environmental lapse rate

Γw: NOT a constant !

Dashed lines: moist parcel’s lapse rate

Solid lines: environmental lapse rate

Are the following 2 statements same (or contradictory)?

- rising air cools
- warm air rises

Hhh…? Let’s think about it for a while!

consider a rising parcel of air -->>

- As the parcel rises, it will adiabatically expand and cool (recall our discussion in chapter 5 about rising parcels of air)
- adiabatic - a process where the parcel temperature changes due to an expansion or compression, no heat is added or taken away from the parcel
- the parcel expands since the lower pressure outside allows the air molecules to push out on the parcel walls
- since it takes energy for the parcel molecules to "push out" on the parcel walls, they use up some of their internal energy in the process.
- therefore, the parcel also cools since temperature is proportional to molecular internal energy

In air, scale reads the weight T1 = W. Immersed in water, the

additional buoyant force reduces the objects weight because of buoyancy Fb . Thereby, T2 = W - Fb

Fb = rfVf g = displaced fluid weight

Fb

T2

T1

W

W

- Archimedes Law: The buoyancy force is equal to theweight of the volume of fluid displaced.
Weight of the volume of fluid* displaced rfVog

Weight (m g) of the object roVo g

The net force on an object is its weight minus the buoyancy

Fn = ro Vo g – rfVog = (ro– rf) Vo g

*The environmental air is treated as a fluid in which a parcel of air

is immersed.

Acceleration = Force/mass

F = Weight - Fb

= ro Vo g - rfVog

Acceleration = F/ (ro Vo )

= g(ro Vo - rfVo)/m

= g(ro - rf)/ro

In air,

replacing densities with temperatures using the ideal gas law, P = r R T, yields an equation for the acceleration of the air parcel, given the temperature and pressure of the parcel (To, Po) and those of the environment (Tf, Pf).

acceleration = g(Po - Pf)/Po = g(Tf- To)/Tf

Therefore,

A large temperature difference means instability. a ~ DT

Your HW#3 Due Nov. 29

Determining stability involves asking what happens to an air parcel if there is a small perturbation (vertical motion).What is its equilibrium like, stable or unstable?

- Expansion and cooling (or compression and heating) without any thermal exchange with the environment
- Adiabatic process does NOT mean isothermal process (What the hell …? Is Q = mcDT wrong?)

Rigid walls: ris constant

Black: isothermal

Blue: adiabatic

Flexible walls: P is constant

Gdry: Temperature change within an unsaturated air parcel.

~-9.8oC/1000m

Gwet : Temperature change within a saturated air parcel.

~ -6oC/1000m

Lapse Rate:

change in Temperature with Altitude G = -DT/Dz

Environmental Lapse Rate:

- Radiosondes yield information about the environmental lapse rate. It is what it is…on average G ~ 6.5oC/1000m.
- This lapse rate is used to estimate the stability after comparing with either the dry adiabatic lapse rate (Gdry) or moist adiabatic lapse rate (Gwet) of our imageryair parcel.

Some things you need to know: The gas laws

- Apply p = r R T to an air parcel with a unit mass,
- p α = R T; α: specific volume (α = r-1)
- p Δα + Δp α = R ΔT
- If this air parcel is in hydrostatic balance,
- r gDz = ptop – pbottom = - Dp
- Δp = - ρ g Δz
- Now, if this air parcel takes in some heat energy Dq while performing some work Dw and causes an amount of internal energy change Du,
- Dq = Dw + Du = pΔ α + Du (1st law of thermodynamics)

The specific heats

Cv : Spceific heat at constant volume

Cv ≡ Dq / DT,

Cv ≡ Du / DT when α is constant

Cp : Spceific heat at constant pressure

Cp ≡ Dq / DT when p is constant

Cp = Cv + R

Dq = pΔ α + Du

= pΔ α+ CvDT

= D (p α) - α Δp + CvDT

= D(RT) - α Δp + CvDT

= RDT - α Δp + CvDT

= (Cv + R)DT - α Δp

= Cp DT - α Δp

= Cp DT + g Δz

p α = RT

Cp ≡ Dq / DT when p is const.

Δp = - ρ g Δz

If dry adiabatic, Dq = 0

Gdry = - DT/D z = g/Cp

Unlike Gdry, Gwet = - DT/Dz is NOT a constant, but varies with temperature and moisture content !

es = eo exp{L/Rv (1/To-1/T)}

Known as Clausius-Clapeyron relation discovered when two engineers were working on the thermodynamics of water vapor to produce a more efficient steam engine.

eo = 611 Pascals

Rv = 461 J/K/kg (e α = RvT)

To = 273o K (i.e., 0oC)

L = { Lv = 2.50 x 106 J/kg}

= { Ld = 2.83 x 106 J/kg}

r = mass of water vapor / mass of dry air = rv/rd

How does the Td change in rising and sinking air?

For an air parcel at T and P, what is its water vapor

content?

Note:

Drs/ΔT > 0

P

T

- Dq = Cp DT + g Δz
- Dq = -LvDrs, for a saturated air parcel (rs: saturation mixing ratio)
- -LvDrs = Cp DT + g Δz
- Gwet = - DT/ Δz = Lv/Cp *Drs/Δz + g/Cp
= Lv/Cp *Drs/Δz + Gdry

=Lv/Cp *Drs/ΔT *ΔT/ ΔZ + Gdry

=- Lv/CpDrs/ΔT Gwet + Gdry

- (1 + Lv/CpDrs/ΔT) Gwet=Gdry
BecauseDrs/ΔT is always positive, Gwet<Gdry

X

Air parcel representation (Pressure, Temperature, mixing ratio)

= Temperature

= Humidity

thin solid: Gd

thick dashed: Gw

thin dashed: r(isohume)

Both air parcel points (temperature, humidity) move

together to the new pressure.

Temperature along

the dry adiabat,

Humidity along

an isohume or

constant moisture

content, because

NO moisture

leaves the parcel.

At some level, the adiabats and isohumes converge.

T and Td then cool along the moist (saturated) adiabat.

The difference

between the actual

mixing ratio in a

cloud and the original

mixing ratio is the

distance between

X and O on the plot.

It is equal to the

amount of condensation produced.

- When unsaturated air is lifted, it cools at the dry adiabatic rate. If lifted high enough, the temperature will drop below the dew point.
- Drier air must be lifted higher than moist air to encounter LCL.
- The height at which saturation just occurs is called the saturation level or the lifting condensation level (LCL).
- This height (in meters) can be estimated for cumulus-type clouds and is approximated by ZLCL = 125/(T - Td), where Td is the dew point temperature derived from the vapor pressure equation encountered before:
e = eo exp{Lv/Rv (1/To-1/Td)}

The parcel of air is

cooler and heavier

than the surrounding

air around it at all

levels.

G < Gdry

When perturbed it

will tend to return to

its original position.

- The atmosphere is
- always stable when
- the environmental
- lapse rate is less
- than the moist
- adiabatic rate.
- G < Gwet

Stratus clouds (cirrostratus, altostratus, nimbostratus) form in stable air.

- Stability favors a small environmental lapse rate.
- Ways to make the lapse rate small….
- Warm the air aloft (Inversions)
- warm advection (warm front)
- slowly sinking air (high pressure)

- Cool the air near the ground (Fogs)
- calm night radiative cooling
- cold advection (cold front)
- air moving over a cold surface

- Warm the air aloft (Inversions)

Dashed: before

Solid: after

When the surface air is saturated in a stable atmos., a persistent fog (or haze) may form.

Fig. 6-5, p. 143

- Descending of a layer of air causes it to warm and shrink via adiabatic compression.
- A temperature inversion can develop (warm air over cool).

Subsidence inversion

Fig. 1, p. 144

The atmosphere is

always unstable when

the environmental lapse

rate is greater than the

dry adiabatic rate.

G > Gdry > Gwet

The parcel of air is

warmer and lighter

than the surrounding

air around it at all

levels.

When perturbed it

will tend to accelerate

away from its

original position.

Cumulus-type clouds

An atmosphere with

an environmental

lapse rate G will be...

- Always Stable if
G < GDry

G < GWet

- Always Unstable if
G > GDry

G > GWet

What About Between?

In this example the dry

air is cooler and heavier

than the air around it at

all levels. It is stable.

The environmental

lapse rate is less than

the dry adiabatic lapse

rate. But,

GDry > G>GWet

A saturated parcel is

warmer than the

surrounding air at

all levels. It is unstable.

With an environmental

lapse rate between the

dry and moist adiabatic

rates, stability depends

upon whether the

air is saturated or

not.

If air can

be lifted to

a level

where it is

saturated,

instability

would

result.

- Instability favors a large environmental lapse rate.
- Ways to increase the lapse rate ….
- Cool the air aloft
- cold advection (jet stream)
- radiative cooling (emitting IR to space)

- Warm the air near the ground
- influx of warm air (warm advection)
- daytime solar heating of the surface
- air moving over a warm surface

- Cool the air aloft

Dashed: before

Solid: after

胞狀層積雲

雲階

Mixing may

occur via

convection

or

turbulence.

Mixing stable air close to saturation can cause stratus-type clouds.

The upper layer cools and saturates while the lower layer warms and

dries out, increasing the environmental lapse rate.

- As a stable layer rises, the change in density spreads it out. If it remains unsaturated, the top cools faster than below.

An inversion layer with

a saturated bottom and

an unsaturated top.

The top layer cools at Gdry

while the bottom layer

cools at Gwet (because

of latent heat release.)

This leads to absolute

instability associated

with severe storms (Ch.14).

- Convective Uplift
- Orographic Uplift
- Convergence Uplift
- Frontal Uplift

Vertical Motion via

Convection: exchange of thermal

energy by mass motion.

Hot air rises because it is

less dense.

Lifting a parcel of air to a

height where condensation occurs,

releases the latent heat stored in the

water vapor as clouds form.

- Cumulus-type cloud height is approximated as ZLCL = 125/(T - Td), where the constant 125 comes from the difference on average between GDry~ 10oK/1000m and GDewPt~ 2oK/1000m.
- With knowledge of the air temperature and dew point, determining cumulus cloud base heights is simple.
- With observation of cloud base height and air temperature, the dew point (hence moisture content) is estimated.

Vertical Motion via

Orographic Uplift:

air that encounters

steep topography

is forced to rise.

Vertical Motion via

Convergence:

advection winds

that encounter each other

force rising motion away

from the surface.

Air rises because there is

nowhere else to go.

Vertical Motion via

Frontal Uplift:

a cold air mass encounters

warm air or a warm air

mass encounters cooler air.

Since colder air is more

dense, it displaces the warm

air upward in a cold front

or a warm front along the

air masses boundary.

Cool air

A warm wet bottom

and a cool dry top.

Convection leads to

large vertical

development

while the sinking

air in between the

clouds is clear.

The stability of air above the condensation level greatly influences the vertical development of the clouds.

Instability may reach to the top of the troposphere where cumulonimbus clouds “anvilout” in response to the stable inversion layer of the stratosphere.

- Entrainment: mixing of environmental air into a current, jet or convection (cloud).
- When mixing of cooler, dry air occurs into convectively
unstable clouds, the clouds cool much more quickly. The rate of cooling can approach the dry adiabatic rate and the convective instability will cease.

- If the air is warm and moist, the instability grows along with the vertical development. Hence, our interest in water vapor (moisture) and infrared (cloud heights) images.

Orographic lifting, adiabatic cooling, heating and loss of moisture content.

- Temperature an air parcel would have if moved dry adiabatically to near surface (p0 = 1000mb).
- θ would remain constant if an air parcel is subjected to only dry adiabatic transformations.
- Dq = Cp DT – αΔp = 0
- p α = R T; => α = R T/p
- Cp R-1DT/T = Δp/p
- θ = T(p0/p)R/Cp, is called Poisson’s equation by meteorologists
- The reversible adiabatic process is called isentropic process

- A parcel of air in stable/unstable equilibrium will return/depart its original position.
- A rising parcel of unsaturated air will cool at the dry adiabatic rate of (~ 10oC/1000m); a descending unsaturated parcel warms at this rate.
- A rising parcel of saturated air will cool at the moist adiabatic rate of (~ 6oC/1000m); a descending saturated parcel warms at this rate.
- The environmental lapse rate is the rate that the actual air temperature decreases with increasing altitude. G = - DT/Dz

- Absolute Stability: Air at surface is cooler than air aloft (inversion), or the environmental lapse rate is greater than the dry adiabatic rate.
- Instability can be initiated if surface air warms, air aloft cools, or vertical lifting occurs (convection, convergence, fronts, topography).
- Conditional Instability: Environmental lapse rate is between the moist and dry adiabatic rates. Unsaturated air is lifted to a point where condensation occurs and becomes warmer than the surrounding air.